Abstract
My ultimate aim in this paper* is to show that conventional wisdom is seriously wrong when it comes to logic. I want to challenge the traditional ideas as to what the most basic part of logic is like and how it ought to be studied. What is more, I will actually prove that these widely accepted ideas are mistaken.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
The gradual development of the conception of first-order logic is an intricate subject of which the final truth has not yet been told. Meanwhile, you can have a glimpse of the problems from studies like Gregory H. Moore, “The Emergence of First-Order Logic”, in William Aspray and Philip Kitcher, editors, History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science, vol. 11, University of Minnesota Press, Minneapolis, 1988, pp.95–135). Moore’s paper is to be read with caution, however, for he is unaware of some of the most important conceptual points concerning the idea of first-order logic. For one thing, he does not even mention Henkin’s distinction between standard and non-standard interpretations of higher-order logic. Yet it is possible to reconstruct a higher-order language, with a suitable non-standard interpretation, as a many-sorted “first-order” language, as far as logic is concerned. Cf. also below, especially sec. 16.
Cf., e.g., Stewart Shapiro, “Second-order Languages and Mathematical Practice”, Journal of Symbolic Logic 50 (1985), pp. 714–42; Georg Kreisel, “Informal Rigor and Completeness Proofs”, in Imre Lakatos, editor, Problems in the Philosophy of Mathematics, North-Holland, Amsterdam, 1967, pp. 138–86.
See here Jaakko Hintikka, “Logical Form and Linguistic Theory”, in Alex George, editor, Reflections on Chomsky, Basil Blackwell, Oxford, 1989, pp. 41–57.
Norbert Hornstein, Logic as Grammar, The MIT Press, Cambridge, 1984; cf. Robert May, Logical Form, The MIT Press, Cambridge, 1985.
See note 3 above and also Jaakko Hintikka and Gabriel Sandu, On the Methodology of Linguistics: A Case Study, Basil Blackwell, Oxford, 1991.
See Jaakko Hintikka and Jack Kulas, Anaphora and Definite Descriptions, D. Reidel, Dordrecht, 1985.
Cf. statements like the following: “Among the many branches of modern mathematics set theory occupies a unique place: with rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.” (See Patrick C. Suppes, Axiomatic Set Theory,Dover, New York, 1972, p. l.)
The way in which first-order logic came to be considered the right foundation for set theory is described in Gregory H. Moore, op. cit.,note 1 above.
See, e.g., S. MacLane, Categories for the Working Mathematician,Springer-Verlag, Berlin, 1971; Jon Barwise, editor, Handbook of Mathematical Logic,North-Holland, Amsterdam, 1977, chapters A8 and D6.
See, e.g., Jean Dieudonné, A Panorama of Pure Mathematics, As seen by N. Bourbaki,Academic Press, New York, 1982 (with further references to the literature).
Warren Goldfarb, “Logic in the Twenties: the Nature of the Quantifier”, Journal of Symbolic Logic 44 (1979), pp. 351–68.
This line of thought actually gives you a general method of deciding as to when a first-order formula with informationally independent quantifiers reduces to the linear form.
Cf. here the brief bibliography of Jaakko Hintikka and Jack Kulas, The Game of Language, D. Reidel, Dordrecht, 1983, pp. 300–3.
For distributive normal forms, see Jaakko Hintikka, Distribute Normal Forms (Acta Philosophica Fennica, vol. 6), Societas Philosophica Fennica, Helsinki, 1953. A good exposition is also contained in Veikko Rantala, Aspects of Definability (Acta Philosophica Fennica, vol. 29, nos. 2–3), Societas Philosophica Fennica, Helsinki, 1977.
See Gabriel Sandu and Jonko Väänänen, “Partially Ordered Connectives”, Zeitschrift für mathematische Logik and Grundlagen der Mathematik 38 (1992), pp. 361–372; Gabriel Sandu, “On the Logic of Informational Independence”, Journal of Philosophical Logic 22 (1993), pp. 29–60.
See the papers referred to in note 15.
According to the story (I was not an eyewitness), Richard Montague was once criticized on the grounds that his universal grammar did not explain the properties of actual languages (humanly possible languages) as distinguished from logical languages and from computer languages. “Why is it that we speak English and not ALGOL?” — Montague’s sincere reply was, “Historical accident.”
Jaakko Hintikka, “Is Scope a Viable Concept in Semantics?”, in ESCOL ‘86: Proceedings of the Third Eastern States Conference on Linguistics, ESCOL, Columbus, OH, 1987, pp. 259–70.
This is in effect the way quantifier scopes are explained in textbooks of logic. In Hintikka and Kulas (note 6 above), it is shown that it embodies a seriously mistaken view of the way pronouns and quantifiers operate in natural languages.
This point is argued in Hintikka and Kulas, note 6 above, especially pp. 137–8.
The information set of a move may be any subset of the set of earlier moves. There is hence no reasons in general why the information sets of successive moves should be ordered even partially.
Bach—Peters sentences were first introduced as prima facie counterexamples to certain types of generative grammars. Independently of this original use of theirs, they can be employed also as counterexamples of certain types of rules for translating from natural languages to a logical notation.
Cf. here Jaakko Hintikka, “Reductions in the Theory of Types”, Acta Philosophica Fennica 8 (1955) pp. 57–115.
H. B. Enderton, “Finite Partially-Ordered Quantifiers”, Zeitschrift für mathematische Logik and Grundlagen der Mathematik 16 (1970), pp. 393–7.
w. V. Quine, Philosophy of Logic, Prentice-Hall, Englewood Cliffs, N.J., 1970, p. 91.
This task of logical theory has not been given its proper due. It is perhaps considered most often in a context of an “experimentalist” and antirealist view of mathematics. But, as Gödel’s example shows, it can be combined with a realist and even Platonist attitude to mathematical truth. Indeed, semantical (and in that sense “realistic”) considerations are especially well suited to sharpen our ideas about possible stronger principles.
These methods go back to Alfred Tarski, “The Concept of Truth in Formalized Languages”, in A. Tarski, Logic, Semantics, Metamathematics,Clarendon Press, Oxford, 1956, ch. 8. (The Polish original was drafted in 1930 and published in 1933; the German version appeared in 1935.)
Cf. here Barbara Partee, “Compositionality”, in F. Landtman and F. Veltman, editors, Varieties of Formal Semantics,Foris, Dordrecht, 1984, pp. 281–313; Barbara H. Partee, Alice ter Meulen and Robert E. Wall, Mathematical Methods in Linguistics,Kluwer, Dordrecht, 1990, section 13.1 (pp. 317–38).
See here Jaakko Hintikka, “Theories of Truth and Learnable Languages”, ch. 10 in Hintikka and Kulas (1983), note 13 above, pp. 259–92.
Cf. here Jon Barwise, “On Branching Quantifiers in English”, Journal of Philosophical Logic 8 (1979), pp. 47–80.
See here Jaakko Hintikka and Gabriel Sandu, “Informational Independence as a Semantical Phenomenon”, in J. E. Fenstad et al.,editors, Logic, Methodology and Philosophy of Science VIII,Elsevier, Amsterdam, 1989, pp. 571–89; Gabriel Sandu, “On the Logic of Informational Independence”, Journal of Philosophical Logic 22 (1993) pp. 29–60.
See here Jaakko Hintikka, “Paradigms for Language Theory”, Acta Philosophica Fennica vol. 49 (1990), pp. 181–209.
Cf. W. V. Quine, Ontological Relativity,Columbia U.P., New York, 1969, pp. 63–7, 104–8; Ruth Marcus, “Modalities and Intensional Languages”, Synthese 13 (1961), pp. 303–22; Saul Kripke, “Speaker’s Reference and Semantic Reference”, in P. A. French et al.,editors, Midwest Studies in Philosophy vol. 2, University of Minnesota Press, Morris, MI, 1977, pp. 255–76; D. Grover, J. Camp and N. Belnap, “A Presentential Theory of Truth”, Philosophical Studies 27 (1975), pp. 73–125; T. Baldwin, “Can There Be a Substantive Theory of Truth?” Récherche sur la philosophie et le langage 10,Université des Sciences Sociales de Grenoble, Grenoble, 1989. Cf. here also D. Gottlieb, Ontological Economy: Substitutional Quantification and Mathematics,Oxford University Press, Oxford, 1980.
See here Hintikka and Kulas (1983), note 13 above; Esa Saarinen, editor, Game-Theoretical Semantics, D. Reidel, Dordrecht, 1979.
Cf. Jaakko Hintikka, “On the Development of the Model-Theoretic Viewpoint in Logical Theory”, Synthese 77 (1988), pp. 1–36.
As such, they are systematizations and further developments of certain types of “language-games” in the sense of Ludwig Wittgenstein. (For them, see Merrill B. Hintikka and Jaakko Hintikka, Investigating Wittgenstein,Basil Blackwell, Oxford, 1986.) The failure of the soi-disant Wittgensteinians to put his important and promising concept to systematic use is but one example of their failure (or refusal) to recognize the constructive potentialities of Wittgenstein’s ideas.
There might seem to be an alternative to (R. false) namely to define S to be false iff it is not true. This would make it impossible to deal with the falsity of propositions directly by means of game rules, however. Below, it will be seen what other consequences the choice between the two conceptions of falsity has; see especially sec. 15.
See Kurt Gödel, “On a Hitherto Unexploited Extension of the Finitary Standpoint”, Journal of Philosophical Logic 9 (1980), pp. 133–42. (Translation of Gödel’s original 1956 article in Dialectica 12, pp. 280–7, with a bibliography.) See also Kurt Gödel, Collected Works vol. 2 (edited by Solomon Feferman et al.), Oxford University Press, Oxford, 1990, pp. 217–53.
This is the leading idea of the applications of game-theoretical semantics to natural languages. For examples of this strategy, see Hintikka and Kulas (1985), note 6 above.
This is the gist of the semantics of model logics originally developed by Kanger, Montague and Hintikka. However, necessity-type operators do not interact with ordinary quantifiers quite in the same way they interact among themselves. The reason is that each possible alternative world (i.e., each value of a necessity-type “universal quantifier”) has a domain of individuals (i.e., range of values of ordinary quantifiers) of its own, possibly different from others. This fact has important consequences. For instance, not all types of inter-and independencies can any longer be reduced to partial orderings.
See note 38 above.
Dana Scott, “A Game-Theoretical Interpretation of Logical Formulae”, McCarthy Seminar, Stanford, 1967, published in the Yearbook 1991 of Kurt Godel Society, Vienna, 1993, pp. 47–48.
See Hintikka and Kulas, note 6 above, especially chapter 2.
Op. cit.,especially pp. 172–9.
Op. cit.,especially pp. 57, 159.
See note 31 above.
In other words, the slash simply temporarily exempts a quantifier, connective or some other expression from the scope of another quantifier (or similar) within whose scope it would otherwise be.
Cf. note 15 above.
For instance, different constructions in terms of knows cannot be analyzed without independence-friendly logic. Cf. here Jaakko Hintikka, “Different Constructions in Terms of `Knows’ ”, in Jonathan Dancy and Ernest Sosa“, editors, Companion to Epistemology, Basil Blackwell, Oxford, 1992, pp. 99–104.
What is true of subordinate questions with knows is true of questions in general (cf. note 49): independence-friendly logic is needed to cope with their logic and semantics.
For instance, (23) is the logical form of such wh-questions in natural languages as have an outside universal quantifier, e.g.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Hintikka, J. (1995). What is Elementary Logic? Independence-Friendly Logic as the True Core Area of Logic. In: Gavroglu, K., Stachel, J., Wartofsky, M.W. (eds) Physics, Philosophy, and the Scientific Community. Boston Studies in the Philosophy of Science, vol 163. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2658-0_18
Download citation
DOI: https://doi.org/10.1007/978-94-017-2658-0_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4436-5
Online ISBN: 978-94-017-2658-0
eBook Packages: Springer Book Archive