Categories of First-Order Quantifiers

• Urszula Wybraniec-Skardowska
Chapter
Part of the Studies in Universal Logic book series (SUL)

Abstract

One well known problem regarding quantifiers, in particular the 1st-order quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functor-function defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility.

Keywords

1st-order quantifiers Categorial languages Syntactic categories Denotation Ontological categories Denotational semantics Compositionality Categorial compatibility

Mathematics Subject Classification (2000)

Primary 03C07 03H05; Secondary 03A99

References

1. 1.
Ajdukiewicz, K.: Die syntaktische Konnexität. Stud. Philos. 1, 1–27 (1935) [English translation: Syntactic connection. In: McCall, S. (ed.) Polish Logic 1920–1939, pp. 202–231. Clarendon Press, Oxford (1967)]Google Scholar
2. 2.
Ajdukiewicz, K.: Zwi ązki składniowe miedzy członami zdań oznajmuj ących [Syntactical relations between constituents of declarative sentences]. Stud. Filozoficzne 6(21), 73–86 (1960) [First presented in English at the International Linguistic Symposium in Erfurt, 27 September–2 October 1958]Google Scholar
3. 3.
Bocheński, J.M.: On the syntactical categories. New Scholasticism 23, 257–280 (1949)
4. 4.
Cresswell, M.J.: Logics and Languages. Mathuen and Co. Ltd, London (1973)Google Scholar
5. 5.
Curry, H.B.: Grundlagen der kombinatorischen Logik. Am. J. Math. 52, 509–536, 789–834 (1930)
6. 6.
Curry, H.B.: Some aspects of grammatical structure. In: Jakobson, R. (ed.) Structure of Language and Its Mathematical Aspects, vol. 12, pp. 57–68. AMS, Providence, RI (1961)Google Scholar
7. 7.
Curry, H.B., Feys, R.: Combinatory Logic, vol. 1. North Holland, Amsterdam (1958)
8. 8.
Frege, G.: Begriffschrift, eine der arithmetischen nachbildete Formalsprache des reinen Denkens. Halle (1879) [English translation in: Geach, P.T., Black, M. (eds.) Translations from the Philosophical Writings of Gottlob Frege. Blackwell, Oxford (1970)]Google Scholar
9. 9.
Frege, G.: Die Grudlagen der Arithmetik: eine logische-mathematische Untersuchungen über den Begriff der Zahl. Breslau (1884) [English translation: The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number. Blackwell, Oxford (1950)]Google Scholar
10. 10.
Frege, G.: Über Sinn und Bedeutung. Zeitschrift für Philosophie und pilosophishe Kritik 100, 25–50 (1892) [English translation in: Feigel, H., Sellars, W. (eds.) Readings in Philosophical Analysis. Appleton-Century-Crofts, New York (1949), and also In: Beaney, B. (ed.) The Frege Reader, pp. 151–171. Blackwell, Oxford (1997)]Google Scholar
11. 11.
Hintikka, J.: Principles of Mathematics Revisited. Cambridge University Press, Cambridge (1996)Google Scholar
12. 12.
Husserl, E.: Logische Untersuchungen. Vol. I, Halle (1900), Vol. II, Halle (1901)Google Scholar
13. 13.
Leśniewski, S.: Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundam. Math. 14, 1–81 (1929)
14. 14.
Leśniewski, S.: Über die Grundlagen der Ontologie. Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe II 23, 111–132 (1930)Google Scholar
15. 15.
Lindström, P.: First-order predicate logic with generalized quantifiers. Theoria 32, 186–195 (1966)Google Scholar
16. 16.
Montague, R.: Universal grammar. Theoria 36, 373–398 (1970)
17. 17.
Montague, R.: Formal Philosophy: Selected Papers of Richard Montague. Yale University Press, New Haven, CT (1974). Ed. and introd. R.H. ThomasonGoogle Scholar
18. 18.
Mostowski, A.: On generalization of quantifiers. Fundam. Math. 44, 12–36 (1957)Google Scholar
19. 19.
Nowaczyk, A.: Categorial languages and variable-binding operators. Stud. Logica 37, 27–39 (1978)
20. 20.
Simons, P.: Combinators and categorial grammar. Notre Dame J. Form. Log. 30 (2), 241–261 (1989)
21. 21.
Simons, P.: Languages with variable-binding operators: categorial syntax and combinatorial semantics. In: Jadacki, J., Paśniczek, J. (eds.) The Lvov-Warsaw School—The New Generation. Poznań Studies in the Philosophy of Sciences and Humanities, vol. 89, pp. 239–268. Rodopi, Amsterdam/New York (2006)Google Scholar
22. 22.
Suszko, R.: Syntactic structure and semantical reference, Part I. Stud. Logica 8, 213–144 (1958)Google Scholar
23. 23.
Suszko, R.: Syntactic structure and semantical reference, Part II. Stud. Logica 9, 63–93 (1960)Google Scholar
24. 24.
Suszko, R.: O kategoriach syntaktycznych i denotacjach wyrażeń w jezykach sformalizowanych [On syntactic categories and denotation of expressions in formalized languages]. In: Rozprawy logiczne [Logical Dissertations to the Memory of Kazimierz Ajdukiewicz], pp. 193–204. PWN, Warsaw (1964)Google Scholar
25. 25.
Suszko, R.: Ontology in the tractatus of L. Wittgenstein. Notre Dame J. Form. Log. 9, 7–33 (1968)
26. 26.
Tarski, A.: The semantic notion of truth: and the foundations of semantics. Philos. Phenomenol. Res. 4(3), 341–376 (1944)Google Scholar
27. 27.
van Benthem, J.: Essays in Logical Semantics. Reidel, Dordrecht (1986)Google Scholar
28. 28.
van Benthem, J.: Quantifiers in the world of types. In: van der Does, J., van Eijck, J. (eds.) Quantifiers, Logic and Language, pp. 47–61. Stanford University, Stanford, CA (1996)Google Scholar
29. 29.
van Benthem, J., Westerståhl, D.: Directions in generalized quantifier theory. Stud. Logica 53, 389–419 (1995)Google Scholar
30. 30.
Wybraniec-Skardowska, U.: Theory of Language Syntax. Categorial Approach. Kluwer Academic, Dordrecht/Boston/London (1991)
31. 31.
Wybraniec-Skardowska, U.: Logical and philosophical ideas in certain approaches to language. Synthese 116(2), 231–277 (1998)
32. 32.
Wybraniec-Skardowska, U.: On denotations of quantifiers. In: Omyła, M. (ed.) Logical Ideas of Roman Suszko, pp. 89–119. Faculty of Philosophy and Sociology of Warsaw University, Warsaw (2001)Google Scholar
33. 33.
Wybraniec-Skardowska, U.: On the formalization of classical categorial grammar. In: Jadacki, J., Paśniczek, J. (eds.) The Lvov-Warsaw School—The New Generation. Poznań Studies in the Philosophy of Sciences and Humanities, vol. 89, pp. 269–288. Rodopi, Amsterdam/New York (2006)Google Scholar
34. 34.
Wybraniec-Skardowska, U.: On language adequacy. Stud. Log. Grammar Rhetor. 40 (53), 257–292 (2015)Google Scholar
35. 35.
Wybraniec-Skardowska, U.: Categories of first order quantifiers. Bull. Symb. Log. 22(3), 427–428 (2015).
36. 36.
Wybraniec-Skardowska, U.: Categorial compatibility of the 1st-order quantifiers. Non Class. Log. Theory Appl. 8, 134–140 (2016)Google Scholar