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LogicalConn012011

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ADVENTURES IN LOGIC FOR UNDERGRADUATES by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University LECTURE 1: LOGICAL CONNECTIVES (this text incorporates minor corrections to the text displayed in the lecture)

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LECTURE 1. LOGICAL CONNECTIVES. Jan. 18, 2011 LECTURE 2. LOGICAL QUANTIFIERS. Jan. 25, 2011 LECTURE 3. TURING MACHINES. Feb. 1, 2011 LECTURE 4. GÖDEL’S BLESSING AND GÖDEL’S CURSE. Feb. 8, 2011 LECTURE 5. FOUNDATIONS OF MATHEMATICS Feb. 15, 2011 SAME TIME - 10:30AM SAME ROOM - Room 355 Jennings Hall
AND, OR, NOT We start with one common way of connecting sentences. Suppose I tell you Mike is a wibel AND Jane is a zibel I don't know about you, but I don't know what this means! But there is something about this sentence that we do know. This sentence, taken as a whole, is true or false according to whether its two constituents are true or false. AND is an example of a logical connective. OR, NOT are also examples of logical connectives:

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AND, OR, NOT Mike is a wibel OR Jane is a zibel Again, this sentence, taken as a whole, is true or false according to whether its two constituents are true or false. Mike is not a wibel This sentence, taken as a whole, is true or false according to whether its one constituent is true or false. What is the rule that determines the truth value of these three example sentences in terms of the truth values of their constituents? TRUTH VALUES: T if true; F if false.
RULES FOR AND, OR, NOT THE TRUTH TABLES Using letters for the constituents, we write A B (A and B) A B (A or B) ¬A (not A) There are four possibilities for A,B. A B A B A B ¬A T T T T F T F F T F F T F T T F F F F T

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LOGICAL EQUIVALENCE: REDUCING CONNECTIVES Two formulas are logically equivalent (written ) if and only if they have the same truth values under the same assignments of truth values to their aggregate letters. E.g., (A B) (B A) (C ¬C) It is also easy to check that A B ¬(¬A ¬B) A B ¬(¬A ¬B) CHALLENGE. Every propositional formula in ¬, is logically equivalent to one in ¬, . Every propositional formula in ¬, is logically equivalent to one in ¬, . (Use induction).
COMPLETENESS OF CONNECTIVE SETS We have been using the connective set ¬, , . This set is logically complete in the following sense. Let S be a set of assignments of truth values to the letters A 1 ,...,A n , n 1. There is a propositional formula in ¬, , ,A 1 ,...,A n which has truth value T under exactly the assignments in S. Because of the reductions from the last slide, we conclude that the connective sets {¬, } and {¬, } are also complete. The connective set { , } is not complete because every propositional formula in { , ,A} is T under some assignment.

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