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Fregean Quantification Theory

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Abstract

Frege’s system of first-order logic is presented in a contemporary framework. The system described is distinguished by economy of expression and an unusual syntax.

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Notes

  1. In contemporary logic, of course, it is standard that the negation of A is definable as A  ⊃  ⊥ (i. e., in the present notation, condAF). The referee has pointed out that the definition of F as (x 1)x 1 is already given by Church [1], p. 9, as I had forgotten. (One [inessential] difference is that for Church, the two truth-values form a type of their own, and are the range of the universal quantifier, D.)

  2. The referee has emphasized that the notation is Polish and that parentheses are not needed as a bracketing device. (The use of parentheses in the universal quantifier is different and of course eliminable by using another notation.) The Polish character of the notation is especially a byproduct of the fact that the conditional, naturally written as cond(t 1,t 2), is here just another special function on the domain, written like any other such function, namely, cond t 1,t 2.

    I shall therefore omit parentheses in the initial official clauses defining the syntax and semantics. Later, I shall freely use them when I think they improve readability.

    Of course, Frege’s own original notation also did not require bracketing but has never been used again because of its peculiar two-dimensional character. (I am not personally fond of Polish notation, in spite of its advantages for certain purposes.)

  3. We have differed from Frege in other ways. Following much (but not all) modern usage, we don’t distinguish in style of letters between free and bound variables, nor, more importantly, do we place restrictions on the occurrence of free and bound variables. (I personally think that there is something to be said for some of these restrictions. Restrictions of this sort are in Hilbert and Ackermann [2] as well as in Frege.)

    Another difference from Frege is the use of variables over an arbitrary domain D, albeit with two distinguished elements. Frege wanted his first-order variables to range over all objects. We haven’t done this here, following contemporary conceptions (why courses-of-values are not included in the domains is explained in the text. Also, we don’t define the definite description operator in terms of the course-of-values function for the reasons given.)

References

  1. Church, A. (1951). A formulation of the logic of sense and denotation. In P. Henle, H. M. Kallen, & S. K. Langer (Eds.), Structure, methods, and meaning: essays in honor of Henry M. Sheffer. New York: The Liberal Arts Press.

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  2. Hilbert, D. & Ackermann, W. (1938). Grundzüge der theoretischen Logik, second edition. Berlin: Springer. (First edition, 1928.) Translated as Principles of Mathematical Logic, R. E. Luce (ed.) and L. Hammond, G. Leckie, and F. Standhardt (trans.). New York: Chelsea Publishing Company, 1958.

  3. Kripke, S. (2008). Frege’s theory of sense and reference: some exegetical notes. Theoria, 74: 181–218. Reprinted in Philosophical Troubles. Collected Papers Volume I. New York: Oxford University Press.

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  4. Parsons, T. (1987). On the consistency of the first-order portion of Frege’s logical system. Notre Dame Journal of Formal Logic, 28, 161–68.

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Acknowledgments

I would like to thank Kevin Klement and the referee for this journal for helpful comments, and Gary Ostertag and Romina Padró for their help in producing this paper. This paper has been completed with support from the Saul A. Kripke Center at The Graduate Center of The City University of New York.

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Correspondence to Saul A. Kripke.

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Kripke, S.A. Fregean Quantification Theory. J Philos Logic 43, 879–881 (2014). https://doi.org/10.1007/s10992-013-9299-x

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