Abstract not found

The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This

We present a foundation for a computational meta-theory of languages with bindings implemented in a computer-aided formal reasoning environment. Our theory provides the ability to reason abstractly about operators, languages, open-ended languages, classes of languages, etc. The theory is based on the ideas of higher-order abstract syntax, with an appropriate induction principle parameterized over the language (i.e. a set of operators) being used. In our approach, both the bound and free variables are treated uniformly and this uniform treatment extends naturally to variable-length bindings. The implementation is reflective, namely there is a natural mapping between the meta-language of the theorem-prover and the object language of our theory. The object language substitution operation is mapped to the meta-language substitution and does not need to be defined recursively. Our approach does not require designing a custom type theory; in this paper we describe the implementation of this foundational theory within a general-purpose type theory. This work is fully implemented in the MetaPRL theorem prover, using the pre-existing NuPRL-like MartinL of-style computational type theory. Based on this implementation, we lay out an outline for a framework for programming language experimentation and exploration as well as a general reflective reasoning framework. This paper also includes a short survey of the existing approaches to syntactic reflection. 1

We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cut-elimination procedure and also no known non-trivial bound on the proof-depth. We then present another infinitary sequent system based on nested sequents that are essentially trees and with inference rules that apply deeply inside of these trees. Thus we call this system “deep ” while we call the former system “shallow”. In contrast to the shallow system, the deep system allows to give a straightforward syntactic cut-elimination procedure. Since both systems can be embedded into each other, this also yields a syntactic cut-elimination procedure for the shallow system. For both systems we thus obtain an upper bound of ϕ20 onthe depth of proofs, where ϕ is the Veblen function. Key words: cut elimination, infinitary sequent system, nested sequents, common knowledge 1.

Justification Logic is a framework for reasoning about evidence and justification. Public Announcement Logic is a framework for reasoning about belief changes caused by public announcements. This paper develops JPAL, a dynamic justification logic of public announcements that corresponds to the modal theory of public announcements due to Gerbrandy and Groeneveld. JPAL allows us to reason about evidence brought about by and changed by Gerbrandy–Groeneveld-style public announcements.

Abstract. Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an epistemic reading of this formula: “t is a possible justification of φ”. Motivated, in part, by this epistemic reading, Fitting introduced a Kripke style semantics for LP in [8]. In this note, we prove soundness and completeness of some axiom systems which are not covered in [8]. 1

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

ABSTRACT. Justification logics are epistemic logics that explicitly include justifications for the agents ’ knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting’s semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya’s minimal bimodal explicit evidence logic, which is a two-agent version of LP. We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic.

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