Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE
aa r X i v : . [ m a t h . L O ] A ug Nonclassical truth with classical strength.
A proof-theoretic analysis of compositional truth over hype . Martin Fischer and Carlo Nicolai and Pablo Dopico Fernandez
Abstract.
Questions concerning the proof-theoretic strength of classical versus nonclassicaltheories of truth have received some attention recently. A particularly convenient case studyconcerns classical and nonclassical axiomatizations of fixed-point semantics. It is known thatnonclassical axiomatizations in four- or three-valued logics are substantially weaker than theirclassical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment – a logic recently studied by Hannes Leitgeb under the label
HYPE . Weshow in particular that, by formulating the theory
PKF over
HYPE , one obtains a theorythat is sound with respect to fixed-point models, while being proof-theoretically on a par withits classical counterpart KF . Moreover, we establish that also its schematic extension – in thesense of Feferman – is as strong as the schematic extension of KF , thus matching the strengthof predicative analysis. Introduction
The question whether there are nonclassical formal systems of primitive truth that can achievesignificant proof-theoretic strength has received much attention in the recent literature. SolomonFeferman [Fef84] famously claimed that ‘nothing like sustained ordinary reasoning can be carriedon’ in the standard nonclassical systems that support strong forms of inter-substitutivity of A and ‘ A is true’. One way of understanding this claim is by measuring how much mathematicscan be encoded in such systems. Since the strength of mathematical systems (whether classicalor nonclassical) is traditionally measured in terms of the ordinals that can be well-ordered bythem, the ordinal analysis of nonclassical systems of truth becomes relevant.We are mainly interested in the proof-theoretic analysis of nonclassical systems inspired byfixed-point semantics [Kri75]. Since fixed-point semantics has nice axiomatizations, both classicaland nonclassical, it represents a particularly convenient arena to measure the impact of weakeningthe logic on proof-theoretic strength. The axiomatization of fixed-point semantics in classicallogic – a.k.a. KF – is known to have the proof-theoretic ordinal ϕ ε Halbach andHorsten have proposed in [HH06] a nonclassical axiomatization, known as
PKF , and showed thatit has proof-theoretic ordinal ϕ ω
0. There have been some attempts to overcome this mismatchin strength on the nonclassical side. [Nic17] showed that even without expanding the logicalresources of the theory,
PKF can be extended with suitable instances of transfinite inductionto recover all classical true theorems of KF . [FHN17] showed that a simple theory featuring Or Γ , depending on whether one focuses on a version of the theory with or without suitable open-endedsubstitution rule schemata. nonclassical initial sequents of the form A ⇒ Tr p A q and Tr p A q ⇒ A can be closed under specialreflection principles to recover the arithmetical strength of PKF and KF . More recently, [Fie20]showed that, by enlarging the primitive concepts of PKF with a predicate for ‘classicality’, onecan achieve the proof-theoretic strength of KF in both the schematic and non-schematic versions.In the paper we explore a different option, which in a sense completes the picture above. Weenlarge the standard four-valued logic of PKF with a new conditional, which is based on thelogic
HYPE recently proposed by [Lei19]. The conditional has several features that resemble anintuitionistic conditional, but its weaker interaction with the
FDE -negation makes it possible tosustain the intersubstitutivity of A and ‘ A is true’ for sentences not containing the conditional.This extended theory, that we call KFL , is shown to be proof-theoretically equivalent to KF .Its extension with a schematic substitution rule, called KFL ∗ , is shown to be proof-theoreticallyequivalent to the schematic extension of KF – called Ref ∗ ( PA ( P )) in [Fef91].In particular, we show that the conditional of the logic HYPE enables one to mimic, whencarefully handled, the standard lower bound proofs by Gentzen and Feferman-Schütte for trans-finite induction in classical arithmetic (Theorem 1) and predicative analysis (Proposition 4),respectively. This enables us to define, in our theories
KFL and
KFL ∗ , ramified truth pred-icates indexed by ordinals smaller than ε (Corollary 4) and Γ (Corollary 7). Moreover, theproof-theoretic analysis of KFL and
KFL ∗ is completed by showing that their truth predicatescan be suitably interpreted in their classical counterparts KF and Ref ∗ ( PA ( P )) without alteringthe arithmetical vocabulary (Propositions 2 and 5).2. HYPE
In this section we will present the logical basis of our systems of truth. We will work witha sequent calculus variant of the logic
HYPE introduced by Leitgeb in [Lei19] by means of aHilbert style calculus. Essentially, the calculus is obtained by extending First-Degree Entailmentwith an intuitionistic conditional and with rules for it in a multi-conclusion style.2.1.
G1h cd . We present a multi-conclusion system based on a multi-conclusion calculus forintuitionistic logic: we call it G1h cd for Gentzen system for the logic HYPE with constantdomains. Sequents are understood as multisets. We work with a language whose logical symbolsare ¬ , ∨ , → , ∀ , ⊥ . For Γ = γ , . . . , γ n a multiset, ¬ Γ is the multiset ¬ γ , . . . , ¬ γ n . The logicalconstants ∧ , ∃ , ↔ can be defined as usual and ⊤ is defined as ¬⊥ . Moreover, we can define‘intuitionistic’ negation ∼ A as A → ⊥ , the material conditional A ⊃ B as ¬ A ∨ B , and materialequivalence A ≡ B as ( A ⊃ B ) ∧ ( B ⊃ A ). For A a formula, we write FV ( A ) for the set of its freevariables, and FV (Γ) for the set of free variables in all formulas in Γ.The system G1h cd consists of the following initial sequents and rules:( ID p ) A ⇒ A ( L ⊥ ) ⊥ ⇒ This system goes back to Maehara’s version used in Takeuti [Tak87] p.52f and Dragalin’s system used in Negriand Plato [NP01] p.108f. onclassical truth with classical strength. 3 Γ ⇒ ∆ , A A, Γ ⇒ ∆( Cut ) Γ ⇒ ∆Γ ⇒ ∆( LW ) A, Γ ⇒ ∆ Γ ⇒ ∆( RW ) Γ ⇒ ∆ , AA, A, Γ ⇒ ∆( LC ) A, Γ ⇒ ∆ Γ ⇒ ∆ , A, A ( RC ) Γ ⇒ ∆ , AA, Γ ⇒ ∆ B, Γ ⇒ ∆( L ∨ ) A ∨ B, Γ ⇒ ∆ Γ ⇒ A, B, ∆( R ∨ ) Γ ⇒ A ∨ B, ∆Γ ⇒ ∆ , A B, Γ ⇒ ∆( L → ) A → B, Γ ⇒ ∆ Γ , A ⇒ B ( R → ) Γ ⇒ A → B, ∆Γ ⇒ ¬ ∆( ConCp ) ∆ ⇒ ¬ Γ ¬ Γ ⇒ ∆( ClCp ) ¬ ∆ ⇒ Γ A ( t ) , Γ ⇒ ∆( L ∀ ) ∀ xA, Γ ⇒ ∆ Γ ⇒ ∆ , A ( y )( R ∀ ) Γ ⇒ ∆ , ∀ xAy / ∈ FV (Γ , ∆ , ∀ xA )We write rk ( A ) for the logical complexity of A , defined as the number of nodes in the longestbranch of its syntactic tree. For a derivation d we let • hgt ( d ) := sup i The sequents ⇒ ⊤ , A ⇒ ¬¬ A , ¬¬ A ⇒ A , are derivable in G1h cd . (ii) The rule of contraposition Γ ⇒ ∆ ¬ ∆ ⇒ ¬ Γ is admissible in G1h cd . (iii) The following rules are admissible in G1h cd : A, B, Γ ⇒ ∆( L ∧ ) A ∧ B, Γ ⇒ ∆ Γ ⇒ A, ∆ Γ ⇒ B, ∆( R ∧ ) Γ ⇒ A ∧ B, ∆ A ( y ) , Γ ⇒ ∆( L ∃ ) y / ∈ FV (Γ , ∆ , ∃ xA ) ∃ xA, Γ ⇒ ∆ Γ ⇒ ∆ , A ( t )( R ∃ ) Γ ⇒ ∆ , ∃ xA onclassical truth with classical strength. 4 (iv) Intersubstitutivity: If χ ⇒ χ ′ and χ ′ ⇒ χ , as well as ψ are derivable in G1h cd , then ψ ( χ ′ /χ ) is derivable, where ψ ( χ ′ /χ ) is obtained by replacing all occurrences of χ in ψ by χ ′ .Proof. Claims (i)-(iii) are direct consequences of the contraposition rules ( ConCp ) and ( ClCp ).(iv) is proved by a straightforward induction on the height of the derivation in G1h cd . (cid:3) We opted for this specific formulation of G1h cd mainly because it substantially simplifies thepresentation of the results of the next sections, which are the main focus of the paper. From aproof-theoretic point of view, the calculus has some drawbacks even at the propositional level,as the rules ConCp and ClCp compromise the induction needed for cut-elimination. In the propo-sitional case, even if one removes ConCp and ClCp and splits the contraposition rule of Lemma1(ii) on a case by case manner, problems for cut-elimination remain [Fis20]. Moreover, whenone moves to the quantificational system, there are deeper problems. The same counterexam-ple that is employed to show that cut is not admissible in systems of intuitionistic logic withconstant domains can be employed for the systems we are investigating. Both problems can beaddressed by employing techniques from Kashima and Shimura [KS94], which however rely onthe extension of the systems with additional resources.Since cut elimination is not the main focus of our paper, we opt for a more compact presen-tation of G1h cd that fits nicely our purpose of extending it with arithmetic and truth rules.2.2. Semantics. In this section we present the semantics of G1h cd (and therefore of HYPE )and sketch its completeness with respect to it. We follow a simplification of the semantics inLeitgeb [Lei19] suggested by Speranski [Spe20]. Speranski connects HYPE -models with Routleysemantics. A Routley frame F is a triple h W, ≤ , ∗i , where:(i) W is a non-empty set of states;(ii) ≤ is a preorder;(iii) ∗ is a function from W to W , which is:- antimonotone , i.e. for all w, v ∈ W , if w ≤ v , then v ∗ ≤ w ∗ ;- involutive , i.e. for all w ∈ W , w ∗∗ = w .A constant domain model M for HYPE is a triple ( F , D, I ) where F is a Routley frame, D isa non-empty set (the domain of the model), and I is an interpretation function. In particular, I assigns to every constant c an element of D and it associates with each state w and n -placepredicate P a set P w ⊆ D n . Constants are interpreted rigidly and, although domains do notgrow, we impose the following hereditariness condition: for all v, w ∈ W , if v ≤ w , then for allpredicates P , P v ⊆ P w . See for example López-Escobar [LE83]. onclassical truth with classical strength. 5 Let M be a constant domain model, w ∈ W and σ : VAR → D a variable assignment on D ,then the forcing relation M , w, σ (cid:13) A is defined inductively: M , w, σ (cid:13) P ( x , ..., x n ) iff ( σ ( x ) , ..., σ ( x n )) ∈ P w ; M , w, σ (cid:13) ¬ A iff M , w ∗ , σ A ; M , w, σ (cid:13) A ∨ B iff M , w, σ (cid:13) A or M , w, σ (cid:13) B ; M , w, σ (cid:13) A → B iff for all v, with w ≤ v , if M , v, σ (cid:13) A , then M , v, σ (cid:13) B ; M , w, σ (cid:13) ∀ xA iff for all x -variants σ ′ of σ, M , w, σ ′ (cid:13) A ; M , w, σ ⊥ . Finally, we define logical consequence. We write, for Γ , ∆ sets of sentences: • M , w (cid:13) Γ ⇒ ∆ iff: if M , w (cid:13) γ for all γ ∈ Γ, then M , w (cid:13) δ for some δ ∈ ∆; • Γ (cid:13) ∆ iff for all M , w : M , w (cid:13) Γ ⇒ ∆.The system G1h cd is equivalent to the following Hilbert-style system QN ◦ featuring theaxiom schemata A → ( B → A ) A → ( B → C ) → (( A → B ) → ( A → C )) A ∧ B → A A ∧ B → BA → A ∨ B B → A ∨ BA → ( B → A ∧ B ) ( A → C ) → (( B → C ) → ( A ∨ B → C )) ¬¬ A → A A → ¬¬ A ∀ xA → A ( t ) A ( t ) → ∃ xA and the following rules of inference: A A → B (MP) B A → B (CP) ¬ B → ¬ AA → B ( x ) x not free in AA → ∀ xB A ( x ) → B x not free in B ∃ xA → B QN ◦ is a neater presentation of HYPE where a few redundant principles are dropped. Theconsequences of the two systems are identical.That our system G1h cd is equivalent to QN ◦ can be seen as follows. G1h cd is an extensionof intuitionistic logic (modulo the definition of ∼ A as A → ⊥ ). Therefore, since all axioms of QN ◦ except for the double negation axioms are intuitionistically valid, Lemma 1 enables us toshow that all axioms of QN ◦ are consequences of G1h cd . Additionally, Lemma 1 shows thatcontraposition is admissible in G1h cd . Rules for quantifiers are easily established in G1h cd . onclassical truth with classical strength. 6 For the other direction a proof on the length of the derivation is sufficient. The fact that thededuction theorem holds in QN ◦ renders the proof particularly simple. Therefore, we have: Lemma 2. G1h cd ⊢ Γ ⇒ ∆ iff QN ◦ ⊢ V Γ → W ∆ . Lemma 2 then entails that G1h cd is equivalent to Leitgeb’s HYPE .Speranski [Spe20] establishes a strong completeness result (for countable signatures) for QN ◦ .Speranski uses a Henkin-style proof similar to the strategy employed in Gabbay et al. [GSS09,§7.2] for intuitionistic logic with constant domains. Leitgeb [Lei19] establishes a (weak) com-pleteness proof for his Hilbert style system based on the work of Görnemann [Gör71]. By Lemma2 we can employ Speranski’s completeness result for our system G1h cd with respect to Routleysemantics: Proposition 1 (Completeness of G1h cd [Spe20]) . Γ (cid:13) ∆ iff there is a finite ∆ ⊆ ∆ , such that Γ ⊢ QN ◦ ∆ . We now turn to investigating how much classical reasoning can be reproduced in our logic.Such questions will turn out to be essential components of the analysis of truth theories over HYPE .2.3. HYPE and recapture. One of the desirable properties of the nonclassical logics employedin the debate on semantic paradoxes is the capability of recapturing classical reasoning in domainswhere there is no risk of paradoxicality, such as mathematics – see e.g. [Fie08]. The following lemma summarizes the recapture properties of G1h cd and extensions thereof. Itessentially states that, in systems based on G1h cd , once we restrict our attention to a fragmentof the language satisfying the excluded middle and/or explosion, the native HYPE -negationand conditional, as well as the defined intuitionistic negation, all behave classically. Lemma 3. (i) The following rules are admissible in extensions of G1h cd : This form of recapture is a slightly different phenomenon from a direct, provability preserving, translation of theentire language of one theory in the other, as it happens for instance in the famous Gödel-Gentzen translation or theS4 interpretations of classical in intuitionistic logic, or intutionistic logic in modal logic respectively. While thosetranslations provide a method to reinterpret the logical vocabulary – by keeping the non-logical vocabulary fixed– in a provability-preserving way, recapture strategies typically show that, for a specific fragment of its language,the nonclassical theory behaves according to the rules of classical logic. For instance, that a nonclassical theoryof truth behaves fully classically if one restricts her attention to the truth-free language. To carry on with theanalogy with the relationships between classical and intuitionistic logic, recapture strategies are much closer tothe identity between the ∆ -fragments of classical and intuitionistic arithmetic. onclassical truth with classical strength. 7 ⇒ A, ¬ A Γ , A ⇒ ∆Γ ⇒ ¬ A, ∆ A, ¬ A ⇒ Γ ⇒ A, ∆Γ , ¬ A ⇒ ∆ ⇒ A, ¬ A Γ , A ⇒ B, ∆Γ ⇒ A → B, ∆ A, ¬ A ⇒¬ A ⇒ A → ⊥ A, ¬ A ⇒ A → ⊥ ⇒ ¬ A ⇒ A, ¬ AA → B ⇒ A ⊃ B ⇒ A, ¬ AA ⊃ B ⇒ A → B (ii) The previous fact can be used to show, by an induction on rk ( A ) , that ⇒ A, ¬ A isderivable for any formula whenever ⇒ P, ¬ P is derivable for any atomic P in A .Proof. We prove the claims for the crucial cases in which a conditional is involved:For (i): ⇒ A, ¬ A Γ ⇒ A, ¬ A, B, ∆ Γ , A ⇒ B, ∆Γ , A ⇒ ¬ A, B, ∆Γ ⇒ ¬ A, B, ∆Γ ⇒ B, ¬ A, A → B, ∆ B, A ⇒ BB ⇒ A → B Γ , B ⇒ ¬ A, A → B, ∆Γ ⇒ ¬ A, A → B, ∆ A, ¬ A ⇒ A, ¬ A ⇒ B ¬ A ⇒ A → B Γ , ¬ A ⇒ A → B, ∆Γ ⇒ A → B, ∆ For (ii): ¬ A, A ⇒ B ¬ A ⇒ A → B ¬ ( A → B ) ⇒ A B, A ⇒ BB ⇒ A → B ¬ ( A → B ) ⇒ ¬ BB, ¬ ( A → B ) ⇒ A → B, ¬ ( A → B ) ⇒ (cid:3) Remark 1. The induction involved in Lemma 3(ii) does not go through in intuitionistic logicwith the HYPE -negation ¬ replaced by the intuitionistic negation ∼ .2.4. Equality. For our purposes it’s important to extend G1h cd a theory of equality. G1h = cd is obtained by adding to G1h cd the following initial sequents for equality. ⇒ t = t ( Ref) s = t, A ( s ) ⇒ A ( t )( Rep) By an essential use of ConCp , we can establish in G1h = cd that identity statements behaveclassically. Lemma 4. G1h = cd derives ⇒ s = t, ¬ s = t and s = t, ¬ s = t ⇒ .Proof. We use the identity sequents: onclassical truth with classical strength. 8 s = t, ¬ s = t ⇒ ¬ t = t ⇒ t = t ¬ t = t ⇒ s = t, ¬ s = t ⇒⇒ ¬ s = t, ¬¬ s = t ⇒ ¬ s = t, s = t (cid:3) Lemma 4 reveals some subtle issues concerning the treatment of identity in subclassical logicsgenerally employed to deal with semantical paradoxes. It tells us that identity is essentiallytreated as a classical notion in G1h = cd . To obtain a similar phenomenon in absence of ConCp and ClCp , one would have to add the counterpositives of Rep and Ref to the system. A nonclassicaltreatment of identity would require some non-trivial changes to Rep and Ref . That identity isa classical notion is perfectly in line with our framework, in which identity is a non-semanticnotion akin to mathematical notions.3. Arithmetic in HYPE Starting with the logical constants introduced above and the identity symbol, we now workwith a suitable expansion of the usual signature { , S , + , ×} by finitely many function symbols forselected primitive recursive functions. Such function symbols are needed for a smooth represen-tation of formal syntax. We call this language L → N . We will also make use of the → -free fragmentof the language of arithmetic, which we label as L N . Our base theory will then be obtained byadding, to the basic axioms for 0 , S , + , × (axioms Q Q HYA − .In the following, the role of rule and axiom schemata will be crucial. It will be particularlyimportant to keep track of the classes of instances of a particular schema, and therefore we willalways relativize schemata to specific languages and understand the schema as the set of all itsinstances in that language. For example, in the case of the induction axioms we use the label IND → ( L ) to refer to the set of all sequents of the form( IND → ( L )) ⇒ A (0) ∧ ∀ x ( A ( x ) → A ( x + 1)) → ∀ xA ( x ) , where A is a formula of L . Similarly, induction rules IND R ( L ) will refer to all rule instancesΓ , A ( x ) ⇒ A ( x + 1) , ∆ ( IND R ( L ))Γ , A (0) ⇒ A ( t ) , ∆for A a formula of L .We call HYA the extension of HYA − by the induction axiom IND → ( L ). HYA is equivalentto Peano Arithmetic PA . This is essentially because of the recapture properties of our logic. Forformulas A containing only classical vocabulary, the properties stated in Lemma 3 entail thatthe rule and sequent formulations of induction are equivalent. Lemma 5. Let L ⊇ L → N . Over HYA − : IND R ( L ) and IND → ( L ) are equivalent when restricted toformulas A such that ⇒ A, ¬ A . onclassical truth with classical strength. 9 Since for A ∈ L → N , ⇒ A, ¬ A and A, ¬ A ⇒ are derivable in G1h cd , we have the immediatecorollary that: Corollary 1. HYA is equivalent to PA . Ordinals and transfinite induction. Our notational conventions for schemata generalizeto schemata other than induction. A prominent role in the paper will be played by transfiniteinduction schemata . In order to introduce them, we need to assume a notation system ( OT , ≺ ) forordinals up to the Feferman-Schütte ordinal Γ as it can be found, for instance, in [Poh09, Ch. 2]. OT is a primitive recursive set of ordinal codes and ≺ a primitive recursive relation on OT thatis isomorphic to the usual ordering of ordinals up to Γ . We distinguish between fixed ordinalcodes, which we denote with α, β, γ . . . , and ζ, η, θ, ξ, . . . as abbreviations for variables rangingover elements of OT . Our representation of ordinals satisfies all standard properties. In particular,we will make implicit use of the properties listed in [TS00], p. 322.We will make extensive use of the following abbreviations. We call a formula progressive if itis preserved upwards by the ordinals: Prog ( A ) := ∀ η ( ∀ ζ ≺ η A ( ζ ) → A ( η ))where ∀ ζ ≺ η A ( ζ ) is short for ∀ ζ ( ζ ≺ η → A ( ζ )). We will use this (standard) notationalconvention in several occasions in what follows. Similarly, we will write ∃ ζ ≺ η A ( ζ ) for ∃ ζ ( ζ ≺ η ∧ A ( ζ )).This formulation of progressiveness is HYA -equivalent to a formulation as a sequent ∀ ζ ≺ η A ( ζ ) ⇒ A ( η ). Moreover, if A ( x ) ∨ ¬ A ( x ) is provable, then Prog ( A ) is HYA -equivalent to:(1) ∀ η ( ∀ ζ ≺ η A ( ζ ) ⊃ A ( η )) . Transfinite induction up to the ordinal α ( ≺ Γ ) will be formulated as the following sequent:( TI α ( A )) Prog ( A ) ⇒ ∀ ξ ≺ α A ( ξ )An alternative would be to use a rule-formulation:Γ , ∀ ζ ≺ η A ( ζ ) ⇒ A ( η ) TI r α ( A ) := Γ ⇒ ∀ ξ ≺ α A ( ξ ) , ∆ TI r α ( A ) differs from the standard rule formulation of transfinite induction (see, e.g. [Hal14]) inthat its premiss features only one formula in the succedent.The two formulations of induction just introduced are equivalent over HYA − , i.e. given TI α ( A ), TI r α ( A ) is admissible, and given TI r α ( A ), TI α ( A ) is derivable. TI α ( L ) is short for TI α ( A ) for every formula A of the language L . TI <α ( L ) is short for TI β ( L )for all β ≺ α . The function ω n is recursively defined in the standard way as: ω = 1, ω n +1 = ω ω n .3.2. Transfinite induction and nonclassical predicates. Our main purpose in this paperis to study the proof-theoretic properties of extensions of HYA with additional predicates that The notion of admissible rule that we employ is the one from [TS00, p. 76]. onclassical truth with classical strength. 10 may not behave classically – i.e. they may not satisfy Lemma 5. In fact, in the case of the purearithmetical language, Lemma 5 gives us immediately that HYA derives TI <ε ( L → N ). In thissection we show directly that Gentzen’s original proof of TI <ε ( L → N ) can be carried out in HYA for suitable extensions of L → N . Theorem 1. Let L + be a language expansion of L → N by finitely many predicate symbols. Then HYA ⊢ TI <ε ( L + ) . The rest of this subsection will be devoted to the proof of Theorem 1, which will involve severalpreliminary lemmata.A key ingredient of Gentzen’s proof – which will also play an important role in subsequentsections – is Gentzen’s jump formula: A + ( θ ) := ∀ ξ ( ∀ η ( η ≺ ξ → A ( η )) → ∀ η ( η ≺ ξ + ω θ → A ( η ))) . Lemma 6. For any A ∈ L + , HYA proves Prog ( A ) ⇒ Prog ( A + ) .Proof. The informal argument is as follows: We assume Prog ( A ) and we want to show Prog ( A + ),i.e. ∀ ζ ≺ θ A + ( ζ ) → A + ( θ ). So we also assume ∀ ζ ≺ θ A + ( ζ ) and ∀ ζ ( ζ ≺ ξ → A ( ζ )) and η ≺ ξ + ω θ to show A ( η ).Informally, we make a case distinction: Either θ = 0 or θ ≻ Case 1 : If θ = 0, then(2) θ = 0 , η ≺ ξ + ω θ ⇒ η ≺ ξ ∨ η = ξ. We have, by the reflexivity sequents and logical rules: ∀ ζ ( ζ ≺ ξ → A ( ζ )) , η ≺ ξ ⇒ A ( η )(3)Again by reflexivity and the identity axioms: Prog ( A ) , ∀ ζ ( ζ ≺ ξ → A ( ζ )) , η = ξ ⇒ A ( η ) . (4)By (2) and Cut , we obtain(5) θ = 0 , Prog ( A ) , ∀ ζ ( ζ ≺ ξ → A ( ζ )) , η ≺ ξ + ω θ ⇒ A ( η ) . Case 2 : θ ≻ 0. Then by a derivable version of Cantor’s Normal Form Theorem:( † ) θ ≻ , η ≺ ξ + ω θ ⇒ ∃ n ∃ θ ≺ θ ( η ≺ ξ + ω θ · n ) . Given that induction for ordinal notations up to ω is provable in HYA , we will show byinduction on n ≺ ω that ∀ ζ ≺ θ A + ( ζ ) , θ ≺ θ ⇒ ∀ ζ ( ζ ≺ ξ + ω θ · n → A ( ζ )) . Troelstra & Schwichtenberg [TS00] already established that the Gentzen proof can be carried out in the minimal → ∀⊥ fragment of IL . onclassical truth with classical strength. 11 The base case is straightforward because the following is trivially derivable (by property ( ord6 )):(6) ∀ η ≺ ξ A ( η ) ⇒ ( ∀ η ≺ ξ + ω θ · A ( η ) . For the induction step, we start by noticing that by instantiating ξ in A + ( θ ) with ξ + ω θ · n ,we obtain:(7) A + ( θ ) ⇒ ∀ ζ ≺ ξ + ω θ · n A ( ζ ) → ∀ ζ ≺ ξ + ω θ · ( n + 1) A ( ζ ) , As mentioned, by letting: B ( x ) := ∀ ζ ≺ ξ + ω θ · x A ( ζ ) HYA proves the ω -induction principle (with n ≺ ω ): B (0) , ∀ n ( B ( n ) → B ( n + 1)) ⇒ ∀ n B ( n ) . Therefore, by a series of cuts, we obtain:(8) A + ( θ ) , ∀ ζ ≺ ξ A ( ζ ) ⇒ ∀ n ∀ ζ ≺ ξ + ω θ · n A ( ζ ) . From (8) we obtain:(9) ∀ ζ ≺ ξ A ( ζ ) , ∀ ζ ≺ θ A + ( ζ ) , θ ≺ θ ⇒ ∀ n ∀ ζ ≺ ξ + ω θ · n A ( ζ ) . Therefore, we can instantiate n and ζ (with η ), and move the antecedent of η ≺ ξ + ω θ · n → A ( η )from the right-hand side to the left hand side of the sequent arrow. Since both n and η are general,we can existentially generalize over them to get:(10) θ ≻ , ∀ ζ ≺ ξ A ( ζ ) , ∀ ζ ≺ θ A + ( ζ ) , ∃ n ∃ θ ≺ θ ( η ≺ ξ + ω θ · n ) ⇒ A ( η ) , which in turn by ( † ) gives us:(11) θ ≻ , ∀ ζ ≺ ξ A ( ζ ) , ∀ ζ ≺ θ A + ( ζ ) , η ≺ ξ + ω θ ⇒ A ( η ) . Now we combine the two cases. Together with our (5) in Case 1, the last sequent enable us toderive: θ = 0 ∨ θ ≻ , Prog ( A ) , ∀ ζ ≺ ξ A ( ζ ) , ∀ ζ ≺ θ A + ( ζ ) , η ≺ ξ + ω θ ⇒ A ( η ) . By the provability of θ = 0 ∨ θ ≻ R → ) and ( R ∀ ) we finally get Prog ( A ) ⇒ Prog ( A + ) . (cid:3) The progressiveness of Gentzen’s jump formula enables us then to establish: Lemma 7. If TI α ( L + ) is derivable in HYA , then TI ω α ( L + ) is derivable in HYA .Proof. We assume TI α ( L + ). Specifically we have(12) Prog ( A + ) ⇒ ∀ ξ ≺ α A + ( ξ ) . onclassical truth with classical strength. 12 By the meaning of Prog ( A + ), we obtain(13) Prog ( A + ) ⇒ A + ( α ) . By the previous Lemma 6 and cut we also have(14) Prog ( A ) ⇒ A + ( α ) . which is(15) Prog ( A ) ⇒ ∀ ξ ( ∀ η ≺ ξ A ( η ) → ∀ η ≺ ξ + ω α A ( η )) . But also(16) ⇒ ∀ η ≺ A ( η ) , and therefore by (15) taking ξ = 0, we obtain Prog ( A ) ⇒ ∀ η ≺ ω α A ( η ) , as desired. (cid:3) Corollary 2. If A is such that HYA proves A ( x ) ∨ ¬ A ( x ) , we have that, if HYA proves theclassical transfinite induction axiom schema for α ( ∀ ζ ≺ ηA ( ζ ) ⊃ A ( η )) ⊃ ∀ ξ ≺ α A ( ξ ) , then HYA proves: ( ∀ ζ ≺ ηA ( ζ ) ⊃ A ( η )) ⊃ ∀ ξ ≺ ω α A ( ξ ) . All is set up to finally prove the main result of this section, the admissibility in HYA of therequired schema of transfinite induction up to any ordinal α ≺ ε . Proof of Theorem 1. The result follows immediately from Lemma 7. Since TI ω ( A ) is triviallyderivable in HYA , the lemma tells us that TI ω n ( A ), for each n , can be reached in finitely manyproof steps. (cid:3) Theorem 1 is key to our proof-theoretic analysis of a theory of truth over HYPE . We nowturn to the definition of such a truth theory.4. The Theory of Truth KFL In this section we introduce the theory of truth KFL , standing for Kripke-Feferman-Leitgeb.The theory is formulated in the language L → Tr := L → N ∪ { Tr } , where Tr is a unary predicate fortruth. KFL is a theory of truth for a → -free language L Tr , which is simply the → -free fragmentof L → Tr . In KFL , the conditional → should be thought of as a theoretical device to articulateour semantic theory, and not as an object of semantic investigation. We elaborate on the role of onclassical truth with classical strength. 13 the conditional in the concluding section 6. Semantically (cf. §4.1), the conditional amounts toa device to navigate between fixed point models of L Tr in the sense of [Kri75]. Definition 1 (The language L Tr ) . The logical symbols of L Tr are ⊥ , ¬ , ∨ , ∀ . In addition, wehave the identity symbol = . Its non-logical vocabulary amounts to the arithmetical vocabulary of L N and the truth predicate Tr . We assume a canonical representation of the syntax of L Tr in HYA . Given the equivalenceof HYA and PA for arithmetical vocabulary stated in Corollary 1, we can assume one of thestandard ways of achieving this (e.g. [Can89]). We apply most of the notational conventions –e.g. Feferman’s dot notation – described in [Hal14, §I.5]. Definition 2 (The theory KFL ) . KFL extends HYA formulated in L → Tr – i.e. with the inductionschema extended to L → Tr – with the following truth initial sequents: Cterm L Tr ( x ) ∧ Cterm L Tr ( y ) ⇒ Tr ( x = . y ) ↔ val ( x ) = val ( y )( KFL ⇒ Tr ( p Tr ˙ x q ) ↔ Tr x ( KFL Sent L Tr ( x ) ⇒ Tr ¬ . x ↔ ¬ Tr x ( KFL Sent L Tr ( x ) ∧ Sent L Tr ( y ) ⇒ Tr ( x ∨ . y ) ↔ Tr x ∨ Tr y ( KFL Sent L Tr ( ∀ . vx ) ∧ var ( v ) ⇒ Tr ( ∀ . vx ) ↔ ∀ y ( CTerm L Tr ( y ) → Tr x ( y/v ))( KFL Tr x ⇒ Sent L Tr ( x )( KFL KFL x ( y/v ) denotes the result of substituting, in the formula with code x , the variable withcode v with the closed term coded by y . In particular, in KFL p Tr ˙ x q stands for the result ofsubstituting, in the code of Tr v , the variable v with the numeral for x .According to Lemma 3 we have that ⊥ , ⊃ and → obey the classical introduction and elimi-nation rules when the antecedent is a formula of L → N . An important property of KFL is that itentails an object-linguistic version of the Tr -schema for sentences that do not contain the con-ditional → . We will return to the philosophical implications of this property in the concludingsection. Lemma 8. The following are provable in KFL : (i) Sent L Tr ( x ) ⇒ Tr p ¬ Tr ˙ x q ↔ Tr ¬ . x (ii) For A ∈ L Tr , ⇒ Tr p A q ↔ A .Proof. (i) is immediate by the axioms of KFL , and (ii) is obtained by an external induction onthe rank of A . (cid:3) Semantics. The intended interpretation of our theory of truth is based on Kripke’s fixedpoint semantics [Kri75] and stems from the HYPE -models presented in Leitgeb [Lei19, §7]. Ourmodel will feature a state space, whose states are fixed-points of the usual monotone operatorassociated with the four-valued evaluation schema as stated in Visser [Vis84] and Woodruff[Woo84]. onclassical truth with classical strength. 14 Let Φ : P ω −→ P ω be the operator defined in [Hal14, Lemma 15.6]. States will have the form( N , S ), where S is a fixed point of Φ. Since we are interested in constant domains and in keepingthe interpretation of the arithmetical vocabulary fixed, we omit reference to N and identify stateswith the fixed points themselves. Therefore, we let: W := { X ⊆ Sent L Tr | Φ( X ) = X } , (17) S ≤ W S ′ : ⇔ S ⊆ S ′ , (18) S ∗ := ω \ S, with X = {¬ ϕ | ϕ ∈ X } , (19) the interpretation of Tr is denoted with Tr S := S. (20)The intended full model M Φ is then the HYPE model based on the frame ( W , ≤ W , ∗ ) with theconstant domain ω . The intended minimal model M min Φ is then given by restricting the set ofstates to the minimal and maximal fixed points. By a straightforward induction on the heightof the derivation in KFL , we obtain: Lemma 9. If KFL ⊢ Γ ⇒ ∆ , then M Φ (cid:13) Γ ⇒ ∆ . Proof Theory: Lower Bound. We show that KFL can define (and therefore prove thewell-foundedness of) Tarskian truth predicates for any α ≺ ε . By the techniques employed inFeferman and Cantini’s analyses of the proof theory of KF [Can89,Fef91], this entails that KFL can prove TI <ϕ ε ( L N ).We first define the Tarskian languages. Definition 3. For ≤ α < Γ , we let: Sent L Tr (0 , x ) : ↔ Sent L N ( x ) , Sent L Tr ( ζ + 1 , x ) : ↔ Sent L Tr ( ζ, x ) ∨ ( ∃ y ≤ x )( x = p Tr ˙ y q ∧ Sent L Tr ( ζ, y )) ∨ ( ∃ y ≤ x )( x = ( ¬ . y ) ∧ Sent L Tr ( ζ + 1 , y )) ∨ ( ∃ y, z ≤ x )( x = ( y ∨ . z ) ∧ Sent L Tr ( ζ + 1 , y ) ∧ Sent L Tr ( ζ + 1 , z )) ∨ ( ∃ v, y ≤ x )( x = ( ∀ . vy ) ∧ Sent L Tr ( ζ + 1 , y )) , Sent L Tr ( λ, x ) : ↔ ∃ ζ < λ Sent L Tr ( ζ, x ) for λ a limit ordinal.We then write: Sent <α L Tr ( x ) : ↔ ∃ ζ ≺ α Sent L Tr ( ζ, x ) , Tr α ( x ) : ↔ Sent <α L Tr ( x ) ∧ Tr ( x ) . As we mentioned, the arithmetical vocabulary behaves classically in KFL . Lemma 10. KFL ⊢ ∀ x ( Sent L N ( x ) → Tr x ∨ Tr ¬ . x ) .Proof. By formal induction on the complexity of the ‘sentence’ x ∈ L N . (cid:3) onclassical truth with classical strength. 15 The next two claims establish that the previous fact can be extended to all Tarskian languageswhose indices can be proved to be well-founded. First, one shows that the claim ‘sentences in Sent <η L Tr are either determinately true or determinately false’ is progressive. Lemma 11. KFL ⊢ ( ∀ ζ ≺ η )( Sent L Tr ( ζ, x ) → Tr x ∨ Tr ¬ . x ) ⇒ Sent L Tr ( η, x ) → Tr x ∨ Tr ¬ . x .Proof. By the definition of OT , KFL proves that η ∈ OT is either 0, or a successor ordinal, or alimit. By arguing informally in KFL , we show that the statement of the lemma holds, therebyestablishing the claim.Lemma 10 gives us the base case. The limit case follows immediately by the definition of Sent L Tr ( λ, x ). For the successor step, one needs to establish (cf. [Nic17, Lemma 7]):(21) Sent L Tr ( ζ, x ) → Tr x ∨ Tr ¬ . x ⇒ Sent L Tr ( ζ + 1 , y ) → Tr y ∨ Tr ¬ . y Claim (21) is obtained by a formal induction on the complexity of y . Crucially, the proof restson the following KFL -derivable claims, which provide the cases required by the induction: Sent L Tr x, Tr x ∨ Tr ¬ . x ⇒ Tr ¬ . x ∨ Tr ¬ . ¬ . x, (22) Sent L Tr ( x ∨ . y ) , Tr x ∨ Tr ¬ . x, Tr y ∨ Tr ¬ . y ⇒ Tr ( x ∨ . y ) ∨ Tr ¬ . ( x ∨ . y ) , (23) Sent L Tr ( ∀ . vx ) , ∀ t Tr x ( t/v ) ∨ ¬∀ t Tr x ( t/v ) ⇒ Tr ( ∀ . vx ) ∨ Tr ( ¬ . ∀ vx ) , (24) Tr x ∨ Tr ¬ . x ⇒ Tr p Tr ˙ x q ∨ Tr p ¬ Tr ˙ x q . (25) (cid:3) By Theorem 1, we obtain: Corollary 3. For any α < ε , KFL ⊢ ∀ x ( Sent L Tr ( α, x ) → Tr x ∨ Tr ¬ . x ) . Since, by Theorem 1, KFL proves transfinite induction up to ordinals smaller than ε , itfollows that we are able to establish the fundamental properties of Tarskian truth predicatesup to any ordinal smaller than ε . For α < Γ , RT <α refers to the theory of ramified truthpredicates up to α , as defined in [Hal14, §9.1]. Corollary 4. KFL defines the truth predicates of RT <α , for α ≺ ε . By the proof-theoretic equivalence of systems of ramified truth and ramified analysis estab-lished by Feferman [Fef64, Fef91], we obtain: Corollary 5. KFL proves TI <ϕ ε ( L N ) . Feferman and Cantini established that KF is proof-theoretically equivalent to RT <ε . Ourresults so far then establish that KFL is proof-theoretically at least as strong as KF . In thenext section, we will show that KFL and KF are in fact proof-theoretically equivalent. onclassical truth with classical strength. 16 Proof Theory: Upper Bound. We interpret KFL in the Kripke-Feferman system KF .For definiteness, we consider the version of KF formulated in a language L T , F featuring truth( T ) and falsity ( F ) predicates. Such a version of KF is basically the one presented in [Can89, §2],but without the consistency axiom that rules out truth-value gluts.In order to interpret KFL into KF , we consider a two-layered translation that differentiatesbetween the external and internal structures of L → Tr -formulas. Essentially, the external transla-tion fully commutes with negation, and translates the HYPE conditional as classical materialimplication. The internal translation treats negated truth ascriptions as falsity ascriptions, andis defined by an induction on the positive complexity of formulas that adheres to the semanticclauses of FDE -style fixed-point models. The internal translation therefore translates truth andnon-truth of KFL as truth and falsity of KF , respectively. Since we want to uniformly translateformulas and their codes inside the truth predicate, we essentially employ the recursion theorem,as described for instance by [Hal14, §5.3]. Definition 4. We define the translations τ : L Tr −→ L T , F , and σ : L → Tr −→ L T , F as follows: (i) ( s = t ) τ = s = t ( s = t ) τ = s = t ( Tr t ) τ = T τ ( t ) ( ¬ Tr t ) τ = F τ ( t )( ¬¬ ϕ ) τ = ( ϕ ) τ ( ϕ ∨ ψ ) τ = ( ϕ ) τ ∨ ( ψ ) τ ( ¬ ( ϕ ∨ ψ )) τ = ( ¬ ϕ ) τ ∧ ( ¬ ψ ) τ ( ∀ xϕ ) τ = ∀ xϕ τ ( ¬∀ xϕ ) τ = ∃ x ( ¬ ϕ ) τ (ii) ( s = t ) σ = s = t ( Tr t ) σ = T τ ( t ) ( ¬ Tr t ) σ = F τ ( t )( ¬ ϕ ) σ = ¬ ϕ σ ( ϕ ∨ ψ ) σ = ( ϕ ) σ ∨ ( ψ ) σ ( ∀ xϕ ) σ = ∀ xϕ σ ( ϕ → ψ ) σ = ¬ ( ϕ ) σ ∨ ( ψ ) σ KFL -proofs can then be turned, by the translation σ , into KF -proofs, as the next propositionshows. Proposition 2. If KFL ⊢ Γ ⇒ ∆ , then KF ⊢ ( V Γ → W ∆) σ .Proof. The proof is by induction on the height of the derivation in KFL and follows almostdirectly from the definition of the translation σ . Only the case of ( KFL 3) is slightly moreinvolved: we require that (with ≡ expressing material equivalence):(26) KF ⊢ Sent L Tr ⇒ T τ ( ¬ . x ) ≡ F τ ( x ) . However, this can be proved by formal induction on the complexity of x . (cid:3) onclassical truth with classical strength. 17 The combination of Proposition 2 and Corollary 4 yields that KF and KFL have the samearithmetical theorems, and in particular they have the same proof-theoretic ordinal – cf. [Poh09,§6.7]. Corollary 6. | KFL | = | KF | = ϕ ε . In the next section we extend our results to schematic extensions of KFL and KF .5. Schematic extension KFL ∗ : Rules and Semantics. In this section we study the schematic extension of KFL in the sense of [Fef91]. This is obtained by extending KFL with a special substitution rule thatenables us to uniformly replace the distinguished predicate P in arithmetical theorems A ( P ) ofour extended theory for arbitrary formulas of L → Tr . More precisely, following Feferman, we willemploy a schematic language L → Tr ( P ) (and sub-languages thereof) featuring a fresh schematicpredicate symbol P , which is assumed to behave classically. Definition 5. The system KFL ∗ in L → Tr ( P ) extends KFL with (i) ∀ x ( P ( x ) ∨ ¬ P ( x )) ; (ii) Disquotational axiom for P : ⇒ Tr ( p P ˙ x q ) ↔ P ( x );( KFL P)(iii) The substitution rule: ⇒ ∀ x ( B ( x ) ∨ ¬ B ( x )) Γ( P ) ⇒ ∆( P ) for B in L → Tr ( P ); Γ , ∆ ⊆ L → N ( P ) . Γ( B/P ) ⇒ ∆( B/P )The properties of KFL expressed by Lemma 8 transfer directly to KFL ∗ , and are proved inan analogous fashion.The semantics given in §4.1 can be modified to provide a class of fixed-point models for KFL ∗ .We call Φ X the result of relativizing the operator from §4.1 to an arbitrary X ⊆ ω . In particular,this means supplementing the positive inductive definition associated with Φ with the clause:a sentence P z , with z a closed term of L Tr , is in the extension of the truthpredicate (relativized to X ) iff val ( z ) ∈ X .This modification clearly does not compromise the monotonicity of the operator. Therefore, let MIN Φ X be the minimal fixed point of Φ X , and MAX Φ X its maximal one. For any X , we then obtainthe minimal HYPE model M min Φ X := ( { MIN Φ X , MAX Φ X } , ⊆ , ∗ )Again in M min Φ X all arithmetical vocabulary is interpreted standardly at its two states (fixed-points). Only the interpretation of the truth predicate varies. Our notation reflects this. Proposition 3. If KFL ∗ ⊢ Γ ⇒ ∆ , then for all X , M min Φ X (cid:13) Γ ⇒ ∆ . Feferman [Fef91] provides a relativized fixed-point construction to arbitrary subsets of natural numbers andestablishes the soundness of KF ∗ . onclassical truth with classical strength. 18 Proof. By induction on the length of the derivation in KFL ∗ .We consider the case of the substitution rule applied to an arithmetical sequent Γ( P ) ⇒ ∆( P ).That is, our proof ends with ⇒ ∀ x ( B ( x ) ∨ ¬ B ( x )) Γ( P ) ⇒ ∆( P )Γ( B/P ) ⇒ ∆( B/P )with B ( x ) an arbitrary formula of L → Tr .By induction hypothesis, for all X , M min Φ X (cid:13) Γ( P ) ⇒ ∆( P ). Since Γ( P ) ⇒ ∆( P ) is arithmeti-cal, for all interpretations Y of P , ( N , Y ) (cid:15) Γ( P ) ⇒ ∆( P ). Then, following [Fef91], we can let Y to be { n | M min Φ X (cid:13) Γ( B ( n ) /P ) ⇒ ∆( B ( n ) /P ) } to obtain that: M min Φ X (cid:13) Γ( B/P ) ⇒ ∆( B/P ) . (cid:3) Proof-theoretic analysis. We first consider the proof-theoretic lower-bound for KFL ∗ .We adapt to the present setting the strategy outlined in [FS00, p. 84]. In particular, Fefermanand Strahm formalize the notion of A -jump hierarchy, which is a hierarchy of sets of naturalnumbers obtained by iterating an arithmetical operator expressed by an arithmetical formula A ( X, θ, y ). An A -jump hierarchy is relativized when the starting point is a specific set of naturalnumbers expressed by some predicate P . The notion of A -jump hierarchy is quite general, andhas as special cases familiar hierarchies such as the Turing-jump hierarchy.For our purposes, it’s useful to consider A -jump hierachies in which membership in second-order parameters is replaced by the notion of satisfaction. In order to achieve this, we employFeferman’s strategy in [Fef91] in which the stages of the Turing jump-hierarchy are representedby means of suitable primitive recursive functions on codes of L Tr -formulas. Specifically, weencode in suitable primitive recursive functions the stages of a hierarchy in which the formula A is the Veblen-jump formula that will be introduced shortly.We denote with A ( Tr f A , η, y ) the result of replacing every occurrence of ( u, v ) ∈ X in A ( X, η, y ) with Tr sub( f Av , p x q , num( u )) , where the functions f Ax ( y ) are recursively defined as follows: f A ( x ) := p P ˙ x q ,f A,ζ ( x ) := p ˙ x ≺ ˙ ζ ∧ Tr f A ˙ x ( ˙ x ) q ,f Aζ ( x ) := p A ( Tr f A, ˙ ζ , ˙ ζ, ˙ x ) q . In the clause for f A,ζ , the input x is intended to be an ordered pair ( x , x ). As in the definitionsof translations σ and τ above, the existence of the function f A can be obtained by employingthe recursion theorem, as it needs to apply to its arithmetical code. onclassical truth with classical strength. 19 Recall the general pattern of the Gentzen jump formula – with → the HYPE conditional: J ( B, ξ ) := ∀ η ( ∀ ζ ≺ η B ( ζ ) → ∀ ζ ≺ η + ξ B ( ζ )) . We build our A - hierarchy on the more complex Veblen-jump formula A , as stated by Schüttein [Sch77, p. 185], which is crucial for the proof-theoretic analysis of predicative systems.One first defines the functions: e (0) = 0 h (0) = 0 e ( ω η ) = η h ( ω η ) = 0 e ( ω η + . . . + ω η n ) = η n h ( ω η + . . . + ω η n ) = ω η + . . . + ω η n − with η n (cid:22) . . . (cid:22) η .The Veblen-jump formula A is then the following: A ( Tr f A , ξ, y ) := ∀ ζ ( h ( ξ ) ζ ≺ ξ J ( Tr f A ζ , ϕ e ( ξ ) y )) . It expresses that, given some ordinal ξ , the A -jump hierarchy in the interval between the ordinal h ( ξ ), and ξ itself is closed under the Gentzen jump relative to ϕ e ( ξ ) y (with y a parameter). Inthe following we will omit the superscripts specifying the formula, since we will keep A fixed.An essential ingredient of the lower-bound proof for KFL ∗ is the ‘disquotational’ behaviourof our truth predicate for stages in the hierarchy that are provably well-founded. Lemma 12. If we have TI α ( L → Tr ) , then for all η , with ≺ η ≺ α Tr f . η ( n ) ↔ A ( Tr f . η , η, n ) Proof. For all η and all n , we can show that Sent η ( f η (( n , n ))) and Sent η ( f η ( n )) by transfiniteinduction on η making use of the properties of the ramified truth predicates such as, for λ ≺ α limit: ∀ ζ ≺ λ (cid:0) Tr λ ( Tr ζ t ) ↔ Tr ζ val ( t ) (cid:1) . Such properties just state that Tarskian truth predicates are fully compositional for ordinals forwhich we have transfinite induction [Fef91], [Hal14, II.9.1].Since all truth predicates in Tr f . η ( n ) are provably compositional by TI α ( L → Tr ), the claim isobtained by the fact that full compositionality entails disquotation as shown by [Tar35]. (cid:3) The disquotational properties allow us to establish some fundamental properties of the A -jumphierarchy. In particular, we show that the A -jump hierarchy can be elegantly expressed by truthascriptions on the functions f α . Lemma 13. If TI α ( L → Tr ) is derivable in KFL ∗ for some α > , then we can derive in KFL ∗ : ∀ y ( P ( y ) ↔ Tr f ( y )) ∧ ∀ ζ [0 ≺ ζ ≺ α → ∀ y [ Tr f ζ ( y ) ↔ ∀ z ( h ( ζ ) z ≺ ζ → J ( Tr f z , ϕ e ( ζ ) y ))]] . Proof. By our disquotational axioms for P , it immediately follows that ∀ y ( P ( y ) ↔ Tr f ( y )). onclassical truth with classical strength. 20 Let’s assume now that 0 ≺ ζ ≺ α . We have Tr f ζ ( y ) ↔ Tr p A ( Tr f ˙ ζ , ˙ ζ, ˙ y ) q the right hand side is equivalent by the disquotational property to A ( Tr f ζ , ζ, y ) . By the definition of f ζ , the latter formula is in turn equivalent to, A ( Tr p ˙ z ≺ ˙ ζ ∧ Tr f ˙ z ( ˙ x ) q , ζ, y ) , which is again equivalent to(27) ∀ z ( h ( ζ ) z ≺ ζ ( J ( Tr p ˙ z ≺ ˙ ζ ∧ Tr f ˙ z ( ˙ x ) q , ϕ e ( ζ ) y ))) . By applying the disquotational property to (27), we obtain: ∀ z ( h ( ζ ) z ≺ ζ J ( Tr f z ( x ) , ϕ e ( ζ ) y )) . (cid:3) We can now show an analogous claim to Schütte’s Lemma 9 in [Sch77, p. 186], establishingthe progressiveness of the stages of the A -hierarchy: Lemma 14. If TI α ( L → Tr ) is provable in KFL ∗ for α < Γ , then KFL ∗ proves: ∀ ζ (0 ≺ ζ ≺ α ∧ ( ∀ θ ≺ ζ Prog ( Tr f θ ) → Prog ( Tr f ζ ))) . Proof. Let l ( · ) be the function that keeps track of the syntactic complexity of an ordinal code.One first shows that the following claims ζ ≺ α (28) ∀ x ≺ ζ Prog ( Tr f x )(29) ∀ y ≺ η Tr f ζ ( y )(30) ∀ y ( l ( y ) < l ( θ ) → ( y ≺ ϕ e ( ζ ) η → ( ∀ z ( h ( ζ ) z ≺ ζ → J ( Tr f z , y ))))(31) θ ≺ ϕ e ( ζ ) η (32) h ( ζ ) (cid:22) ξ ≺ ζ (33)entail J ( Tr f ξ , θ ).First we apply the principle of induction on the syntactic composition of ordinal codes [Sch77,Thm. 20.10, p. 173] (provable in KFL ∗ ):(34) ∀ x ( ∀ y ( l ( y ) < l ( x ) → φ ( y )) → φ ( x )) → φ ( t )to the formula. φ ( u ) : ↔ u ≺ ϕ e ( ζ ) η → ∀ z ( h ( ζ ) z ≺ ζ J ( Tr f z , u )) , onclassical truth with classical strength. 21 Since we can prove that: ∀ x ≺ ζ Prog ( Tr f x ) → (cid:0) ∀ y ≺ θ l ( y ) < l ( θ )) → ( y ≺ ϕ e ( ζ ) θ → ∀ z ( h ( ζ ) z ≺ ζ J ( Tr f z , y )) , we obtain by the above-mentioned induction principle:(35) ∀ x ≺ ζ Prog ( Tr f x ) → ( ∀ y ≺ η Tr f ζ ( y ) → ( θ ≺ ϕ e ( ζ ) η → ∀ z ( h ( ζ ) z ≺ ζ J ( Tr f z , θ )))) . By the definition of progressiveness we obtain:(36) Prog ( Tr f ξ ) → ( ∀ x ≺ ϕ e ( ζ ) η J ( Tr f ξ , x ) → J ( Tr f ξ , ϕ e ( ζ ) η )) . Therefore, by combining the previous two claims, we obtain:(37) ∀ x ≺ ζ Prog ( Tr f x ) → (cid:0) ∀ y ≺ η Tr f ζ ( y ) → ∀ z ( h ( ζ ) z ≺ ζ J ( Tr f z , ϕ e ( ζ ) η )) (cid:1) . Finally, by Lemma 13 applied to (37), we can conclude that(38) ∀ x ≺ ζ Prog ( Tr f x ) → (cid:0) ∀ y ≺ η Tr f ζ ( y ) → Tr f ζ ( η )) (cid:1) . (cid:3) Let’s characterize the fundamental series of ordinals < Γ as functions of natural numbers inthe standard way: γ := ω , and γ n +1 := ϕ γ n 0. Then, we have: Proposition 4. If TI γ n ( P ) is derivable in KFL ∗ , then TI ϕ γn ( P ) is derivable in KFL ∗ .Proof. We assume Prog ( P ). If TI γ n ( P ) is derivable in KFL ∗ , then by Corollary 2 and thedeterminateness of P we can show TI ω γn +1 ( P ). By the substitution rule we get that the hierarchypredicates are well-defined. Additionally we can prove that(39) ∀ ζ ≺ ω γ n + 1 ∀ x ( Tr ζ ( x ) ∨ ¬ Tr ζ ( x )) . Notice that by (39) we can reformulate this fragment of the hierarchy by replacing all the occur-rences of the HYPE -conditional by the material conditional. Therefore, we have:(40) ∀ ζ ≺ ω γ n + 1( Tr f ζ ( x ) ↔ A ( Tr f ζ , ζ, x )) . By the previous lemma, for a ≺ ω γ n + 1,(41) ∀ b ≺ a Prog ( Tr f b ) → Prog ( Tr f a ) , and therefore, by the substitution rule applied to TI γ n ( P ) and (41), we have that Prog ( Tr f ω γn ),which entails Tr f ω γn (0).Using (40), we have:(42) ∀ ζ ( h ( ω γ n ) ζ ≺ ω γ n J ( Tr f ζ , ϕ e ( ω γn ) . However, since h ( ω γ n ) = 0 and e ( ω γ n ) = γ n we can then infer(43) ∀ ζ ≺ ω γ n ( ∀ y ( ∀ x ≺ y Tr f ζ ( x ) → ∀ x ≺ y + ϕ γ n Tr f ζ ( x ))) . By letting ζ = y = 0, we obtain ∀ x ≺ ϕ e ( ω γn ) P ( x ), as desired. (cid:3) onclassical truth with classical strength. 22 Corollary 7. KFL ∗ defines the truth predicates of RT <α , for α ≺ Γ . Following the characterization of predicative analysis in terms of ramified systems given in[Fef64, Fef91], and the relationships between ramified truth and ramified analysis studied there,one can then conclude that the systems of ramified analysis below Γ are proof-theoreticallyreducible to our system KFL ∗ .The argument employed in the previous section to show that KFL can be proof-theoreticallyreduced – w.r.t. arithmetical sentences – to KF can be lifted to KFL ∗ . One can consider thesystem KF ∗ – Ref ∗ ( PA ( P )) in [Fef91] –, and slightly modify the translations σ , τ from Definition4: in particular, we let( P s ) σ = ( P s ) τ = P s ( ¬ P s ) τ = ¬ P s. Then, by induction on the length of proof in KFL ∗ , we can prove: Proposition 5. If KFL ∗ ⊢ Γ ⇒ ∆ , then KF ∗ ⊢ ( V Γ → W ∆) σ . Given the analysis of KF ∗ given in [Fef91], the combination of Propositions 5 and 4 yields asharp proof-theoretic analysis also for KFL ∗ : Corollary 8. | KFL ∗ | = | KF ∗ | = Γ . Laws of Truth and Intensionality The main aim of the paper is to show that KFL and KFL ∗ are proof-theoretically strong.We now conclude by discussing some of their philosophical virtues. We focus on KFL , butour discussion transfers with little modification to KFL ∗ . In particular, we now argue that KFL displays some advantages with respect to its direct competitors in classical logic ( KF ) andnonclassical logic ( PKF ).Even truth theorists that consider classical logic as superior do not question the importanceof the disquotational intuition for our philosophical notion of truth [ ? , p. 189]. Theories such as KF can only approximate such intuition, by restricting it to sentences that are ‘grounded’, inthe sense of being provably true or false. KFL can preserve such intuition in great generality,by validating the Tr -schema for sentences not containing the conditional → . Typically, however,nonclassical theories pay tribute to this greater vicinity to the unrestricted Tr -schema (cf. Lemma8) with a substantial loss in logical and deductive power. This is not so for KFL : its proof-theoretic strength matches the one of KF . KFL appears also to improve on the philosophical rationale behind the fully disquotationaltruth predicate of PKF . Because of their missing conditional, all variants of PKF do not havethe means to express in the object language their basic principles of truth. KFL overcomesthese liminations by replacing this metatheoretic inferential apparatus by truth theoretic lawsformulated by means of the HYPE conditional. This also enables us to formulate fully in thelanguage of KFL principles of ‘mixed’ nature, such as induction principles open to the truth onclassical truth with classical strength. 23 predicate or, in the case of KFL ∗ , additional predicates. This is the root of the increased proof-theoretic strength of KFL . As a consequence, we are also able to speak more fully about thetruth (and falsity) of non-semantic sentences of L → Tr [Lei19, p. 391]: for instance, if comparedto PKF , KFL can prove many more iterations of the truth predicate over basic non-semantictruths such as 0 = 0 (Corollary 4).For a full philosophical defence of KFL – which, however, is not the main aim of this paper –,it is important to say something about the role of the conditional of HYPE and its interactionwith the truth predicate of KFL . There are at least two ways of doing so. One could followLeitgeb in providing a semantic explanation of the intensional nature of the HYPE conditional.According to Leitgeb, truth ascriptions are evaluated locally , at each fixed-point, whereas condi-tional statements are evaluated globally , that is, by looking at the entire structure of fixed points.Therefore, if the Tr -schema held also for conditional claims, a truth ascription that contains theconditional would need to be evaluated both locally and globally, which would amount to acategory mistake in M Φ .Alternatively, one could attempt a direct proof-theoretic explanation of the interaction of thetruth predicate and the HYPE -conditional. Leon Horsten [Hor11] defends PKF on the basis ofinferential deflationism: the basic principles of disquotational truth are given inferentially, andessentially so [Hor11, §10.2]. Horsten claims in particular that the laws of truth can only beexpressed on the background of an inferential apparatus which is not part of the language towhich truth is applied. One might extend Horsten’s inferential approach to the present case,and argue that KFL characterizes truth in a similar fashion. The laws of truth are given on thebackground of a theoretical apparatus that essentially involves the conditional of HYPE . Suchtheoretical apparatus amounts to the inferential structure of the truth laws of PKF , but nowformulated in the language of our theory of truth. This would also explain why the Tr -schemaonly holds for sentences that do not contain the HYPE -conditional. 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