Abstract. The Hintikka-style modal logic approach to knowledge has a well-known defect of logical omniscience, i.e., an unrealistic feature that an agent knows all logical consequences of her assumptions. In this paper we suggest the following Logical Omniscience Test (LOT): an epistemic system E is not logically omniscient if for any valid in E knowledge assertion A of type ‘F is known ’ there is a proof of F in E, the complexity of which is bounded by some polynomial in the length of A. We show that the usual epistemic modal logics are logically omniscient (modulo some common complexity assumptions). We also apply LOT to Justification Logic, which along with the usual knowledge operator Ki(F) (‘agent i knows F ’) contain evidence assertions t:F (‘t is a justification for F ’). In Justification Logic, the evidence part is an appropriate extension of the Logic of Proofs LP, which guarantees that the collection of evidence terms t is rich enough to match modal logic. We show that justification logic systems are logically omniscient w.r.t. the usual knowledge and are not logically omniscient w.r.t. the evidence-based knowledge. 1

This paper presents a game semantics for LP, Artemov’s Logic of Proofs. The language of LP extends that of propositional logic by adding formula-labeling terms, permitting us to take a term t and an LP formula A and form the new formula t:A. We define a game semantics for this logic that interprets terms as winning strategies on the formulas they label, so t:A may be read as “t is a winning strategy on A. ” LP may thus be seen as a logic containing in-language descriptions of winning strategies on its own formulas. We apply our semantics to show how winnable instances of certain extensive games with perfect information may be embedded into LP. This allows us to use LP to derive a winning strategy on the embedding, from which we can extract a winning strategy on the original, non-embedded game. As a concrete illustration of this method, we compute a winning strategy for a winnable instance of the well-known game Nim. 1

Justification Logic is the study of a family of logics used to reason about justified true belief. Dynamic Epistemic Logic is the study of a family of logics obtained by adding various kinds of communication to the language of multi-modal logic, yielding languages for reasoning about communication and true belief. This paper is a first-step in merging these two areas, in that it brings the most basic kind of communication studied in Dynamic Epistemic Logic—the public announcement—over to Justification Logic. This gives us a language for reasoning about public announcements and justified true belief. After giving an overview of Justification Logic, the paper introduces a notion of bisimulation for Justification Logic. Bisimulation allows us to study the affect on language expressivity when we add various kinds of communication to the language. Among a number of expressivity results, we show that adding public announcements to the language of Justification Logic strictly increases language expressivity. This stands in contrast to the Plaza-Gerbrandy Theorem, which states that adding public announcements to multi-modal logic does not increase language expressivity. This leads us to extend the language of Justification Logic in order to provide a Plaza-Gerbrandy analog of multi-modal logic that we can use to reason about justified true belief. 1

Abstract. This paper introduces a notion of bisimulation for Artemov’s logics of evidence-based knowledge. Bisimulation allows us to study the effect of dynamic epistemic operations on language expressivity. It is shown that public announcements, a basic dynamic epistemic operation, add expressivity to the language of evidenced-based knowledge. It is also shown that public announcements are definable in the language of evidence-based knowledge augmented with an evidence admissibility relation. 1

Abstract. The question of whether knowledge is definable in terms of belief, which has played an important role in epistemology for the last 50 years, is studied here in the framework of epistemic and doxastic logics. Three notions of definability are considered: explicit definability, implicit definability, and reducibility, where explicit definability is equivalent to the combination of implicit definability and reducibility. It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief. It is also shown that S5 knowledge cannot be reduced to belief and justification, provided that there are no axioms that involve both belief and justification. §1. Introduction. The observations that knowledge and belief are related goes back to Plato’s dialogue Theaetetus, whose protagonist suggests that knowledge is justified true belief (JTB). Two millennia later, analytic philosophers such as Ayer (1956) and Chisholm (1957) adopted Plato’s slogan. But then a three-page paper by Gettier (1963) refuted the proposed definition by means of a few counterexamples; this started a new area of

under the supervision of Prof.dr J. F. A. K. van Benthem, and submitted to the Board of

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.

Artemov recently presented an ontologically transparent semantics for justifications that interprets justifications as sets of formulas they justify. However, this semantics of modular models has only been studied for the case of the basic justification logic J, corresponding to the modal logic K, and it has been left open how to extend modular models to and relate them to the already existing symbolic and epistemic semantics for justification logics with additional axioms, in particular, for logics of knowledge with factive justifications. We introduce modular models for extensions of J with any combination of the axioms (jd), (jt), (j4), (j5), and (jb), which are the explicit counterparts of standard modal axioms. After establishing soundness and completeness results, we examine the relationship of modular models to the more traditional symbolic and epistemic models. This comparison yields several new semantics, including symbolic models for logics of belief with negative introspection (j5) and models for logics with axiom (jb). Besides pure justification logics we also consider logics with both justifications and a belief/knowledge modal operator of the same strength. In particular, we use modular models to study the conditions under which the addition of such an operator to a justification logic yields a conservative extension. 1

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Halpern and Pass [2011] introduce a logic of justified belief and go on to prove that strong rationalizability is characterized in this logic in terms of common justified belief of rationality (CJBR). Their paper provides semantics for this logic but no axiomatization. We correct this deficiency by reformulating the definition of justified belief and providing a complete axiomatization of this new system. We then prove a result analogous to the characterization of strong rationalizability in terms of CJBR, and analyze the additional assumptions needed to do so. 1