## Abstract

In some logics, anything whatsoever follows from a contradiction; call these logics *explosive*. *Paraconsistent logics* are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go.

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## Notes

I have nothing much to say about which things are and which things are not consequence relations. I certainly have no precise definition in mind; usual precise definitions exclude some things I include. (For example, understanding a consequence relation as a Tarskian closure operation excludes a wide variety of substructural logics.) If it so much as smells consequencey it’s a consequence relation as far as I’m concerned.

For the additive conjunction ∧, we have

*A*∧¬*A*,*A*∧¬*A*⊩*B*, but*A*∧¬*A*⊯*B*; two copies of a contradiction do entail everything, but a single copy does not.You might note that I haven’t said anything about what it takes to count as a negation or a conjunction or a way of combining premises in the first place. No way am I going near

*that*can of worms.Note that there is still something of an indeterministic flavour: if you know that

*A*is designated, that’s not yet enough to say whether ¬*A*is designated or not. But the facts that underlie this are fully deterministic.Any fully structural logic can be seen as preserving

*some*status; this is one upshot of Suszko’s Thesis; see eg [16, 18, 43, 44]. But this is sometimes not the most helpful way to look at it. Moreover, the suggested generalization of this method extends well beyond fully structural logics, as in [22].Typically, this is because we haven’t noticed the contradiction, but there are at least two other kinds of case: 1) cases in which we notice the contradiction, but haven’t yet decided how or whether to resolve it, and 2) cases in which we notice the contradiction, but have decided simply to live with it.

Smooth infinitesimal analysis provides an example. It is a rich topic of mathematical study based on axioms that are inconsistent in classical logic, but not in the intuitionist logic in which it is conducted. See eg [25].

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Ripley, D. Paraconsistent Logic.
*J Philos Logic* **44**, 771–780 (2015). https://doi.org/10.1007/s10992-015-9358-6

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DOI: https://doi.org/10.1007/s10992-015-9358-6