Being realistic about common knowledge: a Lewisian approach

Defined and formalized several decades ago, widely used in philosophy and game theory, the concept of common knowledge is still considered as problematic, although not always for the right reasons. I suggest that the epistemic status of a group of human agents in a state of common knowledge has not been thoroughly analyzed. In particular, every existing account of common knowledge, whether formal or not, is either too strong to fit cognitively limited individuals, or too weak to adequately describe their state. I provide a realistic definition of common knowledge, based on a formalization of David Lewis’ seminal account and show that it is formally equivalent to probabilistic common belief. This leads to a philosophical analysis of common knowledge which answers several common criticisms and sheds light on its nature.     

Deterministic probability: neither chance nor credence

Some have argued that chance and determinism are compatible in order to account for the objectivity of probabilities in theories that are compatible with determinism, like Classical Statistical Mechanics (CSM) and Evolutionary Theory (ET). Contrarily, some have argued that chance and determinism are incompatible, and so such probabilities are subjective. In this paper, I argue that both of these positions are unsatisfactory. I argue that the probabilities of theories like CSM and ET are not chances, but also that they are not subjective probabilities either. Rather, they are a third type of probability, which I call counterfactual probability. The main distinguishing feature of counterfactual-probability is the role it plays in conveying important counterfactual information in explanations. This distinguishes counterfactual probability from chance as a second concept of objective probability.     

Two models of unawareness: comparing the object-based and the subjective-state-space approaches

Over the past 20 years or so, a small but growing literature has emerged with the aim of modeling agents who are unaware of certain things. In this paper we compare two different approaches to modeling unawareness: the object-based approach of Board and Chung (Object-based unawareness: theory and applications. University of Minnesota, Mimeo, 2008) and the subjective-state-space approach of Heifetz et al. (J Econ Theory 130:78–94, 2006). In particular, we show that subjective-state-space models (henceforth HMS structures) can be embedded within object-based models (henceforth OBU structures), demonstrating that the latter are at least as expressive. As long as certain restrictions are imposed on the form of the OBU structure, the embedding can also go the other way. A generalization of HMS structures (relaxing the partitional properties of knowledge) gives us a full converse.     

A note on the relationship between graphs and information protocols

Information protocols (IP’s) were developed to describe players who learn their social situation by their experiences. Although IP’s look similar to colored multi-graphs (MG’s), the two objects are constructed in fundamentally different ways. IP’s are constructed using the global concept of history, whereas graphs are constructed using the local concept of edges. We give necessary and sufficient conditions for each theory to be captured by the other. We find that the necessary and sufficient condition for IP theory to be captured by MG theory, which we call SE, excludes relevant game situations. Hence, we conclude that IP theory remains a vital tool and cannot be replaced by MG theory.     

Characterizing hallucination epistemically

According to the epistemic theory of hallucination, the fundamental psychological nature of a hallucinatory experience is constituted by its being ‘introspectively indiscriminable’, in some sense, from a veridical experience of a corresponding type. How is the notion of introspective indiscriminability to which the epistemic theory appeals best construed? Following M. G. F. Martin, the standard assumption is that the notion should be construed in terms of negative epistemics: in particular, it is assumed that the notion should be explained in terms of the impossibility that a hallucinator might possess a certain type of knowledge on a certain basis. I argue that the standard assumption is mistaken. I argue that the relevant notion of introspective indiscriminability is better construed in terms of positive epistemics: in particular, I argue that the notion is better explained by reference to the fact that it would be rational for a hallucinator positively to make a certain type of judgement, were that judgement made on a certain basis.     

Quantum logic as a dynamic logic

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear “no”. Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth’s proposal of applying Tarski’s semantical methods to the analysis of physical theories, with an empirical–experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operational- realistic tradition of Jauch and Piron, i.e. as an investigation of “the logic of yes–no experiments” (or “questions”). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the “non-classicality” of quantum behavior than any perspective based on static Propositional Logic.     

Competitive equilibrium with intuitionistic agents

This paper studies an economy whose agents perceive their consumption possibilities subjectively, and whose preferences are defined on what they subjectively experience, rather than on those alternatives that are objectively present. The model of agents’ perceptions is based on intuitionistic logic. Roughly, this means that agents reason constructively: a solution to a problem exists only if there is a construction by which the problem can be solved. The theorems that can be proved determine how an agent perceives a set of alternatives. A dual model relates perceived alternatives to a shared language, which the agents use in trading. So perceptions relate objective alternatives to an agent’s subjective view of them, and reporting dually relates an agent’s subjective world to a shared language. It turns out that an appropriately modified notion of competitive equilibrium always exists. However, in contrast with standard results in economic theory, competitive equilibrium need not be efficient.     

Epistemic constraints on practical normativity

What is the relation between what we ought to do, on the one hand, and our epistemic access to the ought-giving facts, on the other? In assessing this, it is common to distinguish ‘objective’ from ‘subjective’ oughts. Very roughly, on the objectivist conception what an agent ought to do is determined by ought-giving facts in such a way that does not depend on the agent’s beliefs about, or epistemic access to, those facts; whereas on the subjectivist conception, what an agent ought to do depends on his beliefs. This paper defends the need for, and explicates, a third category of ‘ought’: ‘warranted oughts’. Section 1 introduces the distinction between objective and subjective ‘oughts’. Sections 2–3 draw attention to some serious problems with each. Section 4 examines, though rejects, a recent attempt to replace subjective ‘oughts’ with objective ‘wide-scope oughts’ operating on belief-action combinations. Section 5 explicates the notion of a warranted ‘ought’ and defends the account against some possible objections. The resulting a picture is one in which an adequate analysis of practical normativity requires both objective and warranted ‘oughts’. Section 6 concludes by responding to a worry about countenancing both.