Logical Omniscience as a Conditionality Issue. A Multi-Modal Approach

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.

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.

Logical omniscience as infeasibility

Logical omniscience may be described (roughly) as the state of affairs in which an agent explicitly believes anything which is logically entailed by that agent's beliefs. It is widely agreed that humans are not logically omniscient, and that an adequate formal model of belief, coupled with a correct semantic theory, would not entail logical omniscience. Recently, two prominent models of belief have emerged which purport both to avoid logical omniscience and to provide an intuitively appealing semantics. The first of these models is due to Levesque (1984b); the second to Fagin and Halpem (1985). It is argued herein that each of these models faces serious difficulties. Detailed criticisms are presented for each model, and a computationally oriented theory of intensions is presented which provides the foundation for a new formal model of belief. This formal model is presented in a decidable subset of first‐order logic and is shown to provide a solution to the general problem of logical omniscience. The model provides for the possibility of belief revision and places no a priori restrictions upon an agent's representation language.

Epistemic logics based on the possible worlds semantics suffer from the problem of logical omniscience, whereby agents are described as knowing all logical consequences of what they know, including all tautologies. This problem is doubly challenging: on the one hand, agents should be treated as logically non-omniscient, and on the other hand, as moderately logically competent. Many responses to logical omniscience fail to meet this double challenge because the concepts of knowledge and reasoning are not properly separated. In this paper, I present a dynamic logic of knowledge that models an agent’s epistemic state as it evolves over the course of reasoning. I show that the logic does not sacrifice logical competence on the altar of logical non-omniscience.

The article discusses the problems of formalizing the content of propositional attitudes and how successfully these problems can be solved in possible worlds semantics. The focus of attention is, firstly, on the phenomena that in the psychological literature are called framing effects and in the semantic literature substitution violation in indirect contexts, and, secondly, the problem of logical omniscience. The first part of the article explains why framing effects create a problem for possible worlds semantics and why the agent is inevitably modeled as logically omniscient in this semantics. Next, a description is given of four approaches to the problem of framing effects in possible worlds semantics: metasemantic, pragmatic, multi-model, and the approach associated with modeling topics. For each of the first three, their main shortcomings are given in relation to this problem and in general. The results of applying the latter approach are considered in more detail in the next part of the article using the example of one of the modern systems of doxastic logic, namely the logic of framing effects by Berto and Özgün. The syntax and semantic rules of this logic are outlined, after which it is explained how, according to these rules, the description of framing effects should look like. Models of situations are given in which an agent has both a belief in some proposition and a lack of belief in another proposition, which is necessarily equivalent to the first one. It is shown that the modeling method used does not lead to problems and is intuitively adequate. In the last part, the axiomatics of the logic of framing effects is presented and some theorems are given that are significant in the context of the problem of logical omniscience. An interpretation of these theorems is given, on the basis of which it is concluded that this logic does not cope with the problem of logical omniscience as successfully as with the framing effects themselves. In conclusion, a direction is proposed in which the logic of framing effects can be developed in order to more fully solve the problem of logical omniscience with its help, and the prospects for such development are discussed.

The traditional possible-worlds model of belief describes agents as ‘logically omniscient’ in the sense that they believe all logical consequences of what they believe, including all logical truths. This is widely considered a problem if we want to reason about the epistemic lives of non-ideal agents who—much like ordinary human beings—are logically competent, but not logically omniscient. A popular strategy for avoiding logical omniscience centers around the use of impossible worlds: worlds that, in one way or another, violate the laws of logic. In this paper, we argue that existing impossible-worlds models of belief fail to describe agents who are both logically non-omniscient and logically competent. To model such agents, we argue, we need to ‘dynamize’ the impossible-worlds framework in a way that allows us to capture not only what agents believe, but also what they are able to infer from what they believe. In light of this diagnosis, we go on to develop the formal details of a dynamic impossible-worlds framework, and show that it successfully models agents who are both logically non-omniscient and logically competent.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

It is common in epistemic modal logic to model the epistemic states of agents via box operators in the normal logic S5. However, this approach treats agents as logically omniscient by requiring their knowledge to be closed under classical logical consequence. A promising way of avoiding logical omniscience consists in extending epistemic models with impossible states, that is states, where complex formulas are not evaluated recursively. However, this approach faces the dual problem of logical ignorance by modeling agents as not even minimally logically competent. In this paper I will outline an epistemic logic that combines impossible states with dynamic realization modalities akin to the dynamic announcement operators from public announcement logic (PAL). I will show that this epistemic logic avoids both the problem of logical omniscience and the problem of logical ignorance. Furthermore, I prove that so-called successful updates in my logic can be characterized in the same way as in PAL and that a similar logic due to Johan van Benthem can be simulated in my logic. Finally, I will compare my epistemic logic with a similar one, which has recently been advanced by JC Bjerring and Mattias Skipper.

Dealing with logical omniscience: Expressiveness and pragmatics

All reasoners described in the most widespread models of a rational reasoner exhibit logical omniscience, which is impossible for finite reasoners (real reasoners). The most common strategy for dealing with the problem of logical omniscience is to interpret the models using a notion of beliefs different from explicit beliefs. For example, the models could be interpreted as describing the beliefs that the reasoner would hold if the reasoner were able reason indefinitely (stable beliefs). Then the models would describe maximum rationality, which a finite reasoner can only approach in the limit of a reasoning sequence. This strategy has important consequences for epistemology. If a finite reasoner can only approach maximum rationality in the limit of a reasoning sequence, then the efficiency of reasoning is epistemically (and not only pragmatically) relevant. In this paper, I present an argument to this conclusion and discuss its consequences, as, for example, the vindication of the principle ‘no rationality through brute-force’. Keywords: finite reasoning, logical omniscience, efficient reasoning, asymptotic analysis, computational complexity.

Formal theories of knowledge, belief, and partial belief usually make the idealizing assumption that an agent's knowledge or beliefs are closed under logical consequence—that knowers or believers are logically omniscient. This paper aims to clarify the nature of and motivation for this idealization. It is argued that it is more difficult than is sometimes supposed to give an adequate representation of belief or knowledge that avoids this idealization. Different ways of distinguishing implicit from explicit belief are distinguished. It is argued that while one can avoid logical omniscience by building linguistic structure into the contents of belief and knowledge, this fails to address the underlying problem, which is a problem about the accessibility or availability of information to influence action.

General Epistemic Logics suffer from the problem of logical omniscience, which is that an agent's knowledge and beliefs are closed under implication. There have been many attempts to solve the problem of logical omniscience. However, according to our intuition, sometimes an agent's knowledge and beliefs are indeed closed under implication. Based on the notion of awareness, we introduce two kinds of negations: general negation and strong negation. Moreover, four kinds of implications, general implication, strong implication, weak implication, and semi-strong implication, are introduced to correspond with the two kinds of negations. In our logics of regular awareness, explicit beliefs are not necessarily closed under general implication, which means that agents are not logically omniscient. However, explicit beliefs are closed under strong implication and semi-strong implication, which captures an intuitive closure property of beliefs.

Intuitionistic epistemic logic introduces an epistemic operator, which reflects the intended BHK semantics of intuitionism, to intuitionistic logic. The fundamental assumption concerning intuitionistic knowledge and belief is that it is the product of verification. The BHK interpretation of intuitionistic logic has a precise formulation in the Logic of Proofs and its arithmetical semantics. We show here that this interpretation can be extended to the notion of verification upon which intuitionistic knowledge is based, thereby providing the systems of intuitionistic epistemic logic extended by an epistemic operator based on verification with an arithmetical semantics too.

Orthodox Bayesianism is a highly idealized theory of how we ought to live our epistemic lives. One of the most widely discussed idealizations is that of logical omniscience: the assumption that an agent’s degrees of belief must be probabilistically coherent to be rational. It is widely agreed that this assumption is problematic if we want to reason about bounded rationality, logical learning, or other aspects of non-ideal epistemic agency. Yet, we still lack a satisfying way to avoid logical omniscience within a Bayesian framework. Some proposals merely replace logical omniscience with a different logical idealization; others sacrifice all traits of logical competence on the altar of logical non-omniscience. We think a better strategy is available: by enriching the Bayesian framework with tools that allow us to capture what agents can and cannot infer given their limited cognitive resources, we can avoid logical omniscience while retaining the idea that rational degrees of belief are in an important way constrained by the laws of probability. In this paper, we offer a formal implementation of this strategy, show how the resulting framework solves the problem of logical omniscience, and compare it to orthodox Bayesianism as we know it.