Lecture_13_2009 - Lecture 13 Lecture 13 Predicate Logic...

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Unformatted text preview: Lecture 13 Lecture 13 Predicate Logic Without Quantifiers Predicate Logic Without Quantifiers Atomic sentences are made of smaller units: individual constants and relation symbols. With this additional structure at the atomic level, the language is constructed using our usual approach (i.e. one inductive clause for each connective). Interpretations Interpretations Interpretations for PC0 Basic concepts for PC0: logical truth, logical falsity, logical implication. Note that every truth value assignment to the atoms can be realized. This means that logical concepts like tautology agree with their sentential counterparts. This will not be the case once we add equality. Adding Equality Adding Equality Syntax Semantics The following are logical truths: a ≈ a a ≈ b → b ≈ a (a ≈ b ∧ ≈ c) → a ≈ c b But are they tautologies? More on Equality More on Equality [EP] For all individual constants c,c’: If A’ is obtained from A by simultaneously substituting in some places c’ for c and/or c for c’, then: c ≈ c’ → (A ↔ A’) is a logical truth Extensionality Principle Extensionality Principle Extensionality Principle: The truth­value of a sentence, in a given interpretation, does not change, if we substitute a term by another term that has the same denotation. ...
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This note was uploaded on 02/27/2011 for the course PHILOSOPHY 101 taught by Professor H during the Spring '11 term at Columbia College.

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