Dialetheism

https://plato.stanford.edu/entries/logic-paraconsistent/

First published Fri Dec 4, 1998; substantive revision Fri Jun 22, 2018.

A dialetheia is a sentence, $A$, such that both it and its negation, $\neg A$, are true. If falsity is assumed to be the truth of negation, a dialetheia is a sentence which is both true and false. Such a sentence is, or has, what is called a truth value glut, in distinction to a gap, a sentence that is neither true nor false.

Dialetheism is the view that there are dialetheias. If we define a contradiction as a couple of sentences, one of which is the negation of the other, or as a conjunction of such sentences, then dialetheism amounts to the claim that there are true contradictions.

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Some Basic Concepts

Why should Russell’s contradiction not be conceived of as something supra-propositional, something that towers above the propositions and looks in both directions like a Janus head? The proposition that contradicts itself would stand like a monument (with a Janus head) over the propositions of logic (1978, III.59).

A dialetheia is a two-way truth, facing both truth and falsity like a Janus-headed figure. In philosophy, there tends not to be a distinction between a view being inconsistentand being incoherent. Both are unacceptable.

Dialetheism is the view that there are dialetheias. If we define a contradiction as a couple of sentences, one of which is the negation of the other, or as a conjunction of such sentences, then dialetheism amounts to the claim that there are true contradictions. As such, dialetheism opposes—contradicts—the Law of Non-Contradiction (LNC), sometimes also called the Law of Contradiction. The Law can be expressed in various ways; fixing the precise formulation is itself a topic of debate (Priest et al 2004, Part II). Thomas Reid put the LNC in the form ‘No proposition is both true and false’. A strong (modal) statement of the LNC is: for any $A$, it is impossible that both $A$ and $\neg A$ be true.

 LP. Dialetheism as set out by Priest takes all instances of the schema $\neg (A\wedge \neg A)$ to be true, as well as taking as true some sentences that are inconsistent with it, namely, true sentences whose negations are true: dialetheias. According to such versions of dialetheism, all contradictions are false and some are true: dialetheism is itself a dialetheia (‘Concluding self-referential postscript’ to Priest 1979, p. 203).

Dialetheism in Western Philosophy

Aristotle takes a number of the Presocratics to endorse dialetheism, and with apparent justification. For example, in Fragment 49a, Heraclitus says: “We step and do not step into the same rivers; we are and we are not” (Robinson, 1987, p. 35). Protagorean relativism may be expressed by the view that man is the measure of all things; but according to Aristotle, since “Many men hold beliefs in which they conflict with one another”, it follows that “the same thing must be and not be” (1009a10–12).

Despite the orthodoxy about the LNC since Aristotle, during the Middle Ages the problem of seemingly true contradictions surfaced in connection to the paradoxes of divine omnipotence—for instance: can God make a stone too heavy for Him to lift? (See Cotnoir forthcoming.) We find St. Pier Damiani getting close to dialetheism in the De divina omnipotentia, by blaming St. Girolamus for having claimed that God cannot overturn the past and twist what happened into something that didn’t happen. Since God lives an eternal present, denying Him power over the past equates to denying Him power over current and future events, which is blasphemous. So God must have the power of making what is done undone. Later on, Nicholas of Cusa placed at the core of his book De docta ignorantia the idea that God is coincidentia oppositorum: as a truly infinite being, He includes all opposite and incompatible properties, therefore being all things, and none of them: God has all properties, including contradictory ones (Heron, 1954, I.4).

“The ultimate truth is that there is no ultimate truth” (Siderits 2007, p.182; Priest 2002, p.260). These utterances, if true, would appear to be self-contradictory, and therefore dialetheias.

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A Simple Case Study: the Liar

In its standard version, the Liar paradox arises by reasoning on the following sentence:

(1): (1) is false

where the number to the left is the name of the sentence to the right. As we can see, (1) refers to itself and tells us something about (1) itself. Its truth value? Let us reason by cases. Suppose (1) is true: then what it says is the case, so it is false. Then, suppose (1) is false: this is what it claims to be, so it is true. If we accept the aforementioned Law of Bivalence, that is, the principle according to which all sentences are either true or false, both alternatives lead to a contradiction: (1) is both true and false, that is, a dialetheia, contrary to the LNC.

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Other Motivations for Dialetheism

Dialetheias produced by the paradoxes of self-reference are confined to the abstract realm of notions such as set or to semantic concepts such as truth. However, the paradoxes of self-reference are not the only examples of dialetheias that have been mooted. Other cases involve contradictions affecting concrete objects and the empirical world, and include the following.

(1) Transition states: when I exit the room, I am inside the room at one time, and outside of it at another. Given the continuity of motion, there must be a precise instant in time, call it $t$, at which I leave the room. Am I inside the room or outside at time $t$? Four answers are available: (a) I am inside; (b) I am outside; (c) I am both; and (d) I am neither. There is a strong intuition that (a) and (b) are ruled out by symmetry considerations: choosing either would be completely arbitrary. (This intuition is not at all unique to dialetheists: see the article on boundaries in general.) As for (d): if I am neither inside not outside the room, then I am not inside and not-not inside; therefore, I am either inside and not inside (option (c)), or not inside and not-not inside (which follows from option (d)); in both cases, a dialetheic situation. Or so it has been argued. For a recent description of inconsistent boundaries using formal mereology, see Weber and Cotnoir 2015.

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A more persuasive worry about dialetheism, relating to rationality, is the claim that if a person could legitimately accept a contradiction, then no one could be forced, rationally, to abandon any view held. For if a person accepts $A$ then, when an argument for $\neg A$ is put up, they could simply accept both $A$ and $\neg A$.

A dialetheist can reply that, again, not all contradictions are equal. Each sentence, including each contradiction, is evaluated on its merits. While a case can be made for the claim that the Liar sentence is both true and false, but this in no way shows that a case can also be made for Brisbane being and not being in Australia. (Of course, if one subscribes to the claim that entailment is explosive, a case for one contradiction is a case for all; but if entailment is paraconsistent, this argument is of no use.)