Connexive logic: new old challenges

After the intense attention the relevance logic community and its friends gave to McCall’s ideas on connexive implication during the late 1960s and nearly all the 1970s, studies on connexive principles faded in the 1980s. By the 1990s, Claudio Pizzi was the only person working systematically on them. (See, e.g. [53], [55], [57], [54].) In fact, the connexive principles seemed forgotten even by relevance logicians, once their main researchers.¹ Moreover, these principles were never seriously considered by paraconsistent logicians outside the Australasian-centered movement, to mention the two brands of non-classical logic more akin to the ideas underlying the connexive principles. Maybe Richard Routley’s vocal comments about the alleged uselessness of connexivity for paraconsistency contributed to that fate of connexive logics. According to him, for any connexive logic X (and with |$N$| standing for a negation), |$A, NA\models _{\tiny \mathbf{X}}B$| is either vacuously valid—for a formula and its negation would cancel each other, making them both untrue—or it is invalid but only at the cost of making contradictions logically inert, that is, they do not entail anything, perhaps only themselves. See [62, pp. 93, 101] and also [61, fn. 13].²

By 2014, Wansing had already been working systematically on connexive principles for around ten years. (See [68], [69], [70], [71].) Like in previous decades, other individuals (e.g. John Cantwell [4], Luis Estrada-González [5], Thomas Ferguson [13], Andreas Kapsner [26], Grigory Olkhovikov [49], Graham Priest [58]) also investigated ideas related to the connexive principles, but, with the only exception of Pizzi, none of them published on them as much as Wansing. And none of them, not even Pizzi or Wansing, was having an impact on the others, or at least an influence strong enough to create a common research program, a community sharing a continued interest on the topic.

Things would change in 2015, when Heinrich Wansing and Hitoshi Omori organized the First Workshop on Connexive Logic in Istanbul, as a part of the World Congress on Universal Logic. Though one can hardly be sure about this, it seems that, without the workshops series, Wansing’s work on the topic would have had the same fate as Pizzi’s work until that moment.

The yearly workshops, most of them held in Bochum, Germany, provided a venue stable enough for people to meet regularly. That indeed contributed to creating a core community in a relatively short time, and it grew because of the wide variety of topics surrounding the connexive principles: historical predecessors, philosophical implications, connections with other kinds of non-classical logics, empirical research on the connexive principles, and the very definition of the phenomenon underlying this field emerging once more in the history of logic.

And here we are, almost 10 years after the first workshop, presenting a selection of papers after the Seventh Workshop on Connexive Logic held in Mexico City in October 2022. At that time, it was only the third occasion the workshop was held outside the limits of Bochum; the previous ones were the kick-off one in Istanbul and the third one, in Kyoto, in 2017. As we write, there have been two more meetings outside Bochum: the eightieth (2023) and ninth (2024) editions of the workshop, in Turin (Italy) and Toruń (Poland), respectively.

What is connexive logic about? According to Damian Szmuc [65], in connexive logic one has ‘high esteem’ for

the idea that there is a certain relation in which some formulas should not stand. To be more specific—while keeping talk of this relation intentionally unspecified—the connexive ideas can be summarized as follows. Firstly, there is a certain relation in which a formula |$A$| and its negation |$NA$| should not stand. Secondly, if the pair of formulas |$A$| and |$B$| stand in this very same relation, then the pair of formulas |$A$| and |$NB$| should not stand in this relation.

(Notation has been adjusted, changing ‘|$\neg $|’ to ‘|$N$|’, to make it fit better with the notation used in this note.)

Most commonly, such relation in which certain formulas are, or are not, is implication, >. Thus, ‘there is a certain relation in which a formula |$A$| and its negation |$NA$| should not stand’ becomes:

• It should not be the case that |$A>NA$|

and ‘if the pair of formulas |$A$| and |$B$| stand in this very same relation, then the pair of formulas |$A$| and |$NB$| should not stand in this relation’ becomes in turn:

• If |$A>B$| then it should not be the case that |$A>NB$|

These ideas may be expressed more formally by requiring that, for every formulas |$A$| and |$B:$|

• |$\nvdash _{\tiny{\mathbf{L}}}(A>_{1}N_{1}A)$|

• |$(A>_{2}B)\nvdash _{\tiny{\mathbf{L}}}N_{2}(A>_{3}N_{3}B),$|

respectively. But usually, the ideas are expressed by requiring that certain schematic formulas are valid:

• |$\vdash _{\tiny{\mathbf{L}}}N_{1}(A>_{1}N_{2}A)$|

• |$\vdash _{\tiny{\mathbf{L}}}(A>_{2}B)>_{3}N_{3}(A>_{4}N_{4}B)$|

Moreover, it has been usually assumed that all the |$>_{i}$|s are in fact different occurrences of the same connective, and likewise for the |$N_{j}$|s.³ That delivers the simpler versions:

• |$\vdash _{\tiny{\mathbf{L}}}N(A>NA)$|

• |$\vdash _{\tiny{\mathbf{L}}}(A>B)>N(A>NB)$|

(In order to avoid cumbersome notation, we will use from now on the simplified versions of the schemas, without any subindices, but deep down we are not assuming that all the occurrences of implications, and negations, are occurrences of the same connective.)

The validities |$N(A>NA)$| and |$(A>B)>N(A>NB)$|⁠, together with

• |$\vdash _{\tiny{\mathbf{L}}}N(NA>A)$|

• |$\vdash _{\tiny{\mathbf{L}}}(A>NB)>N(A>B);$|

plus the requirement that > is not a biconditional, expressed as

• |$\nvdash _{\tiny{\mathbf{L}}}(A>B)>(B>A)$|

constitute a good working definition of connexive logic, employed by some of the most notable scholars in the field. In [74], Wansing and Omori attribute the definition to McCall in [43] and made a call to stick to that definition in the interest of unifying and reducing the terminology, so as to make it easier to survey and access the field from outside the community of connexive logicians.⁴ In this regard, let us offer a word of caution.

It sometimes happens that, when a research field is still new and vibrant, there are several non-equivalent characterizations of its central notions, including the very definition of the field itself and of its target phenomenon. Connexive logic is like that, in spite of being a ‘new’ field with a history of 2,500 years. Nowadays, and contrary to what is claimed in [74], there is no ‘widely agreed upon definition of connexive logic’, and established terminology reaches only a few principles and notions, and sometimes only by chance or for the wrong reasons. For example, most scholars nowadays call the schema |$N(A>NA)$| ‘Aristotle’s Thesis’. But if one wanted to be true to the texts, what Aristotle endorses in the Prior Analytics is the so-called ‘Variant’: |$N(NA>A)$|⁠. Thus, at least historically, Aristotle’s Thesis is what today is called ‘Variant of Aristotle’s Thesis’, and the variant should be what today is identified as the thesis.⁵

In his doctoral dissertation [42, Ch. 5], McCall employed a broad notion of connexive logic. For him, a connexive calculus is a consistent system ‘which serves to catch and domesticate the notion of connexive implication’, where connexive implication is a non-equivalential (in particular, non-symmetric) connective whose ‘mark (necessary and sufficient condition)’ is ‘the truth of one or both of these [Aristotle’s and Boethius’] theses’.⁶ Interestingly enough, such a broad characterization did not disappear in the 1966 paper. There, McCall says [43, p. 416]: ‘a system of connexive logic may range from one in which no proposition implies or is implied by its own negation to one in which Boethius’ thesis is asserted.’ And a couple of pages later [43, p. 418], he says that because a certain system validates both Aristotle’s and Boethius’ Theses and invalidates the symmetry of implication, it is a connexive system. That is, meeting these conditions is sufficient to have a connexive logic; no claim about their necessity is made.

That is important because there are logics with Aristotle’s Theses which nevertheless lack Boethius’ Theses in arrow form, even in Pizzi’s ‘weak’ form, that is, with a material conditional as the main connective. Examples of that are Goddard-Routley’s family of logics PR in [21]; Priest’s in [58], or Francez’s N|$^{\sim _{l}}$| in [16]. Likewise, nothing prevents the existence of logics with Boethius’ Theses but not with Aristotle’s. Since Aristotle’s Thesis can be derived quite easily from Boethius’ through Substitution, Identity (⁠|$A>A$|⁠) and Detachment, a logic with Boethius’ Thesis but not Aristotle’s must lack Substitution, Identity or Detachment.⁷

In another paper of the 1960’s, McCall asks for more than the validity of Aristotle’s and Boethius’ Theses in order to have a connexive logic.⁸ He says:

[Besides Aristotle’s and Boethius’ Theses] Further characteristics of connexive implication include the rejection of the paradoxes of material and strict implication, and the avoidance of what have come to be known as the fallacies of relevance and necessity. [44, p. 350]

In recent times, the need for more constraints was made more vividly by Kapsner. He says:

I take it that these [‘Surely it is not the case that a proposition |$A$| should imply its own negation (or the other way around)’ and ‘Surely if |$A$| implies |$B$|⁠, then |$A$| does not imply not-|$B$| (and if |$A$| implies not-|$B$|⁠, then |$A$| does not imply |$B$|⁠)’] are indeed robust pre-theoretical intuitions, and that it is at least an interesting project to try and do justice to them. However, […] the mere fact that Aristotle and Boethius are obeyed by a logic is not enough to fully answer to these intuitions. [26, p. 2]

In fact, another way to read Szmuc’s passage is to understand the relation in which |$A$| and |$NA$| should not stand as a satisfiable implication. Thus, ‘there is a certain relation in which a formula |$A$| and its negation |$NA$| should not stand’ becomes

• Unsat1: It should not be satisfiable that |$A>NA$| and it should not be satisfiable that |$NA>A$|⁠. (For any |$A$|⁠.)

Similarly, if |$A>B$| implies that it should not be the case that |$A>NB$|⁠, then one could further demand that

• Unsat2: It should not be simultaneously satisfiable that |$A>B$| and |$A>NB$|⁠. (For any |$A$| and |$B$|⁠.)

These ideas, advanced by Andreas Kapsner [26] as characteristic of ‘strong connexivity’, cash out a relation’s not-holding through the metalogical property of satisfaction of a formula. This requirement is not usually demanded from non-classical logics in their ruling out classically valid theses. While an intuitionist, for instance, takes LEM as invalid, they could in principle accept some instances of that schema as valid. Nonetheless, that is not so problematic, since what the LEM demands is that at least one of the disjuncts is true. According to Kapsner, the problem in the case of connexive logic is that, with the advent of Wansing-style connexive logics, one can have non-vacuously true implications of the form |$A>NA$| (resp. |$NA>A$|⁠), which means satisfying both |$A$| and |$NA$|⁠, and then the story to be told about the validity of |$N(NA>A)$| and |$N(A>NA)$| becomes more complicated.⁹

Andreas Kapsner’s contribution to this special issue, ‘Aristotle’s dilemma’, makes the point that a strong connexivist might complain that classical logic not only does not validate theses that should be valid, to wit, the connexive theses, but it also satisfies formulas which should not be satisfiable: |$A>NA$| and |$NA>A$|⁠. So, if neither of these is to be satisfied, then |$(A>NA)\oplus (NA>A)$|⁠, dubbed Aristotle’s dilemma by Kapsner, should not be satisfiable either (provided that the meaning of |$\oplus $| aligns with that of disjunctions from natural languages).¹⁰

In a similar spirit, Federico Pailos, in his ‘Inferential-connexive mixed logics’ presented in this volume, proposes that the relation between |$A$| and |$NA$| should, on the one hand, be represented through a consequence relation rather than a conditional connective, and, on the other hand, that this relation’s not-holding should be interpreted as antisoundness rather than unsatisfiability. Roughly, an inference is antisound if and only if for every valuation |$v$|⁠, if |$v$| satisfies every premise, then |$v$| is also a counterexample to the/every conclusion.

To spell out this proposal, Pailos works with strong Kleene mixed logics, i.e. systems that use the three-valued interpretations of strong Kleene logics but which employ a consequence relation that demands that the premises of a valid inference are designated according to some standard |$S_{1}$| and the conclusions of an inference are designated according to some (possibly different) standard |$S_{2}$|⁠. Thus, in his view,

• It should be antisound to infer |$NA$| from |$A;$|

• It should be antisound to infer |$A>NB$| from |$A>B$|

Note that, while most approaches to connexivity focus on connectives of the object language, like |$N$| and >, Pailos’ approach uses the means of a metalanguage to cash out the intuitions behind the connexive principles. Such a move opens the way to an infinite hierarchy of metainferences expressing the connexive theses.¹¹

Another historical candidate to play the role of the relation alluded to in the Szmuc passage is compatibility. Thus, ‘there is a certain relation in which a formula |$A$| and its negation |$NA$| should not stand’ becomes:

• It should not be the case that |$A$| and |$NA$| are compatible.

And ‘if the pair of formulas |$A$| and |$B$| stand in this very same relation, then the pair of formulas |$A$| and |$NB$| should not stand in this relation’ becomes in turn

• If |$A$| and |$B$| are compatible then it should not be the case that |$A$| and |$NB$| are compatible.

Using ‘|$\circ $|’ for a binary connective of compatibility, the above become, respectively:

• |$\vdash _{\tiny{\mathbf{L}}}N(A\circ NA)$|

• |$\vdash _{\tiny{\mathbf{L}}}(A\circ B)>N(A\circ NB)$|

This may have perplexing consequences, at least for some connexivists. Suppose that |$A\circ B$| iff |$N(A>NB)$|⁠. Then, the two schemas above become, respectively:

• |$\vdash _{\tiny{\mathbf{L}}}NN(A>NNA)$|

• |$\vdash _{\tiny{\mathbf{L}}}N(A>NB)>NN(A>NNB),$|

which, if one assumes Double Negation Elimination, deliver

• |$\vdash _{\tiny{\mathbf{L}}}(A>A)$|

• |$\vdash _{\tiny{\mathbf{L}}}N(A>NB)>(A>B)$|

The first, Identity, is not distinctive of connexive logics.¹² The second one is the contentious converse of Boethius’ Thesis.¹³ Thus, it seems that, to obtain the mainstream connexive theses from something like the Szmuc passage by taking compatibility as the relation it speaks of, negations should be rearranged as follows:

(…) there is a certain relation in which some formulas should stand. To be more specific—while keeping talk of this relation intentionally unspecified—the connexive ideas can be summarized as follows. Firstly, there is a certain relation in which a formula |$A$| and itself should stand. Secondly, if the pair of formulas |$A$| and |$NB$| do not stand in this very same relation, then the pair of formulas |$A$| and |$B$| should stand in this relation.

Then:

• |$\vdash _{\tiny{\mathbf{L}}}(A\circ A)$|

• |$\vdash _{\tiny{\mathbf{L}}}N(A\circ NB)>(A\circ B)$|

Thus, supposing |$A\circ B$| iff |$N(A>NB)$| (and Double Negation Elimination) as we said, one obtains:

• |$\vdash _{\tiny{\mathbf{L}}}N(A>NA)$|

• |$\vdash _{\tiny{\mathbf{L}}}(A>B)>N(A>NB),$|

i.e. the Variant of Aristotle’s Thesis and Boethius’ Thesis, respectively.¹⁴

The notions of compatibility and incompatibility—or consistency and inconsistency, as they were also called—were key in the logical studies, sometimes even more than implication, during around half a century, from at least Christine Ladd [31] to Nelson’s [46] and [47], through Lewis [35] and Lewis and Langford’s [36], among others. Not to mention its importance in later connexivity scholars, like Pizzi, who adopted the term ‘cotenability’ from Goodman; see [56]. Kielkopf [28] also used the term ‘cotenability’.¹⁵

Of the papers in this special issue, Wolfgang Lenzen’s ‘On Nelson’s conception of consistency’ is devoted to scrutinizing the plausibility of Nelson’s arguments in favor of the connexive theses. This work is part of Lenzen’s larger project of critically examining the historiographical claims made in the connexive scholarship. (See [32], [33], [34].) In general, Lenzen is what Routley [62, Ch. 1] would call ‘a stricter’, i.e. someone who thinks that some rather classical species of strict implication can account for some of the logical claims of certain historical figures, including Aristotle, Boethius, Abelard, or Leibniz.¹⁶

For Lenzen, Nelson’s arguments depend crucially on the cogency of the arguments against the validity of both Simplification—|$(A\otimes B)>A$|⁠; |$(A\otimes B)>A$|—and Addition—|$A>(A\oplus B)$|⁠; |$B>(A\oplus B)$|⁠. Lenzen argues that Nelson does not provide good reasons to give up on the validity of those, and thus there would be no convincing argument in Nelson’s text for the validity of connexive theses. To date, the interpretation of Nelson’s work, especially [46], in satisfactory contemporary terms, i.e. with a sound and complete semantics for the logic presented there—is a pending task. Whether one agrees with him or not, Lenzen’s contribution takes an interesting step forward in the direction of providing a better understanding of one of the 20th century forerunners of connexive logic.¹⁷

The relation Francez takes into consideration in the phrase ‘If the pair of formulas |$A$| and |$B$| stand in this very same relation, then the pair of formulas |$A$| and |$NB$| should not stand in this relation’ is the passive is implied by rather than the active implies. Thus, one gets ‘If |$A$| is implied by |$B$|⁠, it is not the case that |$A$| is implied by |$NB$|’, which delivers, more formally:

• |$\vdash _{\tiny{\mathbf{L}}}(B>A)>N(NB>A)$|

Taking advantage of the schematicity of |$B$|⁠, and changing it to |$NB$|⁠, one obtains ‘If |$A$| is implied by |$NB$|⁠, it is not the case that |$A$| is implied by |$NNB$|’, which delivers, more formally:

• |$\vdash _{\tiny{\mathbf{L}}}(NB>A)>N(NNB>A)$|

and, with Double Negation Elimination:

• |$\vdash _{\tiny{\mathbf{L}}}(NB>A)>N(B>A).$|

These schemas, and especially their motivation and their alleged natural language companions, have been received with skepticism in the connexive community. But in a sense they express, in implicative form, two other important ideas in the history of connexive logic, at least from Aristotle to Abelard:

• It cannot be that |$A$| and |$B$| are in the intended relation, and that |$A$| and |$NB$| are in that relation too.

• It cannot be that |$A$| and |$B$| are in the intended relation, and that |$NA$| and |$B$| are in that relation too.

Yet another option to cash out the Szmuc passage is keeping the relation between |$A$| and |$NA$| unspecified and using it to model an implication that delivers the connexive schemas above. Such is the approach of the Torunian school (see for example [23], [24], [29], [37], [30], [10]), and exemplified once more in this volume by the paper ‘Axiomatization of Boolean Connexive Logics (BCLs) with syncategorematic negation and modalities’, by Tomasz Jarmużek, Jacek Malinowski, Aleksander Parol and Nicolò Zamperlin.

In that paper, and motivated by the syncategorematic view on the connectives of negation and modalities, the authors investigate formally the properties of closure under negation and demodalization in the context of BCL. They present three classes of extended BCLs, two for the property of closure under an arbitrary number of negations, and one a modal BCL with the property of closure under demodalization. They present suitable axiomatic systems with the corresponding proofs of soundness, completeness, and decidability.

At the beginning of this note, we said that the relation in the Szmuc passage in which certain formulas are, or are not, is implication. As we have seen, there are many candidates to play that role. Francez has suggested that the other two usual binary connectives, conjunction and disjunction, can play it too. Thus, ‘there is a certain relation in which a formula |$A$| and its negation |$NA$| should not stand’ becomes:

• It should not be the case that |$A$|-and-|$NA;$|

and ‘if the pair of formulas |$A$| and |$B$| stand in this very same relation, then the pair of formulas |$A$| and |$NB$| should not stand in this relation’ becomes in turn

• If |$A$|-and-|$B$| then it should not be the case that |$A$|-and-|$NB.$|

Like in the case of implication, these ideas can be expressed by requiring that certain schematic formulas are valid:

• |$\vdash _{\tiny{\mathbf{L}}}N(A\otimes NA)$|

• |$\vdash _{\tiny{\mathbf{L}}}(A\otimes B)>N(A\otimes NB)$|

where |$\otimes $| stands for some conjunction. (But we do not assume that all the occurrences of |$\otimes $| stand for the same conjunction.) The same can be said, mutatis mutandis, for disjunction.

By putting other binary connectives instead of implication in the most inner places in the usual connexive schemas, one arrives at the phenomenon that Francez calls ‘poly-connexivity’, and that has been studied by him in several places. (Cf. [18], [20], [19].) In [9], it was heavily questioned whether the schemas resulting from Francez strategy can be called ‘connexive’ at all, regardless of their intrinsic interest and possible applications in linguistics.¹⁸

When one starts to ask about the connexivity of other connectives besides implication, and after trying other binary connectives, the next natural step is to ask about the quantifiers. Also, some motivations for connexive logics include matters that contemporary logicians would consider essentially quantificational. (See, e.g. [44].) Despite that, the work on quantified connexive logics is scarce. Weber and Wansing’s contribution to this volume, ‘Quantifiers and connexive logic (in general and in particular)’ mentions some of the few precedents and shows that connexive logics admit not only the standard quantifiers |$\forall x$| and |$\exists x$|⁠, but also the contra-classical variants |$\mathbb{A}x$| and |$\mathbb{E}x$| with the property that |$\sim \!\mathbb{A}xA\leftrightarrow \mathbb{A}x\sim \!A$| and |$\sim \!\mathbb{E}xA\leftrightarrow \mathbb{E}x\sim \!A$|⁠. Such quantifiers, on the one hand, provide a simple solution to the problem of formalizing Aristotle’s and Boethius’ theses in first-order languages; and, on the other hand, address philosophical concerns from restricted quantification and bare plurals. Wansing and Weber thus expand connexive logic QC with these contra-classical quantifiers and provide a coherent proof theory and model theory for the resulting expansion QQC.

The variety of sensible proposals found in the literature and the number of ways of cashing the connexive intuitions, illustrated in the many different readings of Szmuc’s passage presented here, all point to one general conclusion: there is no single way to understand connexivity or the notion of connexive logic, and it is too soon to claim that there is a general agreement on the subject. The field is full of new and old challenges. For the moment, we welcome this conclusion, since it means that more philosophical research is needed and that exciting and novel results are yet to be found. The contributions in this special issue are certainly a further step in this direction. We hope that with this introduction we have conveyed both the charm and the complexity of connexive logic. There is yet much to be discussed and investigated in the area, especially with respect to its scope and its connections to other families of logics, its history, its philosophical foundations, its pervasiveness in ordinary reasoning, and its applications in the sciences. We recommend these questions to our fellow logicians and philosophers, and thank our contributors and reviewers for being part of this endeavor.

Acknowledgements

This work was supported by the PAPIIT projects IG400422 and IN406225; the Conahcyt project CBF2023-2024-55; and the Marsden Fund, Royal Society of New Zealand. We thank Lógica M|$\exists $|X|$\forall $|⁠, Thomas Ferguson, Heinrich Wansing, and Hitoshi Omori for many discussions on connexive logic over the years. We are especially grateful to Jane Spurr and all the editorial team of the Journal of the IGPL for their patience.

Conflict of interest

The authors declare no conflict of interest.

Footnotes

References