An Algebraic View of Logic
نویسنده
چکیده
The standard semantics for classical logic states that each proposition stands for, or denotes, a truth value, either ⊤ or ⊥, according to the usual truth tables for each logical connective. According to this semantics, the categorical judgement P true means that P denotes ⊤, and the hypothetical judgement P1 true, . . . , Pn true ⊢ P true means that P denotes ⊤ whenever each Pi denotes ⊤. The hypothetical judgement induces an ordering relation among propositions defined by taking P ≤ Q to hold iff P true ⊢ Q true. (There is no loss of generality in restricting attention to a binary relation, because P1 true, . . . , Pn true ⊢ P true holds iff P1 ∧ · · · ∧Pn true ⊢ P true. ) This relation is a pre-order. This extends to a partial order on equivalence classes of propositions under mutual entailment. Specifically, define
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