Abstract
American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 ∈ τ(∼A, a) iff 0 ε τ (A, a) and 0 ∈ τ (∼A, a) iff 1 ∈ τ (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost of these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.
Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa.
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Routley, R. The American plan completed: Alternative classical-style semantics, without stars, for relevant and paraconsistent logics. Stud Logica 43, 131–158 (1984). https://doi.org/10.1007/BF00935746
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DOI: https://doi.org/10.1007/BF00935746