Abstract. This paper deals with the problem of giving a principled characterization of the class of logical constants. According to the so-called Tarski-Sher thesis, an operation is logical iff it is invariant under permutation. In the model-theoretic tradition, this criterion has been widely accepted as giving a necessary condition for an operation to be logical. But it has been also widely criticized on the account that it counts too many operations as logical, failing thus to provide a sufficient condition.Our aim is to solve this problem of overgeneration by modifying the invariance criterion. We introduce a general notion of invariance under a similarity relation and present the connection between similarity relations and classes of invariant operations. The next task is to isolate a similarity relation well-suited for a definition of logicality. We argue that the standard arguments in favor of invariance under permutation, which rely on the generality and the formality of logic, should be modified. The revised arguments are shown to support an alternative to Tarski's criterion, according to which an operation is logical iff it is invariant under potential isomorphism.On the traditional semantic account of logical consequence, a sentence φ is said to follow from a set Γ of sentences iff, for every uniform reinterpretation of the extra-logical expressions in Γ and φ, if all sentences in Γ are true, then φ is true. What is nice with this semantic definition of logical consequence is that it is purely extensional. It operates a reduction of logical truth to some kind of general truth, thanks to the quantification over all interpretations, without any need to appeal to metaphysical notions of possibility or necessity. What is more problematic though is that it rests crucially on the distinction between logical and extra-logical expressions. To get the account of logical consequence right, it is thus mandatory to know where the line should be drawn, unless one is ready to accept that we have nothing but a relative concept of logical entailment.

The standard relation of logical consequence allows for non-standard interpretations of logical constants, as was shown early on by Carnap. We may be able to learn the correct interpretations from the standard rules, because the space of possible interpretations is a priori restricted by universal semantic principles. We show that this is indeed the case. The principles are familiar from modern formal semantics: compositionality, supplemented, for quantifiers, with topic-neutrality.Keywords: logical consequence, logical constants, Carnap's Problem, semantic universals, compositionality, topic-neutrality * We are grateful to an anonymous referee for comments that helped us improve an ear-

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson's margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson's luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resourcesensitive logic with implications for the representation of common knowledge in situations of bounded rationality. Inexact knowledge and introspectionStandard modal models for knowledge are commonly S5 models in which the epistemic accessibility relation is an equivalence relation, namely a relation that is reflexive, symmetric and transitive. From an axiomatic point of view, reflexivity corresponds to the fact that knowledge is veridical, symmetry to the idea that if something is true, one knows one will not exclude it, and transitivity to the idea that knowledge is positively introspective, that is the property that whenever I know some proposition, I know that I know it. S5 models can also be described as reflexive models that are euclidean, which also makes them symmetric and transitive. Euclideanness corresponds to the property of negative introspection, namely to the property that whenever I don't know, I know that I don't know. S5 models are commonly used to represent situations of social knowledge, for instance in game theory, due to their well-known correspondence with partitional models of information (Osborne & Rubinstein 1994).An important feature of these models is the fact that they represent a notion of precise or exact knowledge in the following sense: whenever an agent fails to discriminate between two worlds or situations w and w ′ , any other situation which he fails to discriminate from w is also a situation which he fails to discriminate from w ′ , and vice versa. In other words, even though one's knowledge is not necessarily as fine-grained as it should be, it is at least clear cut, since 1 one's uncertainty is partitional. This contrasts with situations of imprecise knowledge, in which the relation of epistemic indiscriminability can fail to be transitive, as in cases of perceptual knowledge in which I can't discriminate between any two adjacent shades of color, and yet such that I can distinguish between shades of color...

The dual character of invariance under transformations and definability by some operations has been used in classical work by for example Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this paper, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in L∞∞ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee's theorem about quantifiers invariant under all permutations and definability in pure L∞∞ as a particular case. We also prove some optimality results along the way, regarding the kind of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.1. Inv(Aut(r)) is the L ∞∞ -closure of r. 2. Aut(Inv(H)) is the smallest subgroup of S Ω including H.

Hintikka makes a distinction between two kinds of games: truthconstituting games and truth-seeking games. His well-known game-theoretical semantics for first-order classical logic and its independence-friendly extension belongs to the first class of games. In order to ground Hintikka's claim that truth-constituting games are genuine verification and falsification games that make explicit the language games underlying the use of logical constants, it would be desirable to establish a substantial link between these two kinds of games. Adapting a result from Thierry Coquand, we propose such a link, based on a slight modification of Hintikka's games, in which we allow backward playing for ∃loïse. In this new setting, it can be proven that sequent rules for first-order logic, including the cut rule, are admissible, in the sense that for each rule, there exists an algorithm which turns winning strategies for the premisses into a winning strategy for the conclusion. Thus, proofs, as results of truth-seeking games, can be seen as effectively providing the needed winning strategies on the semantic games. * Une version très préliminaire des résultats de la section 3 aétép ŕ esentée lors du septième workshop "Games in Logic, Language and Computation" organiséàA msterdam le 28 novembre 2002. Je remercie Serge Bozon et Jacques Dubucs pour leurs suggestions fécondes.