web page: comet.lehman.cuny.edu/fitting

In this paper we introduce a new type of knowledge operator, called evidencebased knowledge, intended to capture the constructive core of common knowledge. An evidence-based knowledge system is obtained by augmenting a multi-agent logic of knowledge with a system of evidence assertions t:ϕ (“t is an evidence for ϕ”) based on the following plausible assumptions: 1) each axiom has evidence; 3) evidence is checkable; 3) any evidence implies individual knowledge for each agent. Normally, the following monotonicity property is also assumed: 4) any piece of evidence is compatible with any other evidence. We show that the evidence-based knowledge operator is a stronger version of the common knowledge operator. Evidence-based knowledge is free of logical omniscience, model-independent, and has a natural motivation. Furthermore, evidence-based knowledge can be presented by normal multi-modal logics, which are in the scope of well-developed machinery applicable to modal logic: epistemic models, normalized proofs, automated proof search, etc. 1

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

The logical omniscience feature assumes that an epistemic agent knows all logical consequences of her assumptions. This paper offers a general theoretical framework that views logical omniscience as a computational complexity problem. We suggest the following approach: we assume that the knowledge of an agent is represented by an epistemic logical system E; we call such an agent not logically omniscient if for any valid knowledge assertion A of type F is known, a proof of F in E can be found in polynomial time in the size of A. We show that agents represented by major modal logics of knowledge and belief are logically omniscient, whereas agents represented by justification logic systems are not logically omniscient with respect to t is a justification for F.

This paper is devoted to the study of self-referential proofs and/or justifications, i.e., valid proofs that prove statements about these same proofs. The goal is to investigate whether such self-referential justifications are present in the reasoning described by standard modal epistemic logics such as S4. We argue that the modal language by itself is too coarse to capture this concept of self-referentiality and that the language of justification logic can serve as an adequate refinement. We consider well-known modal logics of knowledge/belief and show, using explicit justifications, that S4, D4, K4, and T with their respective justification counterparts LP, JD4, J4, and JT describe knowledge that is self-referential in some strong sense. We also demonstrate that self-referentiality can be avoided for K and D. In order to prove the former result, we develop a machinery of minimal evidence functions used to effectively build models for justification logics. We observe that the calculus used to construct the minimal functions axiomatizes the reflected fragments of justification logics. We also discuss difficulties that result from an introduction of negative introspection.

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Contents Preface i 1 Introduction: Unwinding Proofs (`Proof Mining') 1 2 Intuitionistic logic and arithmetic in all finite types 18 3 Modified realizability 27 4 Majorizability and the fan rule 36 5 Semi-intuitionistic systems and monotone modified realizability 41 6 Godel's functional (`Dialectica-')interpretation 42 7 Negative translation and its use combined with functional interpretation 54 8 A non-standard principle of uniform boundedness 63 9 The Friedman A-translation 72 10 Applications to analysis 75 10.1 Applications to the fixed point theory for nonexpansive mappings . . 81 10.2 Applications to uniqueness theorems in approximation theory . . . . 87 11 Final Comments 93 Bibliography 94 ii Chapter 1 Introduction: Unwinding Proofs (`Proof Mining') Proof interpretations of the kind we are going to study in these lectures are tools to extract constructive (computational) data from given proofs by recursion on the proof. Such data

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

intuitionistic fragment of computability logic at