LogicalQuan012611

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ADVENTURES IN LOGIC FOR UNDERGRADUATES by Harvey M. Friedman Distinguished University Professor Mathematics, Philosophy, Computer Science Ohio State University Lecture 2. Logical Quantifiers

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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 WARNING: CHALLENGES RANGE FROM EASY, TO MAJOR PARTS OF COURSES
PROPOSITIONAL CALCULUS - PREDICATE CALCULUS In the first lecture, we worked within the framework of propositional calculus, based on i. Statement letters A 1 ,A 2 ,... . ii. Connectives ¬, , , , . In this talk, we present the far richer framework called predicate calculus . We will approach the full predicate calculus in small bite sized stages, which are important in their own right. Propositional Calculus is NOT powerful enough to support the logic behind mathematics. Predicate Calculus IS. We will have some SPECIAL FUN at the end of the talk.

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VARIABLES AND EQUALITY In this talk, we will scrap the sentential letters A 1 ,A 2 ,... . But we will retain the five connectives ¬, , , , . from Lecture 1. We now introduce the variables v 1 ,v 2 ,..., and the equality symbol =. That’s it. A very gentle first step. The variable equations take the form v i = v j . The variable formulas are built up from the variable equations using the give logical connectives. Examples of variable formulas: v 3 = v 3 ¬(v 3 = v 4 ) v 3 = v 4 v 3 = v 4 v 4 = v 3 (v 2 = v 7 v 9 = v 1 ) ¬(v 8 = v 2 ) Everybody likes to write v i v j instead of ¬(v i = v j ).
VARIABLE FORMULA LOGIC In Variable Formula Logic, we use only variable formulas. To interpret the variable formulas, we must designate a nonempty domain D. The variables are to range over the elements of D. Variable formulas will be true or false in (D,=) depending on which objects from D are assigned to which variables. We use the terminology: D assignments. We say that a variable formula is true in (D,=) if and only if it is true in (D,=) under ALL D assignments. Note that among our example variable formulas, v 3 = v 3 , v 3 v 4 v 3 = v 4 , v 3 = v 4 v 4 = v 3 are each true in (D,=) for all D assignments in (D,=). Hence each of these three formulas is true in (D,=).

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VARIABLE FORMULA LOGIC More holds of these three examples of variable formulas: v 3 = v 3 , v 3 v 4 v 3 = v 4 , v 3 = v 4 v 4 = v 3 are all true in ALL (D,=). Such a variable formula is said to be valid . THEOREM. If a variable formula is true in (Z,=) then it is valid. This claim is false for any (D,=), D finite. CHALLENGE: Prove the above Theorem.
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