Connexive Logic

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.

Questa tesi intende presentare l’implicazione connessiva da due punti di vista: uno logico (parte I) e uno storico (parte II). Nell’introduzione si darà un rapido sguardo ai paradossi dell’implicazione materiale e si indicheranno alcuni tentativi di soluzione: in particolare, si presenteranno i vari approcci connessivi, con particolare attenzione per quello di Graham Priest. Nel capitolo 1 della parte I, si presenterà sintassi e semantica della logica connessiva PrC, così come è esposta in Priest [47], servendosi dei modelli kripkeani. Si mostreranno anche alcune proprietà peculiari della nozione di conseguenza logica connessiva: essa è transitiva e chiusa sotto modus ponens ma non è riflessiva, né monotona né chiusa sotto sostituzione uniforme. I principi connessivi sono validi in tale semantica, però delle tesi di Boezio sono valide solo le versioni deboli, ovvero quelle che hanno per connettivo principale un’implicazione materiale. Verrà poi definita una nuova semantica per PrC, equivalente alla precedente, ma la cui relazione di accessibilità è esplicitata ed è una relazione di equivalenza. Inoltre, si mostrerà come sia possibile, tramite una funzione di traduzione e la sua inversa, tradurre, salva veritate, formule di PrC in formule della logica modale S5 e viceversa. Nel capitolo 2, verrà rapidamente descritto un calcolo di sequenti etichettati per S5 e si accennerà alle proprietà del suddetto calcolo. In seguito, si definirà un calcolo etichettato per PrC (rispetto alla semantica con relazione di equivalenza), chiamato G3.PrC: se ne presenteranno le regole e si dimostreranno i due risultati fondamentali di validità e completezza per G3.PrC. Nella parte II, analizzeremo il passo degli Analitici Primi nel quale alcuni logici rintracciano alcune delle più famose formule connessive. Dopo un commento del passo e una rapida panoramica su alcune delle più autorevoli esegesi del brano, si esporrà una nuova interpretazione del testo, che si basa sui capitoli degli Analitici che fanno da contesto al passaggio in questione: lo Stagirita vorrebbe dimostrare che una conclusione vera non può seguire per necessità da premesse false. Si motiveranno, sulla base della filosofia aristotelica, le due tesi che fungono da base alla nuova interpretazione proposta: (i) le regole logiche cui Aristotele fa riferimento sono semantiche (in termini moderni: lo Stagirita starebbe parlando della nozione di conseguenza logica); (ii) con le due lettere A e B, Aristotele intende le due premesse di un sillogismo (A) e la conclusione (B) e non due enunciati qualsiasi (non sarebbero, dunque, variabili enunciative tout court). Infine, si analizzeranno le tre figure sillogistiche per saggiare la validità del ragionamento aristotelico nella nuova esegesi data, mostrando come, sebbene insorgano alcune difficoltà, sia possibile affermare che la nuova interpretazione valida l’argomento dello Stagirita.

ABSTRACT: This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen's negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall's connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.

In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of what may be thought of as containment logics or Parry systems in general—is a strong relevance property called the ‘Proscriptive Principle’ (PP) described by Parry as the thesis that:

No formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent.

This type of proscription is on its face justified, as the presence of a novel parameter in the consequent corresponds to the introduction of new subject matter. The plausibility of the thesis that the content of a statement is related to its subject matter thus appears also to support the validity of the formal principle.

Primarily due to the perception that Parry’s formal systems were intended to accurately model Kant’s notion of an analytic judgment, Parry’s deductive systems—and the suitability of the Proscriptive Principle in general—were met with severe criticism. While Anderson and Belnap argued that Parry’s criterion failed to account for a number of prima facie analytic judgments, others—such as Sylvan and Brady—argued that the utility of the criterion was impeded by its reliance on a ‘syntactical’ device.

But these arguments are restricted to Parry’s work qua exegesis of Kant and fail to take into account the breadth of applications in which the Proscriptive Principle emerges. It is the goal of the present work to explore themes related to deductive systems satisfying one form of the Proscriptive Principle or other, with a special emphasis placed on the rehabilitation of their study to some degree. The structure of the dissertation is as follows:

In Chapter 2, we identify and develop the relationship between Parry-type deductive systems and the field of ‘logics of nonsense.’ Of particular importance is Dmitri Bochvar’s ‘internal’ nonsense logic Σ0, and we observe that two ⊢-Parry subsystems of Σ0 (Harry Deutsch’s Sfde and Frederick Johnson’s RC) can be considered to be the products of particular ‘strategies’ of eliminating problematic inferences from Bochvar’s system.

The material of Chapter 3 considers Kit Fine’s program of state space semantics in the context of Parry logics. Recently, Fine—who had already provided the first intuitive semantics for Parry’s PAI—has offered a formal model of truthmaking (and falsemaking) that provides one of the first natural semantics for Richard B. Angell’s logic of analytic containment AC, itself a ⊢-Parry system. After discussing the relationship between state space semantics and nonsense, we observe that Fabrice Correia’s weaker framework—introduced as a semantics for a containment logic weaker than AC—tacitly endorses an implausible feature of allowing hypernonsensical statements. By modelling Correia’s containment logic within the stronger setting of Fine’s semantics, we are able to retain Correia’s intuitions about factual equivalence without such a commitment. As a further application, we observe that Fine’s setting can resolve some ambiguities in Greg Restall’s own truthmaker semantics.

In Chapter 4, we consider interpretations of disjunction that accord with the characteristic failure of Addition in which the evaluation of a disjunction A ∨ B requires not only the truth of one disjunct, but also that both disjuncts satisfy some further property. In the setting of computation, such an analysis requires the existence of some procedure tasked with ensuring the satisfaction of this property by both disjuncts. This observation leads to a computational analysis of the relationship between Parry logics and logics of nonsense in which the semantic category of ‘nonsense’ is associated with catastrophic faults in computer programs. In this spirit, we examine semantics for several ⊢-Parry logics in terms of the successful execution of certain types of programs and the consequences of extending this analysis to dynamic logic and constructive logic.

Chapter 5 considers these faults in the particular case in which Nuel Belnap’s ‘artificial reasoner’ is unable to retrieve the value assigned to a variable. This leads not only to a natural interpretation of Graham Priest’s semantics for the ⊢-Parry system S⋆fde but also a novel, many-valued semantics for Angell’s AC, completeness of which is proven by establishing a correspondence with Correia’s semantics for AC. These many-valued semantics have the additional benefit of allowing us to apply the material in Chapter 2 to the case of AC to define intensional extensions of AC in the spirit of Parry’s PAI.

One particular instance of the type of disjunction central to Chapter 4 is Melvin Fitting’s cut-down disjunction. Chapter 6 examines cut-down operations in more detail and provides bilattice and trilattice semantics for the ⊢-Parry systems Sfde and AC in the style of Ofer Arieli and Arnon Avron’s logical bilattices. The elegant connection between these systems and logical multilattices supports the fundamentality and naturalness of these logics and, additionally, allows us to extend epistemic interpretation of bilattices in the tradition of artificial intelligence to these systems.

Finally, the correspondence between the present many-valued semantics for AC and those of Correia is revisited in Chapter 7. The technique that plays an essential role in Chapter 4 is used to characterize a wide class of first-degree calculi intermediate between AC and classical logic in Correia’s setting. This correspondence allows the correction of an incorrect characterization of classical logic given by Correia and leads to the question of how to characterize hybrid systems extending Angell’s AC∗. Finally, we consider whether this correspondence aids in providing an interpretation to Correia’s first semantics for AC.

In this paper, we reflect on some themes related to the formulation of mathematics against the backdrop of a connexive logic. From a positive perspective, we will consider some remarks of the Kneales concerning Aristotle's position on connexive implication and suggest that common themes between the Kneales' Aristotle and the hyper-constructive arithmetic of David Nelson may provide a philosophical basis for connexive mathematics. We will also consider some historical points, including Lukasiewicz' argument that connexive principles may be refuted by appeal to number-theoretic intuitions. Finally, we will take more concrete steps towards the implementation of connexive mathematics by examining how weak subtheories of arithmetic fare when formulated in modest first-order extensions of three connexive logics: Richard Angell's PA1 and PA2 and Graham Priest's P_N. Unfortunately, we will observe that severe pathologies emerge when even extraordinarily weak subsystems of Peano arithmetic are evaluated in these logics, suggesting that Angell and Priest's systems constitute strained, if not unserviceable, bases for arithmetic.

However broad or vague the notion of connexivity may be, it seems to be similar to the notion of relevance even when relevance and connexive logics have been shown to be incompatible to one another. Relevance logics can be examined by suggesting syntactic relevance principles and inspecting if the theorems of a logic abide to them. In this paper we want to suggest that a similar strategy can be employed with connexive logics. To do so, we will suggest some properties that seem to be hinted at in Nelson's work. Following this strategy will ideally shed some light over the notion of content and will also help make a clear comparison between relevance and connexive logics.

In §1 it is shown that Angell, Reichenbach, Chisholm and Goodman agreed in giving legitimacy to the (weak) Boethius' Thesis A → B -> -(A → -B) (BT), but did not perceive the difference between the properties of analytic and context-dependent (synthetic) conditionals, mainly exemplified by counterfactual conditionals. The basic idea of the paper is to define, into a logic which has BT as a basic axiom, synthetic conditionals in terms of the analytic ones and of the so-called "circumstantial" operator *, whose properties are derived from the properties of selection functions in the way shown in §2. Systems of analytic consequential implication, differing from the ones of connexive implication, are translatable into systems of normal modal logic via the definitions A → B =df □(A -> B) & X (A,B) (where X(A,B) stands for □A≡□B & <>A ≡ <> B) and  □ A=df T → A. This fact allows defining the synthetic operators in terms of → and *, or alternatively in terms of □ and *. Plausible definitions of the synthetic operator such as A >' B =df *A → B and A > B =df *A => B (=> being a weak variant of →) are examined and rejected. The definitive proposal is to introduce, in a system named CI.0*Eq, a new operator >> by the definition A >> B = df □(*A -> B) & X (A,B). It is proved that the operator defined in this way satisfies all the properties required for the synthetic consequential operators, in the first place BT in the variant A >> B -> -(A >> -B). § 3 is devoted to identifying the >>-fragment of CI.0*Eq in the following way. A system CI.0>>, written in → and >> , is proved to be complete with respect to models <W, f, R,V>, where f is a selection function. It is then proved that there is an embedding function t from CI.0>> to CI.0*Eq, and beyond this another one, Tr, from CI.0*Eq to KT w , where KT w is a tableaux-decidable extension of the modal system KT written in a language which contains a new operator w which allows translating *A into w(A) & A. So there is a composed embedding function Trt of CI.0>> into KT w. On this basis, via a conversion of CI.0>>-models into KT w-models, it is proved (i) that A is a CI.0-thesis iff Tr t(A) is a KT w-thesis and (ii) tA is a CI.0*Eq-thesis iff A is a CI.0>>-thesis, which establishes the desired result. Thanks to Trt CI.0>> turns out then to be a tableaux-decidable system. The final section contains a discussion of Simplification of Disjunctive Antecedents and of other laws laws which mark important differences between classical and consequential conditional logic.

The paper introduces a variant of connexive logic in which connexivity is
extended from an interaction of negation with implication to an interaction
of negation also with conjunction and disjunction.

The logic is presented by two deductively equivalent methods:
an axiomatic one and a natural-deduction one. Both are shown to be complete
for a four-valued model-theory.

In this paper, a connexive extension of the relevance logic R→ is presented. It is defined by means of a natural deduction system, and a deductively equivalent axiomatic system is presented too. The goal of such an extension is to produce a logic with stronger connection between the antecedent and the consequent of an implication.

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by → and ⇒). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for ⇒, CI⇒, is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for → CI. The main drawback ot the weaker operator ⇒ is that it lacks unrestricted contraposition, but the final section of the paper argues that ⇒ has some properties which make it a valuable alternative to →, turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.

This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance, through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by → and ⇒). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for ⇒, CI⇒, is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for → CI. The main drawback ot the weaker operator ⇒ is that it lacks unrestricted contraposition, but the final section of the paper argues that ⇒ has some properties which make it a valuable alternative to →, turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.

Just as Hartley Slater famously questioned the meaning of paraconsistency, I want to do the same hereby with connexivity, i;e., to talk about about what "connexive logics" are taken to be the name of.

The talk consists of three parts.

Firstly, we consider in which sense connexive logics deal with connexivity in the four main "connexive theses" (two Aristotelian, two Boethian). Several logical concepts are related to these: implication, compatibility, consistency, soudness, contradiction, impossibility.

Second, we see in what sense this special class of formulas requires a special logic: (a) as theorems (or tautologies) of the form |- A -> B; (b) as semantic consequences of the form A |- B; (c) as denied inferences of the form A |-/ B. We claim that the first reading is too strong, the second is trivial for some cases, and the third is too weak.

Thirdly, we propose a logical system and its "modal" extension to validate the connexive theses. The first system is a four-valued logic of acceptance and rejection, AR4, which is a non-classical or deviant extension of Dunn-Belnap's FDE including a logical constant of strong implication. The second system is a doxastic logic AR4B, which includes at least four different sorts of epistemc agents with various criteria of justification. We show that one of these agents makes sense of connexive statements, thereby fulfilling the requirement (a) by a specific translation of the four connexive theses into AR4B. We also show that this belief operator behaves like the modal operator of necessity without falling into the net of Dugundji's Theorem (i.e., there is no finite characteristic matrix of any of the Lewis' systems between S1 and S5).

The final result is an interpretation of connexivity in a fourth sense (d), namely, as a set of valid theorems which rely upon a "non-normal" constant of implication. The difference with "normal" implication lies in the relation between inference rules and consequence as a truth-preserving relation: Modus Ponens is still a truth-preserving relation from premise to consequence in AR4, whereas Modus Tollens is a non-falsity preserving relation.

Two final appendices help to illustrate this added distinction, together with a square of belief oppositions that show the sound behavior of doxastic agents from a connexive perspective.