Cut elimination for systems of transparent truth with restricted initial sequents
aa r X i v : . [ m a t h . L O ] J un Cut elimination for systems oftransparent truth with restricted initial sequents
Carlo NicolaiKing’s College London
Abstract.
The paper studies a cluster of systems for fully disquotational truth based on therestriction of initial sequents. Unlike well-known alternative approaches, such systems displayboth a simple and intuitive model theory and remarkable proof-theoretic properties. We startby showing that, due to a strong form of invertibility of the truth rules, cut is eliminablein the systems via a standard strategy supplemented by a suitable measure of the numberof applications of truth rules to formulas in derivations. Next, we notice that cut remainseliminable when suitable arithmetical axioms are added to the system. Finally, we establisha direct link between cut-free derivability in infinitary formulations of the systems consideredand fixed-point semantics. Noticeably, unlike what happens with other background logics,such links are established without imposing any restriction to the premisses of the truth rules. Introduction
Due to the Liar paradox, fully disquotational approaches to truth – that is, satisfying therules (Tr l ) and (Tr r ) below – require a non-classical logical treatment. Among the nonclassicaloptions, a standard approach is to restrict operational rules for connectives that play a crucialrole in the derivation of the inconsistency, such as negation or material implication. To this familyof approaches belong the various paracomplete or paraconsistent accounts of fully disquotationaltruth defended in the literature (see e.g. [Fie08, Pri05, Kre88, Bea09, HH06]).Formal systems for transparent truth based on restrictions of operational rules and featuringunrestricted rules for semantic notions do not sit well with standard strategies to fully or partiallyeliminate applications of the cut rule. To explain why this is so, let us focus on the case ofunrestricted truth rulesΓ , ϕ ⇒ ∆ (Tr l )Γ , Tr p ϕ q ⇒ ∆ Γ ⇒ ϕ, ∆ (Tr r )Γ ⇒ Tr p ϕ q , ∆In the rules, p · q is a quotation device that yields a canonical name for each sentence of thelanguage. When one wants to eliminate a cut on truth ascriptions Tr p ϕ q that are both obtained Mathematics Subject Classification.
Primary 03F05; Secondary 03A99.Thanks to Peter Schröder-Heister for pointing me to his work on definitional reflection that studies the interactionbetween cut, contraction, and restricted initial sequents studied in this work. Thanks to Andreas Fjellstad forclarifying the role of identity axioms in the systems I discuss. I thank Volker Halbach, Graham Leigh, BeauMount, Luca Tranchini for discussing with me the ideas contained in the paper. Special thanks to Luca Castaldo,Martin Fischer, Lorenzo Rossi for detailed comments. Actually, this generalizes to partial approaches to truth based on supervaluations. We shall elaborate on thispoint later. from ( Tr -l) and ( Tr -r) , a natural thought is to cut on the sentence ϕ in the premises ofthe application of these rules. However it’s clear that Tr p ϕ q is an atomic formula, whereas ϕ may be an extremely (logically) complex sentence. Therefore a simple induction on the logicalcomplexity of the cut formula, on which cut-elimination strategies are traditionally based, willnot work. One has also to keep track of the number of applications of truth rules and inductover it in the main induction hypothesis.There are several alternatives for such tracking devices. One option is to assign a measureto sequents, i.e. to nodes in the derivation tree. However, in the context of logics which restrictthe operational rules, this strategy can only be carried out if one restricts the truth rules, bydisallowing contexts in the premises. Once this restriction takes place, cut can be fully eliminated.Of course, the price to pay is the adoption of properly weaker truth rules. This is for instance thestrategy considered in [Can90, FG18] in the context of Strong Kleene logic and supervaluationallogic respectively. Alternatively, one can restrict the contraction rule, keep the node-basedmeasure of applications of semantic rules, and still obtain a full cut elimination proof [Gri82,Can03]. However, the restriction of contraction has its own drawbacks. For instance, whereasthe systems based on the restriction of operational rules are sound – and, in a suitably controlledenvironment, also complete – with respect to a class of fixed-point models [Kri75], no such linkexists between contraction-free systems and fixed-point semantics or natural alternatives.In this paper I consider a cluster of theories of transparent truth that display both a directlink with fixed-point semantics, but also desirable proof-theoretic properties culminating in theeliminability of cut. Such systems are based on a restriction of initial sequents to formulas notcontaining the truth predicate. The proof-theoretic arguments given below crucially rest on theadoption of a measure for formulas in derivations, called Tr -complexity, that keeps track of thenumber of truth rules applied to ancestors of a single formula in the given proof. While thenotion of Tr -complexity is not new, it is its combination with the restriction of initial sequentsin the context of transparent truth that is the main focus of the paper. Such connection hasbeen studied already for a propositional logic extended with rules for definitional reflection in[SH16]. Plan and structure of the paper.
In section 2, I study the proof-theory of a ‘logic’ of truthLGT, that is a system with no non-logical initial sequents and rules besides ( Tr -l) and ( Tr -r) . The section focuses on the definition of the main measure for application of truth rulescalled Tr -complexity (Definition 2), the proof of the strong invertibility property of the mainLGT-rules (Lemma 3), and culminates with the eliminability of cut in LGT essentially achieved [Zar11] has also presented a cut-elimination argument for an (infinitary) transparent theory of truth over acontraction-free logic. However, both [RR18] and [Fje20] cast some doubts on the logical coherence and applica-bility of the proposal. More on fixed point semantics in §4. Similar measures of complexity have been considered by [Hal99] and [Lei15]. The key ideas of this paper were presented in Tübingen in 2017, where Peter Schröder-Heister pointed to hisindependent work on the idea. The current shape of the paper and results benefited greatly from the studyof Schröder-Heister’s work on definitional reflection. Such exchanges also are at the root of the formulation ofan infinitary Tait system with restricted initial sequents, akin to LPC ∞ below, in Martin Fischer’s HabilitationThesis, Modal Predicates and Their Interaction , Munich, 2018. ut and Initial Sequents 3 in Lemma 5. In the short section 3, we extend the results of section 2 to extensions of LGTwith (geometric) arithmetical axioms by employing the study of the proof-theory of geometricaxioms from [NvP11]. This yields a uniform conservativeness proof of local truth rules overthe base theory (Proposition 1). Section 4 studies the connection of cut-free provability andan infinitary extension of LGT with fixed point semantics (Lemmata 11 and 12). To achievethis, Tr -complexity is extended to the transfinite, and cut elimination is proved for an infinitaryextension of LGT N (Proposition 2).As the reader will notice, the cut elimination strategy introduced in §2 features prominentlyalso in the subsequent sections. Of course, an alternative way of presenting the paper wouldhave been to start with the arithmetical or the infinitary setting, and then inferring the resultsof §2 as immediate corollaries. The current structure of the paper is motivated by the intentionof presenting the main structural lemmata in a simple setting in §2, so that in the subsequentsections the focus could be mainly on the adjustments required by richer frameworks and onother properties such as the connections with fixed-point semantics.2. Logics for transparent truth with restricted initial sequents
We start with a first-order language L with logical constants ¬ , ∧ , ∀ , ⊥ , ⊤ . We let: L Tr := L ∪ { Tr } , for Tr a unary predicate symbol.We write AtFml L Tr for the set of atomic formulas of L Tr , and Sent L Tr for the set of sentencesof L Tr . It is useful not to regard ⊤ , ⊥ as members of AtFml L Tr . The logical complexity | ϕ | ofa formula ϕ of L Tr is defined inductively as the number of nodes in the maximal branch of itssyntactic tree: | ϕ | = , if ϕ is atomic or ⊥ , ⊤ , | ψ | + 1 , if ϕ ≡ ¬ ψ or ϕ ≡ ∀ xψ ,max( | ψ | , | χ | ) + 1 , if ϕ ≡ ψ ∧ χ .To properly formulate our truth rules in the simple setting studied in this section, we followthe standard practice of assuming that for any sentence ϕ ∈ L Tr , there is a term p ϕ q playingthe logical role of its name [Kre88, Can03, Rip12]. In general, there are good reasons to requiremuch more than a simple, essentially metatheoretic quotation device and work with a fullyfledged formal syntax in the background. We will see later on that much of our discussion canbe transferred to such richer settings.In what follows, Γ , ∆ , Θ , Λ . . . stand for finite multisets of formulas of L Tr – and the samenotation will be employed for the different languages considered below. Expressions of the formΓ ⇒ ∆ are sequents . We assume a standard notion of substitution and write Γ( t/x ) for the resultof replacing all free occurrences of x in all formulas in Γ with the term t , which is assumed to befree for x in such formulas. For a formula ϕ , we denote with FV( ϕ ) the set of its free variables.FV(Γ) denotes the set of free variables in formulas in Γ.The system LGT is essentially characterized by all operational rules of classical logic, fullydisquotational truth rules, and crucially by a restriction of initial sequents to principal formulas ut and Initial Sequents 4 that are atomic and do not contain Tr . In the terminology of [TS03], LGT is a G3 system.The label LGT stands for ‘logic of grounded truth’. This choice is informally motivated by thefact that one can read the sequent Γ ⇒ ∆ in LGT as stating that either some member of Γ isdeterminately false, or some member of ∆ is determinately true. This informal picture will berefined by the semantic considerations of Section 4 – and Lemma 11 in particular. Definition . The system
LGT in L Tr features the following initial sequents and rules: (ref − ) Γ , ϕ ⇒ ϕ, ∆with ϕ ∈ AtFml L Γ ⇒ ∆ , ϕ ϕ, Γ ⇒ ∆ (cut) Γ ⇒ ∆ ( ⊤ ) Γ ⇒ ⊤ , ∆ ( ⊥ ) Γ , ⊥ ⇒ ∆Γ , ϕ ⇒ ∆ ( Tr l) Γ , Tr p ϕ q ⇒ ∆ Γ ⇒ ϕ, ∆ ( Tr r) Γ ⇒ Tr p ϕ q , ∆Γ ⇒ ϕ, ∆ ( ¬ l) Γ , ¬ ϕ ⇒ ∆ Γ , ϕ ⇒ ∆ ( ¬ r) Γ ⇒ ¬ ϕ, ∆Γ , ϕ, ψ ⇒ ∆ ( ∧ l) Γ , ϕ ∧ ψ ⇒ ∆ Γ ⇒ ϕ, ∆ Γ ⇒ ψ, ∆ ( ∧ r) Γ ⇒ ∆ , ϕ ∧ ψ Γ , ∀ xϕ, ϕ ( s/x ) ⇒ ∆ ( ∀ l) Γ , ∀ xϕ ⇒ ∆ Γ ⇒ ϕ ( y/x ) , ∆ ( ∀ r) y / ∈ FV(Γ , ∆ , ∀ xϕ )Γ ⇒ ∆ , ∀ xϕ The following measures of complexity are also standard. We employ the usual notions of premisses and conclusion of rules, principal , active , side formulas [Sch77, TS03]:(i) Given rules that are at most α -branching, the length d of a derivation D issup { d γ + 1 | γ < β } where D γ ( γ < β ≤ α ) are D ’s direct subderivations.(ii) The rank of an application of cut on ϕ is | ϕ | + 1. The cut rank of a derivation D is themaximum of the ranks of cut formulas in D .It will sometimes be useful to refer directly to different occurrences of the same (qua syntacticobject) formula in a derivation (cf. [SH16]). When writing, say,(1) γ j , . . . , γ j n n ⇒ δ k , . . . , δ k m m , ϕγ j +11 , . . . , γ j n +1 n ⇒ δ k +11 , . . . , δ k m +1 m , ψ we assume that occurrences of γ ji , with 1 ≤ i ≤ n correspond precisely to occurrences of γ j +1 i – i.e. they are distinct occurrences of the same formula – and similarly for the δ ’s. As anabbreviation, this will be generalized to multisets of sentences: I occasionally write Γ j insteadof γ j , . . . , γ j n n . It should be clear that superscripts are not part of the language. We omitted the standard G3 -rules for ∨ , ∃ , which are nonetheless admissible in the systems below by employingthe usual definitions of ∨ and ∃ in terms of ∧ , ¬ , ∀ . ut and Initial Sequents 5 The idea behind the following measure on proofs, that we call Tr -complexity, plays an im-portant role in recent proof-theoretic studies of primitive truth predicates [Hal99, Lei15]. Itessentially tracks the number of truth rules applied to formulas in derivations. If contraction ispresent, such measure is not easy to define and employ. We will see that the restriction of initialsequents and the absence of explicit contraction enable us to apply the notion of Tr -complexityin the general case of type-free, disquotational truth.
Definition . The ordinal Tr -complexity of an occurrence of a formula ϕ of L Tr in a derivation D in LGT – in symbols, τ D ( ϕ ) is defined inductively as follows: (i) τ D ( ϕ ) = 0 if ϕ ∈ L ; (ii) If D contains only an initial sequent node ( ref , ⊤ , ⊥ ), then τ D ( ϕ ) = 0 for all formulasin it. (iii) If D ends with Γ ⇒ ∆ , ψ Γ ⇒ ∆ , Tr p ψ q then τ D (Tr p ψ q ) = τ D ( ψ )+1 and the Tr -complexity of the formulas in Γ , ∆ is unchanged.Similarly for (Tr l ) . (iv) If D ends with Γ ⇒ ∆ , ϕ ¬ ϕ, Γ ⇒ ∆ then τ D ( ϕ ) = τ D ( ¬ ϕ ) and the Tr -complexity of the formulas in Γ , ∆ is unchanged.Similarly for ( ¬ r) and ( ∀ r) . (v) If D ends with Γ , ϕ, ψ ⇒ ∆ ϕ ∧ ψ, Γ ⇒ ∆ then τ D ( ϕ ∧ ψ ) = max( τ D ( ϕ ) , τ D ( ψ )) and the Tr -complexity of the formulas in Γ , ∆ isunchanged. (vi) If D ends with – cf. notational convention after (1) , Γ j ⇒ ϕ, ∆ k Γ l ⇒ ψ, ∆ p Γ ⇒ ϕ ∧ ψ, ∆ then τ D ( ϕ ∧ ψ ) = max( τ D ( ϕ ) , τ D ( ψ )) , and τ D ( γ i ) = max( τ D ( γ j i i ) , τ D ( γ l i i )) , ≤ i ≤ n ; τ D ( δ i ) = max( τ D ( δ k i i ) , τ D ( δ p i i )) , ≤ i ≤ m. (vii) If D ends with Γ , ∀ xϕ k , ϕ ( t ) ⇒ ∆ ∀ xϕ l , Γ ⇒ ∆ then τ D ( ∀ xϕ l ) = max( τ D ( ∀ xϕ k ) , τ D ( ϕ ( t ))) and the Tr -complexity of the formulas in Γ , ∆ is unchanged. It is in fact the presence of contraction that led to error in its applications in [Hal99], which are rectified by[Lei15], but only for typed truth, not type-free truth. ut and Initial Sequents 6 (viii)
In an application of (cut) , the Tr -complexity of the formulas in the conclusion of therule is treated as in case (vi) above.Finally, the τ -complexity of an LGT -proof D is the maximum of the Tr -complexities for theformulas occurring in it. In what follows, it will be convenient to keep track of all derivation measure in a more compactnotation.
Notation . We write:- LGT m,kn Γ ⇒ ∆ for ‘the sequent Γ ⇒ ∆ has a proof in LGT with length ≤ n , cut-rank ≤ m , and Tr -complexity ≤ k ’.- LGT n Γ ⇒ ∆ for ‘there are m, k such that LGT m,kn Γ ⇒ ∆’, and LGT Γ ⇒ ∆ for‘there is n such that LGT n Γ ⇒ ∆’.- We will omit, when it’s clear from the context, reference to the background system andwrite m,kn instead of LGT m,kn .- We will occasionally also need to refer to the truth complexity of a single formula ina sequent as well. We will keep reference to the proof implicit, and write k ϕ for ‘theoccurrence of ϕ has truth complexity k in the given derivation’.The next lemma states the monotonicity of some of our measures (length and Tr -complexity),some basic properties of ⊥ and ⊤ in derivations, and the fully structural nature of LGT whenformulas of the base language are at stake. Their proofs follow almost immediately from thedefinition of LGT m,kn (monotonicity), or by straightforward inductions on the length of theproof in LGT. Lemma . (i) If LGT m,kn Γ ⇒ ∆ , and k ≤ k and n ≤ n , then LGT m,k n Γ ⇒ ∆ . (ii) If LGT m,kn ⊤ , Γ ⇒ ∆ , then LGT m,kn Γ ⇒ ∆ and the Tr -complexity of the formulas inthe contexts is unchanged. (iii) If LGT m,kn Γ ⇒ ∆ , ⊥ , then LGT m,kn Γ ⇒ ∆ and the Tr -complexity of the formulas inthe contexts is unchanged. (iv) LGT Γ , ϕ ⇒ ϕ, ∆ for all ϕ ∈ L . The usual substitution and weakening lemmata hold for LGT. Crucially for our purposes,they do not entail any increase in the Tr -complexity of the derivation. In the case of weakening,this essentially relies on the fact that, by Definition 2(ii), side formulas in initial sequents haveminimal Tr -complexity.
Lemma . (i) If LGT m,kn Γ ⇒ ∆ , then LGT m,kn Γ( t/x ) ⇒ ∆( t/x ) , where t does not contain variablesemployed in applications of ( ∀ r ) in the proof of Γ ⇒ ∆ . The Tr -complexity of allformulas in Γ , ∆ is unchanged by the substitution. ut and Initial Sequents 7 (ii) If LGT m,kn Γ ⇒ ∆ , then LGT m,kn Γ , Θ ⇒ ∆ , Λ such that formulas in Θ , Λ haveminimal complexity. Moreover, the Tr -complexity of each formula in Γ , ∆ is unchanged. The next lemma contains the key property that differentiates LGT from other nonclassical andsubstructural approaches (cf remark 1 below). Crucially, it states that truth rules are invertiblein a way that does not increase neither the length nor the Tr -complexity of the derivation. Inparticular, when truth ascriptions have non-zero Tr -complexity, inversion actually reduces theirtruth complexity. This property is essential for establishing the admissibility of contraction inLGT and therefore cut-elimination.
Lemma . (i) If LGT m,kn Γ , Tr p ϕ q ⇒ ∆ , then LGT m,kn Γ , ϕ ⇒ ∆ , with τ ( ϕ ) ≤ τ (Tr p ϕ q ) , if τ (Tr p ϕ q ) = 0 ,τ ( ϕ ) < τ (Tr p ϕ q ) , if τ (Tr p ϕ q ) > , and in which the Tr -complexity in the side formulas does not increase.A symmetric claim holds when LGT m,kn Γ ⇒ Tr p ϕ q , ∆ . (ii) If LGT m,kn Γ , ¬ ϕ ⇒ ∆ , then LGT m,kn Γ ⇒ ϕ, ∆ with τ ( ϕ ) ≤ τ ( ¬ ϕ ) and in which the τ -complexity of the side formulas does not increase.A symmetric claim holds when LGT m,kn Γ ⇒ , ¬ ϕ, ∆ . (iii) If LGT m,kn Γ , ϕ ∧ ψ ⇒ ∆ , then LGT m,kn Γ , ϕ, ψ ⇒ ∆ with τ ( ϕ ) , τ ( ψ ) ≤ τ ( ϕ ∧ ψ ) andin which the τ -complexity of the side formulas does not increase. (iv) If LGT m,kn Γ ⇒ ϕ ∧ ψ, ∆ , then LGT m,kn Γ ⇒ ∆ , ϕ and LGT m,kn Γ ⇒ ∆ , ψ with τ ( ϕ ) , τ ( ψ ) ≤ τ ( ϕ ∧ ψ ) and in which the complexity of the side formulas is no greater thantheir τ -maximal occurrence in the premisses. (v) If LGT m,kn Γ ⇒ ∆ , ∀ xϕ , then LGT m,kn Γ ⇒ ∆ , ϕ ( y ) , for any y not free in Γ , ∆ , ∀ xϕ ,with τ ( ϕ ( y )) ≤ τ ( ∀ xϕ ) and in which the complexity of the side formulas does not increase.Proof. We show (i) by induction on n . The other cases are easier.If Γ , Tr p ϕ q ⇒ ∆ – i.e. Γ , Tr p ϕ q ⇒ ∆ is an axiom –, then τ (Tr p ϕ q ) = 0. Therefore, also Γ , ϕ ⇒ ∆ and τ ( ϕ ) ≤ τ (Tr p ϕ q ).If m,kn Γ , Tr p ϕ q ⇒ ∆ with n >
0, then Tr p ϕ q might be principal or not in the last inference.If it’s principal, we haveLGT m,k n Γ , p ϕ ⇒ ∆ n < n, p < k, k ≤ k .(recall that p ϕ signifies: τ ( ϕ ) = p ). The claim is then obtained by monotonicity (Lemma 1(i)).If Tr p ϕ q is not principal, let’s suppose – to consider one of the crucial cases – that the lastinference is an application of (Tr r ). We then have: m ,k n Γ , p Tr p ϕ q ⇒ ∆ , p ψ, ∆ = ∆ , Tr p ψ q , n < n,m = m, k ≤ k, p ≤ k, p < k. ut and Initial Sequents 8 By the induction hypothesis, m ,k n Γ , p ϕ ⇒ ∆ , p ψ , with p ≤ p , and therefore, by ( Tr r) , m,kn Γ , p ϕ ⇒ ∆ , p Tr p ψ q p ≤ k. The remaining cases for this subcase are similarly obtained by induction hypothesis. qed . remark . It is important to observe that, in the presence of initial sequents admitting arbitraryatomic formulas of L Tr , the inversion strategy considered above will not go through. For instance,the derivability of a sequent of the form Γ , Tr p ϕ q ⇒ Tr p ϕ q , ∆ does not guarantee, for instance,the derivability of a sequent Γ , Tr p ϕ q ⇒ ϕ, ∆ with τ ( ϕ ) ≤ τ (Tr p ϕ q ).The absence of explicit contraction – either as a rule or by the assumption of finite setsin sequents – is especially welcome when reasoning with measures such as the Tr -complexity,because it may prove to be difficult to track the Tr -complexity of each formula in a derivation ifit is explicitly allowed to merge with the Tr -complexity other occurrences of the same formulain proofs. However, as it is shown in the next lemma, contraction is an admissible rule in LGT. Lemma τ -admissibility of contraction) . (i) If LGT m,kn Γ , ϕ p , ϕ p ⇒ ∆ , then LGT m,kn Γ , ϕ ⇒ ∆ with τ ( ϕ ) ≤ max( τ ( ϕ p ) , τ ( ϕ p )) and in which the complexity of the side formulas does not increase. (ii) If LGT m,kn Γ ⇒ ϕ p , ϕ p , ∆ , then LGT m,kn Γ ⇒ ϕ, ∆ with τ ( ϕ ) ≤ max( τ ( ϕ p ) , τ ( ϕ p )) and in which the complexity of the side formulas does not increase.Proof. (i) and (ii) are proved simultaneously by induction on n . Let us focus on (i).If Γ , ϕ p , ϕ p ⇒ ∆, then in each case τ ( ϕ p ) = τ ( ϕ p ) = 0 and we have Γ , ϕ ⇒ ∆ inwhich all formulas have Tr -complexity 0. If l +1 Γ , ϕ p , ϕ p ⇒ ∆ and neither ϕ p nor ϕ p areprincipal in the last inference, then l +1 Γ , ϕ ⇒ ∆ – with the expected Tr -complexities – byinduction hypothesis and possibly monotonicity.It remains the case in which l +1 Γ , ϕ p , ϕ p ⇒ ∆ and one of ϕ p or ϕ p is principal in thelast inference. As an example, I treat the crucial case in which ϕ is Tr p ψ q . By assumption, l Γ , ψ p , Tr p ψ q p ⇒ ∆with τ ( ψ p ) < τ (Tr p ψ q p ) ≤ k . By inversion, we have that l Γ , ψ p , ψ p ⇒ ∆ . It can then be that τ ( ψ p ) = τ (Tr p ψ q p ) = 0, or τ ( ψ p ) < τ (Tr p ψ q p ). In both cases, weobtain l +1 Γ , Tr p ψ q ⇒ ∆with τ (Tr p ψ q ) ≤ max( τ (Tr p ψ q p ) , τ (Tr p ψ q p )). It is crucial to observe that without the stronginvertibility property expressed by lemma 3(i) – which in turn relies on the restriction of initial Of course, strictly speaking, inversion may not provide a copy of the proof in which the structure of the oc-currences given by the superscripts is preserved. However, since the only relevant detail here is to distinguishbetween the two occurrences ‘to be contracted’, we keep the same index for the same formulas before and afterthe application of inversion. ut and Initial Sequents 9 sequents –, one would not be able to establish this case. In particular, if τ (Tr p ψ q p ) = k >τ (Tr p ψ q p ), without the special invertibility property of Lemma 3(i) one would not be able tocomplete the proof.It is also worth noticing that the formulation of ( ∀ l) and its associated Tr -complexity rendersthe case of (i) in which one of the ϕ ’s is principal in the last inference and of the form ∀ xϕ straightforward. Also, the simultaneous induction is especially required in the case in which thelast inference is an application of ( ¬ l ) to ϕ p or ϕ p – and symmetrically for (ii) and ( ¬ r ). qed . The reduction lemma can now be proved in a fairly standard way. We let ( α , . . . , α m ) ≺ ( β , . . . , β n ) if α i < β i ( i = 1 , . . . , n ), and for all j < i , α j = β j . Lemma . If LGT m,kn Γ ⇒ ∆ , ϕ l and LGT m,kn ϕ l , Γ ⇒ ∆ , then LGT m,kn + n Γ ⇒ ∆ . In this latter sequent, the occurrences of formulas have Tr -complexity no greater thanthe maximum of their corresponding occurrences in the assumptions of the claim.Proof. The proof is by multiple, complete induction on ( l, m, n + n ), with l = max( τ ( ϕ l ) , τ ( ϕ l )).Our induction hypothesis is thus:(2) m ′ ,kn ′ Γ ⇒ ∆ , ψ l ′ and LGT m ′ ,kn ′ ψ l ′ , Γ ⇒ ∆ entail m ′ ,kn ′ + n ′ Γ ⇒ ∆ , for | ψ | ≤ m ′ , l ′ = max( τ ( ψ l ′ ) , τ ( ψ l ′ )), and ( l ′ , m ′ , n ′ + n ′ ) ≺ ( l, m, n + n ). We only focus oncases in which Tr -complexity plays a crucial role. The rest is standard.If one of Γ ⇒ ∆ , ϕ l or Γ ⇒ ∆ , ϕ l is an axiom, one has to distinguish different subcases: If ϕ l or ϕ l are principal, then depending on whether ϕ is ⊥ , ⊤ , or atomic, we employ Lemma1(i) (in the former cases), or Lemma 4(i). If neither of ϕ l and ϕ l is principal, then Γ ⇒ ∆ isalready an axiom with minimal Tr -complexity.Suppose now that none of Γ ⇒ ∆ , ϕ l or ϕ l , Γ ⇒ ∆ are axioms, but ϕ is not principal in thelast inference of one of their derivations, for instance the derivation of Γ ⇒ ∆ , ϕ l . In such cases,the strategy is analogous for all rules. Let’s consider the case of (Tr l ) as an example; that is,the case in which one has m,kn Γ , p Tr p ψ q ⇒ ∆ , ϕ m,kn ϕ, Γ , Tr p ψ q ⇒ ∆and the leftmost claim is obtained by (Tr l ) from m,k n Γ , p ψ ⇒ ∆ , ϕ with p = p + 1 ≤ k, n < n and k ≤ k , and Γ ≡ Γ , Tr p ψ q . By the weakening lemma, wethen obtain m,k n Γ , Tr p ψ q , ψ ⇒ ∆ , ϕ m,kn ϕ, Γ , p Tr p ψ q , ψ ⇒ ∆with p ≤ k . Since n + n < n + n , the induction hypothesis yields: m,kn + n Γ , p Tr p ψ q , p ψ ⇒ ∆ . ut and Initial Sequents 10 By applying (Tr l ) and lemma 4, one obtains that m,kn +1+ n Γ , Tr p ψ q ⇒ ∆ . This, however, yields the desired claim since n + 1 + n ≤ n + n and τ (Tr p ψ q ) = max( p, p ).We are left with the case in which both ϕ l and ϕ l are principal in the last inferences of therelevant derivations. Here the crucial case in which ϕ ≡ Tr p ψ q follows directly by the maininduction hypothesis, since if our premisses are obtained via applications of the truth rules from m,k n Γ ⇒ ∆ , ψ l m,k n ψ l , Γ ⇒ ∆with τ ( ψ l ) , τ ( ψ l ) < l , the induction hypothesis and the monotonicity properties of LGTimmediately yield m,kn + n Γ ⇒ ∆ with the correct Tr -complexities in Γ , ∆It is worth noting that the case in which ϕ ≡ ∀ xψ is treated standardly as well but onehas first to get rid of the universal quantifier in the premise of ( ∀ l) . This involves an essentialapplication of the substitution lemma that, as we know, leaves Tr -complexities unchanged. qed . As is it clear from the Reduction Lemma, we obtain a cut-elimination theorem with standardhyper-exponential upper bounds.
Corollary . If LGT m,kn Γ ⇒ ∆ , then LGT ,k nm Γ ⇒ ∆ . Cut-elimination obviously entails the consistency of LGT, defined for instance as the non-derivability of the empty sequent in LGT. This may be considered to be a nice feature of LGT qua theory of disquotational truth, as its consistency does not require more substantial notionsof mathematical truth such as the ones involved in model-theoretic consistency proofs. However,often the presence of nice models – even if interpreted in a purely instrumental way – is a sign ofthe conceptual richness of one’s truth predicate. We will see (section 4) that LGT also featuresnice models. 3.
Extension with arithmetical axioms
The cut elimination above can be easily extended to induction-free, arithmetical base theories.For definiteness, we choose our base arithmetical theory to be Robinson’s Q. However, what isrelevant for our discussion is the geometric nature of such arithmetical axioms. We adapt to oursetting the approach to the proof-theory of geometric rules investigated by [NvP11]. Since themain structural lemmata have been introduced, this mainly involves checking that Negri andVon Plato’s extension with geometric axioms interacts well with the truth rules and in particularwith the notion of Tr -complexity and its properties.In this section we work with the language L N of arithmetic. For definiteness, we assumethe language of arithmetic is specified by the signature { , S , + , ×} and let L Tr N := L N ∪ { Tr } .We assume a standard Gödel numbering of L Tr and write e for the Gödel number of the L Tr -expression e and p e q for the corresponding numeral. Numerals are defined as: 0 := 0 and n + 1 = S n . ut and Initial Sequents 11 The axioms of Robinson’s arithmetic Q are the universal closures of the following L N -formulas: ¬ x ) , S( x ) = S( y ) → x = y,x = 0 ∨ ∃ y ( x = S( y )) , x + 0 = x,x + S( y ) = S( x + y ) , x × ,x × S( y ) = ( x × y ) + x. As indicated in [NvP11], a G3-version of Q – equivalent to the axiom based system givenabove – can be defined. In the present context, it will play the role of the base theory of ourtheory of truth, in that it provides us with some explicit machinery for naming sentences of ourlanguage. Unlike what is done in the previous section, we will simultaneously define derivationsin our base system and the relevant measures by means of the relation Q g m,kn . We will includea parameter for the Tr -complexity in this definition to allow for straightforward extensions,although of course if one focuses on purely arithmetical derivations the Tr -complexity of theproof is always 0. Definition g ) . Q g extends the logic of LGT formulated in L Tr – together with a restrictionof (ref) to atomic formulas of L N and by omitting ( ⊥ ) and ( ⊤ ) – with the following rules (= 1) If Q g m,kn Γ , t = t ⇒ ∆ , then Q g m,kn Γ ⇒ ∆ , with n < n . (= 2) If Q g m,kn s = t, ϕ ( s ) , ϕ ( t ) , Γ ⇒ ∆ , then m,kn s = t, ϕ ( t ) , Γ ⇒ ∆ , with ϕ ( v ) an atomicformula of L N and n < n . (Q g
1) Q g m,kn Γ , S x = 0 ⇒ ∆ for any n, m, k . (Q g If Q g m,kn Γ , x = y, S( x ) = S( y ) ⇒ ∆ , then Q g m,kn Γ , S( x ) = S( y ) ⇒ ∆ , with n < n . (Q g If Q g m,kn Γ , x = 0 ⇒ ∆ and Q g m,kn Γ , y = S( x ) ⇒ ∆ , then Q g m,kn Γ ⇒ ∆ , with n , n < n and with y / ∈ FV(Γ , ∆ , x = 0) . (Q g If Q g m,kn Γ , x + 0 = x ⇒ ∆ , then Q g m,kn Γ ⇒ ∆ , with n < n . (Q g If Q g m,kn Γ , x + S( y ) = S( x + y ) ⇒ ∆ , then Q g m,kn Γ ⇒ ∆ , with n < n . (Q g If Q g m,kn Γ , x × ⇒ ∆ , then Q g m,kn Γ ⇒ ∆ , with n < n . (Q g If Q g m,kn Γ , x × S( y ) = ( x × y ) + x ⇒ ∆ , then Q g m,kn Γ ⇒ ∆ , with n < n . In (Q g y acts as an eigenvariable, because it is intended to be playing the role of an existentiallyquantifiable variable.As before, by a straightforward induction on the length of the proof in Q g , we can show that,as far as formulas of L N are concerned, reflexivity holds for them. The next lemma states that,as desired, Q g and Q prove the same theorems. Lemma . Q ⊢ V Γ → W ∆ if and only if Q g ⊢ Γ ⇒ ∆ . ut and Initial Sequents 12 The system LGT N is obtained by extending Q g with fully disquotational truth. The truthrules are only notational variations of ( Tr l) and ( Tr r) . Definition . The relation
LGT N m,kn is defined by means of the direct analogues of clauses (= 1) - (Q g from Definition 3 plus: (Tr r N ) If LGT N m,k n Γ ⇒ ϕ, ∆ , then LGT N m,kn Γ ⇒ Tr l, ∆ , with n < n, k ≤ k , l = ϕ with ϕ a sentence of L Tr , τ (Tr l ) = τ ( ϕ ) + 1 , and the Tr -complexities of the side formulasare unchanged. (Tr l N ) If LGT N m,k n Γ , ϕ ⇒ ∆ , then LGT N m,kkn Γ , Tr l ⇒ ∆ , with n < n, k ≤ k , l = ϕ with ϕ a sentence of L Tr , τ (Tr l ) = τ ( ϕ )+1 , and the Tr -complexities of the side formulasare unchanged. remark . In the rest of the section, we assume that so-called pure variable convention. That is,free and bound variables are always distinct in proofs, and that the eigenvariables of applicationsof (Q g
3) in proofs are distinct.As before, the identity axioms hold unrestrictedly for sentences of L N , so we have(3) LGT N ⊢ Γ , ϕ ⇒ ϕ, ∆ for all ϕ ∈ L N .The substitution lemma for LGT N – compared with its analogue in the previous section –needs a little extra care in dealing with the variables of the geometric rules. Essentially, in therequired induction on the length of the proof in LGT N , the cases of ( ∀ r) and (Q g
3) require theeigenvariables not to occur in the substituens . Similarly, in the weakening lemma one only needsto be careful that the weakened formulas do not contain variables that may appear in geometricrules. In such cases the substitution lemma can be employed. Tr -complexities are handled inprecisely the same way as before.
Lemma . (i) If LGT N m,kn , then LGT N m,kn Γ( t/x ) ⇒ ∆( t/x ) where t is free for x in Γ , ∆ and itdoes not contain any eigenvariables employed in applications of ( ∀ r) , as well as variablesemployed Q g -rules. The substitution does not change the Tr -complexity of the formulasoccurring in Γ , ∆ . (ii) If LGT N m,kn Γ ⇒ ∆ , then LGT N m,kn Γ , Θ ⇒ ∆ , Λ with Θ and Λ not containingvariables appearing in geometric rules and whose formulas have minimal Tr -complexity.Moreover, the Tr -complexity of each formula in Γ , ∆ is unchanged. The invertibility lemma also proceeds with minor variations. Crucially, the kind of τ -invertibilityfor the truth rules involved in lemma 3(i) is preserved. To prove an analogue of Lemma 3(v), oneemploys Remark 2 to ensure that if the last inference involves a geometric rule such as (Q g Lemma . The propositional logical rules of
LGT N are τ -invertible in the way de-scribed by Lemma 3(ii)-(iv). Moreover: ut and Initial Sequents 13 (i) If LGT N m,kn Γ , Tr l ⇒ ∆ with l = ϕ , then LGT m,kn Γ , ϕ ⇒ ∆ , with τ ( ϕ ) ≤ τ (Tr l ) , if τ (Tr l ) = 0 ,τ ( ϕ ) < τ (Tr l ) , if τ (Tr l ) > , and with unchanged Tr -complexity in the side formulas.A symmetric claim holds when LGT m,kn Γ ⇒ Tr p l q , ∆ with l = ϕ . (ii) If LGT N m,kn Γ ⇒ ∆ , ∀ xϕ , then LGT N m,kn Γ ⇒ ∆ , ϕ ( y ) , for any y not free in Γ , ∆ , ∀ xϕ and not among the variables of geometric rules, with τ ( ϕ ( y )) ≤ τ ( ∀ xϕ ) and in which thecomplexity of the side formulas does not increase. The previous lemmata makes it possible to extend in a straightforward way the τ -admissibilityof contraction to LGT N . Lemma . If LGT N m,kn Γ , ϕ k , ϕ k ⇒ ∆ , then LGT N m,kn Γ , ϕ ⇒ ∆ with with τ ( ϕ ) ≤ max( τ ( ϕ k ) , τ ( ϕ k )) and in which the complexity of the side formulas does not increase. Asymmetric claim holds for when the formulas to be contracted appear on the consequent. With these lemmata at hand, we are then able to prove a reduction lemma in the same veinas the previous section. Noticeably, the interaction between truth, identity, and arithmeticalrules is particularly smooth because truth rules only apply to closed terms naming sentences,and therefore no extra-care with variables is needed to deal with cases in which the eliminationof a cut on a non-principal truth ascription is obtained by performing the cut on the premissesof a geometric rule. The cut-elimination procedure in the presence of geometric rules does notchange the hyperexponential upper-bound.
Corollary . Cut is eliminable in
LGT N . The method outlined in this section straightforwardly extends to geometric rules correspondingto the defining equations of other primitive recursive functions. One could also then strengthenthe truth rules, for instance, to pointwise compositional rules such as:Γ ⇒ ∆ , ϕ Γ ⇒ ∆ , ψ Γ ⇒ ∆ , Tr ( l ∧ . m )with ϕ = l, ψ = m and ∧ . the function symbol representing in L N the syntactic operation ϕ, ψ ϕ ∧ ψ ) . Finally, Corollary 2 and subsequent remarks clearly yields conservativity properties of therules (Tr l ) and (Tr r ) over base theories given by geometric axioms. In fact, for ϕ ∈ L N , ifLGT N ⊢ ⇒ ϕ , then there is a cut-free proof D of ⇒ ϕ . All succedents in D must be subformulasof ϕ , and all formulas in the antecedents must be formulas of L N , because they are the only onesthat may disappear due to geometric and identity rules. Therefore, we have: Proposition . LGT N is a conservative extension of Q g . ut and Initial Sequents 14 Infinitary rules and semantics
In this section we first extend LGT to an infinitary system LGT ∞ , and then establish theanticipated links between LGT ∞ and fixed-point semantics.4.1. Infinitary rules.
It is convenient to work with an expansion of L N with function symbolsfor primitive recursive functions, which we call L + N . We then in turn denote with L +Tr theexpansion of L + N with the predicate Tr . L + N will then contain function symbols corresponding tosyntactic operations on Gödel numbers such as ∧ . above and n n ) , n, m p . . . p Tr | {z } m Tr s n q . . . q )(4) n n ϕ ( v ) , t ϕ ( t/v )) ϕ ¬ ϕ ) ϕ, v ∀ vϕ ) s, t s = t )We will employ, respectively, the function symbols Tr. , tr , num , sub , ¬ . , ∀ . , =. to express those op-erations in our language. On occasion we will make reference to a function symbol val for arecursive evaluation function for primitive recursive functions expressing the semantic evaluationfunction t t N taking a closed term and returning its value in the standard model of L N .The infinitary system LGT ∞ is essentially obtained by reformulating LGT in L +Tr , replacingbasic truth and falsities with arithmetical truths and falsities, and supplementing the system withan ω -rule. Later on we will also consider the language L N of second-order arithmetic, extending L + N with second-order (relational) variables and quantifiers. The presence of the ω -rule makes the length of derivation, as well as the associated Tr -complexities, possibly infinite – more precisely,a countable ordinal. In particular, the definition of Tr -complexity needs to be supplementedwith the case in which a derivation ends with an application of the infinitary rule. This can beinformally described as follows. If a derivation D ends with(5) . . . γ j i , . . . , γ j in n ⇒ δ k i , . . . , δ k im m ϕ ( t i ) . . .γ , . . . , γ n ⇒ δ , . . . , δ m , ∀ xϕ then: τ ( γ k ) := sup { τ ( γ j ik k ) | i ∈ ω, ≤ k ≤ n } ,τ ( δ l ) := sup { τ ( δ k il l ) | i ∈ ω, ≤ l ≤ m } ,τ ( ∀ xϕ ) := sup { τ ( ϕ ( t )) | t a closed term of L + N } . Here’s the official definition of the infinitary system LGT ∞ : Definition ∞ ) . LGT ∞ is obtained from LGT by: • Omitting free variables. • Replacing the axioms ( ⊤ ) , ( ⊥ ) with ( T ) m,βα Γ ⇒ r = s, ∆ for any α, β, m and with r N = s N ; ut and Initial Sequents 15 ( F ) m,βα Γ , r = s ⇒ ∆ for any α, β, m and with r N = s N . • Replacing Tr l and Tr r with the more general: (Tr r N ) If LGT N m,βα Γ ⇒ ϕ, ∆ , then LGT N m,δγ Γ ⇒ Tr t, ∆ , with α < γ, β < δ , t N = ϕ , τ (Tr t ) = τ ( ϕ ) + 1 , and the Tr -complexities of the side formulas are unchanged. (Tr l N ) If LGT N m,βα Γ , ϕ ⇒ ∆ , then LGT N m,δγ Γ , Tr t ⇒ ∆ , with α < γ, β < δ , t N = ϕ , τ (Tr t ) = τ ( ϕ ) + 1 , and the Tr -complexities of the side formulas are unchanged. • Replacing ( ∀ r) with: ( ω ) If for all t there are α < γ and β ≤ δ such that m,βα Γ ⇒ ϕ ( t ) , ∆ , then m,δγ Γ ⇒∀ xϕ, ∆ , with τ ( ∀ xϕ ) = sup { τ ( ϕ ( t )) | t a term of L + N } . remark . (i) The general formulation of (Tr r N ) and (Tr l N ) is essential for the claims below. It allowstransfinite iterations of applications of Tr , which are otherwise not available, even in thepresence of the ω -rule. This can be easily seen by considering the function representingthe rightmost operation in (4), which we call tr( n, t ).LGT ∞ ⊢ Tr (tr( n, p q )) for any n ∈ ω .The ω -rule then gives us transfinite iterations of Tr . This process, of course, carries onfor further recursive ordinals by carefully choosing syntactic operations akin to tr( · ).(ii) As before, LPC ∞ proves identity sequents Γ , ϕ ⇒ ϕ, ∆ for all ϕ ∈ L + N .Then the cut-elimination strategy proceeds with only minor variations. We have: Lemma . (i) (Weakening) If LGT ∞ m,βα Γ ⇒ ∆ , then LGT ∞ m,βα Γ , Γ ⇒ ∆ , ∆ with all formulasin ∆ , Γ featuring minimal Tr -complexity. (ii) (Inversion) All rules shared by LGT ∞ and LGT are length-, and τ -invertible as prescribedby Lemma 3, (i) - (iv) . Moreover:if LGT ∞ m,βα Γ ⇒ ∆ , ∀ xϕ , then LGT ∞ m,βα Γ ⇒ ∆ , ϕ ( t ) , for anyclosed term t . In addition, τ ( ϕ ( t i )) ≤ τ ( ∀ xϕ ) and the Tr -complexity ofthe formulas in the contexts of the inverted sequents is unchanged. (iii) Contraction is τ -preserving and length-preserving admissible in LGT ∞ . In particular, the reduction lemma generalizes to ordinals in the expected way. It is obviouslyimportant to employ ordinal addition in the induction to deal with cuts on principal formulas of ω -rules. ut and Initial Sequents 16 Proposition . If LPC ∞ m,βα Γ ⇒ ∆ , ϕ and LPC ∞ m,βα ϕ, Γ ⇒ ∆ , then LPC ∞ m,βα + α Γ ⇒ ∆ .Therefore, If LGT ∞ m,βα Γ ⇒ ∆ , then LGT ∞ ,β Γ ⇒ ∆ . In light of Proposition 2, one can employ LPC ∞ to establish the consistency, via appropriateembeddings, of finitary extensions of LGT N with induction axioms.4.2. Fixed-point models.
We relate the infinitary system introduced above and a fixed-pointmodel for L Tr . The results of this section are intended to compare to [FG18] and, less directly,to [Can90]. We show that the form of invertibility allowed by LGT ∞ enables one to preservefully disquotational truth rules with context while capturing Kripkean grounded truth. In thereferences mentioned, this could only be achieved by disallowing contexts altogether from thepremisses of (Tr l ) and (Tr r ): this, in turn, would render such rules less obviously ‘truth rules’,but open to other interpretations such as the ones based on provability.Consider the following formula of the language L N of second-order arithmetic, where CT( x )and Sent L +Tr are primitive recursive predicate expressing the notions of being a closed term anda sentence of L +Tr respectively:K( X, x ) : ↔ Sent L +Tr ( x ) ∧∃ y ∃ z (Ct L +Tr ( y ) ∧ Ct L +Tr ( z ) ∧ x = ( y =. z ) ∧ val( y ) = val( z )) ∨∃ y ∃ z (Ct L +Tr ( y ) ∧ Ct L +Tr ( z ) ∧ x = ¬ . ( y =. z ) ∧ val( y ) = val( z )) ∨∃ y (CT( y ) ∧ x = sub( p Tr v q , p v q , y ) ∧ val( y ) ∈ X ) ∨∃ y (CT( y ) ∧ x = sub( p ¬ Tr v q , p v q , y ) ∧ ( ¬ . val( y )) ∈ X ) ∨∃ y (Sent L +Tr ( y ) ∧ x = ( ¬ . ¬ . y ) ∧ y ∈ X ) ∨∃ y ∃ z (Sent L +Tr ( x ) ∧ x = ( y ∧ . x ) ∧ y ∈ X ∧ z ∈ X ) ∨∃ y ∃ z (Sent L +Tr ( x ) ∧ x = ( ¬ . ( y ∧ . z )) ∧ ( ¬ . y ) ∈ X ∨ ( ¬ . z ) ∈ X ) ∨∃ v ∃ y (Sent L +Tr ( x ) ∧ x = ( ∀ . vy ) ∧ ∀ u (CT( u ) → sub( x, v, u ) ∈ X )) ∨∃ v ∃ y (Sent L +Tr ( x ) ∧ x = ( ¬ . ∀ . vy ) ∧ ∃ u (CT( u ) ∧ ¬ . sub( x, v, u ) ∈ X )) . By inspection of K(
X, x ), it is clear that X occurs positively in it, in the sense that it does notcontain occurrences of u / ∈ X , and essentially so. We define an operator Φ K : P ( ω ) → P ( ω ) asfollows: Φ K ( S ) := { n | ( N , S ) (cid:15) K( X, n ) } , where ( N , S ) expresses that S is used to interpret the variable X . Since K( X, x ) is X -positive, Φ K is monotone:(6) S ⊆ S only if Φ K ( S ) ⊆ Φ K ( S ) . More precisely, this means that we can translate K(
X, x ) in a Tait-language and no occurrences of u / ∈ X arepresent. ut and Initial Sequents 17 By transfinite recursion, one then sets: Φ K α := Φ K ( Φ K <α ) , with Φ K <α := [ β<α Φ K β . It is clear that Φ K has fixed points , i.e. there are ordinals γ such that Φ K <γ = Φ K γ . We let κ K = min { α | Φ K <α = Φ K α } , and I Φ K := Φ K <α . I Φ K is called the minimal fixed point of Φ K . Itis well-known (see e.g. [Poh09, Ch. 6]), that κ K = ω CK1 .I Φ K is well-known for capturing the concept of grounded truth [Kri75], because any ϕ ∈ I Φ K is either a true atomic sentence of L + N or an atomic truth appears in its ‘dependency’ structure[Lei05]. We will briefly return on the connection between LGT ∞ and grounded truth shortly.For n ∈ I Φ K , its inductive norm is defined as: | n | Φ K := min { α | n ∈ Φ K α } . We also have:(7) n ∈ I Φ K iff ∀ X ( ∀ x (K( X, x ) → x ∈ X ) → n ∈ X ) , so I Φ K is Π -definable in L N . As noticed by [Kri75] (see also [Bur86]), I Φ K is Π -complete. The strict relationships between Π -sets and infinitary cut-free calculi are secured by gen-eral results [Poh09, §6.6]. To witness the link between LGT ∞ and I Φ K , we establish a directcorrespondence between the two frameworks. The existence of a nice semantics for LGT-basedsystems will then immediately follow. Lemma . If LGT ∞ ,βα Γ ⇒ ∆ , then either there is a γ ∈ Γ with | ¬ γ | Φ K ≤ α or there is a δ ∈ ∆ with | δ | Φ K ≤ α .Proof. The proof is by transfinite induction on α ≤ ω CK1 .If α = 0, then the claim follows by definition for ( T ) and ( F ), or from the fact that closedatomic identities of L +Tr are decided by I Φ .If α is successor or limit the claim follows by inductive hypothesis by reflecting on the factthat the disjuncts in K harmonize well with the rules of LPC ∞ . For instance, if Γ ⇒ ∆ is provedby an application of (Tr r N ), we have ,β α Γ ⇒ ∆ , ϕ with β ≤ β, α ≤ α , τ ( ϕ ) < β . If some γ ∈ Γ is such that | ¬ γ | Φ K ≤ α , or some δ ∈ ∆ issuch that | δ | Φ K ≤ α , we are done by the definition of Φ α . Otherwise, by induction hypothesis,we have that | ϕ | Φ K ≤ α and therefore, for t N = ϕ , | Tr t | Φ K ≤ α .Noticeably, even though K does not feature a full clause for negation, I Φ K can still capturetheir behaviour in the absence of initial sequents. Suppose for instance that Γ ⇒ ∆ is such that Otherwise, { n | n ∈ Φ K α \ Φ K <α } would be a subset of N of cardinality ℵ . The idea of the proof: one can uniformly replace y ∈ X by Tr sub( u, p v q , num( y )) in P ( x, X ) – an arbitraryinductive definition – to obtain P ′ ( x, u ). The diagonal lemma then yields a formula ξ ( v ) such that( N , X ) (cid:15) ξ ( v ) iff ( N , X ) (cid:15) P ′ ( x, p ξ ( v ) q ) . Finally, one shows by transfinite induction on the generation of the minimal fixed point I P that n ∈ I P if andonly if ( N , I Φ K ) (cid:15) Tr p ξ ( ˙ n ) q . ut and Initial Sequents 18 ∆ = ∆ , ¬ ϕ and Γ ⇒ ∆ is obtained with an application of ( ¬ r ), so that ,β α Γ , ϕ ⇒ ∆with β ≤ α < α , α ≤ α . By induction hypothesis, either | ¬ γ | Φ K = α for some γ ∈ Γ, or | ¬ ϕ | Φ K = α , or | δ | Φ K = α for δ ∈ ∆. In each case, we obtain the claim by definition of I Φ K .Similarly, if Γ ⇒ ∆ is obtained with an application of ( ¬ l ), so that ,β α Γ ⇒ ϕ, ∆ , in the crucial case in which | ϕ | Φ K ≤ α , one has that | ¬¬ ϕ | Φ K ≤ α , as required. qed . Lemma 11 reveals a non-standard way of thinking about ‘logical’ consequence – or better, sat-isfiability of sequents – in Kripke models which is intrinsic to LGT and extensions thereof. If inthe customary approach – cf. for instance the literature stemming from [HH06] – the satisfiabilityof a sequent Γ ⇒ ∆ is defined as preservation of truth in fixed-point models, the notion of con-sequence underlying the semantics of LGT is based on the existence of appropriate determinatetruth values (false in the antecedent, true in the succedent). In the terminology of [CERvR12],this is a tolerant-strict notion of consequence. As mentioned earlier in the paper, [NR18] pro-posed such notion of consequence and showed that this semantics is compatible with a primitive,self-applicable predicate for consequence which fully internalizes it in the object-language. Amore comprehensive study of such notion, including the formulation of a compositional theoryof a truth whose ω -models are exactly the fixed points of Φ K above, is carried out in [NR20].Conversely, we also have that the extension of I Φ can be characterized in terms of LGT ∞ proofs. Lemma . If | ϕ | Φ K ≤ α , then there is an n ∈ ω such that LGT ∞ ,βα + n ⇒ ϕ , with β ≤ α + n .Proof. The proof is again by induction on α .If α = 0, ϕ can only be s = t or s = t for closed terms s, t and s N = t N or s N = t N respectively.In the latter case, one has , Γ , s = t ⇒ ∆, and therefore Γ ⇒ s = t, ∆. In the former,simply , Γ ⇒ s = t, ∆.If α > α is asuccessor ordinal, one consider the different clauses in Φ K . The mismatch between norm andlength of proof is essentially required when negated formulas are considered. For instance, if | ¬ Tr t | Φ K = α , then | ¬ ψ | Φ K = α < α with t N = ψ . The induction hypothesis then yields ,β α + m ⇒ ¬ ψ for some m , β < β . By the inversion Lemma 10, we obtain ,β α + m ψ ⇒ . Theclaim is then obtained by (Tr l N ), and ( ¬ r ).Similarly, if | ¬¬ ψ | Φ K ≤ α , then | ψ | Φ K ≤ α < α . By induction hypothesis, ,β α + m ⇒ ψ with β ≤ β for some m . By the negation rules, ,β α + m +2 ⇒ ¬¬ ψ . qed . By inspection of the proof of Lemma 12, one notices that the following claim also holds,yielding a symmetric picture to the one depicted by Lemma 11:(8) if | ¬ ϕ | Φ K ≤ α , then there is an n ∈ ω such that LGT ∞ ,βα + n ϕ ⇒ , with β ≤ α + n . ut and Initial Sequents 19 Corollary . (i) | ϕ | ∈ I Φ K if and only if LGT ∞ ,βα ⇒ ϕ for some β ≤ α < ω CK1 . (ii) | ¬ ϕ | ∈ I Φ K if and only if LGT ∞ ,βα ϕ ⇒ for some β ≤ α < ω CK1 . It is clear that, since we can straightforwardly embed LGT N – but, as mentioned in theprevious subsection also an extensions of LGT N with full L Tr -induction – in LGT ∞ , the resultsabove also amount to soundness proofs, with respect to fixed-point semantics, of our systemsmodulo the notion of consequence relation specified by Lemma 11.5. Conclusion
The focus of this paper is on the structural properties of theories of fully disquotationaltruth with restricted initial sequents. If one finds the framework appealing for the basic logicalproperties presented here, there are certainly further philosophical and technical questions to beinvestigated.The kind of reasoning available in theories such as LGT and extensions thereof displays pecu-liar properties. First of all, the rules of inference available are entirely classical. Moreover, thesystems reveal a special relationships occurring between truth ascriptions and the underlying baselanguage which is not available in alternative formal systems for transparent truth. Philosophersoften explain grounded truth in terms of a form of supervenience of truth on the non-truth-theoretic world (cf. for instance [Lei05]). Theories in the style of LGT seem to capture this ideain a particularly strong way. Essentially, the absence of initial sequents featuring the truth pred-icate blocks the possibility of reasoning hypothetically with arbitrary truth ascriptions. Onlyformulas of the base language can be freely assumed in reasoning – cf. Lemma 1(iv), (3), Remark3(ii). Semantically speaking, in the context fully structural approaches, one can perform hypo-thetical reasoning also by employing sentences that may not have a determinate truth value. Inthe present framework this is ruled out, and hypothetical reasoning is only available for sentencesthat are determinately true or false, such as sentences of the base language. This does not amountto say that for no sentence containing the truth predicate some form of hypothetical reasoning isavailable. The framework automatically enables one to iterate the truth predicate over sentencesthat are ‘grounded’. For instance, the inference Tr (tr( n, p q )) ⇒ Tr (tr( n, p q )) is avail-able for any n in LGT N , and this can be iterated into the transfinite in LGT ∞ . Moreover, thisis achieved without assigning any indices to truth predicates: hypothetical reasoning on truth isautomatically grounded in non-truth-theoretic facts, even in the presence of a fully transparenttruth predicate. On the other hand, it’s also clear that blind hypothetical reasoning, given theundefinability of groudnedness, is only available for non-truth-theoretic sentences. It seems in-teresting to explore further the connections between LGT, grounded truth, and the associatednotion of grounded inference stemming from [NR18]. Philosophically, this may be dealt with, for instance, as in [Kri75], by applying Strawson’s analysis that thehypothesis of a truth ascription should be understood as an attempt to make a claim, to express a proposition. ut and Initial Sequents 20
On the logical side, a natural development consists in considering extensions of LGT N withinduction principles and more complex truth rules such as general compositional rules. In par-ticular, since the presence of induction prevents full cut elimination arguments, the main focuswould be on variants of Proposition 1 (conservativity property) for such extensions. The mainstrategy needs to be modified to resemble more closely the conservativity proof-strategy followedin [Hal99, Lei15] for the compositional, Tarskian truth theory known as CT ↾ , in which one doesnot require the strong invertibility properties proper of the G3 systems above. The restrictionof initial sequents in that context looks promising because the the counterexamples found to thegeneral strategy in [Hal99] – cf. [Lei15, §3.7] – involve an essential use of contraction and initialsequents involving truth ascriptions. References [Bea09] J. C. Beall.
Spandrels of Truth . Oxford University Press, 2009.[Bur86] John P. Burgess. The truth is never simple.
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