Categorical PropositionA sentence that relates two classes, or categories. It asserts either all or part of one class is included or excluded from the other class.

Examples:All Nursing students are exposed to hospital setting.

No Nursing course is an engineering course.

Not all Nurses seek employment abroad.Celeste is enrolled in the College of Nursing.

A number of nursing students are planning to take the medical course later.

Categorical Proposition

Components:1. Subject Term

2. Predicate Term

3. Copula

All Nurses are health care providers. ------- ===============

Subject Term

Copula

Predicate Term

Categorical PropositionTypes of categorical propositions:

1. Those that assert that the whole subject class is included in the predicate class.2. Those that assert that whole of subject class is excluded in the predicate class

3. Those that assert that a part of a subject class is included in the predicate class. 4. Those that assert that the part of the subject class is excluded in the predicate class.

All S are P.

No S are P.

Some S are P.

Some S are not P.

Categorical PropositionNote: The words “all”, “no” and “some” are called quantifiers because they specify how a subject class is included or excluded from the predicate class.

Analysis of a standard form categorical proposition.

Quantifier:

Given : All nurses are patient advocates.

Subject Term:

Copula:

Predicate term:

All

nurses

are

patient advocates

Categorical PropositionQUALITY and QUANTITY

Quality: This is an attribute of categorical proposition. It is either affirmative or negative depending on whether it affirms or denies class membership.

Quantity: This is an attribute of a categorical proposition. It is either universal or particular depending on whether the proposition makes a claim about every member or just some member of the class denoted by the subject term.

Categorical PropositionQUALITY and QUANTITY

Propositions Meaning in class Notation

All S are P.

No S are P.

Every member of the S class is a member of the P class; that is, S class is included in the P class.

Some S are P.

Some S are not P.

No member of the S class is a member of the P class; that is, the S class is excluded in the P class.

At least one member of the S class is a member of the P class.

At least one member of the S class is not a member of the P class.

Categorical PropositionQUALITY and QUANTITY

Propositions Letter Name

All S are P.

No S are P.

Universal

Some S are P.

Some S are not P.

SUMMARY

A

E

I

O

Quantity Quality

Affirmative

Universal Negative

Particular Affirmative

NegativeParticular

Categorical PropositionDISTRIBUTION

All S are P.

Some S are not P.

No S are P.

Distribution is an attribute of the terms ( subject and predicate ) of a proposition. A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term otherwise it is undistributed.

A E

I OSome S are P

sP

SS P

P P

*S

*S

Categorical PropositionQUALITY and QUANTITY

Propositions Letter Name

All S are P.

No S are P.

Universal

Some S are P.

Some S are not P.

SUMMARY

A

E

I

O

Quantity Quality

Affirmative

Universal Negative

Particular Affirmative

NegativeParticular

Terms Distributed

S

S and P

none

P

Traditional Square of Opposition

All S are P. No S is P.

Some S are P. Some S are not P.

A E

I O

Contra diction

T

Contra

dicti

on

contrary

sub contrary

subalternsu

balte

rn

F

T

F

At least one is false (but not both true)

At least one is true(but not both false)

Traditional Square of OppositionSUMMARY

E being given as true: E is false, I is true, O is false

I being given as true: E is false, while A and, O are undetermined

A being given as true: E is false, I is true, O is false

O being given as true: A is false, while E and I are undetermined

A being given as false: O is true, E and I are undetermined

E being given as false: I is true, while A and O are undeterminedI being given as false: A is false, E is true, O is true

O being given as false: A is true, E is false, I is true

Square of Opposition1. What is the truth value of the propositions if A proposition is assumed to be true/false.

All successful nurses are college graduates.

No successful nurses are college graduates.

Some successful graduates are college graduates.

Some successful nurses are not college graduates.

If A is true: E is false, I is true, O is false

If A is false: O is True, E and I are undetermined.

Answer

2. If O is false what can we validly infer about A, I, and E?

A is true, E is false, and I is true.

Answer

AEIO

Existential Import and the Interpretation of Categorical Propositions

A proposition has an existential import if typically it is uttered to assert the existence of an object of some kind.

Illustration: (1) All Nurses are passers of the Nursing Licensure Examination.

(2) Some Doctors are nurses.

Discussions

The aforementioned propositions presuppose that Nurses and Doctors exist. In other words the classes referred by these propositions have at least one member.

The aforementioned statements do not assert that unicorns exist.

Example of propositions that do not have existential import.Illustration: (1) All unicorns are one-horned animals.

(2) No unicorns are friendly

Traditional Square of OppositionAll spiders are eight-legged animals. No spiders are eight-legged animals.

Some spiders are eight-legged animals. Some spiders are not eight-legged animals.

A E

I O

Contra diction

T

Contra

diction

subalternsu

balte

rn

T

Note: As shown in the traditional square of opposition, the inferences by sub alternation from A to I and from E to O are valid. Here, A and E have existential import. The contradiction between A and O as well as E and I holds.

Traditional Square of Opposition (TSO)All spiders are eight-legged animals. No spiders are eight-legged animals.

Some spiders are eight-legged animals. Some spiders are not eight-legged animals.

A E

I O

Contra diction

T

Contra

diction

suba ltern

suba

ltern

T

Note: There are cases, however, that TSO does not work. To illustrate: the A proposition All unicorns are single-horned animals. and the corresponding O proposition, Some unicorns are not single-horned animals, are contradictory propositions based on TSO. If both of them, however, are interpreted as having existential import, i.e. if we interpret them as asserting that there are unicorns, then both of them are false if unicorns do not exist. Of course we know that unicorns do not exist. The two propositions have the same truth value and therefore they are not contradictory after all..

All unicorns are single-horned animals.

Some unicorns are not single-horned animals.Some unicorns are single-horned animals.

No unicorns are single-horned animals.

If A and E validly implies their corresponding I and O propositions then it is not correct for it to say that A and O are contradictories. Something must be wrong with TSO!!! Can TSO be saved?

F

F

Traditional Square of Opposition

Is it possible to save TSO?

Two options?

OPTION 1. Treat all the categorical propositions as having existential presuppositions.

OPTION 2. Boolean interpretation of all the categorical propositions.

Traditional Square of Opposition

Is it possible to save TSO?Two options?OPTION 1. Treat all the categorical propositions as having existential presuppositions.

This is actually what is presupposed by the Aristotelian logic. In many cases this is in full accord with ordinary use of modern language like English. However, there are serious limitations, Copi and Cohen explain:

1. We will not be able to formulate propositions that deny that it has members. But propositions about classes that do not have members completely make sense, e.g. All unicorns are single-horned animals.2. Ordinary use of language is not in full accord with this presupposition. Sometimes what we say does not suppose that there are members in the class we are talking about, e.g. All violators of the Sexual Harassment Law will be prosecuted.

3. In science and in other theoretical spheres we often wish to reason without making any presupposition about existence, e.g. Newton’s First law of Motion: bodies that are not acted on by external forces persevere in rest or in their straight line motion.

Modern Square of OppositionOPTION 2. Boolean interpretation of all the categorical propositions.

The modern square of opposition is based on an interpretation of categorical statements introduced by the 19th century logician George Boole. In his interpretation the categorical propositions have the following meaning:

A All S are P. = No members of S are outside P.

E No S are P. = No members of S are inside P.

Note: The interpretation is neutral about existence

I Some S are P. = At least one S exists, and that S is P.

O Some S are not P. = At least one S exists, and that S is not a P.

Note: In this interpretation of I and O there is a positive assertion about existence.

This is the same as in the Aristotelian interpretation

Modern Square of Opposition

All S are P. No S is P.

Some S are P. Some S are not P.

A E

I O

Contra dictionContra

diction

The square of opposition that results from the Boolean interpretation is:

Modern Square of Opposition

All S are P. No S is P.

Some S are P. Some S are not P.

A E

I O

Contra dictionContra

diction

All unicorns are single-horned animals.

Some unicorns are not single-horned animals.

Some unicorns are single-horned animals.

No unicorns are single-horned animals.T

F

T

F

Modern Square of Opposition

When do we use TSO and when do we use MSO?

Although MSO may be used in all categorical propositions, it is preferable to to use TSO on categorical propositions that make assertions about actually existing things, because it provides for more inferences. On categorical propositions that make assertions about things that do not actually exist, the TSO cannot be used.