Does rationality require logical omniscience? Our best formal theories of rationality imply that it does, but our ordinary evaluations of rationality seem to suggest otherwise. This paper aims to resolve the tension by arguing that our ordinary evaluations of rationality are not only consistent with the thesis that rationality requires logical omniscience, but also provide a compelling rationale for accepting this thesis in the first place. This paper also defends an account of apriori justification for (...) class='Hi'>logical beliefs that is designed to explain the rational requirement of logical omniscience. On this account, apriori justification for beliefs about logic has its source in logical facts, rather than psychological facts about experience, reasoning, or understanding. This account has important consequences for the epistemic role of experience in the logical domain. In a slogan, the epistemic role of experience in the apriori domain is not a justifying role, but rather an enabling and disabling role. (shrink)

It would be good to have a Bayesian decision theory that assesses our decisions and thinking according to everyday standards of rationality — standards that do not require logical omniscience (Garber 1983, Hacking 1967). To that end we develop a “fragmented” decision theory in which a single state of mind is represented by a family of credence functions, each associated with a distinct choice condition (Lewis 1982, Stalnaker 1984). The theory imposes a local coherence assumption guaranteeing that as (...) an agent's attention shifts, successive batches of "obvious" logical information become available to her. A rule of expected utility maximization can then be applied to the decision of what to attend to next during a train of thought. On the resulting theory, rationality requires ordinary agents to be logically competent and to often engage in trains of thought that increase the unification of their states of mind. But rationality does not require ordinary agents to be logically omniscient. (shrink)

In this paper, I investigate whether we can use a world-involving framework to model the epistemic states of non-ideal agents. The standard possible-world framework falters in this respect because of a commitment to logical omniscience. A familiar attempt to overcome this problem centers around the use of impossible worlds where the truths of logic can be false. As we shall see, if we admit impossible worlds where “anything goes” in modal space, it is easy to model extremely non-ideal (...) agents that are incapable of performing even the most elementary logical deductions. A much harder, and considerably less investigated challenge is to ensure that the resulting modal space can also be used to model moderately ideal agents that are not logically omniscient but nevertheless logically competent. Intuitively, while such agents may fail to rule out subtly impossible worlds that verify complex logical falsehoods, they are nevertheless able to rule out blatantly impossible worlds that verify obvious logical falsehoods. To model moderately ideal agents, I argue, the job is to construct a modal space that contains only possible and non-trivially impossible worlds where it is not the case that “anything goes”. But I prove that it is impossible to develop an impossible-world framework that can do this job and that satisfies certain standard conditions. Effectively, I show that attempts to model moderately ideal agents in a world-involving framework collapse to modeling either logical omniscient agents, or extremely non-ideal agents. (shrink)

The main goal of this paper is to investigate what explanatory resources Robert Brandom’s distinction between acknowledged and consequential commitments affords in relation to the problem of logical omniscience. With this distinction the importance of the doxastic perspective under consideration for the relationship between logic and norms of reasoning is emphasized, and it becomes possible to handle a number of problematic cases discussed in the literature without thereby incurring a commitment to revisionism about logic. One such case in (...) particular is the preface paradox, which will receive an extensive treatment. As we shall see, the problem of logical omniscience not only arises within theories based on deductive logic; but also within the recent paradigm shift in psychology of reasoning. So dealing with this problem is important not only for philosophical purposes but also from a psychological perspective. (shrink)

The purpose of this paper is to consider the explanatory resources that Robert Brandom‟s distinction between acknowledged and consequential commitments affords in relation to the problem of logical omniscience. With this distinction the importance of the doxastic perspective under consideration for the relationship between logic and norms of reasoning is emphasized, and it becomes possible to handle a number of problematic cases discussed in the literature without thereby incurring a commitment to revisionism about logic. 12.

I discuss three ways of responding to the logical omniscience problems faced by traditional ‘possible worlds’ epistemic logics. Two of these responses were put forward by Hintikka and the third by Cresswell; all three have been influential in the literature on epistemic logic. I show that both of Hintikka's responses fail and present some problems for Cresswell’s. Although Cresswell's approach can be amended to avoid certain unpalatable consequences, the resulting formal framework collapses to a sentential model of knowledge, (...) which defenders of the ‘possible worlds’ approach are frequently critical of. (shrink)

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: an intension, taking care of truth conditions; and 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: the topic-sensitivity of content; the fragmentation of knowledge states; 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. (shrink)

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. (shrink)

This paper looks at three ways of addressing probabilism’s implausible requirement of logical omniscience. The first and most common strategy says it’s okay to require an ideally rational person to be logically omniscient. I argue that this view is indefensible on any interpretation of ‘ideally rational’. The second strategy says probabilism should be formulated not in terms of logically possible worlds but in terms of doxastically possible worlds, ways you think the world might be. I argue that, on (...) the interpretation of this approach that lifts the requirement of certainty in all logical truths, the view becomes vacuous, issuing no requirements on rational believers at all. Finally, I develop and endorse a new solution to the problem. This view proposes dynamic norms for reasoning with credences. The solution is based on an old proposal of Ian Hacking’s that says you’re required to be sensitive to logical facts only when you know they are logical facts. (shrink)

Logical theories for representing knowledge are often plagued by the so-called Logical Omniscience Problem. The problem stems from the clash between the desire to model rational agents, which should be capable of simple logical inferences, and the fact that any logical inference, however complex, almost inevitably consists of inference steps that are simple enough. This contradiction points to the fruitlessness of trying to solve the Logical Omniscience Problem qualitatively if the rationality of agents (...) is to be maintained. We provide a quantitative solution to the problem compatible with the two important facets of the reasoning agent: rationality and resource boundedness. More precisely, we provide a test for the logical omniscience problem in a given formal theory of knowledge. The quantitative measures we use are inspired by the complexity theory. We illustrate our framework with a number of examples ranging from the traditional implicit representation of knowledge in modal logic to the language of justification logic, which is capable of spelling out the internal inference process. We use these examples to divide representations of knowledge into logically omniscient and not logically omniscient, thus trying to determine how much information about the reasoning process needs to be present in a theory to avoid logical omniscience. (shrink)

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. (shrink)

We propose a model for belief which is free of presuppositions. Current models for belief suffer from two difficulties. One is the well known problem of logical omniscience which tends to follow from most models. But a more important one is the fact that most models do not even attempt to answer the question what it means for someone to believe something, and just what it is that is believed. We provide a flexible model which allows us to (...) give meaning to beliefs in general contexts, including the context of animal belief , and of human belief which is expressed in language. (shrink)

This book deals with an old conundrum: if God knows what we will choose tomorrow, how can we be free to choose otherwise? If all our choices are already written, is our freedom simply an illusion? This book provides a precise analysis of this dilemma using the tools of modern ontology and the logic of time. With a focus on three intertwined concepts - God's nature, the formal structure of time, and the metaphysics of time, including the relationship between temporal (...) entities and a timeless God - the chapters analyse various solutions to the problem of foreknowledge and freedom, revealing the advantages and drawbacks of each. Building on this analysis, the authors advance constructive solutions, showing under what conditions an entity can be omniscient in the presence of free agents, and whether an eternal entity can know the tensed futures of the world. The metaphysics of time, its topology and the semantics of future tensed sentences are shown to be invaluable topics in dealing with this issue. Combining investigations into the metaphysics of time with the discipline of temporal Logic this monograph brings about important advancements in the philosophical understanding of an ancient and fascinating problem. The answer, if any, is hidden in the folds of time, in the elusive nature of this feature of reality and in the infinite branching of our lives. (shrink)

Review of Joseph Y. Halpern (ed.), Theoretical Aspects of Reasoning About Knowledge: Proceedings of the 1986 Conference (Los Altos, CA: Morgan Kaufmann, 1986),.

The beliefs of the agents in a multi-agent system have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they commit us to model agents that are logically omniscient. We propose a way of avoiding this problem, using a new kind of entities called subjective situations. We define a new doxastic logic based on these entities and we show how the belief operators (...) have some desirable properties, while avoiding logical omniscience. A comparison with two well-known proposals is also provided. (shrink)

Science advances by means of argument and debate. Based on a formal model of complex argumentation, this article assesses the interplay between evidential and inferential drivers in scientific controversy, and explains, in particular, why both evidence accumulation and argumentation are veritistically valuable. By improving the conditions for applying veritistic indicators , novel evidence and arguments allow us to distinguish true from false hypotheses more reliably. Because such veritistic indicators also underpin inductive reasoning, evidence accumulation and argumentation enhance the reliability of (...) inductive inference, for example, inference to the best explanation. 1 Introduction2 Theory of Dialectical Structures3 Debate Simulations4 From Evidence and Arguments to Truth: The First Route5 From Evidence to Truth: The Second Route6 Conclusion. (shrink)

At least since Aristotle’s famous 'sea-battle' passages in On Interpretation 9, some substantial minority of philosophers has been attracted to the doctrine of the open future--the doctrine that future contingent statements are not true. But, prima facie, such views seem inconsistent with the following intuition: if something has happened, then (looking back) it was the case that it would happen. How can it be that, looking forwards, it isn’t true that there will be a sea battle, while also being true (...) that, looking backwards, it was the case that there would be a sea battle? This tension forms, in large part, what might be called the problem of future contingents. A dominant trend in temporal logic and semantic theorizing about future contingents seeks to validate both intuitions. Theorists in this tradition--including some interpretations of Aristotle, but paradigmatically, Thomason (1970), as well as more recent developments in Belnap, et. al (2001) and MacFarlane (2003, 2014)--have argued that the apparent tension between the intuitions is in fact merely apparent. In short, such theorists seek to maintain both of the following two theses: (i) the open future: Future contingents are not true, and (ii) retro-closure: From the fact that something is true, it follows that it was the case that it would be true. It is well-known that reflection on the problem of future contingents has in many ways been inspired by importantly parallel issues regarding divine foreknowledge and indeterminism. In this paper, we take up this perspective, and ask what accepting both the open future and retro-closure predicts about omniscience. When we theorize about a perfect knower, we are theorizing about what an ideal agent ought to believe. Our contention is that there isn’t an acceptable view of ideally rational belief given the assumptions of the open future and retro-closure, and thus this casts doubt on the conjunction of those assumptions. (shrink)

In this paper several assumptions concerning omniscience and future contingents on the one side, and omniscience and self-reference on the other, areexamined with respect to a classical and a three-valued semantic setting.Interesting features of both settings are highlighted and their basic assumptions concerning omniscience are explored. To generate a context in which the notion of omniscience does not deviate from some basic intuitions, two special futurity operators are introduced in this article: one for what will definitely (...) take place and another one for what is indeterminate as to whether it will take place. Once these operators are introduced, some puzzles about omniscience in combination with future contingents are removed. An analogous solution to some puzzles concerning omniscience and selfreferentiality is also provided. (shrink)

The aim of the book is to clarify the concept of omniscience. This is done first by discussing basic questions on omniscience (chs.1-12) and secondly by offering a theory of omniscience as an axiomatic system in which also a definition of omniscience is given (ch.13). The twelve chapters deal with questions like whether everything is true what God knows, whether God's knowledge is bound to time, whether it concerns singular truths or only laws, whether it extends (...) also to contingent future events ...etc. The book is neither a book about the existence of God nor about proofs for his existence. It is a book about the possibility of a consistent concept of omniscience which can be attributed to God. And it invalidates opposite claims and shows that they are based on wrong or very doubtful premises. The pros and cons at the beginning of each chapter represent different positions and objections which are clarified and discussed in the answer to the objections. (shrink)

According to certain normative theories in epistemology, rationality requires us to be logically omniscient. Yet this prescription clashes with our ordinary judgments of rationality. How should we resolve this tension? In this paper, I focus particularly on the logical omniscience requirement in Bayesian epistemology. Building on a key insight by Hacking :311–325, 1967), I develop a version of Bayesianism that permits logical ignorance. This includes: an account of the synchronic norms that govern a logically ignorant individual at (...) any given time; an account of how we reduce our logical ignorance by learning logical facts and how we should update our credences in response to such evidence; and an account of when logical ignorance is irrational and when it isn’t. At the end, I explain why the requirement of logical omniscience remains true of ideal agents with no computational, processing, or storage limitations. (shrink)

We present a framework for epistemic logic, modeling the logical aspects of System 1 and System 2 cognitive processes, as per dual process theories of reasoning. The framework combines non-normal worlds semantics with the techniques of Dynamic Epistemic Logic. It models non-logically-omniscient, but moderately rational agents: their System 1 makes fast sense of incoming information by integrating it on the basis of their background knowledge and beliefs. Their System 2 allows them to slowly, step-wise unpack some of the (...) class='Hi'>logical consequences of such knowledge and beliefs, by paying a cognitive cost. The framework is applied to three instances of limited rationality, widely discussed in cognitive psychology: Stereotypical Thinking, the Framing Effect, and the Anchoring Effect. (shrink)

The logical omniscience problem, whereby standard models of epistemic logic treat an agent as believing all consequences of its beliefs and knowing whatever follows from what else it knows, has received plenty of attention in the literature. But many attempted solutions focus on a fairly narrow specification of the problem: avoiding the closure of belief or knowledge, rather than showing how the proposed logic is of philosophical interest or of use in computer science or artificial intelligence. Sentential epistemic (...) logics, as opposed to traditional possible worlds approaches, do not suffer from the problems of logical omniscience but are often thought to lack interesting epistemic properties. In this paper, I focus on the case of rule-based agents, which play a key role in contemporary AI research but have been neglected in the logical literature. I develop a framework for modelling monotonic, nonmonotonic and introspective rule-based reasoners which have limited cognitive resources and prove that the resulting models have a number of interesting properties. An axiomatization of the resulting logic is given, together with completeness, decidability and complexity results. (shrink)

Recently, several philosophers have moved from a classical account of divine omniscience according to which God knows all truths to a restricted account of divine omniscience according to which God knows all knowable truths. But an important objection offered by Alexander Pruss threatens to show that if God knows all knowable truths, God must also know all truths. In this paper, I show that there is a way out of Pruss’s objection for the advocate of restricted omniscience (...) if she will define her view in terms of ways of knowing rather than in terms of logical possibilities. (shrink)

A survey of logical problems for the concept of omniscience.

This chapter provides a brief introduction to propositional epistemic logic and its applications to epistemology. No previous exposure to epistemic logic is assumed. Epistemic-logical topics discussed include the language and semantics of basic epistemic logic, multi-agent epistemic logic, combined epistemic-doxastic logic, and a glimpse of dynamic epistemic logic. Epistemological topics discussed include Moore-paradoxical phenomena, the surprise exam paradox, logical omniscience and epistemic closure, formalized theories of knowledge, debates about higher-order knowledge, and issues of knowability raised by Fitch’s (...) paradox. The references and recommended readings provide gateways for further exploration. (shrink)

David Lewis’s property-centered account of belief falls prey to the problem of egocentric omniscience: In self-ascribing the property of being an eye doctor, an agent is thereby self-ascribing the property of being an oculist. It is argued that the problem of egocentric omniscience can be made palatable for Lewis’s property-centered account of belief, at least for the case of linguistic beliefs. Roughly, my solution is as follows: An agent can believe that he or she has the property of (...) being an eye doctor/oculist under the description ‘eye doctor’ without believing that he or she has this property under the description ‘oculist’. Believing that one has a property P under a description D involves the additional self-ascription of the propositional property of inhabiting a world with respect to which that description denotes the property P. This is not the same sort of solution as the one proposed for singular beliefs by Nathan Salmon. Unlike Salmon’s account, belief on the account I am defending is regarded as a two place-relation rather than a three-place relation. Since, on Lewis’s account, self-ascriptive belief subsumes de dicto belief, my solution also sheds light on the problem of logical omniscience. (shrink)

ABSTRACT First, I consider a few motivations to idealize epistemic logics1 in such a degree that brings up the problem of logical omniscience [LOP]. I argue that the main motivation to hold omniscience is of a philosophical-scientific2 background, in the sense philosophers have a not so peculiar way of investigating underlying mechanisms, i.e., the interaction of several different components of complex systems may be better understood in isolation, even if such components are not found isolated in a (...) realistic context. It is defended that the implicit and explicit knowledge distinction is compatible that view since idealizations made by modal epistemic logic are so strong that the agents they describe hardly have anything in common with real agents. I conclude by showing how LOP can be accommodated in the logic of being informed using the Inverse Relationship Principle. 1Epistemic modal logics and the logic of being informed may collapse in many scenarios, as shown in previous articles. 2Does not necessarily imply in a naturalized epistemology a la. For a plea for non-naturalism as constructionism see. (shrink)

This collection of papers in epistemic logic is oriented towards applications to game theory and individual decision theory. Most of these papers were presented at the inaugural conference of the LOFT (Logic for the Theory and Games and Decisions) conference series, which took place in 1994 in Marseille. Among the notions dealt with are those of common knowledge and common belief, infinite hierarchies of beliefs and belief spaces, logical omniscience, positive and negative introspection, backward induction and rationalizable equilibria (...) in game theory. (shrink)

The most widespread models of rational reasoners (the model based on modal epistemic logic and the model based on probability theory) exhibit the problem of logical omniscience. The most common strategy for avoiding this problem is to interpret the models as describing the explicit beliefs of an ideal reasoner, but only the implicit beliefs of a real reasoner. I argue that this strategy faces serious normative issues. In this paper, I present the more fundamental problem of logical (...) omnipotence, which highlights the normative content of the problem of logical omniscience. I introduce two developments of the notion of implicit belief (accessible and stable belief ) and use them in two versions of the most common strategy applied to the problem of logical omnipotence. (shrink)

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The (...) importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn—Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleene's ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense. (shrink)

A very weak omniscience principle is formulated, related omniscience principlesare considered, and the theorem that a function of bounded variation is the difference of two increasing functions is shown to be equivalent to the omniscience principle WLPO. It is a so shown that an arbitrary function with located variation on an interval is the difference of two increasing functions.

Spinoza is a pantheist: he believes that everything that is, is God. Traditional Judaeo-Christian theologians dislike the idea, and Spinoza has always been unpopular for it. Nevertheless, I want here to suggest that, simply by following out the logic of omniscience and omnipotence—two attributes of God on which both Spinoza and his opponents are agreed—it is possible to arrive at a conception of God which is at least very close to Spinoza's own. I do not claim that any of (...) the arguments I use are to be found in Spinoza, but I do think that this way of thinking helps to explain the motives behind his ‘hideous hypothesis’. (shrink)

Many people regard utility theory as the only rigorous foundation for subjective probability, and even de Finetti thought the betting approach supplemented by Dutch Book arguments only good as an approximation to a utility-theoretic account. I think that there are good reasons to doubt this judgment, and I propose an alternative, in which the probability axioms are consistency constraints on distributions of fair betting quotients. The idea itself is hardly new: it is in de Finetti and also Ramsey. What is (...) new is that it is shown that probabilistic consistency and consequence can be defined in a way formally analogous to the way these notions are defined in deductive (propositional) logic. The result is a free-standing logic which does not pretend to be a theory of rationality and is therefore immune to, among other charges, that of “logical omniscience”. (shrink)

Working within Bishop-style constructive mathematics, we derive a number of results relating the nonconstructive LPO and sequential compactness property on the one hand, and the intuitionistically reasonable anti-Specker property on the other.