A Dynamic Solution to the Problem of Logical Omniscience

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

Bayesianism for Non-ideal Agents

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

Hyperfine-Grained Meanings in Classical Logic

This paper develops a semantics for a fragment of English that is based on the idea of `impossible possible worlds'. This idea has earlier been formulated by authors such as Montague, Cresswell, Hintikka, and Rantala, but the present set-up shows how it can be formalized in a completely unproblematic logic---the ordinary classical theory of types. The theory is put to use in an account of propositional attitudes that is `hyperfine-grained', i.e. that does not suffer from the well-known problems involved with replacing expressions by logical equivalents

Knowability Relative to Information

We present a formal semantics for epistemic logic, capturing the notion of knowability relative to information (KRI). Like Dretske, we move from the platitude that what an agent can know depends on her (empirical) information. We treat operators of the form K_AB (‘B is knowable on the basis of information A’) as variably strict quantifiers over worlds with a topic- or aboutness- preservation constraint. Variable strictness models the non-monotonicity of knowledge acquisition while allowing knowledge to be intrinsically stable. Aboutness-preservation models the topic-sensitivity of information, allowing us to invalidate controversial forms of epistemic closure while validating less controversial ones. Thus, unlike the standard modal framework for epistemic logic, KRI accommodates plausible approaches to the Kripke-Harman dogmatism paradox, which bear on non-monotonicity, or on topic-sensitivity. KRI also strikes a better balance between agent idealization and a non-trivial logic of knowledge ascriptions

Logical Omnipotence and Two notions of Implicit Belief

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

Thinking Impossible Things

“There is no use in trying,” said Alice; “one can’t believe impossible things.” “I dare say you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast”. Lewis Carroll, Through the Looking Glass. It is a rather common view among philosophers that one cannot, properly speaking, be said to believe, conceive, imagine, hope for, or seek what is impossible. Some philosophers, for instance George Berkeley and the early Wittgenstein, thought that logically contradictory propositions lack cognitive meaning (informational content) and cannot, therefore, be thought or believed. Philosophers who do not go as far as Berkeley and Wittgenstein in denying that impossible propositions or states of affairs are thinkable, may still claim that it is impossible to rationally believe an impossible proposition. On a classical “Cartesian” view of belief, belief is a purely mental state of the agent holding true a proposition p that she “grasps” and is directly acquainted with. But if the agent is directly acquainted with an impossible proposition, then, presumably, she must know that it is impossible. But surely no rational agent can hold true a proposition that she knows is impossible. Hence, no rational agent can believe an impossible proposition. Thus it seems that on the Cartesian view of propositional attitudes as inner mental states in which proposition are immediately apprehended by the mind, it is impossible for a rational agent to believe, imagine or conceive an impossible proposition. Ruth Barcan Marcus (1983) has suggested that a belief attribution is defeated once it is discovered that the proposition, or state of affairs that is believed is impossible. According to her intuition, just as knowledge implies truth, belief implies possibility. It is commonplace that people claim to believe propositions that later turn out to be impossible. According to Barcan Marcus, the correct thing to say in such a situation is not: I once believed that A but I don’t believe it any longer since I have come to realize that it is impossible that A. What one should say is instead: It once appeared to me that I believed that A, but I did not, since it is impossible that A. Thus, Barcan Marcus defends what we might call Alice’s thesis: Necessarily, for any proposition p and any subject x, if x believes p, then p is possible. Alice’s thesis that it is impossible to hold impossible beliefs, seems to come into conflict with our ordinary practices of attributing beliefs. Consider a mathematical example. Some mathematicians believe that CH (the continuum hypothesis) is true and others believe that it is false. But if CH is true, then it is necessarily true; and if it is false, then it is necessarily false. Regardless of whether CH is true or false, the conclusion seems to be that there are mathematicians who believe impossible propositions. Examples of apparent beliefs in impossible propositions outside of mathematics are also easy to come by. Consider, for example, Kripke’s (1999) story of the Frenchman Pierre who without realizing it has two different names ‘London’ and ‘Londres’ for the same city, London. After having arrived in London, Pierre may assent to ‘Londres is beautiful and London is not beautiful’ without being in any way irrational. It seems reasonably to infer from this that Pierre believes that Londres is beautiful and London is not beautiful. But since ‘Londres’ and ‘London’ are rigid designators for the same city, it seems to follow from this that Pierre believes the inconsistent proposition that we may express as ‘London is both beautiful and not beautiful’

Reasoning about Knowledge in Linear Logic: Modalities and Complexity

In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has no