To the editors:

In his essay, Graham Priest connects, in a quite clear and elegant way, some of the themes dearest to logicians and logically inclined philosophers. Self-reference is a major theme of the essay. More specifically, Priest discusses the difficulties that arise when it comes to results that use self-reference in their derivation. The challenge is to determine which of such results are respectable theorems and which are merely dangerous paradoxes. Priest gives a list of results using self-reference that includes Kurt Gödel’s incompleteness theorems, Martin Löb’s theorem, the liar paradox, and Haskell Curry’s paradox.

If that challenge is the main theme of Priest’s essay, it has its traditional answer, which goes as follows: Gödel’s and Löb’s theorems, as their names already evidence, count as important results of classical mathematics, while the liar and Curry’s paradoxes are the derivations to be avoided. Behind such a classical divide are both Alfred Tarski’s theory of truth, restricting the T-scheme and self-reference—ingredients to derive the liar paradox and Curry’s paradox in classical logic—and the separation of truth from proof, which results from Gödel’s incompleteness theorems. The latter move also accounts for the use of self-reference in the notion of proof, which allows for Löb’s theorem.

The upshot of the separation of truth and proof is that, while every proof is a proof of some truth, some arithmetic truths do not have a proof. This is the classical dividing line separating theorems from paradoxes—at least the ones highlighted here. Motivating such a divide is the need to preserve both the consistency of truth and the formalization of arithmetic. Consistency, then, is obtained at the price of an incomplete image of both truth and arithmetic.

Despite the lasting influence of such an approach, it is not without problems. It is said, for example, to preserve consistency by ad hoc moves. Priest attempts to draw a different dividing line, and with that, he obtains a new story that is in some respects better than the old one. Priest believes that by avoiding the need for Tarskian restrictions on truth and by allowing the use of semantically closed languages in the realm of logic, one can obtain a better theory of truth, which delivers the liar paradox as a natural consequence. The resulting view is called dialetheism.¹ The liar paradox would then become a respectable result according to this alternative theory. To prevent the spread of contradictions, Priest recommends using a paraconsistent logic, one that allows for true contradictions, as delivered by the liar paradox, but does not lead into triviality—i.e., from a contradiction, not every formula follows. But there is more to it.

Dialetheism has another advantage. Consider once again Gödel’s incompleteness theorem, which states that certain formalizations of arithmetic are either inconsistent or incomplete. In a scenario where some contradictions are already derivable, such as the liar paradox, one need not fear inconsistency. The theorem can be taken to show that arithmetic is inconsistent but complete. As Priest insists, every truth of the standard model of arithmetic is then provable, but again, so are some contradictions. As a preliminary result, the liar paradox becomes a derivable result and Gödel’s theorem acquires a new significance.

But due to the loss of modus ponens in the dialetheist setting advanced by Priest, we also lose Löb’s theorem. Wait, no modus ponens? Indeed, that rule of inference will have to go. The rule states that “if A, then B” and “A, B follows.” It may be written in terms of disjunction, as Priest did, granting that from ¬ A ∨ B and A, we derive B. But from an arithmetical point of view, losing modus ponens is not really a problem. Again, as Priest mentions in his essay, it can be shown that there is a version of arithmetic—an inconsistent arithmetic, to be sure—where modus ponens fails but every truth of arithmetic is still provable. Modus ponens is not, after all, necessary for arithmetic. One can do without it. The removal of modus ponens has another major benefit: in the full account of truth and self-reference for which Priest advocates, if it were not for the failure of modus ponens, Curry’s paradox would obtain and we would be led directly to triviality.² But if there is no modus ponens, there is no Curry paradox, and triviality along these lines is avoided.

The new story behind the drawing of the line between paradoxes and theorems makes self-reference fully available, and consequently requires that some contradictions be declared true. As Priest’s essay shows, this is a plea for a dialetheistic and paraconsistent approach to the liar paradox and arithmetic, allowing for controlled inconsistencies. By conceding some contradictions, we gain the completeness of arithmetic. This completeness is obtained at the cost of modus ponens, and the reason, again, is that although in this picture some contradictions can be tolerated, triviality cannot. That is, Curry’s paradox must be blocked.

Some adherents of the classical picture might point out that such a change in view entails further consequences: once one admits that arithmetic is inconsistent, even if non-trivial, our understanding of not only truth, but also proofs, changes. Now one admits that some contradictions may be derived, and, as a result, that there may be a proof of a falsity. The consequences of a contradiction in a theory is an issue taken up by none other than Tarski in a classic paper discussing his theory of truth.³ According to Tarski, the derivation of a contradiction is a sign that the theory in which the inconsistency is derived should be abandoned. He speaks about empirical theories, but the point could be easily carried over to mathematical theories. The problem is not, as paraconsistent theorists claim, that such theories could lead to triviality in a classical setting, but rather that “[w]e know (if only intuitively) that an inconsistent theory must contain false sentences; and we are not inclined to regard as acceptable any theory which has been shown to contain such sentences.”⁴

Proofs acquire some distinctive features under both scenarios. Mathematicians typically determine that their results are true by proving them. But the classical story requires that some truths have no proofs, in order to preserve consistency. As a consequence, the notion of proof has to fall short of delivering all the truths of some interesting theories. According to the dialetheist view advanced by Priest, proofs have their scope restored, given that completeness is granted. Since proofs may now lead one to derive falsities too, this is not an ideal situation in epistemic terms.

What should be done with such distinct accounts of truth, proof, and arithmetic? One would think that both accounts cannot be right, and perhaps that is what Priest has in mind when he compares the classical approach with the dialetheist one. One way to go about choosing between them would be to use intuition. Self-reference and a full theory of truth seem to recommend the dialetheist approach, given their prima facie presence in natural language. But intuition pulls in the other direction too: modus ponens, after all, seems to be the most intuitive inference rule, and consistency is still highly praised by those not initiated in dialetheism. Which of these intuitions is right? There is no easy answer.

I will not argue for the correctness of any of the possible ways of distinguishing between paradoxes and theorems. Given the classical attachment to consistency as the highest priority, it is only to be expected that some restrictions will have to be adopted to preserve consistency. Something similar can be said about the dialetheist approach: given that triviality cannot be accepted, and that modus ponens is the key ingredient to lead to triviality in this setting, then modus ponens has to go. What is fundamentally misguided is the idea that one of the stories is correct and the other wrong. Given that some very fundamental notions are under dispute—truth, proof, and negation—we must evaluate each perspective based on its own commitments. Otherwise, there is always the risk of begging the question against the opposing view. There is simply no ideal perspective.

Once one of the perspectives is adopted, we enter a situation similar to the one described by Wilfrid Hodges, when he said of Alan Turing’s work in computability theory, “Work like Turing’s has a power of creating intuitions [emphasis original]. As soon as we read it, we lose our previous innocence.”⁵ Various logical approaches, I would add, have the power to create intuitions; once one starts using them to explain certain facts, one can no longer return to a state of logical innocence. There is simply no logic-free ground from which to judge the different proposals. The best one can do is to judge them by their own standards and decide whether they are useful for some intended application.

It is natural to see how intuitions can pull to different sides once one starts to examine the issue from one or another perspective. There cannot be contradictions in classical approaches, and so the results obtained by Gödel and Tarski are accommodated as implying the relation of truth and proof that Priest has so clearly discussed. But in a dialetheistic framework, other possibilities become available and consistency is no longer a must. It becomes simple to explain, for instance, why modus ponens should go, as Priest did. At the end of the day, a dialetheistic approach is just a different way of doing arithmetic, like it or not.⁶ With the birth of alternative logics in the twentieth century, deductive theories can be investigated along different lines, following different logical rules and backgrounds. This is no novelty; it already happened before, with intuitionistic mathematics in particular and with constructive mathematics in general.⁷ It would be pointless to dispute which of them is right; they are simply different approaches in their own right.

A lot must be done if we are to justify this picture of the situation. Various approaches to mathematical theories are available, but only a few of them are actually pursued. They must first pass through an institutional filter that weighs the promise of a research program to be fruitful and worth pursuing. Classical mathematics passed through the filter and has been extremely successful. It is incumbent on the alternative approaches to prove their value and then conquer terrain. This is far from easy, and sometimes it just does not happen.⁸

And what about the issue of dividing paradoxes from important results? The idea that there is a correct division seems to be a mirage. What constitutes a paradox or some innocuous result depends on the viewer’s initial framework. In classical mathematics, it is accepted that not every truth can be proven. This is not seen as a shortcoming, but as a fact. Something similar may be said of a dialetheist picture, in which the liar paradox is embraced as a good result, and arithmetic is inconsistent without modus ponens. That is the nature of the beast and there is no way to change it, unless one adopts a different logical background. The important question then becomes, will such inconsistent arithmetic be part of the practice of mathematics? Just as with classical mathematics, the answer will depend on the fruits it can provide and on finding useful applications.⁹