Partiality and games: propositional logic

2009

Bulletin of Symbolic Logic, 1995

The meeting was attended by 150 registered participants from 21 countries. There were 13 one-hour invited lectures: H. Andreka, Recent trends in algebraic logic, language, and information. M. Creswell, Restricted quantification. K. Fine, Progressive reasoning. M. Foreman, An K,-dense ideal on w 2 , small ultrapowers and chromatic numbers. D. Gabbay, Fibred semantics for LDS. M. Gitik, The singular cardinal problem. R. Goldblatt, Varieties of Boolean algebras with operators. Yu. Gurevich, Finite model theory. P. Komjath, Paradoxical decomposition of Euclidean spaces. H. Kotlarski, Automorphisms of countable, recursively saturated models for Peano arithmetic. L. Levin, Randomness and nondeterminism in computing. R. Maddux, On the derivation of identities involving projection functions and direct products. I. Ruzsa, Intensional logic admitting semantic value gaps. In addition 90 contributed papers were accepted in five parallel sections, of which 74 were actually presented at the meeting. Unfortunately, the Organizers had a limited budget, so grants were offered only in a few cases. More available grants would have made possible the attendance of several colleagues with limited financial possibilities. Laszlo Csirmaz, secretary to the meeting Invited addresses H. ANDREKA, Recent trends in algebraic logic, language and information.

Lecture Notes in Computer Science, 2011

SCOPE OF THE SERIES Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science, computer science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject. Kluwer, through its Applied Logic Series, seeks to provide a home for outstanding books and research monographs in applied logic, and in doing so demonstrates the underlying unity and applicability of logic.

Acta Analytica, 2003

Wittgenstein's language games can be put into a wider service by virtue of elements they share with some contemporary opinions concerning logic and the semantics of computation. I will give two examples: manifestations of language games and their possible variations in ...

Journal of Logic, Language and Information

This volume contains extended versions of six papers presented at the Seventh International Conference on Logic, Rationality and Interaction (LORI-VII), which took place at SouthWest University in Chongqing (China) from October 18 to October 21, 2019 (for the complete conference proceedings, see Blackburn et al. 2019). The aim of the LORI conference series is to strengthen the interface between logic, epistemology, game theory, and social theory using perspectives drawn from philosophy, computer science, and artificial intelligence. Its topics of interest include (but are not limited to) epistemic logic, logic of/for games, computational social choice, logic of preference, deontic logic, logic of agency, conditional logic, logic and natural language, and argumentation theory. The papers included in this special issue all put special emphasis on "logic and interaction" and together offer a broad overview of ongoing research on this topic in both philosophy and in computer science. The papers address a number of themes, including the dynamics of distributed knowledge in multi-agent systems, strategic reasoning of rational agents in concurrent games, the formation of friendship and enmity relations in social networks, the connection between logic and supervised learning, the logical theory of causality, and the logic of knowledge and belief. To give a little more detail, the six selected papers are as follows: Logics with Group Announcements and Distributed Knowledge: Completeness and Expressive Power by Thomas Ågotnes, Natasha Alechina, and Rustam Galimullin. This paper extends group announcement logic (GAL) by allowing quantification over announcements made by agents. Intuitively, such announcements (which may be joint) are related to the notion of distributed knowledge. Surprisingly, however, the paper shows that there are no interaction properties between GAL operators and distributed knowledge. The paper also investigates a number of other variants of GAL.

12th International Conference 'Logic Today: Developments and Perspectives' 22-24.06.2016 St Petersburg Russia

International Mathematical Series, 2006

In recent years, a number of 'dynamic epistemic logics' have been developed for dealing with information, communication, and interaction. This paper is a survey of conceptual issues and open mathematical problems emanating from this development. 1 Logical Dynamics The traditional paradigm of logic is drawing a conclusion from some given premises. But derivation from data already at our disposal is just one way in which information can be obtained. We can also observe new facts, or just ask some better-informed person whom we trust. Concomitantly with all this information flow, our knowledge and beliefs change, and this adaptation process may even be triggered by further cues. Such cognitive actions are of logical interest per se, and their explicit study and its various repercussions has been described as a 'Dynamic Turn' in logic (van Benthem 1996). In particular, relevant actions in this broader setting need not be single-agent tasks such as drawing a conclusion or observing some fact. After all, perhaps the simplest logical scenario for getting or giving information is asking a question. But this essentially involves information flow between two agents, and their mutual epistemic and 'social' interactions as the question is asked and an answer is given. An excellent framework for multi-agent dynamic behaviour in communication is epistemic logic (introduced in Section 2), suitably 'dynamified' by using ideas from the dynamic logic of actions. Section 3 is about the best-explored system of this kind, viz. the dynamic logic of public announcements. Section 4 generalizes this to general dynamic-epistemic logic of actions or events whose observation conveys information. The resulting technical questions here blend into issues about more classical logical systems, which are discussed in Section 5 on first-order and fixed-point logics. But knowledge and ignorance are not the only attitudes of participants in a conversation. They also have beliefs about their current situation and expectations about the future. These are revised as observation and communication take place. Thus, epistemic dynamics runs into belief revision. Section 6 is devoted to links between dynamicepistemic logic and belief revision theory as developed in AI and related areas. Definition 1 Standard language. The standard syntax of epistemic logic has a propositional base with modal operators K i φ ('i knows that φ'), C G φ ('φ is common knowledge in group G'): p | ¬φ | φ∨ψ | K i φ | C G φ We write <i>φ for the dual modal existential statement ¬K i ¬φ: which says that agent i considers φ possible. The dual of C G φ is written < C G >φ. ♣ Example 1 Questions and answers. Let Q ask a factual question "P?", where A answers truly: "Yes". A presupposition for giving a normal truthful answer is that A knows that P: K A P. The question itself, if it is a normal cooperative one, also conveys its own presuppositions, such as (i) ¬K Q P ∧ ¬K Q ¬P ('Q does not know if P') and (ii) <Q>(K A P ∨ K A ¬P) ('Q thinks it possible that A knows the answer'). After the whole two-step communication episode, P has become common knowledge among Q, A: C (Q, A} P. Note the crucial role of epistemic iterations: knowledge that agents have about each others knowledge or ignorance, and also the 'group knowledge' achieved at the end. ♣ Another group notion is 'distributed knowledge' D G φ, which holds intuitively when agents in G put their information together. More generally, epistemic logics can be extended by strengthening their operators in many ways, just as in modal logic in general (cf. Blackburn, van Benthem & Wolter, eds., to appear). 2. 2 Semantics Definition 2 Models and truth definition. Models M for the language are triples (W, {~i | i∈G}, V), where W is a set of worlds, the ~i are binary accessibility relations between worlds, and V is a propositional valuation. The major epistemic truth conditions are as follows: M, s |= K i φ iff for all t with s ~i t: M, t |= φ M, s |= C G φ iff for all t that are reachable from s by some finite sequence of ~i steps (i∈G): M, t |= φ Example 2

Annals of Pure and Applied Logic, 1992

Blass, A., A game semantics for linear logic, Annals of Pure and Applied Logic 56 (1992) 183-220. We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard's linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier 'almost' will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective @ is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godel's Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989) fits with game semantics.

2001

VOLUME 48 The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

Philosophical Issues