Paraconsistency and Dialetheism

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Introduction

This article is about paraconsistent logic, logic in which contradictions do not entail everything. Though the roots of paraconsistency lie deep in the history of logic, its modern developments date to just before the middle of the 20th century. Since then, paraconsistent logic — or better, logics, since there are many of them — have been proposed and constructed for many, and very different, reasons. The most philosophically challenging of these reasons is dialetheism, the view that some contradictions are true. Though this article will also discuss other aspects of paraconsistency, it will concentrate specifically on its dialetheic aspects. Other aspects of the subject can be found in the article ‘Paraconsistency: Preservational variations’ in this volume of the Handbook. The subject also has close connections with relevant logic. Many related details can therefore be found in the article ‘Relevant and Substructural Logics’, in volume 4 of the Handbook.

In the following two parts of this article, we will look at the history of the subject before about 1950. We will look at the history of paraconsistency; then we will look at the history of dialetheism. In the next two parts, we will turn to the modern developments, those since about 1950; first paraconsistency, then dialetheism. In the final three parts of the article will look at some important issues that bear on paraconsistency, or on which paraconsistency bears: the foundations of mathematics, the notion of negation, and rationality.

Let us start, however, with definitions of the two central notions of the article. Perhaps the major motivation behind paraconsistency in the modern period has been the thought that there are many situations where we wish to handle inconsistent information in a sensible way — and specifically, where we have to infer from it. (We may also wish to revise the information; but that is another matter. And a knowledge of what does or does not follow sensibly from the information may be necessary for an intelligent revision.)

Let ├ be any relation of logical consequence.1 Let ¬ denote negation. (What, exactly, this is, we will come back to later in this essay.) Then the relation is called explosive if it satisfies the principle of Explosion:

α,¬αβ

or, as it is sometimes called, ex contradictione quodlibet. Explosion is, on the face of it, a most implausible looking inference. It is one, however, that is valid in “classical logic”, that is, the orthodox logic of our day.

Clearly, an explosive notion of logical consequence is not a suitable vehicle for drawing controlled inferences from inconsistent information. A necessary condition for a suitable vehicle is therefore that Explosion fail. This motivates the now standard definition: a consequence relation is paraconsistent if it is not explosive. The term was coined by Miró Quesada at the Third Latin American Symposium on Mathematical Logic in 1976.2

Given a language in which to express premises and conclusions, a set of sentences in this language is called trivial if it contains all sentences. Let Σ be a set of sentences, and suppose that it is inconsistent, that is: for some α, Σ contains both ¬ α and −α. If ├ is explosive, the deductive closure of Σ under ├ (that is, the set of consequences of Σ) is trivial. Conversely, if ├ is paraconsistent it may be possible for the deductive closure of Σ to be non-trivial.3 Hence, a paraconsistent logic allows for the possibility of inconsistent sets of sentences whose deductive closures are non-trivial.

Paraconsistency, in the sense just defined, is not a sufficient condition for a consequence relation to be a sensible one with which to handle inconsistent information. Consider, for example, so-called minimal logic, that is, essentially, intuitionist logic minus Explosion. This is paraconsistent, but in it α,¬α├ ¬β, for all α and β.4 Hence, one can infer the negation of anything from an inconsistency. This is not triviality, but it is clearly antithetical to the spirit of paraconsistency, if not the letter. It is possible to try to tighten up the definition of ‘paraconsistent’ in various ways.5 But it seems unlikely that there is any purely formal necessary and sufficient condition for the spirit of paraconsistency: inconsistent information may make a nonsense of a consequence relation in so many, and quite different, ways.6 Better, then, to go for a clean, simple, definition of paraconsistency, and leave worrying about the spirit to individual applications.

No similar problems surround the definition of ‘dialetheism’. The fact that we are faced with, or even forced into operating with, information that is inconsistent, does not, of course, mean that that information is true. The view that it may be is dialetheism. Specifically, a dialetheia is a true contradiction, a pair, α and −α, which are both true (or equivalently, supposing a normal notion of conjunction, a truth of the form α ∧ − α). A dialetheist is therefore a person who holds that some contradictions are true. The word ‘dialetheism’ and its cognates were coined by Priest and Routley in 1981, when writing the introduction to Priest, Routley, and Norman [1989].7 Before that, the epithet ‘paraconsistency’ had often been used, quite confusingly, for both dialetheism and the failure of explosion.8

A trivialist is a person who believes that all contradictions are true (or equivalently, and more simply, who believes that everything is true). Clearly, a dialetheist need not be a trivialist (any more than a person who thinks that some statements are true must think that all statements are true). As just observed, a person may well think it appropriate to employ a paraconsistent logic in some context, or even think that there is a uniquely correct notion of deductive logical consequence which is paraconsistent, without being a dialetheist. Conversely, though, it is clear that a dialetheist must subscribe to a paraconsistent logic — at least when reasoning about those domains that give rise to dialetheias — unless they are a trivialist.

A final word about truth. In talking of true contradictions, no particular notion of truth is presupposed. Interpreters of the term ‘dialetheia’ may interpret the notion of truth concerned in their own preferred fashion. Perhaps surprisingly, debates over the nature of truth make relatively little difference to debates about dialetheism.9

Section snippets

Explosion in Ancient Logic

Having clarified the central notions of this essay, let us now turn to its first main theme. What are the histories of these notions? Paraconsistency first. It is sometimes thought that Explosion is a principle of inference coeval with logic. Calling the received theory of inference ‘classical’ may indeed give this impression. Nothing could be further from the truth, however. The oldest system of formal logic is Aristotle's syllogistic;10

Contradiction in Ancient Philosophy

Can contradictions be true? At the beginning of Western philosophy it would seem that opinions were divided on this issue. On the face of it, certain of the Presocratics took the answer to be ‘yes’. Uncontroversially, Heraclitus held that everything was in a state of flux. Any state of affairs described by α changes into one described by ¬α. More controversially, the flux state was one in which both α and ¬α hold.31

Background

The revolution that produced modern logic around the start of the 20th century depended upon the application of novel mathematical techniques in proof-theory, model theory, and so on. For a while, these techniques were synonymous with classical logic. But logicians came to realise that the techniques are not specific to classical logic, but could be applied to produce quite different sorts of logical systems. By the middle of the century, the basics of many-valued logic, modal logic, and

Inconsistent Information

As we noted in 1.2, the major motive for modern paraconsistency is the idea that there are situations in which we need to reason in a non-trivial way from inconsistent information. The early proponents of paraconsistent logics mentioned various such situations, but the first sustained discussion of the issue (that I am aware of) is Priest and Routley [1989].107

Introduction: a Brief History

The development of modern logic has been intimately and inextricably connected with issues in the foundations of mathematics. Questions concerning consistency and inconsistency have been a central part of this. One might therefore expect paraconsistency to have an important bearing on these matters. Such expectations would not be disappointed. In this part we will see why. In the process, we will pick up the issue of the set-theoretic paradoxes left hanging in the previous section.

Let us start

What is Negation?

We have now looked at the history of both paraconsistency and dialetheism. No account of these issues could be well-rounded, however, without a discussion of a couple of philosophical notions which are intimately related to both. One of these is rationality, which I will deal with in the next part. The other, which I will deal with in this part, is negation. This is a notion that we have been taking for granted since the start of the essay. Such a crucial notion clearly cannot be left in this

Multiple Criteria

Let us now turn to the final issue intimately connected with paraconsistency and dialetheism: rationality. The ideology of consistency is so firmly entrenched in orthodox western philosophy that it has been taken to provide the cornerstone of some of its most central concepts: consistency has been assumed to be a necessary condition for truth, (inferential) validity, and rationality. Paraconsistency and dialetheism clearly challenge this claim in the case of validity and truth (respectively).

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