Abstract
There are good reasons to want logics, including many-valued logics, to have usable conditionals, and we have explored this in certain logics. However, it turns out that we “accidentally” chose some favourable logics. In this paper, we look at some of the unfavourable logics and describe where usable conditionals can be added and where it is not possible.
Second Reader
L. Humberstone
Monash University
Dedication Alasdair Urquhart has dealt with many-valued logics at a number of places in his illustrious career, starting at least as early as his 1971 “Interpretation of many-valued logic” (Urquhart 1971). Possibly the best known of these places is his 2001 “Basic Many-Valued Logic” entry in the 2nd Edition of the Handbook of Philosophical Logic (Urquhart 2001). Alasdair’s 1973 dissertation was The Semantics of Entailment and his supervisors were Nuel D. Belnap and Alan Ross Anderson. We hope that our discussion of the Anderson-Belnap logic FDE brings back some happy thoughts from those early, heady days.
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Notes
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- 2.
Nelson was a Ph.D. student of Kleene, and his dissertation was devoted to working out details in Kleene’s realizability interpretation of Intuitionistic arithmetic: cf. his Nelson (1947).
- 3.
Being substitution equivalent is not (necessarily) a logical truth about the formulas: two formulas can be substitution equivalent on one assignment of values to the variables but not on others, so there is an interesting sense in which the truth of one formula can imply the substitution equivalence of two others.
- 4.
Abbreviating it as \(\vee _{cmi}\) would be in kee** with our \(\rightarrow _{cmi}\), but the \(\ddot{\vee }\) is established in the literature.
References
Anderson, A., & Belnap, N. (1975). Entailment: The Logic of Relevance and Necessity (Vol. I). Princeton, NJ: Princeton UP.
Belnap, N. (1992). A useful four-valued logic: How a computer should think. In A. Anderson, N. Belnap, & J. Dunn, (Eds.), Entailment: The logic of relevance and necessity, Volume II (pp. 506–541). Princeton UP, Princeton. First appeared as “A Useful Four-valued Logic” Modern uses of multiple-valued logic J.M. Dunn & G. Epstein (Eds.) (pp. 3–37); Dordrecht: D. Reidel, 1977; and “How a Computer Should Think” Contemporary aspects of philosophy G. Ryle (Ed.) (pp. 30–56); Oriel Press, 1977.
Carnap, R. (1943). Formalization of Logic. Cambridge, MA: Harvard University Press.
Dummett, M. (1978). Truth and Other Enigmas. Cambridge, MA: Harvard University Press.
Gibbins, P. (1987). Particles and Paradoxes: The Limits of Quantum Logic. Cambridge, UK: Cambridge UP.
Goldblatt, R. (1974). Semantic analysis of orthologic. Journal of Philosophical Logic, 3, 19–35.
Hazen, A. P. and Pelletier, F. J. (2018a). Pecularities of some three- and four-valued second order logics. Logica Universalis, 12:493–509.
Hazen, A. P. and Pelletier, F. J. (2018b). Second-order logic of paradox. Notre Dame Journal of Formal Logic, 59:547–558.
Hazen, A. P. and Pelletier, F. J. (2019). K3, Ł3, RM3, A3, FDE, M: How to make many-valued logics work for you. In Omori, H. and Wansing, H., editors, New Essays on Belnap-Dunn Logic, pages 201–235. Springer, Berlin.
Humberstone, L. (2011). The Connectives. Cambridge, MA: MIT Press.
Kleene, S. (1952). Introduction to Metamathematics. Amsterdam: North-Holland.
Nelson, D. (1947). Recursive functions and intuitionistic number theory. Transactions of the American Mathematical Society, 61, 307–368. See Errata, ibid., p. 556.
Nelson, D. (1949). Constructible falsity. Journal of Symbolic Logic, 14:16–26.
Nelson, D. (1959). Negation and separation of concepts in constructive systems. In A. Heyting (Ed.), Constructivity in mathematics: Proceedings of the colloquium held at Amsterdam, 1957 (pp. 208–225). Amsterdam: North Holland.
Omori, H. and Wansing, H. (2017). 40 years of FDE: An introductory overview. Studia Logica, 105:1021–1049.
Priest, G. (1997). Inconsistent models of arithmetic, I: Finite models. Journal of Philosophical Logic, 223–235.
Priest, G. (2000). Inconsistent models of arithmetic, II: The general case. Journal of Symbolic Logic, 65:1519–1529.
Priest, G. (2006). In Contradiction: A Study of the Transconsistent (2nd ed.). Oxford: Oxford University Press.
Spinks, M., & Veroff, R. (2018). Paraconsistent constructive logic with strong negation as a contraction-free relevant logic. In J. Czelakowski (Ed.), Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science (pp. 323–379). Berlin: Springer.
Sutcliffe, G., Pelletier, F. J., & Hazen, A. P. (2018). Making Belnap’s ‘useful 4-valued logic’ useful. In Proceedings of the thirty-first international Florida artificial intelligence research society conference (FLAIRS-31). Association for the advancement of artificial intelligence.
Tedder, A. (2014). Paraconsistent logic for dialethic arithmetics. Master’s thesis, University of Alberta, Philosophy Department, Edmonton, Alberta, Canada. Available at https://www.library.ualberta.ca/catalog/6796277.
Tedder, A. (2015). Axioms for finite collapse models of arithmetic. Review of Symbolic Logic, 8:529–539.
Urquhart, A. (1971). An interpretation of many-valued logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 19:111–114.
Urquhart, A. (2001). Basic many-valued logic. In F. Guenthner & D. Gabbay (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 2, pp. 249–294). Dordrecht: Kluwer.
van Fraassen, B. (1966). Singular terms, truth-value gaps, and free logic. Journal of Philosophy, 63:481–495.
van Fraassen, B. (1969). Presuppositions, supervaluations and free logic. In K. Lambert (Ed.), The Logical Way of Doing Things (pp. 67–92). New Haven: Yale UP.
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We gratefully acknowledge the very helpful (and learned!) comments of the second reader.
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Hazen, A.P., Pelletier, F.J. (2022). Some Lessons Learned About Adding Conditionals to Certain Many-Valued Logics. In: Düntsch, I., Mares, E. (eds) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Outstanding Contributions to Logic, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-71430-7_21
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