We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a (...) natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al. (shrink)

Wittgenstein’s language games can be put into a wider service by virtue of elements they share with some contemporary opinions concerning logic and the semantics of computation. I will give two examples: manifestations of language games and their possible variations in logical studies, and their role in some of the recent developments in computer science. It turns out that the current paradigm of computation that Girard termed Ludics bears a striking resemblance to members of language games. Moreover, the kind of (...) interrelations that are emerging could be scrutinised from the viewpoint of logic that virtually necessitates game-theoretic conceptualisations, demonstrating the fact that the meaning of utterances may, in many situations, be understood as Wittgenstein’s language games of ‘showing or telling what one sees’. This provides motivation for the use of games in relation to logic and formal semantics that some commentators have called for. Many of the ideas can be traced to C.S. Peirce, for whom signs were vehicles of strategic communication. The conclusion about Wittgenstein is that the notions of saying and showing converge in his late philosophy. (shrink)

We present a single sequent calculus common to classical, intuitionistic and linear logics. The main novelty is that classical, intuitionistic and linear logics appear as fragments, i.e. as particular classes of formulas and sequents. For instance, a proof of an intuitionistic formula A may use classical or linear lemmas without any restriction: but after cut-elimination the proof of A is wholly intuitionistic, what is superficially achieved by the subformula property and more deeply by a very careful treatment of structural rules. (...) This approach is radically different from the one that consists in “changing the rule of the game” when we want to change logic, e.g. pass from one style of sequent to another: here, there is only one logic, which—depending on its use—may appear classical, intuitionistic or linear. (shrink)

Extending the project of analysis -- Elaborating abilities : the expressive role of logic -- Artificial intelligence and analytic pragmatism -- Modality and normativity : from Hume and Quine to Kant and Sellars -- Incompatibility, modal semantics, and intrinsic logic -- Intentionality as a pragmatically mediated semantic relation -- Afterword : philosophical analysis and analytic philosophy.

Robert Brandom's latest book, the product of his John Locke lectures in Oxford in 2006, is a return to the philosophy of language and is easily read as a continuation and development of the views defended in Making it Explicit. The text of the lectures is presented much as they were delivered, but it contains an ‘Afterword’ of more than 30 pages which responds to questions raised when he gave the lectures, and also when they were subsequently delivered in Prague (...) the following year. The published text also contains relatively technical appendices to two of the lectures.The individual lectures engage with some important and difficult issues, often ones that were explored in detail in the earlier book. However, these discussions are located within a broader meta-philosophical context, and it says something about the abstract and difficult character of these views that they provide the main subject matter of the Afterword. This framework affects how we should understand the relations between this book and Making It Explicit too. Although most of the detailed discussions happily belong within the general project of the earlier book, they are offered as illustrations of a framework that is independent of this project. Indeed, Brandom suggests that defenders of the semantic views of David Lewis, for example, could embrace his main message as well as those who favour Brandom's own form of pragmatism.Neo-pragmatist philosophers such as Brandom's teacher, Richard Rorty, often present themselves as rejecting the analytic tradition in philosophy. When Brandom describes the ‘pragmatist challenge’ to the ‘classical project of analysis’, he appeals to the criticisms found in the work of Wittgenstein and Sellars that are often appealed to by the critics of the analytic tradition. The message of the new book is that the views he has built on this …. (shrink)

We present a game semantics in the style of Lorenzen for Girard's linear logic . Lorenzen suggested that the meaning of a proposition should be specified by telling how to conduct a debate between a proponent P who asserts and an opponent O who denies . Thus propositions are interpreted as games, connectives as operations on games, and validity as existence of a winning strategy for P. We propose that the connectives of linear logic can be naturally interpreted as the (...) operations on games introduced for entirely different purposes by Blass . We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Gödel's Dialectica interpretation , which was connected to linear logic by de Paiva , fits with game semantics. (shrink)

Such a conception, says Dummett, will form "a base camp for an assault on the metaphysical peaks: I have no greater ambition in this book than to set up a base ...

The initial sections of this text deal with syntactical matters such as logical formalism, cut-elimination, and the embedding of intuitionistic logic in classical linear logic. Concluding chapters focus on proofnets for the multiplicative fragment and the algorithmic interpretation of cut-elimination in proofnets.