The scope of provability*

Advances in Modal Logic, 2018

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Synthese, 2020

This paper introduces the logic of evidence and truth LET F as an extension of the Belnap-Dunn four-valued logic F DE. LET F is a slightly modified version of the logic LET J , presented in Carnielli and Rodrigues (2017). While LET J is equipped only with a classicality operator ○, LET F is equipped with a non-classicality operator • as well, dual to ○. Both LET F and LET J are logics of formal inconsistency and undeterminedness in which the operator ○ recovers classical logic for propositions in its scope. Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition α even if α is not true. As well as LET J , LET F is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for LET F where statements P (α) and P (○α) express, respectively, the amount of evidence available for α and the degree to which the evidence for α is expected to behave classically-or non-classically for P (•α). A probabilistic scenario is paracomplete when P (α) + P (¬α) < 1, and paraconsistent when P (α) + P (¬α) > 1, and in both cases, P (○α) < 1. If P (○α) = 1, or P (•α) = 0, classical probability is recovered for α. The proposition ○α ∨ •α, a theorem of LET F , partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory.

Halpern, Moses and Tuttle presented a definition of interactive proofs using a notion they called practical knowledge, but left open the question of finding an epistemic formula that completely characterizes zero knowledge; that is, a formula that holds iff a proof is zero knowledge. We present such a formula, and show that it does characterize zero knowledge. Moreover, we show that variants of the formula characterize variants of zero knowledge such as concurrent zero knowledge ] and proofs of knowledge . and systems framework ). In addition, we introduce some of the notation that will be needed for our new results.

International Journal of Game Theory, 1992

2008

Savage (1954) introduced the sure-thing principle in terms of the de- pendence of decisions on knowledge, but gave up on formalizing it in epistemic terms for lack of a formal deflnition of knowledge. Using simple models of knowledge, we examine the sure-thing principle, presenting two ways to capture it. One is in terms of the union of future events, for which we reserve the original name|the sure-thing principle; the other is in terms of the intersection of kens|bodies of agents&#x27; knowledge|which we call independence of irrelevant knowledge. We show that the two prin- ciples are equivalent and that the only property of knowledge required for this equivalence is the axiom of truth|the requirement that whatever is known is true. We present a symmetric version of the independence of irrelevant knowledge which is equivalent to the impossibility of agreeing to disagree on the decision made by agents, namely the impossibility of agents making difierent decisions being common knowledge

2004

Abstract An issue of a logic of knowledge with justifications has been discussed since the early 1990s. Such a logic along with the usual knowledge operator 2F “F is known” should contain assertions t:F “t is an evidence of F”.

Lecture Notes in Computer Science, 2006

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 K i (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.

Pavel Cmorej has argued that the existence of unverifiable and unfalsifiable empirical propositions follows from certain plausible assumptions concerning the notions of possibility and verification. Cmorej proves, it the context of a bi-modal alethic-epistemic axiom system AM4, that (1) "p and it is not verified that p" is unverifiable; (2) "p or it is falsified that p" is unfalsifiable; (3) every unverifiable p is logically equivalent to "p and it is not verifiable that p"; (4) every unverifiable p entails that p is unverifiable. This article elaborates on Cmorej’s results in three ways. Firstly, we formulate a version of neighbourhood semantics for AM4 and prove completeness. This allows us to replace Cmorej’s axiomatic derivations with simple model-theoretic arguments. Secondly, we link Cmorej’s results to two well-known paradoxes, namely Moore’s Paradox and the Knowability Paradox. Thirdly, we generalise Cmorej’s results, show them to be independent of each other and argue that results (3) and (4) are independent of any assumptions concerning the notion of verification.

Studies in Logic and the Foundations of Mathematics, 1998

This chapter is dedicated to the memory of George Boolos. From the start of the subject until his death on 27 May 1 9 9 6 h e w as the prime inspirer of the work in the logic of provability.