symbolic logic unit 3
TRANSCRIPT
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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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Applying the definition of a well-formedformula:
(((Av B) C) (A (C v D)))
1. A, B, C, and D are well-formed formulas (BaseClause).
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Applying the definition of a well-formedformula:
(((Av B) C) (A (C v D)))
1. A, B, C, and D are well-formed formulas (BaseClause).
2. (A v B) and (C v D ) are well-formed formulas(Recursion Clause).
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Applying the definition of a well-formedformula:
(((Av B) C) (A (C v D)))
1. A, B, C, and D are well-formed formulas (BaseClause).
2. (A v B) and (C v D ) are well-formed formulas(Recursion Clause).
3. ((A v B) C) and (A (C v D)) are well-formedformulas (Recursion Clause)
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Applying the definition of a well-formedformula:
(((Av B) C) (A (C v D)))
1. A, B, C, and D are well-formed formulas (BaseClause).
2. (A v B) and (C v D ) are well-formed formulas(Recursion Clause).
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.