The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction

Advocators of “anti-exceptionalism about logic” —for example, Williamson ([37, 38]) and Hjortland ([23])— can be regarded as methodologists, and their views surely would have implications for the debate here discussed. Note also that Priest appears as a detractor and a demonstrator, a contradiction that, for him, would show that the debate would be better approached through the methodological principles of rational choice. This would be more extensively discussed in Sects. 6 and 7 below.

The only caveat was that Aristotle aimed at establishing a version of PNC stronger than his refutations of detractors allowed to conclude but, as we have mentioned, this is not a problem for Calm supporters in general.

An ancestor of this is found in Plato, Republic [24] 436b: “It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time.”

Also found in Plato. See Republic [24] 602e8-9: “And haven’t we said that it is impossible for the same [person] to think/imagine at once/simultaneously opposite [things] about the same [things]?”

We do not know a simple expression to name this property, not even in our mother-tongue, and certainly we do not dare piss the owners of English off by inventing one, hence the blank space.

And it must not be confused with Non-referential gap: There is no truth value describable as neither-true-nor-false. These are independent principles. Bivalence might not hold because, say, there is a third value, but it is not “neither-true-nor-false” but rather, say, “true-and-false”.

A key ingredient in classical logic is Functionality: (a) Every proposition has one value (b) but only one. In a language with the usual ingredients (conjunction, disjunction, etc. as certain functions) this, together with Bivalence, Falsity of at least one contradictory, No values-gluts and either version of (Interpretational) Non-contradiction, suffices to give all the usual valuations of classical logic.

For more details on this principle and its importance on Aristotle’s overall argument against Detractors of PNC in Metaphysics \(\varGamma \), see [35], where the principle is taken as saying that the meaning of a term is unique, definite and determinate. Nonetheless, we take the mutual exclusiveness of certain semantic properties as a more appropriate reconstruction of Aristotle’s claims on the uniqueness, definiteness and determinateness of meanings and their role in his refutations.

For a guided tour on even more versions of PNC, see [16].

Whereas it can be debated whether Aristotle’s notion of deduction allows reflexivity (A is deducible from A) because of the “something different from [the things] supposed results” clause, demonstration certainly does not allow it: A would have to be both more certain (for it is a premise in the demonstration) and less certain (for it is demonstrated) than itself.

This was a permanent stance:

No truth does have, and no truth could have, a true negation. Nothing is, and nothing could be, literally both true and false. This we know for certain, and a priori and without any exception for especially perplexing subject matters. (...) That may seem dogmatic. And it is: I am affirming the very thesis that [the foes of the PNC] have called into question and —contrary to the rules of debate— I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate. [21, 101]

And in a letter interceding for the publication of Priest’s In Contradiction:

Many people will think that it is an easy thing to refute Priest’s position, decisively and in accordance with customary rules of debate. It is not an easy thing. I myself think that it is an impossible thing: so much is called into question that debate will bog down into question-begging and deadlock. (Quoted in [27, xix].)

In what follows we will use interchangeably ‘x is debated’, ‘x is disputed’, ‘the possibility that x does not hold is entertained’ and even ‘x is defended’ in this reconstruction.

Field [14] and Boghossian [6] call a principle “default reasonable” if it is regarded as being good at leading to true beliefs and avoiding error (hence reasonable) and it is done so without first adducing evidence or argument in its favor (hence default). We do not think PNC would be default reasonable in this sense for Lewis: It is not merely good at leading to true beliefs and avoiding errors, but is the sine qua non to draw a distinction between true beliefs and errors at all; also, it is not merely default, but already the distinction between what counts as evidence and an argument in favor of something presupposes it.

See [9, Ch. 5] for further details.

Thus, pace Priest [26], we take sides with those who say that some of Aristotle’s refutations establish not necessarily PNC itself or at least its Universality, but something else about it, even if that was not Aristotle’s aim. See [34] for further discussion.

Priest [26, 14f] does not regard as refutations some arguments where apparently there is no other agent saying anything on the basis that Aristotle required Detractors to say something. This internalization would help to explain the fact that there could be a refutation even if there were nobody saying anything: Detractors’ saying has been internalized in the supporter’s argument.

We would like to emphasize that the following reconstruction of Aristotle’s refutation strategy is not an attempt to make an exegetically perfect reconstruction —considering, for example, the fact that it is described in terms of a contemporary semantics. The main purpose of this reconstruction is to show in a comprehensive manner how the strategy works. We leave to the Aristotelian scholars to decide whether it matches perfectly the original proposal.

Whether these views should actually be attributed to the historical Heraclitus is a moot point; see [15, Ch. 5]. However, the interest here lies more in the position itself than in the correctness of Aristotle’s scholarship.

The problem is that Priest sometimes conflates trivialism —“Everything is true”— with a version of Heracliteanism —’All contradictions are true’— (cf. [28, 131], although sometimes he says that he is aware that the identification depends on certain assumptions, notoriously ‘and’-elimination (see [26, 56]. Aristotle too thought that semantic Heracliteanism could be equated with trivialism, but he was cautious: “The doctrine of Heraclitus, that all things are and are not, seems to make everything true (...)” (Metaphysics1012a25).

In proof-theoretic terms, Heracliteans would not accept conjunction elimination when the premise is a contradiction, and they would accept conjunction introduction when the conjuncts are not true but only if they are contradictories. In model-theoretic terms, it may be that \(v(A) \ne \top \) or \(v(B) \ne \top \) even if \(v(A\wedge B) = \top \). So, when the conjuncts are contradictories, semantic Heracliteanism’s conjunction resembles the relevance logics’ fusion connective, and thus is not so odd by contemporary lights. Besides, in certain sense, semantic Heracliteanism is dual to non-adjunctivism, which is a thesis found in some paraconsistent logics like the earliest one, Jaśkowski’s: In non-adjunctivism, in general \(v(A\wedge \lnot A) \ne \top \) even if \(v(A) = v(\lnot A) = \top \). Non-adjunctivism was used in some of the earliest attempts to make sense of impossible worlds; see [29].

Note that according to the Aristotelian argument, the option of describing a language in which all contradictions are true but none of their components is true is open to Heracliteans, provided that in its metalanguage either not all contradictions are true or some of the components of a contradiction are true separately from it.

That PNC might hold only for restricted versions has been pointed out several times by scholars. Substances and “essential predicates” are the usual candidates for which oPNC certainly holds (cf. [11, 17, 19]), although sometimes a slippery slope from the restricted oPNC about essential predicates to a general oPNC about any predicates is attempted; see for example [33]. Tahko [32] defends that oPNC holds certainly for concreta and its properties in “genuinely possible worlds with macroscopic objects”, but not necessarily for every object of every possible world, not even for every object of every physically possible world. More recently, Coren [10] has recently provided an interpretation like ours, but attempting to show that Aristotle can succeed in defending a principle much stronger than MSPNC.

See also [30, §19]; [31, Ch. VIII].

In [27, 241ff], Priest explicitly says that some arithmetical statements (in an inconsistent arithmetic) are both provable and not provable.

Lewis mentions this:

It is not an easy thing [to refute Priest’s position, decisively and in accordance with customary rules of debate]. I myself think that it is an impossible thing: so much is called into question that debate will bog down into question-begging and deadlock. (On this point, Priest disagrees with me: he thinks that shared principles of methodology might provide enough common ground.) (Quoted in [27, xix].)

See also [26, 110].

This reveals that Priest underestimated Aristotle’s remark in Posterior Analytics mentioned in Sect. 4. Aristotle says that whenever one wants to demonstrate something but not also something excluded by it, one has to presuppose PNC. If it is presupposed in no demonstration, one always could demonstrate something but also something else excluded by what was supposedly demonstrated, hence all exclusions could be demonstrated. Given MSPNC, the fact that contradictory propositions can be accepted is irrelevant as to the Indemonstrability of (MS)PNC, although perhaps it counts against some form of Universality for especially stronger versions of it.

To be fair, dialetheists and other detractors are right in questioning stronger forms of PNC, like “any sentence of the form \(A\wedge \lnot A\) is untrue in any valuation”, and their assimilation to weaker forms like MSPNC.

Cf. the reconstruction of the debates in [8, 170–173].