Abstract
We present an algebraic-style of semantics, which we call a “content semantics”, for quantified relevant logics based on the weak system BBQ. We show soundness and completeness for all quantificational logics extending BBQ and also treat reduced modelling for all systems containing BB d Q. The key idea of content semantics is that true entailments A→B are represented under interpretation I as content containments, i.e. I(A)⩽I(B) (or, the content of A contains that of B). This is opposed to the truth-functional way which represents true entailments as truth-preservations over all set-ups (or worlds), i.e. (VaεK) (if I(A, a) = T then I(B, a)= T).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. R. Anderson and N. D. Belnap, Jr., Entailment, The Logic of Relevance and Necessity, Princeton U.P., 1975.
R. T. Brady, The simple consistency of a set theory based on the logic CSQ, Notre Dame Journal of Formal Logic, Vol. 24 (1983), pp. 431–449.
R. T. Brady, Depth relevance of some paraconsistent logics, Studia Logica, Vol. 43 (1984), pp. 63–73.
R. T. Brady, Natural deduction systems for some quantified relevant logics, Logique et Analyse, 27 Anńee, 1984, pp. 355–377.
R. T. Brady, The non-triviality of dialectical set theory, in G. Priest, R. Routley and J. Norman (eds.), Paraconsistent Logic, Essays on the Inconsistent, Philosophia Verlag, forthcoming in 1989.
R. T. Brady, Universal Logic, in preparation.
J. M. Dunn, The Algebra of Intensional Logics, Ph.D. Thesis, Uni. of Pittsburgh, 1966.
K. Fine, Models for entailment, Journal of Philosophical Logic, Vol. 3 (1974), pp. 347–372.
K. Fine, Incompleteness of quantified relevance logics, in R. Sylvan (ed.), Directions in Relevant Logics, forthcoming in 1988.
K. Fine, Semantics for quantified relevance logic, typescript, 1985.
D. Gabbay, On 2nd order intuitionistic propositional calculus with full comprehension, Archiv für Mathematische Logik, Vol. 16 (1974), pp. 177–186.
P. Halmos, Algebraic Logic, Chelsea, New York, 1962.
L. Henkin, The completeness of the first-order functional calculus, in J. Hintikka (ed.), The Philosophy of Mathematics, Oxford U.P., 1969, pp. 51–63.
L. Henkin, J. D. Monk and A. Tarski, Cylindrical Algebras, North-Holland, 1971.
P. S. Lavers, Generating Intensional Logics, M. A. Thesis, University of Adelaide, 1985.
R. K. Meyer and R. Routley, Algebraic analysis of Entailment I, Logique et Analyse, Vol. 15 (1972), pp. 407–428.
R. K. Meyer and R. Routley, Algebraic analysis of Entailment II, typescript, 1973.
R. K. Meyer, J. M. Dunn and H. Leblanc, Completeness of relevant quantification theories, Notre Dame Journal of Formal Logic, Vol. 15 (1974), pp. 97–121.
G. Priest, Sense, Entailment and Modus Porens, Journal of Philosophical Logic, Vol. 9 (1980), pp. 415–435.
H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics, Państwowe Wydawnictwo Naukowe, Warsaw, 1963.
R. Routley and R. K. Meyer, The semantics of entailment, in H. Leblanc (ed.), Truth, Syntax and Modality, North-Holland, 1973, pp. 199–243.
R. Routley, R. K. Meyer, V. Plumwood and R. T. Brady, Relevant Logics and their Rivals, Vol. 1, Ridgeview, California, 1982.
Author information
Authors and Affiliations
Additional information
This paper was presented to the Australasian Association for Logic Conference, held at the University of Auckland from 9–12th July, 1986. I wish to thank those present for some helpful comments. I also wish to thank Kit Fine for some useful discussion on some topics of this paper.
Rights and permissions
About this article
Cite this article
Brady, R.T. A content semantics for quantified relevant logics. I. Stud Logica 47, 111–127 (1988). https://doi.org/10.1007/BF00370286
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00370286