Abstract
There is a long-standing debate whether propositions, sentences, statements or utterances provide an answer to the question of what objects logical formulas stand for. Based on the traditional understanding of logic as a science of valid arguments, this question is firstly framed more exactly, making explicit that it calls not only for identifying some class of objects, but also for explaining their relationship to ordinary language utterances. It is then argued that there are strong arguments against the proposals commonly put forward in the debate. The core of the problem is that an informative account of the objects formulas stand for presupposes a theory of formalization; that is, a theory that explains what formulas may adequately substitute for an inference in proofs of validity. Although such theories are still subject to research, some consequences can be drawn from an analysis of the reasons why the common accounts featuring sentences, propositions or utterances fail. Theories of formalization cannot refer to utterances qua expressions of propositions; instead they may refer to sentences and rely on additional information about linguistic structure and pragmatic context.
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Notes
Treating inferences as sequences instead of pairs consisting of a set of premises and a conclusion is more convenient for present purposes because it enables us to easily associate formulas with premises.
I will use standard zero- and first-order logic as examples of logical formalisms since they are paradigmatic for the philosophical tradition I am referring to. However, what follows can be applied to wide range of non-standard logics.
Treating arguments in a publicly accessible medium as paradigm cases is characteristic for most traditions of logic in philosophy, but it has not been uncontested. The main alternative consists in giving theoretical primacy to mental states or acts, as can be found in some approaches which give judgements a central role in logic (cf. Martin-Löf 1996) or take Chomsky’s theories of language as their starting point (cf. Collins 2003). Insofar as mental states or acts are singular events or objects, the points I make about utterances can be transferred to such “mentalistic” conceptions of logic.
This use of “x substitutes for y” has do be distinguished from one that refers to replacing expressions within the same language.
I leave open the question of whether and how the explication given here can be adapted to theories that reject or substantially reinterpret the traditional account of logical proofs outlined above, as for example LePore and Ludwig (2007).
See Iacona (2002, Chap. 5.4) for various explications of primacy.
The argument can be generalized to all inferences of zero-order logic which do not involve constant truth functions. See Hoyningen-Huene (2004, pp. 209–211).
Nolan (1969) has suggested an interpretation of formulas which would treat (2) as a claim about utterances and their parts: every first conjunct of a true conjunction is true. This interpretation of (2) evades the argument about repeatability by assuming that the conclusion-utterance is a part of the premise-utterance and that generally all inferences formalizable by (2) consist of one (conjunctive) utterance only. However, even if one were ready to adopt such an interpretation of (2), this would be of no help for dealing with inferences that consist of two separate utterances, like all the usual examples of conjunction elimination do.
Strictly speaking, there is an asymmetry here. Treating two different Xs as identical may lead to incorrect “proofs” of invalid inferences, whereas treating two occurrences of the same X as different will only result in giving away some proofs of valid inferences.
Even though the complement-definition of content is of little help if applied to utterances, it may be useful if applied to some other Xs.
Frege explicitly formulated this criterion, among others (see Penco 2003).
Whether Stalnaker himself would propose to answer (Q) with reference to propositions is less clear. On the one hand, he takes propositions to have logical structures and to stand in logical relations, such as entailment (1984, p. 10). On the other hand, he treats statements about logical equivalence as a statements about relations between sentences (1984, p. 72).
Some defenders of propositions claim that utterances have truth-values only in virtue of expressing propositions. This turns (T.1) into: A sentence is true iff all its utterances express a true proposition and it is false iff if all its utterances express a false proposition. Since the following arguments do not depend on whether utterances have their truth-value ascribed directly or derived from the propositions they express, I will confine myself to the simpler picture without propositions.
The category of indexicals is here to be understood in a broad sense, so that it includes not only expressions whose meaning is dependent on the circumstances of utterance, but also relative pronouns and other anaphoric or cataphoric expressions.
Just some guarantee that both occurrences have the same truth-value is not enough. Even if all utterances of, for example, “I am here now” must be true regardless of context (Kaplan 1989), it does not follow that all inferences from one utterance of this sentence to another one can be formalized as instances of ϕ ⇒ ϕ. In fact, such a formalization constitutes a fallacy of equivocation unless the two utterances are produced simultaneously by the same person.
We could therefore say, that being an X is a role sentences can play, quite analogous to other roles they may play, such as being a premise or a conclusion.
“Correspondence scheme” is Sainsbury’s term (2001, p. 51). Kalish et al. use “scheme of abbreviation” (1980, p. 8).
Importantly, even though LF is closely related to logical forms of generalized quantifier theory, it is a syntactic structure which is part of Chomsky’s program of naturalizing linguistics. As such it answers to principles which are not logical: properties of LF are “not to be settled in terms of considerations of valid inference and the like” (Chomsky 1986, p. 205n, cf. pp. 67, 156). It therefore is an empirical question, whether and in what respects Logical Forms can be equated with logical forms, in spite of widespread use of LF for addressing logical problems.
There are also theories which hold that propositions themselves are unstructured entities even though logical forms play a crucial role in deciding what proposition(s) an utterance expresses (e.g. Bealer 1993). If this implies that propositions do not have logical forms, they cannot be what formulas stand for according to (LF). If not, such unstructured propositions raise the same problems as those structured by logical forms.
In Brun 2004, Chap. 13, I argue that in a certain sense all adequate formalizations of an utterance in a given logical theory can be interpreted as representing aspects of a single logical form.
Classical starting points for procedures and theories of formalization are Montague (1970) (for an application to extensional first-order logic see Link 1979, pp. 242–245) and Davidson (1984, 1980). A third notable paradigm proceeds from Chomsky’s (1986, 1995) syntactical theories of LF, but see note 17. An important discussion of criteria can be found in Blau (1977, Chap. I.1). Most of what follows can also be applied to informal guidelines for formalizing as presented in e.g. Epstein (2001) and Sainsbury (2001).
References
Bealer, G. (1993). A solution to Frege’s puzzle. Philosophical Perspectives, 7, 17–60.
Blau, U. (1977). Die dreiwertige Logik der Sprache: Ihre Syntax, Semantik und Anwendung in der Sprachanalyse. Berlin: de Gruyter.
Brandom, R. B. (1994). Making it explicit: Reasoning, representing and discursive commitment. Cambridge, MA: Harvard University Press.
Brun, G. (2004). Die richtige Formel: Philosophische Probleme der logischen Formalisierung (2nd ed.). Frankfurt a. M: Ontos.
Carnap, R. (1956). Meaning and necessity: A study in semantics and modal logic (2nd ed.). Chicago: University of Chicago Press.
Chomsky, N. (1986). Knowledge of language: Its nature, origin, and use. Westport: Praeger.
Chomsky, N. (1995). The minimalist program. Cambridge, MA: MIT Press.
Collins, J. (2003). Expressions, sentences, propositions. Erkenntnis, 59, 233–262.
Cresswell, M. J. (1985). Structured meanings: The semantics of propositional attitudes. Cambridge, MA: MIT Press.
Davidson, D. (1963). The method of extension and intension. In P. A. Schilpp (Ed.), The philosophy of Rudolf Carnap (pp. 311–349). La Salle: Open Court.
Davidson, D. (1980). The logical form of action sentences. Criticism, comment, and defence. In Essays on actions and events (pp. 105–148). Oxford: Oxford University Press.
Davidson, D. (1984). Inquiries into truth and interpretation. Oxford: Oxford University Press.
Davidson, D. (1990). The structure and content of truth. Journal of Philosophy, 87, 279–328.
Davidson, D. (1999). Intellectual autobiography. In L. E. Hahn & P. A. Schilpp (Eds.), The philosophy of Donald Davidson (pp. 1–70). Chicago: Open Court.
Epstein, R. L. (2001). The semantic foundations of logic: Predicate logic. Belmont: Wadsworth.
Frege, G. [1879] (1988). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. In Begriffsschrift und andere Aufsätze. Darmstadt: Wissenschaftliche Buchgesellschaft. Translated in van Heijenoort, J. (Ed.). (1967). From Frege to Gödel: A source book in mathematical logic, 1879–1931 (pp. 1–82). Cambridge, MA: Harvard University Press (References are to the original pagination).
Frege, G. [1918] (1967). Der Gedanke: Eine logische Untersuchung. In Kleine Schriften (pp. 342–362). Darmstadt: Wissenschaftliche Buchgesellschaft. Translated in Beaney, M. (Ed.). (1997). The Frege reader (pp. 325–345). Oxford: Blackwell (References are to the original pagination).
Frege, G. (1969). Nachgelassene Schriften, Hamburg: Meiner. Translated as Frege, G. (1979). Posthumous writings. Oxford: Basil Blackwell (References are to the German/English edition).
Goodman, N. (1976). Languages of art: An approach to a theory of symbols (2nd ed.). Indianapolis: Hackett.
Grandy, R. E. (1993). What do “Q” and “R” stand for anyway? In R. I. G. Hughes (Ed.), A philosophical companion to first-order logic (pp. 50–61). Indianapolis: Hackett.
Higginbotham, J. (1991). Belief and logical form. Mind and Language, 6, 344–369.
Hoyningen-Huene, P. (2004). Formal logic: A philosophical approach. Pittsburgh: University of Pittsburgh Press. Translation of Hoyningen-Huene, P. (1998). Formale Logik: Eine philosophische Einführung. Stuttgart: Reclam.
Iacona, A. (2002). Propositions. Genova: Name.
Kalish, D., Montague, R., & Mar, G. (1980). Logic: Techniques of formal reasoning (2nd ed.). Fort Worth: Harcourt, Brace and Jovanovich.
Kaplan, D. (1989). Demonstratives. An essay on the semantics, logic, metaphysics, and epistemology of demonstratives and other indexicals. In J. Almog, J. Perry & H. Wettstein (Eds.), Themes from Kaplan (pp. 481–563). Oxford: Oxford University Press.
Kaplan, D. (1990). Words. Proceedings of the Aristotelian Society supplement, 64, 93–119.
King, J. C. (1995). Structured propositions and complex predicates. Noûs, 29, 516–535.
King, J. C. (1996). Structured propositions and sentence structure. Journal of Philosophical Logic, 25, 495–521.
King, J. C. (2005). Structured propositions. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/archives/sum2005/entries/propositions-structured/.
Lampert, T. (2005). Klassische Logik: Einführung mit interaktiven Übungen (2nd ed.). Frankfurt a. M.: Ontos.
Larson, R. K., & Ludlow, P. (1993). Interpreted logical forms. Synthese, 95, 305–355.
Larson, R. K., & Segal, G. (1995). Knowledge of meaning: An introduction to semantic theory. Cambridge, MA: MIT Press.
Lemmon, E. J. (1987). Beginning logic (2nd ed.). London: Chapman and Hall.
LePore, E. (2000). Meaning and argument: An introduction to logic through language. Malden: Blackwell.
LePore, E., & Ludwig, K. (2007). Donald Davidson’s truth-theoretic semantics. Oxford: Clarendon Press.
Link, G. (1979). Montague-Grammatik. München: Fink.
Martin-Löf, P. (1996). On the meanings of the logical constants and the justifications of the logical laws. Nordic Journal of Philosophical Logic, 1, 11–60. http://www.hf.uio.no/ifikk/filosofi/njpl/vol1no1/meaning/meaning.html. Accessed on 16 April 2008.
Menzel, C. (1993). The proper treatment of predication in fine-grained intensional logic. Philosophical Perspectives, 7, 61–87.
Montague, R. [1970] (1974). Universal grammar. In Formal philosophy: Selected papers (pp. 222–246). New Haven: Yale University Press.
Nolan, R. (1969). Truth and sentences. Mind, 78, 501–511.
Peirce, C. S. (1902). Logic. In J. M. Baldwin (Ed.), Dictionary of philosophy and psychology (Vol. 2, pp. 20–23). New York: MacMillan.
Penco, C. (2003). Frege: Two theses, two senses. History and Philosophy of Logic, 24, 87–109.
Perry, J. (1979). The problem of the essential indexical. Noûs, 13, 3–21.
Quine, W. V. O. (1960). Word and object. Cambridge, MA: MIT Press.
Quine, W. V. O. (1982). Methods of logic (4th ed.). Cambridge, MA: Harvard University Press.
Ramsey, F. P. [1925] (1990). Universals. In Philosophical papers (pp. 8–30). Cambridge: Cambridge University Press.
Richard, M. (1990). Propositional attitudes: An essay on thoughts and how we ascribe them. Cambridge: Cambridge University Press.
Rosenberg, J. F. (1986). Die Autorität der logischen Analyse: Einige Provokationen. Neue Hefte für Philosophie, 26, 69–88.
Sainsbury, M. (2001). Logical forms: An introduction to philosophical logic (2nd ed.). Oxford: Blackwell.
Salmon, N. (1992). On content. Mind, 101, 733–751.
Soames, S. (1987). Direct reference, propositional attitudes, and semantic content. Philosophical Topics, 15, 47–87.
Stalnaker, R. (1976). Propositions. In A. F. MacKay & D. D. Merrill (Eds.), Issues in the philosophy of language: Proceedings of the 1972 Oberlin Colloquium in Philosophy (pp. 79–91). New Haven: Yale University Press.
Stalnaker, R. (1984). Inquiry. Cambridge, MA: MIT Press.
Strawson, P. F. (1952). Introduction to logical theory. London: Methuen.
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. London: Routledge and Kegan Paul.
Zalta, E. N. (1988). Intensional logic and the metaphysics of intentionality. Cambridge, MA: MIT Press.
Acknowledgements
This paper thoroughly revises the points I made in Brun (2004, Chap. 5). Earlier versions have been presented in Bern, Hanover, Lund and Neuchâtel. For discussions and feedback, I am indebted to Michael Baumgartner, Jean-Yves Béziau, Richard Grandy, Paul Hoyningen-Huene, Timm Lampert and Klaus Petrus. Special thanks go to Dominique Kuenzle for collaboration on one of the very first drafts and to two anonymous referees of this journal for extremely helpful comments.
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Brun, G. Formalization and the Objects of Logic. Erkenn 69, 1–30 (2008). https://doi.org/10.1007/s10670-008-9112-3
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DOI: https://doi.org/10.1007/s10670-008-9112-3