Gödel blooming: the IncompletenessTheorems from a paraconsistent perspective

Walter CarnielliCentre for Logic, Epistemology and the History of Science

and Department of Philosophy

University of Campinas - Unicamp, Campinas, SP, Brazil

David FuenmayorDepartment of Mathematics and Computer Science

Freie Universität Berlin

e-mail: walterac@unicamp.br, david.fuenmayor@fu-berlin.de

Abstract

This paper explores the general question of the validity of Gödel’sincompleteness theorems by examining the respective arguments froma paraconsistent perspective, while employing combinations of modallogics with Logics of Formal Inconsistency (LFIs). For this purpose,abstract versions of the incompleteness theorems, employing provabil-ity logic, need to be carefully crafted. This analysis considers distinctvariants of the notion of consistency for formal systems, which, to-gether with the lighter character of the negation operator of the LFIs,enable new formalization variants of the Gödelian arguments, eventu-ally leading to some thought-provoking conclusions. We show that thestandard formulation of Gödel’s theorems is not valid under some weakLFIs: a valid reconstruction requires further premises correspondingto the consistency (in the sense of LFIs) of particular formulas. Thisreadily leads us to a reformulation of Gödel’s theorems as an exis-tence claim. In this paper we also aim at showcasing the convenienceof working with modern proof assistants (in this case Isabelle/HOL),which enable much faster and accurate feedback on verifying or falsi-fying hypotheses during the process of formal proof reconstruction.

Keywords: Gödel’s incompleteness theorems; Provability Logic; paracon-sistency; Logics of Formal Inconsistency; Isabelle/HOL.

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1 How universal are Gödel’s arguments?In a rough and intuitive formulation, Gödel’s first incompleteness theorem(G1) says that, for certain consistent formal systems, there are (true) sen-tences that they cannot decide, i.e., neither prove nor disprove; the secondincompleteness theorem (G2) says that such a system cannot prove its ownconsistency. A formidable amount of papers deals with explanations or in-terpretations of the Gödelian arguments, but few of them touch on the limitsof the Gödelian objection.

Gödel formulated his incompleteness theorems1 employing notions such asaxiomatic systems, primitive recursive arithmetic, arithmetization/numbering,representability/interpretability, consistency, completeness, diagonalization,etc; already a quick literature survey will reveal several different (and in somecases non-equivalent) ‘formalizations’, or more appropriately, explications, ofthese notions. In this paper we focus on the notions of consistency andnegation from the point of view of paraconsistent logic. We aim at examin-ing different, alternative notions of consistency employing both classical andparaconsistent negation, and then investigating their role in Gödel’s argu-ments with the help of the proof assistant Isabelle/HOL (henceforth Isabelle)[NPW02]. For this sake, we will abstract away the complexities of Gödel’sarithmetization procedure and assume the corresponding fixed-point (diag-onalization) lemma as a premise. We will employ for this (a paraconsistentversion of) provability logic [Ver17; Boo95].

It is important to note that we do not aim at a thorough formal recon-struction of Gödel’s proofs (including the arithmetization procedure) usinga proof assistant, as previously done by Paulson [Pau15] in Isabelle andO’Connor in Coq [OCo05]. Special mention deserves the work of John Har-rison, who provides a formal reconstruction of G2 within his proof assistantHOL-Light by employing provability logic in a similar spirit as ours (cf. hisrelated work [Har09, Ch. 7]).

With our formal reconstruction work in Isabelle (Section 6), we aim ratherat introducing a framework for experimenting, in the context of Gödel’sproofs, with different notions of consistency, and from a paraconsistent per-spective. We employ modal logic, both as a paraconsistent and as a provabil-ity logic, drawing upon the shallow semantical embeddings (SSE) approach[Ben19]. SSE rests on the adoption of Church’s simple type theory (STT)[BA19] as an expressive higher-order meta-language into which the logical

1There is some controversy in referring to Gödel’s results as either ‘a theorem’ or‘theorems’. Saul Kripke (in private conversation with the first author) insists that weshould refer jointly to both as ‘Gödel’s theorem’, since G2 is a corollary of G1. We preferto maintain the plural.

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connectives of (a combination of) target logics can be ‘translated’ or ‘embed-ded’, in such a way that the target logic becomes a fragment of STT.2 Indoing this, we can employ the proof assistant Isabelle, whose logic conserva-tively extends STT, to reconstruct and assess different formalized variants ofGödel’s arguments. We make the Isabelle source files for this formalizationwork freely available [CF20] and encourage the interested reader to carryout further experiments (e.g. by varying the encoded logic or the notions ofconsistency), and thus to further expand and improve on this work.

In view of the abundant literature and approaches towards Gödel’s re-sults, we are obliged to restrict ourselves to considering only a few sources.Succinct, but self-contained and fairly detailed, discussions emphasizing mostimportant points in Gödel’s proofs can be found e.g. in the works of Smoryński[Smo77, Sec. 1–2], Epstein & Carnielli [EC08, Ch. 23–24], and in the corre-sponding article in the Stanford Encyclopedia of Philosophy [Raa18]. We willdraw upon them as sources for our formal reconstruction work in Section 6.

Section 2 raises the question about the range of Gödel’s theorems, settingthe stage for an analysis of the dependence of the Gödelian arguments onstandard logical conventions. Section 3 introduces the paraconsistentist pro-gram, and foresees some difficulties as regards the validity of the Gödelianobjection in more subtle logical scenarios. Section 4 discusses the Logics ofFormal Inconsistency, and justifies the choice of the logic RmbC among eli-gible paraconsistent scenarios. Section 5 illustrates the mechanism by whichwe add a provability operator to RmbC, thus obtaining the logic RmbC⊕K,and analyzes several notions of consistency that emerge from this move. Sec-tion 6 is dedicated to the formal reconstruction of the Gödelian argumentswith the help of Isabelle. This task requires a combination of modal andparaconsistent logics embodied in RmbC⊕K. Since such logics are not cur-rently supported off-the-shelf by mainstream proof assistants, we discuss thechanges that turn Isabelle into a flexible, high-level modal and paraconsistentlogic reasoner. This section also contains the most relevant technical results.Finally, Section 7 offers the main conclusions of this paper.

2The SSE technique has been employed successfully in the logical analysis of argu-mentative discourse (e.g. in the formal reconstruction of another, more metaphysicalGödelian argument [BW16; FB17], or in formal ethics [FB19b]), where it has also inspireda computer-supported approach, computational hermeneutics [FB19a], which, adhering tothe slogan: ‘every formalization is an interpretation’, aims at rendering explicit the tacitconceptualizations implicit in argumentative practices. SSE also has applications in AIand normative systems [BPT20], as it supports the reuse of existing reasoning infrastruc-ture for first-order and higher-order logics for seamlessly combining and reasoning withdifferent quantified classical and non-classical logics (including modal, deontic, epistemicand paraconsistent), many of which are well suited for normative reasoning applications.

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2 On the range of Gödel’s theoremsGödel’s incompleteness theorems do not apply unrestrictedly to every math-ematical system. G1 does not apply for instance to Euclidean geometry.Tarski proved in 1948 [Tar98] that the first-order theory of Euclidean ge-ometry is complete and decidable,3 in the sense that every statement in itslanguage is either a theorem (i.e., provable) or its negation is a theorem, andthere is an algorithm to determine which is the case. Even non-Euclideangeometry falls outside the Gödelian barrier: if plane Euclidean geometry isa consistent theory, then so is plane hyperbolic geometry. Nor do Gödel’stheorems apply to systems of arithmetic with the addition operation only,such as Presburger arithmetic. It is necessary to have a certain critical massof mathematical strength to be attacked by the Gödelian objection, or, inother words: to qualify as what Gödel himself defined as ‘a formal system’.

Our aim in this paper is to give some first steps towards the analysis,using non-classical logics, of the applicability of Gödel’s theorems to formalsystems, while exploiting the computing power of modern proof assistants.Very little investigation, if any, touches the validity of Gödel’s proofs in non-classical environments. For instance, [BS14] claims to be studying versionsof Gödel’s arguments under the umbrella of non-classical logics, but just con-centrates on substructural logics: the authors show that the Gödelian rea-soning presupposes a certain amount of contraction in the underlying logic,that is, the validity of the meta-rule ‘Γ, ψ, ψ ` ϕ implies Γ, ψ ` ϕ’. Theythen exhibit a modal system without contraction that invalidates Gödel’sargument. On the other hand, several authors including Kreisel, Feferman,Löb, Jeroslow, Bezboruah–Shepherdson, Pudlák, Wilkie–Paris, Adamowicz–Zdanowski, Willard, Friedman, and Visser, among others (see [BS14] forreferences) have studied abstract conditions that permit the incompletenesstheorems to be derived in a way somewhat independent of logic, but withoutchanging the underlying standard logic. The interested reader may furtherconsult some book-length discussions on Gödel’s results, e.g. [Smi13] and[Smu92]; for a more philosophically-oriented discussion a good reference is[Fra05].

The intuition behind Gödel’s proof ofG1 is basically the following: assumethat the formal system F is consistent (otherwise it proves every sentence bythe Principle of Explosion of classical logic, and thus it is trivially complete).By Gödel’s diagonal (or fixed-point) lemma, one can then construct a sen-tence GF (hinging on F) that is neither provable nor refutable in F, and that

3According to [McN+53], the results were obtained in 1930 and published privately inits full development in 1948.

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can also be shown to be true. Thus F is incomplete, both in the sense thatthere is a sentence that it cannot ‘decide’, and in the sense that there is atrue sentence that it cannot prove. G2, stating that F cannot prove its ownconsistency, follows as a corollary of G1. Alternatively, G2 can be deriveddirectly from Löb’s theorem.

As is well recognized, the notion of consistency employed in G1 is not thesame in the different variants that have been presented; ranging from theso-called ω-consistency originally introduced by Gödel, through the (weaker)1-consistency commonly used in the literature, while also including a moreclassical notion of consistency as in the variant introduced by Rosser in 1936.

But what is the idea of consistency behind G2? At first sight, one can saythat consistency means simply the imperative not to derive a contradiction(absence of contradiction, or universal validity of non-contradiction), whichis the same as non-triviality, in view of the Principle of Explosion (PEx) oftraditional logic. However, the PEx is an unnecessary burden that classicallogic carries uselessly: it is not used, but to mark the ban on contradictions.Contrary to what some unsuspecting people may think, the PEx is not evenused in reductio ad absurdum proofs: a little thought will convince themthat what is at stake in such proofs is the rule of negation introduction.Indeed, any time we have a bottom particle ⊥, the rule α → ⊥ ` ¬α canbe applied, as it involves the introduction of negation, and not any use ofPEx. In classical logic the variant ¬α → ⊥ ` α also holds; notice, however,that this second variant does not hold in intuitionistic logic. This means thatthe beloved, and useful, reductio ad absurdum method of proof acquires itslegitimacy independently of PEx.

Paraconsistent logics, particularly the Logics of Formal Inconsistency(LFI), liberate logical systems from this burden by weakening the PEx, andfor a good reason. As widely acknowledged (see e.g. [DP15]), in many situ-ations, we have no other choice but to reason from contradictory premises,and this is a critical issue, since large knowledge bases or complex argumentsalmost inevitably include contradictions. A consequence of weakening thePEx is that consistency does not coincide with non-contradiction anymore,nor does it coincide with non-triviality. This is already a major philosophicaldifficulty for the classical stance on G2, which also affects, to a lesser extent,G1. As pointed out in the literature, e.g. in [BS14], we cannot easily pin-point a class of formulas that expresses consistency. When we paraphrase G2

as saying that ‘a sufficiently strong consistent theory cannot prove its ownconsistency’ we are forced to remain vague, and more so in the domain ofparaconsistent logics.

A natural question thus emerges: is there a way to avoid Gödel’s resultsby changing the underlying logic? Admittedly, it is not very encouraging

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to know that G1 and G2 are both intuitionistically valid; this is so becausethe usual proof of G1 is entirely constructive. Moreover, the usual proofof G2 consists in coding the proof of G1 using arithmetic, which is, again,constructive. This means that neither classical nor intuitionistic logic arefree of the Gödelian objection. We show that the situation is different forparaconsistent logics, while, at the same time, providing a means to recoverthe Gödelian results by adding further premises that concern the consistency(in the paraconsistent sense to be explained below) of particular formulas.

3 The paraconsistentist programThe Logics of Formal Inconsistency (LFIs) are a broad family of paraconsis-tent logics, which constitute a wide generalization of da Costa’s original hi-erarchy Cn by incorporating operators for consistency (◦) and inconsistency(•). LFIs turn out to be highly flexible logic systems (see e.g. [CCR19] forreferences and discussion).

The paraconsistent program is the investigation of logic systems endowedwith a negation ¬, such that not every contradiction of the form α and ¬αentails everything; in other words, a paraconsistent logic does not suffer fromdeductive trivialism, in the sense that a contradiction does not necessarilytrivialize the deductive machinery of the system by proving everything.4

Formalizing what has been said before, deductive trivialism stems fromthe fact that classical logic cannot stand contradictions, since it endorses theinference rule ex contradictione sequitur quodlibet, or Principle of Explosion:

(PEx) α,¬α ` β,

which authorizes to derive anything from a pair of contradictory propositionsα,¬α.5 The challenge for paraconsistent logics is to shun such an explosivenegation, while still preserving resources for designing an expressive logic.

As mentioned above, the language of LFIs internalizes a notion of consis-tency at the formula-level, independent of (but related to) negation. Consis-tency thus becomes represented in the logic by a new unary connective ◦. Inthe same vein, some LFIs internalize a notion of inconsistency employing theconnective •. In this setting, the notion of inconsistency (•) not necessarilycorresponds to the negation of consistency (¬◦).

4Deductive trivialism should not be confused with trivialism, according to which ev-erything is true.

5This is independent from the fact that classical logic endorses the validity of thePrinciple of Non-Contradiction: ` ¬(α ∧ ¬α), see [CCR18].

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In the LFIs, consistent statements are those too rigid to admit contradic-tions, errors or vagueness, as exemplified by yes–no statements (whether ornot you are pregnant, or have a certain disease). Those statements are rigid,or ‘consistent’, in the sense that they cannot stand any contradiction. Onthe other hand, more flexible, ‘non-consistent’ or ‘inconsistent’ statements(like whether it is hot today) have the ability to resist to contradictions (bynot entailing everything).

This intuitive notion of consistency becomes expressed formally by meansof the connective ◦, whose meaning is governed by axioms and rules. Anal-ogous ideas can be found in the notion of rigidity, as employed in computa-tional ontologies, cf. OntoClean and UFO, as well as in the notion of rigidor stable predicates found in quantified modal logics (cf. [FB17] for an ap-plication in formal argument reconstruction). This indicates that the idea offormally abstracting a notion of consistency is a natural desideratum, andhas instances in other fields.

The basic intuition is that contradictions should not affect all sentences(or all judgments) in the same way, and this is why the sort of Principle ofExplosion employed in the LFIs is restricted to a special set of consistentsentences. Hence a contradictory theory is not necessarily trivial, providedthat the contradictions do not involve statements that have been tagged as‘consistent’, by employing the connective ◦. This flexibility characterizingLFIs is expressed in the so-called Principle of Gentle Explosion, which is anessential part of the definition of LFIs, which we present below.

Definition 3.1. Let L = 〈Θ,`〉 be a Tarskian, finitary and structural logicdefined over a propositional signature Θ, which features a negation ¬ and a(primitive or defined) unary connective ◦. Then, L is said to be a Logic ofFormal Inconsistency (LFI) with respect to ¬ and ◦, if the following holds:

(1) ϕ,¬ϕ 0 ψ for some ϕ and ψ;

(2) ◦ϕ, ϕ,¬ϕ ` ψ for every ϕ and ψ;

(3) there are two formulas α and β such that

(a) ◦α, α 0 β;(b) ◦α,¬α 0 β.

Condition (1) signals the failure of the Principle of Explosion. Condition(2) represents the Principle of Gentle Explosion. Condition (3) is requiredin order to prevent condition (2) from being trivially satisfied.

As a consequence, and in contrast to classical logic, consistency in LFIsis not synonymous with freedom from contradiction, and here the role of

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negation is fundamental. The meaning of consistency in the LFIs is dictatedby its axioms, as occurs with negation (and the other connectives). Forconceptual clarifications the reader is referred to [CCM07] and to [CC16a].The LFI-hierarchy starts from a logic called mbC, which extends positive(i.e. ‘negation-less’) classical logic CPL+ by adding a (paraconsistent) nega-tion ¬ and an unary consistency operator ◦ satisfying some minimal require-ments in order to define an LFI (as in Definition 3.1).

Gödel’s theorems are, of course, crucially dependent on the propertiesof negation, and we will evaluate how this new perspective may affect theirvalidity. It should from now on become clear that the statements ◦α (α isconsistent) and ¬(α ∧ ¬α) (α is non-contradictory) are not equivalent for aparaconsistent negation ¬, that is, for a negation subject to the Principle ofGentle Explosion (instead of the classical Principle of Explosion).

This separation between consistency and non-contradiction, contradictionand non-consistency, as well as inconsistency and non-consistency, togetherwith the consequent distinction between contradiction and triviality, are themain tenets of LFIs. As we shall see in Sections 5 and 6, they will lead todistinct proposals for formalization variants of Gödel’s theorems, and thusaffect their proofs accordingly.6

4 Choosing among paraconsistent scenariosWe start by introducing a ‘negation-less’ fragment of classical logic, or fullclassical positive logic:

Definition 4.1 (Classical Positive Logic). The classical positive logic CPL+

is defined over the language containing {∧,∨,→} by the following axioms andinference rule:

6We will investigate this affectation only at an abstract level and by employing prov-ability logic. In particular, we will not consider Gödel’s arithmetization procedure, northe fixed-point (diagonalization) lemma, which we simply assume. One can thus describeour work as formally reconstructing ‘the last mile’ of Gödel’s proofs.

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Axiom schemas:

α→(β → α

)(Ax1)(

α→(β → γ

))→

((α→ β

)→

(α→ γ

))(Ax2)

α→(β →

(α ∧ β

))(Ax3)(

α ∧ β)→ α (Ax4)(

α ∧ β)→ β (Ax5)

α→(α ∨ β

)(Ax6)

β →(α ∨ β

)(Ax7)(

α→ γ)→

((β → γ)→

((α ∨ β)→ γ

))(Ax8)(

α→ β)∨ α (Ax9)

Inference rule:α α→ β

β(MP)

Starting from CPL+ above as a base logic, we extend it with a (para-consistent) negation ¬ and a (primitive or defined) consistency operator ◦satisfying the conditions stated in Definition 3.1. We thus obtain the para-consistent logic mbC, which is a basic LFI in the sense that its negationand consistency operators enjoy the minimal properties in order to satisfythe definition of LFIs.7

Definition 4.2. The logic mbC, defined over the language containing{∧,∨,→,¬, ◦}, is an LFI obtained from CPL+ by adding the connectives¬, ◦ and the following axiom schemas:

α ∨ ¬α (Ax10)

◦α→(α→

(¬α→ β

))(bc1)

Moreover, we can use ¬ and ◦ to define bottom particles in the languageof mbC, as well as classical negation, also called strong negation.

7Some strong extensions of mbC, such as the logic Cie (and its extensions), do notdistinguish between inconsistency and contradiction as a consequence of their axiomaticpresentation, and some may even allow for the reduction of double negations (this is,however, contrary to what happens in most other LFIs; see [CC16a] for a discussion). Wewill thus restrict ourselves to relatively weak systems starting from the minimal (weakest)Logic of Formal Inconsistency mbC.

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Definition 4.3. For any sentences α and γ:

• ⊥α := ◦α ∧ α ∧ ¬α act precisely as bottom particles, i.e., they satisfy⊥α `mbC β, for every sentence β;

• ∼γα := α→ ⊥γ act precisely as classical (strong) negations, i.e., theysatisfy `mbC α ∨ ∼γα, and α ∧ ∼γα `mbC β for every sentence β.

Since ⊥α are equivalent for all α, and ∼γ are equivalent for all γ (see[CC16a, Ch. 2] for an elaborate discussion), we write simply ⊥ and ∼. Itis worth remarking that mbC (as introduced below) both extends and isextended by classical logic. Indeed, on the one hand mbC is obviously asubclassical logic by definition, and on the other hand the defined connec-tives ⊥ and ∼, added to {∧,∨,→}, completely encode classical logic. mbCcan be (equivalently) expounded as a direct extension of classical logic, by in-corporating an additional negation ¬ to the language, thus defining the logicmbC⊥. The equivalence between mbC⊥ and mbC is shown in [CC16a,Ch. 2].

mbC is the basis for a potentially infinite hierarchy of logics, going upto da Costa logics Cn, paraconsistent many-valued logics [CCM07], paracon-sistent modal logics [Bue10; Bue12], and eventually reaching classical logic,all of them characterized by axiomatic systems extending mbC (see [CC16a,Ch. 3–4] for a detailed discussion).

It has been proved that the logics in the da Costa’s hierarchy Cn (C1

included) are hardly algebraizable: This is partly due to the non-validity of areplacement meta-theorem which would establish the validity of intersubsti-tutivity of provable equivalents (IpE) for such logics. Indeed, Theorem 3.51in [CM02] shows that IpE cannot hold in any paraconsistent extension of thelogic Ci (or, for that matter, in any LFI) in which (¬α ∨ ¬β) ` ¬(α ∧ β)holds; or ¬(α ∧ β) ` (¬α ∨ ¬β) holds.

We will sketch the main lines of RmbC, an extension of mbC (as men-tioned, a minimal LFI extending CPL+) which satisfies the replacementproperty, a meta-property that grants that if α ↔ β is a theorem thenγ[p/α] ↔ γ[p/β] is a theorem, for every formula γ(p). As mentioned, mbCdoes not generally satisfy replacement for sentences containing ¬ and ◦. Byadding replacement for ¬ and ◦ as new global inference rules, full replacementcan be recovered, in the sense that if α ↔ β is a theorem then ¬α ↔ ¬βis also a theorem, and if α ↔ β is a theorem then ◦α ↔ ◦β also is. Asdiscussed in [CCF], this makes RmbC and its extensions fully algebraizablein the standard Lindenbaum-Tarski’s sense, which enables their combina-tion with other similarly algebraizable logics by means of algebraic fibring[Car+08; CC16b]. This is very important for us, since we want to be able to

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combine RmbC with other modal logics (e.g. extending K) for the sake offormally reconstructing Gödel’s arguments.

Definition 4.4. The logic RmbC, defined over the language containing{∧,∨,→,¬, ◦} is obtained from mbC by adding the following inference rules:

α↔ β

¬α↔ ¬β(R¬)

α↔ β

◦α↔ ◦β(R◦)

As observed in [CCF], where RmbC was introduced and from where weborrow its presentation, the rules that grant replacement are global instead oflocal rules; this means that in order to apply each global rule the correspond-ing premise must be a theorem. This is similar to what happens with thenecessitation rule in modal logics. Observe that adding this kind of globalrules requires special care with the definition of derivation from premises.

Definition 4.5 (Derivations in RmbC).

• A derivation of a formula ϕ in RmbC is a finite sequence of formulasϕ1 . . . ϕn such that ϕn is ϕ and, for every 1 ≤ i ≤ n, either ϕi is aninstance of an axiom of RmbC, or ϕi is the consequence of some infer-ence rule of RmbC whose premises appear in the sequence ϕ1 . . . ϕi−1.

• We say that a formula ϕ is derivable in (or a theorem of) RmbC,denoted by `RmbC ϕ, if there exists a derivation of ϕ in RmbC.

• Let Γ ∪ {ϕ} be a set of formulas over Σ. We say that ϕ is derivablein RmbC from Γ, and we write Γ `RmbC ϕ, if either ϕ is derivablein RmbC, or there exists a finite, non-empty subset {γ1, . . . , γn} of Γsuch that the formula (γ1 ∧ γ2 ∧ . . . ∧ γn)→ ϕ is derivable in RmbC.

By the properties of ∧ and→ inherited from CPL+, and by the notion ofderivation from premises just presented, it is easy to see that the deductionmeta-theorem holds in RmbC, and that it is a Tarskian and finitary logic(see [CCF] for details).

As presented in [CCF], a sound and complete semantics for RmbC canbe given by means of a suitable class of Boolean algebras with LFI operators(BALFIs), a (non-additive) generalization of the standard Boolean algebraswith operators (BAOs) used in algebraic semantics for (normal) modal logics.It is important to highlight that the possibility of such a semantic character-ization for paraconsistent logic RmbC opens the door to their combinationwith other logics (e.g. modal logics) by means of algebraic fibring [Car+08,Ch. 3], as well as its use in many other application areas for algebraic methodsin logic.

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Also important to our present purposes: a neighborhood semantics char-acterizing RmbC (and its extensions) has been introduced in [CCF], whereit was shown that RmbC can be defined within the minimal bimodal non-normal logic E [Che80]. In this respect, RmbC can indeed be considered asa (non-normal) modal logic. We will exploit this fact in Section 6, where weconduct automated reasoning with combinations of RmbC and other modallogics, by employing the technique of shallow semantical embeddings (SSE)[Ben19].

Before finishing this section, it is convenient to emphasize some propertiesof the consistency operator and of the paraconsistent negation in the logicmbC, and which are inherited by RmbC. Note that these properties concernparticularly the notions of consistency and inconsistency:

Theorem 4.6. The following properties of mbC also hold in RmbC:

(1) α ∧ ¬α `mbC ¬◦α but ¬◦α 6`mbC α ∧ ¬α;

(2) ◦α `mbC ¬(α ∧ ¬α) but ¬(α ∧ ¬α) 6`mbC ◦α;

(3) ¬α→ β `mbC α ∨ β but α ∨ β 6`mbC ¬α→ β;

(4) ◦α, α ∨ β `mbC ¬α→ β;

(5) α→ β 0mbC ¬β → ¬α but ◦β, α→ β `mbC ¬β → ¬α;

(6) α→ ¬β 0mbC β → ¬α but ◦β, α→ ¬β `mbC β → ¬α;

(7) ¬α→ β 0mbC ¬β → α but ◦β,¬α→ β `mbC ¬β → α;

(8) ¬α→ ¬β 0mbC β → α but ◦β,¬α→ ¬β `mbC β → α.

Proof. Mechanically verified (see Isabelle sources in [CF20]). The proofs canalso be easily adapted from [CC16a], Chapter 2, Propositions 2.3.3, 2.3.4,and 2.3.5.

The above properties help us to understand the connections between con-sistency, non-consistency, contradictions, and non-contradictions, as well asto anticipate some of their effects on Gödelian arguments. Item (1) showsthat contradiction implies non-consistency, but not vice-versa. Item (2)shows that consistency implies non-contradiction, but not vice-versa. Items(3) and (4) show that disjunction cannot be fully recovered from negationand implication, as in the classical case (however this can be done when someparts are consistent). Items (5) to (8) show that some contraposition rulesfor implication do not hold when the paraconsistent negation is considered,

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but will hold under the guarantee of consistency for the consequent in theoriginal conditional form.

The discussion in this section aimed at explaining our choice of a para-consistent scenario to deal with the plethora of variants of Gödel’s theoremsthat emerge by weakening the Principle of Explosion and, in particular, whenconsistency is liberated from being defined as non-contradiction. The follow-ing sections introduce a logic combination featuring both LFI operators anda (normal) modal operator � aimed at capturing the notion of provabilityin formal systems like Peano Arithmetic, drawing upon systems of modallogic extending K. We also discuss some exemplary (though not exhaustive)conclusions achieved through the computer-supported logical analysis of theGödelian arguments in a joint work with the proof assistant Isabelle.

5 A paraconsistent logic of provability: consis-tency abounds

We have so far introduced the logic RmbC as a candidate logic for theformalization of the Gödelian proofs. However, the most important compo-nent for such a logical system is still missing, namely, a provability opera-tor, since we want to encode the notion of being derivable/provable directlyin our object language. Following the tradition of provability logic [Ver17;Boo95], we will employ the modal operator � for this purpose. But firstof all, what is this provability logic? Generally speaking, every time we ap-ply modal logic to the study of formal provability (in some given expressivesystem) it becomes provability logic. However, not every system of modallogic is appropriate for modeling the notion of derivability in a system in-cluding (Peano) arithmetic (e.g. some common modal axioms like T or D areunqualified). Hence, the logics that usually come into consideration whenit comes to provability are normal modal logics extending the well-knownsystem K with either the axiom 4: ` �φ → ��φ (logic K4) and/or theinference rule LR : ` �φ → φ =⇒ ` φ (logic K(4)LR), or the axiomL : ` �(�φ → φ) → �φ (logic GL).

To get an idea of how these modal logics relate to provability, let usconsider a formal system F that includes Peano Arithmetic. We write `F φto indicate that φ is a theorem of F. If φ is an expression of the languageof F (i.e. an F-formula), we shall let dφe denote the corresponding numeralfor the Gödel number of φ (which we henceforth call the Gödel numeral ofφ).8 Given the F-formula PfF (y, x) stating that there is a proof in F with

8Recall that we can establish an injection between the set of F-formulas (and also

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Gödel numeral y for the formula with Gödel numeral x, we can construct theformula PrF (x) := ∃y.PfF (y, x). Hence PrF (dψe) expresses that dψe is theGödel numeral of a sentence ψ that is provable in F.

The link between modal logic and provability in F (both sharing theprimitive logical connectives ⊥ and →)9 becomes explicit by considering thefollowing notion (cf. [Boo95, Ch. 3]).

Definition 5.1 (Realization). A realization r(p) is a function that assignsto each sentence letter p a sentence of the language of F. A realization rinduces a translation (·)r as follows:

1. (p)r = r(p)

2. (⊥)r = ⊥

3. (φ→ ψ)r = (φ)r → (ψ)r

4. (�φ)r = PrF (d(φ)re)

Remark 5.2. Since we aim at obtaining a paraconsistent provability logic,we need to add the following additional items to Definition 5.1 above:

5. (¬φ)r = ¬(φ)r (for LFIs only)

6. (◦φ)r = ◦(φ)r (for LFIs only)

Observe that, in adding the last two items, we assume the existence of coun-terparts for ¬ and ◦ in the language of F. Since F is also assumed to contain(Peano) arithmetic, we can, at least in principle, articulate arithmetic for-mulas featuring ¬ or ◦. This prima facie ability to employ a paraconsistentnegation, as well as a consistency operator, in arithmetic formulas has inter-esting philosophical repercussions. Their analysis is however out of the scopeof this paper. An interesting discussion can be found in [Sha02].

The link between theoremhood in both systems, i.e., between a systemF containing Peano Arithmetic and some extensions of K, is given by thethree results below.

Proposition 5.3. `GL φ if and only if for every realization r, `F (φ)r.

their sequences) and the set of natural numbers, in such a way that each natural numberis recursively associated with at most one formula according to Gödel’s arithmetizationprocedure. Also recall that a numeral corresponding to some natural number n is theF-formula consisting of the symbol 0 preceded by n occurrences of the symbol S.

9Recall that ∼φ can be defined as φ→ ⊥. Other connectives can be defined employing∼ and → as usual.

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Proposition 5.4. If `K4 φ, then for every realization r, `F (φ)r.

Corollary 5.5. If `K φ, then for every realization r, `F (φ)r.

Proof. Proposition 5.3 is Solovay’s arithmetical completeness theorem forlogic GL [Sol76]. Proposition 5.4 (arithmetical soundness theorem for K4)and its Corollary 5.5 (arithmetical soundness for K) are earlier results. See[Ver17] and [Boo95] for a discussion.

We now recall the well-known derivability conditions for PrF (x) (drawingon the ones introduced by M.H. Löb [Löb55]) and their counterpart axiomsin modal logic K4 (note that logic GL is an extension of K4 [Boo95]).

Proposition 5.6 (Derivability conditions). Let F be a formal system con-taining Peano Arithmetic, we have:

1. `F φ implies `F PrF (dφe)

2. `F PrF (dφe) ∧ PrF (dφ→ ψe)→ PrF (dψe)

3. `F PrF (dφe)→ PrF (dPrF (dφe)e)

Proof. Consult e.g. [Löb55] or [Boo95, Ch. 2].

Remark 5.7. Observe that the previous results apply in general to everysystem F which contains (classical) Peano Arithmetic. Hence it also appliesfor arithmetic systems featuring LFI operators ¬ and ◦; recall from Definition4.3 that ⊥ (and in consequence ∼) is definable in the LFIs.

Definition 5.8. The conditions above are encoded in K as follows:

1. Necessitation rule: ` φ implies ` �φ

2. Axiom K: ` �(φ→ ψ)→ (�φ→ �ψ)

Moreover, we have in modal logic K4:

3. Axiom 4: ` �φ→ ��φ

Furthermore, we can enrich our logic with another postulate further re-stricting the behavior of the provability operator �. This postulate drawsupon the following result:

Proposition 5.9 (Löb’s Theorem). Let φ be any sentence of the languageof F. Then:

if `F PrF (dφe)→ φ then `F φ

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Proof. Consult M.H. Löb’s original result [Löb55] (see also [Boo95]).

Definition 5.10. Löb’s theorem can be encoded in (extensions of) modallogic K as follows:

• Löb’s rule LR: ` �φ→ φ =⇒ ` φ

• Löb’s axiom L: ` �(�φ→ φ)→ �φ

The modal logic KLR is obtained by extending K with the inference ruleLR above; similarly, the logic K4LR is obtained by extending K4 with ruleLR. The modal logic GL (Gödel–Löb logic, or provability logic in the strictsense) is obtained by extending K with axiom L. Note that axiom 4 followsfrom L, and also that logics K4LR and GL validate the same formulas(consult [Boo95] for this and other interesting results).

Concerning our present purposes, we define a paraconsistent logic of prov-ability RmbC⊕K by means of algebraic fibring (which generalizes fusion ofnormal modal logics) [Car+08, Ch. 3] between the logic RmbC and the logicK, sharing the connectives {∧,∨,→,⊥}.10 It is evident that RmbC⊕K isan LFI (Def. 3.1), as well as its extensions featuring axioms 4 and L.

As we know, Gödel’s incompleteness results concern (expressive enough)formal systems that are consistent. But what does it mean for a theory to beconsistent (i) classically, and (ii) under the paraconsistent perspective? Aswe will see, we can identify (at least) five ways to answer (i) in the context ofGödel’s proofs. As for (ii) our choices are not fewer. We present an overview:

Definition 5.11. Different notions of consistency:

(1) A system is consistent (simpliciter, in the sense of non-contradictory)when it cannot derive both a sentence and its classical (resp. paracon-sistent) negation, i.e. 6` ∼φ ∧ φ (resp. 6` ¬φ ∧ φ).

(2) A system is ∗-consistent when it cannot derive for the same sen-tence both its classical (resp. paraconsistent) negation and its provabil-ity, i.e. 6` ∼φ ∧�φ (resp. 6` ¬φ ∧�φ).

10Recall that the classical negation ∼ and the bottom particle ⊥ are definable in logicRmbC, see Definition 4.3. Observe that we can combine both logics by means of algebraicfibring, since, as recently shown in [CCF], RmbC is just another (non-normal) modallogic with an algebraic semantics based on Boolean algebras with additional operators.Further modal axioms such as 4 and L can be formulated upon this logic combination (aswe do in Section 6). Results concerning the preservation of meta-properties (soundness,completeness, interpolation, etc.) for combinations of logics employing the algebraic fibringapproach can be consulted in [Car+08, Ch. 2–3].

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(3) A system is S-consistent when it can derive every sentence of whichits provability is derivable, i.e. ` �φ =⇒ ` φ.

(4) A system is P-consistent when it cannot derive a blatantly false or‘impossible’ statement ⊥ (neither in the classical nor in the paraconsis-tent sense). Observe that this vague requirement can be given differentnon-equivalent formalizations: (a) 6` �⊥; (b) ` ∼�⊥ resp. ` ¬�⊥.

(5) A system is ◦-consistent when all of its sentences are (or can beproven) consistent in the sense of the ◦ operator of LFIs, i.e. when` ◦φ. Observe that this corresponds to a whole family of consistencyconditions, depending on which underlying LFI has been chosen.

Remark 5.12. Some observations are worth mentioning:

• φ, ψ, etc. (appearing free in formulas) act as meta-variables over for-mulas of the object logic, and are always implicitly all-quantified.

• The term ‘∗-consistent’ comes from the terms ‘ω-consistency’ and ‘1-consistency’. Observe that at the present level of abstraction (i.e. em-ploying modal logic to model provability) their idiosyncratic propertiesbecome abstracted away.

• In the context of modal logics, S-consistency corresponds to the so-called denecessitation rule (characterizing converse-serial frames). Thiscondition is employed as a further premise in the variant presented bySmoryński [Smo77, Ch. 2] (where it is referred to simply as ‘an addi-tional assumption’). The S in term ‘S-consistent’ comes from there.Observe that the related (stronger) condition ` �φ→ φ is the well-known modal axiom T (characterizing reflexive frames). This condi-tion is in fact undesirable in the context of provability logic; moreoverit is inconsistent (in the most classical and traditional sense) with Löb’scondition L := ` �(�φ→ φ)→ �φ.

• The P in the term ‘P -consistent’ comes from provability (logic), sincethis is the formalization usually employed there for system’s consis-tency [Ver17; Boo95]. Variants (a) and (b) come from the notions ofweak resp. strong representability (see e.g. [Raa18]) when applied to thenegation of �⊥ (which can be understood intuitively as inconsistency).

Some inferential relationships between the notions introduced in Defini-tion 5.11 are presented in Theorems 5.13 and 5.14 below, employing classicaland paraconsistent negation respectively.

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Theorem 5.13. For � as in classical modal logic K (and also K4):11

(i) (1) is a tautology: 6` ∼φ∧φ (consistency simpliciter) is a tautology.

(ii) (2) and (4.a) are equivalent: 6` ∼φ∧�φ (∗-consistency) and 6` �⊥(P -consistency-a) are logically equivalent.

(iii) (3) implies (2)/(4.a): ` �φ =⇒ ` φ (S-consistency) is strongerthan both 6` ∼ψ ∧�ψ (∗-consistency) and 6` �⊥ (P -consistency-a).

(iv) (4.b) implies (2)/(4.a): ` ∼�⊥ (P -consistency-b) is stronger than6` �⊥ (P -consistency-a) and therefore also than 6` ∼φ∧�φ (∗-consistency).

(v) (3) and (4.b) are incomparable: ` �φ =⇒ ` φ (S-consistency)and ` ∼�⊥ (P -consistency-b) are incomparable (i.e. neither impliesthe other).

Proof. Mechanically verified (see Isabelle sources in [CF20]).

As the results in Theorem 5.13 above show, the idea of differentiatingbetween consistency and non-contradiction (which is a main tenet of theLFIs) is indeed readily presupposed in the literature on Gödel’s arguments.Non-contradiction, corresponding to consistency simpliciter, is shown (un-surprisingly) to be a tautology (i). In contrast, all other notions of consis-tency are shown to be strictly stronger (i.e. non-trivial). Indeed, two of them(∗-consistency and P -consistency-a), though quite different-looking, end upbeing equivalent (ii), while the two others (S-consistency and P -consistency-b) are shown to be strictly stronger than them (iii)–(iv), while remainingindependent (v).

Theorem 5.14. We have for the combination RmbC⊕K (also K4):

(i) (1) is not a tautology: 6` ¬φ ∧ φ (consistency simpliciter) is not atautology (anymore).

(ii) (2) implies (1): 6` ¬φ∧�φ (∗-consistency) is stronger than 6` ¬ψ∧ψ(consistency simpliciter).

(iii) (5) implies (1): ` ◦φ (◦-consistency) is stronger than 6` ¬ψ ∧ ψ(consistency simpliciter).

11Note that the results are valid in both logics K and K4. Observe that these statementsare presented from the viewpoint of a (classical) meta-logic, to which belong the symbols:`, 6`,=⇒,∧. It is interesting to note that these results could be automatically verified inmilliseconds by the proof assistant Isabelle (employing the SSE technique [Ben19]).

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(iv) (2) implies (4.a): 6` ¬φ ∧�φ (∗-consistency) is stronger than 6` �⊥(P -consistency-a).

(v) (3) implies (4.a): ` �φ =⇒ ` φ (S-consistency) is stronger than6` �⊥ (P -consistency-a).

(vi) (2) & (3), (3) & (1) and (4.a) & (1) are pairwise incomparable.

(vii) (4.b) incomparable to all others: ` ¬�⊥ (P -consistency-b) is nei-ther implied by nor implies any other.

(viii) (5) incomparable to others excepting (1): ` ◦φ (◦-consistency)is neither implied by nor implies any other, excepting 1 (see iii).

Proof. Mechanically verified (see Isabelle sources in [CF20]).

We can observe from Theorem 5.14 above that, in the paraconsistent set-ting, the inability to derive a contradiction (consistency simpliciter) is nolonger a trivial tautology (i), while still remaining a weaker notion than∗-consistency and than the (LFI) ◦-consistency (ii)–(iii). ∗-consistency,which was equivalent to P -consistency-a in the classical setting (Theorem5.13 (ii)) is now strictly stronger (iv). S-consistency remains strictly strongerthan P -consistency-a (v). However, other inferential relationships are nolonger valid (vi). Notice, in particular, that P -consistency-b (which is thenotion employed in the formalization ofG2) has become, in the paraconsistentsetting, incomparable to P -consistency-a and ∗-consistency (vii). Last butnot least, the newly introduced LFI notion of ◦-consistency is only relatedto consistency simpliciter (viii).

6 Pruning the Gödelian gardenThe formal reconstruction work presented in this section employs the proofassistant Isabelle/HOL [NPW02], whose logic HOL is classic and extendsChurch’s symple type theory STT [BA19]. We will strictly differentiatebetween object-logical and meta-logical expressions. Object logics are: clas-sical modal logic K in Sections 6.2.1 and 6.3.1, and the paraconsistent (LFI)RmbC⊕K in Sections 6.2.2 and 6.3.2. Meta-logical expressions are thosebelonging to Isabelle’s language HOL. The source files for this work havebeen made freely available under [CF20]. We encourage the interested readerto carry out further experiments (e.g. by varying the encoded logic or thenotions of consistency) and also to further expand and improve on this work.

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6.1 Encoding RmbC⊕K in Isabelle/HOL

The formal reconstruction of the Gödelian arguments poses a challenging taskthat requires the combination of modal and paraconsistent logics, which arecurrently not supported off-the-shelf by mainstream proof assistants (e.g. Is-abelle, Coq, Lean). In order to turn Isabelle into a flexible modal and para-consistent logic reasoner, we have adopted the shallow semantical embeddings(SSE) approach, which harnesses the expressive power of higher-order logicas a meta-language allowing the faithful encoding of the semantics of quanti-fied modal logics (among other non-classical logics) in STT/HOL, therebyturning higher-order theorem proving systems into universal reasoning en-gines (see [Ben19] and references therein).

SSE involves the semantical definition of the logical vocabulary of theobject logic (shown below boldface) in terms of the non-logical vocabulary(λ-expressions) of the higher-order meta-logic. In this approach, propositionsbecome encoded via (characteristic functions of) their truth-sets. We thushave for the classical Boolean operators:12

⊥ := λw. False;

ϕ∧ ψ := λw. (ϕ w) ∧ (ψ w);

ϕ→ ψ := λw. (ϕ w) −→ (ψ w);

∼ϕ := λw.∼(ϕ w).

Notice how, in virtue of the shallow nature of SSE, the meta-logical operators(False, ∧, −→, ∼) become reused on the right-hand-side of the definitions.This feature greatly improves computing performance, as well as user experi-ence, by eliminating the need for recursive definitions and proofs. As regardsmodal operators, the embedding meta-logical expressions are constructed bydrawing upon the well-known standard translation into first-order logic:

�ϕ := λw.∀v. (R w v) −→ (ϕ v);

where R denotes the accessibility relation associated with �. LFI opera-tors become embedded as follows (note that without further constraints theycorrespond to those of RmbC):

¬ϕ := λw. (ϕ w) −→ ((S1 ϕ) w);

12Unsurprisingly, there are many different, but equivalent, ways to formulate SSEs forthe object-logical operators. We show a particular alternative for illustrative purposes,which does not (syntactically) correspond to the one provided in the sources [CF20]. Inparticular, we tolerate some redundancy in the formulations when we think it may helpto make the exposition clear.

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◦ϕ := ∼(ϕ ∧ ¬ϕ) ∧ (S2 ϕ);

where S1 and S2 denote (unconstrained) neighborhood functions (see [CCF]).Observe that we reuse previous definitions in the formulation of ◦, which (to-gether with ¬) is a primitive operator, since it features a primitive neighbor-hood function in its semantical definition; this is characteristic for RmbC.

As the previous exposition shows, by employing the SSE technique, object-logical formulas become encoded as STT/HOL-expressions of type w ⇒ bool(where w is a special type for worlds or states), i.e., the type of characteristicfunctions for their corresponding truth-sets. Another important element inSSE is the definition of validity for object-logical formulas. Given a formula ϕ(of type w ⇒ bool) in the object (modal) logic, we can define the expression:

[` ϕ] := ∀w. (ϕ w);

as corresponding to logical validity (and thus theoremhood) in the objectlogic. In particular, in the case of modal logic K, theoremhood follows fromthe completeness of K with respect to Kripke frames, together with thefaithfulness of their embedding into Henkin general frames for STT (seee.g. [BP10] and the references in [Ben19]). As regards RmbC, completenesswith respect to its neighborhood semantics has been shown in [CCF]; thefaithfulness of the embedding of its corresponding neighborhood structuresinto Henkin general frames is conjectured at the present time (similar faith-fulness results for non-normal deontic logics with neighborhood semanticsexist, e.g. [BFP18]). Be that as it may, it is not difficult to manually verifythe proofs reconstructed in this section employing the calculus introduced inSection 4.

A note on Isabelle/HOL setup: As mentioned before, we strictly dif-ferentiate between the (modal) object logics (K and RmbC⊕K) and the(meta-)logic HOL of Isabelle. Object-logical connectives are written usingboldface symbols and appear [` inside turnstile brackets]. Meta-logicalexpressions are those belonging to the (classical) HOL language and includethe connectives ∼ (classical HOL negation), −→ (HOL material implica-tion, particularly useful to model object-logical inference rules), and ∀ (HOLhigher-order quantification). Moreover, Isabelle also provides some proof di-rectives (assume, show, hence, using, by, etc.) which are part of Isabelle’sdomain-specific proof language Isar (see [NPW02]). In particular, observethe use of Isabelle’s keyword by followed by a proof tactic (or the nameof a solver program). Some of the solvers we employ here are: simp (termrewriting engine), blast (tableaux prover), presburger (Presburger arithmeticsolver) and smt (satisfiability modulo theories, SMT ). It is worth mentioning

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that in many cases these tactics were automatically suggested by Isabelle’smeta-prover Sledgehammer [BBP13], which also supports the invocation ofexternal state-of-the-art theorem provers.13 Notably, both Isabelle and the(external) Leo-III higher-order theorem prover [SB18] were able to prove allvariants of Gödel’s theorems fully automatically, i.e. without requiring proofreconstructions (which we carry out for illustrative purposes). Leo-III alsomanaged to prove several other conjectures that eluded Isabelle’s internalsolvers. In most cases, non-theorems could be refuted by Isabelle’s inte-grated model finder Nitpick [BN10]. We will point out the exceptional casesin the below discussion.

6.2 First incompleteness theorem (G1)

We take as starting point the variant of Gödel’s theorem presented in Theo-rem 6.1 below. This variant, as well as its corresponding proof, draws uponthe analysis by Smoryński [Smo77, Sec. 2], Raatikainen [Raa18], and Epstein& Carnielli [EC08, Ch. 24]. It is worth mentioning that these three variants(as well as many others) are indeed very similar, up to an additional premise(which we will discuss below).

Theorem 6.1 (Gödel’s first incompleteness theorem G1). Assume F is aconsistent formal system which contains Peano Arithmetic. Let GF be aformula that satisfies `F GF ↔ ∼PrF (dGF e), we have:

• 6`F GF ; (non-provable)

• under an additional premise, 6`F ∼GF . (non-refutable)

Proof. To be discussed below.

The additional premise in Theorem 6.1 above corresponds to a sort of con-sistency requirement: For [EC08, Ch. 24] it is ω-consistency (as in Gödel’soriginal variant), and for [Raa18] it is the weaker 1-consistency, both of whichare formally modeled here employing the notion of ∗-consistency introducedin Definition 5.11 (see Figures 2 and 5). Regarding the variant by [Smo77,Sec. 2], the consistency requirement (referred to as ‘an additional assump-tion’) is ∀φ. `F PrF (dφe) =⇒ `F φ, which we have formally modeled asS-consistency in Definition 5.11 (see Figures 3 and 6).

The reconstructed proofs forG1, presented in Figures 1 to 3 below, employa classical modal logic K (i.e. featuring only classical negation ∼).

13The provers can be invoked on a local installation or remotely via System on TPTP(http://www.tptp.org/cgi-bin/SystemOnTPTP).

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Figures 4 to 6 below are devoted to reconstructing the proof for thecorresponding paraconsistent variant Gpar

1 employing the logic combinationRmbC⊕K (featuring the paraconsistent negation ¬) as introduced in Sec-tion 5.

6.2.1 Proof reconstruction of G1 (using classical negation)

The reconstructed proof for lemma non_provable in Figure 1 draws mainlyfrom [Raa18] and [Smo77, Sec. 2]. Observe that this is a straightforward proofby reductio ad absurdum,14 where no further assumption is made concerningconsistency. ` �GF is obtained from ` GF (see line 28) by necessitation(which the term rewriting solver simp evidently exploits). The contradictingstatement ` ∼GF is obtained from it (line 29) by using the first (and only)premise (Isabelle notation: assms(1)) corresponding to the fixed-point lemma(for an arbitrary but fixed GF , see lines 23–24).

Figure 1: Proof of non-provability of GF in Isabelle/HOL

The proof for lemma non_refutable_v1 in Figure 2 draws from [Raa18],which employs 1-consistency as additional premise, and which we model as ∗-consistency (Def. 5.11). Similarly to the previous one, this proof is by reductioad absurdum. Here ` �GF is obtained (by the tableaux prover blast, line 41)from ` ∼GF exploiting the fixed-point lemma (line 36), thus obtaining aformula (line 43) which clearly conflicts (line 44) with the instantiation of∗-consistency for GF (line 42).

14Note that all presented proofs are acceptable from the viewpoint of intuitionistic logic.

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Figure 2: Proof of non-refutability of GF employing ∗-consistency

The proof for lemma non_refutable_v2 in Figure 3 draws from [Smo77,Sec. 2]), whose ‘additional assumption’ is modeled as S-consistency or dene-cessitation (formalized as: ` �ϕ =⇒ ` ϕ, cf. Def. 5.11). The proof is similarto the previous one, first obtaining ` �GF (line 55), and then instantiatingS-consistency for GF (line 56) in order to obtain the contradicting formula` GF by modus ponens (line 57).

Figure 3: Proof of non-refutability of GF employing S-consistency

Furthermore, G1 can also be proven employing the other notions of con-sistency. In doing this, lemma non_provable remains unchanged, while theconsistency premise in lemma non_refutable gets replaced by either 6` �⊥(P -consistency-a) or ` ∼�⊥ (P -consistency-b). Employing Isabelle we canverify the validity of G1 in both cases by automated means (see [CF20]).

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6.2.2 Proof reconstruction of Gpar1 (using paraconsistent negation)

In Figures 4 to 6 we reconstruct the proof for Gpar1 , employing ¬ as the para-

consistent negation of RmbC, while ◦ corresponds to the primitive RmbCconsistency operator (note that ∼ remains as the meta-logical classical nega-tion). As discussed in Section 5, we are dealing in this setting with the logiccombination RmbC⊕K (an instance of algebraic fibring between RmbCand modal logic K, cf. [Car+08, Ch. 3]).

The proofs for lemmas non_provable (Figure 4) and non_refutable_v1(2)(Figures 5 and 6) are similar to their classical counterparts (cf. Figures 1–3)but for the additional premises ` ◦�GF and ` ◦GF respectively. They indi-cate that GF and �GF are now to be considered as ‘contradiction-intolerant’,in order for the proof of Gpar

1 to succeed (otherwise we can find counterex-amples employing Isabelle’s integrated model finder Nitpick [BN10]).

Figure 4: Proof of non-provability of GparF (¬ behaves paraconsistently)

Notice in Figure 4 (line 27) that the (LFI) consistency assumption for�GF (line 21) is required as an additional premise for lemma non_provable,this is in order to derive ` ¬GF from ` �GF and the fixed-point lemmaemploying contraposition. Recall from Theorem 4.6 that contraposition isnot valid in logic RmbC (nor generally in LFIs), unless the consequent isconsistent.

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Figure 5: Proof of non-refutability of GparF , employing ∗-consistency

Similarly, observe that for lemmas non_refutable_v1(2) in Figures 5 and6 (in lines 41–42 and 57–58 resp.) the consistency of GF needs to be assumedin order to license the corresponding contraposition step. Moreover, noticein Figure 6 how the final step involves the LFIs’ definition of ⊥ (Def. 4.3).15

Figure 6: Proof of non-refutability of GparF , employing S-consistency

It is worth mentioning that Gpar1 (more specifically the non-refutable

lemma) can also be proved mechanically in Isabelle employing the notionof P -consistency-a 6` �⊥. The variant of Gpar

1 employing ◦-consistency15Recall that, while the rule of negation introduction ` ϕ→ ⊥ =⇒ ` ¬ϕ holds for

LFIs, its converse ` ¬ϕ =⇒ ` ϕ→ ⊥ does not hold in general, unless ◦ϕ is assumed.

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∀ϕ. ` ◦ϕ can be falsified (i.e. a countermodel is found by Nitpick). Inter-estingly, the question concerning the validity of the variant featuring P -consistency-b ` ∼�⊥ has been answered affirmatively by the higher-orderprover Leo-III.16

Another interesting question concerns the validity of Gpar1 in logic com-

binations featuring LFIs stronger than RmbC, after dropping any of theadditional premises ` ◦G or ` ◦�G. We know that both premises are neces-sary when considering the logic RmbC (since Nitpick can find countermodelsotherwise). Moreover, we could also verify that both premises are still neces-sary when considering the extension RmbCciw, which is the weakest LFIin which ◦ is a non-primitive connective, defined as: ◦ϕ := ∼(ϕ ∧ ¬ϕ).However, this question still remains open for stronger LFIs, as we could notyet find proofs or countermodels using automated tools.

6.3 Second incompleteness theorem (G2)

We formalize two variants of the proof for G2, the first one draws from[Smo77] and [Raa18] (see Figures 7 and 9), and the second one draws fromBoolos [Boo95, Ch. 3] (see Figures 8 and 10).

Theorem 6.2 (Gödel’s second incompleteness theorem G2). Assume F is aconsistent formal system which contains Peano Arithmetic. Let ConsF be aformula of F representing the consistency of the system. We also assume thederivability conditions on PrF from Proposition 5.6. We have two variants:

(i) Let GF be some formula which satisfies `F GF ↔ ∼PrF (dGF e). Wehave `F GF ↔ ConsF . As a corollary it follows by (the first part of)G1 that 6`F ConsF .

(ii) From Löb’s theorem (Proposition 5.9), it follows that 6`F ConsF .

Proof. To be discussed below.

Throughout this section we will employ the notion of P -consistency-b asa working formalization for ConsF above, following the established practice(cf. [Boo95]). Other choices (drawing upon Definition 5.11) are possible, butthey require some questionable modifications (see the discussion at the endof this section).

16Leo-III [SB18], when invoked via Isabelle’s meta-prover Sledgehammer [BBP13], in-deed reported a proof, which, however, could not be automatically reconstructed employingIsabelle’s trusted kernel calculus. This situation may (hopefully) change in the future, asthe integration between Isabelle and external theorem proving systems keeps improving.

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6.3.1 Proof reconstruction of G2 (using classical negation)

The reconstructed proofs for G2 presented in Figures 7 and 8 employ classi-cal modal logic K. Observe that under the premises for variant (i) we find∀ϕ. ` �ϕ→ ��ϕ (modal axiom 4), while under the premises for (ii) we find∀ϕ. ` �(�ϕ→ ϕ)→ �ϕ (axiom L). The logic of formalization can thus beseen as K4 and GL for variants (i) (Figure 7) and (ii) (Figure 8) respectively.

Figure 7: Proof of (i) (equivalence of GF and ConsF ) in Isabelle/HOL

The variant (i) in Figure 7 shows that ConsF (defined employing P -consistency-b: ∼�⊥) is equivalent to GF (line 23) as a consequence ofthe fixed-point lemma and the derivability conditions for PrF (i.e. assuminglogic K4, see Section 5). As mentioned before, given the first half of G1

(6`F GF ), the second incompleteness theorem G2 (6`F ConsF ) follows im-mediately (using replacement of equivalents, see Section 4). The proof hasbeen divided into two parts: left-to-right (LtoR, lines 25–29) and right-to-left(RtoL, lines 31–32). The former has been reconstructed formally for illustra-tion purposes (by now the reader should be able to follow the steps guidedby the provided comments); the latter has been proved employing the SMTsolvers integrated into Isabelle (e.g. Z3 and CVC4 ).17

17The required premises have been suggested by Sledgehammer. We take this oppor-tunity to showcase the use of automated tools in the verification of complex proof steps.The concerned reader can easily find the skipped proof steps, e.g. in [Smo77].

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Figure 8: Proof of (ii) (i.e. G2) in Isabelle/HOL

Observe that the variant (ii) in Fig. 8 proves G2 directly in logic GL asa corollary of Löb’s axiom (which formalizes Löb’s theorem in modal logic,see Proposition 5.9).

It is worth mentioning that formulations of the notion of system’s con-sistency (ConsF ) other than ∼�⊥ (which draws upon P -consistency-b, asintroduced in Definition 5.11) could be used to try to formalize G2 (i − ii)above. In doing this, they would need to be rendered as object-logical for-mulas (i.e. inside the turnstile brackets [` . . . ]). For instance, we could tryto rephrase ∀φ. ` �φ =⇒ ` φ (S-consistency) in the object (second-order)modal logic as ∀φ. ��φ → �φ, arguably honoring its intuitive interpreta-tion. Notice, however, that the frames characterized by the former (converse-serial) are quite different from the ones characterized by the latter (dense).In a similar vein, other variants like ∀ψ. 6` ∼ψ ∧�ψ (∗-consistency) and6` �⊥ (P -consistency-a) could become rendered as the object-logical formu-las ∀ψ.∼�(∼ψ ∧�ψ) and ∼��⊥ respectively. However, the adequacy ofsuch paraphrasing is also questionable, and the obtained formulations forConsF do not validate variant (i), as we could verify by generating counter-models with the help of Nitpick. As regards variant G2 (ii), we found thatthe formulations drawing on variants P -consistency-a and ∗-consistency canbe proved automatically by Leo-III (though not by Isabelle, cf. Footnote 16).As for the variant drawing on S-consistency, its status remains unsettled, asit could not be proved or refuted by automated means.18

18At the present time, the conjectures cannot be proved either by Isabelle or by Leo-III,while model finder Nitpick cannot find a counterexample. A manual proof reconstructionmight need to be attempted. Isabelle-acquainted readers are encouraged to download thesources [CF20] and try themselves with higher chances of success.

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6.3.2 Proof reconstruction of Gpar2 (using paraconsistent negation)

The reconstructed proofs for Gpar2 presented in Figures 9 and 10 are formu-

lated in the logic RmbC⊕K (which features the paraconsistent negation¬). They illustrate how some further (implicit) assumptions, concerning the◦-consistency of particular formulas, are required in order to reconstruct theproof as logically valid.

Figure 9: Proof of Gpar2 (i) (equivalence of GF and ConsF )

Like its classical counterpart in Figure 7, the paraconsistent variantGpar2 (i)

in Figure 9 derives the equivalence of ConsF (¬�⊥) and GF in K4. Noticethat, similarly to the proof reconstruction of Gpar

1 , some further (implicit)premises are required in order for contraposition steps to succeed, namely,the ◦-consistency of �GF (lines 25–27) and of �⊥ (required by the SMTsolver, lines 29–30). Note that we can obtain countermodels (employingmodel finder Nitpick) if we drop any of these assumptions.

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Figure 10: Proof of Gpar2 (ii)

Like its classical counterpart (cf. Figure 8), the variant (ii) in Figure 10derivesGpar

2 directly in the logic GL. Observe that this time the ◦-consistencyof �⊥ (line 36) is required as sole further (implicit) premise, in order to getthe step from ` ¬�⊥ to ` �⊥→⊥ off the ground (line 40) as well as thelast step, which hinges on the LFIs’ definition of ⊥ (Def. 4.3).

It is worth mentioning that we can additionally employ ∀ϕ.◦ϕ (drawingupon ◦-consistency) as yet another well-formed object-logical formulation ofConsF in the reconstruction of Gpar

2 (similarly to the discussion at the endof Section 6.3.1 regarding G2). Here, too, we obtain similar results. Noneof the alternative formulations of ConsF (in their paraconsistent versions)can validate Gpar

2 (i), since we could generate respective counterexamples us-ing Nitpick. As regards Gpar

2 (ii), it cannot be validated by employing theformulation of ConsF drawing on ◦-consistency mentioned above, nor byemploying ¬��⊥ (drawing on P -consistency-a). As for the formulations ofConsF : ∀ψ.¬�(¬ψ ∧�ψ) (drawing on ∗-consistency) and ∀φ. ��φ→ �φ(drawing on S-consistency), the status of Gpar

2 (ii) remains unsettled (recallthe discussion in Footnote 18). We refer the reader to the correspondingIsabelle sources in [CF20] for detailed (and updated) information.

Similarly to the discussion at the end of Section 6.2.2, the question con-cerning the validity of Gpar

2 in logic combinations featuring LFIs strongerthan RmbC, after dropping any of the additional premises (` ◦�⊥ or` ◦�G), still remains open, as we could not yet find proofs or countermodelsemploying automated tools.19

19Thus far we haven’t meticulously tried to reconstruct such proofs, so chances are thatIsabelle-acquainted readers will have better luck here.

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7 A long story short: conclusionsThe gist of our analysis appears in Sections 6.2.2 and 6.3.2. Our results showthat a necessary and sufficient condition to validate a paraconsistent ver-sion Gpar

1 of Gödel’s first incompleteness theorem is to assume that both GF

and �GF (i.e. PrF (GF )) are consistent in the LFI sense of ‘contradiction-intolerant’; such conditions are clearly sufficient, and also necessary,20 sinceotherwise we can find counterexamples to Gpar

1 by exploiting the model gen-eration capabilities of Isabelle. This is mainly due to the failure of explosionand contraposition in the logic RmbC (and generally in LFIs), unless the◦-consistency of certain sentences is assumed.

Similarly to the proof reconstruction of Gpar1 , the paraconsistent ver-

sion Gpar2 of Gödel’s second incompleteness theorem requires some further

premises in order for contraposition steps to succeed, namely, the ◦-consistencyof �GF and �⊥ for the first presented variant (i). In this variant, Gpar

2 fol-lows as a corollary of (the first part of) Gpar

1 , and thus the ◦-consistency of�GF and �⊥ constitute sufficient conditions for obtaining Gpar

2 . As regardsthe second variant (ii) of Gpar

2 , we have seen that only the ◦-consistency of�⊥ is required. Recall that, in contrast to the first variant (i), variant (ii)of Gpar

2 does not depend on G1, but relies on Löb’s theorem instead.We may summarize, in a sketchy form, the ideas above as follows:

FP (GF ) and ◦GF and ◦�GF imply G1;

where FP (GF ) stands for ‘GF is a fixed-point for ¬�(·)’ and G1 stands for‘ConsistencyF implies IncompletenessF ’. Reasoning by contraposition:21

FP (GF ) and not G1 imply not ◦GF or not ◦�GF .

A similar reasoning applies to G2, as the reader can verify. Based on this,in an exercise of counterfactual imagination, we can envision that, if thingshad been different in the thirties (e.g. if logics like LFIs already existed), theGödelian results could have been presented along the following lines:

Theorem 7.1 (Gödel’s Existence Theorem). For every consistent ( 6= non-contradictory) and complete formal system F, which includes Peano Arith-metic, a sentence GF can be constructed such that it or its provability PrF (GF )is a non-consistent statement (i.e. it is ‘contradiction-tolerant’). Moreover,if F can prove its own consistency, then PrF (⊥) is also a non-consistentstatement.

20We mean, necessary and sufficient within the present line of reasoning – we don’t claimours is the unique way to formulate the problem.

21We can do this, since our reasoning (meta-logic) is taken to be classical.

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A less technical, more conceptual conclusion is that, by adopting a lighter,more flexible negation (such as the paraconsistent negation featured in RmbC),we genuinely avoid the Gödelian objection, which is mistakenly taken to beuniversal (although Gödel himself never saw it this way). To be sure, Gödel’sargumentation is still sound, it just becomes interpreted, more appropriately,as an existence claim.

Limitations of the Gödelian objection are totally understandable, spe-cially if we take into account that the objection was directed against thefoundations of mathematics, whose notion of negation is the classical one,with its brutal simplification, conflating negation, denial, subtraction andfalsity in just one idea. But the Gödelian objection was not directed againstthe subtle linguistic and pragmatic usage of negation, nor at its usage incontemporary areas like knowledge representation in computer science. Thenon-mathematical usage of negation needs to adhere to some additional pos-tulates to fall prey of Gödel’s arguments; we have shown that the consistency(in the sense of LFIs, namely, ‘contradiction-intolerance’) of the formulasGF and PrF (GF ) is among them. In this respect, an interesting questionis whether there are any ‘natural’ mathematical statements (i.e. not involv-ing the numerical coding of logical notions) which could be shown to beundecidable in our basic paraconsistent systems, as much as the celebratedParis–Harrington theorem is a ‘natural’ undecidable combinatorial statementin the standard case. This and similar issues deserve further investigation.

It is important to note that our experiments in Section 6 have been farfrom exhaustive, and thus further attempts may reveal an adequate formula-tion (or combination) of these, and other, consistency notions which (possiblytogether with some additional premises) validly reconstructs G2 and Gpar

2 . Asimilar observation applies, to a lesser extent, to G1 and Gpar

1 . In particular,we did not investigate formalizations featuring both classical and paracon-sistent negations inside a same formula or appearing in premises of a sameargument. Such experiments are straightforward to realize using our Isabellesources [CF20]. However, the interpretation of the results is not always ap-parent, and a proper analysis for every combination variant (let alone pickingout the relevant or reasonable ones) exceeds the scope of this paper. Anotherinteresting set of experiments involves strengthening our base logic RmbCwith further axioms (i.e. employing stronger logics in the LFI hierarchy) andthen using automated tools to verify whether Gödel’s theorems can be provedin this setting without recurring to additional premises. As mentioned at theend of Sections 6.2.2 and 6.3.2, there are still open questions in this regard,as this task requires quite more effort as well as familiarity with the Isabelleenvironment. The present study aims at paving the way for tackling theseand similar fascinating challenges.

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Acknowledgements: The first author acknowledges support from the Na-tional Council for Scientific and Technological Development (CNPq), Brazil,under research grant 307376/2018-4. We are indebted to Marcelo Conigliofor early discussions on these ideas and to Christoph Benzmüller for adviceon utilizing automated reasoning systems.

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