Many prominent systems of non-classical logic are subsystems of whatis generally called ‘classical logic.’ Systems ofconnexive logic arecontra-classical in the sense that theyare neither subsystems nor extensions of classical logic. Connexivelogics have a standard logical vocabulary and comprise certainnon-theorems of classical logic as theses. Since classicalpropositional logic is Post-complete, any additional axiom in itslanguage gives rise to the trivial system, so that any non-trivialsystem of connexive logic will have to leave out some theorems ofclassical logic. The name ‘connexive logic’ was introducedby Storrs McCall (1963, 1964) and suggests that systems of connexivelogic are motivated by some ideas about coherence or connectionbetween the premises and the conclusions of valid inferences orbetween the antecedent and the succedent (consequent) of validimplications. The kind of coherence in question concerns the meaningof implication and negation (see the entries onindicative conditionals,the logic of conditionals,counterfactuals, andnegation). One basic idea is that no formula provably implies or is implied byits own negation. This conception may be expressed by requiring thatfor every formulaA,

⊬ ~A →A and ⊬A → ~A,

but usually the underlying intuitions are expressed by requiring thatcertain schematic formulas are theorems:

AT: ~(~A →A) and
AT′: ~(A → ~A).

The first formula is often calledAristotle’s Thesis.If this non-theorem of classical logic is found plausible, then thesecond principle, AT′, would seem to enjoy the same degree ofplausibility. Indeed, also AT′ is sometimes referred to asAristotle’s Thesis, for example in Routley 1978, Mortensen 1984,Routley and Routley 1985, and Ferguson 2016. As McCall (1975, p. 435)explains,

[c]onnexive logic may be seen as an attempt to formalize the speciesof implication recommended by Chrysippus:

And those who introduce the notion of connection say that aconditional is sound when the contradictory of its consequent isincompatible with its antecedent. (Sextus Empiricus, translated inKneale and Kneale 1962, p. 129.)

Using intuitionistically acceptable means only, the pair of theses ATand AT′ is equivalent in deductive power with another pair ofschemata, which in established terminology are called(Strong)Boethius’ Theses (cf. Routley 1978) and which may beviewed, in addition with their converses, as capturingChrysippus’ idea:

BT: (A →B) → ~(A →~B) and
BT′: (A → ~B) → ~(A →B).

The names ‘Aristotle’s Theses’ and‘Boethius’ Theses’ are, of course, not arbitrarilychosen. As to AT, it is argued in Aristotle’sPriorAnalytics 57b14 that it is impossible that if not-A,thenA, see Łukasiewicz 1957, p. 50. Note, however, thatŁukasiewicz and Kneale (1957, p. 66) maintain that Aristotle ismaking a mistake here. Moreover, Boethius has been said to hold inDe Syllogismo Hypothetico 843D that ‘ifA thennot-B’ is the negation of ‘ifA, thenB’, (“he said that the negative ofSi est A,est B wasSi est A, non est B,” Kneale and Kneale1962, p. 191). If we look atDe Syllogismo Hypothetico843C-D, we find:

Sunt autem hypotheticae propositiones, aliae quidem affirmativae,aliae negativae […] affirmativa quidem, ut cum dicimus, si esta, estb; si non esta, estb;negativa, si esta, non estb, si non esta, non estb. Ad consequentem enim propositionemrespiciendum est, ut an affirmativa an negativa sit propositiojudicetur. (Note that there is a misprint in Migne’s editionfrom 1860, where instead of the first occurrence of “si non esta, estb” in the above quote there is“si non esta, non estb”.)

Boethius here draws a distinction between affirmative and negativeconditionals and explains that negative conditionals have the form‘ifa, then notb’ and ‘if nota, then notb.’ This statement is quitedifferent from the reading offered by Kneale and Kneale. Note that ATfollows from BT′ and AT′ follows from BT in logics inwhichA →A is theorem and modus ponens is anadmissible inference rule.

LetL be a language containing a unary connective ~(negation) and a binary connective → (implication). A logicalsystem in a language extendingL is called aconnexivelogic if AT, AT′, BT, and BT′ are theorems and,moreover, implication is non-symmetric, i.e., (A →B) → (B →A) fails to be atheorem (so that → can hardly be understood as a bi-conditional).This is the now standard notion of connexive logic. The connective→ in a system of connexive logic is said to be aconnexiveimplication.

Systems of connexive logic have been motivated and arrived at bydifferent considerations. One motivation comes from relevance logicand the idea that semantic consequence is a content relationship, seesection 3.1. Moreover, principles of connexive logic have beendiscussed in conditional logic, see section 3.2 and the entries onindicative conditionals,the logic of conditionals, andcounterfactuals, in different accounts of negation, see section 3.3 and the entry onnegation, and in approaches to contra-classical logics, see section 3.4.Another motivation emerges from empirical research on theinterpretation of negated conditionals in natural language and the aimto adequately model the semantical intuitions revealed by theseinvestigations, see section 3.5.

Richard Angell in his seminal paper on connexive logic (1962) aimed atdeveloping a logic of subjunctive, counterfactual conditionals inwhich what he called a ‘principle of subjunctivecontrariety,’ ∼((A →B) ∧(A → ~B)), is provable. His proof system,PA1, contains BT as an axiom. Also Kapsner and Omori(2017) suggest that a connexive implication is suitable forformalizing counterfactual conditionals, whereas Cantwell (2008), forexample, suggested a system of connexive logic to formalize indicativenatural language conditionals. According to McCall (1975, p. 451),“[o]ne of the most natural interpretations of connexiveimplication is as a species of physical or ‘causal’implication,” and in McCall (2014) he argues that “[t]helogic of causal and subjunctive conditionals is … connexive,since ‘If X is dropped, it will hit the floor’ contradicts‘If X is dropped, it will not hit the floor’.”Boethius’ Thesis BT indeed appears on a list of principles every“precausal” connective should satisfy, see Urchs 1994.McCall (2012, p. 437), however, concedes that “causal logic isstill very much an ongoing project, and no agreed-on formulation of ithas yet been achieved.” Moreover, the characteristic connexiveprinciples are valid for the analysis of conditionals, generics, anddisposition statements presented in van Rooij and Schulz 2019,2020.

Further motivation for systems of connexive logic comes from moreinstrumental studies. In McCall 1967, connexive implication ismotivated by reproducing in a first-order language all valid moods ofAristotle’s syllogistic (see the entry onAristotle’s logic). In particular, the classically invalid inference from ‘AllA isB’ to ‘SomeA isB’ is obtained by translating ‘SomeA isB’ as ∃x(~(A(x) →~B(x))), where → is a connexive implication. InWansing 2007, connexive implication is motivated by introducing anegation connective into Categorial Grammar in order to expressnegative information about membership in syntactic categories (see theentry ontypelogical grammar). Consider, for example, the syntactic category (type) (n→s) of intransitive verbs, i.e., of expressions that incombination with a name (an expression of typen) result in asentence (an expression of types). The idea is that anexpression is of type ~(n →s) iff incombination with a name it results in an expression that is not asentence. In other words, an expression belongs to type ~(n→s) iff it is of type (n → ~s).In the short note Besnard 2011, Aristotle’s thesis AT′ ismotivated as expressing a notion of rule consistency for rule-basedsystems in knowledge representation. A further motivation arises fromthe problem of modelling conditional obligations in deontic logic.Weiss (2019) suggests to understand a certain implication thatvalidates Aristotle’s theses and weak versions ofBoethius’ theses (cf. sections 2 and 3.2) as expressing aconditional obligation operator. Yet another motivation arises fromthe problem of modelling conditional obligations in deontic logic.Another motivation in terms of applications comes from non-classicalmathematics. There is an extended literature on mathematical theoriesbased on non-classical logics, including intuitionistic, fuzzy,relevant, and linear arithmetic and paraconsistent set theory. Earlycontributions in the context of connexive logic are McCall’s1967 connexive class theory, and Wiredu’s 1974 paper onconnexive set theory. Ferguson (2016, 2019) takes up the challenge ofinvestigating the prospects for a connexive mathematics and exploresthe feasibility of a connexive arithmetic.

• 1. Diverging and additional notions of connexivity
• 2. History of connexive logic
• 3. Perspectives on connexive logic
  • 3.1 Connexivity and relevance
  • 3.2 Conditional logic
  • 3.3 Negation
  • 3.4 Contra-classicality
  • 3.5 Empirical perspective
  • 3.6 Proof-theoretical perspective
  • 3.7 Semantical perspective
• 4. Systems of connexive logic
  • 4.1 Algebraic connexive logic
  • 4.2 Connexive logic based on ternary frames for relevance logics (Australian Plan)
  • 4.3 Connexive logic based on subtraction negation
  • 4.4 Boolean connexive relatedness logic
  • 4.5FDE-based connexive logics (American Plan)
  • 4.6 Connexive modal logics
• 5. Connexive logics and consequential logics
• 6. Summary
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Diverging and additional notions of connexivity

There are several further and some diverging notions of a connexivelogic. In particular, the second decade of the 21st century has(unfortunately) seen the introduction of confusingly many new notionsof connexivity and non-uniform terminology. McCall (1966) introducedconnexive logics as systems ranging from logics in which noproposition implies or is implied by its own negation to logics inwhich BT is provable (together with non-symmetry of implication), and,similarly, Mares and Paoli (2019) characterize connexive logics assystems having some or all of AT, AT′, BT, and BT′ amongtheir theorems (without explicitly requiring non-symmetry ofimplication). In McCall 2012, AT′ and BT are said to be thedistinguishing marks of connexive logic, but note that AT andBT′ are valid in the systemCC1 due to Angell(1962) and MacCall (1966) as well. Logics in which some but not all ofAT, AT′, BT, and BT′ are provable (or valid) are called‘demi-connexive’ in Wansing, Omori, and Ferguson 2016(without explicitly requiring non-symmetry of implication), and aresaid to bequasi-connexive in Jarmużek and Malinowski2019a. The identification of the negation of (A →B) with (A → ~B), ascribed to Boethiusby Kneale and Kneale (1962), suggests a strengthening of BT andBT′ to the equivalences:

BTe: (A →B)↔ ~(A→ ~B), and
BTe′: (A → ~B) ↔ ~(A →B).

Sylvan 1989 refers to BTe as a principle ofhyperconnexivelogic. The principles BTe and BTe′ are characteristic of theconnexive logics developed subsequent to the definition of theconnexive logicC and its quantified versionQC in Wansing 2005. According to McCall (2012), theconverse of BT (the right-to-left direction of BTe) is highlyunintuitive in light of what he takes to be counterexamples fromEnglish. For a rejoinder see Wansing and Skurt 2018.

Kapsner (2012, 2019) refers to a logic that satisfies AT, AT′,BT, and BT′ and, moreover, satisfies the requirements

Unsat1: In no model,A → ~A is satisfiable (foranyA), and in no model,~A →A issatisfiable (for anyA), and
Unsat2: In no model,A →B andA→ ~B are simultaneously satisfiable (for anyAandB)

asstrongly connexive, whereas if the conditions Unsat1 andUnsat2 are not both satisfied, the system is only calledweaklyconnexive. Kapsner motivates the extra conditions by twointuitions, namely that it is not the case that a formulaAshould imply or be implied by its own negation, and that ifAimpliesB, thenA does not imply not-B (andifA implies not-B, thenA does not implyB). These intuitions may, however, also be seen to motivate⊬ ~A →A and ⊬A → ~A, respectively (A →B) ⊬(A → ~B) and (A → ~B)⊬ (A →B) instead of Unsat1 and Unsat2.Moreover, imposing Unsat1 and Unsat2 precludes systems that satisfythe variable sharing property (i.e., broadly relevant orsociative logics in Routley’s (1989) terminology, forwhich it holds that ifA →B is a theorem, thenA andB share at least one propositional variable)and satisfy the deduction theorem from being connexive. So far onlyfew strongly connexive logics satisfying the non-symmetry ofimplication condition are known, namely the systemCC1, which, however, “is an awkward system inmany ways” (McCall 2012, p. 429), see section 4.1, and theBoolean connexive relatedness logics from Jarmużek and Malinowski2019a, see section 4.4. In Wansing and Unterhuber 2019, logics thatvalidate Boethius’ theses only in rule form ((A →B) ⊢ ~(A → ~B) and (A→ ~B) ⊢ ~(A →B)) arecalledweakly connexive. Weiss (2019) considers a languagewith classical negation, ¬, classical implication, ⊃, andanother binary conditional, →, (notation adjusted). He calls alogichalf-connexive if it validates the Weak Boethius’Theses:

BTw: (A →B) ⊃ ¬(A →¬B), and
BTw′: (A → ¬B) ⊃ ¬(A→B),

and refers to a logic as connexive if in addition it validates AT andAT′ for → and ¬. The Weak Boethius’ Thesis BTwwas introduced in Pizzi 1977 as “conditional Boethius’thesis” for the connexive implication seen as standing for acounterfactual conditional.

In Kapsner 2019, the demand of strong connexivity is evaluated as“too stringent a requirement,” and the notion ofplainhumble connexivity is introduced by restricting Aristotle’stheses, Boethius’ theses, Unsat1, and Unsat2 to satisfiableantecedents. A survey of the terminology and various notions ofconnexivity used in the literature is presented in theSupplement on Terminology. It remains to be seen whether all the notions in addition to theestablished concept of connexive logic will turn out to beconducive.

If a language is used in which implication is not taken as primitivebut is defined in terms of other connectives, connexive logics couldalso be seen as diverging from the orthodoxy of classical logic bygiving a deviant account of those connectives. A definition of aconnexive implication in terms of negation, conjunction, and necessitycan be found in McCall 1966 and Angell 1967b. More recently, Francez(2020) suggested the notion of “poly-connexivity” tohighlight a modification of the familiar falsity conditions ofconjunctions and disjunctions (in addition to adopting falsityconditions of implications as expressed by BTe′).

2. History of connexive logic

McCall (2012) emphasizes that there is a history of two thousand threehundred years of connexive implication. Historical remarks onconnexive logic may be found, for instance, in Kneale and Kneale 1962,Sylvan 1989, Priest 1999, Nasti De Vincentis 2002, Nasti De Vincentis2004, Nasti De Vincentis 2006, Estrada-González &Ramirez-Cámara 2020, and McCall 2012. In the latter survey,McCall refers to ~((A →B) ∧ (~A→B)) asAristotle’s Second Thesis and,following Martin 2004, to Angell’s principle of subjunctivecontrariety ~((A →B) ∧ (A →~B)) asAbelard’s First Principle, which iscalledStrawson’s Thesis in, for example, Routley 1978and Mortensen 1984. Aristotle’s Second Thesis andAbelard’s First Principle are interderivable with BT and withBT′, respectively, using intuitionistic principles only. BesidesPeter Abelard, another medieval philosopher who discussed and endorsedconnexive principles was Richard Kilwardby, see Johnston 2019.El-Rouayheb (2009, p. 215) reports on a critical discussion inthirteenth-century Arabic philosophy of Aristotle’s thesis ATfor impossible antecedents. Modern connexive logic commenced withNelson 1930, Angell 1962, and McCall 1966, while MacColl (1878) may beregarded as a forerunner. After small numbers of publications from the1960s until the 1990s, with S. McCall, R. Routley, and C. Pizzi as themain contributors, in the 21st century a vigorous new interest inconnexive logic emerged. Remarks on the history of modern connexivelogic can be found in sections 3–5.

One question arising from a historical point of view is that ofexegetical correctness. Can Aristotle’s and Boethius’theses indeed be traced back to Aristotle and Boethius? Lenzen (2020)believes that Aristotle and Boethius intended the theses named afterthem as “applicable only to ‘normal’ conditionalswith antecedents which are not self-contradictory.” He statescorrespondingly restricted versions of Aristotle’s theses in thelanguage of modal propositional logic, principles which according toLenzen (2019) can be found in Leibniz’s writings (after atransformation from Leibniz’s term logic into a system ofpropositional logic) and where, notation adjusted, ↠ stands forstrict implication:

LEIB1 ◊A ↠ ~(A ↠ ~A), and
LEIB2 ◊~A ↠ ~(~A ↠A),

cf. also the modalized versions of AT′ and BT in Unterhuber2016. Lenzen remarks that LEIB1 and LEIB2 are theorems of almost allsystems of normal modal logic and therefore do not lead to anynon-classical system of connexive logic. A similar observation is madein Kapsner 2019. As to Boethius, the question has been raised whetherit is adequate to render his term logic as a propositional logic (seeMartin 1991, McCall 2012), and Bonevac and Dever (2012, p. 192) referto Abelard’s First Principle as the most famous thesisattributed to Boethius but note that they fail to find it in Boethius.Irrespective of these exegetical issues, however, the challenge ofconnexive logic remains, namely to define nontrivial andwell-motivated logical systems that validate both Aristotle’sand Boethius’ theses and satisfy non-symmetry of implication.Another question arising from the long history of connexive logic isin which sense the system nowadays calledclassical logic isindeed classical. A critical discussion of the classicality ofclassical logic from the point of view of paraconsistent and connexivelogic can found in Wansing and Odintsov 2016.

A monograph developing a system of connexive logic in the context ofsolving a broad range of paradoxes is Angell 2002. The first monographdevoted to connexive logic sice its revival in the first two decadesof the 21st century, though not a comprehensive study of connexivelogic, is Francez 2021. Starting from the connexive logicsC and QC, Francez discuses topicssuch as the already mentioned idea of ``polyconnexivity'', certainvariations of Boethius’ theses (cf. section 3.6), and aconnexive theory of classes.

3. Perspectives on connexive logic

Systems of connexive logic can be looked and arrived at from differentperspectives. Although some of these viewpoints are closelyinterrelated, it may be helpful to briefly outline themseparately.

3.1 Connexivity and relevance

Routley (1978), see also Sylvan 1989 (2000, chapter 5), suggested aconception of connexive logic different from McCall’s. If therequirement of a connection between antecedent and succedent of avalid implication is understood as a content connection, and if acontent connection obtains if antecedent and succedent arerelevant to each other, then “the general classes ofconnexive and relevant logics are one and the same” (Routley1978, p. 393), cf. also Sarenac and Jennings 2003, where theconnection between McCall’s connexive systemCC1, presented in section 4.1, and relevancepreservation is studied.

Since every non-trivial system of connexive logic in the vocabulary ofclassical logic has to omit some classical tautologies, and since thestandard paradoxes of non-relevant, material implication can beavoided by rejecting Conjunctive Simplification, i.e., (A∧B) →A and (A ∧B)→B, Routley requires for a connexive logic therejection or qualification of Conjunctive Simplification (orequivalent schemata). Although according to Routley (1978, Routleyet al. 1982) and Routley and Routley (1985) the idea ofnegation as cancellation, see sections 3.3 and 4.3., motivates boththe failure of Conjunctive Simplification and AT’ and BT, themodel-theoretic semantics for connexive logics developed in Routley1978, see section 4.2, makes use of what has later come to be known astheRoutley star negation, see the entry onnegation.

If the contraposition rule and uniform substitution are assumed andimplication is transitive, the combination of ConjunctiveSimplification and Aristotle’s Theses results in negationinconsistency, i.e., there are formulasA such thatA and its negation ~A are both theorems, see, forexample, Woods 1968 and Thompson 1991. Non-trivial negationinconsistent logics (with a transitive consequence relation) must beparaconsistent. Using certain three-valued truth tables, Mortensen(1984) pointed out that there are inconsistent but non-trivial systemssatisfying both AT′ and Simplification. Examples of non-trivialinconsistent systems of connexive logic satisfying ConjunctiveSimplification are presented in section 4.5. The availability of suchconnexive systems may be appreciated in view of the fact thatZermelo-Fraenkel set theory based on a system of connexive logic withSimplification is inconsistent, see Wiredu 1974. Mortensen (1984) alsopointed out that the addition of AT′ to the relevance logicR of Anderson and Belnap has a trivializing effect, afact shown in Routleyet al. 1982 as well.

The relation between connexive logic and relevance logic can also beseen as follows. LetA andB be contingent formulasof classical propositional logic, i.e., formulas that are neitherconstantly false nor constantly true. It is well-known that then thefollowing holds in classical logic:

1. Not: ~A ⊢A
2. IfA ⊢B, then not:A ⊢~B
3. IfA ⊢B, thenA andBshare some propositional variable (sentence letter)

If property (iii) is generalized to arbitrary formulasA andB, it is called thevariable sharing property orvariable sharing principle, which is generally seen as anecessary condition on a logic to count as a relevance logic (see theentrylogic: relevance). So-calledcontainment logics (also calledParrysystems or systems of analytic implication, see Parry1933, Anderson and Belnap 1975, Fine 1986, Ferguson 2015), satisfy thestrong relevance requirement that if ⊢A →B, then every propositional variable ofB is also apropositional variable ofA. The variable sharing propertyindicates a content connection betweenA andB ifB is derivable fromA (or, semantically,AentailsB). The properties (i) and (ii) may be viewed toexpress a content connection requirement on the derivability relationin anegative way. If one wants to express these constraintsin terms of the provability of object language formulas, one naturallyarrives at Aristotle’s and Boethius’ theses.

Connexive relevance logics that combine the ternary frame semanticsfrom relevance logic and the adjustment of falsity conditions alongthe lines of the connexive logicC (see section4.5.1) have been studied in Omori 2016a and Francez 2019, cf. sections4.2 and 4.5.

3.2 Conditional logic

Principles of connexive logic have been discussed in conditional logic(see the entrylogic: conditional), beginning with Ramsey’s (1929) comments on what is now calledtheRamsey Test, as pointed out, e.g., in McCall 2012 andFerguson 2014:

If two people are arguing “Ifp willq?”and are both in doubt as top, they are addingphypothetically to their stock of knowledge and arguing on that basisaboutq; so that in a sense “Ifp,q” and “Ifp,~q” arecontradictories (notation adjusted).

Angell (1966, 1967a, 1978) refers to AT′ as the Law ofConditional Non-Contradiction. Usually, Abelard’s FirstPrinciple, ~((A →B) ∧ (A →~B)) is considered as a principle of conditionalnon-contradiction and as such is endorsed by some philosophers, e.g.,Gibbard (1981, p. 231), Lowe (1995, p. 47), and Bennett (2003, p. 84),without making any reference to connexive logic. Conditionalnon-contradiction fails, however, to be a valid principle in thesemantics suggested by Stalnaker (1968) and Lewis (1973), cf. thediscussion in Unterhuber 2013. The restrictedly connexive logicspresented in Weiss 2019 that validate Aristotle’s theses, BTw,and BTw′ stand in the tradition of Stalnaker and Lewis and aregiven an algebraic semantics that builds on the algebraic semanticsfor conditional logics from Nute 1980.

Another motivation for connexive logic from the perspective ofconditional logic has been presented by John Cantwell (2008) withoutnoting that the introduced propositional logic is a system ofconnexive logic. Cantwell considers the denial of indicativeconditionals in natural language and argues that the denial of, say,the conditional ‘If Oswald didn’t kill Kennedy, Jack Rubydid.’ amounts to the assertion that if Oswald didn’t shootKennedy then neither did Jack Ruby. This suggests that (A→ ~B) is semantically equivalent with ~(A→B). Also Claudio Pizzi’s work on logics ofconsequential implication has been motivated in the context ofconditional logic, see Pizzi 1977 and section 5.

3.3 Negation

As the characteristic connexive principles exhibit an implication anda negation connective, it is not very surprising that connexive logiccan be approached also from considerations on the notion of negation.Two different perspectives emerge with the ideas of negation ascancellation (erasure, neutralization, or subtraction) and negation asfalsity. Negation as cancellation is a conception of negation that canbe traced back to Aristotle’sPrior Analytics and isoften associated with Strawson, who held that a “contradictioncancels itself and leaves nothing” (1952, p. 3). Routley (1978,Routleyet al. 1982), Routley and Routley 1985, and Priest1999 use the notion of subtraction negation to motivate connexiveprinciples. Routley and Routley (1985, p. 205) present thecancellation view of negation as follows:

∼A deletes, neutralizes, erases, cancelsA (andsimilarly, since the relation is symmetrical,A erases∼A), so that ∼A together withAleaves nothing, no content. The conjunction ofA and∼A says nothing, so nothing more specific follows. Inparticular,A ∧ ∼A does not entailA and does not entail ∼A.

Note that if a logic implements the cancellation view of negation, itwill also be paraconsistent because theex contradictionequodlibet principle, (A ∧ ∼A) ⊢B, will not be valid. (The idea ofex contradictionenihil sequitur is discussed in Wagner 1991.) According to theRoutleys, a connection between the subtraction account of negation andAristotle’s thesis AT′ then arises as follows (Routley andRoutley 1985, p. 205):

Entailment is inclusion of logical content. So, ifA were toentail ~A, it would include as part of its content, whatneutralizes it, ~A, in which event it would entail nothing,having no content. So it is not the case thatA entails~A, that is Aristotle’s thesis ~(A →~A) holds.

Accordingly then, for Routley (Routleyet al. 1982, p. 82)connexivism has two leading theses, namely:

1. Simplification (A ∧B →A,A ∧B →B) fails to hold, and itsuse ... is what is responsible for the paradoxes of implication ...
2. Every statement is self-consistent, symbolicallyA ◇A, where the relation of consistency with, symbolised◇, is interconnected with implication in the standard fashion:A ◇B ↔. ~(A →~B).

The cancellation view of negation has been heavily criticized inWansing and Skurt 2018, where it is stressed that connexive logic canbe detached from the notion of negation as erasure and the failure ofConjunctive Simplification.

The notion of negation as definite falsity, in contrast to negation asabsence of truth, does not support the failure of ConjunctiveSimplification but rather the failure ofex contradictionequodlibet if it is coupled with an understanding of inference asinformation flow, because the information thatA ∧~A does not necessarily give the information thatB,for anyB whatsoever. This suggests a separate treatment of(support of) truth and (support of) falsity conditions, which enablesadopting the falsity conditions for implications represented byBTe′.

3.4 Contra-classicality

Humberstone (2000) calls a logiccontra-classical just incase not every formula provable in the logic is provable in classicallogic (and, moreover, considers a more demanding notion of acontra-classical logic by requiring that there is no way oftranslating its connectives in such a way that one obtains a subsystemof classical logic). There are several different kinds ofcontra-classical logics, such as, for example, Abelian logicscontaining the axiom schema ((A →B) →B) →A, connexive logics, and logics of logicalbilattices. The negation, truth order conjunction, weak implication,and information order disjunction fragment of Arieli and Avron’s(1996) bilattice logicBL_⊃, forexample, is a standard propositional vocabulary containing a negation,a conjunction, a disjunction, and a conditional. It is a non-trivialbut inconsistent logic and as such contra-classical.

In Omori and Wansing 2018, a way of obtaining contra-classical logicsis delineated, and in Estrada-Gónzalez 2019 it is discussed inmore detail. Following the pattern of the presentation of theconnexive logicC, cf. section 4.5.1, the generalidea is that of keeping some standard (support of) truth conditionsfor a logical operation and modifying its (support of) falsityconditions. From a bilateralist perspective that treats truth andfalsity as well as provability and disprovability or refutability asseparate semantical, respectively proof-theoretical dimensions thatare on a par, there is also the strategy of keeping some standard(support of) falsity conditions for a logical operation and modifyingits (support of) truth conditions. Connexive logic can be seen ascontributing to the exploration of roads to contra-classicality.

3.5 Empirical perspective

In McCall 2012 one can find some results on testing the endorsement ofconnexive principles (AT′, BT, and BT stated as a rule) given byindicative conditionals in English in concrete form on a group of 89non-expert philosophy students at McGill University in Canada. Thesefindings support the intuition that laymen speakers of Englishsubscribe to those connexive principles to a rather high degree: 88%in the case of AT′, 85% in the case of BT in rule form, and 84%in the case of BT.

Empirical studies on Aristotle’s theses have been carried out byPfeifer (2012), Pfeifer and Tulkki (2017), and Pfeifer and Yama(2017). In one experiment, presented in Pfeifer 2012, the sampleconsisted of 141 psychology students (110 females and 31 males) at theUniversity of Salzburg, Austria. Both AT and AT′ were tested asabstract as well as concrete indicative conditionals. In a secondexperiment, 40 students without training in logic (20 females and 20males) had to solve tasks involving concrete indicative conditionalsin English. In this case, scope ambiguities arising from the negationof conditionals were ruled out. Both experiments provide evidenceagainst the interpretation of indicative conditionals in English asBoolean implication and support the connexive reading of negatedimplications expressed by Aristotle’s theses. Pfeifer sees thesefindings as strong evidence for interpreting indicative conditionalsas conditional events. This interpretation predicts that people shouldstrongly believe that Aristotle’s theses are valid because theonly coherent assessment for them is the probability value 1.

Pfeifer and Tulkki (2017) tested the interpretation of subjunctiveversus indicative conditionals among a group of 60 students of theUniversity of Helsinki, Finland, (30 females and 30 males) and foundno statistically significant differences between the endorsement of ATand AT′ (72%, respectively 77%). Another experiment presented inPfeifer and Yama 2017 found no cultural differences between theWestern samples and an Eastern sample when testing the endorsement ofAT and AT′ among 63 Japanese university students from theGraduate School of Literature and Human Behavioral Sciences at OsakaCity University, with an endorsement of AT and AT′ by 65% and76% of the participants, respectively.

Khemlaniet al. 2014 report on an experiment testing a sampleof 21 native English-speaking participants on denying concrete naturallanguage conditionals (against the background of Johnson-Laird’smental model theory, assuming classical logic). Whereas 28% of theparticipants endorsed denial conditions in accordance with classicallogic, 34% endorsed denial conditions according to Boethius’thesis BT.

Another experiment on the negation of indicative conditionals ispresented in Egré and Politzer 2013. They consider weakeningsof the classical conjunctive understanding of ~(A →B) as (A ∧ ~B) and the connexivereading as (A →~B), namely (A ∧◊~B), respectively (A → ◊~B).Exploiting the flexibility of the “tweaking of the falsityconditions”-approach to connexive logic, presented in sections3.7 and 4.5, Omori 2019 interprets (A →◊~B) in a variant of the modal logicBKfrom Odintsov and Wansing 2010 by suitably adjusting the falsitycondition for implications (A →B), so that~(A →B) is provably equivalent with(A → ◊~B).

3.6 Proof-theoretical perspective

Modern modal logic started as a syntactical enterprise with C.I.Lewis, who defined a series of axiom systems to capture notions ofstrict implication. In a similar vein, Lewis’ student E. Nelsoncame up with an axiom system from which Aristotle’s andBoethius’ theses can be derived. The system is calledNL in Mares and Paoli 2019, where its axioms andinference rules are presented as follows (we here use schematicletters for arbitrary formulas instead of propositional variables anda different symbol for negation):

where ◦ is a primitive binary consistency operator, (A|B)(inconsistency) is defined as ~(A ◦B),A =B is defined as (A →B)∧ (B →A),A ≠B as ~(A =B), andA ≠B ≠C is an abbreviation of (A ≠B) ∧(B ≠C) ∧ (A ≠C).

Providing a sound and complete semantics forNL is anopen problem in connexive logic. Angell’s (1962) axiomatic proofsystemPA1 can also be seen as belonging to theproof-theoretical tradition because it is incomplete with respect thetruth tables presented by Angell. However, Angell provedPA1 to be sound with respect to these truth tables,thereby for the first time presenting a non-triviality proof for aformal system of connexive logic. Providing anintuitivesound and complete semantics forPA1 is another openproblem in connexive logic. (Routley [1978, p. 409] admits that hischaracterization is “not intuitively very satisfying: as itstands the modelling is rather complex, with the modelling conditionsexceeding in number the postulates they model, and basic connexivepostulates like Boethius, instead of being validated in a natural way,have fairly intractable conditions.”)

In proof-theoretic semantics, proof systems of a suitable form areseen as providing a meaning theory, see the entryproof-theoretic semantics. In that spirit, Francez (2016) presents two natural deduction proofsystems for a propositional language with negation and implication,one in which AT, AT′, BT, and BT′ are provable, andanother one in which AT, AT′, and the following variations ofBoethius’ theses are provable:

B3: (A →B) → ~(~A →B) and
B4: (~A→B) → ~(A →B).

Francez motivates these principles by certain natural languagediscourses and a “dual Ramsey Test” that modifies theRamsey test by assuming that in the course of arguing “Ifp willq?,” ~p is hypotheticallyadded to a stock of knowledge. Francez’ natural deduction rulesare straightforwardly obtained by modifying the natural deductionrules for the negation and implications fragment of DavidNelsons’s four-valued constructive logicN4,cf. Kamide and Wansing 2012, in the manner that leads fromN4 to the constructive connexive logicC from Wansing 2005, cf. section 4.5.1. In Francez2019 the natural deduction system that gives AT, AT′, BT, andBT′ is relevantized as in the familiar natural deduction proofsystem for the implication fragment of the relevant logicR by introducing subscripts for book-keeping in orderto avoid empty, irrelevant implication introductions. Omori (2016b)adds conjunction and disjunction to the language of Francez 2016,gives an axiomatization and a characterizing semantics for the naturaldeduction system that allows to prove B3 and B4, and observes thatalthough AT and AT′ are valid, BT, and BT′ are invalid,which prompts him to call the provable equivalence ~(A →B) ↔ (~A →B)“half-connexive”.

The natural deduction proof system in Wansing 2016b can be seen acontribution to a bilateralist proof-theoretic semantics for certainconnexive logics given in terms of provability as well as refutabilityconditions. In addition to a connexive implication that internalizes anotion of provability into the object language, there is also aconnexive co-implication that internalizes a refutability relation.The resulting bi-connexive logic2C is a connexivevariant of the bi-intuitionistic logic2Int fromWansing 2016a, 2018. A natural deduction calculus for a quantum logicsatisfying Aristotle’s theses is presented in Kamide 2017.

According to Schroeder-Heister 2009, Gentzen’s sequent calculusis a “more adequate formal model of hypotheticalreasoning” than natural deduction, and proof-theoretic semanticshas also been developed with respect to various kinds of sequentcalculi. Sequent systems for connexive logics can be found in Wansing2007, Wansing 2008, McCall 2014, Kamide and Wansing 2011, 2016,Kamide, Shramko and Wansing 2017, and Kamide 2019.

3.7 Semantical perspective

A central approach to connexive logic is given by many-valued andmodel-theoretic semantics in terms of truth values or support of truthand support of falsity conditions. As explained in Omori and Wansing2019, the semantics of several connexive logics can be described aseither (i) modifying some standard (support of) truth conditions ofconditionals of a certain kind or keeping standard truth conditions incombination with more complex model structures, or (ii) as tweakingthe standard (support of) falsity conditions of certain familiarimplications. Given the multitude of different connexive logics andthe flexibility of the adjustment of falsity conditions in combinationwith standard (support of) truth conditions, this classificationprovides a general perspective.

A key observation for this classificatory enterprise comes from Omoriand Sano 2015, where a mechanical procedure is described for turningtruth tables using the four generalized truth values of first-degreeentailment logic,FDE, see the entries ontruth values and relevance logic and Omori and Wansing 2017, into pairs of positive and negativeconditions in terms of containing or not containing the classicaltruth values 0 and 1. Then, in McCall’s systemCC1 a connexive conditionalA →B receives a designated value (is true) in a model just incase (i)A does not receive a designated value orBdoesand (ii) 0 belongs to the value ofA iff itbelongs to the value ofB. In this sense, the connexiveimplication of Angell-McCall is obtained by adding a condition to thetruth condition for Boolean implication.

The consequential conditional in the logics of consequentialimplication investigated by Pizzi (1977, 1991, 1993, 1996, 2004, 2005,2008, 2018) and Pizzi and Williamson (1997, 2005) validatesAristotle’s theses but fails to validate Boethius’ theses.It is thus connexive only in a weak sense, but since the consequentialimplication is a strict conditional that is required to satisfy someextra condition, also logics of consequential implication fit into theclassificatory scheme provided by the semantical perspective. Thefollowing table is a slight extension of the summarizing overview fromOmori and Wansing 2019 (with pointers to the relevant sections of thepresent entry), where the approaches above the double line adjust(support of) truth conditions (or add semantical machinery to standardtruth conditions), whereas the approach below the double line tweaks(support of) falsity conditions:

A dialogical semantical treatment of connexive logic can be found inRahman and Rückert 2001.

4. Systems of connexive logic

4.1 Algebraic connexive logic

Whereas the basic ideas of connexive logic can be traced back toantiquity, the search for formal systems with connexive implicationseems to have begun only in the 19th century in the work of H. MacColl(1878), see also Rahman and Redmond 2008. The basic idea of connexiveimplication was spelled out also by E. Nelson (1930), and a morerecent formal study of systems of connexive logic started in the1960s. In McCall 1966, S. McCall presented an axiomatization of asystem of propositional connexive logic semantically introduced byAngell (1962) in terms of certain four-valued matrices. The languageof McCall’s logicCC1 contains as primitive(notation adjusted) a unary connective ~ (negation) and the binaryconnectives ∧ (conjunction) and → (implication). Disjunction∨ and equivalence ↔ are defined in the usual way. Theschematic axioms and the rules ofCC1 are asfollows:

Among these axiom schemata, only A12 is contra-classical. The systemCC1 is characterized by the following four-valuedtruth tables with designated values 1 and 2:

McCall emphasizes that the logicCC1 is only oneamong many possible systems satisfying the theses of Aristotle andBoethius. AlthoughCC1is a system ofconnexive logic, its algebraic semantics appears to be only a formaltool with little explanatory capacity. InCC1, theconstant truth functions1,2,3, and4 can be defined as follows(McCall1966, p. 421):1 := (p →p),2 := ~(p ↔ ~p),3 := (p ↔ ~p),4 := ~(p→p), for somesentence letterp. As Routley and Montgomery (1968, p. 95)point out,CC1 “can be given a semantics byassociating the matrix value 1 with logical necessity, value 4 withlogical impossibility, value 2 with contingent truth, and value 3 withcontingent falsehood. However, many anomalies result; e.g. theconjunction of two contingent truths yields a necessary truth”.Moreover, McCall points out thatCC1 has someproperties that are difficult to justify if the name ‘connexivelogic’ is meant to reflect the fact that in a valid implicationA →B there exists some form of connectionbetween the antecedentA and the succedentB. AxiomA4, for example, is bad in this respect. On the other hand,CC1 might be said to undergenerate, since (A∧A) →A andA → (A∧A) fail to be theorems ofCC1. Routleyand Montgomery (1968) showed that the addition of the latter formulasto only a certain subsystem ofCC1 leads toinconsistency. For a defense of Angell’sPA1against Routley and Montgomery’s critical observations see Bode1979.

These observations may well have distracted many non-classicallogicians from connexive logic at that time. If the validity ofAristotle’s and Boethius’ Theses is distinctive ofconnexive logics, it is, however, not quite clear how damaging theabove criticism is. In order to construct a more satisfactory systemof connexive logic, McCall (1975) defined the notions of a connexivealgebra and a connexive model and presented an axiom systemCFL that is characterized by the class of allconnexive models. In the language ofCFL, however,every implication is first-degree, i.e., no nesting of → ispermitted. McCall refers to a result by R. Meyer showing that thevalid implications ofCFL form a subset of the set ofvalid material equivalences and briefly discusses giving up thesyntactic restriction to first-degree implication. Meyer (1977) showedthat the first-degree fragment of the normal modal logicS5 (and in fact every normal modal logic betweenKT andS5, cf. the entrylogic: modal) andCFL are equivalent in the following sense: alltheorems ofCFL are provable inS5if the connexive implicationA →B is definedas □(A ⊃B) ∧ (A ≡B), where ⊃ and ≡ are classical implication andequivalence, respectively, and every first-degree theorem ofS5 is provable inCFL if□A (“it is necessary thatA”) isdefined as (~p ∨p)→A. In summary,it seems fair to say that as the result of the investigations intoconnexive logic in the 1960s and 1970s, connexive logic, its ancientroots notwithstanding, appeared as a sort of exotic branch ofnon-classical logic.

More recently, Cantwell (2008) presented a truth table semantics for asystem of connexive logic together with a proof-theoreticalcharacterization. The truth tables for negation and implication aretaken from Belnap 1970, but Cantwell's three-valued truth table forthe conditional can already be found in a paper by William Cooper(1968). Like Cantwell, Cooper wanted to formally model how conditionalsentences in the indicative mood and expressed by means ofif-then are used in ordinary conversational English. (WhereasCantwell takes the entire three-element set of truth values as theco-domain of assignment functions for propositional variables, Cooperrestricts assignment functions to mappings from the set ofpropositional variables to the two-element set of classical truthvalues.) Cantwell considers a language containing the constantly falseproposition ⊥ and the following three-valued truth tables fornegation, conjunction, disjunction, and implication with designatedvalues T and − (where ‘T’ stands for truth and‘F’ for falsity):

In this system, introduced as a system of conditional negation,CN, (A → ~B) and ~(A→B) have the same value under every assignment of truthvalues to propositional variables. Cantwell’s system thusvalidates BTe and BTe′, and it turns out to be the connexivelogicMC from Wansing 2005, see section 4.5.3,extended by the Law of Excluded Middle,A ∨ ~A. Acertain expansion ofCN is studied in Olkhovikov2002, 2016 and, independently, in Omori 2016c, see section 4.5.3.

A three-valued logic that validates Aristotle’s theses but notBoethius’ theses and that is subminimally connexive and Kapsnerstrong in the terminology of Estrada-González &Ramirez-Cámara 2016 is the three-valued logicMRS^P that was introduced inEstrada-González 2008. In Estrada-González &Ramirez-Cámara 2016,MRS^P isdiscussed against the background of Cantwell’s three-valuedconnexive logicCN and Mortensen’s (1984)three-valued connexive logic, dubbedM3V by McCall(2012).

McCall (2014) presents a cut-free sequent calculus for a system ofconnexive logic that he calls “connexive Gentzen.” Thecalculus has the non-standard feature of using pairs of axioms thatare not logical truths. An annotation with subscripts is used toenable the elimination of dependencies on such non-standard axioms inthe course of a derivation. The resulting system differs fromCC1 in thatp → (p ∧p) and (p ∧p)→p areprovable, and it is shown to be sound with respect to certainfour-valued matrices. Sound and complete cut-free sequent calculi forcertain constructive and modal connexive logics have been presentedfor the first time in Wansing 2008 and Kamide and Wansing 2011.

4.2 Connexive logic based on ternary frames for relevance logics (Australian Plan)

In the late 1970s and the 1980s, connexive logic was subjected tosemantical investigations based on ternary frames for relevancelogics, making use of the Routley star negation that is distinctive oflogics “on the Australian Plan,” cf. Meyer and Martin1986. Routley (1978) obtained a semantic characterization ofAristotle’s Thesis AT′ and Boethius’ Thesis BT usinga ‘generation relation’G between a formulaA and a possible worlds. The semantics employsmodel structuresF = <T,K,R,S,U,G, *>,whereK is a non-empty set of possible worlds,T∈K is a distinguished world (the ‘realworld’),R,S, andU are ternaryrelations onK,G is a generation relation, and * isa function onK mapping every worlds to its‘opposite’ or ‘reverse’s*. Avaluation is a functionv that sends pairs of worlds andpropositional variables into {0,1}, satisfying the following hereditycondition: ifR(T,s,u) andv(p,s) = 1, thenv(p,u) = 1. Intuitively,G(A,t) issupposed to mean that everything that holds in worldt isimplied byA. A model is a structureM= <F,v>. The relationM,t ⊨A (“A is true attinM”) is inductively defined asfollows:

M,t ⊨p iffv(p,t) = 1
M,t ⊨ ~A iffM,t* ⊭A
M,t ⊨ (A ∧B) iff there ares,u withStsuM,s ⊨A andM,u ⊨B
M,t ⊨ (A ∨B) iff there ares,u withUtsuM,s ⊨A orM,u ⊨B
M,t ⊨ (A →B) iff for alls,u ifRtsu andM,s ⊨A, thenM,u ⊨B

[Note: whenever there is little chance for ambiguity, we replaceR(x,y,z) byRxyz.]

Moreover, it is required that for every formulaA and worldt,G(A,t) impliesM,t ⊨A. A formulaA is true in modelM iffM,T ⊨A, andA is valid with respect to a class of models ifA istrue in all models from that class. AT′ is semanticallycharacterized by the following property of models: ∃t(R(T*,t,t*) andG(A,t)), and BT is characterized by∀w∃s,t,u(R(w,s,t),R(w*,s,u),G(A,s), andR(T,t,u*)).

Mortensen (1984), who considers AT′, explains thatRoutley’s characterization of AT′ is “notparticularly intuitively enlightening” and points out that incertain logics with a ternary relational models semantics anothercharacterization of AT′ is available, namely the condition thatfor every modelM the setC_A := {s :M,s ⊨A andM,s ⊭ ~A} isnon-empty. Like Routley’s non-recursive requirement thatG(A,t) impliesM,t ⊨A,Mortensen’s condition is not a purely structural condition,since it mentions the truth relation ⊨. Mortensen (1984, p. 114)maintains that the conditionC_A ≠∅ “is closest to the way we think of Aristotle,” andemphasizes that for a self-inconsistent propositionA, thesetC_A must be empty, whence AT′ isto be denied. Mortensen also critically discusses the addition ofAT′ to the relevance logicE. In this context,AT′ amounts to the condition that no implication is true at theworldT*.

A more regular semantics for extensions of the basic relevance logicB (not to be confused with the truth valued read as“both true and false”) by either AT′ or BT has beenpresented in Brady 1989. In this semantics, conjunction is defined inthe standard way, and there is a non-empty subset of worldsO⊆K. The setO contains the distinguishedelementT used to define truth in a model. The extended modelstructures contain a function ℑ that maps sets of worlds, and inparticular, interpretations of formulas (alias propositions)I(A) to sets of worlds in such a way that a formulaA is true at a worldt ifft ∈ℑ(I(A)). This allows Brady to state modelconditions capturing AT′ and BT as follows:

AT′: Ift ∈O, then (∃x,y ∈ ℑ(f))Rt*xy* , forany propositionf;

BT: (∃x,y ∈ ℑ(f))(∃z ∈K) (Rtxz andRt*yz*), for any propositionf and anyt ∈K.

Note that these clauses still are not purely structural conditions butconditions on the interpretation of formulas. Also the investigationsinto connexive logics based on ternary frames did not, as it seems,lead to establishing connexive logic as a fully recognized branch ofnon-classical logic.

4.3 Connexive logic based on subtraction negation

Albeit according to Routley (1978), Routleyet al. (1982) andRoutley and Routley (1985) there is a close relation between connexivelogic and the idea of negation as cancellation, Routley suggested asemantics using a generation relation and the star negation in ternaryframes for relevance logics, whereas connexive logics based in astraightforward way on the cancellation view of negation have beenworked out by Priest (1999). Priest (1999) directly translates adefinition of entailment that enforces the null-content account ofcontradictions into evaluation clauses. A model is a structureM= <W,g,v>, whereW is a non-empty set of possibleworlds,g is a distinguished element fromW, andv is a valuation function from the set of propositionalvariables into the set of classical truth values {1, 0}. Two clausesfor evaluating implications at possible worlds are considered(notation adjusted):

(a)M,s ⊨A→B iff (i) there is a worldu withM,u ⊨A and (ii) forevery worldu,M,u⊨A thenM,u⊨B;
(b)M,s ⊨A→B iff (i) there is a worldu withM,u ⊨A, (ii) thereis a worldu withM,u⊭B, and (iii) for every worldu,M,u ⊨A thenM,u ⊨B.

Condition (i) ensures that nothing is implied by an unsatisfiableantecedent. The evaluation clauses for the other connectives areclassical. A formulaA is true in a model (M⊨A) iffM,g ⊨A; andA is valid iffA is true in every model. Condition(ii) ensures that the law of contraposition is valid. A set Δ offormulas is true in a model iff every element of Δ is true inthe model.

There are two notions of entailment (Δ ⊨A), onecoming with clause (a) the other with clause (b):

(a) Δ⊨A iff Δ is true in some model, andevery model in which Δ is true is a model in whichA istrue;
(b) Δ⊨A iff Δ is true in some model,~A is true in some model, and every model in which Δ istrue is a model in whichA is true.

These two connexive logics arise from the idea of negation ascancellation in a straightforward way. They are neither monotonic norclosed under uniform substitution. Proof systems and decisionprocedures for them can be obtained from a straightforward faithfultranslation τ into the modal logicS5, cf. theentrylogic: modal. For implicationsA →B the translation isdefined as follows, where ⊃ is material implication and ¬ isclassical negation:

(a) ◊τ(A) ∧ □(τ(A) ⊃τ(B));
(b) ◊τ(A) ∧ ◊¬τ(B) ∧□(τ(A) ⊃ τ(B)).

Ferguson (2015) observes that the intersection of the semanticalconsequence relations of variant (a) of Priest’s logic and thenegation, conjunction, disjunction fragment of Bochvar’s3-valued logic (cf. the entrymany-valued logic) results in a known system of containment logic, namely the systemRC presented in Johnson 1976.

Although the semantics of Priest’s connexive logics is simpleand transparent, the underlying idea of subtraction negation is notunproblematic. Priest (1999, 146) mentions strong fallibilists who“endorse each of their views, but also endorse the claim thatsome of their views are false”. Their contradictory opinions infact hardly are contentless, so that the cancellation account ofnegation and, as a result, systems of connexive logic based onsubtraction negation appear not to be very well-motivated. In Skurtand Wansing 2018, it is argued that the metaphoric notion of negationas cancellation is conceptually unclear and that Routley’s(Routleyet al. 1982)) suggestion to replace it by a notionof negation as subtraction in generalized arithmetic is unclear atleast insofar as it has not been worked out in detail.

4.4 Boolean connexive relatedness logic

The Boolean connexive logics of Jarmużek and Malinowski 2019a areobtained in the framework ofrelating logic, a generalizationofrelatedness logic. The latter is an instance of whatSylvan (1989, p. 166) calls a “double-barrelled” analysisof implications, an analysis that complements truth conditions with anadditional “sieve” or “filter” that tightensthe relation between antecedent and succedent. If the relation ismeant to be a relevance relation, this is an example of what Schurz1998 calls “relevance post validity” in contrast to“relevance in validity” as investigated in relevancelogic. Boolean connexive logics extend the language of classicalpropositional logic using conjunction, disjunction, and Booleannegation by a relating implication, →^w, the semanticsof which is constrained by a binary relationR on the set ofall formulas. A model then is a pair <v,R>,wherev is a classical valuation function. The truthcondition for relating implication imposes the relatednessconstraint:

<v,R> ⊨A →^wB iff [(<v,R> ⊭A or<v,R> ⊨B) andR(A,B)]

and a notion of validity with respect to a relationR isdefined:R ⊨A iff for every valuationv,v,R ⊨A.

In order to obtain connexive logics, Jarmużek and Malinowskiintroduce the following conditions on binary relationsR:

(a1)R is (a1) iff for anyA: notR(A, ~A)
(a2)R is (a2) iff for anyA: notR(~A,A)
(b1)R is (b1) iff for arbitraryA,B: (i)ifR(A,B) then notR(A,~B) and (ii) R((A →^wB),~(A →^w ~B)) (b2)R is (b2)iff for arbitraryA,B: (i) ifR(A,B) then notR(A,~B) and (ii) R((A →^w ~B),~(A →^wB)).

These conditions suffice to validate Aristotle’s and Boethiustheses. A correspondence between Aristotle’s and Boethius thesesand conditions onR is obtained if the relationsRare required to be closed under negation, i.e., for all formulasA andB,R(A,B) impliesR(~A, ~B). Then,

(a1)R is (a1) iffR ⊨ ~(A→^w ~A)
(a2)R is (a2) iffR ⊨ ~(~A→^wA)
(b1)R is (b1) iffR ⊨ (A→^wB) →^w ~(A→^w ~B)
(b2)R is (b2) iffR ⊨ (A→^w ~B) →^w ~(A→^wB).

However, these correspondences come at a price. Jarmużek andMalinowski point out that imposing negation closure validates theotherwise refutable formula ~((A →^wB) ∧ ~B ∧ ~(~A →^w~B)) with respect to any relationR. Jarmużekand Malinowski also show that these five conditions are independent ofeach other and therefore give rise to 2⁵ different logics.The two connexive ones (alias properly connexive ones in Jarmużekand Malinowski’s terminology), i.e., the logic defined by meansof conditions (a1), (a2), (a3), and (4) and the logic defined by inaddition requiring negation closure, are also Kapsner strong.Moreover, Jarmużek and Malinowski present sound and completetableau calculi for these 2⁵ logics.

4.5FDE-based connexive logics (American Plan)

The basic paraconsistent logicFDE of first-degreeentailment lacks a primitive implication connective and lends itselfto adding an implication connective that validates Aristotle’sand Boethius’ theses by using the falsity conditions ofimplications as expressed by BTe′. This is possible becausenegation is treated according to “the American Plan,”i.e., by making use of four semantical values:T(“told true only”),F (“told falseonly”),N (“neither told true nor toldfalse”), andB (“both told true and toldfalse”), so that support of truth and support of falsity emergeas two independent semantical dimensions:

A receives the valueT at statetifft supports the truth ofA but not the falsity ofA;
A receives the valueF att ifft supports the falsity ofA but not the truth ofA;
A receives the valueN att ifft neither supports the truth ofA nor supports thefalsity ofA;
A receives the valueB att ifft supports both the truth and the falsity ofA.

Negation is then understood as leading from support of truth tosupport of falsity, and vice versa. The method of tweaking the(support of) falsity conditions can be applied to a number ofdifferent conditionals, ranging from constructive, relevant, andmaterial (Boolean) implication to very weak implications studied inconditional logic with the help of so-calledSegerbergframes.

4.5.1FDE-based constructive connexive logic

A system of connexive logic with an intuitively plausible possibleworlds semantics using a binary relation between worlds has beenintroduced in Wansing 2005. In this paper it is observed that amodification of the falsification conditions for negated implicationsin possible worlds models for David Nelson’s constructivefour-valued logic with strong negation results in a connexive logic,calledC, which inherits from Nelson’s logic aninterpretation in terms of information states pre-ordered by arelation of possible expansion of these states. For Nelson’sconstructive logics see, for example, Almukdad and Nelson 1984,Gurevich 1977, Nelson 1949, Odintsov 2008, Routley 1974, Thomason1969, Wansing 2001, Kamide and Wansing 2012.

The key observation for obtainingC is simple: in thepresence of the double negation introduction law, it suffices tovalidate both BT′ and its converse ~(A →B) → (A → ~B). In other words, aninterpretation of the falsification conditions of implications iscalled for, which deviates from the standard conditions. InNelson’s systems of constructive logic, the double negation lawshold, and the relational semantics for these logics is such thatfalsification and verification of formulas are dealt with separately.The systemN4 extendsFDE byintuitionistic implication, however, the falsification conditions ofimplications are the classical ones expressed by the schema~(A →B) ↔ (A ∧ ~B).To obtain a connexive implication, it is therefore enough to assumeanother interpretation of the falsification conditions ofimplications, namely the one expressed by BTe′: (A→ ~B) ↔ ~(A →B).

Consider the languageL := {∧, ∨, →, ~} based ona denumerable set of propositional variables. Equivalence ↔ isdefined as usual. The schematic axioms and rules of the logicC are:

Clearly, a5 is the only contra-classical axiom ofC.The consequence relation ⊢_C(derivability inC) is defined as usual. AC-frame is a pairF =<W, ≤ >, where ≤ is a reflexive andtransitive binary relation on the non-empty setW. Let<W, ≤ >⁺ be the set of allX ⊆W such that ifu ∈Xandu ≤w, thenw ∈X. AC-model is a structureM= <W, ≤,v⁺,v⁻ >,where <W, ≤ > is aC-frameandv⁺ andv⁻ arevaluation functions from the set of propositional variables into<W, ≤ >⁺. Intuitively,Wis a set of information states. The functionv⁺sends a propositional variablep to the states inWthat support the truth ofp, whereasv⁻ sendsp to the states thatsupport the falsity ofp.M =<W, ≤,v⁺,v⁻ > is said to be the model basedon the frame <W, ≤ >. The relationsM,t ⊨⁺A(“M supports the truth ofA att”) andM,t ⊨⁻A (“Msupports the falsity ofA att”) are inductively defined as follows:

M,t ⊨⁺p ifft ∈v⁺(p)
M,t ⊨⁻p ifft ∈v⁻(p)
M,t ⊨⁺(A ∧B) iffM,t ⊨⁺A andM,t ⊨⁺B
M,t ⊨⁻(A ∧B) iffM,t ⊨⁻A orM,t ⊨⁻B
M,t ⊨⁺(A ∨B) iffM,t ⊨⁺A orM,t ⊨⁺B
M,t ⊨⁻(A ∨B) iffM,t ⊨⁻A andM,t ⊨⁻B
M,t ⊨⁺(A →B) iff for allu ≥t(M,u ⊨⁺A impliesM,u⊨⁺B)
M,t ⊨⁻(A →B) iff for allu ≥t(M,u ⊨⁺A impliesM,u⊨⁻B)
M,t ⊨⁺~A iffM,t⊨⁻A
M,t ⊨⁻~A iffM,t⊨⁺A

IfM = <W, ≤,v⁺,v⁻ > is aC-model, thenM ⊨A (“A is valid inM”)iff for everyt ∈W,M,t⊨⁺A.F⊨A (“A is valid onF”)iffM⊨A for every modelMbased onF. A formula isC-valid iff it is valid on every frame. Support oftruth and support of falsity for arbitrary formulas are persistentwith respect to the relation ≤ of possible expansion of informationstates. That is, for anyC-modelM= <W, ≤,v⁺,v⁻ > andformulaA, ifs ≤t, thenM,s ⊨⁺AimpliesM,t⊨⁺A andM,s ⊨⁻A impliesM,t ⊨⁻A. It can easily be shown that a negation normal form theoremholds. The logicC is characterized by the class ofallC-frames: for anyL-formulaA,⊢_CA iffA isC-valid. Moreover,C satisfies thedisjunction property and the constructible falsity property. If⊢_CA ∨B, then⊢_CA or⊢_CB. If⊢_C ~(A ∧B),then ⊢_C ~A or⊢_C ~B. Decidability ofC follows from a faithful embedding into positiveintuitionistic propositional logic.

Like Nelson’s four-valued constructive logicN4,C is a paraconsistent logic (cf.the entrylogic: paraconsistent). Note thatC contains contradictions, for example:⊢_C ((p ∧ ~p)→ (~p ∨p)) and⊢_C ~((p ∧ ~p)→ (~p ∨p)). It is obvious from the abovepresentation thatC differs fromN4only with respect to the falsification (or support of falsity)conditions of implications. As inN4, provable strongequivalence is a congruence relation, i.e., the set {A :⊢_CA} is closed under theruleA ↔B, ~A ↔ ~B /C(A) ↔C(B). Wansing (2005)also introduces a first-order extensionQC ofC. Kamide and Wansing (2011) present a sound andcomplete sequent calculus forC and show the cut-ruleto be admissible, which means that it can be dispensed with.

Whereas the direction from right to left of Axiom a5 can be justifiedby rejecting the view that ifA impliesB andA is inconsistent,A implies any formula, inparticularB, the direction from left to right seems ratherstrong. If the verification conditions of implications are dynamic (inthe sense of referring to other states in addition to the state ofevaluation), then a5 indicates that the falsification conditions ofimplications are dynamic as well. The falsity of (A →B) thus implies that ifA is true,B isfalse. Yet, one might wonder why it is not required that thefalsity of (A→B) implies that ifA istrue,B isnot true. This cannot be expressed in alanguage with just one negation, ~, expressing falsity instead ofabsence of truth (classically at the state of evaluation orintuitionistically at all related states). If one adds toC the further axiom ~A → (A→B) to obtain a connexive variant of Nelson’sthree-valued logicN3, intuitionistic negation ¬is definable by setting: ¬A :=A →~A. Then a5 might be replaced by

a5′: ~(A →B) ↔ (A →¬B).

The resulting system satisfies AT, AT′, BT, and BT′becauseA → ¬~A and ~A →¬A are theorems. For BT, for example, we have:

This logic, however, is the trivial system consisting of everyL-formula (a fact not noticed in Wansing 2005 (Section 6) butpointed out in the online version of that paper).

The systemC is a conservative extension of positiveintuitionistic logic. InC, strong negation isinterpreted in such a way that it turns the intuitionistic implicationof its negation-free sublanguage into a connexive implication.Analogously, strong negation may be added to positive dualintuitionistic logic to obtain a system with a connexiveco-implication, and to bi-intuitionistic logic, or to the logic2Int from Wansing 2016a that also contains animplication and a co-implication connective, to obtain systems withboth a connexive implication and a connexive co-implications, seeWansing 2008, 2016b, and Kamide and Wansing 2016.

The systemsC andQC are connexivebut not Kapsner strong. This is hardly surprising because these logicsare paraconsistent and allow formulasA and ~A to besimultaneously satisfiable in the sense that a state and all itspossible expansions may support the truth of bothA and~A. As a result,A → ~A and~A →A are satisfiable. IfA →~A and ~A →A are unsatisfiable,strong connexivity is in conflict with at the same time satisfying thededuction theorem and defining semantical consequence as preservationof support of truth:A → ~A would entail~(A → ~A), ~A →A wouldentail ~(~A →A), and the formulas (A→ ~A) → ~(A → ~A) and(~A →A) → ~(~A →A)would be valid instead of unsatisifable.

4.5.2FDE-based connexive relevance logic

The starting point for Hitoshi Omori’s (2016a) definition of aconnexive extension of the basic relevance logicBD(see the entrylogic: relevance) is to find a proof theory for extensions ofBD withnegation treated according to the American Plan. Priest and Sylvan(1992) posed this as an open problem, and Omori gives a partialsolution by defining a connexive variantBDW ofBD. The semantics uses models based on ternaryframes. There is a base stateg, the four truth values arerepresented as subsets of the set of classical truth values {0,1}, andinterpretations are defined in the style of Dunn (cf. Omori andWansing 2017). A model is quadruple <g,W,R,I>, whereW is a non-empty set (ofstates),g∈W,R is a three-placerelation onW withRgxy iffx =y,andI is a function that maps pairs consisting of a state anda propositional variable to subsets of {0,1}. The interpretationfunctionI is then extended to an assignment of truth valuesat states for all formulas as follows:

1 ∈I(w, ~A) iff 0 ∈I(w,A)
0 ∈I(w, ~A) iff 1 ∈I(w,A)
1 ∈I(w,A ∧B) iff [1∈I(w,A) and 1 ∈I(w,B)]
0 ∈I(w,A ∧B) iff [0∈I(w,A) or 0 ∈I(w,B)]
1 ∈I(w,A ∨B) iff [1∈I(w,A) or 1 ∈I(w,B)]
0 ∈I(w,A ∨B) iff [0∈I(w,A) and 0 ∈I(w,B)]
1 ∈I(w,A →B) iff forallx,y ∈W: ifRwxy and 1∈I(x,A), then 1 ∈I(y,B)
0 ∈I(w,A →B) iff forallx,y ∈W: ifRwxy and 1∈I(x,A), then 0 ∈I(y,B)

An axiomatization ofBDW is obtained from the axiomsystem forBD by adding BTe′. Like theconstructive connexive logicC, the connexiverelevance logicBDW is negation inconsistent butnon-trivial.

4.5.3 Material connexive logic

Addingex contradictione quodlibet to systemC has a trivializing effect, and adding the Law ofExcluded Middle toC does not result in a logic thathas positive classical propositional logic as a fragment. However, ifimplicationsA →B are understood as material,Boolean implications, then a separate treatment of falsity conditionsagain allows introducing a system of connexive logic. The resultingsystemMC may be called a system of materialconnexive logic. The semantics is quite obvious: a modelMis just a function from the set of allliterals, i.e., propositional variables or negated propositionalvariables, into the set of classical truth values {1, 0}. Truth of aformulaA in a modelM (M⊨A) is inductively definedas follows:

M ⊨p iffv(p) = 1
M ⊨ (A ∧B)iffM ⊨A andM⊨B
M ⊨ (A ∨B)iffM ⊨A orM⊨B
M ⊨ (A →B)iffM ⊭A orM⊨B

M ⊨ ~p iffv(~p) = 1
M ⊨ ~~A iffM⊨A
M ⊨ ~(A ∧B)iffM ⊨ ~A orM⊨ ~B
M ⊨ ~(A ∨B)iffM ⊨ ~A andM⊨ ~B
M ⊨ ~(A →B) iffM ⊭A orM ⊨ ~B

A formula is valid iff it is true in all models. (Alternatively, onecould use the semantics ofC and require the set ofstates of a frame to be a singleton.) The set of all valid formulas isaxiomatized by the following set of axiom schemata and rules:

The logicMC can be faithfully embedded into positiveclassical logic, whenceMC is decidable. Thefollowing truth tables forMC, while considering alanguage with a classical negation, resulting in a system called“dialetheic Belnap Dunn Logic,”dBD, aregiven in Omori 2016c:

The formula ~(A →B) → (A ∧~B) is, of course, not a theorem ofMC. LikeC,MC is a paraconsistent logiccontaining contradictions. The connexive logicMCdiffers from the four-valued logicHBe presented inAvron 1991 by making use of the above clause that guarantees thevalidity of BTe′, i.e.,

M ⊨ ~(A →B) iffM ⊭A orM ⊨ ~B

M ⊨ ~(A →B) iffM ⊨A andM ⊨ ~B.

As already mentioned, Cantwell’s three-valued connexive logicCN can be obtained by extendingMCwith the Law of Excluded Middle and, semantically, by requiring thatfor every propositional variablep and every modelM,M ⊨p orM ⊨ ~p.There is another three-valued connexive logic that is strictlystronger thanCN, namely the “dialetheic Logicof Paradox,”dLP, studied in Omori 2016c, whichturned out to be equivalent with the systemLImp fromOlkhovikov 2002 (published in English translation in 2016). WhilstOlkhovikov uses a unary operator L, understood as a kind of necessityoperator, in the language ofLImp, Omori uses a unaryconsistency operator, ○, in the language ofdLP. The connective L is definable indLP, and the connective ○ is definable inLImp. It is shown in Omori 2016c thatdLP is inconsistent, definitionally complete, andPost complete. Both, Omori (2016c) and Olkhovikov (2016) consider afirst-order extension ofdLP, respectivelyLImp.

4.5.4 Connexive conditional logic

It is quite natural to obtain anFDE-based connexivelogic by starting form David Nelson’s logicN4because the latter system’s intuitionistic implication is theweakest conditional satisfying modus ponens and the deduction theorem.The conditionals studied within conditional logic in the tradition ofRobert Stalnaker and David Lewis, where the conditional is usuallywritten as ‘□→’, are much weaker thanintuitionistic or relevant implication. The project of taking thebasic system of conditional logicCK introduced byBrian Chellas (1975) as the point of departure for obtaining connexiveconditional logics has been carried out in Wansing and Unterhuber2019, and a similar approach is considered in Kapsner and Omori 2017.Whereas the semantics for the Lewis-Nelson models from Kapsner andOmori 2017 uses binary relationsR_A on a non-emptyset of states, for every formulaA, the Chellas-Segerbergsemantics employed in Wansing and Unterhuber 2019 uses binaryrelationsR_X on a non-empty set of states, forsubsetsX of the set of all states. Both versions of thesemantics can be equipped with sound and complete tableau calculi(although Kapsner and Omori only present the models), but theChellas-Segerberg semantics is suitable for developing a purelystructural correspondence theory in terms of properties of relationsthat are language-independent insofar as they are not relativized to aformula.

A pair <W,R> is aChellas frame (orjust a frame) iffW is a non-empty set, intuitivelyunderstood as a set of information states, andR ⊆W ×W × ℘(W), where℘(W) is the powerset ofW. Instead ofRww′X one usually writeswR_Xw′. LetW,R be a frame such that for allX ⊆Wandw,w′ ∈W,wR_Xw′ impliesw′ ∈X. ThenM= <W,R,v⁺,v⁻> is a modelfor the connexive conditional logicCCL iffv⁺ andv⁻ are valuationfunctions from the set of propositional variables into ℘(W), the support of truth and support of falsity conditionsfor propositional variables, negated formulas, conjunctions, anddisjunctions are defined as in the case ofC-modelsand, moreover,

M,w ⊨⁺(A □→B) iff for allu ∈W such thatwR_[[A]]u itholds thatM,u⊨⁺B
M,w ⊨⁻(A □→B) iff for allu ∈W such thatwR_[[A]]u itholds thatM,u⊨⁻B,

where [[A]] is the set of states that support the truth ofA.

If <W,R> is aChellas frame, atriple <W,R,P> is said to be aSegerberg frame (or a general frame) forCCLifP is a binary relation on ℘(W) thatsatisfies certain closure conditions. A quintupleM= <W,R,P,v⁺,v⁻> then is ageneral model forCCL if <W,R,P> is a general frame forCCL,<W,R,v⁺,v⁻> is a model forCCL,and for every propositional variablep,[[p]],[[~p]] ∈P. The closureconditions onP are exactly the conditions guaranteeing thatfor every formulaA, [[A]],[[~A]] ∈P if for every propositional variablep,[[p]],[[~p]] ∈P. If[[A]],[[~A]] is seen as the proposition expressed byA, then a general model forCCL is richenough to guarantee that every proposition expressed by a formula isavailable. This is needed for a purely structural correspondencetheory. The formulaA □→A, for example,is valid on a general frame iff it satisfies the frame condition:

C_{A □→A}: For allX⊆W andw,w′ ∈W,wR_Xw′ impliesw′ ∈X.

General frames forCCL are required to satisfycondition C_{A □→A} in order tomake sure that Boethius’ theses are indeed validated. InUnterhuber and Wansing 2019 sound and complete tableau calculi arepresented forCCL and the weaker systemcCL that validates Aristotle’s theses but notBoethius’ theses and that is obtained by giving upC_{A □→A}. In Wansing andUnterhuber 2019 these results are then extended to systems that areobtained by adding a constructive implication to the language ofcCL andCCL.

McCall (2012) classifies the principles he callsAbelard’sFirst Principle andAristotle’s Second Thesis (cf.section 2) as connexive principles. In Wansing and Skurt 2018 it isargued that since Aristotle’s Second Thesis and Abelard’sFirst Principle both involve conjunction, one may think of obtainingmotivation for them from the idea of negation as cancellation and fromthe failure of Simplification as justified by the erasure model ofnegation. Like the other connexive logics considered in the presentsection,CCL is a system in which Abelard’sFirst Principle and Aristotle’s Second Thesis fail to bevalid.

4.6 Connexive modal logics

There is a growing literature on modal extensions of connexive logics.In Wansing 2005, the language of the connexive logicC is extended by modal operators □ and ◊(“it is possible that”) to define a connexive andconstructive analogueCK of the smallest normal modallogicK. The systemCK is shown tobe faithfully embeddable intoQC, to be decidable,and to enjoy the disjunction property and the constructible falsityproperty.

It is well-known that intuitionistic propositional logic can befaithfully embedded into the normal modal logicS4,which, likeK, is based on classical propositionallogic (cf. the entrieslogic: intuitionistic andlogic: modal). There exists a translation γ, due to Gödel, such that aformulaA of intuitionistic logic is intuitionistically validiffA’s γ-translation is valid inS4. In particular, intuitionistic implication isunderstood as strict material implication: γ(A →B) = □(γ(A) ⊃ γ(B)).Kamide and Wansing (2011) define a sequent calculus for connexiveS4 based onMC. This system,CS4, is shown to be complete with respect to arelational possible worlds semantics. The proof uses a faithfulembedding ofCS4 into positive, negation-freeS4. Moreover, it is shown that the cut-rule is anadmissible rule inCS4 and that the constructiveconnexive logicC stand toCS4 asintuitionistic logic stands toS4. In the faithfulembedding, the modal translation of negated implications is asexpected: γ(~(A →B)) =□(γ(A) ⊃ γ(~B)). A similartranslation is used in Odintsov and Wansing 2010 to embedC into a modal extensionBS4 ofBelnap and Dunn’s four-valued logic.

InCS4 the modal operators □ and ◊ aresyntactic duals of each other: the equivalence between□A and ~◊~A and between ◊Aand ~□~A is provable. Kamide and Wansing (2011) alsopresent a cut-free sequent calculus for a connexive constructiveversionCS4^d– ofS4 without syntactic duality between □ and◊. The relational possible worlds semantics forCS4^d– is not fullycompositional, cf. Odintsov and Wansing 2004.CS4^d– is faithfullyembeddable into positiveS4 and decidable. MoreoverC is faithfully embeddable intoCS4^d–.

Modal Boolean connexive relatedness logics are investigated inJarmużek and Malinowski 2019b, a modal extension of a“bi-classical” paraconsistent connexive logic isintroduced in Kamide 2019, and connexive variants of various modalextensions ofFDE that are extensions ofMC are studied in Odintsov, Skurt, and Wansing2019.

5. Connexive logics and consequential logics

Aristotle’s and Boethius’ theses express, as it seems,some pre-theoretical intuitions about meaning relations betweennegation and implication. But it is not clear that a language mustcontain only one negation operation and only one implication. Thelanguage of bi-intuitionistic logic contains two negations, thelanguage of the bi-intuitionistic connexive logics in Wansing 2016band Kamide & Wansing 2016 contain three negations, and thelanguage of systems ofconsequential implication comprisestwo implication connectives together with one negation, see Pizzi1977, 1991, 1993, 1996, 1999, 2004, 2005, 2008, 2018, Pizzi andWilliamson 1997, 2005. Pizzi (2008, p. 127) considers a notion ofconsequential relevance, namely that “[t]he antecedentand the consequent of a true conditional cannot have incompatiblemodal status,” and suggests to capture consequential relevanceby requiring that in any true conditionalA →B, (i)A strictly impliesB and (ii)A andB have the same modal status in the sense that□A ⊃ □B, □B ⊃□A, ◊A ⊃ ◊B, and◊B ⊃ ◊A are ture, where ⊃ ismaterial implication. Moreover, it is required that □A⊃ ◊A is always true.

In Pizzi and Williamson 1997, a conditional satisfying (i) and (ii) iscalled ananalytic consequential implication and the notionof a normal system of analytic consequential implication is defined.‘Normal’ here means that such a system contains certainformulas and is closed under certain rules. The smallest normalconsequential logic that satisfies AT is calledCI.Alternatively,CI can be characterized as thesmallest normal system that satisfies the Weak Boethius’ Thesis,i.e, (A →B) ⊃ ¬(A →¬B), where → is consequential implication and ¬is classical negation. In Omori and Wansing 2019 the semantics ofCI is presented in a way showing that the semanticsof the consequential conditional is obtained by tweaking the truthconditions of strict implication in Kripke models with a serialaccessibility relation (so that □A ⊃◊A is valid). The standard truth conditions aresupplemented by requiring equal modal status for the antecedent andthe consequent.

Pizzi and Williamson (1997) show thatCI can befaithfully embedded into the normal modal logicKD,and vice versa. Analytic consequential implication is interpretedaccording to the following translation function φ:

φ(A →B) = □(φA ⊃φB) ∧ (□φB ⊃□φA) ∧ (◊φB ⊃◊φA)

As Pizzi and Williamson (1997, p. 571) point out, their investigationis a “contribution to the modal treatment of logics intermediatebetween logics of consequential implication and connexivelogics.” They emphasize a difficulty of regarding consequentialimplication as a genuine implication connective by showing that in anynormal system of consequential logic that admits modus ponens forconsequential implication and contains BT, the following formulas areprovable:

(a) (A →B) ≡ (B →A),
(b) (A →B) ≡ ¬(A →¬B)

where ≡ is classical equivalence. Since (A →B) ↔ ~(A → ~B) is a theorem ofC and other connexive logics, the more problematicfact, from the point of view of this system, is the provability of(a). Pizzi and Williamson also show that in any normal system ofconsequential logic that contains BT, the formula (A →B) ≡ (A ≡B) is provable if(A →B) ⊃ (A ⊃B) isprovable, in other words, consequential implication collapses intoclassical equivalence if (A →B) ⊃(A ⊃B) is provable. The construction ofAristotelian squares of opposition and their combination toAristotelian cubes in systems of consequential implication isconsidered in Pizzi 2008. Two kinds of consequential implication arediscussed and compared to each other in Pizzi 2018.

6. Summary

In summary, it may be said that connexive logic, although it iscontra-classical and unusual in various respects, is not just a formalgame or gimmick. There are several kinds of systems of connexivelogics with different kinds of semantics and proof systems, and in the21st century the subject has been experiencing a renaissance. Theintuitions captured by systems of connexive logic can be traced backto ancient roots, and applications of connexive logics range fromAristotle’s syllogistic to Categorial Grammar, the study ofcausal implications, and connexive mathematics.

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• Angell, R.B., 1966, “Three logics of subjunctive conditionals,” paper presented August 12, 1966 at Colloquium on Logic andFoundations of Mathematics, Hannover, Germany.
• Estrada-González, L., 2019, “The Bochum Plan and the Foundations of Contra-Classical Logics,” CLE e-Prints.
• Vidal, M., 2017, “When Conditional Logic met Connexive Logic,” IWCS 2017 – 12th International Conference on ComputationalSemantics.
• IfCoLog Journal of Logics and their Applications, Vol.(3), No 3 (2016), “Special Issue: Connexive Logics.”
• Logic and Logical Philosophy, Vol. 28, No 3 (2019), “Special Issue: Advances in Connexive Logic.”
• The connexive logic website. This is a website for everyone interested in connexive logic,collecting information about events on connexive logic and keeping anupdated list of publications related to connexive logic.

Related Entries

Abelard [Abailard], Peter |Aristotle, General Topics: logic |Boethius, Anicius Manlius Severinus |conditionals |conditionals: counterfactual |grammar: typelogical |Kilwardby, Robert |logic: classical |logic: conditionals |logic: intuitionistic |logic: many-valued |logic: modal |logic: paraconsistent |logic: relevance |negation

Acknowledgments

The author would like to thank Hitoshi Omori for many stimulatingdiscussions on connexive logic and comments on a draft version of thisentry, Wolfgang Lenzen for making available an excerpt fromBoethius’De Syllogismo Hypothetico, and Hans Rott andAndreas Kapsner for some helpful remarks.