lecture10

lecture10 - Homework due next Wednesday 7.1 A 1 3 5 8 12 B...

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Homework, due next Wednesday 7.1 A 1, 3, 5, 8, 12; B Odd numbers C Odd numbers, D Odd numbers 7.3 A 1, 3, 5, 8, 13, 14, 18 7.5 A 1, 8, 13, 15 7.4 A 3, 5, 6, 8, 10

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The language L The vocabulary of L: Parentheses: ( ) Capital letters: A…Z. The operators: ~, v, ●, →, ↔ The capital letters will be used to stand for statements. The operators stand for ‘not’, ‘or’, ‘and’, ‘if…then’ and ‘if and only if’, respectively. So ‘P Q’ is read ‘if P then Q’; ‘P ● Q’ is read ‘P and Q’, etc
Rules for well-formed formulas A well-formed formula (WFF) of a L is a grammatically correct expression. Let lower case p and q be statement variables. Then we can give the rules for WFFs: Capital letters are WFFs. If p is a WFF, then so is ~ p. If p and q are WFFs, then so is ( p q ). If p and q are WFFs, then so is ( p v q ). If p and q are WFFs, then so is ( p q ). If p and q are WFFs, then so is ( p q). Nothing else is a WFF.

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Some examples Given these rules, statements with scope ambiguities are not WFFs: P → Q v R P ● Q v R But their disambiguations are WFFs: (P → (Q v R)) ((P→Q) v R) ((P ● Q) v R) (P ● (Q v R)) Note that all WFFs except negations and statement letters require parentheses. Thus ~(P) is not a WFF but (~P ● Q) is a WFF.
Permissible departures For the sake of convenience, certain departures from strict grammar are acceptable. We can often drop the outermost parentheses without creating ambiguity: (P v Q) P v Q ((P→ (Q v R)) P → (Q v R) In long expressions, we can alternate parentheses with brackets, for readability: ~ (P v (Q ● R)) ~ [P v (Q ● R)] ((P ● Q) ↔ ~ (R v (S → T))) (P ● Q) ↔ ~ [R v (S → T)]

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Exercises Translate the following, from Layman, p. 260-1: Assuming your test scores are high and you get your paper in on time, you will do well. (T: Your test scores are high, P: You get your paper in on time, W: You will do well) Al wins only if Ed does not win, and Ed wins only if Al does not
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This note was uploaded on 03/14/2010 for the course PHIL 3 taught by Professor Way during the Fall '08 term at UCSB.

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lecture10 - Homework due next Wednesday 7.1 A 1 3 5 8 12 B...

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