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Chapter 1 The Logic of Compound Statements

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Section 1.2 – 1.3 (Modus Tollens) Conditional and Valid & Invalid Arguments

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Conditional Statements A conditional statement is a sentence of the form“if p then q” or p -> q (p implies q). – p is the hypothesis – q is the conclusion

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Example “If you show up for work Monday morning, then you will get the job.” – p = You show up for work Monday Morning. – q = You will get the job. – p -> q When is this statement false?

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Example -> p v ~q -> ~p Order of precedence: 1. ~, 2. ^,v, 3. ->, p v ~q -> ~p (pv~q) -> (~p)

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Logical Equivalence -> p q -> r (p ->r) ^ (q ->r)

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Equivalence -> & or p -> q ~p v q Example – ~p v q = “Either you get to work on time or you are fired.” – ~p = You get to work on time. – q = You are fired. – p = You do not get to work on time. – p -> q = “If you do not get to work on time, then you are fired.”

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Negation of Conditional Negation of if p then q “p and not q” ~(p -> q) p^ ~q Derivation from Theorem 1.1.1 – ~(p -> q) ~(~p v q) – ~(~p) ^ (~q) by DeMorgan’s – p ^ ~q by the double neg law Example – If Karl lives in Wilmington, then he lives in NC. – Karl lives in Wilmington and he does not live in NC.

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Contrapositive of a Conditional The contrapositive of p -> q is ~q -> ~p. Conditional is logically equivalent to its contrapositive: p -> q ~q -> ~p pq~p~qp->q~q -> ~p TTFFTT TFFTFF FTTFTT FFTTTT

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Example Conditional p->q – If Howard can swim across the lake, then Howard can swim to the island. – p = “Howard can swim across the lake.” – q = “Howard can swim to the island.” Contrapositive ~q -> ~p – If Howard cannot swim to the island, then Howard cannot swim across the lake.

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Converse of Conditional Converse of conditional “if p then q” (p->q) is “if q then p” (q->p) Converse is not logically equivalent to the conditional. Example – (conditional) If today is Easter, then tomorrow is Monday. – (converse) If tomorrow is Monday, then today is Easter.

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Inverse of Conditional Inverse of conditional “if p then q” (p->q) is “if ~p then ~q” (~p->q) Inverse is not logically equivalent to the conditional. Example – (conditional) If today is Easter, then tomorrow is Monday. – (inverse) If today is not Easter, then tomorrow is not Monday. However, the converse and inverse are logically equivalent. pq~p~qp->qq->p~p->~q TTFFTTT TFFTFTT FTTFTFF FFTTTTT

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Biconditional Biconditional is “p if, and only if q”. Biconditional is T when both p and q have the same logic value and F otherwise. Symbolically – p q

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Biconditional Truth Table

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Necessary & Sufficient Conditions For statements r and s, – r is a sufficient condition for s (if r then s) means “the occurrence of r is sufficient to guarantee the occurrence of s”. – r is a necessary condition for s (if not r then not s) means “if r does not occur, then s cannot occur”.

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Valid & Invalid Arguments An argument is a sequence of statements. All statements in an argument, except for the final one, is the premises (hypotheses). The final statement is the conclusion. Valid argument occurs when the premises are TRUE, which results in a TRUE conclusion.

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Testing Argument Form Identify the premises and conclusion of the argument form. Construct a truth table showing the truth values of all the premises and the conclusion. If the truth table reveals all TRUE premises and a FALSE conclusion, then the argument from is invalid. Otherwise, when all premises are TRUE and the conclusion is TRUE, then the argument is valid.

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Example If Socrates is a man, then Socrates is mortal. Socrates is a man. :. Socrates is mortal. Syllogism is an argument form with two premises and a conclusion. Example Modus Ponens form: – If p then q. – p – :. q

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Example Valid Form p v (q v r) ~r :. p v q

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Example Invalid Form p -> q v ~r q -> p ^ r :. p -> r

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Modus Tollens – If p then q. – ~q – :. ~p – Proves it case with “proof by contradiction” – Example: – if Zeus is human, then Zeus is mortal. – Zeus is not mortal. – :. Zeus is not human.

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