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Lecture_2 - Lecture 2 Lecture 2 Atoms and Sentences Atoms...

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Unformatted text preview: Lecture 2 Lecture 2 Atoms and Sentences Atoms and Sentences Inductive definitions Atoms versus sentential variables Sentences versus sentential expressions For the most part we can ignore atoms. So why bother with atoms? Tautological Equivalence Tautological Equivalence Logical equivalence versus tautological equivalence Tautological equivalence for sentences Tautological equivalence for sentential expressions A ∧ A ≡ B ∧ B? ¬ ¬ Equivalence Relations Equivalence Relations Reflexivity Symmetry Transitivity ‘≡ ’ is an equivalence relation. ‘≡ ’ respects truth­functional connectives. Disjunction Disjunction Inclusive ‘or’ versus exclusive ‘or’ Semantics for inclusive ‘or’ ‘∨ in terms of the connectives ‘∧ ’ ’ and ‘¬’ Exclusive ‘or’ in terms of the connectives ‘¬’, ‘∧ ’ and ‘∨ ’ Bindings Bindings The following are ambiguous expressions: ¬A ∨ B and A ∨ ∧ B C. What is so bad about these expressions? Binding strength in descending order: ¬, ∧∨ , Tautologies and Contradictions Tautologies and Contradictions Tautologies Logical truths Contradictions Logical falsities Up to tautological equivalence, how many contradictions are there? ...
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