Museum of Outstanding Figures in Ukrainian Culture

This paper deals with a famous problem of epistemic logic – logical omniscience. Logical omniscience occurs in the logical systems where the axiomatics is complete and consequently an agent using inference rules knows everything about the system. Logical omniscience is a major problem due to complexity problems and the inability for adequate human reasoning modeling. It is studied both informal logic and philosophy of psychology (bounded rationality). It is important for bounded rationality because it reflects the problem of formal characterization of purely psychological mechanisms. Paper proposes to solve it using the ideas from the philosophical bounded rationality and intuitionistic logic. Special regions of deductible formulas developed according to psychologistic criterion should guide the deductive model. The method is compared to other ones presented in the literature on logical omniscience such as Hintikka’s and Vinkov and Fominuh. Views from different perspectives such as computer science and artificial intelligence are also provided.

Tableau Systems for Epistemic Positional Logics

The goal of the article is twofold. The first one is to provide logics basedon positional semantics which will be suitable for the analysis of epistemic modalitiessuch as ‘agent ... knows/beliefs that ...’. The second one is to define tableau systemsfor such logics. Firstly, we present the minimal positional logic MR. Then, we changethe notion of formulas and semantics in order to consider iterations of the operatorof realization and “free” classical formulas. After that, we move on to weaker logicsin order to avoid the well known problem of logical omniscience. At the same time,we keep the positional counterparts of modal axioms (T), (4) and (5). For all of theconsidered logics we present sound and complete tableau systems.

Principles of epistemic logic I: Knowledge

All epistemic logics come with some idealizations. Not all such idealizations seem acceptable. A large family of epistemic logics assume that if ⌜φ‎⌝ and ⌜ψ‎⌝ are logically equivalent, so are ⌜One knows that φ‎⌝ and ⌜One knows that ψ‎⌝. This assumption, characteristic of normal epistemic logics but also of many non-normal ones, is acceptable only if the objects of knowledge can be construed as sets of possible worlds known under some mode of presentation or other, where knowledge-ascriptions do not yet make those modes explicit. Unlike fine-grained conceptions that reject the assumption, such coarse-grained conceptions of the objects of knowledge have the untoward consequence that failures of logical omniscience are no longer expressible in the logic. But even on coarse-grained conceptions, epistemic logic cannot be expected to be normal. Fine-grained conceptions allow for failures of logical omniscience to be expressible in the logic. On balance, fine-grained conceptions are to be preferred. Against this backdrop, candidate principles for inclusion in the logic of knowledge are critically reviewed in the light of general epistemological considerations. Very few survive closer scrutiny.

A logic of knowledge based on abstract arguments

We introduce a logic of knowledge in a framework in which knowledge is treated as a kind of belief. The framework is based on a standard KD45 characterization of belief, and the characterization of knowledge undergoes the classical tripartite analysis that knowledge is justified true belief, which has a natural link to the studies of logics of evidence and justification. The interpretation of knowledge avoids the unwanted properties of logical omniscience, independent of the choice of the base logic of belief. We axiomatize the logic, prove its soundness and completeness and study the computational complexity results of the model checking and satisfiability problems. We extend the logic to a multi-agent setting and introduce a variant in which belief is characterized in a weaker system to avoid the problem of logical omniscience.

THOUGHT AS A QUASI-POSSIBILITY

Modal logic determines a lot in modern metaphysics and ontology, which delve deeper and deeper into the realm of the possible, not limited to the analysis of reality. This makes it relevant to study the problems of philosophical argumentation, built on the basis of modal logic. The aim of the work is to prove that thinkability does not necessarily entail a logical possibility. Because of this, many kinds of modal arguments that involve inference from conceivability to possibility can be flawed. Methodology: the author considers the question of the existence of objects impossible from the point of view of classical logical omniscience as a parallel to the idea of the existence of impossible possible worlds by J. Hintikka. The main idea of this article is the assertion that the gap between conceivability and possibility is generated by the intellectual limitations of the epistemic agent. The agent consistently — within the framework of the information available to him, and not in the absolute sense - considers logically possible that which is logically impossible from the point of view of logical omniscience. It turns out that we are able to think not only of something non-existent (to have empty intentions), but even quite capable of thinking the logically impossible. The conceivability of the impossible is somewhat analogous to the conceivability of impossible objects that are constructed by contemporary artists. The paper draws a parallel between the tautological thinkability of such an equation that is not tautological and the thinkability as a theorem of something that is not a theorem (S. Kripke’s modal arguments). As a particular example, the author criticizes the argument of the zombie by D. Chalmers, which is popular in modern philosophy of consciousness. It is shown that the conceivability of a zombie does not exclude the possible inconsistency of a zombie from an absolute point of view. In the second part, various types of the a priori are also considered, the opposition is built between the classical idea of logical omniscience and the agent-based approach using the categories of semantics of possible worlds. The main result of the proposed work is to prove that both formally contradictory and conceptually contradictory can be outside the framework of the epistemic attainability of the final agent. The author introduces a new philosophical category of the quasi-possible.

The Dynamic Epistemic Logic for Actual Knowledge

The dynamic epistemic logic for actual knowledge models the phenomenon of actual knowledge change when new information is received. In contrast to the systems of dynamic epistemic logic which have been discussed in the past literature, our system is not burdened with the problem of logical omniscience, that is, an idealized assumption that the agent explicitly knows all classical tautologies and all logical consequences of his or her knowledge. We provide a sound and complete axiomatization for this logic.

Probability and Decision Theory

This chapter introduces the mathematics of probability and decision theory. The probability calculus is introduced in both a set-theoretic and a propositional context. Probability is also related to measure theory, and stochastic truth-tables are presented. Problems with conditional probability are examined. Two interpretations of the probability calculus are introduced: physical and normative probabilities. The problem of logical omniscience for normative probabilities is discussed. Dutch Book arguments and accuracy-based arguments for Probabilism (the claim that our credences must satisfy the probability axioms) are examined and rejected. Different interpretations of the “idealization” reply to the problem of logical omniscience are considered, and one of them is tentatively endorsed. The expected utility maximization conception of decision theory is introduced, and representation arguments are considered (and rejected) as another reply to the problem of logical omniscience.

The Fundamental Problem of Logical Omniscience

We propose a solution to the problem of logical omniscience in what we take to be its fundamental version: as concerning arbitrary agents and the knowledge attitude per se. Our logic of knowledge is a spin-off from a general theory of thick content, whereby the content of a sentence has two components: (i) an intension, taking care of truth conditions; and (ii) a topic, taking care of subject matter. We present a list of plausible logical validities and invalidities for the logic of knowledge per se for arbitrary agents, and isolate three explanatory factors for them: (1) the topic-sensitivity of content; (2) the fragmentation of knowledge states; (3) the defeasibility of knowledge acquisition. We then present a novel dynamic epistemic logic that yields precisely the desired validities and invalidities, for which we provide expressivity and completeness results. We contrast this with related systems and address possible objections.