    Lewisian Fixed Points I: Two Incomparable Constructions

Our paper is the first study of what one might call "reverse mathematics of explicit fixpoints". We study two methods of constructing such fixpoints for formulas whose principal connective is the intuitionistic Lewis arrow. Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allows several natural semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a well-understood modal system. The second one by de Jongh and Sambin, seemingly simpler, leads to a modal theory that proves harder to axiomatize in an elegant way. Apart from showing that both theories are incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixpoints of simple formulas. We also show that they are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfer to its finite extensions.

Authors

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1 Introduction

Provability logic studies propositional and algebraic aspects of arithmetical theories, their provability predicates and reflection principles. Thanks to Solovay’s arithmetical completeness result [solo:prov76], we know that the provability predicate of Peano Arithmetic [smor:self85, bool:emer91, Boolos1993, lind:prov96, japa:logi98, svej:prov00, arte:prov04, halb:henk14] yields precisely the famous system , also known as the (Gödel-)Löb logic, obtained from the minimal unimodal normal logic by adding the principle . One of the most important facts about  is that it allows definability of explicit fixpoints. That is, given any polynomial where all occurrences of are guarded by , one can use de Jongh-Sambin algorithm to compute a formula not involving and ; [inline,marginclue]we should actually state what stands for! furthermore, thus computed is unique up to propositional equivalence (de Jongh, Sambin [samb:effe76], Bernardi [bern:uniq76]). Actually,  is obtained precisely as the smallest extension of (i.e., the logic of the transitivity axiom ) in which guarded fixpoints are definable. This follows immediately from the fact that, by Löb’s argument, Löb’s principle is entailed by the presence of guarded fixed points in combination with the de Jongh-Sambin result.

This result encodes the algebraic content of the Löb Theorem and Gödel’s Second Incompleteness Theorem. The modal analysis gives us the conceptual resources to say that the consistency statement is the explicit form of the Gödel sentence. More mundanely, it can be seen as elimination of fixpoint operator. The original statement is restricted to guarded fixpoints, but one can indeed extend this classical result to elimination of positive fixpoints of ordinary -calculus [bent:moda06, viss:lobs05] and further beyond (see § LABEL:sec:conclusions). Given the Kripkean meaning of  as the logic of Noetherian (conversely well-founded) transitive frames, such results in turn have found applications, e.g., in characterizing expressivity of XPath fragments [CateFL10:jancl, § 3.1]. The Sambin-de Jongh result has inspired Nakano’s seminal work on modality for recursion [Nakano00:lics, § 7], [Nakano01:tacs, § 5]. Last, but definitely not the least, it can be used to prove other metaresults about , such as the Beth definability property, as observed first by Maximova [Maksimova1989, ArecesHJ98, hoog:defi01, iemh:prop05] (cf. § LABEL:sec:beth).

What happens when we broaden the investigation beyond the classical base and unary provability  ? Regarding the former restriction, already Sambin’s 1976 paper [samb:effe76] noted that the fixpoint theorem works over intuitionistic propositional calculus (IPC). Unfortunately, despite decades of efforts [viss:prop94, Iemhoff01:phd, iemh:moda01, viss:close08, arde:sigm14], there is no known axiomatization of the provability logic of Heyting Arithmetic () and related systems; it is a system much stronger than intuitionistic , including principles such as

 \leavevmode\hbox to9.59pt{\vbox to9.59pt{\pgfpicture% \makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{pgfstrokecolor}{% rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}\pgfsys@color@rgb@fill{0}{0}{0}{% }\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox to 0.0pt{\pgfsys@beginscope{}% \pgfsys@setlinewidth{0.43pt}{}{ {}{} {}{}{}{}{}{}{}\pgfsys@moveto{\text{-}2.580011pt}{\text{-}2.580011pt}\pgfsys@moveto{\text{-}2.580011% pt}{\text{-}2.580011pt}\pgfsys@lineto{\text{-}2.580011pt}{2.580011pt}\pgfsys@lineto{2.580011% pt}{2.580011pt}\pgfsys@lineto{2.580011pt}{\text{-}2.580011pt}\pgfsys@closepath% \pgfsys@moveto{2.580011pt}{2.580011pt}\pgfsys@stroke\pgfsys@invoke{ } } {}{}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss% }\endpgfpicture}}(¬¬\leavevmode\hbox to9.59pt{% \vbox to9.59pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{\pgfsys@beginscope{}\pgfsys@setlinewidth{0.43pt}{}{ {}{} {}{}{}{}{}{}{}\pgfsys@moveto{\text{-}2.580011pt}{\text{-}2.580011pt}\pgfsys@moveto{\text{-}2.580011% pt}{\text{-}2.580011pt}\pgfsys@lineto{\text{-}2.580011pt}{2.580011pt}\pgfsys@lineto{2.580011% pt}{2.580011pt}\pgfsys@lineto{2.580011pt}{\text{-}2.580011pt}\pgfsys@closepath% \pgfsys@moveto{2.580011pt}{2.580011pt}\pgfsys@stroke\pgfsys@invoke{ } } {}{}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss% }\endpgfpicture}}φ→\leavevmode\hbox to9.59pt{% \vbox to9.59pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{\pgfsys@beginscope{}\pgfsys@setlinewidth{0.43pt}{}{ {}{} {}{}{}{}{}{}{}\pgfsys@moveto{\text{-}2.580011pt}{\text{-}2.580011pt}\pgfsys@moveto{\text{-}2.580011% pt}{\text{-}2.580011pt}\pgfsys@lineto{\text{-}2.580011pt}{2.580011pt}\pgfsys@lineto{2.580011% pt}{2.580011pt}\pgfsys@lineto{2.580011pt}{\text{-}2.580011pt}\pgfsys@closepath% \pgfsys@moveto{2.580011pt}{2.580011pt}\pgfsys@stroke\pgfsys@invoke{ } } {}{}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss% }\endpgfpicture}}φ)→\leavevmode\hbox to9.59pt{% \vbox to9.59pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}% \definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{\pgfsys@beginscope{}\pgfsys@setlinewidth{0.43pt}{}{ {}{} {}{}{}{}{}{}{}\pgfsys@moveto{\text{-}2.580011pt}{\text{-}2.580011pt}\pgfsys@moveto{\text{-}2.580011% pt}{\text{-}2.580011pt}\pgfsys@lineto{\text{-}2.580011pt}{2.580011pt}\pgfsys@lineto{2.580011% pt}{2.580011pt}\pgfsys@lineto{2.580011pt}{\text{-}2.580011pt}\pgfsys@closepath% \pgfsys@moveto{2.580011pt}{2.580011pt}\pgfsys@stroke\pgfsys@invoke{ } } {}{}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss% }\endpgfpicture}}\leavevmode\hbox to9.59pt{\vbox to9.59% pt{\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope{}\definecolor{% pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}{}% \pgfsys@color@rgb@fill{0}{0}{0}{}\pgfsys@setlinewidth{0.4pt}{}\nullfont\hbox t% o 0.0pt{\pgfsys@beginscope{}\pgfsys@setlinewidth{0.43pt}{}{ {}{} {}{}{}{}{}{}{}\pgfsys@moveto{\text{-}2.580011pt}{\text{-}2.580011pt}\pgfsys@moveto{\text{-}2.580011% pt}{\text{-}2.580011pt}\pgfsys@lineto{\text{-}2.580011pt}{2.580011pt}\pgfsys@lineto{2.580011% pt}{2.580011pt}\pgfsys@lineto{2.580011pt}{\text{-}2.580011pt}\pgfsys@closepath% \pgfsys@moveto{2.580011pt}{2.580011pt}\pgfsys@stroke\pgfsys@invoke{ } } {}{}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath{}{}{}{}\pgfsys@endscope\hss% }\endpgfpicture}}φ

underivable even in classical , as classically it implies (cf. [LitakV18:im, §5.3] for more examples). The algebraic core of a weak theory can include powerful schemes refutable in a stronger theory enjoying a different provability predicate. This phenomenon is often caused, e.g., by the fact that the weaker theory is closed under some translation method, whereas the stronger one is not.

Allowing non-unary connectives opens up vast new landscapes, especially in the constructive setting. In our paper [LitakV18:im], apart from providing a general framework of schematic logics [LitakV18:im, § 5.1], we have made the case for the constructive strict implication , also called the Lewis arrow. It allows defining as . We list its arithmetical interpretations in § LABEL:sec:arint. The most important one in the study of metatheory of HA is provided by -preservativity for a theory , where is a class of sentences, most commonly taken to be [viss:eval85, viss:prop94, viss:subs02, iemh:pres03, iemh:prop05]. Other ones include contraposed conservativity/interpretability [bera:inte90, shav:rela88, japa:logi98, viss:over98, arte:prov04] [LitakV18:im, § C.3], the logic of admissible schemes or the -provability interpretation. See § LABEL:sec:arint for details. Almost needless to say, constructive arithmetic can be replaced by any other foundational theory rich enough for standard encodings of syntactic notions. There are also computational interpretations originating elsewhere, such as Hughes “classical” arrows in functional programming [Hughes00:scp, LindleyWY08:msfp] (cf. [LitakV18:im, § 7.1]). All these interpretations extend the base system introduced in § 2. Interestingly enough, not all of them validate principles like (cf. § 2) which hold in the standard Kripke semantics of (cf. [LitakV18:im, § 3] and § 3.2). But what axioms do we need to ensure fixpoint results, and how can we compute these explicit fixpoints?

1.1 Our Contributions

[inline,marginclue]We still need to credit IDZ properly.

The classical construction of explicit guarded fixed points proceeds in two stages. One first proves the result for formulas where the main connective is the modal operator and then one shows how to extend the result to all modalized formulas (and possibly beyond that). Our results directly concern the first step. See § LABEL:sec:conclusions for a brief discussion of the second one.

We have two known paradigms for such a construction. First, there is the original de Jongh-Sambin construction (see § LABEL:sec:js) as generalized by Smoryński [smor:self85, Ch. 4]. Secondly, there is the construction given by de Jongh and Visser [dejo:expl91] for the interpretability logic IL (see § LABEL:iglamjv). As it simplifies to the de Jongh-Sambin construction when one adds the principle W (see Figure LABEL:tab:compl), it seemed the master construction. Our results show that de Jongh-Visser construction and the de Jongh-Sambin construction are mutually incomparable. The incomparability result also holds in the classical case.

Our paper is the first study of what one might call “reverse mathematic of explicit fixpoints”. After discussing algebraic and Kripke semantics for extensions of (§ 3), we investigate the effect of adding explicit schemes stating that a given method (de Jongh-Visser or de Jongh-Sambin) indeed yields fixpoints of formulas whose principal connective is . This, however, requires a significant prerequisite: we noted above that the validity of a scheme in the logic (algebraic core) of a given arithmetical theory (§ LABEL:sec:arint) does not need to transfer to the logic of some given finite extension. As we show in a companion paper [tlav19subf], this holds if the base logic enjoys the property of extension stability. After recalling this information (§ LABEL:sec:exsta), we show that the minimal theory in which the de Jongh-Visser construction works is the theory (§ LABEL:iglamjv), which is extension stable. Thus, for the de Jongh-Visser construction we have a precise analogue of Löb’s Logic. In § LABEL:sec:js we show that the case of the de Jongh-Sambin construction is more complex and show the incomparability of its theory (, which also turns out to be extension stable) with . In § LABEL:sec:join we axiomatize the join of both theories. In § LABEL:sec:jscloser we investigate several subtheories of , whose axioms are obtained as de Jongh-Sambin fixpoints of simple formulas. A large part of our results on axiomatizing explicit fixpoints is concisely summarized by Figure LABEL:fig:logics therein. In § LABEL:sec:corresp we present Kripke semantics for some principles investigated in earlier sections and uncover a simple nonconservativity phenomenon.

2 Basics

Our basic system is in the language of intuitionistic propositional calculus (IPC) extended with a binary connective . We write for . The system is given by the following axioms:

prop

axioms and rules for IPC

Tr

We take and for . One can easily derive the intuitionistic version of the classical system K (without ) for the -language from . The system  extends  with

Di

A (-)logic is an extension of  that is closed under modus ponens, necessitation and substitution. Let be a set of formulas. We write for the closure of under modus ponens and necessitation. Note that is not automatically a logic. On the other hand, if is closed under substitution, then so is .

Remark 2.1.

All theorems we claim for  also hold when we omit disjunction from the language, in the sense that we still have all schemes, where the interpretations of the schematic letters are restricted to disjunction-free formulas. Our proofs also work in the disjunction-free setting.

Theorems LABEL:ququatao and LABEL:iamququatao illustrate that is sometimes needed to derive principles not involving . A similar example is provided by the trivialization of in [LitakV18:im, Lemma 4.6]. In the latter case, we know we need Di to make the argument work since the classical interpretability logic IL does not trivialize.    ❍

At some points, we will use a convenient notation for substitution. Suppose a variable, say , of substitution is given in the context. We will write for . We note that is equal to . So we may write .

Lemma 2.2.

Let a designated variable of substitution be given. We have:

Sub1

.

Sub2

Suppose every occurrence of is in the scope of an occurrence in . We have:
.

Proof.

[apptwocalc.tex]sub1sub2    ❑

Lemma 2.3.

(Uniqueness of fixpoints) Fix a designated variable of substitution . Suppose every occurrence of is in the scope of an occurrence in . Suppose does not occur in . We have:

1. .

2. .

3. .

Proof.

[apptwocalc.tex]brilsmurf    ❑

Remark 2.4.

We find a closely related development in Craig Smoryński’s [smor:self85, Chapter 4]. We note that our development is constructive and studies a binary operator, where Smoryński works classically and studies a unary operator. Moreover, since Smoryński aims at a version of the de Jongh-Sambin Theorem, he imposes an additional axiom written (in an adjusted notation) as .   ❍

3 Semantics

Most proofs in this paper are of purely syntactic nature. Nevertheless, we occasionally still need semantics, e.g., for non-derivability results and for better understanding of syntactic systems and notions studied below. For our purposes, it is enough to consider algebraic semantics (§ 3.1) and Kripke semantics (§ 3.2).

3.1 Algebraic Semantics

We briefly recapitulate algebraic semantics as discussed in our companion paper [tlav19subf]. A generic algebraic completeness result after the manner of Lindenbaum and Tarski is obtainable for almost any “natural” logic and extensions of are no exception. We can in fact put it in a general setting: with their The “global consequence relation” of any extension of including both Modus Ponens and is easily seen to be algebraizable [BlokP89:ams, FontJP03a:sl], in fact an instance of what Rasiowa calls an implicative logic [Rasiowa74:aatnl, Font06:sl]. More explicitly, the algebraic semantics looks as follows:

Definition 3.1.

A -algebra[inline,marginclue]a good name? or -algebra is a tuple , where

• (the intuitionistic/Heyting reduct of ) is a Heyting algebra and

• (the strict reduct of ) satisfies

,

,

.

Moreover, is a normalized [inline,marginclue]other names? “fully normal”? “additive”? “-Kripke”? or -algebra if its strict reduct is a weakly Heyting algebra [CelaniJ05:mlq], i.e., it satisfies in addition

.

, , , and are called by Celani and Jansana [CelaniJ05:mlq]  – , respectively. Denote the equational class of -algebras by (haes standing for “Heyting Algebra Expansions”) and the class of normalized ones by s.

[inline,marginclue]Also note that normalized ones are instances of setting by Palmigiano et al?

A valuation in as usual maps propositional atoms to elements of and is inductively extended to defined on all formulas in the obvious way. Write if , if for all , if for every , and if for every . Given any and any set of formulas define

 Mod(Λ) :={A∣∀φ∈Λ.A⊩φ}, Th(K) :={φ∣∀A∈K.A⊩φ}.
Theorem 3.2 (Algebraic Completeness).
• For any , is a logic.

• For any logic , .

Proof.

See our companion paper [tlav19subf] or apply techniques of abstract algebraic logic (AAL) as discussed in standard references [BlokP89:ams, FontJP03a:sl, Rasiowa74:aatnl, Font06:sl].    ❑

3.2 Kripke Semantics

We briefly recapitulate basic information from our overview [LitakV18:im]. A (Lewisian) Kripke frame is a triple , where is a partial order on , and furthermore

-p

if , then   (i.e., ).

Admissible valuations then map propositional atoms to -closed sets and intuitionistic connectives are interpreted as usual using . The clause for in a model is

 k⊩φ\strictifψ if, for all ℓ⊐k, if ℓ⊩φ, then ℓ⊩ψ. (1)

Given such a Kripke frame, it is an easy exercise to define its dual algebra whose Heyting reduct is given as usual by , i.e., upsets of , and the strict reduct (interpretation of ) is induced by (1). The dual algebra is normalized, i.e., the equality corresponding to Di holds. In other words, extensions of which are not extensions of can only be Kripke sound, but not Kripke complete. A fuller discussion of Di and other principles in the Kripke setting (written without employing explicitly algebraic language) can be found in our paper [LitakV18:im]. Definitions of notions such as the finite model property (fmp), i.e., completeness wrt a finite class of frames are standard.

Some modal axioms that we need together with their correspondence conditions are given in Figure LABEL:tab:compl. Its fuller version can be found in [LitakV18:im, § 6], along with an extended version of the following summary of existing results.

Theorem 3.3.
1. has the finite model property [Iemhoff01:phd, Prop. 4.1.1], [iemh:pres03, Prop. 7], [Zhou03, Th. 2.1.10].

2. (, ) correspond to the class of brilliant frames (strong frames, semi-transitive frames) and enjoy the fmp.

3. corresponds to the class of gathering frames and has the fmp [Iemhoff01:phd, Prop. 4.2.1], [iemh:pres03, Prop. 8].

4. corresponds to the class of Noetherian semi-transitive frames and has the fmp [Iemhoff01:phd, Prop. 4.3.2], [Zhou03, Th. 2.2.7].

5. corresponds to the class of Noetherian gathering frames [iemh:pres03, Lem. 9], [iemh:prop05, Lem. 3.10] and has the fmp.

6. corresponds to the “supergathering” property of Figure LABEL:tab:compl on the class of finite frames [Zhou03, Lem. 3.5.1], [iemh:prop05, Th. 3.31].

Open Question 3.4.

Are and related systems involving Kripke complete?    ❍

[inline,marginclue]Mention SL too?