Abstract
We offer a defense of one aspect of Paul Horwich’s response to the Liar paradox—more specifically, of his move to preserve classical logic. Horwich’s response requires that the full intersubstitutivity of ‘ ‘A’ is true’ and A be abandoned. It is thus open to the objection, due to Hartry Field, that it undermines the generalization function of truth. We defend Horwich’s move by isolating the grade of intersubstitutivity required by the generalization function and by providing a new reading of the biconditionals of the form “ ‘A’ is true iff A.”
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Notes
MT is actually formulated as a theory of propositional truth. For the issues addressed in this paper, however, it is simpler to work with sentential truth. Horwich treats sentential truth and propositional truth in a parallel way (see chapter 7 of Truth), and we will feel free to transpose his remarks concerning propositional truth into those concerning sentential truth.
It is not essential to our argument below that truth be treated as a predicate of sentences. The argument could easily be recast into a propositional key, but it would become longer without becoming more illuminating.
We assume the usual conventions necessary for treating truth as a predicate of sentences.
Because of the required modification, our defense of Horwich is only partial; we are unable to defend all aspects of Horwich’s response.
We should note that our interest in mounting a defense of Horwich does not issue from a desire to support deflationism, which animates Horwich’s book. Our interest issues from a desire to support the idea that classical logic is perfectly consistent with the generalization function of truth. Even if we succeed, as we hope, in providing deflationists with a better formulation of MT, they will not thereby be in a better position to answer the main objections to deflationism. For example, the objections given in Anil Gupta’s “Critique of Deflationism” (Gupta 2011) are entirely independent of the proper formulation of MT in face of the Liar.
Field, Saving Truth from Paradox, p. 210; italics added to ‘need’ (Field 2008).
In rough terms, a sentential context ‘... A ...’ counts as extensional iff, necessarily, if \(A\equiv B\) then (... A ... \(\equiv \) ... B ... ); here ‘\(\equiv \)’ expresses material equivalence. A similar scheme explains the extensionality of predicate and name positions.
For the purposes of the discussion in this section, we can assume that the language containing the truth-predicate is a first-order language, equipped with quotation names of sentences. In such a language, all contexts are extensional except those occurring within quotation names.
P. 12; see also Quine’s Pursuit of Truth, §33 (Quine 1992).
See “Critique of Deflationism.”
According to Quine, the function of the truth-predicate is to enable one to express generalizations over sentence positions (e.g., the position of A in ‘if A then A’) using nominal quantification (as in, e.g., “for all sentences x, x is true if x is a conditional whose antecedent and consequent are identical”). The truth-predicate can play this expressive role only if certain intersubstitutivity principles hold. (CI) is one such principle. We are concerned to discover which further principles are required by this expressive role.
Horwich’s response to the problem is to keep classical logic and to yield on the generalization function of truth. Since Horwich restricts even (CI), truth will not serve the generalization function in full. Horwich thinks this is not a cause for concern, since “the utility of truth as a device of generalization is not substantially impaired” (Truth, p. 42, fn. 21). Whether the impairment is substantial or not depends, however, on the details of the minimal theory, which Horwich has not supplied. If the T-biconditionals excluded from MT are few, then the damage may well be minimal; otherwise, the damage could be substantial. There is, in either case, an additional difficulty: MT will not be able to explain how the sentence ‘a conjunction is true only if its first conjunct is true’ generalizes truths of the form ‘if A and B then A’.
Even within a three-valued logic, which Field favors, (FI) implies that truth can serve the generalization function only at the cost of expressive richness. If, for example, exclusion negation is expressible, then the same difficulty arises in the three-valued context as in the classical. If Field is right about the necessity of (FI), then the full generalization function of truth can be obtained only at the expense of some expressive resources. The argument below is aimed at resisting this conclusion. We think that the full generalization function of truth can be had without limiting logical or non-logical resources in any way.
If (2) occurs embedded within quantifiers, then the above technique will work only in some cases, not in all. The technique will not work if one of these quantifiers, say ‘for all x’, binds an occurrence of ‘x’ within ‘x is true’. For such cases, semantic ascent, instituted as outlined above, requires the satisfaction predicate. An analog of (UI) formulated for the satisfaction predicate gives us the needed intersubstitutivities. We do not know what is possible here merely with the truth predicate.
A couple of points of clarification concerning (UI):
(i) Our principal conclusions do not rest on the claim about the sufficiency of (UI). So long as the needed intersubstitutivity principles are consistent with classical logic, our principal conclusions hold.
(ii) We do not take (UI) to be a full theory of truth, one that suffices to explain our uses of the concept of truth. There are principles governing truth that go beyond (UI), for example, the T-biconditionals under the reading proposed below.
Thanks to an anonymous referee for pushing us to clarify the role of (UI) in our argument.
The generalization function remains intact even if some of the \(A_{i}\)’s are paradoxical. You can gain the effect of affirming (3) by affirming (4), even when some of the \(A_{i}\)’s are paradoxical.
Tarski, “Semantic Conception of Truth and the Foundations of Semantics,” §4; italics added (Tarski 2001).
See our “Conditionals in Theories of Truth,” where we define the step biconditional using two step-conditionals. For further information about the logic of these conditionals, see Shawn Standefer’s “Solovay-Type Theorems for Circular Definitions” (Standefer 2015).
In the exposition below, we aim to impart intuitive understanding, without bringing into play the technical details of revision theory. For these details, see Gupta and Nuel Belnap’s Revision Theory of Truth (Gupta and Belnap 1993) and our “Conditionals in Theories of Truth” (Gupta and Standefer 2017).
Technically, the revision process is the collection of sequences one obtains through repeated revisions of arbitrary revision hypotheses. A stage in the process is simply a stage in one of these sequences.
There are rules for assigning G an interpretation at limit stages; see Gupta and Belnap, Revision Theory of Truth.
We are using the rough notion “almost everywhere true” in place of “valid in S \(^{\# }\).”
This is so because we have chosen to work within a classical framework. Revision processes for truth can be constructed for non-classical languages also.
The attributions of truth to these material T-biconditionals are also affirmable.
A relevant theorem: If B and B* meet the conditions laid down in (UI), then B is deducible from the step T-biconditionals in the calculus C \(_{0}^{+}\) iff B* is also so deducible. See our “Conditionals in Theories of Truth” for the details of C \(_{0}^{+}\).
Note that classical revision theory does not provide the resources—more specifically, a suitable sentential connective—for a satisfactory formulation of the Minimal Theory.
References
Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.
Gupta, A. (2011). A critique of deflationism. Reprinted with a postscript in his Truth, Meaning, Experience, pp. 9–52. New York: Oxford University Press. (Originally published in 1993.)
Gupta, A., & Belnap, N. (1993). The revision theory of truth. Cambridge, MA: MIT Press.
Gupta, A., & Standefer, S. (2017). Conditionals in theories of truth. Journal of Philosophical Logic. doi:10.1007/s10992-015-9393-3.
Horwich, P. (1998). Truth (2nd ed.). Oxford: Clarendon Press. (The first edition appeared in 1990.)
Quine, W. V. (1986). Philosophy of logic (2nd ed.). Cambridge, MA: Harvard University Press. (The first edition appeared in 1970.)
Quine, W. V. (1992). Pursuit of truth (Revised ed.). Cambridge, MA: Harvard University Press. (The original edition appeared in 1990.)
Standefer, S. (2015). Solovay-type theorems for circular definitions. Review of Symbolic Logic, 8, 467–487.
Tarski, A. (2001). The semantic conception of truth and the foundations of semantics. Reprinted in Lynch, M. P., The nature of truth: Classical and contemporary perspectives (pp. 331–363). Cambridge, MA: MIT Press. (Originally published in 1944.)
Acknowledgements
We are grateful to the editors, Joseph Ulatowski and Cory Wright, and particularly to an anonymous referee for their helpful comments on our paper. Standefer’s acknowledgement: This research was supported by the Australian Research Council, Discovery Grant, DP150103801.
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Gupta, A., Standefer, S. Intersubstitutivity principles and the generalization function of truth. Synthese 195, 1065–1075 (2018). https://doi.org/10.1007/s11229-017-1318-y
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DOI: https://doi.org/10.1007/s11229-017-1318-y