Abstract
The purpose of this note is to show that a known and natural four-valued logic co-exists with classical two-valued logic in the familiar context of truth tables. The tool required is the power construction. The context of this paper is that of 2-valued truth tables in n propositional variables, say p1, p2, . . . , pn. The two classical truth values are 0 and 1. The truth value assignments (or valuations) are the 2 {0, 1}-vectors of length n. Propositional formulae are freely generated from the propositional variables by application of the usual classical connectives ∼ (not), & (and) and ∨ (or), each of which is defined by a given truth table. Accordingly, each propositional formula has a truth table. Formulae with the same truth table are logically equivalent, and we do not distinguish between them. The valuations form a Boolean algebra Val = (2,∧,∨, ′ ,0,1), where the Boolean operations ∧, ∨ and ′ are defined componentwise from their counterparts in the 2-element Boolean algebra 2 = {0, 1}, and 0 and 1 are respectively the vectors (0, 0, . . . , 0) and (1, 1, . . . , 1) of length n. The elements of 2 are denoted by x,y,z, . . . Up to equivalence, formulae may be regarded as truth functions from 2 to 2, and the connectives are characterised by operations in the image algebra 2. That is, if bool(A,−) : 2 → 2 is the truth function corresponding to a propositional formula A, then bool (∼A,x) = bool (A,x)′ , bool (A & B,x) = bool (A,x) ∧ bool (B,x) , (1) bool (A ∨ B,x) = bool (A,x) ∨ bool (B,x) . 1991 Mathematics Subject Classification: 03B46, 03B05. The paper is in final form and no version of it will be published elsewhere.
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