symbolic logic unit 3

7/29/2019 Symbolic Logic Unit 3

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Language of SL

1. Sentential ConstantsEvery capital letter is a sentential constant.

E.g.: A, B, C, . . .

Sentential constants are used to abbreviate

simple sentences.

E.g.: Peter is tall. abbreviated as -- P


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2. Sentential OperatorsSentential Operator Symbolization(conjunction) (dot)

___and___ ( ___ ___ )(disjunction) (wedge)

___or___ ( ___v___ )(conditional) (horseshoe)If___then___ ( ___ ___ )(biconditional) (triple bar)

___if and only if ___ (___ ___)(negation) (tilde)

not____ ~___


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Terminology

(conjunction)

p and q p and q are conjuncts

(disjunction)

p or q p and q are disjuncts

(conditional) If p then q p is the antecedent and q

the consequent


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3. Syntax: Recursive Definition of a Well-FormedFormula (Sentence-formation Rules for SL):

1. Base Clause: Any statement constant (capital letter) is a wff (i.e.,sentence in SL).

2. Recursion Clause:

If p and q are any wffs, then all the following arealso wffs:

(p q); (p v q); (p q); (p q); ~ p.

3. Closure Clause:

Nothing will count as a wff unless it can be constructedaccording to clauses 1 and 2.


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Applying the definition of a well-formedformula:

(((Av B) C) (A (C v D)))


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(((Av B) C) (A (C v D)))

1. A, B, C, and D are well-formed formulas (BaseClause).


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(((Av B) C) (A (C v D)))


2. (A v B) and (C v D ) are well-formed formulas(Recursion Clause).


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(((Av B) C) (A (C v D)))



3. ((A v B) C) and (A (C v D)) are well-formedformulas (Recursion Clause)


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(((Av B) C) (A (C v D)))



3. ((A v B) C) and (A (C v D)) are well-formedformulas (Recursion Clause)4. (((A v B) C) (A (C v D))) is a well-formed

formula (Recursion Clause).


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Two notions

Major operator: Determines the overall formof a formula. The major operator is the

operator last introduced into a formula.

Subformula: Any well-formed formula that is

part of a larger formula.


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Unit 3: Truth functionalsemantics for our fivesentential operators


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Task

We need to know what our five sententialoperators mean logically.

More precisely, we need to know how theyaffect the truth-values of the compoundsentence formed with them.

We call this: truth-functional semantics


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Truth Tables

We specify themeanings of oursentential operatorswith the aid of truthtables:

p q p op. q

T T

T F

F T

F F


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Truth Tables for the Five SententialOperators

1. Conjunction

and

A conjunction istrue if and only if

both of itsconjuncts are true.

p q (p q)

T T T

T F F

F T F

F F F


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2. Disjunction:

or

A disjunction is true ifand only if at least oneof its disjuncts is true.

In symbolic logic, we usethe inclusive or.

Exclusive

or

:(p v q) ~ (p q)

p q (p v q)

T T T

T F T

F T T

F F F


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3. Biconditional:

If and only if

A biconditional istrue if and only ifboth componentshave the same truth-value.

p q (p q)

T T T

T F F

F T F

F F T


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4. Negation:

not

A negation is true if andonly if the non-negated

sentence is false.

p ~ p

T F

F T


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5. Conditional:(material conditional)

if, then

A conditional isfalse if and only ifthe antecedent istrue and theconsequent false.

p q (p q)

T T T

T F F

F T T

F F T


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Conditional

If I study hard, then I will pass the exam.

1. I study hard I pass the exam.T T T

2. I study hard I do not pass the exam.

T F F

3. I dont study hard I pass the exam.F T T

(I might actually pass the exam, even though I dont study hard.Say, if the exam is easy.)

4. I dont study hard I do not pass the exam.F F T


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Paradoxes of the Material Conditional

(1) If my cat sleeps a lot, then 5+7=12.

Antecedent and consequent are

completely independent. No causalconnection.


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(2) If Gore was elected in 2000, thenNASA will land on Pluto in 2001.

Both, antecedent and consequent are

false, but the conditional is true.


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(3) If cats speak all Western Europeanlanguages, then they do not speakFrench.

Both, antecedent and consequent,contradict each other. But the entireconditional is true.


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Computing Truth-Values of CompoundSentences

1. We write the truth-value of a sentenceimmediately above the sentence letter.

2. We write the truth-values of subformulasunder their major operator, starting withthe smallest subformulas.


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Step 1:

T F T F F T((A v B) (C v D)) (B A)

Note: You dont need to care about how we assign truth -values to the individual sentence letters. Later on,we will develop a procedure that allows us to do thissystematically.


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Step 2:

T F T F F T((A v B) (C v D)) (B A)T T T


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Step 2:T F T F F T

((A v B) (C v D)) (B A)T T T

T


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Step 2:T F T F F T

((A v B) (C v D)) (B A)T T T

T

T


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Compute!

1.T F F T F T

((~A B) (C D)) v (B D)2.

T T F F T T((A B) v A) ((C A) B)


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Homework for W 01/16

Please learn the truth-tables for our fiveoperators.

Do the starred exercises from problemsets 1 to 3 on p. 49/50.

symbolic logic unit 3

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