Abstract
This paper compares two ways of holding that logic is special among the sciences in that it has no restricted class of entities as its subject matter, but instead concerns all entities alike. One way is Williamson’s explanation of how inquiry into logical consequence and logical truth only superficially concerns the linguistic or conceptual entities that bear these properties. Williamson draws on ideas familiar from deflationism about truth, and his account has been called “deflationary.” I argue that the analogy is misleading. While there’s a broad sense in which Williamson offers a deflationary account of logical inquiry, his view differs from deflationism about truth in being best understood as a form of instrumentalism. By contrast, I elaborate a deflationism about logical properties modeled on deflationism about truth. I defend this expressive device deflationism as an explanation of our use of logical predicates.
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06 May 2022
A Correction to this paper has been published: https://doi.org/10.1007/s11229-022-03641-x
Notes
See Hjortland (2017, p. 632) for a list of recent defenses of anti-exceptionalism. The term “anti-exceptionalism” is due to Williamson (2013a), whose opposition to “exceptionalism about logic” (2017, p. 337) is a special case of his opposition to “philosophical exceptionalism” (2007, p. 3). Priest (2006a, p. 165n26) contains a reference to Haack’s chapter.
Cf. Priest (2014, p. 217) and Williamson (2017, p. 334). Martin and Hjortland (2021) criticize “abductivists” about logic for failing to consider predictive success as a measure of explanatory power. Hlobil (2021) argues that “abductivists” err in holding that “abduction can serve as a neutral arbiter in foundational logical disputes,” since what one takes logic’s explananda to be isn’t independent of the logic one advocates or one’s views about logic’s “role and nature.” Methodological anti-exceptionalists can concede these points.
This idea shouldn’t be confused with the view that logically true sentences, logically valid arguments, or logical constants are “topic neutral” (on this see e.g., MacFarlane, 2017). That view is compatible with, and standardly combined with, holding that logic is concerned with properties of linguistic items; topic-neutrality is invoked in characterizing those properties.
Like Williamson, I use “metalinguistic” to mean concerning language. While all types of inquiry conducted using language are linguistic, as are all predicates, only some are metalinguistic. When it comes to properties, by contrast, I’ll speak of those possessed only by linguistic items as “linguistic” properties.
A view that anticipates both deflationisms, and in some ways Shapiro (2011) in particular, is the “modalism” of Bueno and Shalkowski (2009, see note 33 below). Their account of logical consequence “stays in the object language whenever possible when being philosophically serious\(\ldots \). Talk of truth, truth bearers, facts, etc. is talk that is designed to permit us to speak generically” (2009, pp. 311n8, 308). That logical inquiry is fundamentally non-metalinguistic is affirmed by Quine as well: “Logical theory, despite its heavy dependence on talk of language, is \(\ldots \) world-oriented rather than language-oriented; and the truth predicate makes it so” (1970, p. 97). Here I’ll set aside the question of how the two kinds of logical deflationism I’ll be discussing relate to the substitutional account of logical properties Quine uses to substantiate that claim, as it applies to certain first-order languages.
Blake-Turner (2020, p. 552n1) notes that both views have been called “deflationism,” but his topic is Williamson’s view.
His language here calls to mind Quine’s deflationary claim (1992, pp. 80–81) that the truth predicate is needed to generalize along a “dimension of generality” that can’t otherwise be expressed using first-order quantification.
Williamson’s definition of UG(\(\alpha \)) requires using “some fixed order” for the language’s variables of a given type. Here I use the order \(p, q, \ldots \) for sentential variables. Italics will serve as devices of quasi-quotation, and \(\alpha \), \(\beta \), etc. will do double duty as schematic letters and metalinguistic variables.
Williamson notes an important exception: UG(\(\alpha \)) will be metalinguistic if \(\alpha \) contains a truth predicate as one of its logical constants.
I thank a referee for helping me get clearer about this.
Here I ignore as immaterial the fact that inquiry into provability typically avails itself of arithmetized syntax, and for that reason is technically inquiry into arithmetic.
The function \({\dot{\rightarrow }}\) yields a name of the conditional formed from the sentences named by its arguments.
The reasoning depends on the predicate T and the distinguished name \(\langle \lambda \rangle \) being counted as logical constants, so that UG\((\lambda \)) is \(\lambda \). A referee observed that if the logical truth predicate were instead introduced directly using the L-Equivalence, rather than in terms of T, the predicate L would only introduce semantic paradox if L is also a logical constant.
In earlier work, Williamson (2013b, p. 94) had defined logical consequence in terms of the logical truth of conditionals. His reason for rejecting this view is that it places undue constraints on how the consequence predicate relates to the object language’s conditional, constraints that rule out too many non-classical logics (2017, p. 333). Hjortland (2017, pp. 639–640) argues that Williamson’s deflationary account of logical theories can’t make sense of disagreements about the relation of metaconsequence obtaining between arguments. For a critical examination of this objection, see Blake-Turner (2020, pp. 557–583). Here I won’t address the options logical deflationists have for understanding metaconsequence.
Prior to explaining his strategy for extending his non-metalinguistic conception from logical truth to logical consequence, Williamson anticipates an answer to the question I’ve asked. He says that by looking at the “theorems” in Cn\(_{\vdash }\)(\(\emptyset \)), we “can treat different [formal] consequence relations as rival attempts to theorize the typically nonmetalinguistic subject matter” of completely universal generalizations (Williamson, 2017, p. 332). But he never returns to substantiate this claim based on his account of how “rival” formal consequence relations should be evaluated.
The term is introduced in a related sense by Fischer (2015), with reference to a theory of truth. According to his instrumentalist deflationism, however, the truth predicate “does not express a property” (2015, p. 298), whereas Williamson doesn’t deny that the logical truth predicate expresses a property. According to Fischer, moreover, one way the truth predicate is instrumentally useful lies in its “expressiveness,” in how it “allows for the formulation of \(\ldots \) generalizations.” On Williamson’s view, I’m arguing, the instrumental value of logical predicates isn’t a matter of enabling expression, but rather of helping us put forward good theories.
Instrumentalism about logic in Priest’s sense differs from other views given that label in the literature. Kouri Kissel (2019, p. 154) uses “logical instrumentalism” for “the view that norms for deductive reasoning should be evaluated based on one’s aims and goals in reasoning and the domain of investigation.” As she notes, this isn’t the sense in which Haack (1978, pp. 224–225) speaks of “instrumentalism.” Haack’s instrumentalist agrees with Priest’s instrumentalist that logical system’s correctness shouldn’t be explained in terms of its “correspondence” with facts about logical truth and consequence, but goes on to infer that “there is no sense in speaking of a logical system’s being ‘correct’ or ‘incorrect’.”
Stanford has in mind a kind of instrumentalism whose proponents disavow belief in the target theory’s claims being even approximately true, but his point generalizes to the less radical instrumentalism considered here.
This would be an exceptionalism in the philosophy of logic—something that would indeed be unacceptable to Williamson, in view of his opposition to “philosophical exceptionalism” (2007, p. 3).
Illustrating his criticism, Blake-Turner (2020, p. 567) asks how we would respond to a scientist who, “purporting to [explain] the motions of bodies in the solar system,” actually advances a theory that explains the semantic features of words. A closer analogy would be an instrumentalist interpretation of Le Verrier’s term “Neptune”, which wouldn’t be subject to so easy a dismissal.
A referee suggested that the very consideration that makes instrumentalism more plausible for logic than for particle physics (the availability of theory-independently specified target phenomena) reinforces the change-of-subject worry. To the extent that this is a worry, though, it will apply to a plausible instrumentalism in any domain. My point isn’t that logical instrumentalism is invulnerable to such a worry, but just that the instrumentalist reading affords Williamson a prima facie answer to Blake-Turner’s objection.
Dutilh Novaes (2021) argues that “deduction” is fruitfully understood as a “dialogical notion,” as is “deductive validity.” In the present context, it’s important to add that even if predications of logical consequence carry dialectical interest, this needn’t render them fundamentally metalinguistic or metaconceptual. After all, one can hold that conditionals, disjunctions, and negations carry dialectical interest without taking them to be about a linguistic or conceptual subject matter.
For a recent survey evaluating attempts to distinguish realism about a discourse from other views that admit the discourse’s truth-aptness, see Dreier (2018).
A referee suggested that Williamson might be read as a “non-metalinguistic realist” about logical theories, according to whom they describe an objective reality of absolutely fundamental structure. If the idea is that Williamson takes sentences of form UG(\(\alpha \)) to describe such a reality, I agree. But if this reading is applied to ascriptions of logical truth and consequence, it seems “describing” must be understood so broadly that any instrumentalist interpretation of a theory will view it as describing the phenomena for which it plays its instrumental role. If Williamson were saying that ascriptions of logical properties describe a domain of non-linguistic reality, then instrumentalists about atoms would be saying that ascriptions of atomic number describe macroscopic phenomena—they would count as “non-microscopic realists” about atoms.
Instrumentalist and non-instrumentalist deflationism aren’t the only ways to reject representationalist explanations of our talk of logical truth and logical consequence. Another option is the expressivism defended by Field (2015).
Edwards (2013, pp. 290–291) characterizes a substantive property as one that “grounds genuine similarities” between its bearers (cf. Asay, 2014, pp. 149–150). While his criterion agrees with mine in focusing on whether bearers of a property “share anything significant in common,” it differs from my less metaphysically committal criterion in presupposing that it makes sense to speak of a property Fness as itself grounding the similarity between things that are F. Edwards’ criterion has the advantage of allowing primitive properties to count as substantive, where a property is primitive when there’s nothing in virtue of which anything possesses it. One could modify my criterion to include primitive properties: a property would count as substantive provided there’s a uniform answer to the question “In virtue of what do its bearers bear it?”, even if that uniform answer may be “Never in virtue of anything.”
Strictly speaking, this is inaccurate. Part of what it is in virtue of which R“P\(_1\)” \(\ldots \) “P\(_n\)” will presumably be that the sentences “P\(_1\)” through “P\(_n\)” exist, and that they have the meanings they do. But that complication won’t help establish that “R” expresses a substantive property. The more accurate explanation, which invokes the existence and semantic features of bearers of Rness, can’t do any better at locating a feature shared by all such entities in virtue of which they bear Rness.
This remains the case if we allow primitive properties to count as substantive (as explained in note 28): the equivalence at issue gives us no reason to think that the objects that bear Rness do so primitively, unless we think that there’s never anything in virtue of which \(\odot \)P\(_1\) \(\ldots \) P\(_n\).
More accurately, the argument shows that the objects that bear Rness don’t do so in virtue of any fact about language over and above those bearers’ having the meanings they do. Again, though, this qualification won’t make room for a metalinguistic conception of logical inquiry.
Still, I’ve argued elsewhere that the inferentialism of Brandom (1994), according to which implication is a matter of socially instituted norms, involves just this surprising view (Shapiro, 2018). The deflationary argument schema can be applied to any sentential connective: for instance, we gain expressive convenience by introducing a predicate Axy (read “x alternates with y”) such that \(A\langle \alpha \rangle \langle \beta \rangle \) is intersubstitutable with \(\alpha \vee \beta \). The schema can also be modified to apply to quantifiers: we can introduce Sx (read “x is satisfied”) such that where \(\phi \) is an open formula with only x free, \(S\langle \phi \rangle \) is intersubstitutable with \(\exists x\phi (x)\).
Notice that if we include identity as a logical constant, then accepting the above hypothesis would commit one to \(\Box \exists x \exists y x \ne y \leftrightarrow \exists x \exists y x \ne y\). Williamson is indeed prepared to include \(\exists x \exists y x \ne y\) as a logical truth (cf. 2017, p. 329).
However, this understanding of infinite-premise consequence won’t be available for all logics; for substructural logics where collections of premises have more structure than sets, it’s unclear how a quantifier can serve the purpose of aggregating premises into the antecedent of a binary entailment.
For essentially this objection to explaining predications of logical consequence in terms of non-metalinguistic logical compounds, see Zardini (2018, pp. 258, 273n49). In effect, he argues that theorists who see (9) as stating a fact “grounded” in non-metalinguistic conditionals can’t make sense of how multi-premise consequence talk is “illuminating.”
However, contrary to what I had taken for granted (see also Caret and Hjortland, 2015, p. 4), Griffiths claims that “[p]roponents of model- and proof-theoretic accounts of consequence do not try to capture the ‘nature’ of consequence with their definitions,” but merely aim to model consequence formally (2014, p. 179). I won’t be able to address this issue here.
Here are two examples of how such a pluralism may help with issues raised by deflationism itself. The first concerns how modus ponens is treated in the noncontractive relevance logics discussed in Shapiro (2011, 2015) as a way for deflationists to respond to semantic paradox. Here \(\beta \) is an “intensional” consequence of \(\alpha \rightarrow \beta \) together with \(\alpha \), yet fails to be an “extensional” consequence of those premises. According to a deflationary logical pluralism, the extensional consequence predicate might serve to generalize over entailments \(\Rrightarrow _{ext}(\alpha _1, \ldots , \alpha _n, \beta )\) that are equivalent to \(\Rrightarrow _{ext}(\alpha _1, \ldots ,\alpha _{n-1} \wedge \alpha _n ,\beta )\), so that \(\Rrightarrow _{ext}(\alpha \rightarrow \beta , \alpha , \beta )\) is equivalent to the rejected \(\Rrightarrow _{ext}((\alpha \rightarrow \beta ) \wedge \alpha , \beta )\). The intensional consequence predicate might serve to generalize over entailments \(\Rrightarrow _{int}(\alpha _1, \ldots , \alpha _n, \beta )\) that are equivalent to \(\Rrightarrow _{int}(\alpha _1, \alpha _2, \ldots , \alpha _{n-1}, \alpha _n \rightarrow \beta )\), so that \(\Rrightarrow _{int}(\alpha \rightarrow \beta , \alpha , \beta )\) is equivalent to the affirmed \(\Rrightarrow _{int}(\alpha \rightarrow \beta , \alpha \rightarrow \beta )\). On this proposal, contrary to Zardini (2013, p. 581n14), expressive device deflationism doesn’t require a single interpretation of multi-premise logical consequence on pain being “objectionably ad hoc.” For a second example, see Beall (2015) on how an expressive device deflationist can reply to a version of Curry’s paradox involving the entailment connective by stratifying logical consequence, so that different consequence predicates serve to generalize over different connectives \(\Rightarrow _{i}\).
It’s not clear to me which of these options more closely resembles the non-metalinguistic logical pluralism advocated by Bueno and Shalkowski (2009). For an illuminating discussion of the contextualism/relativism contrast as applied to logical vocabulary, see Stewart Shapiro (2014, ch. 4). However, deflating the relativist (“non-indexical contextualist”) form of pluralism discussed there yields the (“indexical”) contextualist form considered here, according to which “entails” expresses differently-behaving connectives in different contexts. That’s because the relativism Shapiro discusses concerns only the language’s ordinary “logical terms” (e.g. “not” and “or”), whereas the logical consequence predicate receives an (indexical) contextualist treatment.
In fact, De (2012, p. 61) claims the extension of deflationism from truth to logical consequence “trivializes,” in that it would (if cogent) extend to “any sentential relation.” But he gives no explanation of how (e.g.) the sentential relation x contains more words than y could be deflated.
One might object that if “Jan believes x” is to behave this way, it needs to be analyzed as “There exists a proposition y expressed by x, and Jan believes y”, whereas De’s objection requires a primitive predicate. However, De’s argument could be recast as concerning a primitive belief predicate that applies to propositions rather than sentences.
This may be how to understand De’s claim, concerning the consequence predicate and the entailment connective, that “each refers in different ways to logical consequence” (De, 2012, p. 68). On the opposition between the internalization thesis about operators and expressive device deflationism about metalinguistic predicates, see Shapiro (2018).
Logical inquiry, understood as concerning logical necessity or entailment, can of course lead to adoption or rejection of claims stated without a logical necessity or entailment connective. When the Liar paradox calls into question the claim \((T\langle \lambda \rangle \rightarrow \lnot T\langle \lambda \rangle ) \rightarrow \lnot T\langle \lambda \rangle \), settling that question can’t be separated from settling questions about entailments.
This contrast was stressed by a referee.
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Acknowledgements
Thanks to audiences at a CUNY-Bergen Workshop, Utrecht University, and the University of Connecticut. Additions and revisions were prompted by feedback from Eduardo Barrio, Heather Battaly, Christopher Blake-Turner, Michael De, Mengyu Hu, Özcan Karabaǧ and, William Lycan, Marcus Rossberg, Stewart Shapiro, Samuel Wheeler, and no doubt others. I’m especially grateful for comments from the anonymous referees.
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Shapiro, L. What is logical deflationism? Two non-metalinguistic conceptions of logic. Synthese 200, 31 (2022). https://doi.org/10.1007/s11229-022-03547-8
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DOI: https://doi.org/10.1007/s11229-022-03547-8