Embed Size (px)
Citation preview
1
CLASSIC LOGIC
A DOORWAY TO TRUTH © 2013 Frank X. Mercurio, Cincinnati, Ohio
All rights reserved. No part of this book may be reproduced or
transmitted in any form or by any means without written
permission from the author.
2
This is not a free e-book.
Purchase of this e-book entitles the buyer to keep one copy on their computer and to print one
copy only. Printing out more than one copy or, distributing it electronically, is prohibited by
international and USA copyright laws and treaties, and would subject the purchaser to penalties
of up to $100,000 per copy.
3
Contents
Lesson 1—The Science Of All Sciences
Lesson 2—How Do We Form Ideas—Concepts?
Lesson 3—Let’s Investigate Terms
Lesson 4—Why Definition and Division?
Lesson 5—Categorical Judgments and Propositions
Lesson 6—Categorical Propositions
Lesson 7—Special Types of Propositions
Lesson 8—Square of Opposition
Lesson 9—Eductions
Lesson 10—Deductive Reasoning
Lesson 11—The Categorical Syllogism, Nature, Structure, & Principles
Lesson 12—Categorical Syllogisms: Rules, Moods, and Figures
Lesson 13—Everyday Discussions and the Syllogism
Lesson 14—Hypothetical Syllogisms
Lesson 15—A Few Select Fallacies
4
INTRODUCTION
The science of logic has been around for thousands of years. The human mind hasn’t changed
much from before the time of, Plato, Socrates, or Aristotle. Classic logic was developed as a
science mostly by Aristotle (384 - 322 BC.). Like mathematics, the science of logic is based on
fact. Its content is as valid today as it was in centuries past.
A good logic course is designed to help you reason more correctly so as to attain truth. Whether
in formulating your own ideas or evaluating those of others, logic will help you analyze and
pinpoint mistakes in reasoning. It helps you keep your judgments error free.
You might object that classic logic is from ancient times and has no relevance today. “Hey, the
modern mathematical logic, symbolic logic, is the way to go,” you might say. The new logic
does have value, especially in designing computer language. It, however, will not help you in
your everyday dealings with concepts, propositions, or arguments. A working knowledge of
classic logic will actually help you think better, write better, and evaluate ideas better. It’s not
electronic reasoning. It’s real human reasoning!
Part of what makes the study of logic challenging, is that you may not have studied a subject like
it. And so, it is completely unfamiliar to you. It’s full of new terminology, like learning a new
language.
The good news is there is nothing to unlearn or revise. To succeed in this discipline it takes
some patience and persistence. A little knowledge of English grammar is helpful.
A short note on the text: Classic Logic: a doorway to truth is divided into three sections and 15
lessons. The three sections correspond to Thomas Aquinas’ three acts of the mind—1. terms, 2.
propositions, and 3. syllogisms. Each lesson has homework questions and many extra exercises
for comprehension. Sections are written in numbered sequence for easier content location.
Highlighted, or areas in red type, indicate statements of greater importance.
I welcome any suggestions for addition or correction that you may see fit to suggest. You can
contact me at [email protected].
5
LESSON ONE — SCIENTIA SCIENTIARUM — THE SCIENCE OF ALL
SCIENCES
1. Logic is a science (and an art) that is essentially intended for use—an initial definition.
2. “For what use,” you might ask?
Answer: Everyday thinking, in study, writing, conversation, speech, and debate.
See Extra 1—Eight Good Reasons Why You Should Study Logic
4. This course is based on the logic of Aristotle who probably was the first person to organize
logic as a science.
5. Other folks have systems as well—the Stoic, Zeno of Citium, Jean Buridan of France, and
the like.
6. Aristotle’s system, refined by Thomas Aquinas, is by far the most reliable and complete.
7. Question for discussion: (answers will vary)
Do you believe people today think differently from the way they did 2300 years ago?
8. Aristotle's logic is as valid today as it was then because it incorporates that which is common
to all people in any age.
9. His ideas contain universal standards of thought that are valid for everyone in all times,
regardless of race, creed, or nationality.
10. Aristotle expressed these ideas with exceptional precision and forethought.
11. Historically speaking, Aristotle was a Greek philosopher who wrote his treatises in the fourth
century B.C.
12. Aristotle (384-322 B.C.) was the first to have constructed, in a series of six treatises, (later
entitled the Organon) a system of logical inquiry.
13. They are: (1) Categories, (2) On Interpretation, (3) Prior Analytics, (4) Posterior Analytics,
(5) The Topics, and (6) On Sophistical Refutations.
6
14. Exercise: Lookup: Aristotle, Socrates, Plato, and Thomas Aquinas. Write a short paragraph
on each one.
CAN LOGIC EXPLAIN EVERYTHING?
1. Logic, in itself, doesn’t solve problems.
2. It’s an instrument, a tool to help you develop error free thinking.
3. Logic puts order in your thought process. However, you have to have something in your mind
to order—you need knowledge in other fields (history, languages, science, math, and the like as a
basis to apply the science of Logic to).
4. The knowledge of logic is not a substitute for thinking through a problem.
5. As a matter of fact, Logic doesn’t solve any problem. Logic will help you investigate an
argument (we are not talking about a fight) in a more orderly fashion. It will help make your
solution to a problem more certain. NOTE: In philosophy, ☺ “arguments” are those
statements a person makes in the attempt to convince someone of something, or present
reasons for accepting a given conclusion.
Examples of logical arguments:
All film stars are celebrities.
Halle Berry is a film star.
Therefore, Halle Berry is a celebrity. Valid but circular (discussed later in the text)
Some film stars are men.
Cameron Diaz is a film star.
Therefore Cameron Diaz is a man. Invalid—doesn’t support the conclusion. (more on this later)
6. Logic can be compared to a blueprint for a building. A blueprint can’t take the place of a
hammer, saw, wood, and nails to build a house. It’s the organizer of the house—it makes it
easier to construct the house correctly. You still need the other stuff!
7
7. So, Logic helps organize your reasoning, like the blueprint organizes the building of a house
or a bridge.
8. If reasoning is all jumbled it will probably turn out to be full of fallacy—mistakes in
reasoning.
Example of a fallacy:
The Brooklyn Bridge is made of atoms.
Atoms are invisible.
Therefore, the Brooklyn Bridge is invisible. It’s a solid bridge, not an atom.
9. Logic helps prevent mistakes in thinking like the above argument.
10. Your process of reasoning should conclude in knowledge just as the drafting of a blueprint
should result in the building of a house.
11. ☺At this point in our study, we can define Logic as: practical knowledge that is useful as
an instrument for other kinds of knowledge—our second definition, but not the last.
THE SCIENCE OF LOGIC?
1. We will add to our definition of Logic the notion that it is a science. So, the question is: What
is science and how is Logic a science?
2. ☺Science is defined as an organized or systematized body of facts—one definition for
science.
3. This definition is too general. A telephone book or a dictionary could also be included in it.
4. Science is more than just organized facts.
5. ☺Science is essentially a habit of the mind—a better definition.
6. Science looks at reality and attempts to explain what is going on. Logic does the same thing.
8
7. Science helps us judge and interpret facts in the light of a uniform set of principles.
8. Also, to know scientifically is to know why things are the way they are.
9. ☺To know scientifically is to know things through their principles and causes.
10. ☺☺Combining: Science is a habit of the mind whereby things are known through their
principles and causes—the best definition of science.
IS SCIENTIFIC KNOWLEDGE AND COMMON SENSE LINKED?
1. You hear the expression, “That’s just common sense,” frequently bantered about. The phrase
is used a lot, but seldom clearly defined.
2. Everyone apparently knows what he/she means by common sense.
3.☺ Common Sense Definition: When you’re confronted with a practical problem or situation,
you can size it up, with a minimum of deliberation, and find the correct solution.
4. Scientific knowledge should not contradict common sense.
5. Common sense has the ability to form certain basic types of judgments.
6. ☺☺In some way, Logic uses common sense as its foundation.
7. You might say, “Logic makes clear what is already hidden in the judgments of common
sense.”
8. Common sense is the starting point of logical thought. It is limited in what it can accomplish
without direction and organization—here’s where Logic steps in.
9. ☺☺☺The purpose of logic, as a science and an art, is to guide one's intellect in the
performance of these operations toward a secure attainment of truth, especially in
those matters where the attainment of truth is difficult and involved.
9
WHAT FACTORS COLOR OUR IDEAS AND OUR ACTIONS?
1. Many people don’t think correctly when they discuss such topics as religion, education, world
problems, and the like.
2. Why? Because of the many factors that make up what is called a psychological barrier.
3. These psychological blocks are brought about, in part, by your particular temperament and
outlook. Your upbringing and your specific environment all color how you view the world
around you.
Some Examples of Psychological Barriers:
1. There are many people who don’t think or who think wrongly because they are too lazy to
think correctly. “Thinking is just too much work,” they say, “let someone else do it for me.”
2. Thought requires a considerable expenditure of effort, which they are not about to engage in.
3. This mental laziness can be brought on by many things: an overdose of media misinformation,
excessive imbibing in alcoholic beverages, too much work, or even TV addiction.
4. Then there are also the folks who pretend to be good thinkers—many academicians would fall
into this category. They have a lot of knowledge in one area to the exclusion of others. “Hey, I
have a PHD—I’m smart!”
5. Socrates is the classical example of this psychological factor working against him.
The Oracle at Delphi told him that he was the wisest man in the whole of Greece.
However, Socrates didn’t believe the Oracle, so he went about looking for the “man”. He
checked high and low but couldn’t find anyone wiser than himself.
Since he couldn’t find anyone, he figured that the Oracle was correct!
He was “in the know,” whereas all of the other folks only thought they were.
Pride kills!
6. Then there are some people, who because they possess a bit of knowledge in a few areas, think
therefore, they can speak on any topic. As they say, a little knowledge can be a dangerous
10
commodity!
7. Some people call themselves “skeptics.” These are the folks willing to admit that they are not
“in the know,” yet their admission is most often coupled with the understanding that “nobody
else knows either.”
8. Gorgias of Leontini, in Sicily, a famous skeptic said: (1) nothing exists; (2) if anything did
exist, no one would know it; (3) even if a person did know, he would not be able to tell it to
others—wonder what his favorite beverage was?
9. Some folks are wishful thinkers. The opinions of wishful thinkers are, in large measure, based
on personal feelings, emotions, and prejudices.
10. A viable opinion must be based upon some solid reason or reasons for holding it. There is a
big difference between an opinion and something that is known to be a fact.
11. Often opinions are prejudices.
12. ☺A prejudice is a judgment one makes before considering the evidence.
13. Prejudice is a rash judgment. The kind of judgment we make in a flash when we meet
someone for the first time. We are quick to size them up without really knowing anything about
them.
14. Some common prejudices might be: All women are bad drivers. All men are numbskulls.
We can’t reason properly if we let our prejudices get in the way.
15. NOTE: The way people think is often determined by the way they live. Also, the way a
person thinks will often determine his behavior. If you know how a person thinks, you can,
most probably, determine how he will act. If you see someone act in a certain way, you can
begin to understand what he believes.
16. On another tack, people often act first and then think up reasons to defend their conduct—
this is called rationalization.
17. ☺Rationalization is lying to ourselves, by excusing our actions.
18. This is especially true in the way many people think about what is right and what is wrong in
matters pertaining to morality. “I only did that because everyone is doing it.”
11
19. Many say that, “Everything depends on how you look at it,” failing completely to realize that
the way they look at things is largely determined by the way they want to look at them. It’s not
based on fact but desire.
20. ☺Opinion can be defined as: a judgment based on a measure of evidence but without real
proof of what is stated.
21. These, and other factors, are what concern the logician.
22. A logician points out the inconsistencies in an argument.
LOGIC DEFINED
1. ☺☺☺Logic may be defined as the science and the art that directs our mental operations in
such a way that they may proceed with order, facility, and consistency, toward the
attainment of truth.—final definition!
2. The mental operations referred to in this definition are the three basic acts of the intellect:
1. Conception—or simple apprehension
2. Judgments
3. Reasoning
12
3. ☺☺The main outlines of this entire course in logic are constructed on the basis of these
three acts. In other words, Logic can be divided into these three areas of investigation.
4. NOTE:
(1) These three acts make up the material object of logic. In other words, they comprise
the subject matter or content of our science; they are that about which our study revolves.
(2) Logic is also concerned with the mental products produced by the mind; terms,
propositions, and arguments (syllogisms).
(3) These are expressed in language as, words, declarative sentences, and paragraphs.
(4) They are finally combined in argument as in this syllogism:
13
(1) First act of the mind: words such as, Socrates, man and mortal
(2) Second act of the mind: a declarative sentence such as, Socrates is a man.
(3) Third act of the mind: a paragraph
All men are mortal,
and Socrates is a man,
so Socrates is mortal.
See Extra 2—Difference between Grammar, Semantics, and Logic.
5. What does Logic study?—It studies the three basic operations of the mind together with their
internal products and the means by which we express them.
6. It’s important to know the material object of any given science. However, it’s even more
important to have an exact knowledge of its formal object.
7. Above, we indicated that the material object of a science is the subject matter or content that
Logic deals with.
8. ☺A formal object is the specific point of view of the person studying the material object.
9. Consider three people looking at the same automobile regarding it from different points of
view. A mechanic is interested in the way the car performs; a designer, in its styling; an
upholsterer, in its interior appointments.
10. Exercise: Consider a decorated birthday cake—how does a baker, chemist, and physicist
look at this object? How would you at your party? See if you can answer this on your own.
11. A logician and psychologist study just about the same material object—our mental
operations.
Both disciplines have different formal objects—logician = how we should think;
psychologist = how we actually do think.
12. Let’s look at the last part of our definition of Logic—logic helps to make thinking orderly,
less difficult, and consistent.
13. ☺ Order can be defined as organizing concepts, propositions, and argument into their
proper form.
14. Disordered thinking:
a. A missing part. Suppose you divide the term “government” according to different
levels of administration.
14
Example: the term “government” divided into Local, and International.
The order here is illogical because your division is missing “national”.
b. A part outside of its relation to a given unit. If you included the United States as a
member of the division, it would lack the proper order.
Example: the term “government” divided into Local, State, National, US, and
International
c. Two (or more) parts not belonging to the same whole or unit. Although the following
example is silly, it will serve to illustrate our point:
Example: the term “schools” divided into colleges, high schools, and schools of fish.
15. Less difficult:
Logic teaches you what general rules to observe and what general fallacies to avoid. In this way
logic makes it easier for you to learn and to know.
16. Logic helps you to be consistent:
As a matter of fact, consistency is so intimately bound up with logic that to be logical means to
be consistent.
17. Logic, together with the order, facility, and consistency that it brings to bear on our thinking
processes is a means toward the attainment of truth.
1. All our reasoning is for the purpose of arriving at new judgments that are true.
2. Although the material truth of a judgment and the validity of a reasoning process
are distinct from each other, they should never be separated in practice.
STAY STRONG!
1. Part of what makes the study of Logic difficult, is that you haven’t ever studied a subject like
it—it’s, for the most part, completely unfamiliar to you. It’s full of new terminology, like
learning a new language.
2. The good news is there is nothing to unlearn or revise. The only pitfall—if there is one—is
your own impatience and possible discouragement as you begin.
15
3. Buck up, relax, and keep plugging ahead. You will learn a lot and you will be a much better
student after this study than you are now. Trust me!
Check out Extra 2a: An Overview of Logic
See Homework Lesson 1
Extra Materials
Extra 1:
What Good Is Logic?—Eight Good Reasons Why You Should Study Logic
1. Order. What good is Logic? Trust me on this one, Logic helps you think in a more orderly
way, and more clearly.
2. Comprehension and Reading. Logic will benefit you with all your other courses. A
knowledge of Logic will help you to read and comprehend any text more clearly and effectively.
3. Writing. If you can think clearly, you will be able to write more clearly. Clear thinking
promotes clear writing.
4. Persuasion. Logic has the power of persuasion. The persuasive power of clear and ordered
thinking is catching. People admire someone with this ability.
5. Wisdom. Philosophy, loosely translated, means the love of wisdom. Logic can help you attain
wisdom.
6. Happiness. Your logical mind can help you become happier in life. By clear and ordered
thinking, you will be more able to attain your goals, making you a happier person. Logic, by
helping you think more clearly, will help you become happier.
7. Democracy. An ordered mind is not so easily swayed by political shenanigans. In a
democracy, it’s important to be informed and to be able to weight a person’s arguments carefully
in order to arrive at the truth or error in the discussion.
8. Religion. Religious beliefs require faith. Some religious beliefs may depend on faith,
however, those beliefs cannot go against Logic. It will help you evaluate a religious argument if
you know logic. BACK
16
Extra 2:
Difference between Grammar, Semantics, and Logic.
It’s important to distinguish Logic from Semantics and Grammar. Many folks confuse the three.
Logic deals with concepts. Semantics and Grammar deal with language, or the communication
of concepts. In particular, Logic deals with mental symbols, how mental entities signify real
entities, how these mental signs relate to each other, and what relationships lead to further
knowledge.
Semantics and Grammar deal with physical signs (written or spoken signs, gestures, and the like,
i.e. something physical, something known through the senses. In particular, Grammar deals with
the order of words within a sentence, and the order of sentences. It studies the order of words
which best can express concepts or thoughts. In other words, Grammar deals with words and
word-order as material through which meaning can be sustained and conveyed.
Semantics deals directly with the meaning of words. Semantics studies the symbolism of
physical things (written words, figures, spoken words, gestures, etc).
Logic asks, “What is the reality signified by this mental symbol?” Semantics asks, “What is the
meaning (the mental entity) signified by this physical symbol?” A physical thing can be a
symbol only in so far as it is charged with meaning—with a mental entity, a concept. Language
directly signifies concept and through the concept it signifies things.
The fact that language is a conventional sign makes Semantics necessary. But Semantics can’t
substitute Logic. No clarity of words and word meaning will help a mind logically
undisciplined. Semantics studies especially the social implications of language. BACK
Extra 2a
An Overview of Logic
The info presented here is the foundation for everything else in logic. If you don’t understand it
clearly at some point in the course, you’ll be hopelessly confused later on. It’s explained in more
detail as we move along through the Lessons.
The term ‘man’ has been, for centuries, defined as a rational animal—one who is able to reason.
We can see common forms, or structures, in all human reasoning, no matter what the contents, or
objects, that we reason about. Logic is the science that studies the forms in human reasoning.
17
The fundamental structure of all reasoning is the movement of the mind from the premises to a
conclusion. We attempt to prove that the conclusion of our argument is true through the truth
found in the premises—the two propositions that lead to the conclusion.
The two basic kinds of reasoning are inductive and deductive. Inductive reasoning reasons from
particular premises (e.g. “I'm mortal” and “You're mortal” and “He's mortal” and “She's
mortal”), usually to a more general or universal conclusion (e.g. “All men are mortal”).
Deductive reasoning reasons from at least one general or universal premise (e.g. “All men are
mortal”) usually to a more particular conclusion (e.g. “I am mortal”). Inductive reasoning yields
only probability, not certainty.
You can’t conclude that all men are mortal on the basis that thirty men are mortal.
Deductive reasoning, when correct, yields certainty—it’s certain that if all men are mortal, and if
I am a man, then I am mortal.
In our study, we found that in order to have a valid argument there are three necessary
conditions.
All the terms of the argument, usually three, must be clear and mean the same thing. Secondly,
the major and minor premise must be true. Thirdly, the argument must be valid.
A “term” in logic is the subject or the predicate of a proposition (a declarative sentence). Terms
are either clear or unclear. Terms are not true or false—they are just what they express. They
can be adequate or inadequate but not true or false.
All propositions are sentences—declarative sentences in English grammar. True means that the
sentence is equivalent to reality. False means that the sentence does not correspond to reality.
The science of Logic will give you the tools to determine the validity of any argument.
The rules of logic conform to common sense. If something doesn’t make sense, the argument
probably is invalid.
Remember, as stated above, terms are either clear or unclear. Propositions are true or false.
Arguments are either valid or invalid. So, ask yourself these three questions when evaluating an
argument: Are the terms all clear and unambiguous? Are the premises all true? Is the reasoning
logically valid?
The argument is both true and valid if your answer to the above three questions is yes. BACK
18
Exercises for Homework:
Homework Lesson 1
1. Write a short statement on Socrates, Plato, Aristotle, and Thomas Aquinas. Two or three lines
is enough. Give their dates, where they lived and something about their lives.
2. Pick out any 5 of the 8 Good Reasons to Study Logic. Write a few sentences about each of
your five choices.
3. Why is Aristotle’s Logic as valid today as it was in his time?
4. Who, in the 13th century, refined Aristotle’s system of Logic and made it Christian?
5. In Logic, what is an “argument”? Could you write a short argument? Hint, first sentence: All
men are mortal.
6. How do you define science?
7. Why do we say that Logic uses common sense as its basis?
8. Explain the psychological barriers; lazy thinking, prejudice, and rationalization.
9. Define Logic. (Its final definition)
10. List the three basic acts of the mind?
11. On the graphic, The Three Acts of the Mind, what is the logical expression and linguistic
expression of each of the three acts—first, second, and third act of the mind?
12. What do we mean by the material object of a study?
13. What do we mean by the formal object of a study? How would a baker, chemist, or a
partygoer view a cake according to its formal object?
14. What is the material object of Logic? What is its formal object? BACK
19
LESSON TWO—HOW DO WE FORM IDEAS—CONCEPTS?
1. All human knowledge begins with our five senses.
2. Science tells us that we humans, begin to sense the world before we are actually born.
3. However, as we grow, our knowledge doesn’t end there.
4. As we grow and develop, we acquire a new kind of knowledge. This new knowledge goes beyond
our powers of sensation.
5. This new knowledge is, in some way, an understanding of the very nature of things.
6. When you said, for the first time, “I see” meaning “I understand,” you transcended sense
knowledge.
7. To know how this happens is to recognize one of the powers of the human intellect—abstraction.
8. ☺To abstract is to know something about the essence (nature) of things, and this abstraction is
one of the basic functions of our concepts.
HOW WE ACQUIRE CONCEPTS FROM OUR SENSES
1. ☺To understand something means to know it beyond the level of sensation.
2. The question is; how do we understand? And where do concepts (ideas) come from?
3. It’s important to have some notion of how we get to know things through our concepts. This is
actually the area for psychology to study and in some degree, philosophy.
4. If we only knew things with our senses, we could never judge or reason about the things we
know—we reason using universals, not particulars. More on this later.
5. Every judgment that we make and every reasoning process we employ presupposes some
understanding of things through concepts.
20
6. So, how do we get our concepts, our ideas? We’ll use the term concept and idea synonymously.
7. To begin: All of our ideas come from our sensory perception of objects.
8. We know that each of our senses has a special job to do.
9. The information that we receive from our senses is specific (the color red, that musical note,
my desk at home and the like.).
10. Our senses can’t give us a unified impression of anything. A sense impression is like a
photographic snapshot. It lets us grasp one thing in one moment of time. Yet, we end up with a
unified idea of a thing—all dogs, all people, all desks etc.
11. Because this is happening, we must have an internal faculty which coordinates these many
impressions into a whole—all the thousands of times we have seen a tree and understood it to be a
tree.
12. This internal faculty is called, among other terms, synthetic sense. NOTE: the synthetic sense is
only one of four of our internal senses of memory, imagination, and the cogitative power (which is
often referred to as “instinct”).
13. Our synthetic sense coordinates all the scattered impressions of the external senses into a unified
whole. The result of this coordinating is a single sense image, technically known as a phantasm.
14. Don’t confuse the phantasm with the concept.
15. A concept is universal. A phantasm would not apply to all the members of one and the same
class or species. The phantasm is concrete and particular—again, like a photo of an object.
sensory perception phantasm concept
21
ABSTRACTION & HOW WE FORM CONCEPTS
1. How do we get our ideas of things? If a sensory perception and a phantasm of a frog is not an idea
of a frog, then how do we get the idea?
2. ☺Abstraction is an activity of the intellect.
NOTE: We discuss all these operations of the mind as though they are separate and
distinct from one another. That’s so we can talk about them. In reality, this process of
forming concepts is almost instantaneous.
3. The intellect focuses its attention upon the phantasm—example: DOG
4. The intellect reveals the intelligible (knowable) parts of the object as they are present in the
phantasm—animal and non-rational.
5. The intellect then disregards the sensible features of the object in question—brown, barking, 20
pounds.
6. The chief concern of the intellect is to learn something of the essence or nature of the object—we
might call the nature of dog in our example dogness.
7. Remember: The essence of a thing is that which makes it the kind of thing that it is. It’s the
nature of a dog, completely stripped of its accidents like color, height, weight, and the like.
8. For example, the essence “man” is that which makes a man a man and not a brute or a plant.
9. The nature of a person does not include the color of her hair or the complexion of her skin, her
height, weight, ethnic group, and so forth.
10. ☺The essence is uniformly present in all members of a species and makes them what they are.
Example:
All persons are rational beings.
All humans are capable of language.
11. ☺☺The point of this analysis is to show that the senses do not actually grasp the essence, but
only the sensible characteristics of the object that they perceive—their accidents.
22
12. Only the intellect, by means of abstraction, can achieve some knowledge of the essence of a
thing.
13. Guess what? You can’t imagine an essence—it’s not sensible. It’s only known by your intellect.
Example: Close your eyes and think of “dog.”
You are thinking of a particular dog, not all dog!
14. When the intellect abstracts, it considers its object from a point of view not shared by the senses.
See Extra 3: Terms, Propositions, Arguments
WHAT EXACTLY IS A CONCEPT?
1. ☺The concept, like the phantasm, is a representation of an object.
2. It represents those aspects of an object which are knowable but escape our sensible knowledge.
3. The concept shows us the essential characteristics that an object has in common with others of the
same class.
4. The concept of “cat,” for example, is not, like a phantasm of an individual cat—the grey, feral cat
taking a short cut through my backyard.
5. The concept “cat” represents what’s common to all cats and can be applied to all cats.
6. The concept itself, as it represents a certain essence or nature, is universal; for this reason it can
be applied to all the individuals of a class.
A LOOK AT COMPREHENSION AND EXTENSION
1. Every object has within itself certain distinguishable features or characteristics that are capable of
being understood.
2. These are called the notes of the object—notes=characteristics.
23
3. These same notes, as actually understood, are the notes of the concept itself.
4. The essential characteristics of the concept “plant” are substance, material, living, nonsentient.
5. These are the notes of the concept “plant” which go to make up its comprehension.
6. ☺ The comprehension of a concept is the complete collection of its essential characteristics.
7. The comprehension of a concept is the concept. It’s the concept’s distinguishable
characteristics.
Example: A plant is a living being. “Living” is part of the concept of a plant. It distinguishes a plant
from a rock.
8. In addition to its comprehension, a concept has extension, or denotation.
9. ☺Extension is not the concept itself. It’s the objects to which the concept refers or applies.
Example: The extension of “plant,” includes all the subclasses and individuals to which the
concept applies—roses, lilies, marigolds, and the like.
10. Summarizing:
a. The comprehension of a concept is the total of its understandable characteristics.
b. The extension of a concept is all of the objects to which it applies.
WHY DO WE NEED TO UNDERSTAND COMPREHENSION AND EXTENSION?
1. For the logician, the importance of distinguishing between the comprehension and the extension of
a term is vital.
2. When we form a judgment or a reasoning process, which of the two predominates, comprehension
or extension?
3. The question is; which point of view predominates, comprehension or extension?
24
4. If we consider a certain object from the viewpoint of its comprehension, then the object that we’re
considering is part of the nature of the thing.
5. To consider a thing’s nature means that we are considering it by abstracting that notion from the
individuals that have that nature.
Example:
A dog is a better pet than a cat. or
A cat is a better pet than a dog.
6. You’re not talking about individual dogs or cats here. But “dogness” and “catness”.
7. You’re not considering its extension but its comprehension—a universal. As one student put it, “its
‘ness’”.
8. If we consider individuals, and not their nature, we are referring to the extension of a concept.
Example:
“All of the boys in the Jones family are heavy eaters.”
9. You are not considering the nature of “boy” or the nature of “family.” Here, the extension of
“boys” and “family” predominates.
10. NOTE: In every proposition there’s a bit of comprehension as well.
See Extra 4: Comprehension and Extension
See Homework for Lesson 2
25
Extra Materials
Extra 3: Term, Proposition, Argument
Identify each of the following as a term, a proposition, an argument, or none of the above.
1. Everyone
2. Everyone in America
3. Everyone in America and in the rest of the world as well
4. I think, therefore I am.
5. Are you mad?
6. You won't pass.
7. You won't pass because you haven't studied.
8. Don't take this logic course.
9. Please be quiet.
10. A falsity so obvious that it couldn't fool a child
11. God exists.
12. It ain't broke, so it don't gotta be fixed.
13. You're weird.
14. Everything in the kitchen sink except the sink itself.
15. Ouch!
BACK
26
Extra 4: Comprehension and Extension
Determine whether the comprehension or the extension is predominant in each of the
underlined terms below. Use C or E.
___1. The petunias in this room are to be shown at the Flower Fiesta.
___2. The whale is not a fish.
___3. Americans are a peace-loving people.
___4. The passengers at gate 29 are waiting for the next flight.
___5. Kleptomaniacs have little regard for the property of others.
___6. Any good joke is worth a laugh.
___7. Uncle Jerry’s jokes no longer amuse me.
___8. A charitable person has regard for the feelings of others.
___9. The water in this tub is too hot for a bath.
___10. Our lawn is all burned out.
(a) Create your own five sentences that have comprehensive subjects.
(b) Create your own five sentences that have extensive subjects.
BACK
27
Exercises for Homework:
HOMEWORK FOR LESSON 2—CONCEPTS AND THEIR ORIGIN
1. How do we first come to know the external world?
2. Do we have concepts at birth? Explain your answer.
4. Are the external senses of themselves sufficient to give a unified impression of any given object?
3. The five external senses are not sufficient to explain how our mind can go from a singular object to
a universal concept. What is the process called that helps us arrive at a universal concept?–hint; it
starts with an “a”.
4. Once we sense a thing, our mind forms a ____________ _______________.
5. From there, we use one of our four internal senses called ________ ________ to form a single
sense impression called a ______________.
[Finally, we come to know some part of the nature of a thing—its “ness”, which is universal and not
particular. All these mental operations deal with singulars. Only the concept is universal.]
6. What does the concept reveal about an object? Explain.
7. What is meant by saying that the concept, unlike the phantasm, is universal?
8. At this point in your study, how would you define a concept?
9. Can you draw the progression of how we first see a specific tree and then end up with the concept
“treeness”? For example, start with TREE—five senses—etc.
10. What is the comprehension of a concept? What is the extension of a concept? Explain both and
give an example of each.
11. Why do we need to be able to distinguish between comprehension and extension? BACK
28
LESSON THREE—LET’S INVESTIGATE TERMS (WORDS)
THE FIRST DIVISION OF LOGIC
1. Because we express our ideas using a language, we must understand language.
2. However, the logician studies language differently from say, a grammarian.
3. The logician studies language from a different point of view than the grammarian—he has a
different formal object.
4. We aren’t interested in such items of language as correct spelling, punctuation, sentence structure,
paragraph arrangement, and the like.
5. What we are interested in is terms, propositions, and syllogisms.
6. So, we are going to first investigate terms and their various uses. Then later, propositions and then
syllogisms.
7. ☺We define a term as an outward expression by which we communicate our ideas (concepts) to
others.
TERMS SIGNIFY SOMETHING
1. A term is a certain kind of sign—it signifies something other than itself.
2. ☺A sign is anything which points to something beyond itself.
See Extra 5: Sign
3. A term is something sensible, that is, perceptible by means of the senses—we can see, hear, or
even (if we use Braille) feel a term.
4. Read the word D-O-G. We not only understand the meaning of that word but see the letters that
29
make it up.
5. In fact, we could not communicate the meaning of this term unless we can make someone else
perceive it.
6. Concepts and terms are both signs, but they are not the same.
7. Concepts, unlike terms, are neither sensory nor sense-perceptible.
8. We cannot imagine a concept, far less see, hear, or feel it.
9. Another difference is; terms are conventional signs, whereas concepts are natural signs.
Conventional=based on the consent of the various peoples.
10. All terms are conventional signs, because their meaning is determined by usage.
11. Although the concept is a natural sign, not every natural sign is a concept.
12. There are many natural signs that are sensible—smoke as a sign of fire, for example.
13. ☺Whether a natural sign is sensible or intellectual (a concept), it is not something that we
arbitrarily construct for ourselves.
14. Terms are rather arbitrary. Concepts are universal.
Example: All peoples come to know DOG in the same way or we couldn’t communicate.
IS “WORD” AND “TERM” SYNONOMOUS?
1. ☺A term is a sensible sign which consists of a word, phrase, or clause that could be a subject or
predicate of a sentence (proposition).
2. Can any word function as a term? NO!
3. Certain words like articles, prepositions, and conjunctions (the, a, of, with, and, or, and so on)
can’t—they don’t signify an object of thought.
30
4. Certain words like nouns, verbs, and adjectives do have an independent meaning, and do function
as terms.
5. Terms are either simple (one word) or complex (a combination of words).
6. Even if a term is complex—if it signifies a single object of thought, we consider it one term.
7. The following are examples of complex terms:
The man who lives next door.
Whoever wins the next election.
The last person who entered this room yesterday.
8. ☺A term is a word, or combination of words, that signifies an object of thought.
9. Looking Back:
a. If a term is simple, it’s a word that stands by itself—”onion”.
b. If a term is complex, it’s a combination of words—The Man in the Iron Mask.
c. “Conventionally signifies” means that a term is a man-made sign as opposed to a natural sign.
d. “Object of thought” is what the term signifies—something understood by the mind in the act of
simple apprehension.
A LOOK AT TERMS AS CATEGORIES—WHAT THEY REPRESENT
1. A term can be considered in two ways; what it represents, and its use—like a subject or
predicate.
2. If we look at a term by what it represents we are considering it as one of ten categories.
3. ☺Categories are the widest possible classifications of terms and concepts under which
predicates can be placed. We call them the ultimate genera or genus.
4. NOTE: A genus is a universal concept or term that includes other things.
5. Example:
The genus “animal” includes brute beasts as well as man—rational and irrational.
31
6. A category—ultimate genus— is one which is not reducible to some other type of genus.
7. Example:
Man is a rational, sentient, living, material, substance.
Substance is the ultimate genus in this division.
8. If you know how to categorize something, you are on your way to making good
definitions.
9. Again, only those terms that are most general are considered categories.
10. Example: Take the term “red.” On the one hand red is a general term, since it includes different
shades of red. Yet, “red” itself belongs to a more general category, “color.”
11. But since “category”, as we understand it in logic, is not any general term, but one so general that
it cannot be fitted into some other category, there is an ultimate category to which both “red” and
“color” belong. This category is called “quality.” It’s an irreducible category.
SUBSTANCE AND ACCIDENTS
1. Terms and concepts refer either to things or to certain aspects of things.
2. There is only one kind of “thing,” and that is called substance.
3. Examples include; dog, plant, stone, desk, antelope and the like.
4. We also consider as real what belongs to a substance—size, shape, color, age, and so on.
5. These characteristics are called accidents.
6. Accidents are a part of the substance and they make it what it is as an individual.
7. An accident can’t exist on its own.
8. Example:
The term, “Old,” doesn’t exist by itself, it exists in a substance. Ex. An old tree.
32
9. We can say that everything exists as either a substance or an accident.
10. Let’s sum up:
a. The categories are a list of terms so general that they cannot be reduced to some other
category.
b. These terms are based on and refer to the order of things themselves.
c. The things to which the categories refer are called either substance or accident.
CATEGORIES EXPLAINED
1. How many categories are there?—10 categories.
See Extra 6: The Categories
2. The basic category is substance. The other nine are all accidents.
3. ☺Substance: is defined as what a thing is.
4. Example:
This dog is a Russian Wolf Hound.
This plant is a marigold.
This desk is a roll top.
5. A substance is the most basic of the ten categories—something which exists in itself.
6. ☺Quantity: can be defined as: “How big?” “How small?” “How heavy?”
7. Example:
The newborn is eight and a half pounds.
Seattle's Floating Bridge weighs over 100,000 tons.
8. ☺Quality: Is defined as any formal characteristics a substance might have.
a. Habits and dispositions: These boys are healthy. Every holy person is virtuous.
b. Different types of capacity: Animals are endowed with instinct. Our team is capable of
scoring a run.
c. Sense qualities: “salty,” “luscious,” “hot,” “pale.” Your chicken is delicious. The surface
of this desk is smooth.
33
d. Figure and shape: “triangular,” “round,” square etc. Our home is shaped like an L. Some
theatres are semi-circular.
9. ☺Relation: It answers to the question: How is one thing or person referred to another?
10. Example:
The person looking over my shoulder is my boss.
Catchers are better hitters than pitchers.
New York is more heavily populated than Chicago.
11. ☺Action: What the substance is doing. Action is doing something to someone or something
else.
12. Example:
The cat is chewing up a grasshopper.
The housekeeper is shaking out the rug.
The fullback is carrying the ball.
13.☺Passion: (passive tense) The substance is being acted upon, is receiving an effect from
another.
14. Example:
The public is being victimized.
My husband is suffering from a cold.
A rocket is being shot to the moon.
15. ☺When: The order of past, present, or future events.
16. Example:
The meeting is tomorrow.
Today is Wednesday.
The child is ten years old.
17. ☺Where: A substance’s position in regard to other things.
18. Example:
My wife is at home.
The convention is in Philadelphia.
My trunk is in the attic.
34
19. ☺Posture: The relative disposition of its parts.
20. Example:
The audience is seated.
The champion is flat on his back.
The students are slouched in their chairs.
21. ☺ Habitus: A substance’s apparel, costume, physical equipment.
22. Example
The horse is saddled.
She is attired in her evening gown.
Our soldiers are wearing bullet-proof Kevlar.
See Extra 7—Categories Exercise
LANGUAGE CAN BE COMPLEX
1. In our daily lives, we use both the category substance and its nine accidents.
2. It’s not always easy to decide which category to use.
3. Words change meaning depending on how we use them.
4. Consider our use of slang.
5. For example, note the following expressions:
“sitting on the fence,” “up a tree,” “dragging his feet,” “stewing in his own juice,” “in the
bag,” “down to earth,” “up in the clouds,” and so on.
6. These notions can’t be translated literally.
7. Terms change their meanings. Language grows and develops.
8. There are some terms (transcendentals) that can’t be categorized at all.
35
9. Example:
being, one, true, and good.
10. When you are trying to determine what category to use, first determine how the term is being
used.
11. What precisely is the term meant to convey?
12. Example:
“sister” implies substance, but it formally signifies “relation”
“being hit by a ball” implies someone or something that does the hitting, but with respect to the one
being hit it formally signifies “passion”, a passive act.
“lying down” implies some kind of place, but it formally signifies “posture.”
13. The following should be considered as substance:
a. Parts of substances, such as hand, heart, hair, tail, blossom, root, and the like.
b. Artifacts, such as house, book, chair, doll.
c. Collective wholes, whether of persons or of things, like team, class, library, bundle,
package.
14. When a term means the absence of something, we place it in the category whose absence it
agrees with.
15. Example:
a. Quality: ignorant, vicious, dishonest, mute, ill, insipid
b. Quantity: weightless, penniless
c. Relation: childless, single, orphaned, widowed, unmarried
d. Habitus: bare, unclothed, uncovered, barefoot, unsaddled
16. The category action is only for things you can see. It doesn’t include thinking, concentrating,
deciding, resolving, and the like. These terms are considered to be qualities.
17. Complex terms signifying one thing alone are considered to be singular.
18. However, the term “brown dog” signifies two distinct natures, that is, a quality (brown) and a
36
substance (dog); or again, “prostrate on the ground” signifies on an equal basis both “posture” and
“where.”
19. Split up these types of terms before you try to categorize them.
HOW ARE TERMS USED IN A PROPOSITION?
1. Terms are used either as a subject of a proposition or as a predicate—remember your grammar?
2. ☺The subject of a proposition is that about which something is said—☺the predicate is that
which gives information about the subject.
3. We have to be able to determine if the meaning of a term changes from one proposition to the
next.
UNIVOCAL, EQUIVOCAL, AND ANALOGOUS COMPREHENSION (meaning) OF
TERMS
UNIVOCAL
1. ☺Univocal: When a term is applied to two or more things in the same way.
2. Example:
Man is an animal.
An ox is an animal.
3. The term “animal” has the same basic meaning with respect to both man and ox.
4. NOTE: All class names are univocal when applied to any member of the class.
5. Example:
A tree is a plant, a shrub is a plant, a rose is a plant.
An elm is a tree, an oak is a tree, a maple is a tree.
6. In these examples, the comprehension (meaning) of the term is the same for all the members of the
group.
37
EQUIVOCAL
1. ☺Equivocal: A term used in two different ways. The meaning of the terms are both different
and unrelated.
2. Example:
I’m writing with a pen.
The pigs are in their pen.
3. Our language is like that. It has many words with different meanings but spelled or pronounced
alike.
4. Example:
A king is a ruler.
This measuring stick is called a ruler.
5. Consider the following:
Every ruler has authority.
Some measuring sticks are rulers.
Some measuring sticks have authority.
6. The equivocal use of terms consists of those terms whose meanings are different.
ANALOGOUS
1. ☺Analogous: Terms whose meanings are different but somehow related.
2. The terms have a partial, not a complete, change in meaning.
3. Example:
Heavy taxes are a burden.
Carrying a load of groceries is a burden.
4. One of the most common instances of the analogous usage of terms is the metaphor.
5. ☺Metaphor: A literary figure of speech that show how two things that are not alike in most
ways are similar in some way.
6. Example:
Her eyes were glistening jewels.
38
7. We like to use metaphors for people: Example: “fox,” “lion,” “tiger”.
8. Not every analogy is metaphorical. There are different kinds of analogy.
9. Consider the following example:
“Democrat” as it refers to a member of the Democratic Party and to one who advocates
democracy as a form of government.
See Extra 8: Univocal, equivocal, and analogous
UNIVERSAL, PARTICULAR, AND SINGULAR EXTENSION OF TERMS
1. ☺The extension of a term is how it is applied to all, some, or only one member of a group.
UNIVERSAL
1. ☺A universal (distributed) term is one which is applied in its complete extension. It is
distributed to all the members of a given class.
2. Example:
All men seek happiness.
No plant is capable of locomotion.
3. Some examples of words that signify universals:
All, whatever, no Each, whoever, none Every, any, no one Everyone, anyone, nobody Everything, anything, nothing
PARTICULAR
1. ☺A term is used as a particular when it’s applied to an indeterminate part of a given class.
2. Example:
Some countries are ruled by dictators.
Certain people do not exercise their right to vote.
39
3. Some examples of words that signify particulars:
a few, very many
almost all, most
many, practically all
SINGULAR
1. ☺A singular term applies to only one specified object.
2. Example:
Abraham Lincoln
Washington, D.C.
The largest telescope in the world
The discovery of America
TERMS; COLLECTIVE AND DIVISIVE
COLLECTIVE
1. ☺A term is collective when it applies to a group or a unit not to the individuals of that unit.
2. Consider this proposition:
All the angles of a triangle are equivalent to two right angles.
3. The term, equivalent to two right angles, applies to the subject as a unit and not to each of the three
angles taken separately.
DIVISIVE
1. ☺A divisive term applies to each individual of the group.
2. For example:
The girls in our family have blond hair.
The members of our team are six feet tall.
3. The predicate here applies to each of the members of the group.
4. Sometimes it’s not so easy to determine if a term applies collectively or divisively.
40
5. For example:
The audience applauded vigorously.
6. Use you common sense here. Consider this use collective—even though each person may not
have clapped at all.
7. Some terms are always collective.
8. Consider the term “organization”:
All organizations have a chairman.
9. It’s important to be able to distinguish between collective and divisive use of terms.
10. Consider this reasoning:
Our organization has ample funds. applied collectively
Since I’m a member of it, I too have ample funds. applied divisively
(Funds applied to the group and then to me)
SUMMARY
I. Meaning of term
A. Terms as signs
B. Concepts as signs
C. Definition of term
II. Terms and concepts as categories
A. Reasons for studying the categories
B. Meaning of category
C. The ten categories
III. Use of terms
A. Univocal, equivocal, and analogous
B. Universal, particular, and singular
C. Collective and divisive
See Extra 9: Collective or Divisive
See Homework Exercises for the first part of Lesson 3
41
FALLACIES—MATERIAL AND FORMAL FALLACIES
FORMAL FALLACIES
1. ☺Formal fallacies are mistakes in reasoning, errors in the operation of the third act of the
mind—illogical argumentation.
2. Example:
Some men are mortal.
Some mortals are fish.
Therefore, some men are fish.
3. “Some men are fish” does not follow from the two premises —”some men are mortal” and
“some mortals are fish,”—even though both premises are true.
4. There is no ambiguity or wrong use of terms in this argument, only bad reasoning.
NOTE: We will look at a few more fallacies in Lesson 15.
Now, however, lets concentrate on seven material fallacies.
MATERIAL FALLACIES
1. ☺Material fallacies are mistakes in understanding the meaning or use of terms, errors in the
operation of the first act of the mind—terms.
2. Material fallacies are not mistakes in logical form but mistakes in the content or matter or
meaning.
3. Most errors and misunderstandings stem from a loose use of language rather than from formal
fallacies—this section is a practical discussion of fallacy.
4. Because we are more often fooled by material fallacies than formal ones, this section is
included with our discussion of terms—called the First Act of the Mind.
5. There are many possible material fallacies, depending on how you classify them.
42
6. We will go a few in order to understand how they work.
7. Like sins, fallacies are easier to avoid if they are labeled.
8. “It would be a very good thing if every trick could receive some short and obviously
appropriate name, so that when a man used this or that particular trick, he could at once be
reproved for it” (Arthur Schopenhauer, The Art of Controversy).
9. It’s less important to “get the answer right” to the question, “Which fallacy is present here?”,
than simply to be aware of them and on the watch for them.
10. For each fallacy, we’ll give a definition, an explanation, and some typical examples of it.
11. The fallacies are grouped under seven different headings, each of which happens to have
seven major fallacies under it; and in each of the seven kinds, the most important and most
common fallacy is placed first.
NOTE: YOU WILL NOT BE ASKED TO MEMORIZE THESE!
43
1. FALLACIES OF LANGUAGE—LINGUISTIC FALLACY
A. Equivocation
B. Amphiboly
C. Accent
D. Slanting
E. Slogans
F. Hyperbole
G. “Straw Man”
2. FALLACIES OF DIVERSION
A. Ad hominem (The Appeal to the Person), including
“Poisoning the Well,”
Tu quoque (“You Too”), and
“The Genetic Fallacy”
B. Ad verecundiam (The Appeal to [Illegitimate] Authority)
C. Ad baculum (The Appeal to Force)
D. Ad misericordiam (The Appeal to Pity)
E. Ad ignominiam (The Appeal to Shame)
F. Adpopulum (The Appeal to the Masses), including
Flattery (or “The Appeal to the Gallery”),
Identification (“I'm one of you!”),
“Everybody Does It,”
“The Polls Say,”
The Appeal to Prejudice,
“Snob Appeal,” and
“The Big Lie”
G. Ad ignorantiam (The Appeal to Ignorance)
3. FALLACIES OF OVERSIMPLIFICATION
A. Dicto simpliciter—(saying something too simply)
B. “Special Case”
C. Composition
D. Division
E. “The Black-and-White Fallacy”
F. Quoting out of Context
G. Stereotyping
4. FALLACIES OF ARGUMENTATION
A. Non sequitur (“It Does Not Follow”)
B. Ignoratio Elenchl (Irrelevant Conclusion, or “Ignorance
of the Chain” connecting premises and conclusion)
C. Petitio principii (“Begging the Question”)
D. “Complex Question”
E. Arguing in a Circle
F. Contradictory Premises
G. False Assumption
44
5. FALLACIES OF INDUCTION
A. Hasty Generalization
B. Post hoc, ergo propter hoc (“After this, therefore caused by this”)
C. Hypothesis Contrary to Fact
D. False Analogy
E. Argument from Silence
F. Selective Evidence
G. Slanting the Question
6. PROCEDURAL FALLACIES
A. “Refuting” an Argument by Refuting its Conclusion
B. Assuming that Refuting an Argument Disproves Its Conclusion
C. Ignoring an Argument
D. Substituting Explanations for Proof
E. Answering Another Argument than the One Given
F. Shifting the Burden of Proof
G. Winning the Argument but Losing the Arguer, or Vice Versa
7. METAPHYSICAL FALLACIES
A. Reductionism or “Nothing Buttery” (especially, Confusing Form with Matter
B. The Fallacy of Accident (Confusing the Accidental with the Essential)
C. Confusing Quantity with Quality
D. The Fallacy of Misplaced Concreteness (Confusing the Abstract with the
Concrete)
E. Confusing the Logical, Psychological, and Physical “Because”
F. The Existential Fallacy (Confusing Essence and Existence)
G. Confusing the Natural with the Common
45
1. FALLACIES OF LANGUAGE
A. EQUIVOCATION
1. We have already seen equivocation when we investigated the equivocal use of terms.
2. It’s the simplest and most common of all the material fallacies.
3. ☺Equivocation is defined as using the same term with two or more different meanings in the
course of an argument.
4. Remember that the term “fallacy” in logic is a mistake in argument, in reasoning.
5. A fallacious argument is one that seems to prove its conclusion but does not. The error lies in
either the terms or the reasoning process.
6. A term is not equivocal in itself. A term is used equivocally when it changes its meaning in
the course of an argument.
7. Concepts are not equivocal, only terms (i.e. the words that express them).
8. To have a concept in your mind is to know what it means. If your mind holds two meanings,
it holds two concepts, not one.
9. You can, however, use one term to express the two concepts—the term “pen” to express both
the concept “ink writing instrument” and also the concept “pig enclosure.”
10. But the two concepts are clear and distinct—the first concept is just what it is and nothing
else; and the second concept is just what it is and nothing else.
11. The problem of equivocation is in the term, not the concept.
12. Quibbling—hair splitting— usually means deliberate equivocation.
46
13. Example:
Since automobile means self-mover,
locomotives must be automobiles.
Mom told us not to go swimming right after lunch,
but I'm not swimming; I'm on a body board.
14. To expose the equivocation or double identity of the equivocal term, use these two steps:
(1)First identify the word or phrase that shifts its meaning.
(2)Then identify the two different meanings by using two different words or phrases.
15. ☺NOTE: To unmake an ambiguity—equivocation—make a distinction.
16. In Medieval times Scholastic Philosophers made many distinctions.
17. Unfortunately, folks today don’t like to take the time to think clearly and exactly.
18. Science is the exception.
19. That’s why science is so successful!
20. ☺The reason we make distinctions is to be able to formulate clear concepts.
21. Many jokes and puns depend on equivocation.
47
Exercises: Explain and remove the ambiguity in the following examples of equivocation by
using the two-step procedure. NOTE: some of these are single propositions rather than
arguments; in this case, identify two possible meanings for the potentially equivocal terms.
1. Those who are the most hungry, eat the most. Those who eat the least, are the most hungry.
Therefore those who eat the least, eat the most.
2. Like most ten year olds, Sam was bored with church services. He overheard his parents
worrying about this: his father said, “I'm afraid Sam is just about bored to death in church.”
Being very literal, this worried Sam. Later that day, Sam asked his minister who were all those
names on the church's war memorial plaque. The minister replied that they were “all the people
that died in the service.” Sam asked, “Was it the morning service or the evening service?” (Find
two ambiguous terms here.)
3. “Who's on first?” (from the famous Abbot and Costello routine)
4. G.K. Chesterton, visiting New York City for the first time, saw two women screaming at each
other from the windows of their 4th floor apartments, which were directly across a narrow alley
from each other. He commented, “Those women will never agree because they are arguing from
different premises.”
5. “What is the highest form of animal life?” “The giraffe.”
6. “Your argument is sound. In fact, it's nothing but sound.”
Foreign language teacher informing a student that he had failed a test on pronunciation: “I will
now pronounce your sentence.”
7. “He cares for her.”
8. “The English don't drive on the right side of the road. Therefore they drive on the wrong side.”
9. The 9th-century philosopher John the Scot Eriugena was sitting at dinner with the rude, racist,
and very drunken king Charles the Bald, who said to him, in a feeble attempt at humor, “Tell me,
John, what separates a Scot from a sot?” The philosopher answered, “Only the width of the table,
48
Sire.” (What word's ambiguity is the point of John's reply?)
10. “Why do you love violence?” “Because I am pious.” “That's ridiculous.” “No, that's logical.
To be like what God is, is to be Godlike, and what God is, is love, therefore to love is Godlike.
To be Godlike is to be pious, therefore he who loves is pious. He who loves violence, loves,
therefore he who loves violence is pious. I love violence. Therefore I am pious.
11. President Clinton, responding to a question from a prosecutor who was investigating whether
he had engaged in illegal activities: “That depends on what you mean by 'is.'“ Could this be a
reasonable, intelligent, and honest answer? Can “is” be equivocal?
12. “Nothing is more expensive than diamonds. But paper is more expensive than nothing.
Therefore paper is more expensive than diamonds.”
49
Amphiboly
1. Amphiboly is not an ambiguous word (or phrase) but ambiguous syntax (word order or
grammatical structure).
2. Simple puns are based on equivocation; slightly longer “language jokes” are usually based on
amphiboly.
Example: What word is pronounced incorrectly by nearly all English people?—Answer:
the word “incorrectly.”
Example: Here is an example of amphiboly that is not a joke, just an ambiguity: “Aristotle
the peripatetic (i.e. the 'walker') taught his students walking.” There are two ambiguities
here. First, who was walking, Aristotle or his students? Second, did he teach them the art of
walking or did he teach them something else while he was walking?
4. The New Yorker magazine used to collect humorous amphibolies, or (as they used to be
called) malapropisms (after Mrs. Malaprop, a character in an old comedy. Malapropism and the
earlier variant malaprop come from Richard Brinsley Sheridan's 1775 play The Rivals, and in
particular the character Mrs. Malaprop. Sheridan presumably named his character Mrs.
Malaprop, who frequently misspoke).
EXAMPLES OF MALAPROPISMS:
1. “WENCH FOR SALE, complete with rope. For further information call 3081.” (advertisement
in the Fairmont West Virginian, reprinted in The New Yorker 3/6/54, p. 104)
2. “SUMMER RENTAL: 5-room house with wood paneling plus small guest and garage in
rear.” (advertisement in Princeton Town Topics, reprinted in The New Yorker 6/12/54)
3. “It won't be a real New England clam chowder unless you put your heart into it.” (New
England Homestead, reprinted in The New Yorker 6/12/54)
4. “GUEST FOR LUNCH ONE WAY TO SOLVE EATING PROBLEM” (headline in the
Providence Bulletin, reprinted in The New Yorker 5/8/54)
50
5. Alice: “Would you - be very good enough - to stop a minute - just to get - one's breath again?”
White King: “I'm good enough, only I'm not strong enough. You see, a minute goes by so
fearfully quick. You might as well try to stop a Bandersnatch.” (Lewis Carroll) A Bandersnatch
is a fictional creature from Lewis Carroll's 1872 novel Through the Looking-Glass.
Exercise: Expose the ambiguity in each of the following examples by the two steps we
learned.
1. “Would you rather a cannibal ate you or a shark?”
2. (title of article in student newspaper): “How To Cook Yourself”
3. (TV commercial): “Drive this 4x4 fully loaded.”
4. (Advertisement): “Dogs bathed, fleas removed and returned to your house for $40.”
5. “And the skies are not cloudy all day.” (“Home on the Range”)
6. “Most men love cigars more than their wives.”
7. (newspaper advertisement): “For Sale: Antique desk suitable for lady with curved legs and
large drawers, also mahogany chest.” (reprinted in the Journal of the American Medical
Association, 9/19/53)
8. “The cook opened the oven stuffed with sausages.”
9. “George, you are being selfish and rude to your sister in calling her stupid. Tell her you're
sorry.” “O.K., Mom. Hey, Sis, I'm sorry you're stupid.”
10. “I'm not myself today.” “I see you have revoked the Law of Non-contradiction.”
51
Accent
1. Accent—the ambiguity comes from voice inflection, ironic or sarcastic tone, or even facial
expression, or innuendo.
2. Accent and amphiboly are the two fallacies that are almost always humorous, usually ironic,
and often sarcastic.
3. Notice, in the following two examples, how the same sentence can have many significantly
different meanings by implication if different words are emphasized:
“I do not choose to run at this time.” (But perhaps he will.)
“I do not choose to run at this time.” (no ambiguity)
“I do not choose to run at this time.” (But I can be forced.)
“I do not choose to run at this time.” (But I can be drafted.)
“I do not choose to run at this time.” (But I may do it tomorrow.)
4. Another, similar example:
“We don't have to tell the whole truth, you know.” (But others do.)
“We don't have to tell the whole truth, you know.” (You only think we do.)
“We don't have to tell the whole truth, you know.” (It's optional.)
“We don't have to tell the whole truth, you know.” (But we should know it.)
“We don't have to tell the whole truth, you know.” (Half truths are OK.)
“We don't have to tell the whole truth, you know.” (Tell myths or evasions instead.)
“We don't have to tell the whole truth, you know.” (But no one else does.)
5. A famous example of accent by sarcasm or irony is Marc Antony's funeral oration in “Julius
Caesar”, from Shakespeare. It’s also a sarcastic undermining of the argument from authority,
the authority of Brutus as an honorable man:
“Friends, Romans, countrymen, lend me your ears;
I come to bury Caesar, not to praise him.”…
“…Come I to speak in Caesar's funeral.
He was my friend, faithful and just to me;
But Brutus says he was ambitious;
And Brutus is an honorable man.
He hath brought many captives home to Rome,
Whose ransoms did the general coffers fill;
Did this in Caesar seem ambitious?
When that the poor have cried, Caesar hath wept:
Ambition should be made of sterner stuff.
52
Yet, Brutus says he was ambitious;
And Brutus is an honorable man.”…
53
Slanting
1. Slanting is sometimes called the fallacy of the “question-begging epithet” because it is
really a form of “begging the question”.
2. “Begging the question” means assuming what you're supposed to prove, and the use of
“slanted” language “begs the question” by telling you whether to like or dislike the thing the
word describes.
3. Instead of proving that the thing it describes is good or bad, it assumes its value or disvalue in
the very description of it—e.g. calling an idea “up-to-date” or “wild,” “traditional” or “stagnant,”
“flexible” or “fickle.”
4. This is equivocation or double meaning because it both describes and evaluates at once, in a
single word. It both denotes a fact and connotes an evaluating attitude toward the fact. That is
why it is classified under the heading of fallacies of equivocation.
5. The most obvious and egregious examples of slanting are in propaganda.
6. But slanting can also be done negatively, by omitting relevant information, and by selecting
only favorable, or only unfavorable, data.
7. This form of slanting is much subtler and harder to detect.
8. The old U.S.S.R.'s official newspaper Pravda was full of blatant propaganda. American
newspapers habitually slant by omission, sometimes as part of deliberate editorial policy, e.g. by
giving candidates of its favored political party more coverage, and the opposing candidates less,
and reversing this when it comes to scandals or accusations.
9. Newspapers also have linguistic policies on divisive issues like abortion: one paper will label
the two sides “pro-choice” and “anti-choice,” another paper will label the same sides “pro-
abortion” and “anti-abortion,” and a third paper will label them “pro-life” and “anti-life.”
10. For papers of the Left, there are no “left-wing extremists,” and for papers of the Right, there
are no “right-wing extremists.” What the Left calls “centrist” the Right calls “left-wing” and
what the Right calls “centrist” the Left calls “right wing.”
54
11. One of the most common forms of slanting is the euphemism: the Holocaust was called “the
final solution to the Jewish problem,” and the slaughter of civilians in war is called “collateral
damage.”
55
Hyperbole
1. Hyperbole means “exaggeration.” This is routinely done by “media hype.”
2. For instance, note every occurrence of the word “crisis” in your daily newspaper.
3. How many happenings are really crises and how many are only “a tempest in a teapot”?
4. Do the same with the word “shocking.” It is highly ironic that this word is used more and
more as Americans find less and less to be shocked at.
5. Another form of hyperbole is the “absurd extension” of the other speaker's claim. Children
often do this:
“You need to clean up your room.” “Oh, so you want me to be your slave.”
“You shouldn't drink so much.” “You're always harping on that.” (A wimpy once-a-
month reminder becomes “harping.”)
“You can't stay out all night. You're only sixteen.” “My life is ruined. I'm a prisoner in my
own home forever.”
6. “Absurd extension” must be distinguished from the legitimate form of argument called
“reduction to absurdity,” or reductio ad absurdum, which consists in proving that if you accept
a certain proposition as a premise, that premise necessarily leads to a conclusion which is
“absurd,” i.e. one which everybody knows is false, and therefore the premise cannot be
accepted.
Example:
Mother (seeing her child take a rock from the Acropolis): You shouldn't do that!
Child: Why not? It is just one rock!
Mother: Yes, but if everyone took a rock, it would ruin the site!
Father: Why did you start smoking?
Daughter: All my friends were doing it.
Father: You're saying that if all your friends jumped off a cliff, you would do that too?
56
Straw Man
1. Straw Man Fallacy consists in refuting an unfairly weak, stupid, or ridiculous version of
your opponent's idea (either his conclusion or his argument) instead of the more reasonable
idea he actually holds.
2. You first set up a “straw man,” or scarecrow, then knock it down, since a straw man is easy
to knock down.
3. One of the rules of medieval debate was designed to block “straw man” arguments: you must
first state your opponent's idea in your own words (to be sure you understand the idea instead of
just parroting the words), to his satisfaction, before you go on to refute it.
4. Even great philosophers often fall into the “straw man” fallacy.
Example: The optimist may “refute” the pessimist by noting that “not everything” is bad,
and the pessimist may “refute” the optimist by noting that “not everything is good.”
Person A: We should liberalize the laws on beer.
Person B: No, any society with unrestricted access to intoxicants loses its work ethic and goes
only for immediate gratification.
5. The proposal was to relax laws on beer. Person B has exaggerated this to a position harder to
defend, i.e., “unrestricted access to intoxicants”. It is a logical fallacy because Person A never
made that claim.
Person A: Our society should be taxed less.
Person B: It is unjust to promote a society that neglects the poor.
6. In this case, Person B has transformed Person A's position from “less taxation” to “neglecting
the poor”, which is easier for Person B to defeat.
57
Extra Materials
Extra 5: Sign
SIGN—That which signifies something other than itself.
Natural Sign:
That in which there is an essential connection between the sign and what is signified. (Smoke and
fire, frown and displeasure)
Artificial Sign (Conventional):
That in which there is no natural connection between the sign and what is signified. (the word CAT
and the animal itself.
Instrumental Sign:
That which is something in itself and indicates something other than itself. This sign is always
known first.
NOTE: Both Natural and Artificial signs are Instrumental Signs.
Formal Sign:
One whose whole essence is to signify. The essence of the concept and sense image is to signify or
mean something else. The formal sign is known last, the thing signified is known first.
BACK
58
Extra 6: The Categories
Taking substance as the first category and then adding the nine accidental Categories, Aristotle's
Categories totaled to ten. He proposed these as the ten irreducibly different ways in which
something can exist. While new discoveries in the future might suggest a change in this enumeration,
his list seems comprehensive enough for everyday experience and our use in logic. The last four can
be named in different ways.
1. Substance - stating a whatness or essence - e.g., dog, man, rose
2. Quantity - stating how much or how many - e.g., numbers,
surfaces, a kilometer, etc.
3. Quality - stating a qualification - e.g., shape, color, happy, hot
4. Relation - stating connections between things - e.g., left, on top of,
before
5. Action - stating what a subject is doing - e.g., pushing, heating
6. Passion - stating what's being done to a subject - e.g., being
pushed, being heated
7. Location (where) - stating the place of a subject - e.g., here, United States,
at home
8. Position (posture)- stating the arrangement of a subject - e.g., sitting,
crouching, prone
9. Time (when) - stating when a subject is operating - e.g., now, last year,
tomorrow
10. Possession (habitus) - stating how the subject is garbed - e.g., armed,
dressed
BACK
59
Extra 7: Categories Exercise
Place each of the following italicized terms in its proper category:
(a) The baby is on his back.
(b) Few persons are untalented.
(c) The picture is on the floor.
(d) The patient is encased in bandages.
(e) Some students are superior to others.
(f) This hound is poised for action.
(g) This steak weighs a pound and a half.
(h) That bird is a blue jay.
(i) The grass is being cut.
(j) I met her last night.
(k) My father is simonizing the car.
(l) His hat is larger than mine.
(m) Some pets are canaries.
(n) This girl is my sister.
(o) The camp is in the mountains.
(p) Your complexion is rosy.
(q) The fish are biting.
(r) The election is next month.
(s) The instructor is looking for some chalk.
(t) The monkey is up a tree.
(u) These portions are unequal.
BACK
60
Extra 8: Univocal, equivocal, and analogous
Determine whether the following italicized terms are used univocally, equivocally, or
analogously:
a. Knot as a unit of nautical measurement; knot in my shoelace.
b. Plato and Aristotle were philosophers.
c. Page applied to a boy and to part of a book.
d. Running for mayor; running a hundred yard dash.
e. Poetry is art; music is art.
f. Water is heavier than air; the air is fresh.
g. To air one's opinions; to air the room.
h. To inherit a fortune; to inherit a small sum.
i. Manufacturing plant; this tulip is a plant.
j. A reddish color; Milwaukee has local color.
k. John is a kind person. What kind of a person is he?
l. Carrots and peas are both vegetables.
m. Hands of a human being; hands of a clock.
n. I can digest my food, but not my logic book.
o. This package is light; this room is light.
p. Bark of a tree; bark of a dog.
q. Litter of pups; litter on the street.
BACK
61
Extra 9: Collective or Divisive
Determine whether the following italicized terms are to be taken collectively or divisively:
a. The books in this room are stacked eight feet high.
b. The pears in this box weigh four pounds.
c. All humans have free will.
d. All of our units are well equipped.
e. My hair is a messy.
f. These flowers make a beautiful bouquet.
g. The United Nations is a world organization.
h. All men have individual rights.
i. The actors of our troop will make a fine cast.
BACK
62
Exercises for Homework:
EXERCISES FOR LESSON THREE—TERMS: THEIR MEANING AND USE
1. How do you define the word, “sign”?
2. Explain the four types of signs. Make sure to give an example of each.
3. Give some of the reasons that make a term different from a concept.
4. Are all words “terms”? Please explain. What kind of words are not terms? Terms can be either
simple or complex. Give an example of each.
5. Categories are considered “ultimate genera”. Can you give two examples of them?
6. Define Categories. Why do we call them the ultimate genera?
7. List the ten categories. Give a short description of each.
8. What is a “substance”, an “accident”?
9. Can an accident exist on its own? Please explain your answer.
10. It can sometimes be difficult to place a term in a specific category. What aspects of our language
can cause this to happen? (three things). Say a little about each of the three. BACK
63
LESSON FOUR—WHY DEFINITION AND DIVISION?
1. There are many things which we know, but we are unable to give a good definition to.
2. For example, almost everyone knows what an airplane is. Try to explain it to someone who
doesn’t know anything about it.
3. ☺The purpose of definition and division is to clarify our knowledge.
THE FIVE PREDICABLES (not as important to study as the categories)(you may skip this
section)
1. ☺A predicable shows us the five ways that a predicate of a sentence relates to its subject.
2. The study of the five predicables and the ten categories go hand in hand—especially if you
intend to develop definitions later on.
3. This area is probably the vaguest aspect of Logic.
4. What do predicables have in common with categories and also how are they different?
5. Predicables and categories are universals that can be applied to many.
6. ☺A universal is a term that can be applied to more than one thing in a class.
For example:
The term “animal” can be applied to men and beasts—animal here is a universal.
7. In a previous lesson, we learned that every term belongs to one of the ten categories either as a
substance or an accident.
8. How do predicables and categories differ?
9. When considering a category, it’s not necessary to see how that term refers to a given
64
subject—you just look at the term itself.
10. Predicables, on the other hand, are predicate terms and our interest here is in how they relate
to the subject.
11. Asking the question, "What does the term stand for?" tells us what category the term belongs
to. And "How does a given term refer to its subject?" tells us how a predicable relates to its
subject.
12. So, the basic question in determining a predicable is: How does the predicate relate to the
subject?
13. Example: Compare the two following statements:
Perjury is something which few people commit.
Perjury is a form of lying.
14. The first statement gives no real clue to the true nature of perjury; whereas the second
statement says something about its essence—almost defines it.
15. ☺It’s this basic point that we must keep in mind throughout the discussion that follows. A
predicable says something about the essence of a subject.
16. If the predicate is of the essence of the subject, it will either give the whole essence or part of
it.
Example:
If we say, "John is man," we actually predicate of "John" his complete essence or nature.
17. When the predicate gives the complete essence of the subject we call it species. Species is the
first of the five predicables—it completely defines the subject.
18. If the predicate gives part of the essence it will be what it shares with other species or what
is different about it.
19. Example: If you say ‘John is an animal” you are predicating the part of his essence that he
shares with other animals. This part is called genus—that part which the species "man" holds in
common with the species "brute." The predicate, animal, belongs to the predicable genus. The
part the subject shares with others.
65
20. Example: If you say “John is rational”, you are predicating the part of his essence that makes
him different from other animals. The predicable here is difference—otherwise known as specific
difference. It’s the part that makes the subject different from others.
21. If the predicate doesn’t state anything about the subject’s essence, it is either a characteristic
or has no connection at all with the subject.
22. Example: If you say “John is able to laugh”, you are giving a characteristic of all persons but
not of brute animals.
23. Able to laugh is the predicable property.
24. Example: If you say “John has brown hair," you are predicating something of the subject
that has no necessary connection with John—a monkey could have brown hair. This type of
predicable is called an accident.
25. ☺There are five predicables: species, genus, (specific)difference, property, and accident.
Note the diagram:
End of the study of predicables.
66
BRIEF DISCUSSION OF SPECIES
1. We start to investigate definition.
2. ☺The best type of definition breaks a thing down into its essential parts—genus and
difference.
3. Often, we define an object by giving its parts that are unique to it and it alone—this type of
definition gives its species.
4. Most definitions contain some type of species but not the strict species.
5. Example: When a zoologist talks about the different species of non-rational animals, he is
using species analogously.
6. The difference between one species of non-rational animal and another—dog and wolf—is
accidental in comparison with the difference between the species man and the species non-
rational animal.
7. We can’t always define things in the strictest sense—such as, man, brute, plant.
8. If we did this, there wouldn’t be very many definitions.
9. However, there are different kinds of definitions. So, we logicians need to take them into
account.
KINDS OF DEFINITION
1. Note the following types of definition:
A. NOMINAL
B. REAL
1. Essential
2. Non-Essential
a. by property
67
b. by extrinsic cause
(1) by efficient cause
(2) by final cause
NOMINAL DEFINITION:
1. The purpose of definition is to set limits to the meaning of the object discussed.
2. In nominal definition the object discussed is called a term.
3. ☺A nominal definition simplifies the meaning of a relatively complex term.
4. A nominal definition may be given:
* by explaining a term by its etymology—word derivation
* by replacing it with a synonym
* by breaking it down into its popular equivalent
* by translating it
* by giving the meaning of a slang term
6. Some examples are:
a. Pachyderm means a thick-skinned animal.
b. A chiropodist is a foot specialist.
c. Cul de sac means bottom of the bag or a blind alley.
d. A matriculation fee is an entrance fee.
REAL DEFINITION—Essential and Nonessential
1. The purpose of a real definition is to give some information about the thing itself (a part of its
essence) that is to be defined.
ESSENTIAL DEFINITION
2.☺Essential definition: Is a combined statement of a thing's proximate genus and essential
difference.
68
3. The first step in the making of an essential definition is to place the term in its proper
category.
4. Let’s define ‘person’. First, place our term in its proper category—substance.
5. Note also the following examples:
Virtue belongs to the category quality.
Poundage belongs to the category quantity.
Motherhood belongs to the category relation.
6. The category of a term is what the subject is—its ultimate genus.
7. The ultimate genus, as opposed to the proximate genus, is the one farthest removed from the
difference—from the essential differentiating feature of the subject that is being defined. The
ultimate genus of both “person” and “brute” is called substance.
8. The proximate genus is the one immediately determined by the difference. The proximate
genus of “person” and “brute” is animal. The difference for the term “person” is rational and in
the case of “brute”, non-rational.
9. We don’t normally use the ultimate genus in a definition.
10. We try to figure it out simply to help us select the proximate genus, which will appear in the
definition.
11. Two other examples of essential definitions:
An organism is a living body.
An animal is a sentient organism.
12. These are definitions of natural substances.
13. What if we wanted to define a zoological or a botanical species.
14. Don’t try an essential definition. Try one of the other types.
15. The reason for this is simple—we just don’t know their exact essence. For example if the
biologist wanted to define a multiflora rose or a Bengal tiger.
69
16. It’s even more impossible to define an artificial object, “house” or “desk”, in an essential
way.
17. This is because an artificial product is a combination of a variety of things—best to define
them according to their purpose.
18. If the term to be defined is one of the nine accidents it’s not a substance and can’t be
defined according to either an ultimate or proximate genus.
19. However, a quasi-essential definition can be given.
For example:
Virtue is a good habit of the mind.
20. The category of the term “virtue” is quality.
21. In saying that virtue is a good habit is to distinguish the term from “vice”.
22. Exercise: Use the chart below and give essential definitions of "body," "organism,"
"animal," "plant," "man," and "brute." In each case state the proximate genus and the essential
difference.
70
NONESSENTIAL DEFINITION
1. ☺Definition by Property: refers to any characteristic of an object which might be found in
other species as well.
2. Example: Chemical properties of a substance—PH, specific gravity, etc.
3. Note these examples of definitions by property:
1. A body is a substance having extension.
2. Man is a food-cooking animal.
3. Mercury is a metallic element remarkable for its fluidity at ordinary temperatures.
DEFINITION BY EXTRINSIC CAUSE—Extrinsic = from outside itself. (two types)
1. ☺Definition by Efficient cause: To set forth an object in terms of the agent which
produces it.
2. Example: The term “egg” is defined: "The reproductive body produced by birds and many
reptiles, especially, in common usage, that of the domestic hen."
3. ☺Definition by Final cause: The reason, end, or purpose for which an object exists or is
made.
4. Definition by final cause is the commonest type of definition and one of the most useful.
5. Example of final cause: “A seismograph is a scientific instrument used for recording the
motions of an earthquake.”
6. Note: Often a definition will combine various elements of the types we have seen.
ACCIDENTAL DESCRIPTION
1. Accidental description is not really a form of definition.
71
2. For example, you can’t define an individual as such. You can only give its accidental
description. This is a substitute for definition.
3. ☺In Accidental Description you end up listing a sufficient number of its specific accidents.
4. Example: Washington, D.C., is the capital of the United States, located on the Potomac
River, and bordered by the states of Maryland and Virginia.
THE RULES OF DEFINITION (there are four)
RULE 1: The definition must be equal with the thing defined—neither too broad nor too
narrow. This has more to do with the extension of the term not its comprehension.
1. This first rule is the most basic test of the soundness of any definition.
2. To define something is to state clearly its comprehension and its extension.
3. A definition should apply exclusively to this one term only.
4. For this reason, a definition should be neither too broad nor too narrow.
5. If the definition is too broad it can be applied to other terms as well. If the term is defined
too narrowly, it won’t completely cover all the aspects of the term defined.
6. Example: If we say, “Education is a process of development”—our definition is too broad.
The predicate term, process of development, applies to other things besides education.
7. Example: If we say, “Science is the causal analysis of physical phenomena”—our
definition is too restrictive, too narrow. Science is much more than a causal analysis of
physical phenomena.
72
RULE 2: A definition should be expressed in univocal terms, and should be kept simple.
1. Don’t use figurative language in the definition. You wouldn’t define the term “eyes”; “The
eyes are the mirrors of the soul.” There is always the danger of constructing a definition by
the use of analogy.
2. You never want a definition to end up more complex than the thing being defined.
3. For this reason, the terms of the definition should be kept relatively simple.
4. The classic example of this type of definition is Samuel Johnson's tongue-in-cheek definition
of a “net” as “... a reticulated fabric, decussated at regular intervals, with interstices and
intersections.”
5. Translated into English: [a network structure of fabric, having the shape of a cross at regular
intervals, with small spaces and intersections.]
RULE 3: The definition should not be circular.
1. One of the most common, and intentionally used violations of logical procedure, is the use of
the circular definition.
2. The purpose of definition is to increase one's knowledge, not the opposite.
3. For example, if you defined the term “dollar” this way, “A dollar is the sum of two half-
dollars”—this would be defining a term in circles.
4. To avoid a circular definition, you would define a dollar in terms of its real and equivalent
purchasing power.
5. ☺The most common circular definition uses synonyms.
6. The use of synonyms is acceptable in a nominal definition. Here its purpose is to confuse the
meaning of a term, not define it.
73
7. Example: If we define the term “doctor,” by stating that, “A doctor is a person who
practices the medical profession,” we are defining in circles.
RULE 4: A definition should be positive, not negative, whenever possible.
1. ☺A definition is a statement of what a thing is, not of what it is not.
2. However, some things are negative in meaning. For example, “A bachelor is defined as an
unmarried man”.
FURTHER REMARKS ON DEFINITION
1. If we are too informal in our speaking and writing, a lack of precision and a confusion of
terminology will occur.
2. To overcome this tendency, use nominal, well thought out definitions whenever possible.
3. This will result in sound logic and good grammar.
4. Some poorly constructed definitions:
A democracy is where the people have a voice in governmental affairs.
Etiquette is how to behave nicely in other people's presence.
A University is where one gets a higher education.
5. A definition should be an objective statement of what a thing is and not a statement of how
you feel about the thing.
See Extra 10: Quotes by Famous Folk
DIVISION
1. If I present you with an outline of the subject matter, you will instinctively realize that this
74
outline—which is really a division—will help you to organize your knowledge.
2. ☺Division is a means of clarifying and organizing our concepts.
3. ☺Division is the break down or analysis of a given whole into its basic parts.
4. Suppose we divide the term “men”, for example, into Europeans, Asiatics, Americans,
Africans, and Australians.
5. In every division there is a whole which is to be divided—men—and there are parts into
which it is divisible.
6. The basis of this division is made according to our continental origins.
7. Consider the following:
PHYSICAL VERSUS LOGICAL DIVISION
1. ☺A physical division is the reduction of any unit into its actual parts.
2. Example: divide the term “engine”, into pistons, bearings, fuel pump, carburetor, and the like.
3. ☺A logical division is the breakdown of a genus into its species or of a species into its
subspecies.
4. Consider the following examples:
Government: divided into executive, legislative, and judicial.
Government: divided into democratic, monarchical, dictatorial.
5. The first division is a physical division, because its parts are the real parts of a government—
75
in this case the branches of government.
6. The second division is a logical division, because its parts are a division of the different kinds
of government.
7. The following rule will help you determine if the division is physical or logical.
a. The whole of a logical division is always predicable of its parts—example; the
term “tree” into elm, oak, maple, etc. We can say an elm is a tree, an oak is a tree, and
the like.
b. The whole of a physical division is not predicable of its parts—example; “tree”
into roots, trunk, branches, leaves and the like. We can’t say that a trunk is a tree, a root
is a tree etc.
8. NOTE: Every physical division is one where its parts are named.
ESSENTIAL AND NON-ESSENTIAL DIVISION
1. Logical division is ordinarily the division of a genus into its species.
2. This means that the whole of a logical division is some kind of genus, and its parts are the
species that belong to that genus.
3. Besides genus and species, there are three other predicables—difference, property, and
accident.
4. If the parts of a thing are divided by reason of a difference in their essence, we would have an
essential division.
5. Example: When we divide the genus “animal” into the species man and brute we are
constructing an essential division, since man and brute are essentially distinct from each other.
6. If the basis of division is either a property or an accident, then the division is a nonessential
division.
76
7. Example: If we divide the term “animal” into tool-using and non-tool-using animal, this is a
nonessential division on the basis of property.
8. If we divide the term animal into hard-skinned and soft-skinned animal, this is a nonessential
division by way of accident.
9. The remote basis of every accidental division of substance is one of the following categories:
quantity, quality, relation, action, passion, when, where, posture, and habitus.
10. Example: If we divide the term "cat" into white, brown, black, and so on, the proximate
basis of our division is color, and the ultimate basis is the category quality.
11. Exercise: Determine first whether the following are essential or nonessential divisions. If
they are nonessential, tell what accident category they belong to (worked out for you).
a. substance: material and immaterial (E)
b. books: large, medium-sized, small (N)(remote)(quantity)
c. people: talented, untalented (N)(proximate)(property)
d. students: well-poised, slouchy (N)(remote)(posture)
e. football players: uniformed, nonuniformed (N)(remote)(habitus)
f. people: married and unmarried (N)(proximate)(relation)
ENUMERATION
1. Enumeration is often confused with division.
2. ☺Enumeration is a listing of individuals, as in naming the Presidents of the United States.
3. As opposed to an enumeration, a division as such gives only classes or subclasses, as when
one classifies Presidents according to whether they had one term of office or more than one
term.
4. Dividing the school faculty according to their departments is a proper division. If we list
each member in the department, that’s enumeration.
77
THE TWO RULES OF DIVISION
1. ☺RULE 1: The parts of a division must be equal to the whole which they divide.
2. It’s a violation of this rule to omit one or more of the parts of a whole.
3. For example, “textbooks” divided into bound and unbound.
4. However, if we divide the term "transportation" into air and land transportation only we are
in error because we omitted transportation by sea.
5. ☺RULE 2: The foundation of the division must be kept uniform throughout.
6. If a division is to be logical, it must remain logical.
7. It would be a violation of rule 2 if you divided the term “climates” into cold, warm, and dry.
Because "dry climate" is a change in the basis of the division.
13. Climates were first divided according to their temperature; then the relative humidity of a
climate was used.
CODIVISION AND SUBDIVISION—A WORD
1. When you divide a term, you can have more divisions than one providing you note them.
2. Example: You can divide the term “science” in various ways; according to subject matter,
according to method, according to recency of discovery, according to relative practical utility,
and so on.
3. All these divisions would be codivisions of science, but each would proceed (logically)
according to its own basis.
4. ☺Codivision is the employment of more than one division, of one and the same logical
unit—each according to a different basis.
78
5. In contrast to codivision, ☺subdivision is the employment of another division which is
subordinate to one of the leading, coordinate members of the original division.
See Extra 11: Defects in Division
See Extra 11a: Type of Definition
See Extra 11b: Would-Be Definitions
See Homework Exercises for Lesson Four
Extra Materials
Extra 10: Quotes by Famous Folk
Directions: See what you can do by identifying the error or truth in these definitions.
1. "Philosophy is unintelligible answers to insoluble problems." (Henry Adams)
2. "A philosopher is a blind man in a dark room looking for a black cat that isn't there." (Lord Bowen)
3. "A philosopher is one who contradicts other philosophers." (William James)
4. "Philosophy is common sense in a dress suit." (Oliver S. Braston)
5. "Religion is the opiate of the people." (Marx)
6. "Law is whatever is boldly asserted and plausibly maintained." (Aaron Burr)
7. “Law is ... an ordinance of reason for the common good, promulgated by him who has care of the
community." (Thomas Aquinas)
8. "A figure is that which is enclosed by one or more boundaries." (Euclid)
9. "Faith is the substance of things hoped for, the evidence of things not seen." (Hebrews 11:1)
10. "Faith is when you believe something that you know ain't true." (Attributed to a schoolboy by
William James in "The Will to Believe")
11. "Economics is the science which treats of the phenomena arising out of the economic activities of
men in society." (John Maynard Keynes, Scope and Methods of Political Economy
12. "Justice is health of soul." (Plato)
BACK
79
Extra 11: Defects in Division
Point out the defects in each of the following divisions:
a. Libraries: school and public.
b. Television programs: sports, entertainment, educational, variety, boxing, and musical.
c. Words: nouns, verbs, adjectives, and swear words.
d. Unemployed persons: hoboes, housewives, and students.
e. Musical instruments: violin, piano, voice.
f. Speeches: political, conversational, oratorical.
g. Taxes: federal, excise, and municipal.
h. Ruler: president, king, yardstick.
i. Mistakes: intentional, unpremeditated, serious.
j. Loans: short term and interest bearing.
k. Paper: stationery, wrapping paper, newspaper.
l. Bark: bark of a tree, bark of a dog.
m. Life: life of an organism, life of the mind, life of a party.
BACK
Extra 11a: Type of Definition
Directions: Identify as best you can the type of definition illustrated by each of the
following:
a. A clock is a device used for indicating time.
b. "Effervescent" means "bubbling over."
c. A juke box is a coin-operated phonograph permitting selection of the records to be played.
d. A honeycomb is a structure consisting of adjoining hexagonal cells of wax made by bees.
e. A tiger is a large feline mammal with transverse, wavy black and tawny stripes on the body.
f. A saddle is a seat for a rider on an animal, bicycle, etc.
80
g. A Mae West is an inflatable life-preserver vest for aviators who may fall into the sea.
h. A plant is a nonsentient organism.
i. A body is a material substance.
j. A body is a substance possessed of extended parts.
k. "Philosophy" means "love of wisdom."
l. Man is an animal endowed with the faculty of articulate speech.
m. Lysol is a clear, brown, oily liquid used as a disinfectant and antiseptic.
n. Sulfur is a nonmetallic element used in the manufacture of gunpowder, matches, and so on, in
vulcanizing rubber, and in medicine.
o. Arteriosclerosis is the hardening of the arteries of the human body.
p. A wound is a physical injury caused by breaking of the skin or other membrane.
BACK
Extra 11b: Would-Be Definitions
Directions: Criticize the following as would-be definitions of their subjects.
a. Security means contentment.
b. Life is when you're still breathing.
c. A separation is not a divorce.
d. Personality is what it takes to have many friends.
e. A circle is something which is round.
81
f. Logic is the science of method.
g. Education is a process involving the interchange of notes from teacher to student.
h. Television is the graveyard of outdated movies.
i. Mental cruelty is when you hurt the feelings of your partner in marriage.
j. A treaty is a verbal agreement.
k. An immigration official is a watchdog of aliens.
l. A contract is something which is binding by law.
m. The newspaper is the workingman's textbook.
n. Life insurance is a means of post-mortem protection.
o. A habit is a disposition which is not easily lost.
p. A college degree is a testimony of academic achievement.
q. Vice is a habit.
r. Economic solvency is when you don't have any debts.
s. A student is someone who studies.
t. A teacher is a pedagogue (educator).
u. Happiness is the sunshine of the soul.
v. Gambling is the sport of the devil.
w. Character is what makes you a man.
x. A secret is a form of pretense whereby nobody is supposed to know what everybody else
knows.
82
y. A high-school teacher is an educated policeman.
BACK
Exercises for Homework:
EXERCISES FOR LESSON FOUR—DEFINITION AND DIVISION
1. Define the term, predicable.
2. What basic point must we keep in mind throughout the discussion of predicables?
3. List the five predicables.
4. Draw the diagram of the five predicables as shown at the top of Section 4, page 4.
5. Using that diagram and this statement, “A plant is a non-sentient being,” could you say which part of the
predicate is its species, specific difference, and its genus?
6. What is the purpose of a nominal definition?
7. List three of the five ways a nominal definition may be given.
8. How would you define definition by final cause?
9. List the four rules of definition and give an example of each.
10. Why do we study division?
11. Define division.
12. Define physical division and give one example of it.
13. Define logical division and give one example of it.
14. What is enumeration and can you give an example of it?
83
15. What is Rule one of division?
16. What is Rule 2 of division? Why is it important?
17. Give an example of an error using Rule 2.
Extra Homework Practice:
I. Explain the basis of each of the following divisions:
a. Science: inductive and deductive.
b. History: ancient, medieval, modern.
c. Toy: plastic, rubber, wooden, and so on.
d. Terms: collective and divisive.
e. People: sanguine, phlegmatic, choleric, melancholic.
II. Determine whether each of the following is a physical or a logical division:
1. Election: local, state, national.
2. Play: prologue, acts and scenes, epilogue.
3. Library: offices, reading room, stacks.
4. Literature: fiction and nonfiction.
5. Homes: bungalows, cottages, ranch homes, and so on.
BACK
84
LESSON FIVE—CATEGORICAL JUDGMENTS AND PROPOSITIONS
(The Second Act of the Mind)
1. Simple apprehension lets us know something about the nature of a thing.
2. If the thing is rather clear, it can be defined and divided.
3. Gaining knowledge doesn’t end with formation of concepts.
4. After we have formed a concept, we make judgments about it.
5. Example: When you formed the concept “train”, you judged that the train is large, gives out
smoke, is noisy, moves on tracks and the like.
HOW WE MAKE JUDGMENTS
1. First question: What does the intellect do when it judges?
2. In this lesson, we’ll consider judgment taken in its most basic sense—the categorical
judgment.
3. To form a judgment, we have to have something that we can judge.
4. We also have to have something that we can assign as predicate to the subject that we are
judging.
5. ☺A subject and a predicate are the only two things we need to form a judgment.
6. Judgment takes place when we mentally affirm or deny that a particular attribute belongs, or
does not belong, to this subject.
7. Example: You are making a judgment about a chair when you mentally say that it is or is not
comfortable.
8. NOTE: When we judge, our intellect unites or disunites two objects of thought in a single
85
immediate act. In an instant, it either affirms or denies one term or the other.
DEFINING CATEGORICAL JUDGMENT
1. ☺A categorical judgment is an act according to which the intellect asserts one object of
thought to be identical or nonidentical with another.
2. NOTE: Judgment is an act of the intellect not any act of sensory knowledge.
3. The nature of judgment involves two distinct objects of thought which is its matter. The
form of the judgment is the mental assertion of identity or nonidentity.
WHEN IS A JUDGMENT TRUE OR FALSE?
1. ☺Only judgments and propositions can be considered, in Logic, to be either true or false.
2. A sense impression is neither true nor false—it just is.
3. Example: Put a straight stick in a stream, part in and part out of the water. What you see is a
bent stick. However, you know that it’s straight despite its submerged appearance. Your
judgment should be, “The stick appears to be bent.” Falsity would occur if you judged, “The
stick is bent.”
4. Is a concept true or false?
5. Not really. You have either a good concept of a thing, or an imperfect concept of the thing.
6. In common, everyday language we say, “That’s a false idea.” In Logic, only a judgment is
considered to be true or false.
7. Concepts are considered to be clear or confused, adequate or inadequate, but not true or
false.
86
8. While we are on the topic of true or false, you hear it stated, “Your reasoning is false.”
9. In Logic, we consider an act of reasoning to be valid or invalid, correct or incorrect, not true
or false.
10. For purposes of logic, it’s important to know that a judgment is true if it affirms of its
subject an attribute that really belongs to it.
11. Example:
Canada is a North American country.
Detroit is not the capital of the United States.
12. A judgment is false if it affirms of its subject something that doesn’t belong to the subject.
13. Example:
France is an Asiatic nation.
Alabama is not a Southern state.
WHAT IS A PROPOSITION?
1. ☺A proposition is a sentence that expresses something that is true or false—a declarative
sentence.
2. A proposition is not:
* A sentence expressing a question. (Where are you going?)
* A sentence expressing a prayer, wish, or hope. (If only I could remember what I said.)
* A sentence expressing an exhortation or a command. (Do this.)
* A sentence expressing an exclamation. (Help!)
3. All these sentences have meaning, but only those sentences characterized as true or
false are propositions.
4. We will be considering only categorical propositions in this lesson.
87
5. ☺A categorical proposition is one that asserts, as a matter of fact, that a certain
predicate does or does not belong to a certain subject.
6. Example:
The doctor is in his office.
My house is not on fire.
THE CATEGORICAL PROPOSITION DISSECTED
1. The matter of a categorical proposition is its subject and predicate. In Logic we refer to the
subject term as “S” and the predicate term as “P”.
2. The form of a categorical proposition is its copula—its joiner.
3. The copula of a categorical proposition is expressed by the verb to be in the present tense,
indicative mood, in the following forms:
* am—am not
* is—is not
* are—are not
4. The idea here is to express any declarative sentence in the above forms.
5. Example:
My father smokes cigars.
6. Expressed in logical form:
My father is a cigar smoker.
88
See Extra 12: Proposition, Yes or No
See Exercises for Lesson Five
Extra Materials
Extra 12: Proposition, Yes or No
Determine which of the following are propositions:
1. Climb over the hill.
2. “Render unto Caesar the things that are Caesar's.”
3. How do you expect to learn?
4. Good day!
5. Many a man is the victim of his own fancy.
6. Uneasy lies the head that wears the crown.
7. Woodman, spare that tree.
8. Please don't go away.
9. India is part of Asia.
10. Is Jerry coming to class today? BACK
89
Exercises for Homework:
EXERCISES FOR LESSON FIVE—CATEGORICAL JUDGMENTS AND PROPOSITIONS
1. What two things do we need to form a judgment?
2. When does a judgment take place?
3. What does our intellect do when we form a judgment?
4. Define a categorical judgment.
5. Is a concept true or false? Is a judgment true or false? Is reasoning true or false? Please explain
your answers.
6. Define a proposition and give an example of one.
7. What is not a proposition?
8. Define a categorical proposition. Give one example.
9. What is the matter and form of a categorical proposition?
10. How is the copula of a categorical proposition expressed? List the three forms. BACK
90
LESSON SIX—QUANTITY, QUALITY, & LOGICAL FORM OF CATEGORICAL
PROPOSITIONS:
1. Let’s look at the most fundamental type of all propositions—the single categorical.
2. This lesson is very important in the study of Logic.
3. ☺☺If you can’t analyze and classify propositions, it’s impossible to have a good
understanding of syllogistic reasoning later on.
6. *NOTE: THIS LESSON IS THE “KEY” TO WHAT FOLLOWS IN THE REST OF THE
COURSE.
PROPOSITIONS—PARTICULAR AND UNIVERSAL
1. The validity or invalidity of an argument depends on the quantity of its propositions—
whether a given premise or conclusion is particular or universal.
2. Much of the time a particular proposition is treated as a universal.
3. ☺It’s absolutely important to be able to tell if a proposition is universal, particular, or
singular.
4. The basic rule for the quantity of proposition:
The quantity of a proposition is determined by the quantity of the subject term.
5. Thus:
A. UNIVERSAL SUBJECT TERM: UNIVERSAL PROPOSITION
Every man is endowed with free will.
No historical event is without significance.
B. PARTICULAR SUBJECT TERM: PARTICULAR PROPOSITION
Some people are not interested in politics.
Certain events are easily forgotten.
91
C. SINGULAR SUBJECT TERM: SINGULAR PROPOSITION
New York is the largest city in the United States.
This man is guilty.
6. NOTE: Words like each, all, whatever, no, & none are signs of universal quantity.
Words like some, certain, and most, are signs of a particular term.
7. ☺As a practical matter, singular propositions are treated as universals—even though it’s
singular, it can be taken in its complete extension.
8. Example: If we say, “John is at home,” we mean “John” in the totality of his person.
9. A singular proposition is not a genuine universal—it stands for only one individual.
10. ☺Every proposition should be taken either as a universal or as a particular statement.
11. It’s not always easy to decide if a term is particular or singular. These are called
indesignates.
12. ☺An indesignate proposition is one that bears no indicated sign of quantity.
13. Example: If we say, “Cars are expensive”—do we mean all or some?
14. Note the following examples:
* A cause is something that produces an effect.
* Man is a moral creature.
* Martyrs are sincere in the profession of their faith.
* Good men are deserving of reward.
* A happy child is a joy to behold.
15. In each of these propositions the predicate has some bearing on the nature of the subject.
These propositions are universal.
16. Note: The subject of each of these propositions stands for a certain nature.
17. When we say, “a happy child is a joy to behold,” we are thinking what it is to be a happy
child.
18. No matter how this judgment is expressed, it is to be understood that the subject is taken as
92
a universal term.
19. For this reason, too, the proposition itself is universal.
20. The following rule applies to indesignate propositions:
☺Every proposition, in which the predicate belongs
to the nature of the subject, is universal.
21. Example: if we say that, “An organism is something endowed with life,” the predication is
universal.
22. Each of the following propositions is universal:
* The telephone is a means of communication.
* Good books deserve to be read.
* A square is a four-sided figure.
* Charity begins at home.
23. However, there are many indesignate propositions that are particular.
24. Note the following are examples:
* Women are fickle.
* Professors are eccentric.
* Students get worried at exam time.
* White collar workers are underpaid.
25. These sentences are considered particular because none of the propositions apply to the
nature of its subject.
26. These kinds of propositions state something that is accidentally true in many instances,
although not in all without exception.
27. ☺☺An indesignate proposition that is understood to be usually true, should be regarded
as particular!!
THE QUALITY OF PROPOSITIONS
1. ☺A proposition is either affirmative or negative, and this is what we mean by its
quality.
93
2. The quality of a proposition is immediately evident.
3. In an affirmative proposition we are applying a certain predicate to a subject.
4. Example:
* A miser is a lover of money.
5. In a negative proposition we are denying a predicate of a subject.
6. Example:
* A socialist is not a conservative.
7. If you’re not sure, refer to the following rule:
8. ☺The quality of a proposition is determined by the quality of its copula.
9. Just because a sentence has a negative subject or predicate, it’s not necessarily negative.
10. ☺A proposition is negative, only if the copula is negative.
11. The following propositions are affirmative:
* Not to be loved is to be lonely.
* Whoever is not with me is against me.
*Some not unfortunate people are inheritors of money.
*Some people who are not interested in performing their duties
are avid defenders of their rights.
12. NOTE: In each of the above examples the predicate is being applied to or assigned to the
subject, even if the term itself is negative.
13. The term “not to be loved” is considered as a unit to which the predicate term “to be lonely”
is applied by way of the affirmation, “is”.
14. If a proposition contains a double negative, disregard it. It will be discussed under obversion
in another lesson.
94
COMBINING THE QUANTITY AND QUALITY OF A PROPOSITION
1. A proposition is either universal or particular—this is its quantity or extension.
2. A proposition is either affirmative or negative—this is called its quality.
3. When we combine quantity and quality, we get four basic types of propositions:
universal: affirmative and negative
particular: affirmative and negative
4. Examples:
universal affirmative: Every car has a motor.
universal negative: No ball is square.
particular affirmative: Some men are talented.
particular negative: Some medicine is not effective.
5. We’ll use a symbol for each of the four types:
A—Universal affirmative
E—Universal negative
I—Particular affirmative
O—Particular negative
6. To simplify further:
A—Every S is P.
E—No S is P. or S is not P.
I—Some S is P.
O—Some S is not P.
7. In an E proposition, the copula appears affirmative (“No S is P”), it’s actually negative
because of the words no or none.
8. ☺When we place the word “no” before the subject term, it signifies universal quantity and
negative quality.
9. Example:
No S is P—No man is an island.
S is not P—Man is not an island.
95
THE PREDICATE AND ITS QUANTITY
1. There are three types of quantity in a proposition: 1. The quantity of the subject term, 2. the
quantity of the proposition (its copula), and 3. the quantity of the predicate term.
2. Ordinarily, we don’t add a sign of quantity to the predicate term.
3. Example: we wouldn’t say, “All men are some mortals.”
4. Every predicate term is either universal or particular.
5. ☺The rules for determining the quantity of the predicate are simple:
a. The predicate term of an affirmative proposition (A or I) is always particular
(undistributed).
b. The predicate term of a negative proposition (E or O) is always universal
(distributed).
6. These two rules will become very important to understand when we study the categorical
syllogism.
7. When two terms like S and P agree with each other from the standpoint of extension, these
same terms also agree in their meaning—comprehension.
8. When two terms disagree in their extension, they also disagree in meaning.
9. The most important notion to grasp here is to understand the agreement or disagreement of
two terms in their meaning or comprehension.
96
EULER’S CIRCLES—Leonhard Euler—18th
century Swiss mathematician:
See handout on Euler’s Circles
“A” PROPOSITION
1. ☺An A proposition affirms that every S includes part of the extension of P.
3. This proposition means that “All textbooks” belong to the class of “things to be studied.”
4. Textbooks, in this proposition, do not exhaust the class of “things to be studied”—you could
also study works of art, maps, etc.
5. The predicate is only using up part of its extension, namely, that part of it which includes
the subject.
6. This is why we say that the predicate is a particular or undistributed term.
7. There is an exception to the rule when the predicate defines its subject:
8. Example:
* Every man is a rational animal.
9. In this example the extension of S and P coincides perfectly—the circles are the same size
and one would fit on top of the other.
10. However, to keep things uniform, in Logic, we still consider the predicate to be particular
(undistributed).
97
“E” PROPOSITION
1. ☺In an E proposition, the subject term is excluded from the all of the extension of P.
3. No part of the extension of S (true soldier) includes any part of the extension of P (coward).
4. P (taken in its complete extension) is denied of S (taken in its complete extension).
5. The predicate is a universal, or distributed, term.
“I” PROPOSITION
1. ☺An I proposition affirms that an indeterminate portion of the extension of S comprises
part of the extension of P.
3. In this I proposition, the predicate is affirmed of the subject, but only as a part of its
extension.
4. The predicate here is particular, or undistributed.
5. NOTE: In stating the I proposition, “Some coffee is imported,” we do not necessarily rule
out the possibility that the A proposition, “All coffee is imported,” is true. Nor do we, for that
matter, imply the truth of the O proposition, “Some coffee is not imported.”
98
“O” PROPOSITION
1. ☺In an O proposition the predicate is denied of the subject (some S).
2. The O proposition states that the predicate is excluded from part of S—that part which we are
considering.
LOGICAL FORM
1. In argumentation, many propositions are loosely construed.
2. To work with them in Logic, we have to reformulate them into one of the patterns of “Every
S is P”; “No S is P”; and so on.
3. ☺So as not to confuse things, it’s of prime importance to know the exact meaning of any
statement before making an attempt to change its form.
4. One of the biggest problems, in any discussion, is not being properly understood.
5. ☺☺Don’t be tempted to interpret what is being said too quickly. It’s easy to insert your
feelings, desires, etc, into the discussion, altering the meaning of the other person’s ideas.
99
6. The best listener tries to be objective in an attempt to get at the intended meaning of a given
statement.
WRITING A PROPOSITION IN LOGICAL FORM
1. ☺There are many ways to put a proposition in its logical form.
2. Some thoughts to consider:
3. Look for the logical subject—what are we talking about?
4. Once you find the subject, it’s easy to point out the predicate.
5. Note the following ideas:
a. Eliminate nonsignificant words.
For example: It is better to work than to starve.
The word “it” has no logical significance and should not be taken as the subject.
The subject here is, “work”. The predicate is, “better than to starve.”
Ask the question, “What’s the predicate referring to?”
To work (S) is (c) better than to starve (P).
b. Transpose an inverted word order.
Example: Greatly to be admired is a person who is genuinely modest.
The logical structure is clearer if we change it to:
A person who is genuinely modest (S) is (c) to be greatly admired (P).
c. Unite a term that is split.
For example: He jests at scars that never felt a wound.
100
He that never felt a wound (S) is (c) one who jests at scars (P).
6. Determine the quantity of the subject term and the quality of the copula.
The form, “Not all (or every) S is P can be confusing.
For example, “Not every man is a genius.”
Here, the word “not” creates a double negative.
“Not every” means “some”. The copula becomes “is not” (instead of “is”).
It should read: “Some S is not P”; “Some man is not a genius”.
This is actually an “O” proposition—known in logic as a “tricky ‘O’”.
7. Exercise: Determine which of the following sentences are “O” propositions and put them in
their logical form ('Some S is not P”).
All men are capable of laughter.
Not everyone knows how to play the flute.
Most people are procrastinators.
Men are not all logicians.
All women are not fickle.
Not all unskilled workers are unintelligent.
No one who loves his wife has an unhappy marriage.
8. When the word “not” follows the copula, the proposition is negative.
Example: “Some men are not unkind.” This is a regular O proposition.
9. If you want to make a universal denial, say 'no S is P”.
Example: “No good sportsman is a cheat.”
101
10. As a means of determining more exactly the form of a proposition we should consider each
of the following four possibilities:
1. S c P
2. non-S c P
3. S c non-P
4. non-S c non-P
11. These four forms are the possible variations that occur within the terms themselves.
12. If a sentence is an A, E, I or O, proposition, it will take one of the forms above.
102
13. Exercise: Classify the following propositions according to the above chart:
a. Some non-miraculous events are hard to explain.
b. Everything that is irrelevant is beside the point.
c. No good person is without honor.
d. Most insignificant events are forgotten.
e. Not everyone over fifty is toothless.
f. All unsung heroes are un-praised.
SOME PROPOSITIONS THAT DON’T FOLLOW THE NORM
1. Some propositions can’t be placed in the form, “S is (or is not) P.”
2. Example:
English is not spoken everywhere. (a relationship of place)
3. If we try to put this proposition in its logical form, the meaning becomes distorted.
Some English is not spoken.
4. We could possibly say this:
Somewhere English is not spoken (Somewhere S is not P.)
5. The following examples are typical:
a. Rosy cheeks are not always a sign of health.
b. Lying is never honorable.
c. Trains are frequently behind schedule.
d. The child is nowhere to be found.
e. Excusing oneself from class is, under certain circumstances, permissible.
8. In all these examples notice that P is assigned to (or denied of) S by some type of
qualification.
9. The following, are some of the verbal indicators of these special types of propositions, in-
order-to determine whether the proposition in which they appear is, an A, E, I, or O proposition:
A Proposition
always, everywhere, anywhere, completely, wholly, entirely, in its completeness, in all
cases, under all circumstances, in all respects, under all conditions, in all ways
103
E Proposition
never, nowhere, in no part, in no case, under no circumstance, in no respect, under no
condition, in no way
I Proposition
sometimes, occasionally, often, at times, frequently, usually, somewhere, for the most
part, partially, in some cases, under some circumstances, in some respects, under
certain conditions, in some ways
O Proposition
sometimes not, not always, frequently not, somewhere not, not everywhere, not
completely, for the most part not, in some cases not
3. NOTE: ☺We do the above proposition juggling in-order-to achieve a certain level of
clarity of expression so there is no misunderstanding.
See Extra 14: A, E, I, O Propositions?
See Extra 14a: Special Types of Propositions
See Homework Lesson 6
104
Extra Materials
Extra 13: Euler’s Circles
105
BACK
Extra 14: A, E, I, O Propositions?
Identify each of the following as A, E, I, or O propositions, and when necessary place the proposition
into its logical form:
1. Prudence is a virtue.
2. Not every cat is a suitable pet.
3. Life is worth living.
4. Growth is essential to progress.
5. Not to move forward is to fall behind.
6. Charity covers a multitude of sins.
7. Whoever is not with me is against me.
8. Not all comedians are funny.
9. There are certain inconsistencies in our foreign policy.
10. Some people who are not wealthy create the impression that they are.
11. Nothing is lost by taking time.
12. There are some languages that take years to master.
13. Unjust aggression is not something to be condoned.
14. All mourners do not weep.
15. A skillful operation is no guarantee that the patient will live.
16. Certain historical events are unparalleled.
17. All's well that ends well.
18. Nothing is well that does not end well
19. All is not well that ends well.
20. Not all campaign literature is educational.
21. Baseball is a sport.
22. Nothing that is material is indestructible.
23. Everything of a spiritual nature is indestructible.
24. Certain things are a matter of taste.
25. Authority is not man-made.
26. Not every contract that is legal is morally justifiable.
106
27. All bureaucrats are not garbage collectors.
28. There are many things that are invisible to the eye of the microscope.
BACK
Extra 14a: Special Types of Propositions
Identify as A, E, I, or O propositions the following special types of propositions:
1. In no part of your term paper is there evidence of any original work.
2. Gambling is not illegal in all its forms.
3. Lying is never honorable.
4. In some ways your appearance resembles that of your father.
5. Students do not always write what they think.
6. The past is never what you imagine it to be.
7. In some ways this home is not designed for practical living.
8. Under no circumstances will your schoolwork be interrupted.
9. One should always respect his neighbor's rights.
10. Somewhere there is peace.
11. Hasty eating frequently causes indigestion.
12. Examinations are sometimes hard to read.
BACK
107
Exercises for Homework:
EXERCISES FOR LESSON SIX— QUANTITY, QUALITY, & LOGICAL FORM OF
CATEGORICAL PROPOSITIONS
1. Why is it absolutely important to be able to tell if a proposition is universal, particular, or singular?
2. What is the basic rule for determining the quantity of a proposition?
3. Why are singulars treated as a universal?
4. What is an indesignate proposition?
5. What rule applies to an indesignate proposition?
6. How do you determine the quality of a proposition?
7. What four types of propositions do we get when we combine the quantity and quality of a
proposition?
8. How do we symbolize the four types? Give an example of each.
9. What are the three types of quantity in a proposition?
10. What are the rules for determining the quantity of the predicate?
11. What are Euler’s Circles? Give an example of each of the four with their appropriate circles.
13. Draw and memorize the chart, Type of Proposition.
14. What is one of the biggest problems in any discussion? Why?
15. Give a few ideas on how we might write a proposition in its logical form. Why is this important?
16. How do we combine A, E, I, and O propositions with their four possible variations? BACK
108
LESSON SEVEN—SPECIAL TYPES OF PROPOSITIONS
1. Let’s take a look at categorical, modal, and hypothetical Propositions.
2. Knowing something about these propositions will help you determine the meanings of more
complex statements.
3. In order to know the best way to contradict a proposition, you have to recognize the kind of
proposition it is.
4. ☺Knowledge of these three propositions will help you identify the type of syllogism you are
dealing with.
5. ☺A syllogism is determined by the kinds of propositions that make it up.
MULTIPLE PROPOSITIONS
1. Terms within a proposition can be simple or complex.
2. Example:
Anyone who fails to learn from his past mistakes (S) is (c) a fool (P).
3. A proposition is a single proposition as long as it has one subject and one predicate. It
doesn’t matter how many words are in the term.
4. At this point in your study, you have been looking at the single categorical proposition.
5. On the other hand, ☺a multiple proposition is a sentence expressing more than one
enunciation.
6. These examples are made up of more than one subject or predicate.
Franklin Roosevelt and Harry Truman were Democratic presidents. (More than one
subject)
Andrew Jackson was both a military general and a president of the United States.
109
(More than one predicate)
Jackson and Grant were both generals and presidents. (More than one subject and
more than one predicate)
7. These propositions express more than one object of agreement.
8. They can be broken down into single categoricals.
9. Example:
Jackson was a military general.
Jackson was a president.
Grant was a military general.
Grant was a president.
10. In a multiple proposition, you don’t need to break it down into its elements, as long as the
meaning is clear and its truth apparent.
11. In complex argumentation, the propositions should be looked at individually—examine their
meaning and truth value.
HALF-TRUTHS
1. In multiple propositions it’s sometimes easy to include a falsehood with a true statement.
2. NOTE: In Logic, it is not possible to have a half truth—a statement is judged to be true or
false, not both.
3. This is why it’s a good idea to examine both propositions to see where the fallacy lies—if
any.
4. ☺If a multiple proposition is false in any one of its parts, the proposition is false.
5. Example: you are taking a test with true-false questions. If any part of a true-or-false
statement is false, mark it “false.”
110
6. Example: “Logic is the study of emotional reaction and reasoning.” False
7. Always be on guard against the package type statement because it could contain a half-truth.
8. When a falsehood is couched in a statement that sounds good, that’s when you can more
easily be taken in.
9. Exercise: On the basis of what you have studied in previous chapters determine whether the
following statements as “packages” are true or false.
1. Brutes and men are endowed with powers of sensation.
2. Brutes and men are rational.
3. Terms and concepts are both conventional signs of their objects.
4. Definition and division alike are means of clarifying our concepts.
5. Truth and falsity apply in their proper sense to judgments and propositions.
6. I and E propositions are negative.
KINDS OF MULTIPLE PROPOSITIONS
1. Since there are many ways in which two or more single propositions can be combined into a
multiple statement, no attempt will be made here to give a complete or near-complete
classification.
2. Logicians, however, distinguish two fundamental types: those propositions that are clearly
multiple in form, for example, “Donald and James are at home”; and those which, though
multiple, have the appearance of a single proposition, for example, “Only you know the
answer.”
3. The first type of proposition may be called an overt, or evident, multiple; that is, it is one
whose multiple character is readily discernible.
4. The second type is a disguised multiple and is spoken of simply as an exponible
proposition—needs some explanation.
111
EVIDENT MULTIPLE PROPOSITIONS
1. ☺A copulative proposition is one that unites or disunites its subject and predicate on a
coordinate basis.
2. In grammar it’s called a compound sentence.
3. ☺A compound sentence contains two independent clauses joined by a coordinator.
4. Example:
I tried to speak Spanish, and my friend tried to speak English.
Mario played football, so Maria went shopping.
5. Commonly used conjunctions: and, both—and, not only—but also, neither—nor.
6. More examples:
Both Mary and Jane attended the party.
Neither the Chief Justice nor his associates are elected. (Both parts are negative.)
7. ☺An adversative proposition is a grammatically complex sentence in which the subordinate
clause is set up in opposition to the enunciation expressed in the main clause.
8. A complex sentence is composed of a main clause and one or more subordinate clauses.
9. Adversative, relative, and causal propositions are all complex sentences.
10. Commonly used conjunctions: although, even though, even if, in spite of, but, despite, while,
whereas.
11. Example:
Even though the sun is shining, it’s raining.
12. ☺Relative (indicating time)—In this type of proposition the subordinate clause
introduces a time relation that has a bearing on what is stated in the main clause.
112
13. Commonly used conjunctions: before, during, after.
14. Example:
After submitting his examination, John realized some of the mistakes he had made.
15. ☺ A causal proposition is one whose subordinate clause assigns the cause or reason for
what is asserted in the main clause.
16. Commonly used conjunctions: because, for, since.
17. Example:
Because some people fail to see the purpose of human existence, their lives are meaningless.
18. The above examples are evident multiple propositions and almost immediately apparent.
19. Normally, these types of propositions don’t need to be broken down into their parts.
20. If they are complicated or lengthy, for example in a legal document, then do it.
21. Their truth depends on the connection or lack of connection between the two clauses.
22. Example:
Because Washington, D.C., is in the east, it’s the capital of the United States.—False
23. Both parts are factually true, but the statement is false because it lacks the causal connection
that it claims.
24. An overt multiple is determined by conjunctions like “and,” “but,” and so on.
25. The word “because” always signifies a causal proposition.
26. There are other conjunctive words that are not as easy to determine.
27. Note the following:
Since you left town everything has been peaceful again.
Shortly after the new President took office the nation went to war.
113
While this young lady served as our baby-sitter Johnny took sick.
28. These three statements are not clear as to their causal relation.
29. Because two things happen at the same time doesn’t mean, necessarily, that one is the cause
of the other.
30. The confusion of time and causality often gives rise to the fallacy, called post hoc ergo
propter hoc (after this, therefore because of this).
See Extra 15: Overt Multiples
A DISGUISED MULTIPLE
Exclusive:
1. ☺An exclusive proposition is easily identified by such words as “only,” “alone,” “none
but,” and so on, and it’s a type of statement that makes a double assertion—one of
affirmation and another of denial.
2. Example:
Only humans wear shoes.
3. This resolves into the following:
Some humans (that is, at least some) wear shoes.
No one else (that is, no nonhuman) wears shoes.
4. This is a typical example of an exclusive proposition, and takes the form “Only S (or non-S)
is P (or non-P).”
5. This type of proposition resolves first into an I-proposition, and then into an E thus:
(I) Some S is P.
(E) No non-S is P.
6. Other examples:
Only professionals are allowed to play in the tournament.
Resolve into:
114
Some (possibly all) professionals are allowed.
No nonprofessional is allowed to play.
Only intelligent beings are free.
Resolve into:
Some (or all) intelligent beings are free.
No nonintelligent being is free.
Only incompetents will be “weeded out.”
Resolve into:
Some (or all) incompetents will be “weeded out.”
No competent person will be “weeded out.”
7. The exclusive proposition also takes the form: “S is only P.”
8. Example:
“Certain things are only a means to an end.”
9. The predicate is assigned to the subject to the exclusion of all other predicates.
10. Exercise: Resolve the following examples:
Courageous people alone climb high mountains.
Only freshmen wear “beanies.”
Exceptive:
11. ☺Closely related to the exclusive proposition, is the exceptive. To see the difference
between the exceptive and the exclusive proposition check out the following examples:
Exclusive:
Only the chairman was on time for the meeting.
The chairman was on time.
No one else—that is, no one not a chairman—was.
Exceptive:
Everyone, except the chairman, was on time.
The chairman was not on time.
Everyone else—that is, anyone not a chairman—was.
12. The exceptive proposition takes the form “All except S is P.”
13. To resolve an exceptive proposition: First remove the exception by means of a negative
115
statement (“S is not P” or “No S is P”), then apply the predicate to all other members that are
not a part of the excepted class (“All non-S is P”).
14. Examples:
All but the Y's are X's.
Resolve into:
None of the Y's are X's. (Remove the exception.)
All non-Y's are X's. (Apply P to the non-excepted classes.)
All except uninvited guests are welcome.
Resolve into:
No uninvited guest is welcome.
Every invited guest is welcome.
15. Exercise: Resolve the following examples:
All except the vaccinated children are to be confined to Ward B.
Everyone except the sheriff knows who committed the crime.
All except dictatorial countries enjoy freedom.
Reduplicative:
16. ☺A Reduplicative Proposition is one in which a predicate is assigned to a subject in a
manner that formally signifies the reason for the predication made. Uses words like
“as”, “as such”, “inasmuch as”.
17. Example:
Every official as such is deserving of public respect.
18 “As such” is a reinforcement of the subject term.
19. Although the following proposition is not strictly reduplicative, it may be assigned to this
category:
As a geographical neighbor of the United States,
Canada has a heavy traffic of American tourists.
Resolve:
Canada is a geographical neighbor of the United States.
She has a heavy traffic of American tourists.
20. Normally, it’s not necessary to resolve a reduplicative into its component parts—the relation
is usually clear.
116
MODALS
1. We’ve dealt with categorical propositions so far—ones that are ultimately reducible to the
form: “S is (or is not) P.”
2. ☺Now we’ll look at the Modal, a type which expresses a certain mode of agreement or
disagreement between two terms—one of necessity, contingency, possibility, or impossibility.
3. Example of a categorical proposition:
Science has all the answers.
4. Example of the following four:
Science must have all the answers. necessity
Science need not have all the answers. contingency
Science may have all the answers. possibility
Science cannot have all the answers. impossibility
5. The difference between these four modal propositions lies in the manner of agreement or
disagreement expressed.
NECESSITY
1. ☺A Proposition of Necessity is one that explicitly asserts that “S must be P.”
2. Example:
Every effect must have a cause.
3. In this type of proposition we are stating, necessarily, that every effect has a cause.
4. At the same time we are ruling out that there is any effect without a cause.
5. The mode of necessity is usually written thus:
S must be P.
S has to be P.
It is necessary that S be P.
S is of necessity P.
117
6. NOTE: The italics here indicate the copula of this type of proposition.
7. The words “must” and “has to” do not always express a strict, logical necessity. The third
and fourth forms are better.
8. Example:
Johnny must be a good boy at the party.
9. Here we are kind of hoping that he will be good at the party. And if he isn’t good, something
unpleasant will happen to him.
10. Example:
One of these days it will have to rain.
That is, if the crops are to grow.
CONTINGENCY
1. ☺A Contingent Proposition is the simple denial of the mode of necessity.
2. Example:
S need not be P.
It is not necessary that S be P.
3. Don’t confuse the denial of necessity with a denial of fact.
4. Example:
A research scholar is not necessarily a good teacher.
5. Here we are saying, he may or may not be a good teacher.
6. What we are denying here, is that a research scholar must be a good teacher.
POSSIBILITY
1. ☺The possibility proposition is one that does not express the agreement of two terms but
affirms the possibility of such an agreement.
118
2. Example:
S can be P.
Men can be virtuous.
3. The proposition only affirms something as a possibility.
IMPOSSIBILITY
1. This mode usually takes the form:
S cannot be P.
It is impossible that S be P.
3. The proposition of impossibility involves the denial of the fact as well.
4. Example:
No man can understand everything.
5. This proposition denies that, “There is some man who understands everything.”
6. NOTE: Any of the modals can be combined, together with a categorical proposition to form
a multiple.
7. Example:
Although it is possible for a man to work eighteen hours a day, doing so is not
advisable.
8. Modal propositions normally take these forms:
Mode of necessity: A proposition—S must be P
Mode of contingency: O proposition—S need not be P
Mode of possibility: I proposition—S can be P
Mode of impossibility: E proposition—S cannot be P
119
HYPOTHETICALS
1. ☺Hypothetical usually means conditional but can include those propositions that are
disjunctive and conjunctive.
2. The copula of this type of proposition, unlike that of a single categorical or a modal, consists
of words used to unite or disunite the enunciations that it contains.
3. Example: if — then, either — or.
CONDITIONAL HYPOTHETICAL
1. A conditional proposition usually takes the following form:
If A is B, then A is C
If a man works, then he deserves the reward of his labor.
2. The conditional hypothetical proposition has two parts—the antecedent and the consequent.
3. ☺The antecedent—the “if clause”—expresses a condition that is sufficient to guarantee
the fulfillment of a given consequent.
4. The truth of this type of proposition lies in the dependence of the consequent upon the
antecedent.
5. A multiple proposition is considered true if it’s true in its separate parts.
6. This rule does not apply to the conditional.
7. Furthermore, you may not resolve a conditional proposition into its separate parts.
8. All the hypotheticals—conditional, disjunctive and conjunctive—must be left as is.
9. A conditional proposition is true if the consequent follows, of necessity, from the antecedent.
10. To deny the truth of a conditional, just point out that the consequent does not necessarily
120
follow from the given antecedent.
11. Example:
If you are an American, you live in the United States.
12. Even though you are an American and you live in the U S, the proposition is false, because
the consequent does not necessarily follow.
13. There are some Americans living outside the country.
14. Consider this proposition:
If man were all-knowing, he would be God.
15. Man is neither all-knowing nor is he God.
16. Both enunciations, separately considered, are false. Yet, the proposition itself is true.
17. Note the following variations not using “if”:
Provided that you study, you will learn.
Unless a man follows his conscience, he will suffer remorse.
Were you a brute, you would not have to study logic.
In the event that the child is lost, he will be hungry.
Assuming what you say to be true, I would have to admit I'm in the wrong.
18. Sometimes a categorical proposition looks like a conditional.
19. Example beginning with “if”:
If men quarrel, they don't always mean to.
20. The meaning of “if” here is “even if” or “although.” It’s not “if”/“then”.
DISJUNCTIVE HYPOTHETICAL
1. You can identify a disjunctive proposition by its copula—either — or.
121
2. Example:
A is either B or C.
The human will is either free or determined.
3. The parts of a disjunctive statement are referred to as “alternatives” or simply as
“enunciations” or “parts.”
4. In the above two statements “B” and “C,” are “free” and “determined.”
5. There are two types of disjunctives—loose and strict.
6. ☺A loose disjunctive sets up two or more parts and asserts that at least one of these parts
must be true.
7. Note that it is possible that both parts are true.
8. Example:
Bill received a low grade either because he was lazy, sick, or untalented.
9. This statement is true if only one of its parts is true.
10. All of the possibilities must be false in order to say that the statement is false.
11. ☺A strict disjunctive says that one part must be true, and the other, or others, are false.
12. This is because the parts of a strict disjunctive are, naturally incompatible with each other.
13. Example:
Every student is either a graduate or a nongraduate.
14. Whichever part is true, the other (or others) must be false.
15. The truth of this type of a statement depends on two conditions:
(1) The parts given must be complete, and
(2) They must mutually exclude each other.
16. We know it’s a strict disjunctive when the parts are related as formal contradictories.
122
17. Example;
“good” and “not good,”—“mortal” and “immortal,”—“on” and “off”.
18. Also, for example:
Every animal is either a male or a female.
19. The statement is true if its parts are complete and mutually exclusive of each other.
SOME FINAL THOUGHTS
1. The statement is a strict disjunctive if the parts are so related that the truth of one rules out
the truth of the other.
2. Some are classified as relative disjunctives:
The light went out either because of a bad bulb or a broken circuit.
3. These parts are not known on a self-evident basis but only in relation to a given situation and
as a result of an inductive process.
4. On the other hand, consider this proposition:
Either the human soul is immortal or it is not.
5. This is an example of an absolute disjunctive.
6. The parts are inherently incompatible with each other.
CONJUNCTIVES
1. ☺A conjunctive proposition is one that involves two or more components, parts, or
enunciations.
2. It plainly asserts that both—or all—of them cannot be true.
123
3. The most common type of conjunctive statement has only two enunciations.
4. Example:
You cannot work and play at the same time.
5. ☺A conjunctive proposition does not declare, as does a disjunctive, that one enunciation is
or must be true but merely that both cannot be true.
6. In a conjunctive proposition, both of its enunciations may be false.
7. Example:
You cannot be both a European and an Asiatic.
8. The truth of this statement depends on the impossibility of being both.
9. Don’t confuse the modal with the conjunctive:
Modal impossible: Nothing can cause itself.
Conjunctive: Nothing can be a cause and effect at the same time, and in the same respect.
CHAPTER SUMMARY
I. Categorical propositions
A. Single
B. Multiple
1. Overt, or evident, multiples
a. Copulative
b. Adversative
c. Relative (indicating time)
d. Causal
2. Exponibles (disguised multiples)
a. Exclusive
b. Exceptive
c. Reduplicative
II. Modal propositions, expressing
A. Necessity
124
B. Contingency
C. Possibility
D. Impossibility
III. Hypothetical propositions
A. Conditional
B. Disjunctive
1. Loose
2. Strict
C. Conjunctive
See Extra 16: Varieties of Propositions
See Extra 17: Summary of Propositions
See Homework Lesson Seven
125
Extra Materials
Extra 15: Overt Multiples
A, single categorical
B, copulative
C, adversative
D, relative
E, causal
1. Neither my parents nor my grandparents were born in Europe.
2. In spite of their being intelligent, there are some students whose marks are below average.
3. Since I did not understand some of the questions, some of my answers were wrong.
4. While he attended college, Bill was a sports enthusiast.
5. If parents are at times impatient with their children, they still love them.
6. Anyone who is truly virtuous is happy.
7. After leaving work, my sister went shopping.
8. I like ice cream, and Jerry likes apple pie.
9. Even though you come to class, you never pay attention.
10. After a good day’s work, Chiara likes to relax by the pool.
BACK
126
Extra 16: Varieties of Propositions
Determine the nature of each of the propositions listed below and assign the corresponding
letter from the key:
a. categorical single h. reduplicative
b. copulative i. conditional
c. adversative j. disjunctive
d. relative (time) k. conjunctive
e. causal l. modal: necessary
f. exclusive m. modal: contingent
g. exceptive n. modal: possible
o. modal: impossible
1. Were man a brute, he would not be morally responsible.
2. All except the older homes in this neighborhood are expensive.
3. Every proposition is either true or false.
4. Neither my parents nor my grandparents were born in Europe.
5. Evil as such is never desirable.
6. In spite of their being intelligent, there are some students whose marks are below average.
7. Since I did not understand some of the questions, some of my answers were wrong.
8. A criminal need not be a thief.
9. Children can be well behaved.
10. While he attended college, Bill was a sports enthusiast.
11. Brutes cannot reason.
12. You cannot watch television and read a book at the same time.
13. Only the Russian delegation is obstructing the progress of the United Nations.
14. Logic need not be difficult.
15. Every scholar as such is devoted to the pursuit of knowledge.
16. If parents are at times impatient with their children, they still love them.
17. A man either loves his wife or he doesn't.
18. All, except those who despair have hope.
127
19 I think it will rain either today or tomorrow.
20. Everything that is moved must be moved by something else.
21. Authority, inasmuch as it establishes order in society, is indispensable.
22. None but a starving man would eat this meal.
23. Anyone who is truly virtuous is happy.
24. After leaving work, my sister went shopping.
25. History does not have to be dull.
EXTRA: Resolve propositions (2), (13), (18), and (22) of the list above into their component
parts.
BACK
128
Extra 17: Summary of Propositions
129
BACK
Exercises for Homework:
Homework Lesson Seven:
1. In Disguised Multiples, what is an exclusive proposition? Give an example and show how it can
be resolved into an “I” and “E” proposition.
2. Contrast the exclusive and the exceptive propositions with an example.
3. What words are key to a reduplicative proposition? Can you give an example of a reduplicative
proposition?
4. Why isn’t it normally necessary to resolve a reduplicative into its component parts?
5. Modal propositions are a bit different from categorical ones. Define a modal proposition.
6. Using your own example, start with a categorical proposition and express it in the four types of a
modal—necessity, contingency, possibility, and impossibility.
7. What is a Hypothetical proposition? Name the three general types of hypotheticals. Give an
example of each.
8. Define the conjunctive proposition.
9. Give an example of a conjunctive proposition.
10. Distinguish between a modal and a conjunctive proposition. BACK
130
LESSON EIGHT—SQUARE OF OPPOSITION
1. Up to now, you have looked at propositions considered in themselves.
2. In this section you will consider some of the relationships that exist, between one proposition
and another.
3. As before, you will investigate, first, the opposition of single categoricals, then the remaining
other types.
WHAT IS OPPOSITION?
1. The first question is this, ☺When are two propositions opposed to each other?
2. ☺Opposition defined: the relation between any two propositions that have the same subject
and predicate but differ in quality, or quantity, or both.
2. To begin with, we are confining our attention here to single categoricals which have the
same subject and predicate terms.
3. Example:
S P
a. No toll bridge is expensive.
S P
b. Some toll bridges are expensive.
4. Both of these propositions are different and opposed in their meaning. However, their subject
and predicate terms are basically the same.
5. How, do these propositions differ?
6. Categorical propositions that are opposed will differ in their quantity, quality, or both.
7. Investigating the above proposition, we can arrive at four possible combinations:
a. Every toll bridge is expensive. (A proposition)
b. No toll bridge is expensive. (E proposition)
131
c. Some toll bridges are expensive. (I proposition)
d. Some toll bridges are not expensive. (O proposition)
8. Proposition “a” and proposition “b” are opposed in quality but not quantity.
9. Proposition “c” and proposition “a” are opposed in quantity, not in quality.
10. Proposition “d” and proposition “a” are opposed in both quantity and quality.
11. The following are the four different kinds of opposition:
Contrary opposition is that between two universals of different quality (A and E).
Subcontrary opposition is that between two particulars of different quality (I and O).
Subaltern opposition is that between a universal and a particular of the same quality
(A and I; E and O)
Contradictory opposition is that between a universal and a particular of different
quality (A and O; E and I)
NOTE: Subalterns are compatible both in truth and in falsity, even though they differ in their
quantity. It’s only in this latter sense that they are opposed, and can therefore be represented on
the square of opposition.
132
12. ☺Normally we don’t distinguish between the types of oppositions—we just say that one
proposition is the "opposite" of another.
13. To see how this works we’ll use a traditional diagram called the square of opposition.
14. Exercise: Determine which of the four types of opposition exists between the following sets:
All politicians are honest. contraries
No politicians are honest.
Some housewives are good cooks. subcontraries
Some housewives are not good cooks.
Some wars are futile. subcontraries
Some wars are not futile.
No civilization is amoral. subalterns
Some civilization is not amoral.
Some harvests are plentiful. subalterns
All harvests are plentiful.
Some poetry is not sentimental. contradictories
All poetry is sentimental.
THE RULES OF OPPOSITION
1. If we say that a given proposition is true or false, it’s possible, by pure logic, to determine
the truth or falsity of any other proposition opposed to it.
2. Begin with a proposition that is given as true or false then test it against the others to see if
they are true, false, or doubtful.
3. Example:
"All good men are virtuous."—given as true.
4. Is its contradictory proposition, "Some good men are not virtuous," true, false, or doubtful?
False
133
CONTRADICTORY OPPOSITION (A—O; E—I)
I. Contradictory propositions cannot both be true.
1. This rule supposes that the initial proposition is given as true.
2. If a proposition is given as true, its corresponding contradictory must be false.
3. Example:
(I) Some homes are expensive. (given as true)
(E) No home is expensive. (must be false)
II. Contradictory propositions cannot both be false.
1. This rule supposes that the initial proposition is given as false.
2. If a proposition is given as false, its corresponding contradictory must be true.
3. Example:
(A) Every nuisance is a bore. (given as false)
(O) Some nuisances are not bores. (true)
6. Summing up: Contradictories are incompatible both in truth and in falsity.
(can’t both be true and can’t both be false)
CONTRARY OPPOSITION (A—E)
Contrary propositions cannot both be true.
1. Here we say that either the A or E is true.
2. If A, is true, what can be implied of its E variant—its contrary?
3. It’s either true, false, or doubtful.
4. But the rule states that contraries cannot both be true.
5. Therefore, if one contrary is given as true, then its corresponding contrary must be false.
134
6. So, if A is true, E must be false.
7. Example:
(A) Every democracy respects human rights. (given as true)
(E) No democracy respects human rights. (false)
(E) No child is a criminal. (given as true)
(A) Every child is a criminal. (false)
Contrary propositions can both be false.
2. If one contrary is given as false must the other be false?
3. NO! If contraries can or may both be false, and one is given as false, the other is doubtful.
4. Example:
No man is a coward.—given as false
5. Its A variant is:
Every man is a coward. doubtful
6. Whenever a rule says "can" or "may," this means it’s "doubtful."
7. Even though we know this proposition is false, we cannot imply that it is.
8. Summarizing: contraries are incompatible in truth, but compatible in falsity.
(Can’t both be true but can both be false)
SUBCONTRARY OPPOSITION (I—O)
Subcontrary propositions can both be true.
1. For subcontraries, both can be true.
3. If one subcontrary is true, the other subcontrary could be true or false.
4. Since subcontraries can be true together, and if one of them is true, then the other may be
either true or false. It’s doubtful.
135
6. Given the truth of the I proposition:
Some candy is hard to chew.
7. Its O variant is doubtful, even though it may be true as a matter of fact.
Some candy is not hard to chew.
8. Given the I proposition as true:
Some logic texts deal with inference (assumption).
9. The O variant is doubtful on the basis of mere implication, even though, in this case, it’s
false.
Some logic texts do not deal with inference.
10. The truth of an I proposition implies the truth of its O variant—and conversely.
11. In the above examples, there is a discrepancy between the point of view of logic and
common sense.
12. Why does common sense favor such an implication, and logic does not?
13. The reason for this difference is because common-sense judgments are practical as opposed
to scientific.
14. A practical judgment settles doubt on the basis of what is considered more or less probable
or likely.
15. Concluding: If I is true, it is likely or probable that O also is true.
16. It usually turns out that both propositions are true.
Subcontrary propositions cannot both be false.
1. Contraries may both be false. This doesn’t apply to subcontraries.
2. If a subcontrary is false, its corresponding subcontrary must be true.
3. If an O proposition is given as false, "Some men do not have emotions," we imply the truth of
its I variant—“Some men do have emotions.”
136
4. If an I proposition is given as false, "Some monkeys are human," the O variant is necessarily
true. “Some monkeys are not human.”
5. To sum up: Subcontraries are compatible in truth, but incompatible in falsity.
(Can both be true but can’t both be false)
SUBALTERN OPPOSITION (A—I; E—O)
1. If the universal is true, its subalternate (particular) is likewise true; if its subalternate
(particular) is true, the universal is doubtful.
2. We pass from the truth of the universal to the truth of the particular, but not conversely.
3. If an A proposition is true, its I variant is necessarily true.
4. Example:
All snow is white. (given as true)
Some snow is white. (true)
5. The truth of the particular is contained in the truth of the universal.
6. You can’t imply the truth of the universal from the truth of the I proposition.
7. For example:
Some convicts are innocent. (given as true)
Every convict is innocent. (doubtful)
8. If the universal is false, its subalternate, (particular), I is Doubtful; if its subalternate
(particular) is false, the universal is false.
9. This rule, in effect, states that, if the universal is false, the particular may be either true or
false; you can’t tell.
10. If the particular is false, it follows that the universal is false.
11. If an A proposition is false, its I variant is doubtful.
137
12. Example:
All weather is pleasant. (given as false)
Some weather is pleasant. (doubtful)
13. If an I proposition is false, then its A variant is certainly false.
14. Example:
Some dogs are centipedes. (given as false)
All dogs are centipedes. (false)
See Extra 18: Square of Opposition Exercises
15. The following chart is a summary of conclusions derivable from the square of opposition.
A. Examine each conclusion of this chart and cite the appropriate rule that governs it.
B. Then, close the text and reconstruct the chart on the basis of the rules just studied.
PRACTICAL OBSERVATIONS
1. So far we have investigated the fourfold scheme of opposition as it relates to single
categoricals.
138
2. Now, we’ll look at various other types of statements.
3. ☺The purpose of your study of the square of opposition is to enable you to know how to
refute a given proposition.
4. This is important because it’s during refutation that the most deliberate blunders in a
discussion or a debate are committed.
5. ☺To avoid error, know that the only propositions that are incompatible in their truth are
contraries and contradictories.
6. ☺☺☺This means that the only way to refute any proposition is to establish the truth of
either its contrary or its contradictory.
7. If you can do this, then the proposition of your opponent is proven false.
8. Note that: You do not refute an I or an O proposition by setting up its subcontrary,
because both can be true.
9. Example of what some try to pull:
You claim that some of the athletes on our team are not good sports. Well, it's my own
contention that most of them are, and if you want me to prove it to you, I can.
10. The above argument is not enough to disprove the subcontrary statement against which he is
arguing, for subcontraries may both be true.
11. It is possible to disprove a proposition by its contrary. However, doing so is neither
necessary nor advisable.
12. If we can show that our proposition (which is the contrary of our opponent's) is true, then
we have refuted our opponent's position, since contraries cannot both be true.
13. It’s also true that contraries can both be false.
14. Sometimes an arguer will oppose one false proposition with another that is also false.
139
15. What if a Socialist is attempting to establish the following proposition:
All economic activity should be in the hands of the government.
16. Someone may try to oppose it with its contrary:
No economic activity should be in the hands of the government.
17. This proposition is as false as the original, since contraries can both be false.
18. ☺You should be very certain of your ground before attempting to refute one contrary
by means of another.
19. In this regard, the old saying is, "He who proves too much, proves nothing."
LET’S SUM UP:
1. To attempt to refute one subcontrary proposition by means of another is no
refutation at all, because subcontraries can both be true.
2. To attempt to refute one contrary by means of another is to resort to a risky
means of refutation, because contraries can both be false.
3. NOTE: THE CARDINAL RULE OF LOGIC:
The only thing required to refute any proposition
is to establish the truth of its contradictory.
4. Example to refute the proposition:
“All economic activity should be in the hands of the government.”
5. Just establish the following (contradictory):
"Some economic activity should not be in the hands of the government."
6. ☺Under most circumstances the only sane, practical way to refute a proposition is to
establish its contradictory.
7. ☺Under all circumstances this is all that is necessary.
140
OPPOSITION OF SINGULARS
1. A singular proposition is refuted by a simple denial.
2. *The only way of opposing a singular proposition is with its contrary.
3. Example:
John is at home.
John is not at home.
The weather is disagreeable.
The weather is not disagreeable.
4. What is true of singular propositions is equally true of those propositions whose subject term
is an infinitive (begins with “to”) or a gerund (ends in “ing”).
5. Example:
To exist is to live.
To exist is not to live.
Eating soup without wearing a bib is a hazardous experience.
It is not a hazardous experience.
OPPOSITION OF MODALS AND OF SPECIAL TYPES
1. In an earlier lesson, four types of modals were examined.
2. Check out these propositions according to the square of opposition:
141
3. ☺In modals, all the same rules apply as those which we have already stated for single
categoricals.
4. Example: The following statement is given as true:
Learning need not be an unpleasant experience.
5. This is a contingent proposition.
6. As an O proposition, you get the following results:
(A) Learning is necessarily an unpleasant experience. (false)
(E) Learning cannot be an unpleasant experience. (doubtful)
(I) Learning can be an unpleasant experience. (doubtful)
7. These propositions are treated simply according to their mode.
8. Occasionally a proposition will combine one of the usual signs of quantity and mode.
9. Note the following examples:
(A) Everyone must make mistakes. (given as true)
(E) No one can make mistakes. (false)
(I) Someone can make mistakes. (false)
(O) Someone need not make mistakes. (false)
10. Note: If a singular proposition is expressed as a modal, it may be treated as any other
modal, that is, by simply varying its modes.
SPECIAL TYPES
142
11. There are propositions that are not reducible either to a strict type of categorical or to a strict
type of modal:
12. Example:
Boys are sometimes mischievous.
13. This type of proposition must be treated as a class by itself.
14. Check your chart—Special Types
15. Suppose the following (E) proposition is true:
Lying is never permissible.
16. The following are obtainable:
(A) Lying is always permissible. (false)
(I) Lying is sometimes permissible. (false)
(O) Sometimes lying is not permissible. (true)
17. Note: Any singular proposition containing the Special Type words or phrases can be treated
in the same way.
OPPOSITION OF MULTIPLES
1. We’ll treat multiples by using simple denial.
2. Keep in mind two basic points that have already been explained:
A. If a multiple proposition is false in any one of its enunciations, the proposition as
it stands is false.
B. To establish the contradictory of any proposition, it is necessary to give only the
minimum required to deny it.
3. Example:
Because Jim was interested in history, he took the course.
143
4. To contradict this or, for that matter, any kind of multiple proposition, show that one of its
enunciations is false.
5. Example:
either: Jim was not interested in history.
or: He did not take the course.
or: He did not take it for the reason given.
6. The sense of a loose disjunctive is that at least one part of it is true.
7. In opposing this type of statement, you must posit that at least one part of the statement is
true and that the original proposition is therefore false.
EXAMPLES FOR STUDY
1. Neither graduates nor undergraduates are required to take this course.
Contradictory:
Either graduates are required to take this course or undergraduates are (required to take it).
2. Only hunters carry guns.
Resolved:
(I) Some hunters carry guns.
(E) No non-hunter carries a gun.
Contradictory:
Either no hunters carry guns or some non-hunters do.
3. Because Jones had the flu he did not attend the meeting.
Contradictory:
Either Jones didn't have the flu or he did attend the meeting or (if he didn't) he missed it for
some other reason.
4. All except the A-students are required to take the exam.
Resolved:
(E) No A-student is required to take the exam.
(A) All non-A's are required to take it.
Contradictory:
Either some A-student is required to take the exam or some non-A student is not.
144
5. After we ate lunch we discussed politics.
Contradictory:
Either we didn't eat lunch or talk politics or (if we did) we didn't do so at that time.
OPPOSITION OF HYPOTHETICALS (a simple denial refutes a hypothetical)
1. To deny a conditional you point out that the consequent does not necessarily follow from its
antecedent.
2. Example:
If a person is poor, he must be virtuous.
Even though (that is, although) a person is poor, he need not be virtuous. (denial)
3. In this example, one part of the statement admits a possibility, the other contradicts the mode
of the consequent.
4. Another example:
If you are intelligent, you cannot be prejudiced.
Although you may be intelligent, you can be prejudiced. (denial)
5. To deny a disjunctive statement consider the following.
6. A loose disjunctive requires that at least one of its parts be true. To deny it, you have to
show that all the enunciations are false.
7. ☺The denial of a conjunctive proposition consists merely in showing that the alternatives
given can both be true:
One cannot be a teacher and a research scholar at the same time.
One can be both. (denial)
8. Note the summary:
1. Single categoricals
2. Singulars
3. Modals and special types
4. Multiples
5. Hypothetical
145
a. Conditionals
b. Disjunctives
c. Conjunctives
See Homework Lesson 8
146
Extra Materials
Extra 18: Square of Opposition Exercises
I. Decide whether the following are correct or incorrect:
a. Given an O proposition as false, its A variant is doubtful.
b. Given an O proposition as true, its I variant is false.
c. Given an E proposition as false, its A variant is doubtful.
d. Given an I proposition as true, its A variant is doubtful.
e. In stating that "Some politicians are local officials," I imply of necessity that "Some are not."
f. In order to disprove the proposition, "Every American is a Communist," I must establish the
proposition, "No American is a Communist."
g. The contradictory of the proposition, "All men are not happy" is "All men are happy."
II. With respect to the examples given below, determine in each case whether the corresponding
variants (A, E, I, or O) are true, false, or doubtful.
a. Given as false: "Some trees do not have branches."
b. Given as true: "Many a drug is harmful."
c. Given as false: "No holy person is happy."
d. Given as false: "No monkey lives in a cage." Is the following proposition true, false, or doubtful?
"All monkeys do not live in cages."
e. Given as true: "Every examination isn't difficult." Is the following proposition true, false, or
doubtful? "Some examinations are difficult."
III. Determine in each case whether the corresponding variants (A, E, I, or O) are true, false, or
doubtful:
a. Given as true: "An organism must have life."
b. Given as false: "Miracles can't happen."
c. Given as false: "A scientist must be a genius."
147
IV. With respect to the examples given below, determine in each case whether the corresponding
variants (A, E, 1, or O) are true, false, or doubtful:
a. Given as true: "Professors are sometimes punctual."
b. Given as false: "Students always say what they mean."
V. What proposition do you think must be established to refute each of the following:
a. Only fish can swim.
b. Politicians never live up to their campaign promises.
c. Men are in no respect more clever than the brutes.
d. What you say is for the most part true.
e. Nothing of what you told me is false.
f. If a person is dishonest, he must be a liar.
g. Man must be either a brute or a god.
h. You cannot talk and chew your food at the same time.
i. A good traveler must always be "on the move."
j. Unless a person studies logic, he cannot reason correctly.
k. If you have read tonight's newspaper, you must have seen the weather report.
L. New York and Chicago are both eastern cities.
m. All sins are forgivable.
n. All students, except the A students, must take the final examination.
o. Some people can give what they do not have.
BACK
148
Exercises for Homework:
Homework Lesson 8:
1. When are two propositions opposed to each other? Give an example of two opposed propositions.
2. Using your above example, work out the four possible combinations.
3. What are the four types of opposition? Explain each of them.
4. Draw the classical Square of Opposition. You should know this by heart.
5. Give an example of each type of opposition.
6. How do we test the truth or falsity of a proposition?
7. How would you explain the statement, Contradictories are incompatible both in truth and in falsity.
8. Why do you think we study the square of opposition?
9. What is the purpose of studying the square of opposition?
10. Why wouldn’t you want to refute one subcontrary with another?
11. Why is refuting one contrary with another risky?
12. What is the cardinal rule of logic? Learn this by rote.
13. How do you oppose a singular proposition?
14. Can you give an example of a proposition beginning with an infinitive—with a gerund?
15. What is the general rule for dealing with the opposition of modals?
16. In regard to the opposition of multiples, state two basic points to keep in mind.
17. How do you refute a hypothetical? BACK
149
LESSON NINE—EDUCTIONS
1. Here we’ll look at the logic of implication—kind of like a person’s interpretation.
2. ☺All types of implication are known as eductions—ways of drawing out the implied
meanings of a given proposition.
3. Eduction is not deduction. (deduction is a conclusion reached by applying the rules of logic
to a premise)
4. In eduction we’re not examining a new truth but merely the same truth from a different point
of view.
5. ☺Eduction can be defined as; a process of making explicit, in a second proposition, a
meaning that is virtually contained in the original from which it is derived.
6. This study is important because people most often attach an interpretation (that is, an
implication) to a statement that is not justified by the statement itself.
WHAT IS OBVERSION?
1. ☺☺The purpose of obversion is to change the quality (positive or negative) of the
proposition, without changing the meaning.
2. NOTE: The second proposition must be consistent in meaning with the original.
3. The original statement is called the obvertend; the derived statement, the obverse.
4. Example:
obvertend: None of his arguments is irrefutable.
obverse: All of his arguments are refutable.
5. In obversion, if the obvertend is true, then the obverse is necessarily true.
150
6. In the above example the proposition is stated in a negative form: “None of his arguments is
irrefutable.”(uses a double negative).
7. Its positive form must have the same meaning. “All of his arguments are refutable.”
RULES OF OBVERSION
RULE 1: Contradict the copula
1. To contradict the copula, make an affirmative copula negative (change is to is not) and a
negative copula affirmative (change is not to is).
2. NOTE: In an E proposition (No S is P), the copula, although apparently affirmative, is really
negative because of the word “no.”
3. In obverting an A proposition to an E, the copula is automatically negated by the use of words
like “no,” “none,” “nobody.”
RULE 2: Contradict the predicate term.
1. The contradictory of any given term is its denial.
2. You can use a prefix or a suffix to contradict a term. For example: in-, im-, ir-, -less,
(endless), and so on.
3. Make sure that the prefixes and suffixes—especially prefixes—really involve a denial.
4. Note the following list of terms:
true contradictories not contradictories
adequate—inadequate valuable—invaluable
expensive—inexpensive tense—intense
mortal—immortal loosened—unloosened
moral—amoral flammable—inflammable
5. The terms in the first list are true contradictories; those in the second are not.
151
6. The prefix in-, as in “invaluable” doesn’t contradict the meaning of the original word, it
intensifies it!
7. Careful with the following:
religious—irreligious (?)
advantageous—disadvantageous (?)
moral—immoral (?)
legal—illegal (?)
There’s a big difference between a book that is nonreligious and one that is irreligious.
There’s a big difference between an act that is amoral and one that is immoral.
There’s a big difference between a non-legal action and one that is illegal.
8. If in doubt, attach the prefix non- to the original word to contradict it.
9. In some propositions, the predicate term is too confusing. It’s better not to attempt an
obversion in this case.
KINDS OF OBVERSION
1. Before obverting a proposition, put it into its logical form.
2. Applying the rules to each of the four types of single categoricals, you get the following:
a. An A proposition obverts to an E proposition:
obvertend: (A) Every natural event is explainable.
obverse: (E) No natural event is unexplainable.
b. An E proposition obverts to an A proposition:
obvertend: (E) No just act is unrewarded.
obverse: (A) Every just act is rewarded.
c. An I proposition obverts to an O proposition:
obvertend: (I) Some people are unemployed.
obverse: (O) Some people are not employed.
d. An O proposition obverts to an I proposition:
152
obvertend: (O) Some workers are not members of a union.
obverse: (I) Some workers are nonunion members.
3. ☺☺The reason we may want to obvert a proposition is to see the positive implications in a
statement that is negative in form.
4. NOTE: Some modern writers and speakers like to use the double negative because it can be
confusing.
5. Example:
War is not without its ethical implications.
War has ethical implications.
6. The double negative can be used successfully—as it is here. Just be aware of its occasional
abuse.
SAMPLE FORMS
See Extra 19: Square of Opposition and Eductions
153
CONVERSION
1. ☺Conversion is a process of interchanging the subject and predicate terms in a way that
the result (the converse) is consistent with the meaning of the original (the convertend).
2. There are rules to do this. You can’t arbitrarily switch the terms.
3. Example:
“Snakes eat pigs”, therefore, “Pigs eat snakes.”
4. Even if you correctly convert the proposition, “Some snakes are pig-eaters” to “Some pig-
eaters are snakes,” the process still appears trivial (consider the boa).
5. If the conversion is done properly, you can avoid incorrect implications.
6. Example:
convertend: (A) All true art has an ennobling effect.
7. From this statement you may not imply that anything which has an ennobling effect is true
art. The appropriate converse would be:
(I) Something that has an ennobling effect is true art.
8. So, if the original proposition (the convertend) is true, the derived proposition
(the converse) is necessarily true.
CONVERSION RULES
1. Before trying to do a conversion you have to determine if the proposition is A, E, I, or O so
you can put the convertend in its logical form.
RULE 1: Keep the quality of the converse the same as that of the convertend. (Quality
means positive or negative)
2. If the convertend is affirmative, the converse must likewise be affirmative; if the convertend
is negative, the converse must also be negative.
RULE 2: In converting a proposition do not overextend a term.
154
3. Before converting a proposition it’s necessary to determine the quantity of each of the terms
in the convertend.
4. ☺☺REMEMBER: The predicate of an affirmative proposition is particular.
The predicate of a negative proposition is universal
(Check your chart for all possible forms of A, E, I and O propositions.)
5. Suppose that one of the terms of the convertend is particular (undistributed).
6. If you make this term universal (to distribute it) in the converse you would violate Rule 2.
7. NOTE: Nothing in the rule forbids a reduction in the quantity of a term. You just can’t
overextend it.
8. If a term that is universal in the convertend becomes particular in the converse, there is no
rule violation.
9. GENERAL RULE: It is never legitimate (in an implication or an inference) to overextend
a term. However, there is nothing that forbids a reduction of a term.
See Extra 20: Eductions Exercise
TYPES OF CONVERSION
“A” Proposition Conversion
1. Conversion rule #2 is more often violated when converting an A proposition.
2. Example:
All scientific knowledge is organized.
Therefore all organized knowledge is scientific.
3. This may seem logical, as do most fallacies of implication, but let’s check it out.
Su P
p
convertend: (A) All scientific knowledge is organized (knowledge).
155
Pp S
p
converse: (I) Some organized knowledge is scientific.
4. Since the predicate term of the convertend (an A proposition) is particular, it must remain
particular.
5. Since the subject of the converse is a particular term, the converse itself is a particular (I)
proposition.
6. Thus: All S is P becomes Some P is S.
7. NOTE the fallacy in the following implication:
Every good Republican is a good American.
8. Therefore:
Every good American is a good Republican.
9. This type of fallacy is common in everyday life!
“E” Proposition Conversion
1. ☺In converting an E proposition it’s impossible to overextend a term, because both S and
P are universal.
convertend: (E) No scientist is an ignorant person.
converse: (E) No ignorant person is a scientist.
2. If one (or both) of the terms is negative, check your conversion twice. It’s easy to make an
error here.
3. Example:
A person who is not virtuous is not happy.
4. This proposition does not convert to
No happy person is virtuous.
5. But it does convert to:
No happy person is non-virtuous.
156
“I” Proposition Conversion
1. An I proposition converts to another I.
convertend: (I) Some ignorant people are prejudiced.
converse: (I) Some prejudiced people are ignorant.
“O” Propositions DO NOT CONVERT.
1. Example:
Some detectives are not policemen.
2. Therefore:
Some policemen are not detectives.
3. Each of these propositions are true but the second is not the converse of the first.
4. The above example is as invalid as the following:
Some Americans are not New Yorkers.
Some New Yorkers are not Americans.
Some engines are not gas-propelled.
Some gas-propelled objects are not engines.
5. Rule 2 of conversion forbids the conversion of an O proposition.
6. Suppose an O proposition as convertend and another O as the converse.
7. NOTE below the overextension of the subject term of the original proposition when it
appears as predicate of the converse:
S P
Supposed convertend: particular universal
S P
Supposed converse: particular universal
157
SAMPLE FORMS
8. In the above examples the original forms of the terms are retained as given; only their
position is changed.
LET’S COMBINE THE FORMS
1. You can combine obversion and conversion.
2. At first glance the combined forms look more difficult than they are.
3. We’ll only deal with three types:
a. Partial contraposition (PC)
b. Full contraposition (FC)
c. (full) Inversion (INV)
158
Partial Contraposition
1. To do a partial contrapositive of a proposition you obvert then convert as a continuous
process:
2. NOTE: The partial contrapositive (PC) of an A is E; of an E is I; of an O is I.
3. You can’t partial contrapositive an I proposition.
4. Note in partial contraposition:
(A) the terms are reversed (as in conversion)
(B) the quality is changed (as in obversion)
(C) the original P-term is denied.
5. Thus by direct process:
All S is P becomes No non-P is S
All non-S is non-P becomes No P is non-S.
6. Exercise. The following are correctly worked out forms of partial contraposition. Go over
these examples on paper to see how they were derived.
159
(A) All expected events are foreseen.
(E) Nothing unforeseen is an expected event. (PC)
(E) No buttered popcorn is inexpensive.
(I) Something expensive is buttered (popcorn). (PC)
(O) Some human being is not insensitive to pain.
(I) Some being that is sensitive to pain is human. (PC)
Full Contraposition
1. For full contraposition use the formula OCO. (obvert, convert, obvert)
2. Note: The full contrapositive of an A is an A; of an E, is an O; of an O is an O. I has no full
contrapositive.
160
3. A quicker way to do an FC, is by reversing the terms of the original statement and then
contradicting both of the original terms.
4. Example:
All S is P becomes All non-P is non-S.
5. NOTE that an E proposition reduces to an O.
6. Example:
(A) All imaginative persons are resourceful.
(A) All unresourceful persons are unimaginative. (FC)
(E) No finite being is perfect.
(O) Some imperfect being is not infinite. (FC)
(O) Some involuntary act is not deliberate.
(O) Something indeliberate is not a voluntary act. (FC)
(A) You say that all the graduate students are required to take this course.
(A) In that case, all students who are not required to take this course are nongraduate students.
(FC)
Inversion
1. The (full) inverse of an A is worked out by a double process of obversion and conversion
(OCOC).
2. The inverse of an E is worked out by a double process of conversion and obversion (COCO).
I and O propositions cannot be inverted.
161
3. NOTE: The inverse of an A is I. The inverse of an E is an O proposition.
4. Inversion can be done by contradicting both of the original terms, and leave them in their
original position.
5. Remember, in full contraposition we contradict both terms and reverse them.
6. In inversion we contradict both terms but do not reverse them.
7. Example:
1. (A) All inattentive students are undeserving of good marks.
(I) Some attentive students are deserving of good marks. (INV)
162
2. (E) No mature person is unstable.
(0) Some immature person is not stable. (INV)
3. (A) You claim that all Communists are radicals.
(1) I imply from this that; some non-Communists are non-radicals. (INV)
8. Check out the summary charts regarding PC, FC, and INV.
9. Chart A shows how each of the combined forms can be arrived at by the direct process.
10. Chart B is a summary of the kind of proposition (A, E, I, O) that is obtained from each of
the combined forms.
A REVIEW OF THE LOGIC OF IMPLICATION
1. You would want to review the various types of implication from the square of opposition,
obversion, conversion, and the combined forms.
2. One way of doing this review is to take each of the four types of propositions (A, E, I, and O)
and work out whatever implications are derivable from them.
3. For example, given a true A in the form All S is P, we get the following results:
163
4. For the square of opposition, the terms are retained in their original form and position. Only
the quantity and quality is changed.
5. In obversion, the quality is changed and the P-term is denied.
6. In conversion, the quality is the same, but the terms are reversed.
7. In partial contraposition, the quality is changed, the terms reversed, and the P-term is denied.
8. In full contraposition, the quality is the same, and the terms are reversed and both are denied.
9. In inversion, the quality is the same, and the original terms are both denied.
10. By determining the type of implication, you can assess the validity of a proposition.
See Homework Lesson 9
164
Extra Materials
Extra 19: Eductions Exercise
I. Obvert the following propositions. (change the quality [positive or negative] of the proposition,
without changing the meaning). If you are in doubt as to the usual means of contradicting a given
predicate term, use the prefix “non-”.
a. Everyone who is registered is a freshman.
b. No poor man is wealthy.
c. Every material thing is corruptible.
d. Not every non-philosopher is a scientist.
e. Some scientists are unprejudiced.
f. Every unknown hero is deserving of praise.
g. Most scientists are non-philosophers.
h. No authenticated manuscript is false.
i. Not all tobacco is filtered.
j. Most television programs are non-educational.
II. Put the following propositions in their logical form if necessary, and then convert them if they are
convertible. (interchanging the subject and predicate terms).
NOTE: (a) a converted term must have the same meaning that it had in the convertend; (b) in
converting a proposition you do not merely change it to the passive voice: for example, "Every
donkey has long ears" to "Some long ears are had by donkeys"; and (c) in converting a proposition it
is permissible to supply a new word where necessary: for example, "Some professors are learned" to
"Some learned men are professors." REMBER: an “O” proposition is not convertible.
a. Some unforgettable events are unforeseen.
b. Whoever loves God loves his neighbor.
c. Every stenographer has a boss.
d. Not every political observer is a disinterested one.
e. Every eagle has claws.
f. Most teachers are non-wealthy men.
g. No university is a high school.
h. Physical exercise requires energy.
165
i. Some of my neighbors are "crabs."
j. Margaret loves Mike.
k. Some that don't talk do.
l. The United Nations is working toward the cause of peace.
m. Some immoral actions are not illegal.
n. A liar is not an honest man.
o. Not every unexpected guest is unwelcome.
p. Every Communist is a radical.
q. Some cars are not convertibles.
r. All wise men are knowledge seekers.
s. All businessmen are taxpayers.
t. Every man is not worthy of honor.
III. Evaluate the following eductions.
a. If every learned person is a reader of books, it follows that anyone who reads books is learned.
b. Whatever you dream you imagine, and conversely whatever you imagine you dream.
c. Some insane people are not morally culpable. It follows from this that some persons who are
morally culpable are not insane.
d. If it is true that all virtuous deeds are not unrewarded, then some virtuous deeds are rewarded.
e. If every good Christian loves his neighbor, then, clearly, anyone who loves his neighbor is a good
Christian.
f. You say that no good American is a radical. In that case, no radical is a good American.
g. To state that some people are not unconcerned about their health is to imply that they are concerned.
BACK
166
Extra 20: Square of Opposition and Eductions
A. To disprove each of the following propositions, what proposition must be proved? In other
words, give the contradictory of each of the following propositions.
1. "Man was born free and is everywhere in chains." (Rousseau)
2. "There is nothing new under the sun." (Ecclesiastes)
3. "I alone have escaped to tell you." (Job, quoted in Moby Dick)
B. Evaluate the following inferences:
1. Since it's true that some savings and loans are not safe, it must be true that some savings and
loans are.
2. It's false to say that no man ever lost money misjudging the acumen of consumers. Therefore
it must be true that some men did lose money misjudging consumer acumen.
3. It must be true that not all truths are true, because it's false that every truth is true.
4. If love is never unsuccessful, then it can't be true that some love is unsuccessful.
5. If it's false that some teachers are insane, it must be false that all teachers are insane.
6. Since some persons are good at arguing, some persons are not good at arguing.
C. If you know that the proposition in column 1 is true or false as indicated, what do you know
about the corresponding proposition in column II? Is it true, false, or unknown? NOTE: We are
using nonexistent nouns here; “nobs” and “palats”.
Column I Column II
1. All nobs are palats: false. 1. Some nobs are palats.
2. Some nobs are palats: false. 2. All nobs are palats.
3. No nobs are palats: false. 3. All s nobs are palats.
4. Some nobs are not palats: false. 4. Some nobs are palats.
5. Some nobs are not palats: true. 5. No nobs are palats.
6. Some nobs are palats: true. 6. No nobs are palats.
7. No nobs are palats: true. 7. Some nobs are not palats.
8. All nobs are palats: true. 8. All palats are nobs.
BACK
167
Exercises for Homework:
Homework Lesson 9:
1. How can you define Eduction?
2. Why is the study of eductions important?
3. What is the purpose of obversion?
4. List the two rules of obversion.
5. What must you pay attention to when using a prefix or a suffix to contradict a term?
6. What is the difference between the term “religious” and “irreligious”?
7. List the four types of obversion?
8. Make up your own examples to illustrate all four types of obversion.
9. Why do we obvert a proposition in the first place?
10. What is Conversion?
11. List the two rules of conversion.
12. Give the general rule of conversion.
13. What type of proposition does an “A” proposition convert to? Can you give an example?
14. Why does the conversion of an “A” proposition often result in a violation of Rule #2?
15. What is wrong with this conversion of an “A” proposition?
“Every good Republican is a good American.”
“Every good American is a good Republican.”
16. Give the proper conversion of the above example.
17. Why is it impossible to overextend a term when converting an “E” proposition? Give an example
of a properly converted “E” proposition.
18. How is an “I” proposition converted? Can you give your own example of a converted “I”
proposition?
19. Why can’t you convert an “O” proposition?
20. Using the following example show why the given conversion is incorrect.
“Some detectives are not policemen.”
“Some policemen are not detectives.”
21. List the three forms of conversion when you combine obversion and conversion.
22. What is the cardinal rule of Logic? (repeat question from past lesson) BACK
168
LESSON TEN—DEDUCTIVE REASONING—THE THIRD ACT OF THE MIND
1. The most important part of the study of Logic is reasoning and the laws that govern it.
2. All the pages that you’ve studied are a preparation for the study of reasoning—the third act
of the mind.
REASONING vs. JUDGMENT
1. ☺Reasoning is the most complex of the three acts of the mind.
2. Through reasoning, the intellect knows whether something is valid or not.
3. REMEMBER: it’s only in the act of judgment that we actually attain truth.
4. That’s why we say that the act of judgment is considered to be the most significant of the
three acts of the mind.
5. ☺Reasoning is never an end in itself, but merely a means toward an end—the end always
being some new truth that is attained in an act of judgment.
6. Remember, reasoning is made up of judgments (in reasoning, we call a judgment a premises).
7. The judgments (propositions/premises) are either true or false. The reasoning process itself is
either valid or invalid.
8. If we say that reasoning is true or false, we say it only by way of analogy.
FIRST PRINCIPLES—SELF EVIDENT TRUTH
1. ☺If you could know truth by intuition, there would be no need for reasoning.
(Intuition in Latin is intueri, to look into.)
2. What we know intuitively, we know directly and immediately.
169
3. Intuitive truths are relatively few, but they are fundamental.
4. Our intuitions are objective and universal—they are not hunches, suppositions, or
assumptions.
5. ☺☺These intuitions are called self-evident truths (first principles)—truths that are
common to all people.
6. Example: “A thing can-not be and not be at the same time and in the same respect.” Aristotle
said, “The most certain of all basic principles is that contradictory propositions are not true
simultaneously."
7. This is called the principle of noncontradiction.
8. It’s impossible to think of a person both as a father and not a father with respect to his own
son.
9. First principles are important because they’re the starting point of all human knowledge and
the starting point of every act of reasoning that proceeds from them.
10. ☺☺Without these truths we could not reason at all!! (Something can’t come from
nothing)
11. If you didn’t have first principles, consider how a person might reason:
Every man has the ability to laugh.
Jerry is a man.
Jerry does not have the ability to laugh.—nonsense, for sure!
See Extra 21: The defense of noncontradiction
ACQUIRING NEW KNOWLEDGE
1. Reasoning is important because it allows us to advance human knowledge beyond intuition—
beyond the first principles.
170
2. Keep in mind that what we know by reasoning is not self-evident.
3. ☺Reasoning is a process by which we come to know something that was, up until now,
unknown. (TO US)
4. How does this work?
5. To learn something new, we relate the new element to what was known previously. We are
building on old knowledge.
6. We never come to know something that is completely new.
7. From certain known truths, we advance to a knowledge of some new truth.
8. ☺What we actually come to know, in a conclusion, was potentially known in the premises
that produced it.
9. You can’t put two different truths together and somehow arrive at a new judgment.
10. Example: There’s no logical connection between the two following propositions:
Brazil is one of the world's greatest coffee centers.
The baby is crying for its bottle.
11. Since there is no relationship between these propositions, no conclusion can be made.
12. To draw a conclusion, the two propositions must be related to each other.
13. Example:
Reading material, which deals only with matters of current interest, will be of little use
to future generations.
Most magazines are exclusively concerned with matters of current interest.
Most (of today's) magazines will be of little use to future generations.
14. We learned that judgment is an act of the mind that unites or disunites two objects of
thought.
171
15. ☺☺Reasoning is an act of the mind by which the intellect unites
two distinct judgments and comes to know some new truth.
VALIDITY AND TRUTH
1. Under what conditions does reasoning lead to new knowledge?
2. There is a connection that exists between validity and truth.
3. .3☺☺We come to the knowledge of a third proposition from two previous propositions that
are true and are logically connected.
4. NOTE: This third proposition (the conclusion) must be true if the propositions are true and
if the reasoning process is valid.
5. Given the fulfillment of these two conditions, the propositions are true and the reasoning
process is valid, the conclusion must be true.
6. On the other hand, if either of these two conditions is lacking there can be no guarantee of
the truth of the conclusion.
7. If you reason from true premises, and your reasoning is invalid, your conclusion will be
doubtful.
8. Even if you reason validly from premises, one or both of which are false, your “conclusion”
will again be doubtful.
172
See Homework Lesson 10
Extra Materials
Extra 21: The defense of noncontradiction
Some philosophers have denied the principle of non-contradiction—thus the following repartee.
As an indirect defense of the principle of Noncontradiction, consider the following fun but
insightful dialogue between A and B, two fictional disciples of the philosopher Georg Wilhelm
Friedrich Hegel (1770-1831). (This dialogue is from H. Gensler’s book Formal Ethics
[Routledge 1996], pp. 36-37.)
A: Are you still a follower of Hegel?
B: Of course! I believe everything that he wrote. Since he denied the law of noncontradiction
[a.k.a. PNC], I deny this too. On my view, P [where P is any proposition] is entirely compatible
with not-P.
A: I'm a fan of Hegel myself. But he didn't deny the law of noncontradiction.
B: You read the wrong commentators!
A: Don't get so upset! You said that he did deny the law, and I said that he didn't. Aren't these
compatible on your view? After all, you think that P is compatible with not-P.
B: Yes, I guess they're compatible.
A: No they aren't!
B: Yes they are!
A: Don't get so upset! You said that they are compatible, and I said that they aren't. Aren't these
two compatible on your view? Recall that you think that P is compatible with not-P.
B: Yes, I guess they're compatible. I'm getting confused.
A: And you're also not getting confused! Right?
(Author’s note; You can be confused and not confused at the same time!)
BACK
173
Exercises for Homework:
Homework Lesson 10:
1. Explain why reasoning is the most complex part of the study of Logic.
2. Why is judgment considered the most important aspect of Logic?
3. What is the purpose or end of reasoning?
3. Explain what intuition is in Logic. How is it different from saying, “woman’s intuition”?
4. If intuition is not a hunch or an assumption, what is it?
5. Intuitions are called self-evident truths (first principles). Give one example of a self-evident truth.
6. Why are first principles important?
7. Define reasoning.
8. How do we learn through reasoning?
9. Under what conditions does reasoning lead to new knowledge?
10. If both premises are true, and the reasoning valid, what can you say about the conclusion?
BACK
174
LESSON ELEVEN—THE NATURE, STRUCTURE, & PRINCIPLES OF THE
CATEGORICAL SYLLOGISM
1. We use categorical syllogisms every day, in some way.
2. ☺According to Aristotle's definition: A syllogism is a discourse in which, certain things
being stated [that is, the premises], something other than what is stated [that is, the
conclusion] follows of necessity from their being so. (Prior Analytics, Bk. I, Ch. I, 24b, 18)
3. The categorical syllogism is comprised of categorical propositions.
4. ☺☺We may define a categorical syllogism as an argumentation in which two terms, by
virtue of their identity or nonidentity with a common third, are declared to be identical or
nonidentical with each other.
5. The following are examples of categorical syllogisms. In each of these examples the
“common third”, or “middle term”, is italicized:
All cats are sentient. No bug is a giant.
A lion is a cat. Some man is a giant.
A lion is sentient. Some man is not a bug.
Every giraffe has a long neck. All geese are good swimmers.
Every giraffe has spots. Every good swimmer is a floater.
Some animal with spots has a long neck. Some floater is a goose.
6. ☺Matter: (composition) The categorical syllogism consists of three propositions and three
terms within the propositions.
7. In our examples (both above and below), the first two propositions are the premises
(sometimes called the “antecedent”); the third is the conclusion (the “consequent”).
8. Example:
All human contracts are subject to violation.
A peace treaty is a human contract.
Therefore, all peace treaties are subject to violation.
175
9. ☺NOTE: although there are only three distinct terms, each of them appears twice within
the syllogism.
All human agreements (1) are subject to violation. (3).
A peace treaty (2) is a human agreement. (1).
All peace treaties (2) are subject to violation (3).
10. The three terms are as follows:
A. The Middle Term (M)
1. This term appears in each of the premises but never in the conclusion.
2. The middle term is the heart of the syllogism, since it is the means by which it is possible
to connect the other two terms in the conclusion.
3. The middle term is the common standard of reference for the relation that exists between
the terms of the conclusion.
4. In the example above the middle term is “human agreements”.
5. ☺NOTE: The middle term is always the reason for the conclusion.
6. The conclusion is made evident through the middle term.
B. The Subject Term (S)
1. This term, also called the minor term of the syllogism, appears once in the premises and
once in the conclusion.
2. ☺It’s called the subject term because it’s the subject of the conclusion.
3. “Peace treaties” is the subject term of the syllogism above.
C. The Predicate Term (P)
1. The predicate, or major term, of the syllogism likewise appears once in the premises and
once in the conclusion.
176
2. ☺It’s the predicate of the conclusion.
3. The predicate term of our example is “subject to violation”.
4. From now on, we will use the letters S and P to signify, not the subject or predicate of a
proposition, but the subject or predicate terms of the syllogism.
MAJOR AND MINOR PREMISES
1. From now on, we will call the premises of a syllogism the major and the minor premise.
2. ☺The major premise is the one in which the predicate, or major term, appears. (the P
term of the conclusion).
3. ☺The minor premise is the one in which the subject, or minor term, appears. (the S term
of the conclusion).
4. Example:
MAJOR PREMISE:
M P
All human contracts are subject to violation.
MINOR PREMISE:
S M
All peace treaties are human contracts.
CONCLUSION:
S P
All peace treaties are subject to violation.
5. Note: in the example above the major premise is first, and the minor, second.
6. This is the normal way to set up the syllogism—it’s not always followed in practice.
177
THE FORM OF A SYLLOGISM
With respect to form, we distinguish in every categorical syllogism both its mood and its
figure.
1. ☺The mood of a syllogism is the respective designation of the premises and conclusion as
A, E, I, or O propositions.
2. In the example given above, both the premises and the conclusion are A propositions;
therefore, we signify its mood thus: AAA (major A, minor A, conclusion A).
3. ☺The figure of a syllogism is the position of the middle term, in the premises.
4. Example:
Major M—P
Minor S—M
Conclusion S—P
5. Here the middle term is the subject of the major premise and the predicate of the minor.
6. A categorical syllogism will appear in one of four possible combinations:
7. In Figure 1 the M-term is the subject of the major premise and predicate of the minor.
8. In Figure 2 it’s the predicate of both premises.
178
9. In Figure 3 it’s the subject of both premises.
10. Figure 4 is the reverse of 1—the M-term is the predicate of the major and the subject of the
minor premise.
11. Learn to identify these figures by number.
12. Exercise: Specify both the mood and the figure of each of these examples.
1) All cats are sentient. 2) No bug is a giant.
A lion is a cat. Some man is a giant.
A lion is sentient. Some man is not a bug.
3) Every giraffe has a long neck. 4) All geese are good swimmers.
Every giraffe has spots. Every good swimmer is a floater.
Some animal with spots has a long neck. Some floater is a goose.
See Extra 22: First Matter and Form Exercise
See Extra 23: Second Matter and Form Exercise
SYLLOGISTIC REASONING’S UNIVERSAL PRINCIPLE
1. The universal principle has two parts:
☺ A. Two things that are identical with a common third are identical with each other.
☺ B. Two things, of which one is identical with a third and the other nonidentical,
differ from each other.
2. We use the words in the above statements: “identical” and “nonidentical” rather than “equal”
and “unequal” so as not to confuse logical identity or nonidentity with mathematical equality or
inequality.
3. When the Logician uses the word “identity” it means likeness of comprehension or meaning,
and not a mathematical equivalence.
Figure 1
Figure 3
179
4. Note how the above principle is applied to the theory of the syllogism:
MAJOR: 1. Every organism is endowed with life.
MINOR: 2. A cat is an organism.
CONCLUSION: 3. A cat is endowed with life.
5. The middle term is organism.
6. The term organism is “identified” in the major premise with endowed with life (Step 1).
7. In the minor it is identified with cat (Step 2).
8. Having separately identified cat and endowed with life with a common third (organism), we
identify them with each other in the conclusion:
A cat is endowed with life (Step 3).
9. The three steps are diagramed:
10. We can illustrate the second part of this principle with the following syllogism:
MAJOR: 1. No purely material substance is an organism.
MINOR: 2. A rock is a purely material substance.
CONCLUSION: 3. No rock is an organism.
11. Here we designate (in the major) the nonidentity of the P term, organism, with the M term,
purely material substance.
12. In the minor we declare the S term, rock, to be identical with the M term, material
180
substance.
13. In the conclusion we declare the two extremes (S and P) rock and organism to be non-
identical with each other. Each step is traced in the following diagram:
AXIOMS
1. Of the above four figures, figure 1 is the most common and typical.
2. REMBER: In Figure 1, the M term is placed as subject of the major premise and as predicate
of the minor (M-P, S-M).
3. Figure 1 will always take the form “All (or no) M is P; S is M; therefore, S is (or is not) P.”
4. The next two axioms only apply to Figure 1.
5. These two axioms are important because Figure 1 is more basic than any of the other figures.
☺AXIOM 1: Whatever can be affirmed of a logical whole can be affirmed of its logical parts.
☺AXIOM 2: Whatever can be denied of a logical whole can be denied of its logical parts.
6. Example of Axiom 1:
Every science is knowledge building.
Mathematics is a science.
181
Mathematics is knowledge building.
7. The “logical whole” of the above example is science and the “logical part” is mathematics.
8. The procedure we have followed in constructing the syllogism is this:
1) Affirm something of a logical whole
(knowledge building is affirmed of science);
2) Include the part within the whole
(mathematics is included within the area of science);
3) Affirm of the part what has already been affirmed of the whole
(knowledge building is affirmed of mathematics).
9. NOTE that Step 1 represents the major premise, Step 2, the minor, and Step 3, the conclusion.
10. Exercise. Let’s construct our own syllogisms.
11. Supply the appropriate terms in the blank spaces by following the three steps indicated
above:
12. Draw up four examples of your own. Begin with a “logical whole,” and continue as stated
above.
13. An example of Axiom 2:
No person is a piece of property.
A slave is a person.
No slave is a piece of property.
14. The “logical whole” of this example is person; the “logical part” is slave.
15. The procedure we followed in constructing the syllogism is this:
1) Deny something of a logical whole
(piece of property is denied of person)
182
2) Include the part within the whole
(slave is included within the extension of person)
3) Deny of the part what has already been denied of the whole
(piece of property is denied of slave).
16. Again note that 1) stands for the major; 2) for the minor; and 3) for the conclusion.
17. Exercise. Complete the following example.
Construct another two examples on your own.
18. A thought: It would be wrong to suppose that because General Motors (“logical whole”) is a
wealthy organization and because I am a GM stockholder (“logical part”) that I am wealthy.
19. The problem above is that the “logical whole” must be taken to mean a genuine universal
and not a collective totality.
20. A genuine universal:
1) Signifies some nature that is common to many.
2) Is found in every member belonging to its extension.
3) Is known by way of abstraction.
21. Example:
Every true genus is common to more than one species;
It is realized in every species that belongs to it;
And it is known by way of abstraction.
22. Furthermore:
Every species (taken as a genuine universal)
Is common to many individuals;
Is completely realized in each member;
And it is known by abstraction.
23. By contrast, a collective unit (like “General Motors”) is not a true universal.
183
24. Not even the president of the organization is General Motors.
25. However, it can always be said of any individual man (like Joe Brown) that he is a man
(that he possesses this nature as his own).
26. ☺A realistic logic is based on an abstractive knowledge (however imperfect) of the
natures of the things that we know.
27. REMBER: If man didn’t have the power of abstraction, it would be impossible for him to
judge or to reason.
APPARENT VS REAL SYLLOGISMS
1. Some people today doubt the real value of syllogistic reasoning as a means of gaining new
knowledge.
2. Since the time of Francis Bacon (1561—1626), there has been a tendency either to minimize
the importance of syllogistic reasoning or to deny its value altogether—they say that this type of
reasoning is circular.
3. It’s true that we have to distinguish between an apparent syllogism and a genuine one—one
that leads a person to new knowledge.
4. ☺The principles, upon which a categorical syllogism is based, are true principles and are
not the result of a collection of singulars.
5. True principles are genuine “starting points” from which further knowledge can proceed.
6. Consider the following example:
1. All the books in my library bear my signature.
2. Gone with the Wind is one of the books in my library.
3. Therefore, this book (Gone with the Wind) bears my signature.
7. In this example there is no progression of thought. It’s a circular argument.
184
8. After examining each book in my library (including Gone with the Wind) I know that it bears
my signature. I have to know the conclusion before I state the first proposition.
9. Let’s take another example:
1. All the New England states voted Republican in the presidential election of 1952.
2. Main is a New England state.
3. Therefore Main voted Republican in the presidential election of 1952.
10. In this example, the knowledge of proposition 3 is not dependent upon proposition 1. No
new knowledge.
11. I had to know that Main, as well as all the New England states, voted Republican before I
constructed the syllogism—this is not new knowledge gained from syllogistic reasoning.
12. Compare the examples above with the following one:
1. No action that is merely a means to an end is performed for its own sake.
2. Walking is an action that is merely a means to an end.
3. Therefore, walking (is an action that) is not performed for its own sake.
13. In this example, proposition 1 is not an enumeration of singulars.
14. It represents a judgment independent of the knowledge that walking, eating, sleeping, and so
on, are not actions performed for their own sake.
15. Consider the following:
1. No action that is merely a means to an end is performed for its own sake. And
knowing that
2. Walking is merely a means to an end. A person comes to know as a result of the
connection that exists between these two propositions that
3. Walking (is an action which) is not performed for its own sake.
16. Proposition 3 represents a genuine movement of thought, a real advance in knowledge, and
is not a mere reaffirmation of what had been known “all along.”
See Extra 24: Genuine Syllogisms Yes or No
See Homework Lesson 11
185
Extra Materials
Extra 22: First Matter and Form Exercise
DIRECTIONS:
1. Mark off the matter of each syllogism—M, S, P.
2. Indicate the major premise and minor premise.
3. Mark the mood of each premises—A, E, I, O.
4. Indicate the figure of each syllogism. (see above chart)
A All cats are sentient. No bug is a giant.
A A lion is a cat. Some man is a giant.
A A lion is sentient. Some man is not a bug.
Every giraffe has a long neck. All geese are good swimmers.
Every giraffe has spots. Every good swimmer is a floater.
Some animal with spots has a long neck. Some floater is a goose.
186
BACK
Extra 23: Matter, Mood and Figure
Directions: Mark the matter of each syllogism (M, S, P) – Indicate the major and minor premise
– Mark the mood of each premise (A E, I, O) – Indicate the figure of each syllogism (1, 2, 3, 4).
1. All baffling subjects are subjects I dislike.
And math is a baffling subject.
Therefore math is a subject I dislike.
2. Treating a human person like an animal is wrong.
Selling human persons is treating one like an animal.
Therefore selling human persons is wrong.
3. All the books in my library are first editions.
That book is a book in my library.
That book is a first edition.
4. All men are mortal.
St. Thomas was a man.
Therefore St. Thomas is mortal.
5. Whatever is composed of parts is can be destroyed.
Whatever is material is composed of parts.
Therefore whatever is material can be destroyed.
187
6. Hitler was a tyrant.
A tyrant is not a good person.
Therefore Hitler was not a good person.
7. All men are mortal.
All teachers are men.
Therefore all teachers are mortal.
8. All teachers are men.
No men are immortal.
Therefore no immortals are teachers.
9. No humans are gods.
And all men are humans.
Therefore no men are gods.
10. Calculators are machines.
And humans are not machines.
Therefore humans are not calculators.
BACK
188
Extra 24: Genuine Syllogisms Yes or No
Directions: Indicate which of the following examples are genuine syllogisms and which
are not. Explain your answers.
1. Every president has the power to execute the nation's laws.
Abraham Lincoln was a president.
Abraham Lincoln had the power to execute the nation's laws.
2. All of the freshmen in our school are graduates of the local high school.
Jacobs is one of the freshmen in our school.
Jacobs is a graduate of one of the local high schools.
3. No good businessman is in the habit of making poor investments.
Jones is a good businessman.
Jones is not in the habit of making poor investments.
4. Damp climates are not conducive to sound health.
Some sections of the US are characterized by the dampness of their climate.
Some sections of the United States are not conducive to sound health.
5. All of the trucks in our fleet have Diesel engines.
This semitrailer is one of the trucks in our fleet.
This semitrailer has a Diesel engine.
6. No one living in our block is a member of the Chamber of Commerce.
Smith lives in our block.
Smith is not a member of the Chamber of Commerce.
7. Every automobile in this parking lot is American-made.
Smith's automobile is in this parking lot.
Smith's automobile is American-made.
BACK
189
Exercises for Homework:
Homework Lesson 11:
1. State the two universal principles of syllogistic reasoning.
2. Why do we use the terms “identical” and “nonidentical” rather than “equal” and “unequal”
when comparing terms?
3. Diagram the syllogism:
All dogs are sentient.
A Dachshund is a dog.
A Dachshund is sentient.
4. What is the most common figure? Show how the M term is placed as subject of the major
premise and as predicate of the minor in an example of your own.
5. State the two axioms that apply to figure 1.
6. Draw up four examples of your own. Begin with a “logical whole and proceed as in the
example.
7. Draw up two examples of your own making a universal denial.
8. Why is the following syllogism considered circular?
1. All the books in my library bear my signature.
2. Tolstoy's War and Peace is one of the books in my library.
3. Therefore, this book (Tolstoy's War and Peace) bears my signature.
Mark the matter of each syllogism (M, S, P) – Indicate the major and minor premise – Mark the
mood of each premise (A E, I, O) – Indicate the figure of each syllogism (1, 2, 3, 4).
9. All that is material is mortal.
Man is material.
Therefore man is mortal.
10. Every dog barks.
A constellation is a dog.
Therefore a constellation barks.
BACK
190
LESSON TWELVE—CATEGORICAL SYLLOGISMS: THEIR RULES, MOODS,
AND FIGURES
1. ☺☺You need to become seriously familiar with the rules of syllogistic reasoning.
2. It’s not enough to learn the rules by rote. You have to know how to put the rules into practice.
3. You will learn six rules and two corollaries.
1. The syllogism should consist of no more than three terms.
2. The middle term must be universal in at least one premise.
3. No term which is particular in a premise may be made universal
in the conclusion.
4. No conclusion can be drawn from two negative premises.
5. Two affirmative premises require an affirmative conclusion.
6. A negative premise requires a negative conclusion.
4. NOTE: Rule 1 pertains to the structure of the syllogism.
Rules 2 and 3 pertain to the quantity of the terms.
Rules 4, 5, and 6 pertain to the quality of its propositions.
5. Corollaries: 1. No conclusion can be drawn from two particular premises.
2. If one premise is particular, the conclusion must be particular.
HOW THEY WORK
RULE 1: The syllogism should consist of no more than three terms.
1. Check out the two following propositions for their number of terms.
To err is human.
To forgive is divine.
191
2. There are four terms here and no one would mistake it for a syllogism.
3. Sometimes you can have an arrangement which contains four terms but seems like there are
only three.
major: Every square is four-sided.
minor: Your head is square.
conclusion: Your head is four-sided.
4. This example looks like a three-term construction: “four-sided” (P), “square” (M), and
“head” (S).
5. However, this is really no syllogism at all since the meaning of the intended middle term
“square” is different in each of the premises.
6. The argument is really a four-term construction, and therefore invalid.
7. ☺The name of the fallacy that violates Rule 1 is the fallacy of four terms.
8. Most typically, this fallacy has a middle term with an inconsistent meaning—known as the
fallacy of the ambiguous middle.
9. The fallacy of the ambiguous middle is often used in jokes.
10. Be careful when evaluating a syllogism—the middle term, in the course of an argument, can
have a significant change in meaning.
11. Note the following example:
major:
A division in the government of a nation is unfavorable to that nation's welfare.
minor:
The legislative, judicial, and executive branches of the United States government are a
division in the government of a nation.
conclusion:
The legislative, judicial, and executive branches of the United States are (a division)
unfavorable to that nation's welfare.
192
12. The more abstract the terms of an argument, the more difficult it is to detect this fallacy.
13. When necessary define the meaning of the terms—especially the following:
progress education
democracy peace
religion free enterprise
14. Note the following example:
The citizens of our community contributed a million dollars to the local charity drive.
Joe Jones did not contribute a million dollars to the local charity drive.
Joe Jones is not a citizen of our community.
15. The problem here is that in the major premise it is collectively applied to its subject, but in
the minor it is divisively applied.
16. There are, however, syllogisms which, having the appearance of four terms, are resolvable
into three.
17. Note the example:
Anyone who knows the science of reasoning is a competent judge of a sound argument.
Some students in our class know logic.
Some students in our class are competent judges of a sound argument.
18. Although the middle term of this syllogism is not worded the same, its meaning is the
same—the argument is valid.
19. It’s often mistakenly assumed that differently worded terms are the same in their meaning.
20. Consider the example:
major: Every man is rational.
minor: John is a man.
conclusion: John is intelligent.
21. This is an example of a four-term construction, without an ambiguous middle.
193
22. In this example the P term of the conclusion is different from the P term as it appears in the
major premise.
23. Rational, means having the capacity to be intelligent—it doesn’t follow that John, as a man,
is actually intelligent.
24. The terms “rational” and “intelligent” are two different terms, and so the syllogism is
invalid.
25. In actual argumentation, an error of this sort is far less easily detected, because few
arguments appear in a strictly logical form.
26. ☺☺NOTE: In actual discourse, there’s a lot of verbiage that appears between one
premise and another and between the premises and the conclusion.
27. To conclude the treatment of Rule 1, note another example of a four-term construction:
major:
Anyone who weakens his physical constitution is endangering his health.
minor:
Drunkards weaken their physical constitution.
conclusion:
People who take intoxicating drinks are endangering their health. invalid
28. It’s true that all drunkards are people who take intoxicating drinks, but it’s not true that
anyone who takes intoxicating drinks is a drunkard.
RULE 2: The middle term must be universal in at least one premise.
1. You know that it’s the function of the middle term to serve as a common point of reference
for uniting or disuniting S and P in the conclusion.
2. Yet, if the middle term is taken particularly—that is, undistributed—in both premises, there is
no guarantee that S and P are referring to the same part of M.
194
3. Example:
Everyone who has the flu is sick.
Everyone who has the measles is sick.
Therefore, whoever has the measles has the flu.
4. People afflicted with the flu and people afflicted with measles belong to two different parts of
the extension of those who are sick.
5. In order to unite or disunite the two terms of the conclusion (S and P), the middle term must
be taken universally—it must be distributed—in at least one of the premises.
6. Without this, no valid conclusion can be drawn.
7. ☺NOTE: The predicates of A and I propositions are particular and those of E and O
propositions are universal.
8. Note the quantity of the M term—signified by the small letter p—in both premises of the
following example:
Pu Mp
(A) Everything worthwhile is difficult of attainment.
Su Mp
(A) All virtue is difficult of attainment.
Su Pp
(A) All virtue is worthwhile.
9. M is marked “p” in both of the premises above because in both premises it’s the predicate of
an affirmative proposition (an A).
10. Keep in mind that the predicate of an affirmative statement (A or I) is particular.
11. The fallacy in the above example is that of an undistributed middle (Mp Mp).
12. The conclusion seems reasonable because it happens to be true.
13. ☺☺However, the logician doesn’t judge the validity or invalidity of a syllogism by the
possible truth of the conclusion.
14. The conclusion is invalid because of the fallacy of the undistributed middle.
195
15. In the above example S and P—virtue and worthwhile—happen to fall within the same part
of the extension of M—difficult of attainment.
16. Note the following fallacy:
Pu Mp
Everything worthwhile is difficult of attainment.
Su Mp
Carrying coals to Newcastle is difficult of attainment.
Su Pp
Carrying coals to Newcastle is worthwhile.
17. Even a non-logician would suspect that there is something wrong because the conclusion is
ridiculous.
18. The plausibility or non-plausibility of the conclusion is no basis for judging the validity of
an argument.
19. In order for a syllogism to be valid, the middle term must be universal in one of the
premises—either the major or the minor.
20. Often the middle term is universal in both premises—however, it’s not necessary for it to
be.
21. Exercise: Mark off the quantity of each term in the following arrangements,
and decide which of these do and which do not conform to Rule 2.
1. No M is P 2. All P is M 3. All M is P
Some S is M Some M is S Some M is S
Some S is not P Some S is P Some S is P (see answer booklet)
RULE 3: No term that is particular in a premise may be made
universal in the conclusion.
1. The fallacy that violates this rule is called the fallacy of overextension.
2. NOTE: the terms S and P are related to each other in the conclusion because they are related
to M in the premises.
196
3. If either of these terms (S or P) is particular in a premise, it must remain particular in the
conclusion.
4. To make the term universal in the conclusion would be to overextend it.
5. If you overextend a term in the conclusion, you attribute to it more than what is presented in
the premises.
6. REMEMBER: The conclusion is the effect of the premises, and no effect can be greater
than its cause.
7. This violation takes the form of the overextension of S or of P in the conclusion.
8. ☺☺These forms are respectively designated as the fallacy of the illicit minor and the
fallacy of the illicit major.
9. This is why you need to mark off the quantity of each of the terms in the syllogism.
10. You can’t forget the propositions A, E, I, or O and the rules governing the quantity of
predicate terms.
Fallacy of the Illicit Minor
1. The S term of the syllogism is always the subject of the conclusion.
2. Since the fallacy of the illicit minor is an overextension of the S term in the conclusion, this
violation can occur only when the conclusion is a universal proposition (A or E).
3. For example:
Mu Pp
All sticky substances are adhesive.
Mu Sp
All sticky substances are messy.
Su Pp
All messy substances are adhesive.
197
4. Here the S term (of the syllogism) as predicate of an affirmative premise (the minor) is
particular (Sp).
5. It must remain particular as subject of the conclusion.
6. The above syllogism would be valid if the conclusion was stated:
Sp
Some messy substances are adhesive.
Fallacy of the Illicit Major
1. This fallacy occurs when a P term that is particular in the major premise is made universal in
the conclusion.
2. Since negative propositions have universal predicates, this fallacy can occur only when the
conclusion is a negative proposition (E or O).
Mu Pp
All pre-meds are required to take biology.
Su Mu
No graduate student is a pre-med.
Su Pu
No graduate student is required to take biology.
3. Exercise: Mark off the quantity of each term in the following arrangements. Decide whether
there is a violation either of Rule 2 or 3. If so, name the fallacy.
1. All M is P 2. All P is M 3. No P is M
All M is S Some S is not M Some S is M
All S is P No is P. Some S is not P
4. Not all M is P 5. All M is P
All S is M Some S is not M
No S is P Some S is not P (see answer booklet)
198
RULE 4: No conclusion can be drawn from two negative premises.
1. If both premises are negative, S and P are both excluded from the extension of the intended
middle term.
Some river fish are not tasty.
The sailfish is not a river fish.
Conclusion?
2. The following example seems okay on the surface:
No nongraduate student is eligible for an M.A.
No freshman is a graduate student.
No freshman is eligible for an M.A.
3. This example is formally invalid not only because of four terms, but also because of two
negative premises.
4. If you obvert the E proposition of the minor to an A—Every freshman is a nongraduate
student—you will have a valid syllogism: (contradict both the copula and predicate term)
No nongraduate student is eligible for an M.A.
Every freshman is a nongraduate student.
No freshman is eligible for an M.A.
RULE 5: Two affirmative premises require an affirmative conclusion.
1. The human mind can take a peculiar turn for the worse in the course of an argument.
2. REMEMBER: ☺☺If two terms agree with a common third, then of necessity they agree
with each other.
3. This means that if P is M and if S is M, then it follows that S is P, and the conclusion is
affirmative.
4. Example of a fallacy of Rule 5:
Milk is healthful for children.
Some dairy product is milk.
Some dairy product is not healthful for children.
5. The conclusion here should obviously be affirmative:
Some dairy product is healthful for children.
199
6. Another example:
Whatever is inexpensive is cheap.
Some shoes are inexpensive.
Some shoes are not (?) cheap.
7. Again the conclusion should read affirmatively:
Some shoes are cheap.
8. Occasionally it may appear that a negative conclusion is justified, even though two
affirmative premises are given.
9. Consider the following examples:
Whatever is immortal is imperishable.
The human soul is immortal.
The human soul is not something perishable.
Whoever is just is responsible.
Every virtuous person is just.
No virtuous person is irresponsible.
10. NOTE: Apart from the fact that the above examples are in violation of Rule 5, they also
violate Rule 1 (four terms) and Rule 3 (illicit major).
RULE 6: A negative premise requires a negative conclusion.
1. If one of the terms agrees with M and the other disagrees, the two extremes (S and P) must
disagree with each other.
2. Note the following:
No dude rancher is an expert at roping steers.
Some Indians are experts at roping steers.
3. One has no choice but to conclude that:
Some Indians are not dude ranchers.
4. NOTE: There are no special names of fallacies in violation of Rules 4, 5, and 6. If a violation
occurs, cite the rule that is violated.
200
COROLLARIES
1. No conclusion can be drawn from two particular premises.
2. If one premise is particular, the conclusion must be particular.
1. These two corollaries serve a practical purpose—a violation of either one of them is an
indication of the presence of one of the three following fallacies:
Fallacy of the undistributed middle.
Fallacy of the illicit minor.
Fallacy of the illicit major.
2. NOTE: You may not need to study the following proofs. Just memorize the two corollaries.
COROLLARY 1: No conclusion can be drawn from two particular premises.
1. Any combination of two particular premises automatically involves a violation of either
Rule #2 or Rule #3.
2. A Combination of Two ‘I’ Propositions
3. Suppose you constructed two I propositions as the major and minor premises.
4. Since the subject and predicate terms of these propositions are both particular, you would end
up with a premise arrangement in which all of the terms are particular (undistributed).
5. Regardless, then, of where the M term appears in either of these premises, it would be
particular in both of them.
6. Therefore, you have the fallacy of the undistributed middle—a violation of Rule 2.
7. A Combination of an I and an O Proposition:
8. With this combination of premises you would have only one universal term, namely, the
predicate of the O proposition.
9. Next, the conclusion of the syllogism must be negative, since one of the premises—the O
201
proposition—is negative.
10. The predicate of the conclusion would be universal (Pu).
11. Yet, the only universal term in the premises has already been used for the placement of M.
12. This means that the P term, whether it appears as subject or predicate of the major premise,
will be particular—the fallacy of the illicit major.
13. Any attempt to conclude from an I and an O combination would result in the fallacy of
either the undistributed middle or the illicit major.
COROLLARY 2: If one of the premises is particular, the conclusion must be particular.
1. A Combination of Two Affirmative Premises:
2. With this combination, if one premise is universal—an A proposition—and the other
particular—an I proposition, there is only one universal term, namely, the subject of the
universal premise—the A proposition.
3. To avoid the undistributed middle you would have to use the subject of the A proposition for
the placement of M.
4. Since the remaining terms are particular, the S term is particular—in the minor premise.
5. In order to avoid an illicit minor, you would have to keep it particular as subject of the
conclusion (Sp).
6. The conclusion would be an I proposition.
7. A Combination in Which One Premise Is Affirmative and the Other Negative:
8. If one premise of this combination is universal and the other is particular (both cannot be
particular), these possibilities arise: An E and an I proposition, or an A and an O.
202
9. In either of these combinations you have two universal terms, the subject of the universal
premise and the predicate of the one that is negative.
10. One of these terms must be used for the placement of M to avoid an undistributed middle.
11. Since one of the premises is negative, the conclusion must be negative, and the predicate of
the conclusion universal.
12. To void an illicit major the remaining universal term must distribute the P term in the major
premise.
13. The S term must be kept particular in the conclusion, otherwise you would have an illicit
minor.
14. To avoid an illicit minor, the conclusion must be particular—an O proposition.
15. A careful analysis of the proof just given reveals that a violation of Corollary 2 involves a
violation of either Rule #2 or Rule #3 of the syllogism and, the fallacy of an undistributed
middle, an illicit major, or an illicit minor.
See Extra 25: Syllogisms for practice #1
If you need more practice: See Extra 26: Syllogisms for practice #2
VALID MOODS
1. In an earlier lesson, syllogistic mood was defined as the designation of the premises and
conclusion as A, E, I, or O propositions.
2. If you use the letters A I I—in that order—the purpose is to indicate the following type of
syllogism:
major: A proposition
minor: I proposition
203
conclusion: I proposition
3. If you use the letters A I, you intend merely to indicate the major and minor premises without
considering the conclusion.
4. The question: How many different kinds of moods are there?
5. Theoretically, with respect to only the major and minor premises, there are sixteen possible
moods:
AA EA IA OA
AE EE* IE* OE*
AI EI II* OI*
AO EO* IO* OO*
6. Of these sixteen, how many are valid?—only eight.
7. The moods that are marked by an asterisk (*) are invalid because they contain either two
negative premises—see Rule 4—or two particular premises—see Corollary 1.
8. The following eight valid moods are all that remain:
AA EA IA OA
AE
AI EI
AO
9.NOTE: Don’t conclude that when a syllogism is in one of these eight moods, it’s
automatically valid—there could be something else wrong with it.
204
FIGURES
1. Remember, the figure of a syllogism is the position of the middle term in the premises.
2. The chart is a reminder of the four figures.
3. As an EXERCISE, you could combine the eight moods with the four figures.
4. For example, is the A A mood valid in Figure 1?
5. If so, which conclusion does it generate? (Take a universal conclusion whenever you
can).
6. In working out the valid moods for each figure, be sure to mark off the quantity of each
of the terms.
7. Whatever the figure an AE mood would be marked off like this:
A u p
E u u
8. NOTE: there are four valid moods for Figure I, four for Figure 2, six for Figure 3,
and five for Figure 4.
205
HOW TO DETERMINE A VALID SYLLOGISM
1. The following ideas have been presented to date:
A. Fundamental theory and principles.
B. The matter of the syllogism, its three terms (S, M, P) and three propositions
(major, minor, and conclusion).
C. The six rules and their corresponding fallacies.
D. The two corollaries.
E. The eight valid moods.
F. The four figures.
2. Even if you have a good grasp on the above areas, there are no guarantees that you
will have a genuine working knowledge of the syllogism.
3. A good working knowledge of the syllogism takes time and practice.
See Extra 27: Steps to Analyze a Categorical Syllogism
4. Let’s look at a few relatively simple syllogisms in which the major, minor, and conclusion
are each expressed—and expressed in their proper order.
5. Example:
Not every public official is elected.
A mayor is a public official.
A mayor is not elected.
6. First ask the question: Is there a fallacy of four terms? In other words, is there a fallacy of
the ambiguous middle?
7. In this example the M term “public official” has the same meaning in both of the premises—
206
no ambiguous middle.
8. The next step is to, put the syllogism in its logical form, marking off each proposition as A, E,
I, or O in order to determine the mood.
9. Example:
O Some public officials are not elected.
A Every mayor is a public official.
E No mayor is elected.
10. An (OAE) syllogism should clue you that there is something wrong—it presents a violation
of Corollary 2—particular premise, particular conclusion.
11. Next specify, in addition to the mood, the figure and the quantity of each of the terms:
O—Mp Pu
A—Su Mp
E—Su Pu
12. Now that the syllogism is in skeletal form, any defects become apparent.
13. Now examine the syllogism for one or more of the following fallacies in the order given:
a. Undistributed middle
b. Illicit minor
c. Illicit major
14. The M term is particular in both premises (Mp Mp). It’s an undistributed middle and,
therefore, an invalid syllogism.
15. Another example:
A slave is not allowed the free exercise of his natural rights.
Most Americans are not slaves.
Most Americans are not allowed the free exercise of their natural rights.
16. Is there a possible violation of Rule 4?—two negative premises?
17. The middle-term “slave” has an identical meaning in both premises.
18. What’s the mood?—E, O, O.
19. The example is invalid because of its mood—no conclusion from two negative premises
207
(Rule 4).
20. Again:
All true art is beautiful.
Some contemporary sculpture is not true art.
Some contemporary sculpture is not beautiful.
21. Ambiguous middle? No, the term “true art” has the same signification.
22. Is the mood, generally speaking, valid? Yes, AOO is generally valid.
23. Skeletonize the syllogism:
A—Mu Pp
O—Sp Mu
O—Sp Pu
24. It contains an illicit major (Pp Pu).
25. The syllogism is invalid, because of the over-extension in the conclusion of the P term
“beautiful.”
26. Once you form the habit of properly analyzing a syllogism, you shouldn’t have difficulty
determining whether or not it’s valid, and if invalid, what fallacy occurs.
27. In examining a syllogism for its validity, it’s necessary to do two things:
A. Give attention to the middle term.
28. This term is easy to locate, since it’s the only term that appears in both premises and never
in the conclusion.
29. Once the middle term is located, the next step is to look for a possible violation of Rule I.
(The syllogism should consist of no more than three terms.)
30. If the middle term is ambiguous, you don’t need to look any farther to know that the
syllogism is invalid.
B. Set forth the mood, the figure,
and the quantity of each of the terms.
208
31. In other words, determine the form of the syllogism.
32. Any error in form will indicate one of the remaining fallacies—an undistributed middle, an
illicit minor or major, two negative premises, and so on.
See Extra 28: Syllogisms for practice #3
HOW TO CONSTRUCT A SIMPLE SYLLOGISM
1. REMBER the two following fundamental facts:
A. The conclusion is what you are attempting to prove.
B. The premises are the means whereby it is proved.
2. In constructing a syllogism, then, you must know first what you want to prove.
3. Example:
Conclusion: Some nations are not suited for self-rule.
4. Since this proposition is your conclusion, you have the S and P terms of the syllogism.
5. To find a middle term, ask the question: Why?
6. There can normally be many possible middle terms—choose a true one.
7. Suppose you decide that the incapacity for producing responsible leadership is the reason
why some nations are not suited for self-rule.
8. This, then, is your middle term.
9. When you identify the S term, “some nations”, with the middle term, you have your minor
premise:
Some nations are not capable of producing responsible leadership.
10. To construct the major premise, simply identify the same middle term of the minor with the
predicate of the conclusion.
209
Any nation incapable of producing responsible leadership is not suited for self-rule.
11. Your syllogism should read:
Any nation incapable of producing responsible leadership is not suited for self-rule.
Some nations are not capable of producing responsible leadership.
Some nations are not suited for self-rule.
12. The syllogism is in the first figure, and it’s valid.
13. Check out this example as a possible conclusion:
Some marriages are not happy.
14. “Some marriages” is the subject of the syllogism and “happy” is its predicate term.
15. To find the middle term, ask the question: “Why aren’t some marriages happy?”
16. Maybe because of too much quarreling.
17. You might decide that “excessive quarreling” is your middle term.
Some marriages are characterized by excessive quarreling.
18. The remaining assumption is that:
No marriage, that is characterized by excessive quarreling, is a happy one.
19. This is your major premise.
20. Note: a negative conclusion must have a negative premise.
21. The syllogism would be constructed in this way:
E—No M is P.
I—Some S is M.
O—Some S is not P.
See Extra 29: Supply Appropriate Middle Term
See Extra 30: Construct Valid Syllogism from Conclusion
See Extra 30a—for more practice
210
See Homework Lesson 12
211
Extra Materials
Extra 25: Syllogisms for practice #1—start with # 11
Examine each of the following examples for their validity. Decide in the case of an invalid
syllogism which rule is violated, and name the fallacy (or cite the rule that is violated). The
propositions in each example are to be treated as major, minor, and conclusion, respectively.
Note:
a. If necessary, place each proposition into its logical form.
b. In working out this exercise do not confuse "words" with "terms." Frequently two different
terms in an argument contain one or more of the same words, but this of itself does not make
the terms identical. Terms are identical only if they are identical in meaning.
c. By the same token, if a term appears in the conclusion with some new word that it did not
contain in the premise, this of itself is not enough to invalidate the argument. Often it is
necessary in order to make a syllogism read intelligibly to fill in a word or two. So, again, the
question is whether the meaning is the same.
Hold numbers 1 -10 for Homework #Lesson 12
1. No irresponsible person is worthy of high position.
Most squanderers are irresponsible.
Most squanderers are not worthy of high position.
2 All art is not religious.
Raphael's "Madonna and the Child" is a work of art.
It is not a religious work of art.
3. Not every science is practical.
Logic is practical.
Logic is not a science.
4. No pessimist is a realist.
Some optimists are not realists (either).
So some optimists are not pessimists.
5. Every patriot has a genuine love of his homeland.
But some opportunists do not have a genuine love of their homeland.
212
So some opportunists are not true patriots.
6. All headaches are troubling.
But many "cross-word" puzzles are headaches.
So many "cross-word" puzzles are troubling.
7. No qualified physician practices medicine without a license.
No qualified physician is a "quack.
So no "quack" practices medicine without a license.
8. Anyone able to dance is able to walk.
Some husbands are not able to dance.
Some husbands are not able to walk.
9. Nothing that is doubtful is certain.
Some propositions are doubtful.
Some propositions are certain.
10. Wild animals do not make for good pets,
And a coyote is a wild animal.
So a coyote is not a good pet.
BACK
Start here for Extra 25
11. Anyone who deceives himself is a fool.
Anyone who deceives others is a liar.
So some liar is a fool.
12. Anything which makes sense is logical.
But some explanations are not logical.
So some explanations do not make sense.
13. All prejudiced persons are opinionated,
And all prejudiced persons are partial.
So anyone who is partial is opinionated.
14. All wantonly harmful acts are wrong,
And everything that is wrong is contrary to the moral law.
So something that is contrary to the moral law is not a wantonly harmful act.
15. Every genius is an exceptional person.
But some exceptional persons are mentally handicapped.
So some mentally handicapped persons are geniuses.
16. Every literary masterpiece is worth reading well.
All good textbooks are worth reading well.
213
All good textbooks are literary masterpieces.
17. No unjust act is commendable.
Some unjust acts are crimes.
Some crimes are commendable.
18. A bad habit is easy to acquire.
And a bad habit is hard to uproot.
So something that is hard to uproot is easy to acquire.
19. No invalid syllogism is a sound argument.
No true act of reasoning is an invalid syllogism.
No true act of reasoning is a sound argument.
20. No fast talker is easy to understand.
Some “fast” talkers are hard to believe.
Some people who are hard to believe are not easy to understand.
21. Whatever is worth doing is worth doing well.
Some useless tasks are (obviously) not worth doing.
Some useless tasks are not worth doing well.
22. Men are not beasts,
And beasts are not rational.
So beasts are not men.
23. No self-evident truth is demonstrable.
But first principles are self-evident truths.
So, first principles are demonstrable.
24. All good opera is good music,
And all good music is pleasing to the ear.
So something pleasing to the ear is good opera.
BACK
214
Extra 26: Syllogisms for practice #2
A few Syllogisms for Practice #2: See what you can do with these. Usually, errors come
because of four terms, 2 negative premises, 2 particular premises, undistributed middle (middle
terms are both particular), illicit major (overextended major term), and illicit minor
(overextended minor term) premises.
1. No dogs are spirits.
And no mammals are spirits.
Therefore all dogs are mammals.
2. Some rocks are green.
And some green things are alive.
Therefore some rocks are alive.
3. Whatever is in sense experience is material.
All knowledge comes from sense experience.
Therefore all knowledge is material.
4. No person can know everything.
You are a person.
Therefore you can’t know everything.
5. Sympathy is a virtue.
Justice is not sympathy.
Therefore justice is not a virtue.
6. A vigorous life is worth living.
Some lives are not vigorous.
Therefore some lives are not worth living.
7. Rabbits are not carrots.
Humans are not carrots.
Therefore humans are not rabbits.
8. Some sheep are not white.
And some white things are animals.
Therefore some sheep are not animals.
9. All hamsters are mammals.
And all cats are mammals.
Therefore all hamsters are cats.
10. No philosophers are god-like.
And no lawyers are god-like.
Therefore all philosophers are lawyers.
BACK
215
Extra 27: Steps to Analyze a Categorical Syllogism
1. Find the conclusion.
2. Put the conclusion in logical form if necessary.
3. The subject of the conclusion is the subject of the syllogism.
a. It’s the minor term of the syllogism and appears normally in the second premise.
4. The predicate of the conclusion is the predicate of the syllogism.
a. It’s the major term of the syllogism and appears normally in the first premise.
5. Having scoped out the conclusion, ask yourself, “What’s the reason for this conclusion?” The
answer to the question gives you the middle term. The middle term is the reason given for the
conclusion.
All headaches are troubling.
But many crossword puzzles are headaches.
So many crossword puzzles are troubling.
Ask the question, “Why are many crossword puzzles troubling?”
The answer: Because they are a headache.
4 term—headaches is used analogously in the second premise (it’s not a pain in the head, it’s an
annoyance).
6. Once you discover all three terms, mark them off in the syllogism.
M P
All headaches are troubling.
S M
But many crossword puzzles are headaches.
S P
So many crossword puzzles are troubling.
7. Remember: The middle term only appears in the premises never in the conclusion!
8. At this point, determine whether the terms are particular or universal and mark them so.
Mu Pp
All headaches are troubling.
Sp Mp
But many crossword puzzles are headaches.
Sp Pp
So many crossword puzzles are troubling.
9. Remember:
a. A predicate term is particular if it follows a positive copula.
b. A predicate term is universal if it follows a negative copula.
c. Singular subjects are considered universal.
10. Now test your syllogism against the 6 rules.
1. The syllogism should consist of no more than three terms—Four Terms
216
2. The middle term must be universal in at least one premise—Undistributed Middle
3. No term which is particular in a premise may be made universal in the conclusion
—Illicit Major or Illicit Minor
4. No conclusion can be drawn from two negative premises—violation of Rule Four.
5. Two affirmative premises require an affirmative conclusion—violation of Rule Five.
6. A negative premise requires a negative conclusion—violation of Rule Six.
BACK
217
Extra 28: Syllogisms for practice #3
A few more syllogisms for practice. See what you can do with these. Usually, errors come
because of four terms, 2 negative premises, 2 particular premises, undistributed middle (middle
terms are both particular), illicit major (overextended major term), and illicit minor
(overextended minor term) premises.
1. All power has the potential for dishonesty.
Knowledge is power.
Therefore knowledge has the potential for dishonesty.
2. All material objects are mortal.
Man is a material object.
Therefore man is mortal.
3. Young children are not learned.
But professors are learned.
Therefore young children are not professors.
4. All socialists are in favor of distribution of wealth.
This candidate is in favor of distribution of wealth.
Therefore this candidate must be a socialist.
5. No violence is just.
All violence is aggression.
Therefore no aggression is just.
6. Some geese are white.
And some white things are beautiful.
Therefore some geese are beautiful.
7. Some cheese is not smelly.
And some smelly things taste good.
Therefore some cheese does not taste good.
8. Odd numbers are not even numbers.
Five is not an even number.
Therefore five is not an odd number.
9. All squares have four sides.
This object is a square.
Therefore this object has four sides.
10. All dogs bark.
The constellation Canis Major is a dog.
Therefore the constellation Canis Major barks.
BACK
218
Extra 29: Supply Appropriate Middle Term
Supply an appropriate middle term for each of the following syllogisms. Construct premises
that are true. Set forth the form (figure and mood) of each syllogism as well.
1. Every _________________is trustworthy.
Every honest person is__________________
Every honest person is trustworthy.
2. Every _______________________ is clever.
Some students are ___________________
Some students are clever.
3. No one ______________________is a traitor to his country.
Every good citizen is ______________________
No good citizen is a traitor to his country.
4. No _____________________is just.
Some wars are ____________________
Some wars are not just.
5. Every conservative person is ________________________
No gambler is _____________________________
No gambler is conservative.
6. Whatever is educational is __________________________
Some entertainment is not __________________________
Some entertainment is not educational.
7. No pleasant conversationalist is ______________________
Every bore is ___________________________
No bore is a pleasant conversationalist.
8. No well-patronized establishment is ________________________
Some stores are.________________________
219
Some stores are not well patronized.
9. Every _______________________is a permanent source of enjoyment.
Every _______________________is a literary masterpiece.
Some literary masterpiece is a permanent source of enjoyment.
10. Every _____________________is useful.
Some _____________________is science.
Some science is useful.
11. No ____________________is a man-eater.
Every ____________________is an animal.
Some animals are not man-eaters.
12. No ____________________is man-made.
Some ___________________are beautiful.
Some beautiful things are not man-made.
13. Some _______________________are wealthy.
All _________________________are generous.
Some generous people are wealthy.
BACK
220
Extra 30: Construct Valid Syllogism from Conclusion
Construct a valid syllogism for each of the following conclusions.
1. Some business enterprises are not a success. (
2. Some of the students in our class deserve good marks.
3. Big-game hunting is a dangerous sport for amateurs.
4. Most libraries are suitable places for study.
5. Some drivers are careless.
6. Cancer is an insidious disease.
7. No good deed is unrewarded.
8. Most children are interested in cowboy movies.
9. Some syllogisms are not valid.
221
10. Some food is not easily digested.
BACK
Extra 30a—Test each of the following syllogisms using the six rules.
1. Some children smoke. Some who smoke get cancer. Therefore some children get cancer.
2. Saints are never joyless. Some joyless people are popes. Therefore some popes are not saints.
3. Peace is good. War is not peace. Therefore war is not good.
4. Some people are magnets for trouble. No ombudsmen are magnets for trouble. Therefore some ombudsmen are not people.
5. Everything in Massachusetts is in America. Everything in Boston is in America.
Therefore everything in Boston is in Massachusetts.
6. No one who does not wonder philosophizes. Conformists do not wonder. Therefore conformists are non-philosophers.
7. Whatever is divine is supernatural. Whatever is supernatural can perform miracles. Therefore whatever is divine can perform miracles. Man cannot perform miracles. Therefore man is not divine.
8. All mortal things are made of parts. The soul is not made of parts. Therefore the soul is not mortal.
9. Nothing immortal is made of parts. Man is made of parts. Therefore man is not immortal. 10. Whatever doesn’t listen to reason can’t be governed by reason. Falling in love does not listen to reason. Therefore falling in love cannot be governed by reason. BACK
11. Fools are never blessed. But some fools are lucky.
Therefore not all lucky people are blessed.
12. All bachelors are unmarried men. No married woman is a bachelor. Therefore no married woman is an unmarried man. 13. Constancy is confidence. Constancy is a virtue. Therefore confidence is a virtue.
14. Foolhardiness is virtuous. Foolhardiness is confidence.
Therefore some confidence is not virtuous.
15. Some books are sources of amusement. All logic books are books. Therefore some logic books are sources of amusement.
16. Whoever intentionally kills another should die.
Soldiers intentionally kill another.
Therefore soldiers should die.
17. John loves Jesus. Jill loves John. Therefore Jill loves Jesus.
18. Gorillas are nonhumans. Humans can speak. Therefore gorillas can't speak.
222
Exercises for Homework:
Homework Lesson 12:
1. Use the moods AA, AE, AI, and AO in combination with figure #1 and create four syllogisms.
Which conclusion does each generate? Indicate the terms and whether they are particular or
universal. Then determine which ones are valid and which ones are invalid.
2. Examine the following 10 syllogisms. Check them for validity.
1. No irresponsible person is worthy of high position.
Most squanderers are irresponsible.
Most squanderers are not worthy of high position.
2 All art is not religious.
Raphael's "Madonna and the Child" is a work of art.
It is not a religious work of art.
3. Not every science is practical.
Logic is practical.
Logic is not a science.
4. No pessimist is a realist.
Some optimists are not realists (either).
So some optimists are not pessimists.
5. Every patriot has a genuine love of his homeland.
But some opportunists do not have a genuine love of their homeland.
So some opportunists are not true patriots.
6. All headaches are troubling.
But many "cross-word" puzzles are headaches.
So many "cross-word" puzzles are troubling.
7. No qualified physician practices medicine without a license.
No qualified physician is a "quack.
So no "quack" practices medicine without a license.
8. Anyone able to dance is able to walk.
Some husbands are not able to dance.
Some husbands are not able to walk.
9. Nothing that is doubtful is certain.
Some propositions are doubtful.
Some propositions are certain.
223
10. Wild animals do not make for good pets,
And a coyote is a wild animal.
So a coyote is not a good pet.
6. A predicate term is particular if it follows a copula.
7. A predicate term is universal if it follows a copula.
8. Singular subjects are considered .
BACK
224
LESSON THIRTEEN—EVERYDAY DISCUSSIONS AND THE SYLLOGISM
1. NOTE: Different subjects can have different degrees of certainty—consider the topics
politics vs. mathematics.
2. ☺☺Even with the best application of reasoning, there are certain fields of knowledge that
don’t have the same kind and degree of certainty that’s found in the stricter types of
science.
3. Because of this, you’ll make a distinction between demonstrative and probable reasoning.
4. REMBER: Logic is more concerned with the correctness of the form of an argument than the
truth of its propositions.
5. Having said that, you must also consider an argument in regard to the truth of its premises
and conclusion.
6. REMEMBER: The third rule of syllogistic reasoning states that the premises are the cause of
the conclusion, and the conclusion is the effect of the premises.
7. Accordingly, if you reason from premises that are only probably true, you can only arrive at a
probable conclusion.
8. Note the difference between the two following syllogisms with respect to their content:
No conscientious citizen will fail to cast his vote in an important election.
Smith is a conscientious citizen.
Therefore, Smith will not fail to cast his vote in an important election. (probable)
No nonliving substance is capable of organic growth.
Mercury is a nonliving substance.
Therefore, mercury is not capable of organic growth. (demonstrative)
9. According to their form both syllogisms are identical and equally valid.
10. Compare the truth expressed in the major premise of both syllogisms.
11. The major premise of the first syllogism is probable because it’s not something a person
must do according to his nature.
225
12. It’s true, but it’s not true in such a way that it could not be otherwise.
13. In the second syllogism, the major premise expresses something that is a part of the nature
of a nonliving substance and is therefore demonstrative.
14. ☺☺Most argumentation is nonscientific in nature.
HOW CERTAIN IS CERTAIN?
1. ☺☺In general, remember that any act of reasoning is more certain and scientific the
closer it gets to being a genuine universal.
2. Universals such as “man,” “animal,” “plant,” and the like, can be characterized by their
essential differences—a part of their natures.
3. It is this fact that makes it possible to reason about natural species with a much higher degree of
certainty than when you reason about “Americans,” “Southerners,” different types of cars,
climates, and so on.
4. These terms are called universal only by way of analogy.
5. Americans as Americans are not strictly a “class by themselves,” since the only stable nature
they have is as men and not as Americans.
ENTHYMEMES—ARGUMENTS FROM SIGNS
1. ☺☺In its original Aristotelian usage, an enthymeme is a syllogism which, in its
suppressed major premise, is based upon truths that are only probable and nonscientific—
arguments from sign.
2. Some examples of arguments from sign:
*That person seems shy or embarrassed. (the suppressed major is—because of the redness of
the person’s cheeks).
226
*It looks like it will rain (suppressed major—because of the cloudiness of the sky).
*That guy is uncultured. (suppressed major—because of his messy appearance).
*From the discoloration of a piece of meat that it is old.
*From the hardness of a loaf of bread that it is stale.
3. There are many arguments from sign that you use in everyday discussion. You continually
conclude about something because of what you sense.
4. Under ordinary circumstances an argument from sign leads to no necessary conclusion, and is
the source of an incredible number of mistakes which people commit in their everyday life.
Especially in forming first impressions.
5. At best, most sign arguments only provide a clue to a problem which must be solved by the
use of induction—i.e. by further observation, hypothesis, and the experimental testing of the
hypothesis.
6. NOTE: If a sign argument is based on a natural sign, there is less of a chance of error.
However, a sign is still only a clue for the solving of a problem.
Gray hair is, under certain conditions, a sign of advancing age.
A cloudy sky is, under certain conditions, a sign of rain.
A persistent cough is, under certain conditions, a sign of tuberculosis.
THE ENTHYMEME—AN ABBREVIATED SYLLOGISM
1. ☺☺Nowadays, an enthymeme is any kind of syllogism in which one of its propositions is
suppressed.
2. ☺☺An enthymeme may be defined as an abbreviated syllogism that implies either one of
its premises or its conclusion.
3. In daily argumentation, you won’t normally find a properly constructed syllogism with a
major, minor, and conclusion.
4. Example of what you won’t read in the paper or on some blog:
227
No Incompetent official is deserving of re-election.
Mr. X is an incompetent official.
Therefore, Mr. X is not deserving of re-election.
5. It would be more likely expressed in some such way as the following, with a lot of fluff in
between points:
A. Being an incompetent official, Mr. X does not deserve to be re-elected.
B. Incompetent officials are hardly deserving of re-election.
C. Mr. X should be rejected at the polls.
D. Incompetency in office is hardly a recommendation for re-election.
E. Is there anyone in this community who has a reasonable doubt as to the incompetency of Mr.
X?
6. NOTE: The logical structure of the above enthymemes is as follows:
1. SUPPRESSED MAJOR—An incompetent official is not deserving of re-election.
Mr. X is an incompetent official.
Mr. X is not deserving of re-election.
2. No incompetent official is deserving of re-election.
SUPPRESSED MINOR—Mr. X is an incompetent official.
Mr. X is not deserving of re-election.
3. No incompetent official is deserving of re-election.
Mr. X is an incompetent official.
SUPPRESSED CONCLUSION—Mr. X is not deserving of re-election.
7. NOTE: Syllogisms with a suppressed major, minor, or conclusion are called enthymemes of
the first, second, or third order respectively.
IDEAS ON HOW TO SUPPLY A SYLLOGISM’S MISSING PART
1. ☺☺The first thing to do is to write down the conclusion—i.e. what is the argument trying
to prove.
2. If the conclusion is given, it’s usually preceded by “hence,” “therefore,” “consequently,” “as
a result,” “it follows that,” and so on.
3. If the conclusion is suppressed (third order enthymeme), determine the point of the argument
in question.
228
4. Example:
No poorly written work is easy to read.
Some term papers, however, are poorly written.
Ergo?
No first-rate newspaper is biased in its editorial opinions. Our local newspaper is notorious for
its editorial bias.
Ergo?
An excessive fee is an unjust fee.
Some doctors’ fees are outrageous (that is, excessive).
Ergo?
5. It’s more difficult to supply a missing major or minor premise.
6. Check out the following example:
Mere memorization of subject matter is no guarantee of one’s having learned it.
For this reason, one can’t be sure, when cramming for an examination, of learning the
subject matter.
7. The first step is to write the conclusion in clear, explicit terms.
S c P
Cramming for an examination is not a guarantee of one’s having learned the subject
matter.
8. The conclusion of the argument will always give you the minor (S) and major (P) terms of
the syllogism.
M P
Mere memorization of the subject matter is not a guarantee of one’s having learned it.
9. In supplying the missing minor premise identify the subject of the conclusion (S) with the
middle term (M) of the major premise:
S c M
Cramming” for an examination is merely to memorize the subject matter—the suppressed
premise (an enthymeme of the second order).
10. The syllogism should read as follows:
MAJOR:
M P
Mere memorization of the subject matter is not a guarantee of one’s having learned it.
229
MINOR:
S M
Cramming for an examination is merely to memorize the subject matter.
CONCLUSION:
S P
Cramming for an examination is not a guarantee of one’s having learned one’s subject
matter.
11. First-order enthymemes are more frequent—the major premise is implied.
12. Example of implied major premise:
Because some men are truly virtuous, they are genuinely happy.
13. Conclusion:
S c P
They (that is, some men) are genuinely happy.
14. The S term “some men”, as subject of the conclusion, indicates it’s the minor term of the
syllogism.
15. Now, identify the middle term (M) of the minor premise (“truly virtuous”) with the
predicate term (P) of the conclusion (“genuinely happy”).
16. However, where do you place the middle term—figure 1 or figure 2?
M P
All truly virtuous men are genuinely happy.
17. Or:
P M
All genuinely happy men are truly virtuous.
18. If you select the figure 1—it’s valid.
major: A Mu—Pp
minor: I Sp—Mp
conclusion: I Sp—Pp
19. If you select figure 2—it’s invalid—undistributed middle.
major: A Pu—Mp
minor: I Sp—Mp
conclusion: I Sp—Sp
230
20. Try to select the figure that produces a valid construction as long as you don’t create a false
premise.
major: All truly virtuous men are genuinely happy.
minor: Some men are truly virtuous.
conclusion: Some men are genuinely happy.
21. At times an enthymeme of the first order is based on an assumption that is questionable or
false.
22. So, it’s important to know how to determine the hidden premise.
23. You do this so you can determine the truth of the premise.
24. Check out this argument with a false hidden premise.
John: Sue, you’ll never make a good driver.
Sue: I like that! Whatever makes you think that I won’t?
John: Because you’re a woman, that’s why.
Sue: So that’s it. Then no woman makes a good driver, in which case I don’t stand a chance.
John: Well, I’d hardly say that.
Sue: You’d better not, because that happens to be your hidden major, and it’s downright false.
John: As Alexander Pope said: “A man convinced against his will is unconvinced still.”
Sue: In that case, besides being illogical you’re stubborn. Anyway, you’ll find out if I can’t drive,
because I’ve just signed up for my first lesson today.
25. In refuting an argument with a person who doesn’t know Logic, you can’t use technical
language—they wouldn’t understand you.
26. Rather, you would point out the absurdities of their argument.
27. You can do this by substituting one of the terms of the argument as it was originally
given—called reductio ad absurdum (reduced to the absurd).
28. For example:
Argument: Because this substance is hard, it must be made of steel.
Refutation: I grant that anything made of steel is hard, but so are my teeth. Should I conclude,
then, that they too are made of steel?
231
29. Ordinary argumentative discourse is usually narrative, descriptive, or expository
construction.
30. Examine:
An economy that is based on excessive taxation is fundamentally an unsound economy.
Every government has, of course, the right to tax both individuals and corporations,
inasmuch as taxation is the only means that a government has of maintaining itself in
existence. When, however, the costs of taxation are so high that they are prejudicial to a
free economy, a situation has developed in which the government is no longer serving the
interests of the citizenry at large. Is there any doubt about the fact that the economy of
our government (based as it is on a system of excessive taxation) is fundamentally an
unsound economy?
31. What is the main point of the argument?
The economy of our government is fundamentally an unsound economy.
32. In the above argument the conclusion is in the form of a rhetorical question—one stated but
not expecting an answer.
33. What supports this conclusion?
The economy of our government is one that is based on excessive taxation.
34. Putting the argument in a proper syllogism, it should read:
MAJOR:
Any economy that is based on excessive taxation is fundamentally an unsound economy.
MINOR:
The economy of our government is one that is based on excessive taxation.
CONCLUSION:
The economy of our government is fundamentally an unsound economy.
35. Once set up, you can examine it from the standpoint of both its logical structure and the
truth of its premises.
232
EPICHEIREMA
1. ☺☺An epicheirema is an argument that consists of one basic syllogism, but has a reason
attached to one or both of its premises.
2. Example:
MAJOR:
Every realistic system of education is based on the needs of the student, because the failure
to cope with these needs defeats the very purpose for which education is intended.
MINOR:
Some systems of education, however, are not based on the needs of the student, because
they do not include adequate programs for vocational guidance, which, clearly, is one of
the student’s most basic needs.
CONCLUSION:
Some systems of education are not realistic.
3. As a starting point to analyzing an argument, begin by examining the leading points of the
argument.
4. Example:
The United States can no longer afford the risk of an isolationist policy.
5. This is a conclusion of the argument. Now give a reason for this conclusion—a middle term.
6. Example:
No major government can any longer afford the risk of an isolationist policy.
7. You can now put the basic syllogism together:
MAJOR:
No major government can any longer afford the risk of an isolationist policy.
MINOR:
The United States is a major government.
CONCLUSION:
The United States can no longer afford the risk of an isolationist policy.
8. In an argument, it’s not necessary to put your ideas in the form of major, minor, and
conclusion.
233
9. NOTE: In the following paragraph, the conclusion is in the first line:
The United States can no longer afford the risk of an isolationist policy. There was a time
in the history of our country when, in the struggle for independence, it was neither
desirable nor possible for the government of the United States to involve itself in either
European or Asiatic affairs. Today, however, the situation is immensely changed. The
present international situation is such that no major government can afford to close its
eyes to the world-wide threat of terrorism. That is to say, no major government can any
longer afford the risk of an isolationist policy. Who can deny the fact that the United
States, with its natural resources, industrial potential, and capacity for world-wide
leadership is a major government?
THE CHAIN ARGUMENT
1. ☺☺An argument involving a connected series of syllogisms is a polysyllogism (“poly” meaning
“many”).
2. A polysyllogism is a series of syllogisms that are connected so that the conclusion of one
doubles as a premise (usually as the major premise) for the one that follows.
3. Example:
Anything that has life has a soul.
All things that breathe have life.
*All things that breathe have a soul. (Conclusion 1)
Every animal is a thing that breathes.
*Every animal has a soul. (Conclusion 2)
Man is an animal.
*Man has a soul. (Conclusion 3)
Frank is a man.
*Frank has a soul. (Conclusion 4)
4. Each conclusion, marked by an asterisk (*), is as the major premise for the following
234
syllogism.
5. A special form of polysyllogism is the Sorites.
6. ☺☺A Sorites is a chain argument in which all the conclusions are suppressed except the
last.
7. The Aristotelian Sorites is the best known form:
8. The argument takes on this form:
A is B
B is C
C is D
D is E
E is F
A is F
235
9. Watch out for ambiguity in the use of terms:
Whatever is likely is probable.
Whatever is probable is uncertain.
Whatever is uncertain is unknown.
Whatever is unknown is unpredictable.
Whatever is unpredictable is impossible to guess.
Whatever is likely is impossible to guess.
10. In this argument there is ambiguity in the use of such terms as “uncertain”, “unknown,” and
“unpredictable”—they need qualification or an explanation.
11. For example in the statement, “Whatever is unknown is unpredictable,” the question is
whether something is absolutely unknown or only relatively so.
12. The statement as it stands would be true only if something were completely (absolutely)
unknown.
ARGUMENTS FROM ANALOGY
1. Some arguments can’t be placed into syllogistic form.
2. ☺Arguments of this sort, to mention only a few, are those involving an appeal to belief or
authority, the employment of statistics, circumstantial-evidence, and the use of analogy.
3. Let’s take a look at arguments from analogy.
4. Every analogy is based on some type of comparison—either of the parts or of one complete whole
or unit as compared to another.
5. Basically, an analogy is valid from a logical point of view depending upon the aptness and
relevancy of the comparison that’s made.
6. Example:
236
City X decides that because one-way streets has proved helpful in the downtown
area of its neighbor, City Y, they should therefore adopt the same system.
6. Here you have an example of an analogy that may or may not prove useful, depending on a
careful evaluation of a variety of circumstances, such as the comparative rate of traffic, its
density, the over-all size of the area, and the like.
7. In general, there is no set logical prescription for the valid use of the argument from analogy.
8. It’s interesting to note that the argument from analogy is something that even a child’s mind
can grasp.
“Mommy, you bought Janie a lollipop this afternoon while I was taking my nap.
Won’t you get one for me too because I’ve been just as good as Janie—even better.”
See Extra 31: Tough Syllogisms for Thought
See Homework Lesson 13
237
Extra Materials
Extra 31: Tough Syllogisms for Thought
1. No one philosophizes who cannot wonder, and computers cannot wonder, therefore
no computer philosophizes.
2. Objection 1: It seems that mercy cannot be attributed to God, for mercy is a kind of
sorrow, as Damascene says, but there is no sorrow in God; and therefore there is no
mercy in Him." (St. Thomas Aquinas, ST 1,21,3)
3. Not all syllogisms are silly, for only persons are silly, and nothing but non-persons are
syllogisms. (Hint: reword the more complex proposition, which is more flexible and
changeable, to conform to the terms of the more simple propositions, which are less
flexible and changeable. Remember that you may never change the meaning of a
proposition, only the wording.)
4. Most subjects that tend to withdraw the mind from pursuits of a low nature are useful.
But classical learning does not do this, and therefore it is not useful.
5. Bacon was a great statesman. He was also a philosopher. We may infer from this that
any philosopher can be a great statesman.
6. Happiness is always desired for its own sake and never as a means to anything else.
Pleasure is always desired for its own sake and never as a means to anything else.
Therefore happiness is identical with pleasure.
7. All immoral companions should be avoided, and some immoral companions are
intelligent persons, therefore some intelligent persons should be avoided.
8. Apes are never angels, and philosophers are never apes, therefore philosophers are
never angels.
9. Mathematics improves the reasoning powers. But logic is not mathematics. Therefore
238
logic does not improve the reasoning powers.
10. "Is a stone a body? Yes. Then is not an animal a body? Yes. Are you an animal? I
think so. Therefore you are a stone, being a body." (Lucian) (There are two syllogisms
here.)
11. "His imbecility of character might have been inferred from his proneness to
favorites, for all weak princes have this failing." (De Morgan)
12. Rational beings are responsible for their actions. Brute animals are not responsible.
Therefore they are not rational.
13. Any honest man admits his rival's virtues. Not every scholar does this. Therefore
some scholars are not honest.
14. "Since all knowledge comes from sensory impressions, and since there's no sensory
impression of substance itself, it follows logically that there is no knowledge of
substance." (Robert Pirsig, Zen and the Art of Motorcycle Maintenance, paraphrasing
Hume.)
15 "Because intense heat is nothing else but a particular kind of painful sensation; and
pain cannot exist but in a perceiving being; it follows that no intense heat can really exist
in an unperceiving corporeal substance." (George Berkeley, Three Dialogues between
Hylas and Philonous)
16 "Since fighting against neighbors is an evil, and fighting against the Thebans is
fighting against neighbors, it is clear that fighting against the Thebans is an evil."
(Aristotle, Posterior Analytics)
17. "Whenever I'm in trouble, I pray. And since I'm always in trouble, there is not a day
when I don't pray." (Isaac Bashevis Singer) (Hint: you will need to change the wording
of these three propositions, without changing their meaning, to put the argument into the
form of a three-term syllogism.)
18. "The after-image (idea) is not in physical space. The brain process is. So the after-
239
image is not a brain-process." (J.J.C. Smart, "Sensations and Brain Processes,"
Philosophical Review 4/59)
19. "According to Aristotle, none of the products of Nature are due to chance. His proof
is this: That which is due to chance does not reappear constantly nor frequently, but all
products of Nature reappear either constantly or at least frequently." (Moses
Maimonides, Guide of the Perplexed)
20. "No man can be a rhapsode who does not understand the meaning of the poet. For
the rhapsode ought to interpret the mind of the poet to his hearers, but how can he
interpret him well unless he knows what he means?" (Plato, Ion) (This syllogism too
needs rewording first.)
21. "What is simple cannot be separated from itself. The soul is simple; therefore, it
cannot be separated from itself." (Duns Scotus, Oxford Commentary)
22. "Since morals have an influence on the actions and affections, it follows that they
cannot be derived from reason, because reason alone, as we have already proved, can
never have any such influence." (David Hume, A Treatise of Human Nature)
23. "No man should fear death, for death is according to nature, and nothing is evil
which is according to nature." (Marcus Aurelius) (Hint: can you get these four terms
reworded into three without changing the meaning? Is this the fallacy of four terms? Is it
two syllogisms?)
24. "We define a metaphysical sentence as a sentence which purports to express a
genuine proposition but does, in fact, express neither a tautology nor an empirical
hypothesis. And as tautologies and empirical hypotheses form the entire class of
significant [meaningful] propositions, we are justified in concluding that all
metaphysical assertions are nonsensical [not meaningful]." (A.J. Ayer, Language, Truth
and Logic)
25. "You cannot recognize non-being, nor speak of it, for that which can be thought and
that which can be are the same." (Parmenides) (Is there an implied premise at work
240
here?)
26. "Since the act of friendship is love, there are thus three kinds of friendship,
according to the three objects of love: (1) friendship because of virtue, which is true and
essential good, (2) friendship because of something pleasing to the senses, (3) friendship
because of utility." (Aquinas)
27. "Then what are we to say about that which is holy, Euthyphro? According to you, is
it not that which is loved by all the gods?" "Yes." "Just because it is holy, or for some
other reason?" "No, it is just for that reason." "And so it is because it is holy that it is
loved; it is not holy because it is loved." "So it seems." "On the other hand, a thing is
beloved and pleasing to the gods just because they love it." "No doubt of that." "So what
is pleasing to the gods is not the same as what is holy, Euthyphro, nor, according to what
you say, is the holy the same as what is pleasing to the gods. They are two different
things." "How is that so, Socrates?" "Because we are agreed that the holy is loved just
because it is holy, and is not holy just because it is loved. Isn't that so?" "Yes." "Whereas
what is pleasing to the gods is pleasing to them just because they love it." (Plato,
Euthyphro) (Hint: this is just one single syllogism, with just three terms.)
28. God must exist because God, by definition, lacks no perfection, and existence is a
perfection.
29. "And how do you know that you're mad?" "To begin with," said the Cat, "a dog's not
mad. You grant that?" "I suppose so," said Alice. "Well, then," the Cat went on, "you see
a dog growls when it's angry and wags its tail when it's pleased. Now I growl when I'm
pleased and wag my tail when I'm angry. Therefore, I'm mad." (Lewis Carroll)
30. You hear 'south-west' or 'south-east' but never 'south-north', because that would be a
contradiction. So south cannot be north. But if what is south can at the same time be
west, and west can be north, it logically follows that south can be north.
BACK
241
Exercises for Homework:
Homework Lesson 13:
1. Consider these two syllogisms. Which one is considered probable and which one is
demonstrative. Explain your answer.
A. No conscientious citizen will fail to cast his vote in an important election.
Smith is a conscientious citizen.
Therefore, Smith will not fail to cast his vote in an important election.
B. No nonliving substance is capable of organic growth.
Mercury is a nonliving substance.
Therefore, mercury is not capable of organic growth.
2. When is an act of reasoning more certain and scientific?
3. Define Enthymeme. Give one example of an enthymeme.
4. What is the first thing you must do in trying to build a syllogism knowing only the
conclusion?
5. In refuting an argument with a person who doesn’t know Logic, what should you do to show
them their error?
6. Define Epicheirema. Give one example of an epicheirema.
7. What is a chain argument? What is a Sorites?
8. Explain the argument from analogy.
BACK
242
LESSON FOURTEEN—HYPOTHETICAL SYLLOGISMS
1. Much of our reasoning is hypothetical in nature, rather than categorical.
2. Hypothetical syllogisms have their own set of rules. The rules for the categorical syllogism
don’t apply.
THE CONDITIONAL HYPOTHETICAL SYLLOGISM
1. The most basic type of hypothetical is the conditional syllogism. Its major premise is always
a conditional proposition. In this lesson, we’ll only investigate the conditional hypothetical.
2. Example:
If a person is farsighted, he needs glasses.
3. The condition, or antecedent, of any conditional proposition is usually written in the form of
an if clause.
4. The part that follows is called the consequent.
5. ☺☺If the consequent follows necessarily from the antecedent, the proposition is true.
6. Check out this example of a simple conditional syllogism:
major: If a person is farsighted, he needs glasses.
minor: John is farsighted.
conclusion: John does need glasses.
7. ☺☺Structurally, a conditional syllogism has as its major premise a conditional
proposition. The minor and conclusion are categoricals.
8. The above example is a conditional syllogism in the positing mood.
9. In other words, if it is affirmative, keep it affirmative; if it is negative, keep it negative.
10. Keep in mind too that to posit something is to accept it; to sublate something is to reject or
243
to contradict it.
11. The minor premise posits the major affirmatively, “John is farsighted.” The conclusion
posits the consequent “John does need glasses”.
12. Using the same major premise in the example above, it’s possible also to construct a
syllogism in the sublating mood:
major: If a person is farsighted, he needs glasses.
minor: Tom does not need glasses.
conclusion: Tom is not farsighted.
RULES FOR THE CONDITIONAL SYLLOGISM
1. Positing mood
a. Minor posits antecedent
b. Conclusion posits consequent
2. Sublating mood
a. Minor sublates consequent (contradicts it)
b. Conclusion sublates antecedent (contradicts it)
3. According to RULE 1 it is wrong, in the positing mood, to posit the consequent in the minor
and the antecedent in the conclusion.
4. Example:
major: If you do not eat, you will get a headache.
minor: You have a headache.
conclusion: So you did not eat.
Invalid: fallacy of positing the consequent (that is, in the minor)
5. To correct the syllogism, make the minor and conclusion read:
minor: You are not eating.
conclusion: So you will get a headache.
6. The syllogism assumes that the major premise is true—that there is a connection between not
eating and getting a headache.
7. According to the rule for the sublating mood it’s incorrect to sublate the antecedent in the
minor and the consequent in the conclusion.
244
major: If you work hard, you will be tired.
minor: You did not work hard.
conclusion: You are not tired.
Invalid: fallacy of sublating the antecedent (that is, in the minor)
8. In the sublating mood, make the minor and conclusion read:
minor: You are not tired.
conclusion: So you did not work hard.
See Extra 32: Hypothetical Syllogisms
See Extra 33—More Practice If Necessary
THE THEORY
1. Why is it correct to proceed from the truth of the antecedent (in the minor) to the truth of the
consequent (in the conclusion) and not conversely?
2. You need the rule because, for most conditional propositions, there can be any number of
conditions that lead to a given consequent.
3. Example: Consider the statement, “I will be late for class.” There can be many conditions
that cause this to happen:
If I oversleep,
If I read the morning newspaper,
If I miss the bus,
I will be late for class.
If I get involved in an accident,
If I go to the wrong building, etc.
4. If you posit any one of the above antecedents, you are correct in positing also the consequent;
I will be late for class.
5. Suppose, however, that I AM late for class.
245
6. In this case you can’t conclude that any of the above reasons are correct.
7. The conditional proposition usually takes one of four forms:
1. Both enunciations affirmative:
If a is b, then a is c.
If this flower grows, it will bloom.
2. Negative antecedent, positive consequent:
If a is not b, then a is c.
If a child is not sick, it will play.
3. Positive antecedent, negative consequent:
If a is b, then a is not c.
If a nation is prosperous, it does not need financial aid.
4. Both enunciations negative:
If a is not b, then a is not c.
If you do not eat, you will not recover.
8. When you construct a syllogism in the positing mood, keep its original quality.
9. For example using form 3 above:
If a nation is prosperous, it does not need financial aid.
This nation is prosperous.
It does not need financial aid.
10. Example using the sublating mood:
If a nation is prosperous, it does not need financial aid.
This nation does need financial aid.
It is not prosperous.
11. Some examples not beginning with “if”.
Had you not been at home, you would not have received my call.
Unless (that is, if—not) you know how to study, you will not learn.
Provided that she knows how to cook, she’ll be a good wife.
12. Sometimes the consequent is given first:
You cannot study if you do not know how to read.
13. Let’s use this example and construct a conditional syllogism in the subletting mood.
minor: You can study.
conclusion: You do know how to read.
246
14. A conditional syllogism can take the form of an enthymeme.
15. Example:
FIRST ORDER
major: SUPPRESSED
minor: This child is singing.
conclusion: It must be happy.
SECOND ORDER
major: If a child sings, it is happy.
minor: SUPPRESSED
conclusion: So this child must be happy.
THIRD ORDER
major: If a child sings, it is happy.
minor: But this child is singing.
conclusion: SUPPRESSED
16. With the first-order enthymeme above, you have a choice of reconstructing it either as a
categorical or as a conditional syllogism:
Categorical Conditional
Any child that sings is happy. If a child sings, it is happy.
This child is singing. This child is singing.
This child is happy. This child is happy.
17. Exercise: Construct a valid syllogism, using the 5 major premises below, in both the
positing and sublating moods.
1. If A is M, then A is also N.
2. If non-A is B, then X is Y.
3. If every X is a Y, then no non-X is a Z.
4. If None of the D’s are E’s, then all of the E’s are F’s.
5. If some of the non-P’s are Q’s, then all of the non-Q’s are R’s.
18. You can construct hypothetical syllogisms with DISJUNCTIVE and CONJUNCTIVE
propositions as well.
247
19. Remember that much of our everyday reasoning is hypothetical.
20. A mother might reason to herself:
Either Jimmy went with his father to the grocery store or he is romping about somewhere
in the neighborhood. But he can’t be [that is, is likely not] with his father, because his
father was in a hurry [it being assumed that he never takes the child when pressed for
time]. So I’ll see if I can locate him at the Smith’s [the most likely neighborhood hangout].
21. The success of these techniques is dependent on your ability to “size up” likely alternatives
which in relation to a given set of circumstances are complete and mutually exclusive.
22. Failure normally happens by omitting some alternative.
See Homework Lesson 14
Extra Materials
Extra 32: Hypothetical Syllogisms
NOTE: Whether the syllogism is in the positing or sublating mood is determined solely by what
the minor premise does. So, first determine the mood, and only after that whether the syllogism
is valid or not. Incidentally, all the dilemmas are formally valid; however, their weaknesses can
be exposed by discussing their unwarranted assumptions, failure to make distinctions where
necessary, and so on.
Identify the type of syllogism in each of the following examples, and evaluate. If the syllogism
is a dilemma, be careful to examine it on the grounds of its matter as well as its form.
Whenever part of the syllogism is implied, supply the missing part.
1. If you fall, you will get hurt. But you are hurt, so you must have fallen.
2. Provided that you look for a bargain, you will find one. You did not find any bargains, so you
could not have looked.
248
3. If a person works hard, the time passes quickly. The time is passing quickly, so I must be
working hard.
4. One cannot be a good student without reading books. But I am reading plenty of books. So I
must be a good student.
5. You will not have any friends unless you are kind to people. But you do have friends.
6. If Smith sues Jones and loses the case, he has nothing to gain. If he sues Jones and wins, the
money will go to the lawyer. In either case he has nothing to gain.
7. If this operation is not performed, the patient will die. The patient died, so the operation was
not performed.
8. If you are interested in helping your constituents (I know that you are), you will oppose any
further tax increase.
9. Mary will not go to the dance unless she has a partner. But she does have a partner, so she
will go.
10. Either you eat or you starve. Since you are not eating, you are starving. (Define the word
"eat.")
11. If a boat sinks, it is no longer seaworthy. This submarine is sinking. (Any ambiguity?)
12. This tooth either should be extracted or filled. But it is useless to fill it.
13. If a dog barks, it will not bite. This dog does not bark, so it will bite.
14. If you wear a hat, you will not catch cold. But you are not wearing your hat, so you did
catch cold.
15. You will pass if you know how to study. But you failed your course. Therefore, you do not
know how to study.
16. If you do not study, you will not pass. But you do study so you will pass.
249
17. If a worker is discontented with his pay, he will be inefficient. But most workers are
inefficient.
18. If you are consistent, you are logical. If you are inconsistent, you are human. Therefore,
you are either a logician or a human being.
19. If a government economizes on its military expenditures, it weakens its defense. If it does
not, it must maintain a high scale of taxes. Our government then either must weaken its defense
or maintain a high scale of taxes.
20. If a speech is effective, the audience will respond. But the audience is not responding.
Draw your own conclusion.
21. Scene: Student taking an examination in logic.
Question: True or false: a syllogism is an act of inference.
Student's Reasoning: The answer to this question must be false. If the answer were true, the
instructor would be asking too easy a question. No instructor of logic asks easy questions.
22. If you think about what you are writing, you cannot think about your writing; if you think
about your writing, you cannot write about what you are thinking. Now, either you think about
what you are writing, or you think about your writing. Draw your own conclusion.
23. Socrates to Meno:
"See what a tiresome dispute you are introducing. You argue that a man cannot inquire either
about that which he knows, or about that which he does not know; for if he knows, he has no
need to inquire; and if not, he cannot; for he does not know the very subject about which he is to
inquire."— Meno, 80.
24. Either effective legal measures must be taken to reduce smog in some of our metropolitan
areas or public health will be in jeopardy. Now it is quite evident that in certain areas where
smog is a serious problem that effective legal measures have not been taken. Draw your own
conclusion.
25. Either the good are rewarded or the sayings of the Bible are false. But the sayings of the
Bible are not false. So? BACK
250
Extra 33—More Practice If Necessary
26. Men become wise if they learn from experience. Some people, however, never learn from
experience.
27. “If there is no payoff, there is no gambling,” said the Chief of Police. “It is as simply as all
that.” In Des Moines there is no payoff. In Chicago there is gambling. Draw your own
conclusion.
28. Recently, a friend of mine (a business executive) told me that most fortunes are made, not
found. In other words, if you want to succeed in the business world, you have to try and keep
trying. Now, I know a number of people who, after spending years in their respective fields of
business, ended up complete failures. It must be that they did not try hard enough.
29. If you are in love, you will find it hard to study. But you told me yourself that you are not
in love. So you should not have any trouble studying.
30. The accused is either lying or he is ignorant of the law. In any event, he can be prosecuted.
31. A country cannot enjoy peace and wage war at the same time. Since our country is not
waging war, it is enjoying peace.
32. Provided that an intelligent student applies himself, he will pass his courses. There is no
doubt of the fact that John has intelligence. Yet, he failed some of his courses.
33. You cannot be at home and at school at the same time. Since you were not at school
yesterday, you must have stayed home.
34. You cannot at one and the same time live in California and miss the enjoyments of nature.
There are many people, however, who do miss the enjoyments of nature. It must be that these
people do not live in California.
35. Either this man does not know what he is talking about, or he is not telling the truth. But he
does know what he is talking about, so he is not telling the truth.
36. If you are in St. Louis, you are not at home. But you are in St. Louis.
251
37. It is ridiculous for anyone to read Mortimer Adler's How to Read a Book. If you know how
to read, you do not need to read it; if you do not know how to read, there is no point in trying.
38. Either you study what you know or you do not know what you are studying. Either way,
there is no advantage to studying.
39. If reason is infallible, error is impossible. But error is possible.
40. A person either will profit by his mistakes or fail to correct them. Unfortunately, most
people do not profit by their mistakes.
41. Scene: Student taking an examination in logic.
Question: True or false: a syllogism is an act of inference.
Student's Reasoning: The answer to this question must be false. If the answer were true, the
instructor would be asking too easy a question. No instructor of logic asks easy questions.
42. If you think about what you are writing, you cannot think about your writing; if you think
about your writing, you cannot write about what you are thinking. Now, either you think about
what you are writing, or you think about your writing. Draw your own conclusion.
"See what a tiresome dispute you are introducing. You argue that a man cannot inquire either
about that which he knows, or about that which he does not know; for if he knows, he has no
need to inquire; and if not, he cannot; for he does not know the very subject about which he is to
inquire."— Meno, 80.
44. Either effective legal measures must be taken to reduce smog in some of our metropolitan
areas or public health will be in jeopardy. Now it is quite evident that in certain areas where
smog is a serious problem that effective legal measures Have not been taken. Draw your own
conclusion.
45. Either the good are rewarded or the sayings of the Bible are false. But the sayings of the
Bible are not false. So? BACK
252
Exercises for Homework:
Homework Lesson 14: Hypothetical Syllogisms
1. Explain what is meant by the positing and sublating moods of the conditional
syllogism. Give an example of each type.
2. What do you understand by the fallacy (a) of positing the consequent? (b) of
sublating the antecedent?
3. Explain the theory (using your own example) behind the positing mood of the
conditional syllogism.
4. Explain the theory (using your own example) behind the sublating mood of the
conditional syllogism.
5. Give two examples of conditional propositions whose antecedent is not
expressed by the conjunction "if." From these examples construct two conditional
syllogisms in the positing mood and two in the sublating mood.
6. What logical procedures are to be followed in the positing and sublating moods
of a disjunctive syllogism?
7. Show why a loose disjunctive is not a suitable basis for a syllogism in the
positing mood. Please exemplify.
8. May a loose disjunctive be employed as the basis of a disjunctive syllogism in
the sublating mood? Explain and exemplify.
9. Give an example of your own of a valid conjunctive syllogism.
10. Explain why a conjunctive syllogism is invalid in the sublating mood.
253
11. What is the difference between a simple and a complex dilemma? Give your
own example of each type.
12. What is meant by "taking a dilemma by the horns"? What is meant by
"escaping between the horns of a dilemma"?
BACK
254
LESSON FIFTEEN—A FEW SELECT FALLACIES
1. Most folks understand the term “fallacy” as a kind of error, or mistaken impression.
2. For the Logician: ☺☺A fallacy is not a mistake in judgment, but a mistake in reasoning,
whether intentional or not.
3. The error, inherent in reasoning, leads to some type of false judgment.
4. You have already been introduced to a number of fallacies—the undistributed middle, illicit
minor, illicit major, and so on.
5. We’ll look at some linguistic and nonlinguistic fallacies in this lesson.
LINGUISTIC FALLACIES
1. Let’s take a look at five Linguistic Fallacies—Equivocation, Amphiboly, Composition and
division, Accent, and Parallel Word Construction.
EQUIVOCATION
1. Equivocation is the same as the fallacy of the ambiguous middle.
2. It occurs when one term is given more than one meaning in an argument.
3. It refers to both the analogous and equivocal use of terms in a syllogism.
4. Equivocation is not so much the problem here. Rather the analogous use of terms is the most
dangerous and the more easily disguised.
5. An example of equivocation:
Every philosopher is a scientist.
Every man is a philosopher.
Every man is a scientist.
255
6. The term “philosopher” has two meanings. In the major premise it means “a professional
philosopher”. In the minor, it means “any person who thinks deep thoughts”.
AMPHIBOLY
1. ☺☺An AMPHIBOLY is a defect in grammatical construction.
2. This apology was made to another person:
“I apologize for the misstatement I made concerning the fact that Congressman X was
guilty of accepting a bribe.”
3. The person made the apology, but because of the construction, you get the idea that
Congressman X was guilty of accepting a bribe.
4. At best, it’s a dubious apology.
FALLACY OF COMPOSITION AND DIVISION
1. ☺☺The fallacy of composition and division occurs when, a term is taken divisively in
one proposition and collectively in another, or vice versa.
2. For example:
Because there are many wealthy capitalists living in the United States, the United States
itself is a wealthy nation.
3. “Wealthy” is applied divisively to capitalists and collectively to nation.
4. Another example.
A student thinks himself to be intelligent because, according to the testimony of his
instructor, the class to which he belongs is intelligent.
5. Here, “intelligent” is applied collectively to class and divisively to student.
256
ACCENT
1. ☺☺The fallacy of accent occurs because of a difference of interpretation by a misplaced
emphasis upon a syllable, word, or phrase in a sentence.
2. Meaning can be changed by the way a speaker stresses certain words.
3. Example:
You may think as you please. [But others may not]
You may think as you please. [It is permissible, but]
You may think as you please. [But you may not act as you please]
4. We also consider anything that is taken out of context to belong to this group—the fallacy of
accent.
5. The most common complaint is being either misquoted or our words taken out of context.
6. This happens continuously in politics.
Example: In an earlier speech the Senator was opposed to a military buildup,
whereas now he advocates it.
7. Possibly, in the earlier statement, there was a need for the military buildup, now there isn’t.
PARALLEL WORD CONSTRUCTION
1. ☺☺Parallel Word Construction occurs when a person reasons that because two words are
similar in their structure, they are similar in meaning.
2. If, for example, two words have the same prefix or suffix, they have the same meaning.
3. A term ending in ‘ism’ often indicates that that term represents a specific school of thought.
From that you might conclude that the term ‘syllogism’ is also some school of thought.
257
NONLINGUISTIC FALLACIES
1. Let’s take a look at six nonlinguistic fallacies—Accident, Special Case, Ignoring the Issue,
Begging the Question, False Cause and Complex Question.
2. These fallacies come from a source other than vagueness of language.
ACCIDENT
1. ☺☺The fallicay of accident is one that rests on the assumption that a general rule is
rigidly applicable to every instance that falls under its extension.
2. Example:
Circumstances alter cases.
Rules have their exceptions.
Youth is inexperienced.
3. These propositions are true for the most part, but not always.
4. If you don’t make allowance for exceptions to these general rules, you have committed the
fallacy of accident.
SPECIAL CASE
1. ☺☺The Special Case fallacy says, what is true of one or more special cases is true of all
cases without exception.
2. Example:
Thus, it is assumed that in some cases the use of alcohol leads to ruin, therefore all use of
alcoholic liquor is wrong (that is, for everybody).
Because in some instances almsgiving contributes to laziness, therefore almsgiving should,
in all instances, be discouraged.
Because in some cases this medicine prevents patients from coughing, therefore it should
258
be prescribed for all patients, regardless of the nature or cause of their cough.
3. Two of the commonest forms of this fallacy are:
(1) Concluding from the abuse of a thing in a few particular cases to its complete abolition.
(2) Concluding from the effectiveness of a certain remedy in a few special cases to its
effectiveness for all similar cases without exception.
IGNORING THE ISSUE
1. ☺☺The Ignoring the Issue fallacy occurs when you try to disprove what has not been
stated or attempting to prove a point other than the one in question.
2. A person, with no real argument in their favor, will try all sorts of distracting techniques to
evading the issue. One is called straw man.
3. ☺☺The straw-man argument consists in refuting an opponent by making up some point
other than what was stated and arguing against that. You are attacking a ‘straw man’—a
nonexistent point.
4. Politicians are good at this. Example:
SENATOR A: I cannot agree with you, Senator B, in your claim that all of the Republican
senators are opposed to the administration’s policy.
SENATOR B: Sir, that is not what I said. What I did say was that some Republican
senators are opposed to the policies of the present administration.
SENATOR A: I don't care what you said, because by your previous statement you implied
that no honest Republican could (in accordance with his principles) support the policies of
this administration.
SENATOR B: Yes, I did say that, Senator, but I didn't imply, as you claim I do, that all
Republicans are actually opposed to administration policy. To say that none of them (as
Republicans) should support the administration is not to imply that all of them are really
opposed to it.
259
5. Another way to ignore the issue is the appeal to the man fallacy.
6. ☺☺Appeal to the man (argumentum ad hominem). The arguer attempts to discredit an
opponent’s ideas by discrediting the opponent himself.
7. Name-calling is a good example of this fallacy.
8. You can argue against this fallacy by simply considering the source.
9. Example: If someone is acting like an expert in an area they clearly are not, question their
“authority.”
10. ☺☺Appeal to the populace (argumentum ad populum). This fallacy occurs when the
speaker tries to “sell” his cause to the people by addressing himself to their
prejudices, emotions, and so on.
11. A rabble-rousing leader can incite the instincts of the mob using this technique.
12. During election time, this is a common argument.
13. There are also the Appeal to physical force (argumentum ad baculum), Appeal to pity
(argumentum ad misericordiam), Appeal to reverence (argumentum ad verecundiam), and the
Appeal to ignorance (argumentum ad ignorantiam)—otherwise known as a “snow job.”
BEGGING THE QUESTION
1. ☺☺By “begging the question” is meant assuming, in one way or another, as proved the
very point that is to be established.
2. For example:
The color of your eyes is an hereditary factor
because it is a characteristic handed down to you by your parents.
3. The question still remains: What proof is there for the statement?
4. Another common form of this fallacy is called the vicious circle.
260
5. A circular argument takes place when someone:
(1) Gives a reason for some proposition and
(2) Proceeds to “prove” it by means of the conclusion.
6. Example:
(1) All radicals should be deported from our country,
because they are a constant menace to good government.
(2) These men are a constant menace to good government, because they are radicals. (!)
7. Another fallacy of begging the question is called the question-begging epithet.
8. ☺☺The question-begging epithet is defined as one that either commends its object to the
listener or efficiently condemns it.
9. Example:
the fair-play amendment
the people's candidate
the favorite of millions
the century of enlightenment
the dark ages
a do-nothing Congress
10. This kind of statement can lead to a permanent impression on the unreflecting mind.
FALSE CAUSE
1. **The fallacy of false cause involves an attempt to draw from the conclusion of an
argument an absurd or untrue consequent that does not follow from it.
2. Example:
The individual citizen, living in a democracy,
does not actually enjoy political freedom.
3. In order to show indirectly that this proposition is false, you must posit a consequent that the
opponent will have to admit is false.
4. Example:
In that case, the individual citizen (living in a democracy) is not allowed to vote.
261
5. In admitting the falsity of this second proposition, you would also have to admit as false, the
original statement.
6. This is a legitimate type of proof, provided that the second proposition follows from the first.
Otherwise, it is the fallacy of false cause.
7. Example:
Bill: I maintain that the government should increase its welfare benefits.
Tom: On your own recommendation, then, the country might just as well go socialist.
8. Now-a-days, it is customary to include the inductive fallacy known as post hoc, ergo propter
hoc—after this, therefore on account of this.
9. ☺☺In the Inductive fallacy you assume that two events which are related to each other in
time or place are related also as cause and effect.
10. Example:
Because Smith was standing next to me when I first noted the disappearance of my
wallet, I accuse him of having stolen it.
Because I observed that my instructor was looking at me during the examination period,
I assume that his reason for doing so was that he suspected me of cheating.
All superstitious practices are, in effect, instances of this fallacy,
assuming as they do a causal relation between mere chance events.
COMPLEX QUESTION
1. ☺☺A question can be worded in such a way that the person answering it by a simple
“yes” or “no” unsuspectingly “incriminates” himself.
2. For example:
Are you still cheating on your wife?
When did you decide in favor of this nonsense of going to college?
How long have you been stealing your neighbor's apples?
3. Of course, the answer to a “loaded” question of this sort lies in a denial of the double
supposition upon which it rests.
262
4. Suppose, for instance, that some “high-pressure” salesman asks the following question:
Have you placed an order yet for your new refrigerator?
5. I might promptly reply that:
I neither placed an order for a refrigerator nor do I consider it mine until I buy it (unless, of
course, he chooses to give it away!)
263
Exercises for Homework:
Homework Lesson 15: A Few Select Fallacies
Some of the examples below are clear-cut reproductions of the fallacies discussed in this lesson;
others are not. Assign the proper name to the fallacies below and give your reason. If you are
not sure as to the precise nature of the fallacy, analyze the example for any unsupported
assumptions that underlie it.
1. Because it is wrong to use swear words, it is wrong to take an oath in court.
2. Pipe smoking, they say, is an aid to concentration. Most scientists, I notice, are pipe
smokers.
3. The accused is insane, because he lacks sufficient mentality to be guilty of crime.
4. Only one person will win the prize. Since I am only one person, I shall win the prize.
5. The doctor says I have ulcers. This confirms my conviction that my wife never was a good
cook.
6. This man is a skeptic. He does not agree with any of my arguments.
7. There is too close a tie-up between gambling and politics. For that reason all gambling
should be abolished and so too should politics.
8. You poor child! Did your mother spank you again? How can a woman be so cruel as to
torture her own flesh and blood?
9. A little knowledge is a dangerous thing. As an undergraduate student of logic, I have only a
little knowledge of it. So what I know about logic must be dangerous. (Discuss meaning of
word "little" in this argument.)
10. This person is nearsighted because he has myopia (nearsightedness).
11. Lincoln, who was born in a log cabin, became president. Why can't you?
264
12. Don't you ever tire of repeating that nonsense?
13. Women are better drivers than men, because statistics prove that women have fewer
accidents than men.
14. How long have you been nursing along your imaginary ailments?
15. An hour ago, when I said, "It is five o'clock," my statement was true. If I say that it is five
o'clock now, my statement is false. This proves that any statement you make is true at one time
and false at another, so that all truth is relative.
16. Marks are no necessary sign of a student's intelligence. Then, why not end the whole
system of the assignment of grades?
17. Psychiatrists are very presumptuous people to think that they can read a person's mind.
18. There is nothing more certain than the fact that nothing can be known with certainty. (Is this
self-contradictory?)
19. Einstein's theory of relativity does not make much sense. After all, there are few people
who claim to understand it.
20. According to the Scriptures it is wrong for a man to judge his fellow men. Should we not,
then, do away with the practice of judging a fellow man in court?
21. There is only one way to convince a reactionary; send him to a concentration camp.
22. There is no more effective method of punishing crime than the death sentence. Sentence
every criminal to death, and crime will soon disappear.
23. Most of the time spent in reading books is wasted because a person forgets more of what he
reads than he remembers.
24. Religion is merely a form of hypocrisy. I know many people who are regular church-goers,
but their lives are anything but models of good conduct.
25. Having completed four years of high school, you must have a thorough knowledge of the
265
classics.
26. Haste makes waste. Since efficiency experts encourage haste, they encourage waste.
27. Wife to husband: "Dear, you can't imagine how much money I saved for you this week. I've
been out shopping for bargains."
28. The best things in life are free. Since going to school takes money, education is not one of
the best things in life.
