Wednesday, June 29th

Independence friendly (IF) logic of Jakko Hintikka and Gabriel Sandu provides a logical representation of a game-theoretical phenomenon of imperfect information. This phenomenon is reflected in the syntax by extending the language of predicate logic with co called “slashed” representing independence of a variable in an existentially quantified formula in the scope of a universal quantifier. Semantics for this kind of formulas is then provided by evaluation games of imperfect information. Truth of a formula is, as in the classical case, identified with existence of a winning strategy for the initial player (Eloise, Proponent, …) and its falsity is defined dually. As in some games of imperfect information no player has a (pure) winning strategy, there are some IF formulas which are neither true nor false, hence IF logic can be understood as a three valued logic. Moreover, in the framework of equilibrium semantics allowing players to play mixed strategies we can identify the value of an IF formula with the equilibrium payoff of the initial player which is a rational number between 0 and 1.

The phenomenon of imperfect information can be applied at the propositional level as well. Propositional fragment of IF logic was studied by Sandu and Pietrinen who showed that this fragment is functionally complete with respect to functions from {0,1}n to {0, ½, 1}. We extend Sandu-Pietarinen results for the case of equilibrium semantics: first we show that we can obtain all rationals in [0,1] as values of a propositional IF formula and as a main result we prove that propositional IF logic is functionally complete with respect to all functions from {0,1}n to rationals in [0,1].

Constructive modal logics are obtained by adding to intuitionistic logic a minimal set of axioms for the box and diamond modalities.

In this talk we first define arenas, which are graphs encoding modal formulas, and winning innocent strategies for games played on these arenas.

We then characterize the winning strategies corresponding to proofs in the constructive modal logic CK and we show the correspondence between these strategies and sequent calculus derivations.
We conclude by proving that our strategies compose and provide a denotational semantics for this logic.

Systems for first-order classical dialogical logic are known since Lorenzen’s seminal paper Logik und Agon. However, completeness for such system is considered folklore, and no satisfactory proof for first-order classical dialogical logic can be found in the literature.
In this talk, we present a completeness proof for first-order classical dialogical logic.
Such a proof is obtained by inductively defining a mapping between formal E-strategies and derivations in a cut-free complete sequent calculus.
For this purpose, we use a polarized sequent calculus obtained by extending to the first-order the sequent system LKQ introduced by Herbelin in his PhD dissertation.

DNA-logics (DNA stands for Double Negation Atoms) have been recently introduced as generalizations of inquisitive logic. DNA-logics are negative variants of intermediate logics. In this talk, I will discuss a systematic method of designing new types of logics that can be seen as “close relatives” of DNA-logics since they share main semantic features with them. Many questions about the theory of these logics are open including their game semantics.

Game semantics provides an alternative view on basic logical concepts. Provability games, i.e., games characterising the validity of a formula provide a link between proof systems and semantics. We present a new type of provability game, namely the Mezhirov game, for various propositional and modal logic and give a proof sketch of the adequacy result. The games are finite resulting in a finite search for winning strategies.

Wilfried Hodges’s criticism of dialogical logic and even more generally of  the contribution of the games perspective to logic has been influential and echoes of it are ubiquitous in the literature in the field. His perspective is grounded in the strategical level, the level that provides the dialogical stance on logical validity and wonders what is the “story” that might motivate a game theoretical approach. The conclusion of Hodges is that dialogical logic and other game theoretical approaches might have some psychological – if any – rather than a philosophical or formal result.

Some defenders of the dialogical approach bit the bullet and expanded on the psychological features of the dialogical stance. This lead for example to a dialogical interpretation of the structural rules, which yields a table of rankings from most plausibly revisable to least plausibly revisable.

Hodges’s criticism misses the very point of the dialogical approach, namely the play level: the level where meaning is constituted by dialogical meaning explanations. In fact, Paul Lorenzen and Kuno Lorenz, the inceptors of the dialogical approach conceived structural rules as emerging from rules governing the interaction of challenge and defences that constitute a play, from the thesis up to the elementary statements that result from such an interaction – not the other way around. This bottom up procedure, which is a crucial trait of the dialogical framework, and which distinguishes it from the top-down (axiomatic) construction of Gentzen-style inferences, has been overlooked by many of the criticisms and particularly so in relation to the understanding of the dialectical stand on structural rules. An alternative response to Hodge’s criticism that seems to be compatible with a play-level perspective is one where structural rules are understood as shaping different form of a human-server interaction.

In the present paper I will recall the “old-ways” to structural rules and explore new further developments that integrated the expressivity of fully interpreted languages in the style of Per Martin-Löf’s Constructive Type Theory within the bottom up play level perspective for material and formal dialogues that we call Immanent Reasoning. If time I will also present some critical remarks on Martin-Löf’s recent take on dialogical logic.

Thursday, June 30th

Semantic Games, in the spirit of Hintikka, offer a fruitful dynamic approach to the concept of truth in a model. In contrast, provability games like Lorenzen’s dialogue game, characterize logical validity. In this talk, I will present a technique of lifting semantic games to provability games in a natural way, via disjunctive games. Intuitively, this game corresponds to playing all semantic game for a fixed formula simultaneously over all models. Logical validity is thus characterized by winning strategies of the disjunctive game, which in turn can be rewritten as proofs in an analytic calculus. I will illustrate this technique for various logics, discuss some potential future work and briefly touch on the direction back.

In this talk, I offer a précis of my book The Dialogical Roots of Deduction (CUP, 2020). The book offers an account of the concept and practices of deduction by bringing together perspectives from philosophy, history, psychology and cognitive science, and mathematical practice. I drawn on all of these perspectives to argue for an overarching conceptualization of deduction as a dialogical practice: deduction has dialogical roots, and these dialogical roots are still largely present both in theories and in practices of deduction. The account also highlights the deeply human and in fact social nature of deduction, as embedded in actual human practices.

In this talk we present an overview of different types of bisimulation games and its applications. We will focus on connections between bisimilar equivalence and modal equivalence for standard and infinitary modal languages.

One of classic models of argumentation is based on attack graph. Its idea was introduced by S. Toulmin in “The Uses of Argument” in 1958. Nodes are arguments and a directed edge between arguments presents iff one argument weakens another. There is a modal logic description for agents games on such graphs, see a course “Abstract Argumentation and Modal Logic” on ESSLLI 2021 by Davide Grossi and Carlo Proietti. We introduce modification which is based on an idea that these arguments and their relations came from sources and these sources are not equally trustable by agents. It directly gets us to a consideration of several subjective attack graphs and epistemic structure, especially if we assume that agents might have only partial information about relation between arguments.  In our presentation we are going to show how this assumptions changes properties of games and what are winning strategies then.