Ondrej Majer

Games: Unifying Logic, Language, and Philosophy

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Towards Evaluation Games for Fuzzy Logics

The article provides two kinds of game-theoretical semantics for fuzzy logics with special attention to Łukasiewicz logic. The first one is a generalization of the evaluation games for classical logic. It is shown that it provides an interesting contribution to the model theory of fuzzy logics as, unlike the standard semantics, it can deal with the so-called non-safe models. The second kind of semantics makes explicit the intuition about fuzzy logics as logics of partial truth and provides a semantics in the form of a bargaining game. Finally, a basic kind of logic of informational independence of a Hintikka-Sandu style is introduced.

Logic of questions and public announcements

This paper aims to explore the role of questions in communication in a group of cooperative rational agents. Using epistemic representation of questions proposed in [6] we employ the framework of public announcement logic to explore the flow of information in the process of asking and replying questions in a group. We show that some of the erotetic notions we introduce nicely correspond to the standard epistemic ones.

Epistemic logics for sceptical agents

In this article, we introduce an epistemic modal operator modelling knowledge over distributive non-associative full Lambek calculus with a negation. Our approach is based on the relational semantics for substructural logics: we interpret the elements of a relational frame as information states consisting of collections of data. The principal epistemic relation between the states is the one of being a reliable source of information, on the basis of which we explicate the notion of knowledge as information confirmed by a reliable source. From this point of view it is natural to define the epistemic operator formally as the backward-looking diamond modality. The framework is a generalization and extension of the system of relevant epistemic logic proposed by Majer and Peliš (2009, college Publications, 123–135) and developed by Bílková et al. (2010, college Publications, 22–38). The system is modular in the sense that the axiomatization of the epistemic operator is sound and complete with respect to a wide class of background logics, which makes the system potentially applicable to a wide class of epistemic contexts. Our system admits a weak form of logical omniscience (the monotonicity rule), but avoids stronger ones (a necessitation rule and a K-axiom) as well as some closure properties discussed in normal epistemic logics (like positive and negative introspection). For these properties we provide characteristic frame conditions, so that they can be present in the system if they are considered to be appropriate for some specific epistemic context. We also prove decidability of the weakest epistemic logic we consider, using a filtration method. Finally, we outline further extensions of our framework to a multiagent

Logic of change, change of logic

Understanding human behaviour, and indeed human beings more generally, requires an understanding of their attitudes: their beliefs, desires and preferences of course, but also a plethora of other attitudes or attitudinal factors, such as their presuppositions, their assumptions, their intentions, their attention, their awareness and their emotional states. But people’s attitudes change over time: to take the least controversial example, their beliefs change on learning new facts. It is not sufficient to understand the role human attitudes play at any particular moment; an understanding of how different attitudes change is also required. Formal models—be they from logic or mathematics—have historically played an important role in the modelling and study of some of these attitudes. Notable examples include the Bayesian model of (partial) beliefs as probability functions (de Finetti 1937; Ramsey 1931), the modal logician’s representation of belief and knowledge in terms of accessibility relations or partitions on sets of possible worlds (Hintikka 1962; Aumann 1976), typical models of preferences as order relations over sets of

On Łukasiewicz’s Theory of Probability

Łukasiewicz’s theory of probability has been neglected. This is perhaps due to the fact that it was written at an early stage in the development of formal logic, and so Łukasiewicz’s intentions are at times obscure. We argue that this neglect is unjust: by restating the theory in modern terms, and paying careful attention to Lukasiewicz’s motivations, the theory is of interest, especially within its historical setting. We point out a novel notion of interpretation which can be used to explicate the theory, and we also note that the theory is a pragmatic one. We conclude by raising three puzzles, and hopefully answering them.

Eliciting Uncertainties: A Two Structure Approach

We recast subjective probabilities by rejecting behaviourist accounts of belief by explicitly distinguishing between judgements of uncertainty and expressions of those judgements. We argue that this entails rejecting that orderings of uncertainty be complete. This in turn leads naturally to several generalizations of the probability calculus. We define probability-like functions over incomplete algebras that reflect a subject’s incomplete judgements of uncertainty. These functions can be further generalized to (partial) inner and outer measures that reflect approximate elicitations.

On Semantic Games for Łukasiewicz Logic

We explore different ways to generalize Hintikka’s classic game theoretic semantics to a many-valued setting, where the unit interval is taken as the set of truth values. In this manner a plethora of characterizations of Łukasiewicz logic arise. Among the described semantic games is Giles’s dialogue and betting game, presented in a manner that makes the relation to Hintikka’s game more transparent. Moreover, we explain a so-called explicit evaluation game and a ‘bargaining game’ variant of it. We also describe a recently introduced backtracking game as well as a game with random choices for Łukasiewicz logic.

Equilibrium Semantics for IF Logic and Many-Valued Connectives

We connect two different forms of game based semantics: Hintikkas game for Independence Friendly logic IF logic and Giless game for Łukasiewicz logic. An interpretation of truth values in [0,i¾?1] as equilibrium values in semantic games of imperfect information emerges for a logic that extends both, Łukasiewicz logic and IF logic. We prove that already on the propositional level all rational truth values can be obtained as equilibrium values.

Introduction to the Special Issue Epistemic Aspects of Many-Valued Logics

The papers in this special issue are based on presentations delivered at the conference Epistemic Aspects of Many-valued Logics, held at the Institute of Philosophy of the Academy of Sciences of the Czech Republic, in Prague, 2010. All papers consequently revolve around the application of non-classical logical tools— mathematical fuzzy logic and/or probability theory—to epistemological issues. Timothy Williamson employs a modal epistemic logic enriched with probabilities to generalize an argument against the KK-principle. He argues that we can know a proposition even if our evidential probability for that proposition is low. In fact he argues that the evidential probability of a known proposition can be (arbitrarily) close to 0. The argument is first presented with a basic idealized model, which is then extended to much more complicated and realistic models. This then raises a problem for decision theory, since you can know that p, while your evidence tells you (strongly) that not p. Williamson argues that this problem should be understood as a failure of luminosity, i.e., that if you are in a state then you are in a position to know that you are in that state. Williamson also argues that a version of deductive closure of knowledge is defensible as long as we drop the KK-principle. Since knowledge states aren’t luminous, standard arguments against closure don’t work. Colin Howson argues the subjective interpretation of the probability calculus should be understood in the framework of many-valued higher-order logics. Howson argues that the standard objections to second-order logic rely on a misguided conception of logic, and argues that we should instead concentrate on the model-theoretic notions of consistency and consequence. He then interprets the finitely additive probability calculus using these notions, linking fair bets with probabilities a la de Finetti. The restriction to finite additivity yields two logical properties: that a consistent probability assignment is extendable to a total assignment, and that an assignment is consistent if and only if all of its restrictions