Abstract
The paper is a critique of the widespread conception of logic as a neutral arbiter between metaphysical theories, one that makes no `substantive’ claims of its own (David Kaplan and John Etchemendy are two recent examples). A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds (however mistaken), with a consequent proliferation of alternative logics. However, this apparent contentiousness of logic is often treated as though it were neutralized by the possibility of studying all these alternative logics within an agreed metalogical framework, typically that of first-order logic with set theory. In effect, metalogic is given the role of neutral arbiter. The paper will consider a variety of examples in which deep logical disputes re-emerge at the meta-level. One case is quantified modal logic, where some varieties of actualism require a modal meta-language (as opposed to the usual non-modal language of possible worlds model theory) in order not to make their denial of the Barcan formula self-defeating. Similarly, on some views the intended model theory for second-order logic can only be given in a second-order metalanguage—this may be needed to avoid versions of Russell’s paradox when the first-order quantifiers are read as absolutely unrestricted. It can be shown that the phenomenon of higher-order vagueness eventually forces fuzzy logical treatments of vagueness to use a fuzzy metalanguage, with consequent repercussions for what first-order principles are validated. The difficulty of proving the completeness of first-order intuitionistic logic on its intended interpretation by intuitionistically rather than just classically valid means is a more familiar example. These case studies will be discussed in some detail to reveal a variety of ways in which even metalogic is metaphysically contested, substantial and non-neutral.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Ibid., p. 24.
Ibid., p. 23.
Russell (1914, p. 50). The passage occurs in a chapter called ‘Logic as the Essence of Philosophy’. The title of the book indicates that Russell’s conception of the relation between science and philosophy was less exclusive than von Wright’s.
Of course, computer science has become almost as important as mathematics qua home for research in logic; sociologically, philosophy comes third.
von Wright (1993, p. 24).
See Eklund (1996), and references therein.
According to von Wright, “When viewing the history of modern logic as a process of ‘rational disenchantment’ in areas of conceptual crisis or confusion, one is entitled to the judgement that the most exciting development in logical theory after the second world war has been the rebirth of modal logic” (1993, p. 19).
See Shapiro (1991).
For many purposes in modal logic it is unnecessary to designate a particular member @ of W in a model; it is designated here for expository purposes.
Using modal considerations, one argues that the formulas true under that assignment and the intended interpretations of the connectives constitute a maximal consistent set in K, the weakest normal modal logic. Thus there is a point in the canonical model (in a sense of model on which no point is designated the ‘actual world’) for K at which all and only those formulas are true (see Hughes and Cresswell 1996, pp. 112–120). That point can be taken as the actual world for the canonical model.
Just take the union of the classes of models associated as before with particular assignments.
Shapiro (1996, p. xi).
Although Hartry Field (2008) uses something closely related to continuum-valued semantics in his treatment of the paradoxes, its role is purely instrumental, not explanatory. He does not privilege metalogic as the level at which scientific order is restored. Thus he does not exemplify the approach under discussion in the text.
See Machina (1976).
Adolf Lindenbaum proved what was in effect a very general result along these lines, by showing that any given logic S for a language L is sound and complete with respect to a semantics in which the values assigned to formulas of L are equivalence classes of formulas of L under the relation of logical equivalence in S and the equivalence class of theorems is the designated value, provided that logical equivalence in S is a congruence relation with respect to the operators of L (he constructed what is now known as the Lindenbaum algebra for S). See for example, Dummett (2000).
See Dummett (2000, pp. 154–204), for a detailed discussion and further references. The first notion of completeness discussed here is what he calls ‘internal completeness’; see particularly Theorems 5.36 and 5.37. The new models are Wim Veldman and Harry de Swart’s generalized Beth trees, in which the falsity constant \( \bot \) can be verified at a node provided that all atomic formulas are too. D.C. McCarty (2007, pp. 372–373) gives an argument that even intuitionistic propositional logic is incomplete for infinite sets of premises.
See Field (2008, pp. 108–114), for related considerations, although Field’s preferred logic is not the continuum-valued one.
Let f(S) be the system of all principles validated using S as the metalogic. Thus Sα+1 = f(Sα). If α is a limit ordinal, Sα is the system of all principles in every Sβ for β < α; consequently Sα+1 ⊆ Sα. Note that if S ⊆ T then f(S) ⊆ f(T). Of course, S1 ⊆ S0 because any bivalent model is a special case of the continuum-valued semantics in which all formulas are assigned either 0 or 1. One then proves by induction on α that if β ≤ α then Sα ⊆ Sβ, so the sequence is monotonically decreasing. Since the total number of principles (in any reasonable precise sense) will be bounded by the size of the language, a fixed point will eventually be reached. The construction is reminiscent of that in Kripke (1975).
For more discussion see Williamson (1994, pp. 127–131), which defends a classical alternative to continuum-valued logic.
The principle is named after Ruth Barcan Marcus, who discovered and was the first to formalize it, but it was already known to Ibn Sina (Avicenna, 980–1037); see Movahed (2006).
See Kripke (1963).
Friedman (1999).
For a more detailed argument along the lines of this paragraph, see Williamson (2007). Similar considerations tell against the (independently implausible) idea that the disputing parties are using the logical connectives with the same context-sensitive meaning but in different contexts, so that their reference shifts. As in other debates, the cooperative norms of communication tend to create a unified context in relevant respects, precisely in order to avoid equivocation. Claims of non-equivocation or sameness of meaning in the text should be read as tacitly including claims of sameness of reference. A further objection to the charge of equivocation is that in many cases, including the dispute between classical and intuitionistic logic, the attempt to combine two sets of connectives, one set conforming to one logic and the other to the other, leads to the collapse of one logic into the other; see Williamson (1987/88). As emphasized in Schechter (2011), the collapse results are sensitive to differences in the way the logics are axiomatized. The collapse result still goes through for a classical logician committed to the classical natural deduction rules and an intuitionist committed to the intuitionistic natural deduction rules, provided that neither party’s commitment is qualified by a tacit restriction on the vocabulary of the language.
Given an interpreted higher-order language and a decision as to which of its atomic expressions are to count as non-logical on which all of the latter may be replaced by distinct quantifiable variables of the corresponding type, there is a good Tarskian notion of logical truth according to which a formula is logically true if and only if its universal closure is simply true. Nothing in the text undermines the claim that in such a setting there is a unique set of the logical truths of the language.
Kaplan (1995, p. 42)
Etchemendy (1990, p. 143)
Etchemendy (1990, p. 111).
Appeals to the concept of analyticity will not help here; see Williamson (2007, pp. 48–133).
References
Beall, J. C. (Ed.). (2003). Liars and heaps: New essays on paradox. Oxford: Clarendon Press.
Beall, J. C., & Restall, G. (2006). Logical pluralism. Oxford: Oxford University Press.
Boolos, G. (1985). Nominalist platonism. Philosophical Review, 94, 327–344.
Davies, M. (1978). Weak necessity and truth theories. Journal of Philosophical Logic, 7, 415–439.
Dummett, M. (2000). Elements of intuitionism (2nd ed.). Oxford: Clarendon Press.
Eklund, M. (1996). On how logic became first-order. Nordic Journal of Philosophical Logic, 1, 147–167.
Etchemendy, J. (1990). The concept of logical consequence. Cambridge, Mass: Harvard University Press.
Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.
Field, H. (2009). Pluralism in logic. The Review of Symbolic Logic, 2, 342–359.
Fine, K. (1977). Prior on the construction of possible worlds and instants. In A. N. Prior (Ed.), Worlds, times and selves. London: Duckworth. Reprinted in K. Fine, Modality and tense: philosophical papers (Oxford: Clarendon Press, 2005).
Friedman, H. (1999). A complete theory of everything: Validity in the unrestricted domain. http://www.math.ohio-state.edu/friedman/
Gupta, A. (1980). The logic of common nouns: An investigation in quantified modal logic. New Haven: Yale University Press.
Hughes, G. E., & Cresswell, M. J. (1996). A new introduction to modal logic. London: Routledge.
Humberstone, L. (1996). Homophony, validity, modality. In J. Copeland (Ed.), Logic and reality: Essays on the legacy of Arthur Prior. Oxford: Clarendon Press.
Jané, I. (1993). A critical appraisal of second-order logic. History and Philosophy of Logic, 14, 67–86.
Kaplan, D. (1995). A problem in possible-world semantic. In W. Sinnott-Armstrong, D. Raffman, & N. Asher (Eds.), Modality, morality, and belief: Essays in honor of Ruth Barcan Marcus. Cambridge: Cambridge University Press.
Kripke, S. A. (1963). Semantical considerations on modal logic. Acta Philosophica Fennica, 16, 83–94.
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.
Machina, K. F. (1976). Truth, belief and vagueness. Journal of Philosophical Logic, 5, 47–78.
McCarty, D. C. (2007). Intuitionism in mathematics. In Shapiro (2007).
McGee, V. (2003). Universal universal quantification: Comments on Rayo and Williamson. In Beall (2003).
Movahed, Z. (2006). Ibn-Sina’s anticipation of Buridan and Barcan formulas’. In A. Enayat, I. Kalantari & M. Moniri (eds.), Logic in Tehran: Proceedings of the workshop and conference on logic, algebra and arithmetic, held October 18–22, 2003, ASL Lecture Notes in Logic 26. Natick, MA.: A.K. Peters.
Peacocke, C. (1978). Necessity and truth theories. Journal of Philosophical Logic, 7, 473–500.
Prawitz, D., Skyrms, B., & Westerståhl, D. (Eds.). (1994). Logic, methodology and philosophy of science, IX (Uppsala 1991). Amsterdam: North-Holland.
Quine, W. V. (1970). Philosophy of logic. Prentice-Hall, N.J.: Englewood Cliffs.
Rayo, A., & Uzquiano, G. (1999). Towards a theory of second-order consequence. Notre Dame Journal of Formal Logic, 40, 315–325.
Rayo, A., & Uzquiano, G. (Eds.). (2006). Absolute generality. Oxford: Clarendon Press.
Rayo, A., & Williamson, T. (2003) A completeness theorem for unrestricted first-order languages. In Beall (2003).
Russell, B. A. W. (1914). Our knowledge of the external world, as a field for scientific method in philosophy. London: Allen and Unwin.
Schechter, J. (2011). Juxtaposition: A new way to combine old logics. The Review of Symbolic Logic, 4, 560–606.
Shapiro, S. (1991). Foundations without foundationalism: A case for second-order logic. Oxford: Clarendon Press.
Shapiro, S. (Ed.). (1996). The limits of logic: Higher-order logic and the Löwenheim-Skolem theorem. Dartmouth: Aldershot.
Shapiro, S. (Ed.). (2007). The Oxford handbook of philosophy of mathematics and logic. Oxford: Oxford University Press.
von Wright (1993) The tree of knowledge and other essays. Leiden: E.J. Brill.
Williamson, T. (1987/88). Equivocation and existence. In Proceedings of the Aristotelian Society 88, pp. 109–127.
Williamson, T. (1994). Vagueness. London: Routledge.
Williamson, T. (1998). Bare possibilia. Erkenntnis, 48, 257–273.
Williamson, T. (2003). Everything. In J. Hawthorne & D. Zimmerman (Eds.), Philosophical perspectives 17: Language and philosophical linguistics. Oxford: Blackwell.
Williamson, T. (2007). The philosophy of philosophy. Oxford: Blackwell.
Williamson, T. (2013). Modal logic as metaphysics. Oxford: Oxford University Press.
Author information
Authors and Affiliations
Corresponding author
Additional information
Thanks to participants in the 2008 Barcelona workshop on metametaphysics, to lecture audiences at South-western University Chongqing and St Petersburg State University, and to Matti Eklund, Peter Pagin, and an anonymous referee for helpful comments on earlier versions of various parts of this material.
Rights and permissions
About this article
Cite this article
Williamson, T. Logic, Metalogic and Neutrality. Erkenn 79 (Suppl 2), 211–231 (2014). https://doi.org/10.1007/s10670-013-9474-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-013-9474-z