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On the succinctness of some modal logics
No full textOne way of comparing knowledge representation formalisms that has attracted attention recently is in terms of representational succinctness, i.e., we can ask whether one of the formalisms allows for a more 'economical' encoding of information than the other. Proving that one logic is more succinct than another becomes harder when the underlying semantics is stronger. We propose to use Formula Size Games (as put forward by Adler and lmmerman (2003) [1], but we present them as games for one player, called Spoiler), games that are played on two sets of models, and that directly link the length of a play in which Spoiler wins the game with the size of a formula, i.e., a formula that is true in the first set of models but false in all models of the second set. Using formula size games, we prove the following succinctness results for m-dimensional modal logic, where one has a set I = {i(1),..., i(m)} of indices for m modalities: (1) on general Kripke models (and also on binary trees), a definition [for all(Gamma)]phi = Lambda(i is an element of Gamma) [i]phi (with Gamma subset of I) makes the resulting logic exponentially more succinct for m > 1; (2) several modal logics use such abbreviations [for all(Gamma)]phi, e.g., in description logics the construct corresponds to adding role disjunctions, and an epistemic interpretation of it is 'everybody in Gamma knows'. Indeed, we show that on epistemic models (i.e., S-5-models), the logic with [for all(Gamma)]phi becomes more succinct for m > 3; (3) the results for the logic with 'everybody knows' also hold for a logic with 'somebody knows', and (4) on epistemic models, Public Announcement Logic is exponentially more succinct than epistemic logic, if m > 3. The latter settles an open problem raised by Lutz (2006) [18]. 2013 Elsevier B.V. All rights reserved.</p
Restructuring Searle's Making the Social World
No full textInstitutions are normative social structures that are collectively accepted. In his book Making the Social World, John R. Searle maintains that these social structures are created and maintained by Status Function Declarations. The article's author criticizes this claim and argues, first, that Searle overestimates the role that language plays in relation to institutions and, second, that Searle's notion of a Status Function Declaration confuses more than it enlightens. The distinction is exposed between regulative and constitutive rules as being primarily a linguistic one: whereas deontic powers figure explicitly in regulative rules, they feature only implicitly in constitutive rules. Furthermore, he contends that Searle's collective acceptance account of human rights cannot adequately account for the fact that people have these rights even when they are not recognized. Finally, It is argued that a conception of collective intentionality that involves collective commitment is needed in order to do justice to the normative dimension of institutions.</p
Explanation, understanding, and unrealistic models
No full textHow can false models be explanatory? And how can they help us to understand the way the world works? Sometimes scientists have little hope of building models that approximate the world they observe. Even in such cases, I argue, the models they build can have explanatory import. The basic idea is that scientists provide causal explanations of why the regularity entailed by an abstract and idealized model fails to obtain. They do so by relaxing some of its unrealistic assumptions. This method of 'explanation by relaxation' captures the explanatory import of some important models in economics. I contrast this method with the accounts that Daniel Hausman and Nancy Cartwright have provided of explanation in economics. Their accounts are unsatisfactory because they require that the economic model regularities obtain, which is rarely the case. I go on to argue that counterfactual regularities play a central role in achieving 'understanding by relaxation.' This has a surprising implication for the relation between explanation and understanding: Achieving scientific understanding does not require the ability to explain observed regularities. (C) 2012 Elsevier Ltd. All rights reserved.</p