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# Semantics for relevant logics

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In what follows there is presented a unified semantic treatment of certain “paradox-free” systems of entailment, including Church's weak theory of implication (Church [7]) and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6]). We shall refer to these systems generally as relevant logics.

The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice—interpreted informally as the semilattice of possible pieces of information. Completeness theorems can be given relative to these semantics for the implicational fragments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems—in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the “paradoxes of material implication”.

The systems discussed are formulated in fragments of a first-order language with → (entailment), &, ⋁, ¬,(x) and (∃x) primitive, omitting identity but including a denumerable list of propositional variables (p, q, r, p1,…etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, A1,… are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The semantics of the systems are given in informal terms; it is an easy matter to turn the informal descriptions into formal set-theoretical definitions.

References
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[1]Ackermann, Wilhelm, Begründung einer strengen Implikation, this Journal, vol. 21 (1956), pp. 113128.
[2]Anderson, Alan Ross, Completeness theorems for the system E of entailment and EQ of entailment with quantification, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 16 (1960), pp. 201216.
[3]Anderson, Alan Ross, Some open problems concerning the system E of entailment, Acta Philosophica Fennica, Fasc. 16 (1963), pp. 718.
[4]Anderson, Alan Ross and Belnap, Nuel D. Jr., The pure calculus of entailment, this Journal, vol. 27 (1962), pp. 1952.
[5]Anderson, Alan Ross and Belnap, Nuel D. Jr., Tautological entailments, Philosophical studies, vol. 13 (1962), pp. 924.
[6]Belnap, Nuel D. Jr., Intensional models for first degree formulas, this Journal, vol. 32 (1967), pp. 122.
[7]Church, Alonzo, The weak theory of implication, Kontrolliertes Denken, eds. Menne-Wilhelmy-Angsil, , Kommissions-Verlag Karl Aber, Munich, 1951, pp. 2237.
[8]Church, Alonzo, Minimal logic (abstract), this Journal, vol. 16 (1951), p. 239.
[9]Church, Alonzo, Introduction to mathematical logic, vol. 1, Princeton University Press, Princeton, 1956.
[10]Dunn, J. M., Algebraic completeness results for R-mingle and its extensions, this Journal, vol. 35 (1970), pp. 113.
[11]Johansson, Ingebrigt, Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus, Compositio mathematica, vol. 4 (1937), pp. 119136.
[12]Kripke, Saul A., Semantical analysis of modal logic. II. Non-normal modal propositional calculi, The theory of models, eds. Addison-Henkin-Tarski, , North-Holland, Amsterdam 1965, pp. 206220.
[13]Kripke, Saul A., The problem of entailment (abstract), this Journal, vol. 24 (1959), p. 324.
[14]Kripke, Saul A., Semantical analysis of modal logic. I. Normal propositional calculi, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 6796.
[15]Routley, Richard, Semantical analysis of entailment and relevant implication, unpublished manuscript.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
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