Possibilistic Justification Logic

We explore two ways of formalizing Kreisel’s addendum to the Brouwer-Heyting-Kolmogorov interpretation. To do this we compare Artemov’s justification logic with simply typed \(\lambda \) calculus, by introducing a map from justification terms into \(\lambda \) terms, which can be viewed as a method of extracting the computational content of the justification terms. Then we examine the interpretation of Kreisel’s addendum in justification logic along with the image of the resulting justification terms under our map.KeywordsJustification logicType theoryIntuitionistic logic\(\lambda \) calculusBHK interpretation

Justification logics are epistemic logics that include explicit justifications for an agent’s knowledge. In the present paper, we introduce a justification logic \(\mathcal{JALC}\) over the description logic \(\mathcal{ALC}\). We provide a deductive system and a semantics for our logic and we establish soundness and completeness results. Moreover, we show that our logic satisfies the so-called internalization property stating that it internalizes its own notion of proof. We then sketch two applications of \(\mathcal{JALC}\): (i) the justification terms can be used to generate natural language explanations why an \(\mathcal{ALC}\) statement holds and (ii) the terms can be used to study data privacy issues for description logic knowledge bases.

Several justification logics have been created, starting with the logic LP, (Artemov, Bull Symbolic Logic 7(1):1---36, 2001). These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open.

In multi-agent systems, an agent generally forms her belief based on information from multiple sources, such as messages from other agents or perception of the external environment. While modal epistemic logic has been a standard formalism for reasoning about agent's belief, it lacks the expressive power for tracking information sources. Justification logic (JL) provides the missing expressivity by using justification terms to keep track of the belief formation process. However, because JL does not make a clear distinction between potential and actual evidence, the interpretation of justification formulas in JL turns out to be ambiguous. In this paper, we present a justification-based multi-source reasoning formalism built upon second-order propositional modal logic. Our framework not only inherits the source-tracking advantage of JL but also allows the distinction between the actual observation and simply potential admissibility of evidence.

Justification logics were introduced by Artemov in 1995 to provide intuitionistic logic with a classical provability semantics, a problem originally posed by Gödel. Justification logics are refinements of modal logics and formally connected to them by so-called realization theorems. A constructive proof of a realization theorem typically relies on a cut-free sequent-style proof system for the corresponding modal logic. A uniform realization theorem for all the modal logics of the so-called modal cube, i.e., for the extensions of the basic modal logic K with any subset of the axioms d, t, b, 4, and 5, has been proven using nested sequents. However, the proof was not modular in that some realization theorems required postprocessing in the form of translation on the justification logic side. This translation relied on additional restrictions on the language of the justification logic in question, thus, narrowing the scope of realization theorems. We present a fully modular proof of the realization theorems for the modal cube that is based on modular nested sequents introduced by Marin and Straßburger.

Previous intuitionistic justification logics included explicit justifications for all admissible rules of intuitionistic logic in order to get completeness with respect to provability semantics. We present the justification logic iJT4, which does not have these additional justification terms. We establish that iJT4 is complete with respect to modular models and that there is a realization of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit version of intuitionistic S4.

Ten years ago I wrote a paper in honor of Rohit Parikh’s 60th birthday, [4]. Now I am honored to write another for his 70th. In this paper I will make use of my paper from 10 years ago, to help provide a simple, constructive proof of the Realization Theorem for the modal logic S5. The Realization Theorem is a fundamental result in a developing area known as justification logics. Since these are not (yet) standard in the way that modal logics are, I will begin by sketching what justification logics are, saying why they are significant, and saying what the Realization Theorem is about. Then we can get to the more technical material, which essentially amounts to combining work from two of my earlier papers, [4] and [9].

Justification logic is an epistemic framework that provides a way to express explicit justifications for the agent's belief. In this paper, we present OPAL, a dynamic justification logic that includes term operators to reflect public announcements on the level of justifications. We create dynamic epistemic semantics for OPAL. We also elaborate on the relationship of dynamic justification logics to Gerbrandy-Groeneveld's PAL by providing a partial realization theorem.

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.

The aim of this paper is to explore the advantages deriving from the application of relating semantics in epistemic logic. As a first step, I will discuss two versions of relating semantics and how they can be differently exploited for studying modal and epistemic operators. Next, I consider several standard frameworks which are suitable for modelling knowledge and related notions, in both their implicit and their explicit form and present a simple strategy by virtue of which they can be associated with intuitive systems of relating logic. As a final step, I will focus on the logic of knowledge based on justification logic and show how relating semantics helps us to provide an elegant solution to some problems related to the standard interpretation of the explicit epistemic operators.

Faroldi argues that deontic modals are hyperintensional and thus traditional modal logic cannot provide an appropriate formalization of deontic situations. To overcome this issue, we introduce novel justification logics as hyperintensional analogues to non-normal modal logics. We establish soundness and completeness with respect to various models and we study the problem of realization.

There has been a long-lasting misunderstanding in the literature of artificial intelligence and uncertainty modeling, regarding the role of fuzzy set theory and many-valued logics. The recurring question is that of the mathematical and pragmatic meaningfulness of a compositional calculus and the validity of the excluded middle law. This confusion pervades the early developments of probabilistic logic, despite early warnings of some philosophers of probability. This paper tries to clarify this situation. It emphasizes three main points. First, it suggests that the root of the controversies lies in the unfortunate confusion between degrees of belief and what logicians call “degrees of truth”. The latter are usually compositional, while the former cannot be so. This claim is first illustrated by laying bare the non-compositional belief representation embedded in the standard propositional calculus. It turns out to be an all-or-nothing version of possibility theory. This framework is then extended to discuss the case of fuzzy logic versus graded possibility theory. Next, it is demonstrated that any belief representation where compositionality is taken for granted is bound to at worst collapse to a Boolean truth assignment and at best to a poorly expressive tool. Lastly, some claims pertaining to an alleged compositionality of possibility theory are refuted, thus clarifying a pervasive confusion between possibility theory axioms and fuzzy set basic connectives.

In recent years there has been a major trend in uncertainty (more specifically, partial belief) modelling emphasizing the idea that the degree of confidence in an event is not totally determined by the confidence in the opposite event, as assumed in probability theory. Possibility theory belongs to this trend that describes partial belief in terms of certainty and plausibility, viewed as distinct concepts. The distinctive features of possibility theory are its computational simplicity, and its position as a bridge between numerical and symbolic theories of partial belief for practical reasoning. The name ‘possibility theory’ was coined by L. A. Zadeh in the late seventies [Zadeh, 1978a] as an approach to uncertainty induced by pieces of vague linguistic information, described by means of fuzzy sets [Zadeh, 1965]. Possibility theory offers a simple, non-additive modelling of partial belief, which contrasts with probability theory. As we shall see, it provides a potentially more qualitative treatment of partial belief since the operations ‘max’ and ‘min’ play a role somewhat analogous to the sum and the product in probability calculus.

Uncertainty is often dealt with by simulating probability, possibility, or belief theories with string-symbolic languages implemented on digital computers; heuristic algorithms of interference are based on different kinds of binary or multivalued logic. In this paper the author suggests that computation and uncertainty may be managed naturally by synergetic, self-organizing dynamical physic systems, which are trained and are regulated to function as 'computing machines'.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The uncertainty which is involved in inductive inference, is often dealt with by simulating probability, possibility, or belief theories with digital computers, and by using heuristic methods of inference based on different kinds of logic (binary, multivalued or fuzzy). We suggest that uncertainty may be managed naturally by synergetic (co-operative), self-organizing, dynamic physical systems, which are trained and are regulated to function as pattern-association "computing machines".