15718_Symbolic Logic Intro

8/4/2019 15718_Symbolic Logic Intro

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Symbolic Logic

CPT 120 ~ Quantitative Analysis I

Professor Robert H. Orr


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1999 Trustees of Indiana

University

What is logic?

logic is a process people use to infer one thingfrom another

logic is the root of all mathematics and is the rootof all rational thought (truth)

Butlogic is not thinking


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University

Philosophers &Mathematicians Philosophers studied logic:

Socrates Plato

Aristotle Mathematicians studied logic:

George Boole Georg Cantor

Augustus De Morgan Kurt Gdel

Modern philosophers Bertrand Russell and AlfredNorth Whitehead attempted to derive allmathematical concepts directly from symboliclogic (Principia Mathematica)


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University

Why study logic?

Logic helps people to:

clarify and analyze information understand what others are saying in written and

oral communication

recognize complex ideas which may be masked

in simple statements perform well in academic courses


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1999 Trustees of IndianaUniversity

Logic Definitions

Premise:a statementwhich can be either true orfalse

argument:one or more premises which are logicallyconnected followed by a conclusion

conclusion:the result that follows the application ofthe laws of logic to the premises of an argument

tautology:an argument that is always trueExample:Either the sun shines or it does not

contradiction:an argument that is always false

Example:I can fly and I cannot so do


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Categorical Logic

Uses syllogisms to state the arguments

Consists ofa major premise, minor premise

and a conclusion Example:

All cats are black

All black things are plastic.

All cats are plastic.

Major premise

Minor premise

Conclusion


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Truth-Functional Logic

Decomposes statements into simple clauses andconnecting words

Example:

Roses are red and violets are blue.

can be broken into 2 simple primitives:1) Roses are red. 2) Violets are blue.

Along with the connecting word and


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Conjunction: The andConnective some other words & phrases that are

equivalent to and

also, too both in addition (to)

plus including all

but however as well as

with along with together with


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Disjunction: The orConnective the connective or is inclusive; it means:

oneor the otheror both

some other words or phrases that areequivalent to or inclusive or exclusive

either otherwise unless and/oralternatively on the other hand

nevertheless instead (of) else


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Symbolizing Sentences

Complex sentences can be broken downinto simpler sentence structures

Structures that cannot be furtherdecomposed are calledprimitives

Use lower case letters (p, q, r,) to

represent each primitive Use the logic symbols to represent the

connectives


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Symbolizing Sentences:PrimitivesSentence Representation

John likes pizza. p

John doesnt like pizza. ~p

The sky is blue. qThe sky is not blue. ~q


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Symbolizing Sentences:Compound SentencesSentence Representation

Bob likes jazz and pizza.

Bob doesnt like jazz or pizza.

Bob doesnt like jazz and pizza.

Bob doesnt like jazz but likes pizza.

Bob likes jazz butnot pizza.

qp

q~p~

qp~

q~p

q~p~

In the above examples, letpdenote Bob likesjazz and letqdenote Bob likes pizza.


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Symbolizing Paragraphs

Identify simple sentences(primitives)

If necessary, decompose compound and

complex sentences into their simplercomponents

Repeat process until all primitives have beenidentified

Select the appropriate logic operator Assign a variable to represent each primitive

Express all primitives in symbolic form


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What is an Argument?

An argument consists of a set of premisesfollowed by a conclusion

A premise is a statement that is either trueor false

Likewise, a conclusion is also a statementthat is either true or false


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Symbolic Representation ofan Argument

Q)PP(P n21

Where the Pi represent the individual premisesfrom a set of npremises, and Q represents thearguments conclusion.

Note that all premises must be anded together.


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Building Truth Tables:1 or 2 Variables One Variable Truth Table Two Variable Truth Table

P

T

F

P q

T T

T F

F T

F F

Cells under the rightmost variable

always alternate truth values


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Building Truth Tables:3 Variablesp q r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F


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Truth Tables: Logical And

p q

T T T

T F F

F T F

F F F

qp true when both pandqare true


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Truth Tables: Inclusive Or

p q

T T T

T F T

F T T

F F F

qp

false when both pandqare false


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Truth Tables: if-then

p q

T T T

T F F

F T T

F F T

qp

false when pistrue and qis false

P is theantecedent;q is theconsequent


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p ~p

T F

F T

Truth Tables: Not

has the oppositevalue of p

To be read as not p or its not the case that p


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Truth Tables: if-and-only-if

p q

T T T

T F F

F T F

F F T

qp

true when: both pand qare true both pand qare false

Also called abiconditional


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Truth Table Example: break compound proposition into simpler

arguments

p q

T T F F T

T F T T F

F T F F TF F T F T

q)~(pq~pq

q)~(p~


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p q ~

T T T T F F T

T F F T T T F

F T T F F F T

F F T F F T T

break compound proposition into variables andconnectives

Truth Table Example:

(p

q)

q)~(p~

~


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Laws of the Algebra of Logic

Idempotent Laws

Associative Laws

Commutative Laws

ppp ppp

r)(qprq)(p r)(qprq)(p

pqqp pqqp


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Distributive Laws

Identity Laws

r)(pq)(pr)(qp

r)(pq)(pr)(qp

pfp

ttp

ptp

ffp


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Complement Laws

DeMorgans Laws

tp~p fp~p

pp~~ ft~ tf~

q~p~q(p~ ) q~p~q(p~ )

Documents

15718_Symbolic Logic Intro