Studia Logica

, Volume 43, Issue 1–2, pp 131–158 | Cite as

The American plan completed: Alternative classical-style semantics, without stars, for relevant and paraconsistent logics

  • Richard Routley


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.


Mathematical Logic Relational Semantic Classical Form Computational Linguistic Classical Feature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Polish Academy of Sciences 1984

Authors and Affiliations

  • Richard Routley
    • 1
    • 2
  1. 1.Research School of Social Sciences Australian National UniversityAustralia
  2. 2.Environmental Studies University of VictoriaCanada

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