In this programmatic talk, I will talk about three major examples of paraconsistency and paradoxes in games. Paraconsistent logic is a family of non-classical logics in which contradictions do not entail everything. In this talk, first, I will discuss game theoretical semantics for paraconsistent logic and observe how semantic games change in different, especially in paraconsistent logics. Then, I will consider a self-referential paradox of epistemic game theory, called Brandenburger-Keisler Paradox, and present a model for it. Following, I will shift my attention non-self-referential paradoxes and suggest a non-self-referential paradox in games. These three major cases, I will argue, will be a call for the necessity of the use of non-classical logics in game theoretical reasoning.

In a forthcoming paper, Walter Carnielli and Abilio Rodriguez propose a Basic Logic of Evidence (BLE) whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodriguez understanding of evidence is informal. We provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which box-X can be read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit evidence interpretations in a formal sense.

In this talk I will critically discuss some widely shared methodological assumptions about paraconsistent logic. I will argue that there exist several reasons not to consider classical logic as the reference logic for developing systems of paraconsistent logic and will suggest to weaken a certain maximality condition. Moreover, I will argue that the guiding motivation for the development of paraconsistent logics should be neither epistemological nor ontological, but informational, and I will, from this perspective, discuss the idea of “ex contradictione nihil sequitur”.

Solutions to the paradoxes of semantic self-reference which endorse the unrestricted T-schema (and cognate principles) normally assume that the conditional involved in the schema is a detachable one (i.e., one that satisfies modus ponens). In a paraconsistent setting, there is another possibility: to take it to be the material conditional. In this talk, I will discuss some of the ramifications of this possibility.