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notes_15_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & \$ % Announcements & Such • Explosions in the Sky : The Only Moment We Were Alone • Administrative Stuff – HW #2 solutions posted. [Don’t sweat symbolizations too much.] + Hints to controversial problems were on my “tips” handout. – I have posted two handouts: (1) solutions to problems from lecture on logical truth, equivalence, etc., and (2) three examples of the “short” truth-table method for validity (to be discussed today). + Make sure you study my handouts. They tend to be useful. – HW #3 due Friday, usual drill (truth-table methods for validity). • Today: Chapter 3, Finalé — Two Final Topics on LSL Semantics – Some facts about semantic the semantic consequence relation ( ). – Expressive Completeness. • Next: Chapter 4 — Natural Deduction Proofs for LSL – Natural deductions are the most challenging topic of the course. UCB Philosophy Chapter 3 Finalé 10/01/08 Branden Fitelson Philosophy 12A Notes 2 ' & Properties of the Semantic Consequence Relation: • The following four metalinguistic statements are synonymous: – The argument p 1 , p 2 , . . . , p n ∴ q is valid . – q follows from p 1 , p 2 , . . . , p n . – p 1 , p 2 , . . . , p n (jointly) entail q . – p 1 , p 2 , . . . , p n q • Here are some important properties of with explanations: – p p * Every interpretation on which p is true is an interpretation on which p is true. That is, all p-interpretations are p-interpretations. – If p q and q r , then p r . * If all p-interpretations are q-interpretations and all q-interpretations are r-interpretations, then all p-interpretations are r-interpretations. UCB Philosophy Chapter 3 Finalé 10/01/08 Branden Fitelson Philosophy 12A Notes 3 ' & \$ % * Remember: the following argument is valid (but not sententially!). All P s are Q s. All Q s are R s. ∴ all P s are R s. * More on arguments like this in the second half of the course . . . – If p r , then p & q r . * If all p-interpretations are r-interpretations, then all (p & q)-interps are r-interpretations [since all (p & q)-interpretations are p-interpretations!]. – (p & q) r if and only if p,q r * If all p & q-interpretations are r-interpretations, then all { p,q }-interpretations are r-interpretations (pretty obviously). – p q if and only if p → q * If all p-interpretations are q-interpretations, then all interpretations (whatsoever) are (p → q)-interpretations. * p → q is a tautology [ p → q ] iff there is no interpretation on which p is true and q is false, which is just the definition of p q ! UCB Philosophy Chapter 3 Finalé 10/01/08 Branden Fitelson Philosophy 12A Notes 4 ' & Expressive Completeness • In LSL, we have five connectives: h∼ , & , ∨ , → , ↔i . But, we don’t “need” all five. We can express all the same propositions with fewer connectives....
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