aa r X i v : . [ m a t h . L O ] J un KF , PKF , and Reinhardt’s Program
Luca Castaldo ∗ & Johannes Stern † University of Bristol
Abstract
In “Some Remarks on Extending an Interpreting Theories with a Partial Truth Predicate”Reinhardt (1986) famously proposed an instrumentalist interpretation of the truth theoryKripke-Feferman ( KF ) in analogy to Hilbert’s program. Reinhardt suggested to view KF as atool for generating “the significant part of KF ”, that is, as a tool for deriving sentences of theform T p ϕ q . The constitutive question of Reinhardt’s program was whether it was possible“to justify the use of nonsignificant sentences entirely within the framework of significantsentences”? This question was answered negatively by Halbach and Horsten (2006) but weargue that under a more careful interpretation the question may receive a positive answer.To this end, we propose to shift attention from KF -provably true sentences to KF -provablytrue inferences , that is, we shall identify the significant part of KF with the set of pairs h Γ , ∆ i , such that KF proves that if all members of Γ are true, at least one member of ∆ istrue. In way of addressing Reinhardt’s question we show that the provably true inferencesof KF coincide with the provable sequents of the theory Partial Kripke-Feferman ( PKF ). Kripke’s theory of truth (Kripke, 1975) is a cornerstone of contemporary research on truth andthe semantic paradoxes. The theory provides us with a strategy for constructing, that is defining,desirable interpretations of a self-applicable truth predicate, so-called fixed points. These fixedpoints can serve as interpretations of the truth predicate within non-classical models of thelanguage, as in Kripke’s original article, but can also be used in combination with classicalmodels, so called closed-off models. Feferman (1991) devised an elegant axiomatic theory ofthe Kripkean truth predicate of these closed-off fixed-point models. The theory is known asKripke-Feferman ( KF ) and is still one of the most popular classical axiomatic truth theories inthe literature. Nonetheless, KF displays a number of unintended and slightly bizarre features,which it inherits from the behavior of the truth predicate in the closed-off fixed-point models.While in the non-classical fixed-point models the truth predicate is transparent, i.e., ϕ and T p ϕ q will always receive the same semantic value, this no longer holds in the closed-off models. Ratherfor each closed-off model there will be sentences ϕ , e.g. the Liar sentence, such that either ϕ and ¬ T p ϕ q will be true in the model, or ¬ ϕ and T p ϕ q will be true in the model. As a consequenceone can prove this counterintuitive disjunction in KF for the Liar sentence λ , i.e. ( ∗ ) KF ⊢ ( λ ∧ ¬ T p λ q ) ∨ ( ¬ λ ∧ T p λ q ) . ∗ [email protected] † [email protected] This was also suggested by Kripke (cf. Kripke, 1975, p.715). KF as an acceptable theory of truth. However, reasoning within the non-classical logic ofthe Kripkean fixed-points seems a non-trivial affaire or, as Feferman would have it, “ nothing likesustained ordinary reasoning can be carried on ” in these non-classical logics (Feferman, 1984,p. 95). Giving up on KF thus hardly seems a desirable conclusion.In reaction to the counterintuitive consequences of KF , Reinhardt (1985, 1986) proposed aninstrumentalist interpretation of the theory in analogy to Hilbert’s program. Famously, Hilbertproposed to justify number theory, analysis and even richer mathematical theories by finitarymeans. Without entering into Hilbert-exegesis, the main idea was of course to provide con-sistency proofs for these mathematical theories in a finitistically acceptable metatheory. Froma finitist perspective this would turn the mathematical theories into useful tools for producingmathematical truths. But the fate of Hilbert’s program, at least on its standard interpretation, iswell known: Gödel’s incompleteness theorems are commonly thought to be the program’s coffinnail. Nonetheless Reinhardt (1985, 1986) was optimistic that his program had greater chancesof success. Reinhardt proposed to view KF as a tool for deriving Kripkean truths in the sameway Hilbert viewed, say, number theory as a tool for deriving mathematical truths. A Kripkeantruth is a sentence that is true from the perspective of the Kripkean fixed-points: if KF ⊢ T p ϕ q ( T p ¬ ϕ q ), then ϕ is true (false) in all non-classical fixed-point models, that is, we are guaranteedthat ϕ receives a semantic value from the perspective of Kripke’s theory of truth. This is not ageneral feature of the theorems of KF but peculiar to those sentences that KF proves true (false).The latter sentences Reinhardt called “ the significant part of KF ” (Reinhardt, 1986, p. 219) andlabelled the set of KF -significant sentences KFS := { ϕ | KF ⊢ T p ϕ q } . In light of this terminologythe constitutive question of Reinhardt’s program is whether it was possible “ to justify the useof nonsignificant sentences entirely within the framework of significant sentences ” (Reinhardt,1986, p. 225)?But what would such a successful instrumentalist interpretation of KF and the use of non-significant sentences amount to? Reinhardt himself is scarce on the exact details, however, atthe end of Reinhardt (1985) he asks the following question:If KF ⊢ T p ϕ q is there a KF -proof ϕ , . . . , ϕ n , T p ϕ q such that for each ≤ i ≤ n , KF ⊢ T p ϕ i q . (cf. Reinhardt, 1985, p. 239) If we can answer this question positively it seems that we can justify each Kripkean truth prov-able in KF by appealing solely to the significant fragment of KF : even though we have reasoned in KF , each step of our reasoning is part of KFS and hence “remains within the framework of signifi-cant sentences”. This interpretation of Reinhardt’s program is adopted by Halbach and Horsten On p. 225 Reinhardt (1986) writes:“I would like to suggest that the chances of success in this context, where the interpreted or significantpart of the language includes such powerful notations as truth, are somewhat better than in Hilbert’scontext, where the contentual part was very restictred.” KFS is sometimes also called the inner logic of KF (cf. Halbach and Horsten, 2006, p.638). We decided tostick with Reimhardt’s original terminology. To be precise, Reinhardt (1985) asked whether there was a KF -proof ϕ , . . . , ϕ n , ϕ rather than a KF -proof ϕ , . . . , ϕ n , T p ϕ q . But this presupposes that all KF -provably true sentences are also theorems of KF . While this istrue in the variants of KF Reinhardt considers, this is not the case in all versions of KF discussed in the literature.However, all remarks concerning our version of the question generalize to Reinhardt’s original question. Reinhardt’s Problem . Unfortunately, as Halbach and Horsten(2006) convincingly argue, if understood in this way the instrumentalist interpretation of KF willfail. We refer to Halbach and Horsten (2006) for details but, in a nutshell, the reason for thisfailure is that the truth-theoretic axioms of KF will not be true in the non-classical fixed-pointmodels and hence not be part of KFS , e.g., if ϕ i := ∀ x ( Sent ( x ) → ( T ( x ) ↔ T ( ¬ . ¬ . x ))) , then ϕ i KFS . Indeed, Halbach and Horsten take this to show “ that Reinhardt’s analogue of Hilbert’sprogram suffers the same fate as Hilbert’s program ” (Halbach and Horsten, 2006, p. 684).However, we think that this conclusion is premature and argue that, to the contrary, if suitablyunderstood Reinhardt’s program can be deemed successful. Our key point of contention is thatHalbach and Horsten (2006), arguably following Reinhardt, employ the perspective of classicallogic when theorizing about the significant part of KF . But the logic of the significant part of KF is not classical logic but the logic of the non-classical fixed-point models, that is, a non-classicallogic. This observation has two interrelated consequences for Reinhardt’s program. First, contraReinhardt and, Halbach and Horsten we should not identify the significant part of KF exclusivelywith the set of significant sentences . Rather it also seems crucial to ask which inferences areadmissible within the significant part of KF . Of course, in classical logic this difference collapsesdue to the deduction theorem but not so in non-classical logics. For example, the three-valuedlogic Strong Kleene , K , has no logical truth, but many valid inferences—if, in this case, we wereto focus only on the theorems of the logic, there would be no logic to discuss. Moreover, sincethe significant sentences can be retrieved from the significant inferences, that is the provablytrue inferences, we should focus on the latter rather than the former in addressing Reinhardt’sProblem . To this end, it is helpful to conceive of KF as formulated in a two-sided sequentcalculus rather than a Hilbert-style axiomatic system. Let Γ , ∆ be finite sets of sentences andlet T p Γ q be short for { T p γ q | γ ∈ Γ } . The admissible inferences of the significant part of KF ,which we label KFSI , can then be defined as follows: KFSI := {h Γ , ∆ i | KF ⊢ T p Γ q ⇒ T p ∆ q } Second,
Reinhardt’s Problem , according to the formulation of Halbach and Horsten (2006),which admittedly was inspired by Reinhardt’s (1985) original question, conceives of KF -proofs assequences of theorems of KF . But by focusing on sequences of theorems, we cannot fully exploitthe significant part of KF , that is, KFSI for precisely the reasons Halbach and Horsten (2006)used to rebut Reinhardt’s program: while double negation introduction is clearly a member of
KFSI , proving this fact by a sequence of theorems would take us outside of
KFS since it woulduse the truth-theoretic axiom ∀ x ( Sent ( x ) → ( T ( x ) ↔ T ( ¬ . ¬ . x ))) , which is not a member of thesignificant part of KF . This suggest a reformulation of Reinhard’s Problem in terms of anotion of proof that focuses on inferences rather than theorems. To this end, it proves usefulagain to formulate KF in a two-sided sequent calculus and to conceive of proofs as derivationtrees, where each node of the tree is labeled by a sequent. As a matter of fact in this case wecan distinguish between two versions Reinhard’s Problem :1. For every KF -theorem of the form T p ϕ q , is there a KF -derivation tree with root ∅ ⇒ T p ϕ q such that for every node d of the tree, d ∈ KFSI ? See §2 for details on notation. The provable true sentences can be viewed as inferences where the truth of the sentence follows from an emptyhypothesis. Notice that moving to a two-sided sequent formulation of KF is not essential. Due to the deduction theoremwe can also define KFSI be appeal to KF formulated in an axiomatic Hilbert-style calculus. In this case, thedefinition would amount to KFSI := {h Γ , ∆ i | KF ⊢ ^ T p Γ q → _ T p ∆ q } .
3. For every KF -derivable sequent of the form T p Γ q ⇒ T p ∆ q , is there a KF -derivation treewith root T p Γ q ⇒ T p ∆ q such that for every node d of the tree, d ∈ KFSI ?The first question is a reformulation of Halbach and Horsten’s (2006)
Reinhardt’s Prob-lem . The second question, which we label
Generalized Reinhardt Problem , asks whetherall provably true inferences can be justified by appealing to the significant inferences only. Ar-guably, to deem Reinhardt’s program successful we need to give an affirmative answer to the
Generalized Reinhardt Problem . Otherwise, a proof of T p ϕ q could still rely on inferencesthat, whilst part of KFSI , cannot themselves be justified by appealing only to the significantinferences of KF . Perhaps surprisingly, we shall show that an affirmative answer to the Gener-alized Reinhardt Problem can be given. It follows that on this more careful formulationReinhardt’s program can be deemed successful.Arguably, one may still take issue with this conclusion and argue that our answer to the
Generalized Reinhardt Problem is at best a partial completion of Reinhardt’s program:what is still required is an independent axiomatization of the significant part of KF , for thiswould prove KF dispensable. We postpone a discussion of this view to the conclusion. Rather wewill now take a fresh look at the question of an independent axiomatization, which Halbach andHorsten called Reinhardt’s Challenge (Halbach and Horsten, 2006, p. 689). This will proveinstrumental in answering the
Generalized Reinhardt Problem . Reinhardt (1985) askedfor an independent axiomatization of the significant part of KF . More precisely, (Reinhardt, 1985,p. 239) asked:a) “Is there an axiomatization of { σ | KF ⊢ T p σ q } which is natural and formulated entirelywithin the domain of significant sentences,. . . .”b) “Similarly for the relation Γ ⊢ S σ defined by KF + { T p γ q | γ ∈ Γ } ⊢ T p σ q .”Halbach and Horsten (2006) proposed their theory Partial Kripke-Feferman ( PKF ) in way ofanswering to
Reinhardt’s Challenge . PKF is formulated in a non-classical, two-sided sequentcalculus and thus fits neatly with our observation that one should focus on the provably trueinferences of KF rather than the provably true sentences. Moreover, PKF is arguably a naturalaxiomatization of Kripke’s theory of truth. However, Halbach and Horsten (2006) observedthat there are sentences ϕ such that KF ⊢ T p ϕ q but PKF ϕ , which led them to conclude that Reinhardt’s Challenge cannot be met. The reason for this asymmetry is due to the differencein proof-theoretic strength of KF and PKF : while KF proves transfinite induction for ordinalsbelow ε , PKF only proves transfinite induction for ordinals smaller than ω ω . As a consequence,there will be arithmetical sentences that KF proves true that PKF cannot prove. The story doesnot end there however. First, as Halbach and Nicolai (2018) observe, the discrepancy between KF and PKF arises only if the rule of induction is extended beyond the arithmetical language,that is, if we restrict induction to the language of arithmetic—call the resulting theories KF ↾ and PKF ↾ —then KF ↾ ⊢ T p ϕ q , if and only if, PKF ↾ ⊢ ϕ . This highlights that the asymmetrybetween KF and PKF is not due to the truth-specific principles but the amount of induction thatis assumed in the respective theories.Second, corroborating the latter observation, Nicolai (2018) showed that the asymmetry be-tween KF and PKF is indeed solely due to the amount of induction available within the respectivetheories: Nicolai shows that if transfinite induction up to < ε is added axiomatically to PKF —call this theory
PKF + — then KF ⊢ T p ϕ q , if and only if, PKF + ⊢ ϕ . Moreover, Nicolai (2018)shows that independently of which version of induction is assumed in KF there will be a suitable PKF -style theory, which will have exactly the provably true sentences of the relevant KF -styletheory as theorems. Nicolai took these results to “ partially [accomplish] a variant of a programsketched by Reinhardt ” (cf. Nicolai, 2018, p. 103).4owever, the work by Halbach and Nicolai provides at best a positive answer to Question a),it does not—at least not immediately—yield an answer to Question b). Indeed, it seems as if,despite working in a non-classical, two-sided sequent calculus Halbach and Horsten (2006) havelargely neglected Question b) of Reinhardt’s Challenge and so have subsequent publicationson this issue. Indeed the version of
PKF originally proposed by Halbach and Horsten (2006)failed to yield a positive answer to Question b) for rather banal reasons—even for theories withrestricted induction: the version of KF Halbach and Horsten (2006) consider assumes the truthpredicate to be consistent, which, as we shall explain in due course, means that the logic of
KFSI is K . But Halbach and Horsten formulate PKF in symmetric strong Kleene logic KS and as a consequence KF ⊢ T p ϕ q , T p ¬ ϕ q ⇒ T p ψ q while PKF ϕ, ¬ ϕ ⇒ ψ . The main technicalcontribution of this paper is to clarify the situation and to show how, building on Nicolai’s (2018)work, a positive answer to Question b) of Reinhardt’s Challenge can be provided. To thisend, we show how to pair the different variants of KF with a suitable PKF -style theory suchthat the provable sequents of the latter theory constitute exactly the significant inferences ofthe former theory. Moreover, it turns out that once we have an independent axiomatization of
KFSI , the
Generalized Reinhardt Problem can be answered rather immediately: it is easyto show that if a sequent Γ ⇒ ∆ ∈ KFSI , then there will be a KF -derivation tree such that eachnode of the tree can be derived in PKF + . But the provable sequents of PKF + constitute exactlythe significant inferences of KF , that is, KFSI —every node of the KF -derivation is a member of KFSI . The derivation remains within the significant part of KF . The paper starts by fixing some basic terminology and notation. More specifically, Section 2introduces the language and the logical systems underlying
PKF and its variants. That is, weintroduce the logics
FDE , KS , K and LP . In the next section, Section 3, we introduce the relevantfamilies of KF - and PKF -like theories and observe some basic properties of these
PKF -systems.In Section 4 we prove the central technical results of this paper. We show that for each KF -liketheory we can find a PKF -counterpart such that the latter is an independent axiomatization ofthe significant inference of the former. In other words, we show that the set of pairs h Γ , ∆ i suchthat T p Γ q ⇒ T p ∆ q is derivable in a KF -like theory coincides with the set of pairs h Γ , ∆ i suchthat Γ ⇒ ∆ is derivable in a corresponding PKF -like theory. It turns out a positive answer to
Generalized Reinhardt Problem is but an immediate corollary of the existence of such anindependent axiomatization. L T denotes the language of first-order Peano arithmetic ( PA ) extended by a unary predicate T . L PA := L T \{ T } is the T -free fragment of L T . Terms and formulae are generated in the usualway. By an L T -expression we mean a term or a formula of L T . n is the numeral correspondingto the number n ∈ ω ; = is the formal identity symbol of L T . We fix a canonical Gödel numbering of L T -expressions. If e is an L T -expression, the Gödel number (= gn) of e is denoted by e and p e q is the term representing e in L PA . We introduce some primitive recursive relations, inpractice working in a definitional extension of PA : Term ( x ) ( ClTerm ( x )) := x is the gn of a (closed) term ; Var ( x ) := x is the gn of variable ; ml n ( x ) ( Sent ( x )) := x is the gn of a formula with at most n (0) free distinct variables ; Eq ( x ) := x is the gn of an equality between closed terms ; Ver ( x ) := x is the gn of true closed equality . We also assume to have standard representations of the following primitive recursive operationson Gödel numbers: ∧ . : ϕ, ψ ϕ ∧ ψ ) ¬ . : ϕ ¬ ϕ ) ∀ . : v k , ϕ ∀ v k ϕ ) = . : t, s t = s ) T . : t T ( t ) num : n n ( the gn of the n -th numeral ) sub : e, t, v k e [ t/v k ] e [ t/v k ] is the expression obtained from an expression e by replacing each free occurrence of v k bya term t . We occasionally write e ( t ) , when it is clear which variable is being substituted. We alsohave a recursive function val ( x ) such that val ( p t q ) = t for closed terms t . We write ∃ . a.b insteadof ∃ . ( a, b ) . We often abbreviate num ( x ) with ˙ x . We write ϕ ( ~v ) to denote a formula ϕ whose freevariables are contained in ~v , and p ϕ ( ˙ x ) q for sub ( p ϕ ( v ) q , num ( x ) , p v q ) . This definition extendsto the case of multivariables in the obvious way, and we write p ϕ ( ˙ ~x ) q for p ϕ ( ˙ x , . . . , ˙ x n ) q . TheGödelnumbering is canonical, so in particular we require that the following are provable in (afragment of) PA : PA ⊢ val ( ˙ x ) = x ∧ ClTerm ( ˙ x ) PA ⊢ Fml ( x ) → ∀ z Sent ( x [ ˙ z/v ]) PA ⊢ ClTerm ( x ) ∧ ClTerm ( y ) → ( Ver ( x = . y ) ↔ val ( x ) = val ( y )) Terminology and notation for Gentzen-systems.
A sequent is a pair Γ ⇒ ∆ of finite sets of L T -formulae. For a finite set of formulae Γ( ~x ) := { ϕ ( ~x ) , . . . , ϕ n ( ~x ) } with free variables contained in ~x , ¬ Γ( ~x ) denotes the set of negated formulaeof Γ , i.e. {¬ ϕ ( ~x ) , . . . , ¬ ϕ n ( ~x ) } , and V Γ( ~x ) denotes the iterated conjunction ϕ ( ~x ) ∧ · · · ∧ ϕ n ( ~x ) .If Γ = ∅ we identify V Γ with = . For t free for v in Γ (i.e. t is free for v for all members ϕ , . . . , ϕ n of Γ ), we write Γ[ t/v ] for { ϕ [ t/v ] , . . . , ϕ n [ t/v ] } . For Γ a finite set of L T -sentences,we let T p Γ q := { T p γ q | γ ∈ Γ } .A derivation of a sequent Γ ⇒ ∆ is a tree with nodes labelled by sequents. Given a derivation D and a node Γ ⇒ ∆ of D , call it d , we write Γ ′ | d | ∆ ′ for Γ ′ , Γ ⇒ ∆ , ∆ ′ .The height of a derivation D is the maximum lenght of the branches in the tree, where the lenghtof a branch is the number of its nodes minus 1.In a rule of inference • formulae in Γ , ∆ are called side formulae , or context , • the formulae not in the context in the conclusion are called principal formulae , • the formulae in the premises from which the conclusion is derived (i.e. the formulae in thepremises not in the context) are called active formulae .A literal is an atomic formula or the negation of an atomic formula. The cut rank of a formula which is eliminated in a cut-rule is the positive complexity of the formula. The supremum of thecut ranks of a derivation D is called the cut rank of D .6 .2 Sequent calculi for FDE and some of its extensions
In this section we introduce the various logics underlying the systems of truth employed in thepaper. We start with the two-sided sequent calculus of F irst D egree E ntailment ( FDE ). For ageneral overview of the different non-classical logics employed in this section see Priest (2008).
Definition 2.1 ( FDE ) . The logic of
FDE consists of the following axioms and rules. ϕ, Γ ⇒ ∆ , ϕ Γ ⇒ ∆ , ϕ ϕ, Γ ′ ⇒ ∆ ′ Γ ′ , Γ ⇒ ∆ , ∆ ′ ( Cut )( ∧ L ) ϕ, ψ, Γ ⇒ ∆ ϕ ∧ ψ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ Γ ⇒ ∆ , ψ Γ ⇒ ∆ , ϕ ∧ ψ ( ∧ R )( ∀ L ) ϕ [ t/v ] , Γ ⇒ ∆ ∀ vϕ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ [ u/v ]Γ ⇒ ∆ , ∀ vϕ ( ∀ R )( ¬¬ L ) ϕ, Γ ⇒ ∆ ¬¬ ϕ, Γ ⇒ ∆ Γ ⇒ ∆ , ϕ Γ ⇒ ∆ , ¬¬ ϕ ( ¬¬ R )( ¬∧ L ) ¬ ϕ, Γ ⇒ ∆ ¬ ψ, Γ ⇒ ∆ ¬ ( ϕ ∧ ψ ) , Γ ⇒ ∆ Γ ⇒ ∆ , ¬ ϕ, ¬ ψ Γ ⇒ ∆ , ¬ ( ϕ ∧ ψ ) ( ¬∧ R )( ¬∀ L ) ¬ ϕ [ u/v ] , Γ ⇒ ∆ ¬∀ vϕ, Γ ⇒ ∆ Γ ⇒ ∆ , ¬ ϕ [ t/v ]Γ ⇒ ∆ , ¬∀ vϕ ( ¬∀ R ) Conditions of application: ϕ literal in initial sequents; u eigenparameter. Let Γ ⇒ ∆ , ϕ ( ¬ L) ¬ ϕ, Γ ⇒ ∆ ϕ, Γ ⇒ ∆ ( ¬ R) Γ ⇒ ∆ , ¬ ϕ ψ, Γ ⇒ ∆ ¬ ψ, Γ ⇒ ∆ ( GG ) ϕ, ¬ ϕ, Γ ⇒ ∆ for ϕ, ψ atomic. Definition 2.2. • Classical Logic, CL , is the system given by FDE without ¬◦ M, for ◦ ∈ {¬ , ∧ , ∀} , M ∈ { L, R } ,and with the addition of unrestricted ( ¬ L) and ( ¬ R) . • Strong Kleene, K , is the system FDE + ( ¬ L) . • Logic of Paradox, LP , is the system FDE + ( ¬ R) . • Kleene’s Symmetric Logic, KS , is the system FDE + ( GG ) . We now extend the base logics with rules for identity. Let t = t, Γ ⇒ ∆ Ref Γ ⇒ ∆ ϕ ( t ) , Γ ⇒ ∆ RepL s = t, ϕ ( s ) , Γ ⇒ ∆ Γ ⇒ ∆ , ϕ ( t ) Γ ⇒ ∆ , s = t RepR Γ ⇒ ∆ , ϕ ( s ) For similar calculi defining the same logic see, for instance, Scott (1975) and Blamey (2002). ϕ literal. Definition 2.3. • CL = is CL + (Ref) + (RepL) • FDE = is FDE + (Ref) + (RepL) . • K = is K + (Ref) + (RepL) . • LP = is LP + (Ref) + (RepR) . • KS = is KS + (Ref) + (RepL) . Remark 2.4. • (RepL) and (RepR) are equivalent over FDE , and they both yield the replacement schema s = t, ϕ ( s ) , Γ ⇒ ∆ , ϕ ( t ) • It can easily be shown that
FDE ( KS ) and BDM ( SDM ), that is, the system(s) defined byNicolai (2018), are equivalent. The system KS , however, has the advantage of enjoying acut elimination theorem. • The reason for formulating K = and KS = with (RepL) , and LP with (RepR) is to obtaina syntactic proof of full Cut elimination. KF -like and PKF -like theories
This section introduces the KF -like and PKF -like truth theories. The theory Kripke-Feferman KF was developed by Feferman (cf. Feferman, 1991) and further studied by, e.g., Reinhardt(1986); McGee (1990) and Cantini (1989). The theory Partial Kripke-Feferman ( PKF ) may beseen as the non-classical counterpart to KF . It was developed by Halbach and Horsten (2006)and further studied by, e.g., Halbach and Nicolai (2018) and Nicolai (2018), who introduced thetheory PKF + . We first introduce different rules of induction employed in the formulation of the theories. Tothis end we fix a standard notation system of ordinals up to Γ . We use a, b, c . . . to denotethe code of our notation system whose value is α, β, γ · · · ∈ On (with the exception of ω and ε -numbers, for which we use the ordinals themselves), and we use ≺ to denote the p.r. ordering We remark that Cut elimination holds for the sequent calculi introduced above. The key observation is thatthe calculi defined in this article are designed so that if the cut formula is principal in both premises of Cut,then the complexity of ϕ is >0, i.e. ϕ cannot be a literal. In fact, In FDE = , K = , and KS = , there is no ruleintroducing a literal on the right—that is the reason why we formulated K = and KS = with (RepL), as theyboth have one rule introducing literals on the left, i.e. ( ¬ L) and ( GG ), respectively—; in LP = there is no ruleintroducing a literal on the left—that is the reason why we formulated LP = via (RepR), as this calculus has onerule introducing literals on the right, i.e. ( ¬ R). Two points are worth emphasizing. The first is that, as far as weknow, the rule GG excluding the simultaneous occurrence of gaps and gluts is new and KS is the first sequentcalculus for symmetric Strong Kleene logic admitting a syntactic proof of Cut elimination. The second is thatwe are also not aware of Gentzen style calculi for first-order FDE , K , LP , or KS using so called geometric rulesfor identity—at least in the literature on truth and paradoxes, the calculi above might be the first axiomatizingidentity with rules instead of axioms. A notable exception is Picollo (2018), however, Cut is not eliminable in thecalculus defined. See Halbach (2014) for a presentation and discussion of both theories. See for instance Feferman (1968), Pohlers (2009). ∀ z ≺ y ( ϕ [ z/v ]) is short for ∀ z ( ¬ ( Ord ( z ) ∧ Ord ( y ) ∧ z ≺ y ) ∨ ϕ [ z/v ]) , where Ord represents the set of codes of ordinals. For α < Γ and a formula ϕ ( v ) ∈ L T we let TI <α ( ϕ ) denote ∀ z ≺ y ϕ ( z ) , Γ ⇒ ∆ , ϕ ( y )Γ ⇒ ∆ , ∀ x ≺ a ϕ ( x ) ( TI <α )We have the following induction schemata ( ϕ ( v ) ∈ L T ): Γ , ϕ ( u ) ⇒ ϕ ( u ′ ) , ∆Γ , ϕ ( ) ⇒ ϕ ( t ) , ∆ u / ∈ F V (Γ , ∆ , ϕ ( )) ( IND)Fml ( x ) , T x [ ˙ u/v ] , Γ ⇒ ∆ , T x [ ˙ u ′ /v ] T x [ num ( ) /v ] , Γ ⇒ ∆ , T x [ ˙ z/v ] ( IND int )where u is an eigenvariable and z is an arbitrary term.We now introduce the basic truth principles employed in the systems of truth we discuss inthe paper. Definition 3.1 (Truth Axioms) . The following truth-theoretic initial sequents are called truthaxioms.
Reg1-2 are called regularity axioms . ( T =) ClTerm ( x ) , ClTerm ( y ) , val ( x ) = val ( y ) , Γ ⇒ ∆ , T ( x = . y ) ClTerm ( x ) , ClTerm ( y ) , T ( x = . y ) , Γ ⇒ ∆ , val ( x ) = val ( y )( T ¬ =) ClTerm ( x ) , ClTerm ( y ) , ¬ (cid:0) val ( x ) = val ( y ) (cid:1) , Γ ⇒ ∆ , T ( ¬ . ( x = . y )) ClTerm ( x ) , ClTerm ( y ) , T ( ¬ . ( x = . y )) , Γ ⇒ ∆ , ¬ (cid:0) val ( x ) = val ( y ) (cid:1) ( ¬ T ¬ )( i ) Sent ( x ) , T ( ¬ . x ) , Γ ⇒ ∆ , ¬ T ( x )( ii ) Sent ( x ) , ¬ T ( x ) , Γ ⇒ ∆ , T ( ¬ . x )( T Sent ) T x, Γ ⇒ ∆ , Sent ( x )( Reg1 ) Var ( z ) , ClTerm ( y ) , Sent ( ∀ . z.x ) , T x [ y/z ] , Γ ⇒ ∆ , T x [ ˙ val ( y ) /z ] ( Reg2 ) Var ( z ) , ClTerm ( y ) , Sent ( ∀ . z.x ) , T x [ ˙ val ( y ) /z ] , Γ ⇒ ∆ , T x [ y/z ] The following rules are called truth-rules:
ClTerm ( x ) , T ( val ( x )) , Γ ⇒ ∆ T rp L ClTerm ( x ) , T ( T . x ) , Γ ⇒ ∆ ClTerm ( x ) , Γ ⇒ ∆ , T ( val ( x )) T rp R ClTerm ( x ) , Γ ⇒ ∆ , T ( T . x ) ClTerm ( x ) , T ( ¬ . val ( x )) , Γ ⇒ ∆ ClTerm ( x ) , ¬ Sent ( val ( x )) , Γ ⇒ ∆ T nrp L ClTerm ( x ) , T ( ¬ . T . x ) , Γ ⇒ ∆ ClTerm ( x ) , Γ ⇒ ∆ , T ( ¬ . val ( x )) , ¬ Sent ( val ( x )) T nrp R ClTerm ( x ) , Γ ⇒ ∆ , T ( ¬ . T . x ) Sent ( x ) , T ( x ) , Γ ⇒ ∆ T dn L Sent ( x ) , T ( ¬ . ¬ . x ) , Γ ⇒ ∆ Sent ( x ) , Γ ⇒ ∆ , T ( x ) T dn R Sent ( x ) , Γ ⇒ ∆ , T ( ¬ . ¬ . x ) Sent ( x ∧ . y ) , T ( x ) , T ( y ) , Γ ⇒ ∆ T and L Sent ( x ∧ . y ) , T ( x ∧ . y ) , Γ ⇒ ∆ Notice that ˙ val ( y ) := num ( val ( y )) . ent ( x ∧ . y ) , Γ ⇒ ∆ , T ( x ) Sent ( x ∧ . y ) , Γ ⇒ ∆ , T ( y ) T and R Sent ( x ∧ . y ) , Γ ⇒ ∆ , T ( x ∧ . y ) Sent ( x ∧ . y ) , T ( ¬ . x ) , Γ ⇒ ∆ Sent ( x ∧ . y ) , T ( ¬ . y ) , Γ ⇒ ∆ T nand L Sent ( x ∧ . y ) , T ( ¬ . ( x ∧ . y )) , Γ ⇒ ∆ Sent ( x ∧ . y ) , Γ ⇒ ∆ , T ( ¬ . x ) , T ( ¬ . y ) T nand R Sent ( x ∧ . y ) , Γ ⇒ ∆ , T ( ¬ . ( x ∧ . y )) Sent ( ∀ . v.x ) , ∀ yT ( x [ ˙ y/v ]) , Γ ⇒ ∆ T all L Sent ( ∀ . v.x ) , T ( ∀ . v.x ) , Γ ⇒ ∆ Sent ( ∀ . v.x ) , Γ ⇒ ∆ , ∀ zT ( x [ ˙ z/v ]) T all R Sent ( ∀ . v.x ) , Γ ⇒ ∆ , T ( ∀ . v.x ) Sent ( ∀ . v.x ) , ¬∀ z ¬ T ( ¬ . x [ ˙ z/v ]) , Γ ⇒ ∆ T nall L Sent ( ∀ . v.x ) , T ( ¬ . ∀ . v.x ) , Γ ⇒ ∆ Sent ( ∀ . v.x ) , Γ ⇒ ∆ , ¬∀ z ¬ T ( ¬ . x [ ˙ z/v ]) T nall R Sent ( ∀ . v.x ) , Γ ⇒ ∆ , T ( ¬ . ∀ . v.x ) The following definitions introduce the various KF - and PKF -style theories.
Definition 3.2 ( KF ) . KF is obtained from CL = by adding sequents ⇒ ϕ for ϕ axiom of PA ,( IND ), truth axioms and truth-rules of Df 3.1, except the axiom ( ¬ T ¬ ). Definition 3.3 ( KF -variants) . We introduce variants of KF :(i) KF cs is obtained from KF by adding Cons , i.e.,
Sent ( x ) , T ( ¬ . x ) , Γ ⇒ ∆ , ¬ T ( x ) .(ii) KF cp is obtained from KF by adding Comp , i.e.,
Sent ( x ) , ¬ T ( x ) , Γ ⇒ ∆ , T ( ¬ . x ) . (iii) KF S is obtained from KF by adding Sent ( x ) , Sent ( y ) , T ( x ) , T ( ¬ . x ) , Γ ⇒ ∆ , T ( y ) , T ( ¬ . y ) ( GoG )Let KF ⋆ ∈ { KF , KF cs , KF cp , KF S } . Then(iv) KF ⋆ ↾ is obtained from KF ⋆ by restricting ( IND ) on L PA -formulae.(v) KF int ⋆ is the theory obtained from KF ⋆ ↾ by replacing the restricted version of ( IND ) with ( IND int ) . Fact 3.4.
Let KF ⋆ ∈ { KF , KF cs , KF cp , KF S } . Then KF ⋆ ↾ is contained in the subtheory KF int ⋆ ↾ of KF int ⋆ , obtained by allowing ( IND int ) only in the following restricted version:
Fml L PA ( x ) , T x [ ˙ u/v ] , Γ ⇒ ∆ , T x [ ˙ u ′ /v ] ( IND int ↾ L PA ) T x [ /v ] , Γ ⇒ ∆ , T x [ ˙ z/v ] for u eigenvariable and z arbitrary. Remark 3.5.
Let KF int ⋆ ∈ { KF int , KF intcs , KF intcp , KF intS } . Every axiom of KF int ⋆ has the form Θ , Γ ⇒ ∆ , Λ . Formulae in Θ , Λ are called active . Every active formula has positive complexity . Call a derivation D quasi-normal if D has cut rank 0. By application of standard techniquesfor Cut-elimination for predicate calculus, we get Note that
Cons is ( ¬ T ¬ )( i ) and Comp is ( ¬ T ¬ )( ii ) . Of course, over the nonclassical logics studied in thispaper, ( ¬ T ¬ ) does not imply that the truth predicate is consistent and complete. ( ¬ T ¬ ) is just axiomatizingthe well known property of fixed-point models that the anti-extension A can be defined via the extension E as A := { ϕ | ¬ ϕ ∈ E } . Literals have positive complexity = 0 . roposition 3.6. Let KF int ⋆ be as in Remark 3.5. Then every KF int ⋆ -derivation D can be trans-formed into a quasi-normal derivation D ′ with the same end sequent. Proposition 3.7 (Inversion) . Let KF int ⋆ be as in Remark 3.5. Then(i) If KF int ⋆ ⊢ n Γ ⇒ ∆ , ∀ vϕ , then KF int ⋆ ⊢ n Γ ⇒ ∆ , ϕ [ u/v ] for any u / ∈ F V (Γ , ∆ , ϕ ) .(ii) If KF int ⋆ ⊢ n ∀ vϕ, Γ ⇒ ∆ , then KF int ⋆ ⊢ n ϕ [ t/v ] , Γ ⇒ ∆ for some term t . We move on to
PKF -like theories.
Definition 3.8 ( PKF ) . PKF is obtained from
FDE = by adding sequents Γ ⇒ ∆ , ϕ and ¬ ϕ, Γ ⇒ ∆ for ϕ axiom of PA , ( IND ), truth axioms and truth-rules of Df. 3.1, and the following two rulesrequiring identity statements to behave classically: Γ ⇒ ∆ , s = t = ¬ L ¬ ( s = t ) , Γ ⇒ ∆ s = t, Γ ⇒ ∆ = ¬ R Γ ⇒ ∆ , ¬ ( s = t ) Definition 3.9 ( PKF -cluster) . We introduce variants of
PKF (i)
PKF cs is obtained by adding ( ¬ L) to PKF .(ii)
PKF cp is obtained by adding ( ¬ R) to PKF .(iii)
PKF S is obtained by adding ( GG ) to PKF .Let
PKF ⋆ ∈ { PKF , PKF cs , PKF cp , PKF S } . Then(iv) PKF ⋆ ↾ is obtained from PKF ⋆ by restricting ( IND ) on L PA -formulae.(v) PKF + ⋆ is obtained by extending PKF ⋆ with ( TI <ε ) . Since our formulation of
PKF deviates from the formulation in Halbach and Horsten (2006),we now show that
PKF behaves classically on the T -free fragment of L T and that ψ ( ~x ) and T p ψ ( ˙ ~x ) q are interderivable. Lemma 3.10.
Let
PKF ⋆ be one of the PKF -like theories introduced in Df. 3.9, ϕ ∈ L PA , and ψ ( ~x ) ∈ L T . Then(i) PKF ⋆ ⊢ Γ ⇒ ∆ , ϕ, ¬ ϕ and PKF ⋆ ⊢ ϕ, ¬ ϕ, Γ ⇒ ∆ ;(ii) PKF ⋆ ⊢ Γ ⇒ ∆ , ψ ( ~x ) iff PKF ⋆ ⊢ Γ ⇒ ∆ , T p ψ ( ˙ ~x ) q ;(iii) Unrestricted ( ¬ L) and ( ¬ R) are admissible for ϕ ∈ L PA .Proof. (i) and (ii) are shown by a straightforward induction on ϕ . For (iii), observe more generallythat, if ϕ, ¬ ϕ, Γ ⇒ ∆ and Γ ⇒ ∆ , ϕ, ¬ ϕ are both derivable, then ( ¬ L) and ( ¬ R) are derivedrules Γ ⇒ ∆ , ϕ ϕ, ¬ ϕ, Γ ⇒ ∆ Cut ¬ ϕ, Γ ⇒ ∆ Γ ⇒ ∆ , ¬ ϕ, ϕ ϕ, Γ ⇒ ∆ Cut Γ ⇒ ∆ , ¬ ϕ Finally, to complete the picture we note that contraposition is admissible in
PKF and
PKF S . Adding ¬ ϕ, Γ ⇒ ∆ for ϕ axiom of PA to PKF -like theories makes contraposition admissible in
PKF and
PKF S (see Lemma 3.11). emma 3.11 (Contraposition) . Contraposition, i.e. the rule Γ ⇒ ∆ ¬ ∆ ⇒ ¬ Γ is admissible in PKF ( PKF ↾ , PKF + ) and PKF S ( PKF S ↾ , PKF + S ).Proof. The proof is by induction on the height of derivations. We show two crucial cases involvingthe rules ( GG ) and (RepL). Suppose the derivation ends with D ψ, Γ ⇒ ∆ D ¬ ψ, Γ ⇒ ∆ GG ϕ, ¬ ϕ, Γ ⇒ ∆ We have i.h. ¬ ∆ ⇒ ¬ Γ , ¬ ψ i.h. ¬ ∆ ⇒ ¬ Γ , ¬¬ ψ Inv ¬ ∆ ⇒ ¬ Γ , ψ ϕ, ¬ ∆ ⇒ ¬ Γ , ϕ, ¬ ϕ ¬ ϕ, ¬ ∆ ⇒ ¬ Γ , ϕ, ¬ ϕ GG ψ, ¬ ψ, ¬ ∆ ⇒ ¬ Γ , ϕ, ¬ ϕ Cut ¬ ψ, ¬ ∆ ⇒ ¬ Γ , ϕ, ¬ ϕ Cut ¬ ∆ ⇒ ¬ Γ , ϕ, ¬ ϕ ¬¬ R ¬ ∆ ⇒ ¬ Γ , ¬ ϕ, ¬¬ ϕ If the derivations ends with D ϕ ( t ) , Γ ⇒ ∆ RepL s = t, ϕ ( s ) , Γ ⇒ ∆ we reason as follows. We first derive i.h. ¬ ∆ ⇒ ¬ Γ , ¬ ϕ ( t ) ¬ ϕ ( s ) , ¬ ∆ ⇒ ¬ Γ , ¬ ϕ ( s ) RepL s = t, ¬ ϕ ( t ) , ¬ ∆ ⇒ ¬ Γ , ¬ ϕ ( s ) Cut s = t, ¬ ∆ ⇒ ¬ Γ , ¬ ϕ ( s ) = ¬ R ¬ ∆ ⇒ ¬ Γ , ¬ ϕ ( s ) , ¬ ( s = t ) Notice that in
PKF we have invertibility of ( ¬¬ R). Reinhardt’s Challenge
In this section we address
Reinhardt’s Challenge and show that given a KF -like theory thereis a corresponding PKF -like theory such that the set of inferences T p Γ q ⇒ T p ∆ q provable inthe PKF -like theory coincides with the set of significant inferences of the KF -like theory. Thisobservation may be considered as a positive answer to Question b) discussed in the Introduction,that is, as providing an independent axiomatization of the significant inferences of the KF -liketheories. Moreover, by axiomatizing the set of significant inferences of KF -like theories we obtaina positive answer to the Generalized Reinhardt Problem as an immediate corollary. Moreprecisely, as shown in Proposition 4.18, every significant inference of a KF -like theory has asignificant derivation, i.e., whenever the theory proves T p Γ q ⇒ T p ∆ q , we can find a derivation D of T p Γ q ⇒ T p ∆ q such that every node of D is itself a significant inference.To begin with define the notion of significant inferences for arbitrary truth theories. Definition 4.1 (Significant Inferences) . Let Th be an axiomatic truth-theory formulated in L T ,and Γ , ∆ be finite sets of L T -sentences. The set of significant inferences of Th is defined as Th SI := {h Γ , ∆ i | Th ⊢ T p Γ q ⇒ T p ∆ q } . Since the truth predicate of
PKF -like theories is transparent (cf. Lemma 3.10(ii)) the signifi-cant inferences of any
PKF -like theory will simply amount to the set of provable inferences of thetheory. We also note in passing that the significant part of a truth theory ( Th S ) in the sense ofReinhardt (1985, 1986), that is the provably true sentences of the theory, can be retrieved from Th SI by setting Th S := { ϕ ∈ Sent | h∅ , ϕ i ∈ Th SI } . Let us now show that for each KF -like theory there is a PKF -like theory such that the provablesequents of the latter constitute exactly the significant inferences of the former.
Definition 4.2 ( PKF ◦ , KF ◦ ) . The pair ( PKF ◦ , KF ◦ ) is a variable ranging over the followingtheory-pairs ( PKF ↾ , KF ↾ ) , ( PKF cs ↾ , KF cs ↾ ) , ( PKF cp ↾ , KF cp ↾ ) , ( PKF S ↾ , KF S ↾ )( PKF , KF int ) , ( PKF cs , KF intcs ) , ( PKF cp , KF intcp ) , ( PKF S , KF intS )( PKF + , KF ) , ( PKF + cs , KF cs ) , ( PKF + cp , KF cp ) , ( PKF + S , KF S ) . Moreover, let Th ∈ { KF , KF int , KF ↾ , PKF + , PKF ↾ } . Then Th ⋆ ∈ { Th , Th S , Th cs , Th cp } . We can now start proving the principal result of this section, i.e., we can prove that
PKF ◦ = KFSI ◦ . We first show that PKF ◦ ⊆ KFSI ◦ . Proposition 4.3.
Let ( PKF ◦ , KF ◦ ) as defined in Definition 4.2. ThenIf PKF ◦ ⊢ Γ( ~x ) ⇒ ∆( ~y ) , then KF ◦ ⊢ T p Γ( ˙ ~x ) q ⇒ T p ∆( ˙ ~y ) q . Proposition 4.3 is essentially due to Halbach and Horsten (2006), Halbach and Nicolai (2018)and Nicolai (2018), who proved the claim for theories without index cs or cp , that is, for pairsof theories that do not assume the truth predicate to be consistent or complete. It thus sufficesto extend their result to these theories. 13 roof of Proposition 4.3. The proof is straightforward. For pairs extended with a consistencyprinciple, it suffices to show that the KF -theory “internalizes” the soundness of ( ¬ L). That is, itsuffices to show that, e.g., if KF cs ⊢ T p Γ q ⇒ T p ∆ q , T p ϕ q , then KF cs ⊢ T p ¬ ϕ q , T p Γ q ⇒ T p ∆ q .Symmetrically for theories extended with a completeness principle one needs to show that the KF -theory “internalizes” the soundness of ( ¬ R).In light of the definition of KF ◦ SI , Proposition 4.3 immediately yields that the provableinferences of PKF ◦ are a subset KF ◦ SI : Corollary 4.4.
PKF ◦ ⊆ KF ◦ SI . KF ◦ -significant inferences to PKF ◦ -provable sequents The proof of the converse directions of Proposition 4.3 and Corollary 4.4 constitutes the maintechnical contribution of this article. The basic idea underlying the proof is to show by inductionon the height of KF ◦ -derivations that if KF ◦ ⊢ T p Γ q ⇒ T p ∆ q then PKF ◦ ⊢ T p Γ q ⇒ T p ∆ q . Thiswill be shown by proving a stronger claim, i.e. it will be shown that, whenever Γ , ∆ contain onlyliterals KF ◦ ⊢ Γ ⇒ ∆ implies PKF ◦ ⊢ Γ + , ∆ − ⇒ Γ − , ∆ + where for Θ ∈ { Γ , ∆ } and At the set of atomic sentences, Θ + := { ϕ ∈ At | ϕ ∈ Θ } Θ − := { ϕ ∈ At | ¬ ϕ ∈ Θ } This transformation is motivated by (i) the fact that identity behaves classically in
PKF ◦ and (ii)the following semantic consideration: if a formula of the form ¬ T t is classically false (true), then
T t is either true (false) or both (neither) from the perspective of the non-classical theory of truth.As a consequence, moving
T t in the succedent (antecedent) of the sequent will not interfere withthe validity of the sequent from the perspective of the non-classical logics at stake. Moreover, if Γ ⇒ ∆ is of the form T p Γ ′ q ⇒ T p ∆ ′ q , the transformation leaves the sequent unaltered, hence,if we prove that for each KF ◦ -provable sequent its transformation is PKF ◦ -provable, we obtainour desired result as a corollary. Lemma 4.5 (Main Lemma) . For Γ , ∆ sets of literals, for all ⋆ KF int ⋆ ⊢ Γ ⇒ ∆ implies PKF ⋆ ⊢ Γ + , ∆ − ⇒ Γ − , ∆ + . Remark 4.6.
In the following proof we implicitly use the facts that • the replacement schema s = t, ϕ ( t ) , Γ ⇒ ∆ , ϕ ( s ) is derivable in PKF -like systems; • if KF int ⋆ ⊢ n Γ ⇒ ∆ , ¬∀ vϕ , then KF int ⋆ ⊢ n ∀ vϕ, Γ ⇒ ∆ , and if KF int ⋆ ⊢ n ¬∀ vϕ, Γ ⇒ ∆ , then KF int ⋆ ⊢ n Γ ⇒ ∆ , ∀ vϕ . Proof of Lemma 4.5.
The proof is by induction on the height n of KF int ⋆ -derivations. n = 0 Suppose Γ ⇒ ∆ is a KF int ⋆ -initial sequent. For the pair KF int - PKF , we first notice thatevery axiom of KF int is an axiom of PKF . Hence for initial sequents not involving negated atomicformulae, the proof is immediate. The only truth-theoretic axiom of KF int involving a negatedatomic formula is ( T ¬ = ). We obtain the desired conclusion by application of Lemma 3.10(iii),e.g. Notice that the use of rules for identity instead of identity axioms does not impact the arguments due toHalbach and Horsten (2006), Halbach and Nicolai (2018) and Nicolai (2018). In other words, ψ ∈ Θ − iff ¬ ψ ∈ Θ . Note that ¬∀ vϕ can be principal on the right (left) only if it has been derived via ¬ R ( ¬ L). lTerm ( x ) , ClTerm ( y ) , ¬ ( val ( x ) = val ( y )) , Γ + , ∆ − ⇒ Γ − , ∆ + , T ( ¬ . ( x = . y )) Lemma 3.10(iii)
ClTerm ( x ) , ClTerm ( y ) , Γ + , ∆ − ⇒ Γ − , ∆ + , T ( ¬ . ( x = . y )) , val ( x ) = val ( y ) Cons , Comp , GoG
As for theories with specific T -axioms, we want to show (we omit contextfor readability) PKF cp ⊢ Sent ( x ) ⇒ T ( x ) , T ( ¬ . x ) (1) PKF cs ⊢ Sent ( x ) , T ( x ) , T ( ¬ . x ) ⇒ (2) PKF S ⊢ Sent ( x ) , Sent ( y ) , T ( x ) , T ( ¬ . x ) ⇒ T ( y ) , T ( ¬ . y ) (3)For (1) Sent ( x ) , ¬ T ( x ) ⇒ T ( ¬ . x ) ¬ R Sent ( x ) ⇒ T ( ¬ . x ) , ¬¬ T ( x ) Sent ( x ) , T ( x ) ⇒ T ( x ) ¬¬ L Sent ( x ) , ¬¬ T ( x ) ⇒ T ( x ) Cut
Sent ( x ) ⇒ T ( x ) , T ( ¬ . x ) For (2) we have
Sent ( x ) , T ( x ) ⇒ T ( x ) ¬ L Sent ( x ) , T ( x ) , ¬ T ( x ) ⇒ Sent ( x ) , T ( ¬ . x ) ⇒ ¬ T ( x ) Cut
Sent ( x ) , T ( x ) , T ( ¬ . x ) ⇒ Finally, for (3)
Sent ( x ) , Sent ( y ) , T ( y ) ⇒ , T ( y ) ¬ T ( y ) Sent ( x ) , Sent ( y ) , ¬ T ( y ) ⇒ T ( y ) , ¬ T ( y ) GG Sent ( x ) , Sent ( y ) , T ( x ) , ¬ T ( x ) ⇒ T ( y ) , ¬ T ( y ) ¬ T ¬ Sent ( x ) , Sent ( y ) , T ( x ) , T ( ¬ . x ) ⇒ T ( y ) , T ( ¬ . y ) n = m + 1 Suppose Γ ⇒ ∆ has been derived. We distinguish two cases: either Γ ⇒ ∆ containsa principal formula, or it contains no principal formula. If the latter, then it has been derivedeither by (Ref), in which case we just apply i.h. and (Ref) in PKF ⋆ , or it has been derived byCut For this to work, it is crucial that we are dealing with quasi-normal derivations. In thiscase (Cut) is applied to literals only and we can just apply (Cut) in PKF ⋆ . If ϕ ∈ At , we have i.h. Γ + , ∆ − ⇒ Γ − , ∆ + , ϕ i.h. ϕ, Γ + , ∆ − ⇒ Γ − , ∆ + Cut Γ + , ∆ − ⇒ Γ − , ∆ + If ϕ ≡ ¬ ψ ∈ At − , where At − is the set of negated atomic formulae, then we use i.h. and cut on ψ .Now suppose Γ ⇒ ∆ contains a principal formula. The rules having literals as principal formulaeare the following: ( ¬ L), ( ¬ R), (RepL), (
IND int ), and truth-rules. ¬ L , ¬ R Immediate, e.g. suppose the KF int ⋆ -derivation ends with Γ ′ ⇒ ∆ , ϕ ¬ L ¬ ϕ, Γ ′ ⇒ ∆ ϕ ∈ At , then by induction we have PKF ⋆ ⊢ Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + , ϕ , which is our desiredconclusion. The case where ϕ ≡ ¬ ψ ∈ At − need not be taken into account, as we are dealingwith derivations containing only literals in the end-sequent.RepL Suppose the KF int ⋆ -derivation ends with ϕ ( t ) , Γ ′ ⇒ ∆ s = t, ϕ ( s ) , Γ ′ ⇒ ∆ Suppose first that ϕ ∈ At . For the pairs KF int - PKF , KF intcs - PKF cs , and KF intS - PKF S , we just applyi.h. and (RepL) in the PKF -variant. For the pair KF intcp - PKF cp , we reason in PKF cp as follows Rep Schema s = t, ϕ ( s ) , Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + , ϕ ( t ) i.h. ϕ ( t ) , Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + Cut s = t, ϕ ( s ) , Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + Now suppose ϕ ≡ ¬ ψ ∈ At − . We first reason in an arbitrary PKF ⋆ i.h. Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + , ψ ( t ) Rep Schema s = t, ψ ( t ) , Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + , ψ ( s ) Cut s = t, Γ ′ + , ∆ − ⇒ Γ ′− , ∆ + , ψ ( s ) IND int
Suppose the KF int ⋆ -derivation ends with Fml ( x ) , T ( x [ ˙ u/v ]) , Γ ⇒ ∆ , T ( x [ ˙ u ′ /v ]) IND int T ( x [ /v ]) , Γ ⇒ ∆ , T ( x [ ˙ z/v ]) with u eigenvariable and z arbtrary term. We use ( IND ) in
PKF ⋆ as follows i.h Fml ( x ) , T ( x [ ˙ u/v ]) , Γ + , ∆ − ⇒ Γ − , ∆ + , T ( x [ ˙ u ′ /v ]) IND T ( x [ /v ]) , Γ + , ∆ − ⇒ Γ − , ∆ + , T ( x [ ˙ z/v ]) Logical rules for ∧ and ∀ need not be taken into account as they cannot be the last rule of aquasi-normal derivation D that has in the end-sequent. We end the proof by dealing with truth-rules. We first note that, with the exception of truth-rules for the universal quantifier, all othertruth-rules have only literals as active formulae. Hence, it suffices to apply i.h. and the rule itselfin PKF ⋆ . As for rules involving ∀ , we exploit height-preserving inversion (see Proposition 3.7).For example T all R Suppose the derivation end with Γ ⇒ ∆ ′ , ∀ zT ( x [ ˙ z/v ]) T nall R Γ ⇒ ∆ ′ , T ( ∀ . v.x ) By inversion, from KF int ⋆ ⊢ n Γ ⇒ ∆ , ∀ zT ( x [ ˙ z/v ]) , we get KF int ⋆ ⊢ n Γ ⇒ ∆ ′ , T x [ ˙ u/v ] for u eigenvariable. Using i.h., we then reason in PKF ⋆ as follows16 .h. Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , T x [ ˙ u/v ] ∀ R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , ∀ zT x [ ˙ u/v ] T all R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + T ( ∀ . v.x ) A similar argument works for T all L . T nall R Suppose the derivation ends with Γ ⇒ ∆ ′ , ¬∀ z ¬ T ( ¬ . x [ z/v ]) T nall R Γ ⇒ ∆ ′ , T ( ¬ . ∀ . v.x ) From KF int ⋆ ⊢ n Γ ⇒ ∆ ′ , ¬∀ z ¬ T ( ¬ . x [ ˙ z/v ]) , we get KF int ⋆ ⊢ n ∀ z ¬ T ( ¬ . x [ ˙ z/v ]) , Γ ⇒ ∆ ′ and byinversion we get KF int ⋆ ⊢ n ¬ T ( ¬ . x [ ˙ y/v ]) , Γ ⇒ ∆ ′ , for some y . Using i.h., we then reason in PKF ⋆ as follows Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , T ( ¬ . x [ ˙ y/v ]) ¬¬ R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , ¬¬ T ( ¬ . x [ ˙ y/v ]) ¬∀ R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , ¬∀ z ¬ T ( ¬ . x [ ˙ z/v ]) T nall R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , T ( ¬ . ∀ . v.x ) By inspecting the proof of the Main Lemma it is immediate that we can lift the lemma tothe pair KF int ⋆ ↾ and PKF ⋆ ↾ , that is, we obtain the following Corollary 4.7.
For Γ , ∆ sets of literals, for all ⋆ KF int ⋆ ↾ ⊢ Γ ⇒ ∆ implies PKF ⋆ ↾ ⊢ Γ + , ∆ − ⇒ Γ − , ∆ + . We also note that the Lemma 4.5 can be formalized in
PKF ⋆ , which will prove importantwhen we consider KF -systems with full induction. The relation KF int ⋆ ⊢ nρ Γ ⇒ ∆ expressing that Γ ⇒ ∆ is derivable in KF int ⋆ with a derivation of height ≤ n and cut rank ≤ ρ can be canonicallyrepresented in arithmetic via a recursively enumerable predicate Bew KF int ⋆ ( n , r , p Γ ⇒ ∆ q ) . Corollary 4.8.
For Γ , ∆ sets of literals, for all ⋆ PKF ⋆ ⊢ Bew KF int ⋆ ( n , , p Γ ⇒ ∆ q ) implies PKF ⋆ ⊢ Γ + , ∆ − ⇒ Γ − , ∆ + . We have shown how to transform provable sequents in KF -systems with restricted or internalinduction into provable sequents of appropriate PKF -like theories. It remains to show the claimfor KF -like theories with full induction. For these theories, cut elimination fails. However, itis well known that axioms and rules KF -like theories can be derived in a sequent calulus withinfinitary rules replacing the schema of induction. Hence we can take a detour via the infinitarysystem KF ∞ , which contains KF and enjoys partial cut elimination. The key observation thatmakes this detour possible is that for KF ∞ quasi-normal derivations up to height ε the strategyemployed in our Main lemma (Lemma 4.5) can be used to provide suitable PKF + -derivations.This proves sufficient for lifting Lemma 4.5 to the KF -theories with full induction. The techniquefor embedding of KF in KF ∞ is well known, however, we give a succinct presentation of KF ∞ forthe sake of completeness. See, for instance, Takeuti (2013) or Rathjen and Sieg (2018). efinition 4.9 ( PA ω ) . The language of the infinitary sequent-style version of PA , PA ω , isobtained by omitting free variables. The axioms of PA ω are ∅ ⇒ ϕ if ϕ is a true atomic sentence; ϕ ⇒ ∅ if ϕ is a false atomic sentence; ϕ ( t ) ⇒ ϕ ( s ) if ϕ is an atomic sentence and t i and s i evaluate to the same numeral. PA ω has all the inference rules of PA except for ( ∀ R ) and ( IND) . Instead, it has the ω -rule Γ ⇒ ∆ , ϕ ( t ) Γ ⇒ ∆ , ϕ ( t ) . . . Γ ⇒ ∆ , ϕ ( t n ) . . . ω Γ ⇒ ∆ , ∀ vϕ Definition 4.10 ( KF ∞ ⋆ ) . KF ∞ ⋆ extends PA ω with truth-theoretic axioms and truth-theoretic rulesof KF ⋆ (formulated in the new language). KF ∞ -derivations, due to ( ω ), are possibly infinite. Notions about derivations introducedabove, including height and cut rank, carry over without modifications. In particular, KF ∞ -derivations are well-founded trees, where at each node there is either the root, or instances ofaxioms, or there is a 1-fold branching (corresponding to unary rules), or a two fold branching(corresponding to binary rules), or an ω -fold branching (corresponding to the ω -rule). Note thatif KF ∞ ⋆ ⊢ α Γ ⇒ ∆ , ∀ vϕ , then KF ∞ ⋆ ⊢ α Γ , ϕ ( t ) for any closed term t .Every KF ∞ -derivation can be transformed into a KF ∞ -derivation with height < ω and finitecut rank. It is also well-known that the cost of lowering the cut-rank from k + 1 to k isexponential with base ω , that is if PA ω ⊢ αk +1 Γ ⇒ ∆ , then PA ω ⊢ ω α k Γ ⇒ ∆ . It follows thatevery KF ∞ -derivation of height α < ε and cut rank m can be transformed into a quasi-normalderivation of height ϕ m α , where ϕ m α stands for m iterations of the Veblen function ϕ on α . We can thus restrict our attention to KF ∞ -derivations of finite cut rank and length < ε .These derivations can be primitive recursively encoded by natural numbers, and the codes willcontain information about the derivation. In particular, if u codes a derivation D we canprimitive recursively read off from u a bound for the length of D and a bound for its cut rank;additionally, we can read off the name of: the last inference, its principal/side formulae and itsconclusion. This enables us to find a predicate, say Bew ∞ ( a, r , p Γ ⇒ ∆ q ) , expressing the relation KF ∞ ⊢ αρ Γ ⇒ ∆ , i.e., Γ ⇒ ∆ is derivable in KF ∞ with a derivation of length ≤ α and cut rank ≤ ρ . Due to the amount of transfinite induction available in PKF + the embedding of KF in KF ∞ and partial cut-elimination for KF ∞ can be formalized in PKF + . Lemma 4.11.
Let Γ , ∆ ⊆ L T . Then for all ⋆ (i) For all n, r ∈ ω , PKF + ⋆ ⊢ Bew KF ⋆ ( n , r , p Γ ⇒ ∆ q ) → Bew ∞ ( ω , r , p Γ ⇒ ∆ q ) .(ii) For α < ε , PKF + ⋆ ⊢ Bew ∞ ( a, r , p Γ ⇒ ∆ q ) → Bew ∞ ( ϕ r a, , p Γ ⇒ ∆ q ) . Lemma 4.12.
For Γ , ∆ sets of literals, for all ⋆ , and for all α < ε , PKF + ⋆ ⊢ Bew ∞ ( a, , p Γ ⇒ ∆ q ) implies PKF + ⋆ ⊢ Γ + , ∆ − ⇒ Γ + , ∆ − . Proof.
The proof is by transfinite induction on α < ε and to a large extent a formalization ofthe proof of Lemma 4.5. For α = 0 it suffices indeed to formalize the proof of Lemma 4.5. Thecase where α is a limit ordinal involves an application of the ω -rule. These cases need not be For sake of readability we omit ⋆ -index for the KF ∞ theories. Recall the well-known identities ϕ α = ω α and ϕ ε . See Schwichtenberg (1977, §4.2.2) for details. α = β + 1 , the crucial cases are thetruth-rules involving quantifiers, as the remaining cases are again immediate by formalizing theproof of Lemma 4.5. We discuss ( T all R)—the other rules can be treated along the same lines.Suppose the KF ∞ -derivation ends with ⊢ β Γ ⇒ ∆ ′ , ∀ zT ( s [ ˙ z/v ]) T all R ⊢ α Γ ⇒ ∆ ′ , T ( ∀ . v.s ) By inversion on the upper sequent we get, for all closed terms t KF ∞ ⊢ β Γ ⇒ ∆ ′ , T ( s [ num ( t ) /v ]) . This can be formalized within
PKF + , that is, PKF + ⊢ ∀ y ( ¬ ClTerm ( y ) ∨ Bew ∞ ( b, , p Γ ⇒ ∆ ′ , T ( s [ ˙ y/v ]) q )) Now, let u be an eigenparameter and recall that PA ⊢ ClTerm ( ˙ x ) . Then PA ⇒ ClTerm ( ˙ u ) ⇒ ¬ ClTerm ( ˙ u ) , Bew ∞ ( b, , p Γ ⇒ ∆ ′ , T ( s [ ˙ u/v ]) q ) Lemma 3.10(iii)
ClTerm ( ˙ u ) ⇒ Bew ∞ ( b, , p Γ ⇒ ∆ ′ , T ( s [ ˙ u/v ]) q ) Cut ⇒ Bew ∞ ( b, , p Γ ⇒ ∆ ′ , T ( s [ ˙ u/v ]) q ) We apply the induction hypothesis and reason in
PKF + ⋆ as follows i.h. Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , T ( s [ ˙ u/v ]) ∀ R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , ∀ yT ( s [ ˙ y/v ]) T all R Γ + , ∆ ′− ⇒ Γ − , ∆ ′ + , T ( ∀ . v.s ) Since, as mentioned, KF ⋆ can be embedded in KF ∞ without exceeding quasi-normal derivationof length < ε , we obtain: Corollary 4.13.
For Γ , ∆ sets of literals, for all ⋆ KF ⋆ ⊢ Γ ⇒ ∆ implies PKF + ⋆ ⊢ Γ + , ∆ − ⇒ Γ − , ∆ + . This immediately yields that the set of significant inferences of KF -like theories is containedin the set of provable sequents of appropriate PKF -like theories.
Lemma 4.14. KF ◦ SI ⊆ PKF ◦ Proof. If KF ◦ ⊢ T p Γ q ⇒ T p ∆ q , then PKF ◦ ⊢ T p Γ q ⇒ T p ∆ q by Lemma 4.5 and Corollaries 4.7,4.13. But the truth predicate of PKF ◦ is transparent and thus PKF ◦ ⊢ Γ ⇒ ∆ .Lemma 4.14 in combination with Corollary 4.4 show that PKF ◦ yields precisely the significantsentence of KF ◦ . In other words we have answered Question b) of Reinhardt’s Challenge . Proposition 4.15 ( Reinhardt’s Challenge) . PKF ◦ = KF ◦ SI . Notice that
PKF + ⊢ ∀ vϕ, Γ ⇒ ∆ , ϕ ( t ) for arbitrary t . Generalized Reinhardt Problem we remark thatan answer to Reinhardt’s Question b) yields an answer to Question a) as a corollary, that is,our result subsumes the results provided by Halbach and Horsten (2006), Halbach and Nicolai(2018), and Nicolai (2018).
Corollary 4.16. { ϕ ∈ Sent L T | PKF ◦ ⊢ ϕ } = KF ◦ S . Of course, this result also implies that by means of a very simple observation connectingprovably true KF -sequents to provable PKF -sequents we have reduced questions regarding theproof-theoretic strength of
PKF -like theories to questions regarding the proof-theoretic strengthof KF -like theories. Corollary 4.17. KF ◦ and PKF ◦ are proof-theoretically equivalent, i.e., KF ◦ [[ PA ]] ≡ PKF ◦ [[ PA ]] . We promised to give a positive answer to the
Generalized Reinhardt Problem , that is, thequestion whether for any pair h Γ , ∆ i ∈ KF ◦ SI there is a KF ◦ -derivation such that each node ofthe derivation tree is a member of KF ◦ SI . It is now time to make good on our promise. Makingessential use of Proposition 4.15 we show that if h Γ , ∆ i ∈ KF ◦ SI , then there is a KF ◦ -derivationsuch that each node of the derivation tree is a PKF ◦ -provable sequent. This implies the desiredconclusion in virtue of Proposition 4.15. Proposition 4.18 ( Generalized Reinhardt Problem) . If h Γ , ∆ i ∈ KF ◦ SI , then there is a KF ◦ -derivation D of Γ ⇒ ∆ such that for each node d of D , d ∈ KF ◦ SI .Proof. If h Γ , ∆ i ∈ KF ◦ SI , then by Proposition 4.15 PKF ◦ ⊢ Γ ⇒ ∆ and hence PKF ◦ ⊢ T p Γ q ⇒ T p ∆ q . Now let D ′ be an arbitrary KF int ⋆ -derivation of T p Γ q ⇒ T p ∆ q , e.g. δ Γ ⇒ ∆ δ Γ ⇒ ∆ R Γ ⇒ ∆ δ T p Γ q ⇒ T p ∆ q Let d . . . d k − be the nodes of D ′ . In order to obtain D , it suffices to replace each d i with T p Γ q | d i | T p ∆ q , i.e.: T p Γ q | δ | T p ∆ q T p Γ q Γ ⇒ ∆ , T p ∆ q T p Γ q | δ | T p ∆ q T p Γ q , Γ ⇒ ∆ , T p ∆ q R T p Γ q , Γ ⇒ ∆ , T p ∆ q T p Γ q | δ | T p ∆ q T p Γ q ⇒ T p ∆ q KF ◦ is closed under weakening, D is a KF ◦ -derivation of T p Γ q ⇒ T p ∆ q . But every node T p Γ q , Γ i ⇒ ∆ i , T p ∆ q of D is derivable in PKF ◦ = KF ◦ SI .It may good to put our answer to the Generalized Reinhardt Problem in perspective:we have shown that for every significant inference there is a way to classically derive the sequentsuch that every node of the derivation is itself a significant inference and hence that every nodeof the proof is acceptable to the significant reasoner, i.e., the non-classical logician. This does notimply that the non-classical logician can follow the classical reasoning, i.e., that the KF ◦ -proofis also a PKF ◦ -proof. Our result only shows that the classical reasoning is acceptable to thenon-classical logician and that KF -style theories can be used instrumentally. It does not showthat one can always reason non-classically in KF ◦ . But if the latter were the case, it seemsthat KF ◦ would deliver an independent axiomatization of its significant part in its own right.Surely—while it is certainly an interesting question whether for every KF ◦ -significant inferencethere is a KF ◦ -derivation, which is also a PKF ◦ -derivation—, such a result is not required for aninstrumental interpretation of KF ◦ and left for future research. In this paper we had a fresh look at Reinhardt’s program and proposed to focus on the provablytrue inferences of KF -like theories rather than the provably true sentences only. We showed thatif we conceive of the significant part of KF -like theories as the set of provably true inferencesthen we can remain within the significant part of the theory in proving its significant inferences.This answers the Generalized Reinhardt Problem and also shows that we need not stepoutside the significant part of KF in proving theorems of the form T p ϕ q , which was the contentof the original formulation of Reinhardt’s Problem . From the perspective of KF -derivationsthe use of the nonsignificant part of KF is hence dispensable and in this sense an instrumentalistinterpretation of KF is certainly available. However, should we conclude that we have justifiedthe use of nonsignificant sentences entirely within the framework of significant sentences?One may think that to answer the latter question affirmatively an independent characteriza-tion of the significant part of KF needs to be provided and this is precisely the content of Rein-hardt’s challenge . Building on results by Halbach and Horsten (2006); Halbach and Nicolai(2018) and Nicolai (2018) we have shown how to provide axiomatizations of the significant partof KF -like theories in non-classical logic. The only remaining question is whether these axioma-tizations are fully independent. We take it that there is no doubt in this respect concerning theaxiomatizations of the significant part of KF -like theories with internal or restricted induction.Turning to KF -like theories with full induction the crucial question is whether the rule ( TI <ε )is available from within the significant framework. Ultimately, an answer to this question willdepend on the role the theory of truth is supposed to play within one’s theoretical framework. If,for instance, one takes the theory to play an important role in the foundations of mathematicsand, for instance, to play a role in singling out the limits of predicativity (cf. Feferman, 1991),then one should arguably refrain from thinking that ( TI <ε ) can be assumed without furtherjustification from within the significant framework. But, to the contrary, if the theory of truthis to play no role in the foundations of mathematics and classical mathematical theorizing isfreely available from within the significant framework, then it is hard to see why ( TI <ε ) shouldnot be considered as fully justified from within the significant perspective. In this case it wouldseem that Reinhardt’s program needs to be deemed successful. However, a proper philosophi-cal assessment of the rule of transfinite induction up to ε from the perspective of Reinhardt’sprogram is beyond the scope of the paper. Nonetheless, as we hope to have established in thispaper—contra Halbach and Horsten (2006) and Halbach and Nicolai (2018)—there is no major21echnical obstacle preventing the success of Reinhardt’s program and, in this sense, Reinhardtwas certainly right in claiming that “the chances of success in this context (. . . ) are somewhatbetter than in Hilbert’s context” (Reinhardt, 1986, p.225). Funding acknowledgement
Luca Castaldo’s research was supported by the AHRC South, West and Wales Doctoral TrainingPartnership (SWW DTP). Johannes Stern’s research is funded by the ERC Starting GrantTRUST 803684. We thank Carlo Nicolai for very helpful comments and, in particular, forpointing to a substantial simplification of our proof of Proposition 4.15.
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