We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a (...) robust system of justifications. This renders a new, evidence-based foundation for epistemic logic. As a case study, we offer a resolution of the GoldmanRed Barns in Justification Logic. Furthermore, we formalize the well-known Gettier example and reveal hidden assumptions and redundancies in Gettier’s reasoning. (shrink)

Justification Logic provides an axiomatic description of justifications and delegates the question of their nature to semantics. In this note, we address the conceptual issue of the logical type of justifications: we argue that justifications in the logical setting are naturally interpreted as sets of formulas which leads to a class of epistemic models that we call modular models . We show that Fitting models for Justification Logic naturally encode modular models and can be regarded as convenient pre-models of the (...) former. (shrink)

In this paper individual proofs are integrated into provability logic. Systems of axioms for a logic with operators “A is provable” and “p is a proof of A” are introduced, provided with Kripke semantics and decision procedure. Completeness theorems with respect to the arithmetical interpretation are proved.

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...) resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

§I schematises the evidence for an understanding of ‘know’ and other terms of epistemic appraisal that embodies contextualism or subject-sensitive invariantism, and distinguishes between those two approaches. §II argues that although the cases for contextualism and sensitive invariantism rely on a principle of charity in the interpretation of epistemic claims, neither approach satisfies charity fully, since both attribute metalinguistic errors to speakers. §III provides an equally charitable anti-sceptical insensitive invariantist explanation of much of the same evidence as the result of (...) psychological bias caused by salience effects. §IV suggests that the explanation appears to have implausible consequences about practical reasoning, but also that applications of contextualism or sensitive invariantism to the problem of scepticism have such consequences. §V argues that the inevitable difference between appropriateness and knowledge of appropriateness in practical reasoning, closely related to the difference between knowledge and knowledge of knowledge, explains the apparent implausibility. (shrink)

Contextualism treats ‘knows’ as an indexical that denotes different epistemic properties in different contexts. Contrastivism treats ‘knows’ as denoting a ternary relation with a slot for a contrast proposition. I will argue that contrastivism resolves the main philosophical problems of contextualism, by employing a better linguistic model. Contextualist insights are best understood by contrastivist theory.

In Artemovʼs Justification Logic, one can make statements interpreted as “t is evidence for the truth of formula F.” We propose a variant of this logic in which one can say “I have degree r of confidence that t is evidence for the truth of formula F.” After defining both an axiomatic approach and a semantics for this Logic of Uncertain Justifications, we will prove the usual soundness and completeness theorems.

Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus.

This paper presents a partial analysis of perceptual knowledge, an analysis that will, I hope, lay a foundation for a general theory of knowing. Like an earlier theory I proposed, the envisaged theory would seek to explicate the concept of knowledge by reference to the causal processes that produce (or sustain) belief. Unlike the earlier theory, however, it would abandon the requirement that a knower's belief that p be causally connected with the fact, or state of affairs, that p.

Justification logics provide a framework for reasoning about justifications and evidence. In this article, we study a fuzzy variant of justification logics in which an agent’s justification for a belief has certainty degree between 0 and 1. We replace the classical base of justification logics with Hájek’s rational Pavelka logic. We introduce fuzzy possible world semantics with crisp accessibility relation and also single world models for our logics. We establish soundness and graded-style completeness for both kinds of semantics. We also (...) introduce extensions with product conjunction. Finally, we offer a solution to a variant of the sorites paradox in our fuzzy justification logics. (shrink)

A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.

A propositional logic of explicit proofs, LP, was introduced in [S. Artemov, Explicit provability and constructive semantics, The Bulletin for Symbolic Logic 7 1–36], completing a project begun long ago by Gödel, [K. Gödel, Vortrag bei Zilsel, translated as Lecture at Zilsel’s in: S. Feferman , Kurt Gödel Collected Works III, 1938, pp. 62–113]. In fact, LP can be looked at in a more general way, as a logic of explicit evidence, and there have been several papers along these lines. (...) A major result about LP is the Realization Theorem, that says any theorem of S4 can be converted into a theorem of LP by some replacement of necessitation symbols with explicit proof terms. Thus the necessitation operator of S4 can be seen as a kind of implicit existential quantifier: there exists a proof term such that…. In this paper, quantification over evidence is introduced into LP, and it is shown that the connection between S4 necessitation and the existential quantifier becomes an explicit one. The extension of LP with quantifiers is called QLP. A semantics and an axiom system for QLP are given, soundness and completeness are established, and several results are proved relating QLP to LP and to S4. (shrink)

This book presents an attempt to develop a theory of knowledge and a philosophy of mind using ideas derived from the mathematical theory of communication developed by Claude Shannon. Information is seen as an objective commodity defined by the dependency relations between distinct events. Knowledge is then analyzed as information caused belief. Perception is the delivery of information in analog form for conceptual utilization by cognitive mechanisms. The final chapters attempt to develop a theory of meaning by viewing meaning as (...) a certain kind of information-carrying role. (shrink)

This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses. This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or (...) graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature. (shrink)

This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses. This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or (...) graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature. (shrink)