In our introduction we make some remarks on the main topics of this issue: assertion and proof. We briefly describe how each of the papers in the present publication has contributed from either different or complementary perspectives to the logical reflection on assertion and proof, while also specifying the relation between them.

In this paper we first develop a Dialetheic Logic with Exclusive Assumptions and Conclusions, DLEAC. We adopt the semantics of the logic of paradox (LP) extended with a notion of model suitable for DLEAC, and we modify its proof theory by refining the notions of assumption and conclusion, which are understood as speech acts. We introduce a new paradox – the rejectability paradox – first informally, then formally. We then provide its derivation in an extension of DLEAC contanining the rejectability (...) predicate. (shrink)

Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.

Douglas Walton’s dialectical theory of argumentation, developed in a swath of papers and several monographs, most central of which are The New Dialectic and, co-authored with Eric Krabbe, Commitment in Dialogue, is one of the most thoroughly developed, detailed and fruitful theories of argumentation available. But Walton’s dialectical theory of argumentation is valuable not only as a comprehensive framework. It also contains important insights fit to illuminate and answer questions that arise outside of his framework and that he may or (...) may not have intended to address. Here, I will sketch how Walton’s concepts of dialogue types and dialogue shifts, which allowed him to build a highly original fallacy theory, can be used for constructing a method of argument-evaluation from a moral perspective. Walton has the goal of making it possible to evaluate argumentation without needing to attempt a determination of the arguer’s intentions, which he thinks would require too much psychological speculation. It may seem that the moral evaluation of argumentative behavior requires exactly this. However, I argue that Walton’s method of grounding argument evaluation in the goals of argumentative dialogues can be adapted by identifying moral goals of argumentation associated with harm avoidance and respect for dignity. This makes it possible to determine whether argumentative behavior is desirable or undesirable from a moral point of view without needing to inquire into an arguer’s intention. Crossing that line is only necessary if we want to determine whether moral blame should be put on the arguer. -/- . (shrink)

This brief note corrects an error in one of the reduction steps in my paper 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks' published in the Journal of Applied Logics 8/2 (2021): 531-556.

Bilateralists hold that the meanings of the connectives are determined by rules of inference for their use in deductive reasoning with asserted and denied formulas. This paper presents two bilateral connectives comparable to Prior's tonk, for which, unlike for tonk, there are reduction steps for the removal of maximal formulas arising from introducing and eliminating formulas with those connectives as main operators. Adding either of them to bilateral classical logic results in an incoherent system. One way around this problem is (...) to count formulas as maximal that are the conclusion of reductio and major premise of an elimination rule and to require their removability from deductions. The main part of the paper consists in a proof of a normalisation theorem for bilateral logic. The closing sections address philosophical concerns whether the proof provides a satisfactory solution to the problem at hand and confronts bilateralists with the dilemma that a bilateral notion of stability sits uneasily with the core bilateral thesis. (shrink)