Abstract
The resemblance of the theory of formal consequence first offered by the fourteenth-century logician John Buridan to that later offered by Alfred Tarski has long been remarked upon. But it has not yet been subjected to sustained analysis. In this paper, I provide just such an analysis. I begin by reviewing today’s classical understanding of formal consequence, then highlighting its differences from Tarski’s 1936 account. Following this, I introduce Buridan’s account, detailing its philosophical underpinnings, then its content. This then allows us to separate those aspects of Tarski’s account representing genuine historical advances, unavailable to Buridan, from others merely differing from—and occasionally explicitly rejected by—Buridan’s account.
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Notes
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The treatments of truth in a model and consequence given here are substantially those of Fitting and Mendelsohn 1998. Alternatively, one might let I assign a denotation for variables and introduce x-variants in the definition of quantifiers, or use a number of other approaches. See the various approaches discussed in Garson 2013.
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The proof of this is simple: both the identity and non-identity functions are candidate values for the second-order variable X in X(x, y). Since these partition the class of ordered pairs (i.e. every ordered pair satisfies one or the other of these), there is always some function the variable X may be mapped to to include an ordered pair as its arguments mapping to Trm.
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Alonzo Church had already read Tarski as varying the domain across models in 1956. The first paper to have suggested domain variation was absent in Tarski’s original account appears to have been Corcoran 1972, 43. That Tarski assumed a fixed-domain in his account of formal consequence was then defended at length by Etchemendy 1988, 1990, 2008, and later taken up by Sagüillo 1997, 2009; Corcoran and Sagüillo 2011; Bays 2001; and Mancosu 2006, 2010b. A variable-domain reading of Tarski 2002 was accepted by Sher 1991, 1996; Ray 1996; and in Stroińska and Hitchcock’s introduction to Tarski 2002, each broadly on grounds of interpretive charity. The most sophisticated proponent of a variable-domain interpretation in Tarski’s pre-WWII work is Gómez-Torrente 1996, 2009. Mancosu has summarized the status of the current debate in Mancosu 2010a.
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This same attitude persists in Tarski 1986, where logic is regarded as the most general of the sciences, and logical notions are accordingly identified as those remaining invariant for “all one-one transformations of the space, or universe of discourse, or ‘world’, onto itself” (49).
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Gödel’s original proof only applied to languages strong enough to formulate Peano Arithmetic, and hence including rules for mathematical induction (or their equivalent). Later, Rosser 1936 and others extended Gödel’s incompleteness results, showing they were replicable for all extensions of the weaker system Q, not including induction rules.
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In this, Tarski is following Russell, Ramsey, and Wittgenstein. See Ramsey 1931, 59ff.
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For the influence of Buridan on Prior’s work, see Uckelman 2012.
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This difference in definition hints at a much deeper one. For Buridan, consequences are always individual sentence tokens, i.e. actually written or spoken hypothetical expressions, which are evaluated by determining whether the connections they express hold in all possible situations (including those where the expressions themselves do not exist, and hence are neither true nor false). For Tarski and the modern approach, by contrast, consequences are never actual sentences, both because of the aforementioned abstraction at the level of the models of a sentential function, and because the antecedent Γ of a classical consequence Γ ⊨ ϕ is always at least countably infinite, since it is closed under entailment.
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It is not. See Archambault 2017, 55–60.
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Cf. Burleigh 1955, 66.
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A note on the language of “syncategoremata”: to my knowledge, the phrase “syncategorematic terms” does not occur in Buridan. Terms are those words in which every sentence “bottoms out” (hence the name “term,” i.e. end or limit), and so are just those words against which syncategoremes are divided.
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This is also true on Tarski’s account, though it is not so on the received classical analysis. The basic reason for the latter is the decision to regard the constant symbols as uninterpreted.
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I thank Milo Crimi for bringing this problem to my attention.
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Even if Tarski’s division exacerbates a tendency, already found in Buridan, to prescind from treating the meaning of terms prior to their propositional role.
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- 21.
Cf. Barcan Marcus 1978.
Bibliography
Aquinas, Thomas. 1989. Expositio libri Peryermeneias. Edited by R.A. Gauthier. Translated by Jean T. Oesterle. Leonine. Vols. I*-1. Opera Omnia. Paris: Cerf.
Archambault, Jacob. 2017. The development of the medieval Parisian account of formal consequence. PhD thesis, Fordham University.
Barcan Marcus, Ruth. 1978. Nominalism and the substitutional quantifier. The Monist 61: 351–362.
Baron, Sam. 2015. Tensed truthmaker theory. Erkenntnis 80: 923–944.
Bays, Timothy. 2001. On Tarski on models. Journal of Symbolic Logic 66: 1701–1726.
Benétreau-Dupin, Yann. 2015. Buridan’s solution to the liar paradox. History and Philosophy of Logic 36: 18–28.
Bigelow, John. 1996. Presentism and properties. Philosophical Perspectives 10: 35–52.
Blok, W.J., and Don Pigozzi. 1988. Alfred Tarski’s work on general metamathematics. Journal of Symbolic Logic 53: 36–50.
Bulthuis, Nathaniel. 2016. The motivations for Walter Burley’s theory of the proposition. British Journal for the History of Philosophy 24: 1057–1074.
Buridan, John. 1976. Tractatus de Consequentiis. In Philosophes Médiévaux 16, ed. Hubert Hubien. Louvain: Publications Universitaires.
———. 2001. Summulae de Dialectica. Translated by Gyula Klima. New Haven: Yale University Press.
———. 2013. Summulae de Locis Dialecticis. Edited by Niels Jørgen Green-Pedersen. Turnhout: Brepols.
———. 2015. Treatise on consequences. Translated by Stephen Read. Bronx: Fordham University Press.
Burleigh, Walter. 1955. De puritate artis logicae. Edited by Philotheus Boehner. St Bonaventure: Franciscan Institute.
Cameron, Ross. 2008. How to be a truthmaker maximalist. Noûs 42: 410–421.
———. 2011. Truthmaking for presentists. Oxford Studies in Metaphysics 6: 55–100.
———. 2013. Changing truthmakers: Reply to Tallant and Ingram. Oxford Studies in Metaphysics 8: 362–373.
Chalmers, David. 1999. Is there synonymy in Ockham’s mental language? In The Cambridge companion to Ockham, ed. Paul Vincent Spade, 76–99. Cambridge: Cambridge University Press.
Corcoran, John. 1972. Conceptual structure of classical logic. Philosophy and Phenomenological Research 33: 25–47.
Corcoran, John, and José M. Sagüillo. 2011. The absence of multiple universes of discourse in the 1936 Tarski consequence-definition paper. History and Philosophy of Logic 32: 359–374.
Crisp, Thomas M. 2007. Presentism and the grounding objection. Noûs 41: 90–109.
Dumitriu, Anton. 1974. The logico-mathematical antinomies: Contemporary and scholastic solutions. International Philosophical Quarterly 14: 309–328.
Dutilh Novaes, Catarina. 2005. Buridan’s Consequentia: Consequence and inference within a token-based semantics. History and Philosophy of Logic 26: 277–297.
———. 2011. Lessons on sentential meaning from mediaeval solutions to the liar paradox. The Philosophical Quarterly 61: 58–78.
———. 2012. Form and matter in later Latin medieval logic: The cases of suppositio and consequentia. Journal of the History of Philosophy 50: 339–354.
———. 2020. Medieval theories of consequence. In The Stanford encyclopedia of philosophy. https://plato.stanford.edu/archives/fall2020/entries/consequence-medieval
Etchemendy, John. 1988. Tarski on truth and logical consequence. Journal of Symbolic Logic 53: 51–79.
———. 1990. The concept of logical consequence. Cambridge: Harvard University Press.
———. 2008. Reflections on consequence. In New essays on Tarski and philosophy, ed. Douglas Patterson, 263–299. Oxford: Clarendon Press.
Fitting, Melvin, and Richard L. Mendelsohn. 1998. First-order modal logic. Dordrecht: Kluwer.
Garson, James W. 2013. Modal logic for philosophers. 2nd ed. Cambridge: Cambridge University Press.
Gómez-Torrente, Mario. 1996. Tarski on logical consequence. Notre Dame Journal of Formal Logic 37: 125–151.
———. 2009. Rereading Tarski on logical consequence. Review of Symbolic Logic 2: 249–297.
Jané, Ignacio. 2006. What is Tarski’s common concept of consequence? Bulletin of Symbolic Logic 12: 1–42.
Kemeny, John G. 1956a. A new approach to semantics–part I. Journal of Symbolic Logic 21: 1–27.
———. 1956b. A new approach to semantics–part II. Journal of Symbolic Logic 22: 149–161.
King, Peter. 1985. Jean Buridan’s philosophy of logic: The treatise on supposition, the treatise on consequences. Dordrecht: D. Reidel.
Klima, Gyula. 2004. Consequences of a closed, token-based semantics: The case of John Buridan. History and Philosophy of Logic 25: 95–110.
———. 2008. Logic without truth: Buridan on the liar. In Unity, truth and the liar: The modern relevance of medieval solutions to the liar paradox, ed. S. Rahman, T. Tulenheimo, and E. Genot, 87–112. Berlin: Springer.
Kneale, William, and Martha Kneale. 1962. The development of logic. Oxford: Oxford University Press.
Kretzmann, Norman, Anthony Kenny, and Jan Pinborg, eds. 1982. The Cambridge history of later medieval philosophy. Cambridge: Cambridge University Press.
Mancosu, Paolo. 2006. Tarski on models and logical consequence. In The architecture of modern mathematics, ed. José Ferreirós and Jeremy Gray, 318–470. Oxford: Oxford University Press.
———. 2010a. Fixed- versus variable-domain interpretations of Tarski’s account of logical consequence. Philosophy Compass 5: 745–759.
———. 2010b. The adventure of reason: Interplay between mathematical logic and philosophy of mathematics. Oxford: Oxford University Press.
Moody, E.A. 1952. Truth and consequence in medieval logic. Amsterdam: North-Holland.
Ockham, William. 1974. Summa logicae. In Opera philosophica, ed. Philotheus Boehner, Gedeon Gàl, and Stephen Brown, vol. Vol. 1. St. Bonaventure: Franciscan Institute.
Parsons, Terence. 2014. Articulating medieval logic. Oxford: Oxford University Press.
Ramsey, Frank P. 1931. The foundations of mathematics. In The foundations of mathematics and other essays, ed. R.B. Braithwaite, 1–61. London: Kegan Paul.
Ray, Greg. 1996. Logical consequence: A defence of Tarski. Journal of Philosophical Logic 25: 617–677.
Read, Stephen. 2007. William of Ockham’s The sum of logic. Topoi 26: 271–277.
Rosser, Barkley. 1936. Extensions of some theorems of Gödel and Church. Journal of Symbolic Logic 1: 87–91.
Sagüillo, José M. 1997. Logical consequence revisited. Bulletin of Symbolic Logic 3: 216–241.
———. 2009. Methodological practice and complementary concepts of logical consequence: Tarski’s model-theoretic consequence and Corcoran’s information-theoretic consequence. History and Philosophy of Logic 30: 21–48.
Schiemer, Georg, and Erich H. Reck. 2013. Logic in the 1930s: Type theory and model theory. Bulletin of Symbolic Logic 19: 433–472.
Sher, Gila. 1991. The bounds of logic. Cambridge: MIT Press.
———. 1996. Did Tarski commit Tarski’s fallacy? Journal of Symbolic Logic 61: 653–686.
Spade, Paul Vincent. 1980. Synonymy and equivocation in Ockham’s mental language. Journal of the History of Philosophy 18: 9–22.
Tarski, Alfred. 1941. Introduction to logic and to the methodology of deductive sciences. Oxford: Oxford University Press.
———. 1953. A general method in proofs of undecidability. In Undecidable theories, 1–35. Amsterdam: North-Holland.
———. 1986. What are logical notions? History and Philosophy of Logic 7: 143–154.
———. 2002. On the concept of following logically. Trans. Magda Stroińska and David Hitchcock. History and Philosophy of Logic 23: 155–196.
Trentman, John. 1970. Ockham on mental. Mind 79: 586–590.
Uckelman, Sara L. 2012. Arthur prior and medieval logic. Synthese 188: 349–366.
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Archambault, J. (2023). The Semantic Account of Formal Consequence, from Alfred Tarski Back to John Buridan. In: Hochschild, J.P., Nevitt, T.C., Wood, A., Borbély, G. (eds) Metaphysics Through Semantics: The Philosophical Recovery of the Medieval Mind. International Archives of the History of Ideas Archives internationales d'histoire des idées, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-031-15026-5_15
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