Higher-Order Logic and Disquotational Truth

Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature of this function. We offer a series of formal results in support of the thesis that disquotational truth is a device to simulate higher-order resources in a first-order setting. More specifically, we show that any theory formulated in a higher-order language can be naturally and conservatively interpreted in a first-order theory with a disquotational truth or truth-of predicate. In the first part of the paper we focus on the relation between truth and full impredicative sentential quantification. The second part is devoted to the relation between truth-of and full impredicative predicate quantification.


Introduction
Truth predicates can serve a certain logical or quasi-logical function. Examples of this function are easily found in the literature. Suppose you agree with Einstein's remarks about relativity but can't quite remember what he said. Then you can express your agreement by saying 'What Einstein said about relativity is true'. Or suppose you want to express your disagreement with a theory that isn't finitely axiomatised but you can't put your finger on the particular axioms that are to blame. Then you can say 'Not every theorem of the theory is true'. However, it is not entirely clear how this function can be best understood or more precisely characterised.
Frequently, the truth predicate is described as a device for expressing infinite conjunctions and disjunctions (e.g. Putnam [1], Gupta [2]); sometimes it is said to provide finite axiomatisability (e.g. Halbach [3]); others have characterised it as a device for sentential quantification (e.g. Azzouni [4], Grover et al. [5], Parsons [6]). In [7] we have argued that the first two accounts are only partially correct and fall short of providing a full account of the logical function of truth. In [8] we have proposed a positive account based on the third proposal: the truth predicate, together with ordinary first-order quantifiers, enables us to emulate quantification into sentence and predicate position within a first-order framework -i.e. we can emulate all inferences. In other words, the truth predicate provides us with the same inferential power as higher-order quantifiers but in a grammatically conservative way, that is, without abandoning the convenient framework of first-order logic. In [8] we offered philosophical arguments in support of this thesis; in the present one we provide a series of formal results to this effect.
We focus first on the relation between truth and quantification into sentence position -'sentential quantification', for short. Assume we want to express the factivity of knowledge as a single claim, this is, to universally generalise on ϕ's position in the schema Kϕ Ñ ϕ where K is a sentential operator for knowledge. This is usually done with the aid of sentential quantifiers, as follows: @α pKα Ñ αq (Factivity α ) We submit that the following principle formulated in terms of a truth predicate T instead does exactly the same job as far as the inferential behaviour is concerned: @v pSentpvq Ñ pKpTvq Ñ Tvqq (Factivity T ) provided that Sentpvq applies to the appropriate range of sentences, and each sentence in this range is intersubstitutable with its truth ascription also in the context of the knowledge operator. 1 That is, we maintain that the inferential behaviour of (Factivity T ) is equivalent to that of (Factivity α ). Similarly, if one wishes to express that some statements made by Einstein hold, one can turn to a one-place operator E, for 'Einstein said that', and sentential quantifiers and write Dα pEα^αq (Einstein α ) 1 Alternatively, one could deploy a one-place predicate K˚for knowledge and simply write @v pSentpvq Ñ pK˚pvq Ñ Tvqq (FactivityT ) We submit that the same effect can be achieved by means of a truth predicate instead, by Dv pSentpvq^pEpTvq^Tvqq (Einstein T ) with the same proviso as in the previous example.
In general, sentential quantifiers allow us to quantify into sentence position over a given class of sentences (possibly containing sentential quantifiers themselves), namely those for which we have instances of a comprehension principle for the new quantifiers (cf. Section 3.1). We would like to suggest that disquotational truth predicates work in a similar way: they allow us to generalise into sentence position precisely over the class of sentences for which we have an instance of disquotation.
To support our claims, we show that every admissible inference in a language with (full impredicative) sentential quantifiers can be translated into an admissible inference in a suitable disquotational truth theory without sentential quantifiers in a natural way, that is, along the lines of the given examples. Roughly, expressions like (Factivity α ) are translated as (Factivity T ), and expressions like (Einstein α ) as (Einstein T ). This is a natural translation inasmuch as (Factivity T ), (Einstein T ), and expressions of the like are straightforward formalisations of the natural language readings we usually make of (Factivity α ), (Einstein α ), and other expressions involving sentential quantification. We typically deploy a truth predicate to pronounce these latter expressions in natural language, since a more direct reading would be ungrammatical. The idea behind the truth theory is to include an instance of disquotation for each translation of a formula in the range of comprehension.
Moreover, we show that the truth theory does not license any more inferences between translations than sentential quantifiers between the original formulae. In other words, sentential quantifiers allow for the same inferences as disquotational truth, modulo the given translation.
We propose an analogous analysis of the relation between satisfaction and (full impredicative) second-order quantification, understood as a theoretical device for quantification into predicate position. 2 Assume we wish to express (the following version of) Leibniz's Law as a single claim by universally generalising on ϕ's position in @u@v pu " v Ñ pϕpuq Ñ ϕpvqqq (LL-schema) Using second-order quantifiers, we can write @V @u@v pu " v Ñ pV u Ñ V vqq (LL) We submit that a disquotational truth-of or satisfaction predicate, Sat, affords us the same inferential power. For one can formulate Leibniz's Law as a single claim as follows: @w pForm 1 pwq Ñ @u@v pu " v Ñ pSatpw, uq Ñ Satpw, vqqqq (LL T ) where Form 1 pvq holds exactly of all relevant formulae with just one free individual variable.
To support our thesis, we show that every admissible inference in a language with (full impredicative) second-order quantification can be translated into an admissible inference in a suitable disquotational truth or satisfaction theory without secondorder quantification in a natural way, that is, along the lines of the given examples. Expressions like (LL) are roughly translated as (LL T ). Again, this is a natural translation, as (LL T ) and expressions of the like are straightforward formalisations of the natural language readings we typically make of (LL) and other expressions involving second-order quantification. We usually employ a truth-of or satisfaction predicate to pronounce the latter into natural language, as more direct readings would be ungrammatical. As before, the idea behind the truth or satisfaction theory is to include an instance of (uniform) disquotation for each naturally translated formula in the range of second-order comprehension.
Furthermore, we show that the disquotational theories do not license any more inferences between translations than second-order quantification does between the original formulae. In other words, second-order quantifiers afford us exactly the same inferences as disquotational truth or satisfaction, modulo the given translation.
In fact, the results mentioned above can be generalised to higher orders: every inference in a language with (full impredicative) n-th-order quantifiers can be translated into an admissible inference in a suitable disquotational truth or satisfaction theory and no more inferences are valid in the latter.
In the more formal literature on the subject one can find many results relating theories of truth to theories formulated in a higher-order language. For example, it is well known that the theory of arithmetical comprehension ACA and the theory of typed compositional truth CT are mutually interpretable (cf. Takeuti [10, chap. 3.18] and Halbach [11, chap. 8]). A similar result holds between the system of ramified truth RT ă 0 (essentially a formalisation of Tarski's hierarchical truth theory) and ramified analysis RA ă 0 , and between the Kripke-Feferman theory of truth KF and RA ă 0 (cf. Feferman [12] and Halbach [13,14]).
However, these results are somehow partial and limited in scope. First, they often relate predicative fragments of higher-order logic to theories of truth, 3 leaving the wrong impression that full impredicative higher-order quantification cannot be handled by a truth predicate. Second, they often relate these fragments of higher-order logic to compositional theories of truth, suggesting that they cannot be handled by a disquotational truth predicate. 4 On the other hand, in the philosophical literature there is an almost exclusive focus on the relation between sentential quantifiers and the truth predicate, neglecting the fact that the truth predicate can actually allow us to mimic quantification into predicate position as well.
By contrast, the results offered in this paper show that full impredicative sentential and predicate quantification (for all finite orders) can be emulated by suitable disquotational theories of truth or satisfaction. We would like to emphasise, however, that we do not necessarily endorse these as our best truth theories, as the choice of axioms is a bit derivative. It would be desirable if more natural systems, from which these axioms follow, could be devised.
Having a proper grasp of just how powerful disquotational truth can be is philosophically important. For example, in [8] and [18] we have argued that our account of the function of truth helps to dispel the objection that a deflationary theory of truth needs to be conservative: 5 if it is the job of the truth predicate to emulate higher-order quantification, then we shouldn't be surprised that the addition of a truth predicate can lead to non-conservative extensions, because the same holds for higher-order quantification. For a more detailed philosophical analysis of our results we refer the reader to our aforementioned papers.
Before we move to the formal part of the paper, let us briefly address an objection to our account raised by Nicolai [21]. He argues that, if disquotational truth were a tool to simulate higher-order quantification in a first-order setting, a strong form of theoretical equivalence should hold between disquotational truth theories and corresponding higher-order systems. 6 As Nicolai correctly points out, mere mutual interpretability, proof-theoretic equivalence, or mutual truth-definability do not suffice to support the view defended in this paper. The reason is that in general these forms of theoretical equivalence are unreliable. They hold between obviously disparate systems, as they can be based on unnatural translations. 7 As Nicolai [21, p. 20] argues "a natural correspondence between truth and quantification" is required. However, stronger forms of theoretical equivalence that would avoid this issue -e.g. bi-interpretability and definitional equivalence -fail to hold even in the simplest cases (cf. Nicolai [21,Prop. 4]). Nicolai concludes that the absence of an adequate form of theoretical equivalence between higher-order systems and truth theories to support the view that truth is a device to mimic higher-order quantification strongly suggests that the view is mistaken.
In this paper we propose an alternative criterion for theoretical equivalence between truth and higher-order resources that relies on natural translations. Thus, despite being weaker than bi-interpretability and definitional equivalence, it avoids the charges held against mutual interpretability and other insufficient forms of theoretical equivalence. We believe that our criterion successfully supports our claim that disquotational truth (or satisfaction) is a device to simulate higher-order resources in a first-order setting.
The paper is structured as follows. In Section 2, we give some technical preliminaries for the remainder of the paper. In Section 3 we focus on the relation between truth and sentential quantification. Section 4 is devoted to truth and second-order quantification in languages that contain a name for every object in the intended domain of the first-order quantifiers, and Section 5 to satisfaction and second-order quantification, without the previous qualification. 5 See, for instance, Shapiro [19] and Ketland [20]. 6 See Halbach [11, chap. 6] for an overview of the different kinds of theoretical equivalence deployed in the literature and Fujimoto [22] for a more in-depth discussion. 7 For example, PA is mutually interpretable with PA` ConpPAq (i.e. PA together with an axiom stating that PA is inconsistent), but few would consider these theories to have the same theoretical status. Let L be any effectivised 8 (standard) first-order language, with logical symbols ", ,^, and @ and individual variables v 1 , v 2 , . . . Other logical symbols should be understood as the usual abbreviations. L may also include individual constants a 1 , a 2 , . . . and predicates F n 1 , F n 2 , . . . of each arity n, and other non-logical symbols to be specified. To simplify our presentation we don't allow function symbols in L, but our results can be rather easily extended to languages with them. L is intended as (a formal representation of) the language we use prior to the addition of expressive resources that may allow us to generalise into sentence or formula position.
In what follows, let u, v, . . . stand for arbitrary individual variables, c for arbitrary constants, t for arbitrary terms, F for arbitrary predicates, ϕ, ψ, . . . for arbitrary formulae, and Γ, Δ, . . . for arbitrary sets of formulae, possibly with subindexes. We identify each language with the set of its formulae and terms.
Let L T expand L with a fresh copy of the signature of the language of Peano arithmetic (PA), L PA , and monadic predicates D, for the domain of discourse of L, and T, for truth. L PA will be used to express syntactic notions of L and the expansions thereof with which we will work in Section 3 and Section 4. 9 (By contrast, in Section 5 we will use a set-theoretic language to talk about the syntax of L.) We assume a fixed effective coding # of expressions of L and these expansions by numbers. If n is the code of an expression , we write x y for the numeral of n in L T -i.e. the term that consists of n occurrences of the successor function symbol S followed by the constant 0.
Note that neither 'L' nor 'L T ' denote particular languages; they are both variables ranging over languages with the aforementioned characteristics. By contrast, 'L PA ' is a proper name for the language of PA. In subsequent sections we will introduce further assumptions about L and L T as appropriate.
We assume L PA contains predicate and function symbols for primitive recursive (pr) sets and functions to be specified, including a monadic predicate N for the set of natural numbers. 10 As is well known, finite sequences of numbers can be recursively coded by numbers themselves. Let xv 1 , . . . , v n y n be an n-place function symbol for the pr function that maps each n-tuple of natural numbers to its code. We often omit the superscript, for readability. Also, let subpu, vq define the pr function substitution, 8 Very roughly, an effectivised language is a language whose symbols are gödelised -entailing that the set of its symbols is countable and the set of its codes computable. For further details, see Monk [23, def. 10.2]. 9 To prove general results to the effect that the apparatus of higher-order quantification can be consistently and conservatively substituted with that of truth, the signatures of L and L PA must be disjoint (and the truth predicate of L T must not occur in L either). This is why we work with fresh copies of the arithmetical and truth-theoretic vocabulary, thereby excluding theories that say something about the (same) syntactic objects -i.e. the natural numbers. Note, however, that these theories are consistently extensible with our truth-theoretic machinery all the same, although the resulting systems might have more (translated) consequences than their higher-order counterparts. For, if any such theory Γ is indeed about the natural numbers (or truth), then it must be sound, so it should not contradict our syntactic or truth-theoretic principles (provided, of course, that the latter are themselves sound). Nonetheless, Γ might entail some (sound) syntactic or truth principles already, which is why conservativeness isn't guaranteed. 10 We assume that in the formulation of the axioms of Peano arithmetic all quantifiers are relativised to N. that takes the code u of a formula ϕ and the code v of a finite sequence of numbers k 1 , . . . , k n and returns the code of the formula that results from replacing in ϕ the free variable with the lowest subindex with the numeral of k 1 , the free variable with the second lowest subindex with the numeral of k 2 , and so on. Finally, if v 1 , . . . , v n (with increasing subindexes) occur free in ϕ, we abbreviate subpxϕy, xv 1 , . . . , v n yq by xϕp 9 v 1 , . . . , 9 v n qy. For any expressions 1 , 2 of the same grammatical category, let ϕr 1 { 2 s be the result of substituting 1 for every occurrence of 2 (which should be free in case 2 is a variable), where 2 is free for 1 in ϕ. Finally, let any sequence of distinct expressions 1 , . . . , n be abbreviated by .
All the calculi we work with throughout the article extend a standard natural deduction calculus for first-order logic with identity, FOL, given by the following axiom and rules of inference: if v doesn't occur in an undischarged assumption (@vE) @v ϕ $ ϕrt{vs

Truth and Sentential Quantification
In this section we show that truth can mimic full impredicative sentential quantification in a natural way, as indicated in the introduction -i.e. we can simulate all inferences. Adding sentential quantifiers to any first-order language can be perfectly simulated by the addition of a truth predicate governed by a restricted form of disquotation instead. Furthermore, we show that adding a truth predicate of those characteristics doesn't add any inferential power to our language, above and beyond those already afforded by sentential quantification. Interestingly, although the disquotational truth theories in question are formulated in a language containing arithmetical vocabulary, no arithmetical axioms are needed. In more detail, we first expand L to a language L S with sentential quantifiers and extend FOL to a calculus SQL for L S with inference rules for the new quantifiers. Then, in Section 3.2, we provide a semantics for L S and show that SQL is sound and complete with respect to it. In Section 3.3 we give a natural translation η of expressions of L S into L T , which will help us formulate a truth theory UTBrηs in L T , roughly consisting of all instances of (uniform) disquotation for translations. We then show in Section 3.4 that any theory Γ formulated in SQL can be relatively interpreted in the L T -theory ηpΓ q, formulated over UTBrηs. Moreover, in Section 3.5 we show that if an inference between translations is admissible in UTBrηs then the inference between the corresponding original formulae of L S is admissible in SQL, relying on the completeness result proved in Section 3.2. Hence, the inferential power of our truth theory doesn't exceed the inferential power of sentential quantification.

Sentential Quantification
Throughout this section we allow L to contain any (computable) subset of sentential constants s 1 , s 2 , . . ., which play the syntactic role of atomic sentences of the language, and finitely many non-logical operators l n 1 , l n 2 , . . . of various arities n. Our definitions and proofs, however, are given for monadic operators only, to facilitate readability. (When it isn't entirely obvious how to generalise to arbitrary arities, we provide precise formulations in footnotes.) Non-logical operators are used to express modalities such as 'Einstein said that . . .', 'it's a theorem of arithmetic that . . .', or '. . . grounds . . .'. Unlike predicates, operators of this kind can interact directly with sentential quantifiers, allowing us to generalise on schematic principles as we've seen in the examples in the introduction. 11 Let L S expand L with sentential -also known as 'propositional' -variables α 1 , α 2 , . . . and a λ-term λ.ϕ for each sentence ϕ of L S . Sentential variables are (open) formulae of L S . By contrast, λ-terms are sentences of this language. They are introduced to the language solely for technical reasons, as we explain after Definition 7.
It should be noted that, in the absence of non-logical operators (and quotation functions or functions of the like), sentential quantification can be simulated in firstorder logic alone, without the need of a truth predicate. For, in that case, @α ϕ can be defined simply as ϕrJ{αs^ϕrK{αs and Dα ϕ as ϕrJ{αs _ ϕrK{αs, where J stands for any logical truth and K for any logical falsity. By contrast, non-logical operators bring forth an intensional aspect. For instance, we would like to keep open the possibility that, for some sentence ϕ that is neither a logical truth nor a logical falsity, l ϕ but neither l J nor l K -e.g. if l is to express 'Einstein said that . . . '.
In what follows, let α, β, . . . stand for arbitrary sentential variables, s for arbitrary sentential constants, and l for arbitrary non-logical sentential operators.
Definition 1 Let SQL formulated in L S extend FOL with the following rules of inference: Note that p@αEq only allows us to instantiate the sentential universal quantifier with a formula without free individual variables. This is for quantificational hygiene purposes. Otherwise, sentential quantifiers would allow us to quantify not only into sentence but also into formula position. We will deal with quantification into predicate position in subsequent sections.
(SLE) states that logically equivalent expressions are intersubstitutable in all modal contexts. It entails a coarse-grained account of the kind of "objects" to which the modalities apply. This is highly undesirable, for instance, if one wishes to deploy an operator l G to express a hyperintensional notion such as the logical grounding relation, as it is usually done. According to most theories of logical grounding, while for each true sentence ϕ, l G pϕ, ϕq, i.e. ϕ grounds ϕ, it's not the case that l G pϕ, ϕq, i.e. ϕ doesn't ground itself. Note, however, that the relative interpretability result we offer in Section 3.4 doesn't rely on (SLE). It is only when we prove the converse direction -that our truth theory doesn't outstrip the inferential resources available in SQL -that substitution under logical equivalents is required. Thus, every inference that can be carried out in SQL minus (SLE) can be naturally simulated in a disquotational truth theory without (SLE), whereas it's an open question whether more inferences can be drawn in the truth theory without (SLE) than in SQL in absence of this rule.
Finally, note that SQL entails the following: Note that free as well as bound sentential variables may occur in ϕ, i.e. this is a principle of full impredicative comprehension. To see why it follows, consider the following derivation in SQL: 1. @α pα Ø ϕq assumption 2. pϕ Ø ϕq p @ αEq 3. @α pα Ø ϕq p Iq, 1-2

A Semantics for Sentential Quantification
In this section, we provide a semantics for L S and show that SQL is sound and complete with respect to it. In Section 3.5, the completeness result will help us show that every inference between translations of formulae of L S that can be drawn in our truth theory can be also drawn in SQL between the corresponding original expressions. Fine [25] and Kaplan [26] have provided a semantics for languages with sentential quantifiers and intensional operators that generalises Kripke's possible-world semantics for modal logic. Unfortunately, both of them focus on propositional rather than first-order languages. Moreover, neither can be extended to a semantics for first-order languages that is adequate for our present purposes. Whilst in Kaplan's semantics the sentential operators are only intended to express alethic modalities, in Fine's the truth conditions of sentences of the form lϕ at a world are given in terms of the accessibility relation, as is usual. As a consequence, in both cases the interpretations of the sentential operators are closed under logical consequence so, if l were to express, e.g. 'Einstein said that . . .', we would be committed to the idea that Einstein said -the content of -every logical consequence of 'The most incomprehensible thing about the universe is that it is comprehensible'. But this seems undesirable. Note also that, in Kaplan's semantics, the range of the sentential quantifiers in each model is the whole power set of the set of possible worlds. Hence, if we extended this semantics to a first-order language, no calculus would be complete with respect to it.
By contrast, we provide a Henkin-style semantics, in which sentential quantifiers are allowed to range over just a subset of the power set of the set of possible worlds, and the interpretation of the sentential operators is only closed under logical equivalence. It should be emphasised that our semantics is intended merely as a formal tool to establish our results: we wouldn't like to imply that this is the intended semantics for L S .

Definition 2 Let an
M`i s an interpretation function for the non-logical vocabulary of L S satisfying the following conditions: Informally, D M is the universe of discourse for the first-order variables to range over, W M is a set of possible worlds, P M is a set of propositions (where a proposition is simply a set of possible worlds) for the sentential variables to range over, and each }¨} w M`a ssigns an interpretation to the non-logical vocabulary built from elements of D M as usual, except sentential constants take truth values t or f and non-logical operators take sets of propositions in P M as their interpretation. Since individual constants are rigid designators (they receive the same interpretation in each world), let us simply write }c} M`. Definition 3 An assignment g for an L S -structure M maps each individual term to D M and each sentential variable to P M so that gpcq " }c} M`.
For each variable ξ of any type, let gro : ξ s be an assignment identical to g except it maps ξ to o.

Definition 4
Let M be an L S -structure, w P W M , and g an assignment for M. Truth in M, w, g is given as follows: 13 In general, if l is an n-place operator, we have }l} w M`P ℘pP n M q.
A formula of the form lϕ is true in M, w, g just in case the set of possible worlds in which ϕ is true under g belongs to the interpretation of l in w. Simply put, the proposition expressed by ϕ belongs to the interpretation of l in w.
If M is an L S -structure, let us write M, w, g ( Γ as short for 'M, w, g ( ψ for every ψ P Γ '. The following definition of a faithful L S -structure is modelled after the Henkin semantics for second-order languages. Definition 5 M is a faithful L S -structure iff every instance of (Sentential Comprehension) is true in M.
The rest of this section is devoted to proving the following adequacy theorem: We only prove this result for finite Γ as, first, this is all we need for the purposes of this paper and, second, the proof of the more general result is considerably longer and more convoluted. 15 The right-to-left direction -i.e. the soundness of SQL with respect to the class of faithful L S -structures -follows easily by induction on the length of proofs. For the proof of the left-to-right direction, that is, the completeness of SQL, we will follow Henkin's standard strategy for the most part. We first extend every consistent set of sentences of L S to a maximally consistent set with the witness property. Then, we show that any such set has a model. This latter result will become useful in Section 3.5 as well. 14 In general, if l is an n-place operator, M, w, g ( lpϕ 1 . . . , ϕ n q iff xtw 1 P W M | M, w 1 , g ( ϕ 1 u, . . . , tw 1 P W M | M, w 1 , g ( ϕ n uy P }l} w M`. 15 The difficulty lies on the left-to-right direction, that is, the completeness of the calculus with respect to SQL-consequence, for which we rely on a version of Henkin's Lemma -our Lemma 1. For reasons that will become clear later on (cf. fn 19), if we let Γ be infinite in Theorem 1, we should generalise Lemma 1 to sets of formulae, not just sentences. But this complicates the proof of the latter a great deal (cf. fn 18).

Lemma 1 Every consistent set of sentences of L S has a faithful model M in which
Proof Let LS expand L S with fresh sentential constants s1 , s2 , . . . and fresh individual constants a1 , a2 , . . .. These fresh constants will play the role of witnesses in our maximally consistent sets. Let ϕ 1 , ϕ 2 , . . . be an enumeration of the sentences of LS s.t., for each n, both an and sn occur in sentences only after the n-th place.
Let Γ Ď L S be a consistent set of sentences. We first extend Γ to a maximally consistent set of sentences Δ Ď LS with the witness property, that is: 16 Thus, in Δ every existential claim ϕ n has a witness, an or sn . Let: Ť iPω Γ i Δ can be easily seen to satisfy conditions 1-3 and to be a subset of Γ . Δ is also guaranteed to be consistent by construction, as adding together with each existential statement a corresponding witnessing claim results in inconsistency only if adding the existential statement alone does so too. If ϕ n :" Dv ψ, given our enumeration of sentences of LS , an cannot occur in Γ n , ϕ n . Thus, if Γ n Y tϕ n , ψran {vsu is inconsistent an does not play an essential role in the derivation of a contradiction: everything that can be derived from ψran {vs in this context follows equally from Dv ψ (i.e. ϕ n ). Therefore, Γ n , ϕ n must be inconsistent too. Similar remarks hold if ϕ n :" Dα ψ.
Let rΔs be the class of maximally consistent sets of sentences of LS with the witness property that share with Δ the same identity statements and the same sentences of the form psn Ø ϕq, where Dα pα Ø ϕq -which belongs to every maximally consistent set -is the n-th sentence in our enumeration. Note that the relation between two individual constants c 1 and c 2 of LS that holds just in case c 1 " c 2 P Δ -or, what is the same, c 1 " c 2 belongs to every member of rΔs -is easily seen to be an equivalence relation. Let rcs be the equivalence class to which c belongs.
Next, we define a faithful LS -structure with countable domains for both firstorder and sentential quantifiers and a world in which all members of Δ -and, thus, all members of Γ -are true. Let M rΔs " xD rΔs , W rΔs , P rΔs , }¨} rΔs y be given as follows: i. D rΔs " trcs | c P LS u ii. W rΔs " rΔs iii. P rΔs " ttΣ P rΔs | s P Σu | s P LS u iv. For each Σ P W rΔs , rΔs " t iff s P Σ Thus, the first-order domain of M rΔs , D rΔs , consists of the equivalence classes of individual constants that co-denote according to all sets in rΔs, and each member of the latter set serves as a possible world in the model -i.e. belongs to W rΔs . The domain for the sentential quantifiers, P rΔs , contains exactly one proposition for each sentential constant s, namely, the set of worlds in which s is true. Thus, P rΔs must be countable. Finally, in each world Σ, the interpretation of predicate letters and nonlogical operators is grounded on sentences belonging to Σ in which those symbols occur.
Note that, for clause iv.(c) to work, we must have that, for each sentence ϕ, tΠ P rΔs | ϕ P Πu P P rΔs . By iii, there must be a constant s s.t. tΠ P rΔs | ϕ P Πu " tΠ P rΔs | s P Πu, that is, ps Ø ϕq must belong to every Π P rΔs. This is guaranteed by the fact that Dα pα Ø ϕq P Δ and so, if this existential claim occurs in the n-th place of our enumeration, by the way in which Δ is constructed, psn Ø ϕq P Δ too. By definition of rΔs, psn Ø ϕq must also belong to every member of rΔs.
Next, we show that, for every sentence ϕ and every Σ P rΔs, by induction on the logical complexity of ϕ. ϕ :" s: The result follows directly from iv(d). ϕ :" F pc 1 , . . . , c n q: The result follows directly from iv(a) and iv.(b). Assume the result holds of every sentence of lower complexity than ϕ. We reason by cases: ϕ :" lψ: ϕ :" @α ψ: The remaining cases can be established in an analogous way. This concludes our induction. 18 Since Δ P rΔs, we can conclude that all members of Δ are true in M rΔs , Δ. Let M be the reduct of M rΔs to L S . Since W M " W rΔs , we have that M, Δ ( Γ . And, since P M " P rΔs , P M must also be countable. Finally, note that M is a faithful L Sstructure as, for every formula ϕ P L S with only β β β free, @β β βDα pα Ø ϕq must belong to every Σ P rΔs, so it follows from our previous result that, for every Σ P rΔs, M, Σ ( @β β βDα pα Ø ϕq. Hence, M ( Dα pα Ø ϕq too. This concludes the proof of Lemma 1. To finish the proof of Theorem 1, let us assume that Γ & SQL ϕ and let α and v be the only free variables occurring in Γ, ϕ (recall Γ must be finite). Thus, 19 By Definition 6, Γ * SQL ϕ. This completes our proof.

Translation and Truth
Let LS expand L S with fresh sentential constants s1 , s2 , . . .. Similarly, let LT expand L T with s1 , s2 , . . .. We now provide a translation η of LS into LT and formulate a corresponding disquotational truth theory UTBrηs that will allow us to simulate the inferential resources of SQL. The fresh sentential constants are needed to ensure that for every consistent set of sentences Γ of L S we can find a faithful model M in which P M is countable (as in Lemma 1). If there were no such model, we wouldn't 18 If we wish to generalise Lemma 1 to consistent sets of formulae (not of just sentences), we must not only find a model M rΔs and a world w P W rΔs but also an assignment g under which all members of our new Δ are true. To establish the induction case for ϕ :" @α ψ, it's important that every set in P rΔs corresponds to a sentential constant, as given by iii. So we have to make sure that g assigns to each sentential variable α a set tΣ P W rΔs | s P Σu. This is not trivial, as we could in principle have, for instance, that pα Ø sq P Δ (or plα Ø lsq P Δ) for each sentential constant s of LS . Making sure this doesn't happen adds a thick layer of complication we would rather avoid. 19 Note that the standard completeness proof-strategy is blocked here. We can infer that Γ Y t ϕu is consistent from Γ & SQL ϕ, but Lemma 1 doesn't allow us to conclude that the set has a model (and, so, that Γ * SQL ϕ). Lemma 1 is restricted to sets of sentences, whilst Theorem 1 isn't. This is important for the proof of Theorem 3, which (very roughly) shows that a disquotational truth predicate does not license any more inferences between formulae (not just sentences) than do sentential quantifiers. This is why we focus on the sentence @α@v p Ź ψPΓ ψ Ñ ϕq instead of Γ Y t ϕu. However, this is only possible if Γ is finite, as otherwise the antecedent would be infinitely long. For infinite Γ , a generalisation of Lemma 1 to formulae is required, but the proof is considerably more complicated (cf. fn 18), as noted in fn 15. be able to translate Γ into the language of truth while preserving the meaning of the arithmetical vocabulary, because the translation maps statements involving sentential variables into statements involving truth ascriptions, and of course there are only countably many sentences to which we can ascribe truth. We could avoid the use of these fresh sentential constants if we used (structured) propositions instead of sentences as truth bearers. However, at least as far as the present paper is concerned, we prefer to use sentences as truth bearers, as there are well-established theories of sentences available in the literature (e.g. Peano arithmetic) but no widely accepted, formal theory of structured propositions. 20 In order to provide the translation, let us (recursively) split the variables v 1 , v 2 , . . . of LT into two infinite sequences x 1 , x 2 , . . . and y 1 , y 2 , . . . Variables belonging to the first sequence are called 's-variables', as they are the variables to which we will map the sentential variables of LS . Variables in the second sequence are reserved for translating the individual variables of LS .
Via our fixed coding of expressions by numbers, Kleene's Recursion Theorem (cf. Hinman [27]) entails the existence of a pr function η : LS Ñ LT such that: 21,22 ηptq " if ϕ :" ψ^χ @y i pDpy i q Ñ ηpψqq, if ϕ :" @v i ψ @x i pTrsl η px i q Ñ ηpψqq, if ϕ :" @α i ψ where D is a monadic predicate of L T intended to express the domain of discourse of L (cf. Section 2) and Trsl η pvq abbreviates Du pSent LS puq^ϕ η pu, vqq, where Sent LS pvq is a predicate of L PA for the set of (codes of) sentences of LS , ϕ η pu, vq a formula of L PA expressing the function η itself, and u is a fresh variable. In other words, Trsl η pvq defines the set of expressions of LT that are translations of LS -sentences.
Our translation behaves as we anticipated in the introduction: sentential variables α i are translated into singular truth ascriptions to the corresponding s-variable x i and logical structure is otherwise preserved, except quantifiers are relativised 20 Note that the fresh sentential constants play no role in the proofs of Theorems 2 and 3; they are only used in Corollaries 4 and 5. 21 If P L S , we write ηp q instead of ηp# q, for perspicuity. 22 If l is an n-place operator and ϕ :" lpϕ 1 , . . . , ϕ n q, ηpϕq is lpηpϕ 1 q, . . . , ηpϕ n qq.
appropriately. Moreover, the non-logical vocabulary of LS remains unmodified. Thus, η is a natural translation.
Next, we introduce a theory formulated in LT with a disquotational truth predicate for translations. To simulate inferences of SQL, our truth theory must contain some principles that allow it to tell between syntactic objects and the objects that LS -sentences are about. In addition, a syntactic principle governing translations and a version of (SLE) governing the non-logical operators is be required. Note that although LT contains the signature of L PA , no axioms of PA governing these symbols will be included in our truth theory.
The last point deserves some emphasis: in order to simulate higher-order quantification, none of the usual arithmetical axioms are required in the truth theory. (Nor does our truth theory include any truth-theoretic axioms for the language L PA .) On the one hand, this supports the idea that the function of the truth predicate is indeed of a quasi-logical nature: our results do not rely on a strong base theory. On the other hand, since the formulation of our truth theory involves function symbols and other expressions from L PA (as is standard practice in the literature), one may expect the truth theory to prove the standard principles governing these functions. Otherwise, those symbols can completely deviate from their intended meanings, making it unclear whether the language can be truly understood as talking about expressions. 23 Fortunately, all our results -in particular, the conservativeness results in Section 3.5 -still hold if the usual arithmetical axioms, together with truth-theoretic axioms for L PA , are added to our truth theory, as we'll see later on (cf. Corollaries 4 and 5).
For the following definition, note that if ϕ P LS , then α i occurs free in ϕ just in case x i is free in ηpϕq and, similarly, v i occurs free in ϕ just in case y i is free in ηpϕq.

Definition 7
UTBrηs Ď LT is the theory formulated in FOL given by the following axioms and rule: pDq Dc, for each individual constant c of L pTrsl η q @x k p Ź iďn Trsl η px k i q Ñ Trsl η pxϕp 9 x k qyqq, if ϕ is in the range of η with only s-variables free, x k (SLE η ) If ϕ and ψ are in the range of η with only x j and y k free and $ UTBrηs @x j @y k p ľ iďm Trsl η px j i q^ľ iďn Dpy k i q Ñ pϕ Ø ψqq, then x k qy Ø ϕpx k qqq, if ϕ is in the range of η with only s-variables free, x k Following the notation introduced in the preliminaries, Trsl η pxϕp 9 x k qyq is short for Trsl η psubpxϕpx k qy, x k yqq, and Txϕp 9 x k qy is short for Tsubpxϕpx k qy, xx k yq. Axiom (D) guarantees that every individual constant of L denotes an object in D. (Trsl η ) states that, if we substitute translated sentences for the free s-variables in a translation, the result is also a translated sentence. Note that this axiom, needed for proofs later, wouldn't be true if we didn't have λ-terms in LS : while Tx i and ηpϕq are translations, Txηpϕqy isn't unless λ.ϕ is in the domain of η. (SLE η ) is just a translated version of (SLE), where the corresponding ranges of the variables occurring in ϕ and ψ are indicated by the assumptions in the premises and in the conclusion of the rule. Finally, (UTB η ) is a uniform version of disquotation for sentences in the range of η, with s-parameters only, ranging accordingly over translated sentences.
As already mentioned in the introduction, we do not necessarily recommend UTBrηs as our ultimate truth theory, as the choice of axioms is parasitic on LS . However it would be a mistake to view them as cherry-picked; they are precisely what is required to make our philosophical point: While sentential quantifiers allow us to quantify into sentence position over the class of sentences for which we have comprehension, the truth predicate allows us to quantify into sentence position over the class of sentences for which we have disquotation. If we restrict our attention to the simplified case of a class C of sentences which don't themselves contain sentential variables, then the whole machinery of UTBrηs would not be needed in order to prove inferential equivalence results; 24 all that is needed on the side of the truth theory are the instances of disquotation for the members of C -a natural, non-ad hoc, set of truth-theoretic axioms. The reason why UTBrηs involves a more complex (and perhaps less perspicuous) set of truth-theoretic axioms is to handle the more general case where C may contain sentences in which sentential quantifiers occur. Here we need to add instances of disquotation for the translations of the sentences in C instead, as members of C may contain vocabulary alien to the language of truth. But as we have argued, our translation is a natural one; if that is so, then there is nothing ad hoc about adding the disquotation instances for the translations of LS -sentences. Of course, if we opt for a truth predicate rather than sentential quantifiers to generalise on sentence positions to begin with, the class of translations of sentences of the language with sentential quantifiers would not be the most natural class we might wish to generalise over, but other classes will be.

Preservation of Inferences
The main result of this section establishes that any inference one can draw in SQL from some premises to some conclusion can also be drawn in UTBrηs from the translations of the premises to the translation of the conclusion, together with assumptions fixing the respective range of the variables that occur free in these translations. If only sentences are involved in premises and conclusion, no extra assumptions are needed. Simply put, we show that η preserves all valid inferences from SQL to UTBrηs. 25 If Γ Ď L S , let ηpΓ q " tηpψq | ψ P Γ u, Trsl η pΓ q " tTrsl η px i q | x i occurs free in ηpΓ qu and D η pΓ q " tDpy i q | y i occurs free in ηpΓ qu.
For perspicuity, we will often drop the subindexes of the variables v i and α i and assume that ηpvq " y and ηpαq " Tx.
To prove this result we first need the following lemmata. This result follows easily from the definition of η by induction on the logical complexity of ϕ. The proof of the following is slightly more convoluted.

Lemma 3 If the only variables occurring free in
Roughly, this lemma establishes (a parametrised version of) the following: applying η to the result of substituting α for ψ in ϕ is equivalent in UTBrηs to applying η to α, ϕ, and ψ first and then performing the substitution on the resulting expressions.
Proof The proof is by induction on the logical complexity of ϕ. If α doesn't occur free in ϕ, the result follows trivially, as there's nothing to substitute. Thus, we assume α occurs free in the formula and reason in UTBrηs from the assumption that Ź iďn Trsl η px k i q: ϕ :" α: Assume that the result holds for every formula of lower complexity than ϕ. We reason by cases: 26 ϕ :" lχ : The other cases can be proved in a similar fashion.
We can now go back to Theorem 2. We prove it by induction on the length of the derivation of ϕ from Γ in SQL. Assume for now that no λ-terms occur in this derivation. We will later show how to modify our proof to account for the occurrence of λ-terms. Throughout the proof we only reason in UTBrηs, so let '$' stand for derivability in this theory.
If the proof is one-step long, then either ϕ P Γ , in which case the result follows trivially, or ϕ is a logical axiom. In that case, we have the following: (IL) ϕ :" @v v " v. Then, ηpϕq is @y pDpyq Ñ y " yq, which is a logical truth and, thus, can be derived in UTBrηs without assumptions.
Assume that the result holds for every Δ and every ψ if ψ can be derived from Δ in fewer than n steps. We consider the possible ways in which ϕ could have been inferred from Γ : (LL) t 1 " t 2 , ψrt 1 {vs $ ψrt 2 {vs, where ϕ :" ψrt 2 {vs. We have: that is, ηpΓ q, Trsl η pΓ, ϕq, D η pΓ, ϕ, t 1 " t 1 q $ ηpt 1 q " ηpt 2 q def. η ηpΓ q, Trsl η pΓ, ϕq, D η pΓ, ϕ, t 1 " t 1 q $ ηpψqrηpt 1 q{ηpvqsq Lemma 2 as we can replace both t 1 " t 2 and ψrt 1 {vs on the left-hand side of the turn-style with t 1 " t 1 , since the only variables that occur in t 1 " t 2 and ψrt 1 {vs but not in ϕ occur in t 1 . Thus, we have the following: ηpΓ q, Trsl η pΓ, ϕq, D η pΓ, ϕ, t 1 " t 1 q $ ηpψqrηpt 2 q{ηpvqsq (LL) If all free variables in t 1 occur in Γ, ϕ, the proof is complete. Otherwise, let v k be the variables occurring free in t 1 but not in Γ, ϕ: Since y k don't occur free in ηpϕq and axiom (D) entails that there is at least one object satisfying D, one can easily get rid of the unwanted assumption that the free variables in t 1 range over D by an application of the Deduction Theorem and a logical principle governing quantifiers. We use this trick to prove other subsequent cases as well.
This completes the proof of Theorem 2 for Γ, ϕ without λ-terms. If λterms λ.ψ 1 , . . . , λ.ψ n occurred in Γ, ϕ, then one could replace each of them with ψ 1 , . . . , ψ n , translate the resulting inference into an admissible inference in UTBrηs, and then replace back each relevant occurrence of ηpψ 1 q, . . . , ηpψ n q with Txψ 1 y, . . . , Txψ n y. Since UTBrηs entails the equivalence between each sentence ψ and Txψy even in the scope of the non-logical operators, the resulting inference is admissible in the theory.

Corollary 1 Let Γ be a set of sentences and ϕ a sentence of L S :
If Γ $ SQL ϕ, then ηpΓ q $ UTBrηs ηpϕq.
Slightly abusing terminology, one could take this result to show that η is a relative interpretation of each theory Γ formulated in SQL in the first-order theory ηpΓ qÙ TBrηs. 27 Note, however, that η isn't just any relative interpretation but one that preserves the non-logical vocabulary of LS and, in addition, maps every sentential variable α to a truth ascription Tx, i.e. η is a natural relative interpretation, as we argued in the introduction.

Associated Models
The purpose of this section is to show that disquotational truth for a class of (translations of) expressions doesn't afford us more inferential power than sentential quantification over this class. In other words, we show that if the inference from some premises to a conclusion is not admissible in SQL, then the inference from the translations of the premises to the translation of the conclusion is not admissible in UTBrηs either. Together with the results proved in Section 3.4, this allows us to conclude that disquotational truth and sentential quantification are equivalent means of achieving one and the same goal, that is, to enable quantification into sentence position over a given class of sentences.
To establish the main result of this section, we first provide a semantics for L T and extensions thereof with respect to which FOL is sound. This is needed because L T is not a standard first-order language but contains non-logical sentential operators. (Again, we do not necessarily endorse this semantics as the most natural one, but it is sufficient for our purposes -i.e. the proof of Theorem 3 below.) We then show how to transform any faithful LS -structure M with a countable domain for the sentential quantifiers into a model M η for UTBrηs, preserving truth under η (Lemma 5). Our main result -that UTBrηs is conservative over SQL (Theorem 3) -then follows easily. Indeed, since M η interprets the arithmetical vocabulary in the standard way, we have as a corollary that UTBrηs`PA -actually, UTBrηs together with any sound "syntax theory" formulated in L PA -is conservative over SQL as well. Moreover, a simple modification of our argument shows that our results still hold if we add disquotation axioms for L PA to UTBrηs. That is, UTBrηs`UTBrPAs is conservative over SQL as well, where UTBrPAs denotes the result of extending PA with axioms of uniform disquotation for the language of L PA .

Definition 8 An
M`i s an interpretation function for the non-logical vocabulary satisfying the following conditions: Predicates, sentential constants, and non-logical operators may receive a different interpretation in each possible world of an L T -structure, just like in L S -structures. Non-logical operators, however, are no longer assigned sets of sets of possible worlds as their interpretation but rather subsets of members of the first-order domain, D M . Thus, to evaluate the truth of formulae of the form lϕ in each world w and assignment g, instead of looking for a set in }l} w M`c ontaining exactly those worlds in which ϕ is true under g, we will look for a member of D M in }l} w M`t hat is related via R M to exactly those worlds in which ϕ is true under g. This is the purpose of R M .

Definition 9
Let M be an L T -structure, w P W M , and g an assignment for M. This lemma follows easily from Definition 9 by induction on the length of the derivation of ϕ from Γ in FOL. Note that there are no rules or axioms for non-logical operators in the calculus, so FOL imposes no special constraints on the truth values formulae of the form lϕ may take. 28 If l is an n-place operator, we have that }l} w M`Ď D n M . 29 In general, if l is an n-place operator, M, w, g ( lpϕ 1 . . . , ϕ n q iff, for some p 1 , . . . , p n s.t. xpy P }l} w M`, it is the case that @i ď n@w 1 P W M , xw 1 , p i y P R M iff M, w 1 , g ( ϕ i .
The following important lemma shows that we can transform any faithful L S -structure M with a countable domain for the sentential quantifiers into a LT -structure, preserving truth under η. Let M η be given as follows:

Lemma 5 For every faithful L S -structure M in which P M is countable, there is an
if is a non-logical symbol of LS other than a sentential operator 30 We assume w.l.o.g. that ω and D M are disjoint, as we can always work with a set different from but isomorphic to ω instead. 31 Note that our domain contains in addition to the elements of ω also some non-numerical objects. We assume that, whenever is a function symbol of L PA , } } w M η yields some dummy value if applied to a member of D M . 32 More generally, if l is an n-place operator, }l} w M η " tx#ηpϕ 1 q, . . . , #ηpϕ n qy | x}ϕ 1 } M`, . . . , }ϕ n } M`y P }l} w M`u .
The existence of M η is guaranteed by our assumption that L and L PA don't share any non-logical vocabulary, as stated in the preliminaries.
Informally, R η M relates each world w with the (codes of) translations of sentences that express true propositions in w, and }l} w M η is the set of (codes of) translations of sentences that express a proposition in }l} w M`. Moreover, in each world w, T is interpreted as the set of (codes of) translations of sentences that express a true proposition in w. Note that the same proposition might be expressed in M by more than one sentence, so more than one code might correspond to the same proposition.
Conditions 1 and 2 are clearly satisfied. Regarding condition 3, for each assignment g for M, let g η be defined as follows: Since every proposition in P M is expressed by a sentential constant of LS , for each α i there must always be a sentence ϕ P LS s.t. gpα i q " }ϕ} M`. This guarantees that, for every s-variable x i , g η px i q " #ηpϕq, for some sentence ϕ P LS .
Next, we will prove a slightly stronger result that directly entails condition 3. This will become handy later, in the proof of condition 4. We show by induction on the build-up of ϕ that, for each assignment g for M -or, what's the same, M`there is an assignment g η for M η s.t., for every w P W M and ϕ P LS : ( 1 ) ϕ :" α: Let g η pxq " #ηpψq, so gpαq " tw P W M | M`, w ( ψu. Then: The results for ϕ :" F pt 1 , . . . , t n q and ϕ :" s follow trivially from iii.(a). Assume the result holds for every formula of lower complexity than ϕ. We reason by cases: ϕ :" lψ: Assume, first, that M η , w, g η ( lηpψq. By Definition 9, there must be a d P }l} w M η s.t. By iii.(d), there must be a sentence χ of LS s.t. #ηpχq P }l} w M η and By iii.(d), ii, and the inductive hypothesis, we have that there's a sentence χ of LS s.t. }χ} M`P }l} w M`a nd can be derived in UTBrηs by only n`1 applications of (SLE η ), where x, y are the only variables occurring free in ηpϕq, ηpψq. As before, our proof can be easily generalised to any number of free variables. This entails that @x@y pTrsl η pxq^Dpyq Ñ pηpϕq Ø ηpψqqq (5) must be derivable in UTBrηs by fewer than n`1 applications of (SLE η ), which means that (5) must be true in M η , by inductive hypothesis. Assume, for contradiction, that (4) is not true in M η . Thus, there must be a w P W M and an assignment g η for M η s.t. M η , w, g η * lηpϕq Ø lηpψq. By (1), M`, w, g * lϕ Ø lψ and, by Definition 4, Again, by (1), we have that is, there is a w 1 P W M s.t. M η , w 1 , g η ( ηpϕq and M η , w 1 , g η * ηpψq or vice versa. In other words, M η , w 1 , g η * ηpϕq Ø ηpψq, so (5) isn't true in M η after all, contradicting our initial assumption.
This completes the proof of Lemma 5. Now, together with Lemmata 1 and 4, Lemma 5 entails a conservativeness result. As we will show next, UTBrηs does not allow us to draw more inferences between translations than SQL already allows between the original formulae.

Corollary 2 Let Γ be a set of sentences and ϕ a sentence of L S :
If ηpΓ q $ UTBrηs ηpϕq, then Γ $ SQL ϕ.
A theory T 2 extending another theory T 1 is said to be proof-theoretically conservative over T 1 just in case, if T 2 proves a sentence of the language of T 1 , T 1 proves it as well. This notion can be naturally extended to theories T 1 and T 2 such that T 2 does not extend T 1 , provided that T 2 is an extension of a translation of T 1 . In that case, let us say that T 2 is proof-theoretically conservative over T 1 modulo the given translation just in case, if T 2 proves the translation of a sentence ϕ of the language of T 1 , then T 1 proves ϕ. Thus, Corollary 2 establishes that, for each set of sentences Γ of L S , ηpΓ q`UTBrηs is proof-theoretically conservative over Γ`SQL modulo η, as η is a translation of L S into LT and ηpΓ q`UTBrηs extends ηpΓ`SQLq (cf. Corollary 1).
Put together, Corollaries 1 and 2 entail the following:

Corollary 3 Let Γ be a set of sentences and ϕ a sentence of L S :
As a consequence, since η is a natural translation, Corollary 3 allows us to conclude that the addition of a uniformly disquotational truth predicate for a given class of sentences affords us exactly the same inferential power as sentential quantifiers. The generalising function of a truth predicate governed by this kind of disquotation is no more and no less than the simulation of quantification into sentence position in a first-order setting.
Earlier we raised the question whether our results still hold if we add the usual axioms of PA to our truth theory (cf. Section 3.3). It follows from Lemma 5(1) that this is the case indeed. 33 Let TrpNq be the set of sentences of L PA that are true in the standard model.

Corollary 4 UTBrηs`TrpNq is conservative over SQL modulo η.
A fortiori, UTBrηs combined with any sound theory formulated in L PA to serve as a syntax theory (e.g. PA) is also conservative over SQL modulo η. As the following corollary shows, Lemma 5 also entails that, if Γ Ď L S is consistent, then ηpΓ qÙ TBrηs together with any sound "syntax theory" formulated in L PA has a model in which the arithmetical vocabulary is interpreted standardly.

Corollary 5
If Γ Ď L S is consistent, then ηpΓ q`UTBrηs has a model in which the arithmetical vocabulary is interpreted by the standard model of arithmetic.
Note that a simple modification of our argument shows that we can replace UTBrηs with UTBrηs`UTBrPAs in Corollaries 4 and 5, where UTBrPAs is the result of extending PA with uniform disquotation for the language of L PA . 34

Truth and Predicate Quantification
In this section our goal is to establish that quantification into predicate position in a language with a name for each object in its intended first-order domain can be simulated by means of a truth predicate, as anticipated in the introduction. We give a natural translation of an arbitrary second-order language of those characteristics into 33 Recall that the axioms of PA are relativised to the predicate N (cf. fn 10). 34 One simply needs to add all L PA -sentences that are true in N to the extension of the truth predicate }T} η M in the proof of Lemma 5. a corresponding first-order language with a truth predicate and show that the inferences licensed by full impredicative second-order quantification translate exactly into the admissible inferences in a truth theory whose truth predicate is governed by a restricted version of uniform disquotation. (As before, it is interesting that although the truth theories in question are formulated in a language containing arithmetical vocabulary, no arithmetical axioms are needed.) To facilitate readability, we focus on second-order quantification, but the results in this section are easily generalised to all higher orders.
More fully, we first expand L to L 2 , a language with quantification into predicate position, and extend FOL to a second-order calculus SOL with inference rules and axioms for the new quantifiers. We then provide a natural translation τ of L 2 into L T and formulate a corresponding truth theory UTBrτ s, containing, roughly, an instance of uniform disquotation for each formula in the range of τ . We then show that any admissible inference from Γ to ϕ in SOL can be translated into an inference from τ pΓ q to τ pϕq in UTBrτ s. In addition, we show that UTBrτ s does not license any inference between translations that SOL doesn't already allow for between the corresponding original expressions. We do so by showing that every model of SOL can be transformed into a model of UTBrτ s that preserves truth under τ . In other words, we show that SOL and UTBrτ s have the same inferential power. 35

Quantification into Predicate Position
Let L be as in the preliminaries. We further assume it contains a canonical namei.e. a designated closed term -for each object in its intended domain. Let L 2 expand L with predicate variables V n 1 , V n 2 , . . . of each arity n, and a λ-term tλv. ϕu (playing the syntactic role of an n-place predicate) for every formula ϕ with exactly v free.
Let U, V stand for arbitrary predicate variables.
Definition 10 Let SOL formulated in L 2 extend FOL with the following rules and axioms: Note that both free and bound predicate variables may occur in ϕ in (Comprehension), i.e. this is a fully impredicative principle. 35 Schindler [28,Proposition 25] essentially shows that all theorems of SOL can be translated into theorems of a disquotational truth theory. The results in the present section do not only establish that inferences can be preserved as well, but also that UTBrτ s doesn't license any more inferences than SOL. Moreover, in [28] it hasn't been noticed that essentially no arithmetical or syntax axioms are needed for that result.

Translation and Truth
Analogously to what we did in Section 3.3, let L2 expand L 2 with a denumerable stock of fresh predicate constants Q n 1 , Q n 2 , . . ., for each arity n. Similarly, let LT extend L T by Q n 1 , Q n 2 , . . . 36 We now provide a translation τ of L2 into LT and formulate a corresponding disquotational truth theory UTBrτ s that will allow us to simulate the inferential resources of SOL.
In order to do this, let us (recursively) split the variables v 1 , . . . , v n of LT into infinite sequences x 1 1 , x 1 2 , . . ., x 2 1 , x 2 2 , . . . and y 1 , y 2 , . . .. Variables belonging to all sequences but the last one are called 'p-variables', as they are the variables to which we map the predicate variables of L2 .
Given our fixed coding, by Kleene's Recursion Theorem we obtain a pr function τ : L2 Ñ LT that translates the expressions of L2 into LT as follows: . . , τ pt n qq, if ϕ :" F pt 1 , . . . , t n q Tsubpx n i , xτ pt 1 q, . . . , τ pt n qyq, if ϕ :" V n i pt 1 , . . . , t n q Tsubpxτ pψqy, xτ pt 1 q, . . . , τ pt n qyq, if ϕ :" tλv. ψupt 1 , . . . , t n q τ pψq, if ϕ :" ψ τ pψq^τ pχq, if ϕ :" ψ^χ @y i pDpy i q Ñ τ pψqq, if ϕ :" @v i ψ @x n i pTrsl n τ px n i q Ñ τ pψqq, if ϕ :" @V n i ψ Again, D is a monadic predicate of L T intended to express the domain of discourse of L (cf. Section 2) and Trsl n τ pvq abbreviates the formula Du pFor L2 puq^ϕ τ pu, vq^nvffpvq " nq of L PA , where ϕ τ pu, vq expresses τ in L PA , nvffpvq is a function symbol for the function that maps the code of each formula of LT to the number of non-p-variables that occur free in it, For L2 pvq defines the set of formulae of L2 without free occurrences of predicate variables, and u is a fresh variable. In other words, Trsl n τ pvq is true exactly of all formulae of LT in the range of τ with just n free variables, none of which are p-variables. In the remainder of this section, let us refer to these formulae as 'n-place translations'.
Our translation behaves as we anticipated in the introduction: predications of the form V n i t 1 , . . . , t n are translated as 'the result of substituting τ pt 1 q, . . ., τ pt n q for the free variables in x n i is true' (i.e. x n i is true of τ pt 1 q, . . . , τ pt n q) and logical structure is otherwise preserved, except quantifiers are relativised appropriately. Moreover, the non-logical vocabulary of L2 remains unmodified. Thus, τ is a natural translation. 36 See the remarks at the beginning of Section 3.3 for the rationale behind this.
Next, we will formulate a uniformly disquotational truth theory in LT for the formulae of this language that fall in the range of τ . The theory will contain, in addition, principles that allow it to distinguish between syntactic objects and the objects that belong to the first-order domain of L2 . Again, note that although LT contains the signature of L PA , no axioms governing these symbols will be included in our truth theory. Since no strong base theory is required for our results, this may be taken to support the idea that the function of the truth predicate is of a quasi-logical nature. On the other hand, we emphasise that all results proved here still hold if our truth theory is expanded with the usual arithmetical axioms and truth-theoretic axioms for the sentences of L PA (cf. Corollaries 9 and 10).
To increase readability, we often write x i instead of x n i and Trsl τ px i q instead of Trsl n τ px n i q, if the arity is clear from the context. In what follows, let t l k abbreviate x l 1 j 1 , . . . , x l m j m .

Definition 11
UTBrτ s is the theory formulated in LT extending FOL with the following axioms: pDq Dc, for each individual constant c of L pTrsl τ q @x l j @y k p Ź iďm Trsl τ px l i j i q^Ź iďn Dpy k i q Ñ Trsl r τ pxϕp 9 x l j , 9 y k qyq, if ϕ is in the range of τ and its only free variables are x l j , y k , and y p 1 , . . . , y p r pUTB τ q @x j @y k p Ź iďm Trsl τ px j i q^Ź iďn Dpy k i q Ñ pTxϕp 9 x j , 9 y k qy Ø ϕpx j , y k qqq, if ϕ is in the range of τ and its only free variables are x k and y k 37 (Trsl τ ) establishes that replacing in a translation ϕ each free l i -place p-variable with the canonical name of an l i -place translation and n-many of the n`r free non-pvariables with canonical names for members of the domain of L results in an r-place translation. In turn, (UTB τ ) provides an instance of disquotation for each formula in the range of τ with parameters from the set of translations and the domain of L.

Preservation of Inferences
The main result of this section establishes that any admissible inference in SOL from some premises to some conclusion can also be drawn in UTBrτ s from the translations of the premises to the translation of the conclusion, together with assumptions fixing the respective range of the variables that occur free in these translations. If only sentences are involved in premises and conclusion, no extra assumptions are needed. Put simply, we show that τ preserves all valid inferences from SOL to UTBrτ s.
If Γ Ď L 2 , let τ pΓ q " tτ pψq | ψ P Γ u, Trsl τ pΓ q " tTrsl n τ px n i q | x n i occurs free in τ pΓ qu and D τ pΓ q " tDpy i q | y i occurs free in τ pΓ qu. 37 To facilitate proofs, we require that at least one non-p-variable occurs free in ϕ.
Since all variables occurring free in @V n i ψ are also free in ϕ, by weakening: The other rules are proved in a similar fashion as in Section 3.4, appealing to Lemma 6 and the axioms of UTBrτ s. This completes the proof of Theorem 4 for Γ, ϕ without λ-terms. The latter can be easily dealt with deploying a similar trick as in the proof of Theorem 2 at the end of Section 3.4.

Corollary 6
Let Γ be a set of sentences and ϕ a sentence of L 2 : If Γ $ SOL ϕ, then τ pΓ q $ UTBrτ s τ pϕq.
As before, abusing our terminology slightly, this result could be taken to show that τ is a relative interpretation of each second-order theory Γ in the first-order theory τ pΓ q`UTBrτ s. 38 But note that τ isn't just any relative interpretation but one that preserves not only the non-logical vocabulary of L 2 but also maps every second-order formula of the form V pt 1 , . . . , t n q to a truth-of ascription Tsubpx, xτ pt 1 q, . . . , τ pt n qyq. In other words, τ is a natural relative interpretation, as we argued in the introduction.

Associated Models
The purpose of this section is to show that disquotational truth for a class of (translations of) expressions does not carry more inferential power than second-order quantification over this class. More specifically, we show that if the inference from some premises to a conclusion is not admissible in SOL, then the inference from the translations of the premises to the translation of the conclusion is not admissible in UTBrτ s either. Together with the results we obtained in Section 4.3, the result of this section suggests that disquotational truth and second-order quantification provide equivalent ways of achieving one and the same goal, that is, to enable quantification into predicate position over a given class of formulae. Again, it should be noted that this conservativeness result still holds if we expand UTBrτ s with the usual arithmetical axioms and disquotation axioms for the formulae of L PA .
To establish the main result of this section, we first introduce a Henkin (or general) semantics for L 2 , define faithful structures, and show how to obtain an LT -structure from a countable faithful L 2 -structure that preserves truth under τ . Again, we do not claim that our Henkin models are the intended models of L 2 ; we simply use them to establish our results. D M is the universe of discourse for the first-order variables to range over and each S n M is the set over which the n-place predicate variables range over. If M is a Henkin L 2 -structure, an assignment g for M maps each individual variable of L 2 to a member of D M and each n-place predicate variable to a member of S n M . Truth in M, g and truth in M simpliciter are defined as usual.

Definition 13
A Henkin L 2 -structure M is faithful iff every instance of (Comprehension) is true in M.

Definition 14
If ϕ P L 2 and Γ Ď L 2 , then ϕ is an SOL-consequence of Γ (Γ ( SOL ϕ) iff, for every faithful Henkin L 2 -structure M and assignment g for M, if M, g ( Γ , then M, g ( ϕ. It is well known that SOL is sound and complete with respect to the class of faithful Henkin structures (cf. Shapiro [29]).
In what follows we show how to transform each countable faithful Henkin L 2structure M into an LT -structure M τ that interprets the arithmetical vocabulary in the standard way, preserves truth under τ , and makes all axioms of UTBrτ s true. As in the previous section, a simple modification of our argument shows that we can actually pick M τ so that M τ ( UTBrτ s`UTBrPAs.
Here, a Henkin L 2 -structure M is countable iff D M and S n M , for each n ą 0, are all countable. If τ pΓ q $ UTBrτ s τ pϕq, then Γ $ SOL ϕ.

Lemma 8 For every countable faithful Henkin
Put together, Corollaries 6 and 7 entail the following: Corollary 8 Let Γ be a set of sentences and ϕ a sentence of L 2 : Corollary 7 shows that each theory τ pΓ q`UTBrτ s formulated in L T is prooftheoretically conservative over Γ`SOL modulo τ , as τ pΓ q`UTBrτ s is an extension of τ pΓ`SOLq (cf. Corollary 6). Thus, since τ is a natural translation, we conclude from Corollary 8 that the addition of a uniformly disquotational truth predicate for a given class of sentences affords us the same inferential power as predicate quantifiers (provided that the base language has a canonical term denoting each object in the first-domain of its intended interpretation). The generalising function of a truth predicate governed by this kind of disquotation is no more and no less than the simulation of quantification into predicate position in a first-order setting.
As in the previous section, we note the following two consequences of Lemma 8, where TrpNq is the set of sentences of L PA that are true in the standard model.
A fortiori, UTBrτ s combined with any sound "syntax theory" (e.g. PA) formulated in L PA is also conservative over SOL modulo τ . As the following corollary shows, Lemma 8 also entails that, if Γ Ď L 2 is consistent, then τ pΓ q`UTBrτ s together with any sound "syntax theory" formulated in L PA has a model in which the arithmetical vocabulary is interpreted standardly.

Corollary 10
If Γ Ď L 2 is consistent, then τ pΓ q`UTBrτ s has a model in which the arithmetical vocabulary is interpreted by the standard model of arithmetic.
Again, note that a simple modification of our argument shows that we can replace UTBrτ s with UTBrτ s`UTBrPAs in Corollaries 9 and 10, where UTBrPAs denotes the result of extending PA with axioms of uniform disquotation for the language of L PA : one simply needs to add all L PA -sentences that are true in N to the extension of the truth predicate }T} τ M in the proof of Lemma 8.

Satisfaction and Predicate Quantification
In Section 4 we showed that quantification into predicate position can be mimicked by means of a truth predicate, provided that the base language has a name for each object in its intended first-order domain. This restriction was in place to allow for the formulation of uniform disquotation in terms of a truth predicate. One can define 'true of' in terms of the standard truth predicate with the aid of the substitution function, i.e. ϕpuq is true of d if and only if the sentence that results from replacing in ϕpuq each free variable u i with a name for d i is true. Thus, to define truth-of in terms of truth every object in the range of the quantifiers that lead uniform disquotation must have a name. The goal of this section is to point to results analogous to the ones established in the previous section that apply to base languages without restrictions. For this purpose, instead of using a truth predicate and defining truth-of in terms of it, we appeal directly to a primitive truth-of or satisfaction predicate. Since the proofs are very similar to the ones offered in the previous section, we mostly state our results in this section without proof, to avoid repetition.
There is one difference though. In order to talk about satisfaction of predicates by sequences of objects, we need to be able to talk about sequences of objects. In the previous section, we assumed the domain of objects to be countable. This enabled us to talk about sequences of objects within PA, because (given a coding of those objects by numbers) one can code finite sequence of objects by numbers. This strategy is no longer available as we now allow our domain of objects to be uncountable. Thus, in the present section, we use a fragment of set theory as our syntax theory. The basic set theory BS described in Devlin [30, p. 36] is more than sufficient. 39,40 Let L Sat expand L with (a fresh copy of) the signature of L P (the language of set theory) and a binary predicate Sat, for satisfaction, instead of truth. On its intended meaning, Satpu, vq holds of the code u of a formula ϕ and the code v of a sequence xd 1 , . . . , d n y just in case ϕ has exactly n free variables and is true of d. As before, we (recursively) split the variables of L Sat into p-variables, x 1 1 , x 1 2 , . . ., x 2 1 , x 2 2 , . . ., and non-p-variables, y 1 , y 2 , . . ..
Let Seqpvq be a formula of L P that says that v is a finite sequence. Moreover, let lnpvq be a function symbol for the function that maps each finite sequence to its length and u v for the function that takes two sequences and returns the result of concatenating them in order. Finally, let Sat˚pu, vq be short for Dw pSeqpwq^Satpu, w vq^lnpwq`lnpvq " nvffpuqq. 39 BS comprises the axioms of Extensionality, Pairing, Union, Fnfinity, Cartesian Products, the Σ 0 -Comprehension schema, and a certain induction schema. This theory is weaker than Kripke-Platek set theory. 40 As with the previous cases, the axioms of the syntax theory are not really required for our results. However, in the present case, we decided to include them into the truth theory as their disentanglement would require more word space than in the previous cases.
In plain words, Sat˚pu, vq holds just in case there is a sequence w such that, concatenated with v, satisfies u, provided that the length of w plus the length of v is exactly the number of non-p-variables that occur free in u.
As before, we define a translation σ : L 2 Ñ L Sat and formulate a disquotational theory UTBrσ s in terms of it. Whilst we deploy Sat˚pu, vq in the definition of σ , the disquotational axioms are given for Satpu, vq. This is to deal with possible first-order parameters in the derivation of the translation of (Comprehension) in UTBrσ s (cf. footnote 41 in Theorem 6).
We formulate our theory of uniform satisfaction in terms of σ as follows: Definition 15 UTBrσ s is the theory formulated in L Sat extending BS with (D) and the following axiom scheme: pUTB σ q @x j @y k p Ź iďm Trsl σ px j i q^Ź iďn Dpy k i q Ñ pSatpxϕp 9 x j , y k qy, xy k yq Ø ϕpx j , y k qqq, if ϕ is in the range of σ and only x j and y k are free in ϕ The results we state in what follows are analogous to those we offered in the previous section. Their proofs are analogous too, with the exception perhaps of Theorem 6, which we sketch briefly.
Proof The proof is just like the proof of Theorem 4, by induction on the length of the derivation of ϕ from Γ in SOL. The only difference is in the case where the derivation