1. The Problem

It may be said that when Pilate asked, “Are you the king of the Jews?” and Jesus replied “you say that I am” (Lucas XXIII, 3), what was at stake was not the truth of the sayings about Jesus but their agreement about these. There is a tradition of logicians that see formal validity as a matter of such agreements. They are the dialogue logicians or dialogicians.

Dialogicians have developed a broad number of investigations along the last decades, and many subjects and problems treated by other logicians have been also explored by them (^{Clerbout and McConaughey 2022}). Still, one such problem that is severely underdeveloped in its dialogical trend is the solution to semantic paradoxes. The purpose of this paper is to fill in this gap by presenting two dialogical truth theories which are non trivial and substructural, in a sense to be specified.

One may ask why it is that dialogicians have payed little attention to semantic paradoxes. A plausible explanation is that the key concept among the latter, “truth”, has an unclear dialogical meaning. As in all game-theoretic traditions, in this trend validity is captured by the existence of winning strategies; but unlike proposals such as Hintikka’s, in Dialogics one does not have a model in sight. Therefore, there is not a straightforward way to relate winning strategies to truth-preservation.

Yet it seems reasonable to expect that words like “truth” were amenable to a dialogical explanation, because we use such words in language games. My bet is that for some of these games it makes sense to think of dialogical truth in the following terms:

Dialogical Truth: to claim that a proposition is true amounts to declare oneself as being able to defend that claim.

The following well-known observation about truth may help to illustrate the adequacy of this definition:¹ in most contexts, when making a claim about anything, it is redundant to add the predicate “... is true” to that claim. And it may be argued that this is because to claim that a proposition is true amounts to claim that proposition. So, from a model-theoretic point of view, the propositions “The sky is blue” and

“The sky is blue” is true

Will come out true in the exact same cases (models). And from a dialogical point of view, someone should be able to defend the claim that

I can defend the claim “The sky is blue”

In the exact same cases in which she can defend the claim “the sky is blue”. So it looks like a plausible suggestion to state that both notions, i.e., model-theoretic truth and dialogical ability-to-defend a claim, are two different explications of the same informal notion (truth).

An understanding of truth along these lines is useful for two reasons. The first is that it is at even with the Principle of Equality,² because it states the conditions of truth in terms of challenges and defenses of claims. The second is that it is amenable to exhibit paradoxical features. Consider the ordinary statement made by Chadwick:³

Say Chadwick is facing a trial for his responsibility in the management of the revolt. The role of the public attorney is to challenge this stance and so Chadwick will have to defend it, and this defense will depend on the sayings of the president. Now suppose that the only thing Piñera had ever said about the revolt is:

If we understand “true” according to the proposed definition, then (1) expresses solely the commitment of Chadwick to defend (2); and (2) expresses the commitment of Piñera to defend that Chadwick cannot defend (2).

As it is easy to see, a defense of (2) amounts to show that (1) can be challenged. Thus a dialogical version of the yes-no paradox obtains: to successfully defend (1), Chadwick needs to successfully defend (2); but, to successfully defend (2), Chadwick must not be able to successfully defend (1)... and so on.

The truth theory that I will introduce in section 3 will capture this sense of dialogical truth, leaving open place for paradoxes like the one from this example. Then, two solutions with classical recapture results will be given. But first, in section 2 I will comment on one antecedent to my proposal and why it falls short as the dialogical truth theory I am looking for. A brief comparison between the two is given at the end of section 3. A final section recapitulates the conclusions.

2. Previous Proposals

Thus far the only dialogical approach to the semantic paradoxes (that I know of) is ^{Dutilh Novaes and French 2018}. There are two features of this proposal that are worthy of attention. First, we will look at the dialogical system itself. The authors say about this system that it

borrows elements from previous dialogical approaches in logic (in particular Lorenzen’s dialogical logic (^{Lorenzen and Lorenz 1978}; ^{Keiff 2009}) and Hintikka’s game-theoretical semantics (^{Hintikka and Sandu 1997})) but it differs significantly from these earlier proposals in many respects. (^{Dutilh Novaes and French 2018, p. 131})

These “significant differences” will constitute severe drawbacks for the kind of truth theory we are looking for and will be our main reason to depart from this framework. Second, we will focus on the solution to the semantic paradoxes they endorse and the general conclusions they draw about the family of the so-called substructural solutions. As we will see at the end of the next section, our proposal will also disagree with their final diagnose on the paradoxes.

The common feature in all game-theoretical approaches to logic is that arguments are depicted as games between two players. Validity (and invalidity) is captured by the notion of winning strategy: the possibility to play in such an intelligent way that victory is attained no matter what the other does. The dialogical interpretation of logic defended by ^{Dutilh Novaes (2013}, ^2015a, ^2015b) considers dialogues between Prover and Skeptic. Prover has to explain to Skeptic that the conclusion follows from the premises, while Skeptic must resist to this conclusion as much as reasonably possible. So their dialogues are cooperative in some respects and adversarial in others.

One salient feature of this proposal is that it pays “no special attention to logical connectives; instead, [it focuses] on very general features of logical reasoning and argumentation” (^{Dutilh Novaes and French 2018, p. 131}). In other words, this is a dialogical interpretation of a proof-theoretical system. A closer look on the details may help to better illustrate this claim.

The logical system used by Dutilh Novaes and French is a “sequent-to-sequent style natural deduction system” ^{Dutilh Novaes and French (2018)} introduced in ^{Sø rensen and Urzyczyn (2006)}, enriched with a paradoxical operator from ^{Read (2000)}. From the point of view of pure logic, the system behaves just like a proof-theoretic device: there are rules of introduction and elimination, derivations take the form of oriented graphs, and so on. Prover and Skeptic are called to justify such elements, as their interaction will provide the required information to produce valid derivations.

The game proceeds as follows:

Our dialogues begin with Prover offering a sequent, and play proceeds with Prover and Skeptic making alternating moves. A dialogue is won by Prover whenever the only sequents which Skeptic has available to challenge are instances of (Id).⁴ To see why this is the case, note that in any situation where Prover can do this, then all the formulas involved must have, at some earlier stage, appeared in a Skeptic offer, and thus have been granted by Skeptic. (^{Dutilh Novaes and French 2018, p. 137})

And the moves of each player are expected to be the following:

Here’s the example they present in pages 137-138 (slight notational changes):

This informal dialogue is formalized as the following derivation:

The authors provide the following explanation as how to relate the dialogue with the derivation:

The sequents offered by Skeptic are ones which entail the challenged sequent because the challenged sequent can be derived from it using only our introduction rule (⊃ I); by contrast, the sequents offered by Prover are ones which entail the challenged sequent either because (a) the sequent offered by Skeptic can be derived from them using only elimination rules (⊃ E) and (⊥ E) and the structural rule (KL), or (b) they can be used to derive a sequent Γ  ≻  ⊥ using only elimination rules or our structural rule for which there is some formula C such that from Γ  ≻  C the challenged sequent can be derived using only introduction rules. (^{Dutilh Novaes and French 2018, p. 137})

Therefore, the natural deduction derivation conveys information about the strategy for Prover. Still, it does not represent an actual play; at most, it is a representation of the wisest possible play for the Prover. To see why, let us consider again the informal presentation of the dialogue. As the Prover claims that A⊃B,  B⊃C  ≻  A⊃C, the Skeptic challenges it with A,  A⊃B,  B⊃C  ≻  C. Why did (s)he decide to do that? The choice is completely mysterious, unless we grant that they have the rules of the calculus in sight. In other words, they are playing to produce a valid proof-theoretic derivation.

I grant that this gives the proposal a wide scope and makes it easy to relate the dialogical framework with the proof-theoretic one, highlighting their similarities and confluences. But it has the severe drawback that their understanding of what is for a particular dialogue to exhibit paradoxical features (as in the Chadwick example) is very imprecise. One does not see what happens when Prover makes a claim such as the liar paradox, or what the difference is between a paradox being claimed by the Prover or by the Skeptic. This makes rather obscure everything that they can say about paradoxes and their proposed solutions, to which we will now turn.

Dutilh Novaes and French are interested in the substructural solutions to the semantic paradoxes. Say that ⊢ is a consequence relation over a certain language, in the sense that Γ⊢A states that the argument with premises Γ and conclusion A is valid. We say that ⊢ is structural if and only if the following conditions hold:⁵

Any logic whose consequence relation fails to meet some of these conditions is called substructural. As the ongoing literature on semantic paradoxes has shown, there are non-trivial truth theories than can be defined in some substructural logics with nice classical recapture results (i.e., everything besides paradoxical sentences behaves as is expected in a classical theory).⁶ But “whatever restrictions on structural rules we may want to enforce, it is highly desirable that such restrictions be accompanied by independent motivation, not directly related to paradoxes” (^{Dutilh Novaes and French 2018, p. 130}). The purpose of their paper is to provide such an independent motivation, based on the dialogical framework described above.

The authors consider four possible structural constraints: non-reflexivity, non-transitivity and two more that are of no interest here. After evaluating the relevant structural rules as games between Prover and Skeptic, they conclude that “nonreflexive solutions are to be preferred over the others” (^{Dutilh Novaes and French 2018, p. 147}). The reason is that they found Reflexivity already suspicious:

[O]nce we move to an explicitly multi-agent context such as the Prover-Skeptic dialogues we described above, Reflexivity acquires a different meaning, as it amounts to Prover putting forward A for Skeptic to grant twice. [...] To “force” Skeptic to grant A once he has granted A is a rather pointless dialogical move, indeed a violation of pragmatic conversational norms against redundant moves [...]. In an adversarial context, such a move would most likely make Prover look silly rather than triumphant; in a cooperative context, such a move would be utterly uninformative and superfluous (again, a breach of the Gricean maxim of quantity). (^{Dutilh Novaes and French 2018, p. 146})

So, once they turn to the consideration of the proof system with the paradoxicality operator, they argue for a constraint in the use of the (Id) rule:

As is clear from the arguments above, we think that what has gone wrong here is the reliance on structural reflexivity. Indeed, the justification given for why Skeptic cannot challenge (Id) earlier-namely that instance of (Id) must involve only formulas which Skeptic has already granted-no longer holds in this setting. This suggests that we should simply be more explicit about which formulas have been granted by the participants beforehand, instead of having this information be carried implicitly by which formulas appear in the antecedent of Skeptic offers. On this revised account, Prover and Skeptic must at least implicitly settle on a collection of sentences which constitute conversational bedrock-sentences for which no further challenge is relevant. On this picture the validity of principles like (Id) is not a matter of logic per-se, but rather one of negotiated agreement. (^{Dutilh Novaes and French 2018, p. 149})

I have two problems with this argument. For a start, as we have seen, the instances of (Id) are the sole condition of victory for the Prover in this setting. As quoted above, “a dialogue is won by Prover whenever the only sequents which Skeptic has available to challenge are instances of (Id)”. If this is the case, and Prover is supposed to find a way to settle these instances, it is a bit odd that later on the authors recognize that this “is a rather pointless dialogical move”, especially if logical validity is supposed to be conveyed by winning strategies for Prover. My second concern is with the final conclusion of the preceding quote: “On this picture the validity of principles like (Id) is not a matter of logic per-se, but rather one of negotiated agreement”. If we must seek a previous negotiated agreement on what instances of (Id) in which we should (or should not) rely on (or, equivalently, which are the propositions we both agree), then the subsequent dialogue (and therefore, its logic) is rather pointless. It seems that the only way to block this odd conclusion would be to expect this agreement to stem from the dialogue itself, i.e., that the interaction between the players settles which propositions are grounded and which are not. This is precisely what our account will accomplish.

3. The Present Proposal

I do not regard any proposal, including the one to be advanced here, as the definitive interpretation of the dialogical use of “true”, or the definitive solution to the semantic paradoxes. On the contrary, I see many open questions regarding the philosophical justification of the proposal, and its metalogic is still severely underdeveloped. I do hope that the dialogical theories given here have two virtues: first, that they provide some good ideas and interesting insights to push forward some further developments; second, that the formal expression of these ideas captures important intuitions at the level of the plays and not only of the strategies.

As Dutilh Novaes and French, I am interested in the substructural solutions to the paradoxes. But unlike them, I shall work in a full-fledged dialogical framework and not a proof-theoretic presentation interpreted in a dialogical way. Substructurality then will emerge as a consequence of reasonable decisions at the player level.

I shall begin by exposing the kind of Dialogics we will be working with. Then I shall turn to the truth theory I propose and the way in which it handles the paradoxes. The framework to be briefly exposed in the first part follows the line of Lorenzen and Lorenz (^{Lorenzen and Lorenz 1978}, ^{Lorenz 2001}, ²⁰¹⁰) as continued by Rahman and his colleagues and students. For a more detailed exposition, see ^{Rahman et al. (2018, ch. 4-5)} and ^{Clerbout and McConaughey (2022)}.

4. Conclusions

There are few works in the literature concerning Dialogics and truth theories. To the best of my knowledge, the only straightforward paper that addresses this issue is ^{Dutilh Novaes and French 2018}. Still, their proposal is not as dialogical as it can be. In this paper I presented a full-fledged dialogic truth theory, with two solutions to the semantic paradoxes: the Strict and the Tolerant games for truth. These solutions came out to be substructural, as the Strict game is non-reflexive and the Tolerant is non-transitive.

It is not the purpose of the present work to make any particular recommendation among substructural solutions to paradoxes. If such a choice is context-dependant, as we suggested in the last paragraph of the preceding section, this at most inclines us towards a certain kind of dialogical pluralism (in the sense of ^{Keiff 2007}). But to defend this point is not the intention of this paper. My purpose is rather to provide a family of flexible instruments which can be explored simultaneously and whose fertility and consonance with intuition can be checked.

There are mathematical applications and purely technical problems which I have not mentioned in this sketch. For instance, there is the question of soundness and completeness between the strict and tolerant games and the logics TS and ST. All this will have to be the aim of forthcoming work. For the time being, the current objectives of this inquiry have been fulfilled. Unlike Dutilh Novaes and French’s, my proposal does tell what is the dialogical meaning of truth, obtains the paradoxes as a side-effect of other well-motivated definitions, and finds a way to block them while having substructurality as a consequence of it. I think that this is the major virtue of this proposal over theirs: that I do not need to “justify” substructurality, I cope with it (in any of its forms) as a not-so-high price to pay in exchange for a theory that is intuitive and non-trivial.

Disclaimer As a literary trait, the title, overall structure and some parts of the text in this paper mimic Saul Kripke’s Outline of a Theory of Truth. The fragments of similar text are non-substantial though. There was no plagiarism intended.