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Unformatted text preview: Introduction to Logic Lecture 5 Brian Weatherson, Department of Philosophy September 16, 2009 Logic 201 (Section 5) Lecture 5 1 Two Definitions of Validity 1 An argument is valid if it’s true that: if the premises are true, the conclusion must be true. 2 An argument is valid if it is impossible for the premises to be true and the conclusion false. These definitions are equivalent, so you can use either depending on which is more convenient. We’ll say that a valid argument with true premises is sound . An argument that is either invalid, or has false premises, is unsound . Logic 201 (Section 5) Lecture 5 2 Validity An argument is valid if, when the premises are true, the conclusion must be true. That is, there is no world where the premises are true and the conclusion false. If we can build a world where the premises are true and the conclusion false, then we have shown the argument is invalid. The next slide shows a demonstration in Tarski’s World that this argument is invalid. Cube ( a ) Larger ( a , b ) Cube ( b ) Logic 201 (Section 5) Lecture 5 3 Invalidity Because there is a world where the premises are true and the conclusion false, the argument is invalid . Logic 201 (Section 5) Lecture 5 4 What are Proofs A proof that an argument is valid is: 1 A sequence of statements; 2 The first statements are the premises of the argument; 3 The last statement is the conclusion of the argument; 4 Each statement follows directly from earlier statements; 5 And we say exactly how each statement follows from earlier statements. Logic 201 (Section 5) Lecture 5 5 Rules Rules are what let us move from earlier lines to later lines in a proof. For each of the logical symbols we introduce, we will have two rules. These are, for reasons that may be a little mysterious at first, called introduction and elimination rules. Logic 201 (Section 5) Lecture 5 6 Rules for Identity =Introduction Infer x = x at any time, with no support steps needed. =Elimination From x = y , and a sentence with x in it, derive the sentence you get by replacing x with y . Logic 201 (Section 5) Lecture 5 7 Properties of Identity Identity Is: 1 Symmetric; and 2 Transitive. A symmetric relation is one that, if it holds between x and y , also holds between y and x . For instance, is married to is symmetric, and loves is not. A transitive relation is one that, if it holds between x and y , and holds between y and z , holds between x and z . For instance, taller is transitive, and is a friend of is not. Logic 201 (Section 5) Lecture 5 8 AnaCon Logic 201 (Section 5) Lecture 5 9 AnaCon Fitch has one rule in it that isn’t strictly part of logic. That rule is AnaCon AnaCon is short for Analytic Consequence....
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 Fall '11
 JonWinterbottom
 Logic

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