27 & 28 September 2024
a workshop on truth, definability and quantification into sentence position
at the University of Vienna

Does Sentential Quantification Tell Us What Truth Is?
Traditional theories of truth attempt to answer the question “What is truth?” This
question can be taken as a request for a real definition or a conceptual analysis. A real definition would give an account, in terms of features that are metaphysically more basic, of what makes something true when it is—a description, if you like, of the essence of truth. A conceptual analysis would specify the manner in which our concept truth is composed out of simpler concepts; or, less demandingly, it would put forward as logically equivalent to our concept truth a concept that is composed of simpler or better-understood concepts. Either of these—a real definition or a conceptual analysis—would be worth having, if we could get them. In this paper, I will consider how an explicit definition of truth in terms of sentential quantification fares as a real definition and as a conceptual analysis. For concreteness, I consider the definition offered by Künne 2003, and I argue that it falls short on both counts.

Künne’s definition looks like this: (D) ∀x(True(x) ↔ ∃p(x = [p] ∧ p)) where “[ ]” is the nominalization operator (pronounced “the proposition that”) that takes a sentence as an argument and yields a singular term that refers to the proposition expressed by that sentence. In this paper, I focus on the sentential quantificational apparatus in (D): the propositional quantifiers and open-formulae containing propositional variables. I want to know whether, conceptually or metaphysically, this apparatus points to something more basic than truth.

I begin by assessing Künne’s definition as a conceptual analysis. First, I set aside as implausible the suggestion that (D) specifies the internal structure of the concept truth. If it did, then competence with the concept truth would require competence with the apparatus of sentential quantification. But our best empirical evidence (in particular, evidence about quantification in natural language) suggests this doesn’t hold. More promising is the suggestion that (D) succeeds as a conceptual analysis by providing an explication of truth in terms of concepts that are (in some sense) better understood. The best case for this, in my view, is the suggestion that Künne’s definition offers an explanation of the “denominalization” inferences associated with the concept truth: p ⟚ [p] is true. Künne’s definition allows us to derive these inference rules from more general formal principles of higher-order logic, and, in this sense, offers a unifying explanation of the inference rules that many have suggested are constitutive of the concept truth.

In response, I offer a general argument that Künne’s sort of explanation presupposes truth, because there is no way to understand a rule as formally valid without deploying the concept truth. We might say that, for all p q, an argument from [p] to [q] is formally valid if and only if it is formally necessary that p → q. But to distinguish formal necessitation from mere necessitation, we need an account of which elements of a proposition are logical constants. If we suppose, as is common, that having a permutation invariant extension is part of what makes something a logical constant, then the notions of truth, reference, and satisfaction are ineliminable in the explanation of what makes a given rule or schema formally valid. Even if (as I think is doubtful) the notion of a proposition and the apparatus of sentential quantification can be understood independently of the concept truth, Künne’s explanation of the “denominalization” inference rules provides us with little reason to think his analysans is conceptually more fundamental than his analysandum.

I turn to assessing Künne’s proposal as a real definition. Does the definition characterize truth in terms of features of the world that are metaphysically more basic? We must ask first: what are the features of the world that answer to the apparatus of propositional quantifiers? I consider two sorts of answers, one corresponding to a (broadly) Tarskian perspective on quantifiers and variables, the other corresponding to a (broadly) Fregean perspective.

On a broadly Tarskian approach, the semantic value of a variable is a function from variable assignments to objects in some domain. In Künne’s case, the natural domain would be the domain of propositions. Traditionally, binding quantifier expressions (e.g. “∃x”) aren’t themselves given a semantic value but are interpreted syncategorematically—by describing the contribution they make to truth-value of complex sentences—but, alternatively, we can interpret the quantifier expressions as functions that map functions from assignments to propositions to propositions. In either of these cases, the Tarskian understanding of quantification introduces variable assignment functions to the ontology as the worldly correlate of variables. But if this is correct, I argue it is doubtful that the property expressed by Künne’s analysans is more metaphysically basic than truth. If there were no languages containing variables, there would be no variable assignment functions, but there could very well still be truths.

On a broadly Fregean approach, variables as such do not have denotations, but a
quantifier associated with the semantic type τ will have as a semantic value a property of properties of things in the domain associated with τ . So for instance, “∃x” denotes the property of being an instantiated first-order property. In the case where τ is the semantic type of a sentence, there is a question of what property is being characterized when “∃p” is affixed directly to the variable “p,” as in “∃p.p.” I argue that, given the logical properties of sentential quantification, the only serious candidate is the property of being true. On this understanding, the worldly underpinning of a claim like “∃p(c = [p] ∧ p),” Künne’s analysis of “c is true,” is that the property of being true is coinstantiated with the property of being mapped by the nominaliziation function to something identical with c. Clearly, if the property of being true figures directly in the characterization of what features of the world are expressed by the apparatus of sentential quantification, a definition of truth in terms of sentential quantification does not explain what truth is in more metaphysically fundamental terms.

Plural Propositional Quantification and Truth
Recent metaphysics has seen a resurgence in the use of the languages of higher-order logic. In such a setting, talk of propositions is regimented using propositional quantifiers, i.e., quantifiers binding propositions which take the position of sentences. In recent work, I have also employed plural analogs of such propositional quantifiers, which I call “plural propositional quantifiers”.
Here, I discuss a recent objection to plural propositional quantifiers. The objection argues that given the way I introduce such quantifiers, plural propositional variables must be able to function as sentential terms. It follows that an (unambiguous) sentence can express multiple propositions. The objection continues by claiming that this consequence is untenable, which can be brought out by considering what it would take for a sentence expressing multiple propositions to be true.
In my reply, I reject both the suggestion that plural propositional variables must be able to function as sentential terms, and the suggestion that there is a problem concerning the truth of (unambiguous) sentences expressing multiple propositions.

Ramsey on the Indefinability of Truth
It seems, on first glance, that F.P. Ramsey held that truth is definable. Early in his short career, he put it this way: ‘p is true’ and ‘p’ are equivalent. Later he expressed it this way: a statement is true if and only if there is a way it says things are and they are thus. But he argued that the definition of truth is of limited interest. An account of truth needs to be mostly an account of belief and an account of belief will be mostly about the downstream effects of belief—the myriad of actions that cannot be swept up into a definition. My argument will be that if we take a second look at Ramsey’s extant papers and manuscripts, it becomes clear that he thought that the concept of truth is not capable of being precisely defined or analysed. Since definability is a matter of being subject to precise analysis, my suggestion will be that Ramsey, despite trotting out a famous definition of truth, thought that truth was indefinable.

Deflationism and propositional quantification
There are several natural ways of using propositional quantifiers to generalize over the basic deflationist idea that truth-ascriptions are equivalent to appropriately “de-nominalized” sentences, most notably,

∀x (x is true iff ∃p (x = <p> & p)) (Ramsey (unpublished?), Baldwin (1989), Künne (2003))
or
∀p (<p> is true iff p) (Hill (2002), Båve (2013))

Paul Horwich objects against these formulations that they cannot allow us to infer the wanted instances without circularity. If the quantifiers are characterized in terms of truth, then, I will assume, this is correct. But Horwich also argues that the same holds if they are characterized inferentially, arguing that the inference rules would have to be specified using propositional quantifiers, leading to a regress. I argue that this argument is an instance of Carroll’s regress, and that, accordingly, the circularity is merely apparent. He also argues that the above formulations would be in tension with the deflationist idea that the truth predicate helps us “dispense with” propositional quantifiers. I consider several clarifications of this objection and argue that it fails on each of them. A general conclusion is that since, on pain of circularity, we cannot use the truth predicate in order to generalize over the instances, we need some other means of accomplishing this task. But this task, by the deflationist’s own lights, is of course performed by propositional quantifiers.

I then point out that an inferential characterization of propositional quantifiers is the one deflationists should adopt, given what most believe, that deflationism is inconsistent with characterizing the meanings of logical constants (and other expressions) truth-theoretically. Building on Stewart Shapiro (2001), I propose a formal inferentialist characterization of the propositional quantifiers, and then argue, firstly, how this account can underpin the idea that Horwich’s circularity objection resembles Carroll’s regress in relevant respects, and, secondly, that it provides a more precise sense in which the truth-predicate allows us to “dispense with” propositional quantifiers.

I next go on to argue that propositional quantifiers, taken as defined by Shapiro-style inference rules, make them intelligible and coherent if first-order objectual quantifiers are. That is, I point to ways in which they parallel each other, with the upshot that various objections against the use of propositional quantifiers—generally or by deflationists about truth specifically—are sound only if standard first-order quantifiers are disqualified on the same grounds (which I am assuming is absurd).

Finally, I show how theories of truth stated with propositional quantifiers can explain general facts about truth, which is a task that Horwich’s Minimal Theory has notorious difficulties with. This presupposes certain uncontroversial inference rules for first-order quantifiers, but also some less obvious, if plausible, assumptions about that-clauses functioning as terms instantiating universal first-order quantifiers. I argue that this is still less contentious than Horwich’s proposal of using an omega-rule to infer the desired generalizations (cf. Raatikainen (2005)).

References
Baldwin, T. (1989), “Can There Be a Substantive Theory of Truth?”, Recherche sur la Philosophie et le Langage 10, pp. 99-118.
Båve, A. (2013), “Formulating Deflationism”, Synthèse 190, pp. 3287–3305.
Hill, C. (2002), Thought and World: An Austere Portrayal of Truth, Reference, and Semantic
Correspondence, Cambridge, Cambridge University Press.
Horwich, P. (1990/2000), Truth, Oxford, Clarendon Press.
Künne, W. (2003), Conceptions of Truth, Oxford, Oxford University Press.
Raatikainen, P. (2005), “On Horwich’s way out”, Analysis 65, pp. 175–177.
Shapiro, S. (2001), “Classical Logic II: Higher-Order Logic”, in: L. Goble, ed., Blackwell Companion to Philosophical Logic, Oxford, Blackwell, pp. 33–54.

Sentential-Variable Deflationism and Adverbial Quantification
We explain and motivate a specific version of what we call “sentential-variable deflationism” (SVD). This deflationary approach, suggested by Ramsey’s (1929) later work on truth, starts with sentential variables and quantifiers and uses these logical devices to explain the logical role and meaning of truth-talk. So, a sentence like

(1) Everything the Oracle says is true

gets explained via an appeal to a “quantified sentential-variable” (QSV) string like

(2) For all p, if the Oracle says that p, then p.

Since SVD makes sentential variables and quantifiers the explanatory basis of truth-talk, a theory of this sort must provide an independent account of these “higher-order” logical devices. Many theorists have assumed that the sentential devices illustrated in (2) should be understood as substitutional quantifiers and variables, but there are grave concerns about this assumption. Depending on how these substitutional devices are explained, this approach would either raise circularity worries for SVD, by explaining substitutional quantification in terms of the truth of the substitution results (cf. Parsons (1971), Kripke (1976), David (1994, pp. 85-90), Horwich (Ibid., pp. 25-6)), or would generate “fragmentation” concerns by explaining substitutional quantification as a means for encoding infinite conjunctions or infinite disjunctions of substitution instances (cf. David (Ibid., p. 98-9)), rather than as a means for expressing genuine generalizations (cf. Gupta (1993)).

Azzouni (2001, 2006) and Picollo and Schindler (2018, 2022) avoid the problems that arise for a substitutional approach by stipulating alternative, technical accounts of the ‘For all p’ and ‘p’ employed in (2) and its ilk, which are non-substitutional. They advocate this “stipulated formalism” approach because, in their views, there is no way to explain the generalizing role that truth-talk implements using natural language. The question for SVDists is whether they could and should co-opt any of their proposals as the prior basis for an account of truth-talk. If Azzouni and Picollo and Schindler are right about what is required to explain the generalizing role indicated by the ‘For all p’ in (2), then SVDists would need to go this route. But if it is possible to capture the kind of sentential generalizing indicated in (2) by ‘For all p’ directly in a natural language, then this would obviate any need for going this “stipulated formalism” route and would thus provide a sufficient reason for not taking this approach.

We maintain that the natural-language interpretation (NLI) route is available if one adapts an approach that emerges from A.N. Prior’s work. Prior (1956, 1971) understands sentential variables and quantifiers as sui generis formal devices that function as non-nominal, adverbial sentential variables and quantifiers. Accordingly, we call this type of deflationary account of truth-talk “adverbial sentential-variable deflationism” (ASVD). An ASVDist need not follow Prior and embrace his informal adverbial quantifier and variable neologisms, ‘anywhether’, ‘somewhether’, and ‘thether’. As stipulated devices, these expressions provide only an impressionistic grasp of an adverbial reading of sentential variables and quantifiers. For a more informative account of these formal devices that we can use to explain (rather than merely model) the generalizing role of truth-talk, we prefer the NLI approach, and for this we need to go beyond stipulated expressions. Prior himself suggests how to do this, when he notes that one can understand ‘For some p, p’ in terms of ‘Things are somehow’ and can understand ‘For all p, if he says that p, then p’ in terms of ‘However he says things are, thus they are (or, that’s how they are)’. (Cf. Prior (1967, p. 229) and (1971, p. 38)).

We maintain that there are ways of extending Prior’s “how-talk” suggestions to provide an adverbial reading of sentential quantification in general. Taking ‘this is how things are’, or, more typically, ‘that is how things are’, as English adverbial sentential-variable expressions, we interpret the QSV strings employing sentential quantifiers and variables via English-language how-talk, using the ordinary expressions ‘somehow’, for ‘For some p’, and ‘however’, for ‘For all p’. Thus, we understand a QSV string like

(3) For some p, p and the Oracle said that p

in terms of

(4) For somehow things are, that is how things are, and the Oracle said that that is how things are.

Sentence (4) is formulated in a way that tracks the structure of (3), but it bears noting that quantifying with respect to “how things are” should not require that there always are such things that are thus, as (4) suggests. Rayo and Yablo (2001) handle this by reading how-talk quantifiers (governing predicate variables) disjunctively, in terms of somehow and however some things are or are not related. We can modify their move to apply to quantifiers governing sentential variables by expanding the how-talk quantifier interpretations of them to ‘For however things are or are not’ and ‘For somehow things are or are not’. Applying this to (4) yields,

(4+) For somehow things are or are not, that is how things are, and the Oracle said that that is how things are.

With the disjunctive element of how-talk quantification thus made explicit, we can provide an NLI of a universal generalization like (2) as

(5) For however things are or are not, if the Oracle says that that is how things are, then that is how things are.

A striking aspect of the how-talk NLI approach is that it deploys a form of non-nominal quantification, which does not involve a domain of entities that serve as the values of the variables (no “hows”), or even a class of linguistic items that serve as the substituends of the sentential variables. These sui generis quantifiers are neither objectual nor substitutional and are not to be explained by an appeal to “ways” (contra Künne (2003)). The understanding they provide of sentential variables and quantifiers provides a deflationary account of truth-talk in terms of quantification into sentence position. As we explain, our approach renders truth-talk dispensable since, as we show, its expressive roles can be captured in a natural language like English by how-talk quantification.

Deflationary Accounts of Truth
My talk will be divided into three parts. To start with, I’ll begin to address the question of which is the better deflationary account of the meaning of “true” (when predicated of propositions) — as between

(i) My use-theoretic, so-called ‘minimalist’ suggestion, whose heart is the view that the English meaning of “true”, when applied to (or denied of) propositions, consists in our disposition of English speakers to accept instances of the schema: “The proposition that p is true <–> p”.

(ii) And Wolfgang Künne’s proposed definition of “x is true” as 
“(Ep)(x = <p>  & p)”, which invokes quantification into sentence positions.

After that – in the relatively short 2^nd part of my talk – I’ll turn to the question of whether either of these proposals can be adequate, since, arguably, neither of them is able to do justice to the plausible view that a statement that’s untranslatable into our language might nonetheless be true.

And finally. I will – in light of those discussions, I’ll offer my answers to the 5 question that are posed in the workshop announcement:
1) Is truth definable?
2) Is propositional quantification coherent?
3) Do natural languages involve propositional quantification, and in what sense?
4) What do the answers to these questions mean for philosophical attempts to define or explain truth?
5) Is truth redundant if explicitly definable? And, not redundant if not explicitly definable?