44 lines
1.2 KiB
Markdown
44 lines
1.2 KiB
Markdown
---
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categories:
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- Logic
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tags: [propositional-logic]
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---
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# Logical equivalence
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> Two sentences, P and Q, are truth-functionally equivalent if and only if there
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> is no truth assignment in which P is true and Q is false
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### Informal expression
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P: If it is raining then the pavement will be wet.
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Q: The pavement is not wet unless it is raining.
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### Formal expression
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$$
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(P \rightarrow Q) \longleftrightarrow (\lnot P \lor Q)
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$$
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### Truth-tables
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| $P$ | $Q$ | $ P \rightarrow Q $ | $ \lnot P \lor Q$ |
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| --- | --- | ------------------- | ----------------- |
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| T | T | T | T |
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| T | F | T | F |
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| F | T | T | T |
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| F | F | F | T |
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### Derivation
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> Propositions $P$ and $Q$ are equivalent in a system of
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> [derivation](Formal%20proofs%20in%20propositional%20logic.md) for
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> propositional logic if $Q$ is derivable from $P$ and $P$ is derivable from
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> $Q$.
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Note that the property of equivalence stated in terms of derivablity above is
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identical to the derivation rule for the
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[material biconditional](/Logic/Proofs/Biconditional_Introduction.md):
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