## Welcome!

This is a website for everyone interested in connexive logic. The purpose of this website is to offer a very brief overview of connexive logic, collect some information about events on connexive logic, and keep an updated list of publications related to connexive logic.

What is connexive logic?

In short, connexive logic is a logic in which the following formulas are counted as theorems:

• Aristotle's Theses: ~(~A→A), ~(A→~A)
• Boethius' Theses: (A→B)→~(A→~B), (A→~B)→~(A→B)

As one can see, none of these formulas are theorems in the so-called classical logic, and thus connexive logic is highly non-classical. In the following paragraphs, you can find a quick overview of connexive logic through some major figures who contributed to the development of connexive logic. For a more detailed introduction, see the entry for Connexive Logic in the Stanford Encyclopedia of Philosophy.

Origins of connexive logic: Aristotle and Boethius

Aristotle

Boethius

As is shown by the fact that the characteristic theses of connexive logic are named after Aristotle and Boethius, one of the motivations for connexive logic is due to these two influential philosophers. For example, Aristotle in Prior Analytics 57b14, assumes that "It is impossible that if A, then not-A." Moreover, it has been thought that Boethius holds that " ‘if A then ~B’ is the negation of ‘if A, then B’ ", though this has been challenged by Christopher Martin (his paper is listed in Publications page).

Modern founders of connexive logic: Richard Angell and Storrs McCall

Richard Angell

Storrs McCall

As is well-known, logic in the modern form took off by the contributions of logicians such as George Boole, Gottlob Frege, Bertrand Russell, and David Hilbert. Frege developed what is now known as classical logic in 1879, and soon after that various nonclassical logics were developed based on various motivations. Those include modal logic, intuitionistic logic, many-valued logic, etc. And it was in the 1960s when connexive logic finally got established and attracted some attention. The important logicians here are Richard Angell and Storrs McCall. More specifically, Angell aimed at devising a formal system that realizes what he calls the principle of subjunctive contrariety, i.e. the principle that `If p were true then q would be true' and `If p were true then q would be false' are incompatible. Based on this work of Angell, McCall introduced the terminology "connexive logic", and studied extensively the formal system presented by Angell. McCall also explored the possibility of applying connexive implication to reproduce all valid moods of Aristotle's syllogistic in a first- order language. Note, finally, that Angell has a monograph in which he develops a system of connexive logic in order to solve a broad range of paradoxes.

Further developments of connexive logic

R. Routley (later Sylvan)

Claudio Pizzi

Graham Priest

For further developments of connexive logic, we may raise four more philosophers, namely Richard Routley (later Sylvan), Claudio Pizzi, Graham Priest and Heinrich Wansing. Let us briefly review their contributions.

After the publications of Angell and McCall, Richard Routley (later Sylvan), together with H. Montgomery, published a critical piece on the contributions of Angell and McCall. More specifically, Routley and Montgomery pointed out some implausible consequences of systems of Angell and McCall. Ten years later, Routley suggested a semantic framework to characterize the theses of Aristotle and Boethius, based on the semantic framework for relevant logics, and this was followed by some of the relevant logicians, such as Robert Meyer, Chris Mortensen and Ross Brady who also published their results on connexive logic. These results are highly interesting since relevant logics already pay attention to the relevance between the antecedent and the consequent of implications. However, the semantic framework for relevant logics are already not so simple, and extending the semantic framework to accommodate connexive theses resulted in a rather complicated semantics.

Another intensive study that is closely related to connexive logic is carried out by Claudio Pizzi. Motivated by the works of Angell and McCall, Pizzi adds some motivations from the investigations on counterfactuals. Pizzi then turns to develop some formal systems based on conditional logic. The distinct feature of Pizzi's systems is marked by having two implications; one is the material implication and the other is what he calls the consequential implication. Note however that Pizzi, together with T. Williamson, emphasizes a disturbing difficulty of regarding consequential implication as a genuine implication connective by showing the following result. Namely, in any normal system of consequential logic that admits modus ponens for consequential implication and contains one of the Boethius' theses, the formula (A→B)≡(B→A) is derivable where → is the consequential implication and ≡ is the material equivalence.

The theses of Aristotle and Boethius involve two connectives, namely implication and negation. And so far, the approaches to connexive logic have more emphasis on implication. By contrast, the system of connexive logic devised by Graham Priest has a distinctive feature in the understanding of negation. That is, Priest examines the idea of negation as cancellation (also known as subtraction negation) which can also be traced back to Aristotle's Prior Analytics. This view of negation is often associated with P. Strawson, who held that a "contradiction cancels itself and leaves nothing". The semantic framework presented by Priest is essentially based on the modal logic S5, and thus quite simple. However, the underlying idea of negation as cancellation is not unproblematic, and seems to be quite difficult to justify as a motivation for connexive logic, as concluded by Priest. (For more on negation in general, see the entry for Negation in the Stanford Encyclopedia of Philosophy.)

Finally, Heinrich Wansing developed systems of connexive logic based on David Nelson's constructive logics. This approach has the advantage of being philosophically intuitive, and technically simple. Indeed, it is intuitively plausible since the semantic framework for the negation-free fragment inherits the intuitionistic framework. Moreover, the simple move we make to reach the connexivity amounts to having a non-standard falsity condition (which needs to be stated independently of the truth condition) for implications. That is, in Nelson's logics, the implication is false if and only if the antecedent is true and the consequent is false. But in Wansing's logics, the implication is false if and only if it holds that if the antecedent is true then the consequent is false. Note also that Wansing, in an earlier work, motivates connexive implication by introducing a negation connective into categorial grammar in order to express negative information about membership in syntactic categories.