Abstract
In §93 of The Principles of Mathematics, Bertrand Russell (1903) observes that “the variable is a very complicated logical entity, by no means easy to analyze correctly”. This assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are constructed out of variable assignments. But this has disquieting consequences, including Fine’s antinomy of the variable and an undue dependence of meanings on language (representationalism). Here I develop an alternative, Fregean version of predicate logic that uses the traditional quantifier–variable apparatus for the expression of generality, possesses a fully compositional, non-representational semantics, and is not subject to the antinomy of the variable. The advantages of Fregean over Tarskian predicate logic are due to the former’s treating variables not as meaningful lexical items, but as mere marks of punctuation, similar to parentheses. I submit that this is indeed how the variables of predicate logic should be construed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74, 47–73.
Bourbaki, N. (1954). Éléments de Mathématique: Théorie des Ensembles. Paris: Hermann.
Chomsky, N. (1982). Some concepts and consequences of the theory of government and binding. Cambridge, MA: MIT Press.
Chomsky, N. (1995). The minimalist program. Cambridge, MA: MIT Press.
Curry, H., & Feys, R. (1958). Combinatory logic (Vol. I). Amsterdam: North Holland.
Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.
Evans, G. (1977). Pronouns, quantifiers, and relative clauses (I). Canadian Journal of Philosophy, 7(3), 467–536.
Fine, K. (2000). Neutral relations. Philosophical Review, 109(1), 1–33.
Fine, K. (2003). The role of variables. Journal of Philosophy, 100(12), 605–631.
Fine, K. (2007). Semantic relationism. Oxford: Blackwell.
Frege, G. (1893). Grundgesetze der Arithmetik, Volume I, Jena: Hermann Pohle. English translation in P. Ebert and M. Rossberg (editors and translators), Gottlob Frege: Basic Laws of Arithmetic. Oxford: Oxford University Press (2013).
Geurts, B., Beaver, D., & Maier, E. (2016). Discourse representation theory. In E. Zalta (Ed.), Stanford encyclopedia of philosophy (Spring 2016). http://plato.stanford.edu/archives/spr2016/entries/discourse-representation-theory/.
Heim, I., & Kratzer, A. (1998). Semantics in generative grammar. Oxford: Blackwell.
Higginbotham, J. (1983). Logical form, binding, and nominals. Linguistic Inquiry, 14(3), 395–420.
Hilbert, D., & Ackermann, W. (1938). Grundzüge der theoretischen Logik. Berlin: Springer.
Hilbert, D., & Bernays, P. (1934). Grundlagen der Mathematik I. Berlin: Springer.
Humberstone, L. (2000). What \(Fa\) says about \(a\). Dialectica, 54(1), 3–28.
Jacobson, P. (1999). Towards a variable-free semantics. Linguistics and Philosophy, 22, 117–185.
Jacobson, P. (2003). Binding without pronouns (and pronouns without binding). In R. Oehrle & G.-J. Kruiff (Eds.), Resource-sensitivity, binding, and anaphora (pp. 57–96). Dordrecht: Kluwer.
Janssen, T. (1997). Compositionality. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (pp. 417–473). Amsterdam: Elsevier.
Janssen, T. (2011). Compositionality. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (2nd ed., pp. 495–554). Amsterdam: Elsevier.
Janssen, T. (2012). Compositionality: Its historic context. In M. Werning, W. Hinzen, & E. Machery (Eds.), The Oxford handbook of compositionality (pp. 19–46). Oxford/New York: Oxford University Press.
Kaplan, D. (1986). Opacity. In L. E. Hahn & P. A. Schilpp (Eds.), The philosophy of W. V. Quine (pp. 229–289). La Salle: Open Court.
Klein, U., & Sternefeld, W. (2017). Same same but different: An alphabetically innocent compositional predicate logic. Journal of Philosophical Logic, 46(1), 65–95.
Kracht, M. (2011). Lectures on interpreted languages and compositionality. Berlin: Springer.
Kratzer, A. (1995). Stage-level and individual-level predicates as inherent generics. In G. N. Carlson & F. J. Pelletier (Eds.), The generic book (pp. 125–175). Chicago: University of Chicago Press.
Lemmon, E. J. (1965). Beginning logic. Princeton: Van Nostrand.
Marsh, W., & Partee, B. (1987). How non-context free is variable binding? In W. Savitch, et al. (Eds.), The formal complexity of natural language (Studies in linguistics and philosophy 33) (pp. 369–386). Dordrecht: Reidel.
May, R. (1977). The grammar of quantification. Doctoral dissertation, MIT.
May, R. (2006). The invariance of sense. Journal of Philosophy, 103(3), 111–144.
Montague, R. (1973). The proper treatment of quantification in ordinary English. In K. J. J. Hintikka, J. M. E. Moravcsik, & P. Suppes (Eds.), Approaches to natural language (pp. 221–242). Dordrecht: Reidel.
Pagin, P., & Westerstahl, D. (2010). Compositionality I: Definitions and variants. Philosophy Compass, 5(3), 250–264.
Partee, B. (2013). The starring role of quantifiers in the history of formal semantics. In V. Punčochář & P. Švarný (Eds.), The logica yearbook 2012 (pp. 113–136). London: College Publications.
Pickel, B. (2010). Syntax in basic laws §§29–32. Notre Dame Journal of Formal Logic, 51(2), 253–277.
Pickel, B., & Rabern, B. (2016). The antinomy of the variable: A Tarskian resolution. Journal of Philosophy, 113(3), 137–170.
Potts, C. (2002). No vacuous quantification constraints in syntax. In M. Hirotani (Ed.), Proceedings of the North East Linguistic Society (Vol. 32, pp. 451–470).
Quine, W. (1940). Mathematical logic. New York: Norton.
Quine, W. (1960). Variables explained away. Proceedings of the American Philosophical Society, 104, 343–347.
Russell, B. (1903). The principles of mathematics. Cambridge: Cambridge University Press.
Schütte, K. (1977). Proof theory. Berlin: Springer.
Smith, P. (2003). An introduction to formal logic. Cambridge: Cambridge University Press.
Tennant, N. (1978). Natural logic. Edinburgh: Edinburgh University Press.
van Benthem, J. (1987). Logical syntax. Theoretical Linguistics, 14(2–3), 119–142.
Williamson, T. (1985). Converse relations. Philosophical Review, 94(2), 249–262.
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. London: Kegan Paul.
Zimmermann, T. E., & Sternefeld, W. (2013). Introduction to semantics. Berlin: Mouton de Gruyter.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wehmeier, K.F. The proper treatment of variables in predicate logic. Linguist and Philos 41, 209–249 (2018). https://doi.org/10.1007/s10988-017-9224-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10988-017-9224-9