Semantic Proofs

# Semantic Proofs - Review of Semantic Concepts...

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Review of Semantic Concepts Truth-functional truth: Statement true on every truth-value assignment Truth-functional falsity: Statement false on every truth-value assignment Truth-functional indeterminacy: Neither truth-functionally true nor truth-functionally false Consistency: A truth-value assignment where all the members of the set are true Equivalence: No truth-value assignment where one is true and the other false (i.e. They have identical truth-values) Entailment: Every truth-value assignment that makes the members of the set true makes the further sentence true

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Set Terminology 2200 Γ = Represents a set of sentences of SL { P }: Set whose only member is P . Sometimes referred to as the “unit set of P ” or “singleton P 2200 = Used to represent the joining together of the members of two sets 2200 = Represents the “empty set” (the set which has no members)
Consistency and Truth- functional Truth Prove: If {~ P} is inconsistent, P is truth- functionally true Strategy: Assume inconsistency of {~ P }, show that the truth-functional truth of P follows

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functional Truth Explanation: Assume that {~ P } is inconsistent. A set is inconsistent just in case there is no truth-value assignment on which all the members of that set are true. Since ~ P is the only member of this set, the inconsistency of the set shows that ~ P must be truth-functionally false (as it would be false on every truth-value assignment). Since ~
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Semantic Proofs - Review of Semantic Concepts...

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