Abstract
A system FDQ of first degree entailment with quantification, extending classical quantification logic Q by an entailment connective, is axiomatised, and the choice of axioms defended and also, from another viewpoint, criticised. The system proves to be the equivalent to the first degree part of the quantified entailmental system EQ studied by Anderson and Belnap; accordingly the semantics furnished are alternative to those provided for the first degree of EQ by Belnap. A worlds semantics for FDQ is presented, and the soundness and completeness of FDQ proved, the main work of the paper going into the proof of completeness. The adequacy result is applied to yield, as well as the usual corollaries, weak relevance of FDQ and the fact that FDQ is the common first degree of a wide variety of (constant domain) quantified relevant logics. Finally much unfinished business at the first degree is discussed.
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References
N. D. Belnap, Jr., Intensional models for first degree formulas, Journal of Symbolic Logic 32 (1967), pp. 1–22.
R. Routley, A rival account of logical consequence, Reports on Mathematical Logic 3 (1974), pp. 41–51.
R. Montague and L. Henkin, On the definition of “formal deduction”, Journal of Symbolic Logic 21 (1956), pp. 129–136.
A. N. Prior, Formal Logic, Second Edition, Clarendon Press, Oxford, 1962.
A. R. Anderson and N. D. Belnap, Jr., First degree entailments, Technical Report No. 10, Office of Naval Research, New Haven. Reprinted in Mathematische Annalen 149 (1963), pp. 302–319.
A. R. Anderson, Completeness theorems for the systems E, of entailment and EQ of entailment with quantification, Zeitschrift für Mathematische Logic und Grundlagen der Mathematik 6 (1960), pp. 201–216.
R. Routley and R. K. Meyer, Relevant Logics and their Rivals, RSSS, Australian National University, 1979.
A. R. Anderson and N. D. Belnap, Jr., Entailment: the Logic of Relevance and Necessity, Volume 1, Princeton University Press, Princeton, 1975.
R. K. Meyer, Coherence Revisited, unpublished monograph (copies available from the author, RSSS, Australian National University).
A. Church, Introduction to Mathematical Logic, Volume 1, Princeton University Press, Princeton 1956.
R. Routley and R. K. Meyer, Semantics of entailment I, in Truth, Syntax and Modality (edited H. Leblanc) North Holland, Amsterdam, 1973, pp. 199–243.
Z. Vendler, Each, all, every and any, Mind 71 (1962), pp. 145–160.
E. Routley, A semantical analysis of implicational system I and of the first degree of entailment, Mathematische Annalen 196 (1972), pp. 58–84.
R. Routley, Constant domain semantics for quantified non-normal modal logics and for certain quantified quasi-entailment logic, Reports on Mathematical Logic, forthcoming 1978.
R. K. Meyer, J. M. Dunn and H. Leblanc, Completeness of relevant quantification theories, Notre Dame Journal of Formal Logic 15 (1974), pp. 97–121.
R. Routley, Ultralogic as universal ? Relevance Logic Newsletter 2 (January and May 1977) pp. 50–90 and 138–175.
R. Routley, Dialectical logic, semantics and metamathematics, Erkenntnis, forthcoming 1978.
R. K. Meyer and N. D. Belnap, Jr., A Boolean-valued semantics for R, unpublished; typescript, Canberra, 1976.
R. Carnap, Introduction to Semantics, Harvard University Press, Cambridge, Mass., 1942.
R. Carnap, Logical Foundations of Probability, Second Edition, University of Chicago Press, Chicago, 1962.
E. Mendelson, Introduction to Mathematical Logic, Van Nostrand, Princeton, 1964.
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Routley, R. Alternative semantics for quantified first degree relevant logic. Stud Logica 38, 211–231 (1979). https://doi.org/10.1007/BF00370443
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DOI: https://doi.org/10.1007/BF00370443