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References [1] D. Propositional relevance through letter-sharing. Journal of Applied Logic, —, Beliefs, belief revision, and splitting languages. Logic, language, and computation, —, The price to be paid is the non-semi-recursiveness of the discussed theories. This entails that the theories are not suitable as absolute foundations for mathematics.

The dynamic proofs for adaptive logics warrant that the theories can nevertheless perfectly function as pragmatic foundations for mathematics. I argue that a pragmatic foundation is all we can hope for. This thesis, named CS, is intuitively wrong. The explanation is that we generally interpret a conditional as putting a connection between its antecedent and its consequent. Illustrated by the Wason selection task, we can also note that this type of conditional allows contra- position.

Which core logic? Joseph Vidal-Rosset Archives H. It is this doctrine that I will argue against in the proposed paper. Decidable many-valued logic for the application in empirical sciences Paul Weingartner University of Salzburg Austria paul.

It contains a modal logic and can be extended with deontic operators. Material implication v. It has a well-known defect: material implicational paradoxes. In order to inherit its merit, overcome its defects, the author proposes mutually inverse implication, which is composed of the mutually inverse processes of inductive composition and decomposition.

This notion is anchored to foundational research in philosophical logic. One example is formal semantics and its research into topics like linguistic meaning, syn- onymy, reference, and expressive power. In particular, we wish to investigate how, relative to a given framework, hyperintensions relate logically to intensions and extensions. This investiga- tion presupposes in turn an investigation into what to make of the notions of intension and extension.

But it is not a matter of course that either intensions or extensions are to be preserved. The overall purpose informing the symposium is to lend further impetus to the accelerated research into hyperintensions characteristic of various quarters of analytic philosophy. While the need for hyperintensionality is widely ap- preciated in logically oriented quarters, this is less so with the more informal portions of analytic philosophy, including philosophy of language and epistemol- ogy.

We see this symposium as an important move in that direction. In the following we provide philosophical motivation for exploring the notion of hyperintensionality and some conceptual background. The enterprise of philosophical logic has been through an extensional and an intensional phase.

The intensional phase was ushered in by the arrival of possible-world semantics, which established itself as a respectable paradigm of formal seman- tics in the early s. Much technical and philosophical mileage has since been had out of this intensional logic. Needless to say, though, the possible-world conception of intensions left Quine and other advocates of a frugal logical ontology unimpressed, since possible worlds were invoked as functional arguments.

Possible worlds were deemed too abstract and obscure. The problem, in essence, is that what makes possible-world semantics into an ex- tensional logic of intensions is its individuation of intensions rather than the validation of the principles of extensional logic.

Explorations into hyperintensional logic have often run in tandem with the re- search paradigm of so-called structured meanings. However, the exact nature of the relationship between hyperintensions and structured meanings remains an open research question, and is one we would be happy to see addressed in our symposium. Therefore, perhaps the fundamental question to raise when it comes to hyperin- tensions is whether it is feasible, or desirable, to aim for an extensional logic of hyperintensions, or whether the ambition to validate the principles of extensional logic also for hyperintensions is either unattainable or misguided.

And what is a hyperintensional context, anyway? Must the adop- tion of such contexts perhaps come with a semantics that alters the semantic properties of terms and expressions?

Should we perhaps adopt a range of hyperintensional criteria of individuation, tailored to particular contexts, rather than privileging one particular individu- ation? A function-based approach is potentially a fruit- ful paradigm of hyperintensional individuation, but ultimately just one among several options.

Finally, a deeper perspective with potentially far-reaching con- sequences is that the quest for hyperintensions may exceed the bounds of model- theoretic semantics and set theory. Of course, it is always a technically feasible option to introduce hyperintensions, of various sorts like hyperpropositions, hyperproperties, etc.

The era of hyperintensionality may conceivably see a revival of structured, or complex or compound, logical entities whereas the received view has tended to be that structure is syntactic structure only. Thus, the relationship between hyperintensions and structured logical objects adverted to above may turn out to run deep.

But the hyperintensional quarters are as yet far from having arrived at anything like a consensus or a dominant view on this topic. Contributions – Algorithmic intensionality and underspecification of verb-phrase ellipsis Roussanka Loukanova Uppsala Sweden rl. This aspect is strictly connected to the ideal of syntactical completeness. For example, propo- sitional classical logic had been considered as the axiomatic theory complete for the class of Boolean-algebras, while propositional intuitionistic logic as the one whose structures correspond to Heyting-algebras.

Also non-classical logics have been described following this idea; a neat example are modal logics, where to every modal axiom corresponds a frame-condition and vice versa 2 cf.

What counts is not the way in which a sentence is de- rived by the axioms, but just that it is derived or not. This is clearly a too idealized image. In particular, two problems emerge. On the other, axioms hide structural properties, such as contraction and weakening, that are obstacles to the reconstruction of proofs and their combina- torial analysis. A solution to these problems is possible if a dynamical character is assigned to the axioms. This implies a radical change of the standard notion of axiom.

This is the solution proposed by S. Negri and J. Axioms are no more considered as the starting point of proofs, but become part of them: they are transformed into inferential rules that can interact with the logical ones.

In this manner the computational properties are maintained and the structural rules are shown to be admissible. First, it is possible to prove independence and decidability results [21], [24]. Secondly, completeness theorems can be proved in a quasi-syntactic way, by constructing a counter- model from the non-terminating branches of a derivation i. Thirdly, it is possible to operate a proof- analysis on axiom-like sentences so to detect which structures are singled out by those sentences.

The standard relationship between axioms and structures is 2. Under the proviso that the frame condition is expressed in second-order logic. Contributed Symposia 71 therefore inverted. A second solution is the so-called deduction modulo, [4]. Its basic idea is to transform axioms into decidable rewrite rules. In this way, the notions of axiom and of axiomatic theory radically change: an axiom becomes a rule for calculating, while an axiomatic theory is no more a coherent set of sentences, but it becomes an algorithm.

Moreover, it is possible to maintain an epistemological distinction between the mathematical part of a proof and the logic-inferential one. By working with the usual logical inferential rules modulo the mathemat- ical rewrite rules the constructive properties of proofs are saved; in particular, normalization is obtained via the construction of a model whose truth values are formed by the sets of proof-terms i.

This means that the semantical objects are built starting from syntactical proof- objects: structures are not something prior to the axioms but are constructed from proofs. A third way is to extend the Curry-Howard paradigm to axiomatic theories.

Roughly speaking, instead of changing the set of inferential rules or enlarging the set of types corresponding to a lambda-term, the idea consists in enriching the notion of proof-term. Thus, the usual lambda-terms are replaced by binary entities, the realizers or processes , formed by a proof-term and a stack. Secondly, each axiom is associated to a particular set of realiz- ers, i. More pre- cisely, the notion of counter-model is replaced by that of counter-proof, which is in turn an extension of the notion of proof.

The completeness theorem then becomes an internal completeness theorem because of this homogeneity between proofs and models: for every sentence i. Ludics is a general framework for studying the notion of interaction between compu- tational processes. Even axioms are no more the point of departure of proofs and loose their pri- macy: in ludics there is an operation the daimon that allows to decide that an arbitrary sentence can close a derivation tree and therefore become an axiom, cf.

An interesting one is linguistics. In particular, a ludics-based approach to linguistics and for- mal grammars permits to model some features that axiom-based theories are 3. Philosophical Logic not able to do in a satisfactory way e. To sum up, these three solutions seems to impose a radical change on the traditional notion of axiom when constructive or dynamical properties of proofs are taken as crucial aims.

A fundamental point is to understand if this change is really indispensable or if it is possible to conceive alternative proposals that allow the coexistence of the traditional notion of axiom with constructive and dynamical approaches to mathematics. Following the line of this summary, another aim of the symposium is to investigate the evolution of the notion of axiom in relation to other notions be- longing to logical practice, as those of semantics and of proof.

Correspondence theory. Gabbay D. E Guenthner, editor, Handbook of Philosophical Logic. Reidel, Dordrecht, David Hilbert. In Encyclopedia of Philosophy, volume 3, pages — MacMillan, New York, Hermann, Paris, Theorem proving modulo.

Journal of Automated Reasoning, —72, Truth values algebras and proof normalization. In Types for Proofs and Programs, pages — Lecture Notes in Computer Science, Fleury, M. Quatrini, and S. Dialogues in ludics. In Quatrini M. Pogodalla, S. Locus Solum. Mathematical Structures in Computer Science, —, From foundations to ludics.

The Bulletin of Symbolic Logic, —, Hilbert and logic. Kluwer, Dordrecht, Geometry and empirical science. American Mathematical Monthly, —17, Contributed Symposia 73 [12] D. Logische Principien des mathematischen Denkens. What is the axiomatic method? Synthese, published online:1— 17, Typed lambda-calculus in classical Zermelo-Fraenkel set theory. Archive for Mathematical Logic, —, Theoretical Com- puter Science, —, Realizability in classical logic.

Lecomte, M. Quatrini, and M. Ludics and anti-realism. North-Holland Publishing Co. Classical realizability with forcing and the axiom of countable choice. Kripke completeness revisited. Primiero and S. Co, London, Cut elimination in presence of axioms. Springer, Berlin, Annals of Pure and Applied Logic, —, Videnskapsselskapet Skrifter, I. Matematisk- naturvidenskabelig Klasse, —36, In other words, what would happen of the ar- guments put forth by anti-realists in support of their intuitionistic revision of classical logic?

A good question is where exactly is the place of the antirealist approach to meaning in the dialogical framework and how does it relate to the dialogical approach to intuitionistic logic. The second topic is about argumentation and ontological commitments.

With our third topic, we will explore the relations between argumentation and Law. For example, we can cite [3]. A good overview can be found in [2] 3.

Dubucs and B. College Publications, To appear. Dialogical logic. In Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Dialogue foundations: Dialogue logic revisited. Logik und Agon. Pac versus demal. In Logic of Knowledge. Theory and Applications. Game semantics and the manifestation thesis.

In Anti Realism. Springer, To appear. Namely, given a theoretical problem with which both approaches are expected to deal, does one of the two have more to say than the other?

Among many possibilities, we choose to engage the theme of semantic para- doxes. Paradoxes have accompanied the development of modern logic—and of the semantic theories built thereon—ever since its birth. In the inferentialist case, one could say that a paradox tied to such semantic picture arose even before the formulation of the semantics itself.

Philosophical Logic agreement on which constraints, if any, can be imposed on a given set of rules. The situation has indeed become worse due to the incredible development of proof-theoretic means of characterizing logical systems. The result is that the attempt at characterizing what is meaningful risks of becoming a formalism-dependent issue. Either more implicitly or in a more explicit manner, in every type of formalism it is possible to trace some parameters governing structural properties of the deducibility relation.

The degree of freedom in tuning such pa- rameters yielded the characterization of a wide variety of sub-structural logics. Most of them focused on restrictions that a semantic theory should satisfy in order to avoid paradoxical consequences, typical examples of which are the restrictions on the possibility of fully displaying self-reference.

More recently, an alternative approach to paradoxes has been developed, aiming at an analysis of the principles needed to actually infer the paradoxical consequences from the semantic theory [14], [3],[11], [8], [1]. The pursue of this strategy has awaken the inferentialist interest in these more traditional semantic paradoxes. For, it turned out that in order to get paradoxes at all, crucial is the appeal to structural principles governing deducibility.

In his talk, Julien Murzi, will discuss recent views that focus on the role played by contraction in formalizing paradoxes. According to Schroeder-Heister [15, 16], one of the dogmas of standard se- mantics is the idea that the notion of logical consequence, which underlies the one of the validity of an argument, should be analyzed in terms of transmission, or preservation, of a more basic notion truth, provability. In particular, he argues that, by relaxing one of its fundamental assumptions—namely, that valid arguments must always normalize—an account of at least certain types of se- mantic paradoxes could be given.

Rather, one needs to weaken the structural rules of the logic, i. In particular, Murzi will suggests that a natural way of revisionary logicians to cope with the validity paradoxes, and with the semantic paradoxes more generally, is to weaken the structural rule of contraction. In his contribution, Murzi will raise some worries about contraction-free logics. The talks will approach the theme from distinct perspectives: as a result, we hope that the complexity of the problems at stake could be best appreciated and that unexpected convergences may possibly be traced.

Spandrels of Truth. Oxford University Press, Tonk, plonk and plink. Universal Logic. Truth and Other Enigmas. Duckworth, London, Truth and Other Enigmas, chapter Realism, pages — Duckworth, The Logical Basis of Metaphysics. Harvard University Press, Harvard, Saving Truth from Paradox. Oxford University Press, Oxford, Theoria, —40, Philosophical Logic [10] D.

Dummett on a theory of meaning and its impact on logic. Taylor, editor, Michael Dummett: contributions to philosophy. In Contradiction. The runabout inference-ticket. Analysis, 21 2 —39, Harmony and autonomy in classical logic.

Some remarks on extending and interpreting theories,with a partial predicate for truth. Hypothetical reasoning: A critique of dummettprawitz-style proof-theoretic semantics. College Publications, The categorical and the hypothetical: A critique of some fundamental assumptions of standard semantics. Palmgren, and D. Springer, Forthcoming.

Proof and paradox. Paradox without self-reference. Analysis, 55 3 —, Proposi- tional equality is interpreted as homotopy and type isomorphism as homotopy equivalence. Logical constructions in type theory then correspond to homotopy- invariant constructions on spaces, while theorems and even proofs in the logical system inherit a homotopical meaning. In parallel, Vladimir Voevodsky IAS has recently proposed a comprehen- sive, computational foundation for mathematics based on this homotopical in- terpretation of type theory.

The program is currently being implemented with the help of the automated proof assistant Coq. This talk will survey some of these recent developments. Berger swansea. Applying program extraction to proofs in this setting leads to represen- tations of real numbers and continuous real functions by non-wellfounded trees, and to implementations of new algorithms in exact real arithmetic.

It is well known that, since its origin 75 years ago , computation theory is intimately related to logic. This is particularly true for computational complexity, a theory born in the sixtieth and seventieth that tries to classify problems according to the machine resources—time and space, mainly—that their computations require.

In fact, the most important classes of computational complexity have descriptive character- izations. In this talk, I shall argue that not only the main classical complexity classes polynomial time, polynomial space, logarithmic space, etc. Typically, for sequential resp. The uniformization problem asks, given a class C of relations and a class F of functions, whether each relation in C can be uniformized in F. We present classical results and recent developments concerning the existence and computability of uniformizations in various settings.

Zucker [3] and Pottinger [2] on a question raised in by Kreisel [1]. References [1] Georg Kreisel. A survey of proof theory ii. Normalisation as a homomorphic image of cut-elimination. Annals of Mathematical Logic, —, The correspondence between cut-elimination and normalisation. We estimate also the complexity of complete extensions of the arithmetic with addition and multiplication.

The research of the third author was partially supported by grant No N of the Polish Ministry of Science and Higher Education. Primiero UGent. To unfold these problems, we present the role of modalities in type theories and show a procedural semantics where these notions are treated in terms of input accessibility via code mobility rules. We get these new proofs of old results from translating some statements and techniques of computability theory into the study of logical theories.

Constructive Analysis, volume of Grundlehren der mathematischen Wissenschaften. Non-standard analysis. North-Holland Publishing Co, Amsterdam, Journal of Sym- bolic Logic, Inpress, A tale of three Reverse Mathematics.

Submitted, Reverse mathematics and non-standard analysis; a treasure trove for the philosophy of science. Keio University Press, Subsystems of second order arithmetic. Perspectives in Mathematical Logic. Recursively enumerable sets and degrees. We consider this to be an ideal opportunity for addressing one of the most recent and challenging subjects in the area of logic: the link between proof systems and computer science. The origins of this connection go back to the study of proof systems in constructive logics and their notion of mathematical proof.

In this purely the- oretical setting, the socalled CurryHoward isomorphism, establishing the iden- tity between proofs and programs, represents the starting point for an extremely fruitful research area. The list can be easily extended. The most striking aspect of this evolution is certainly that proof systems have been shown to be central not only as logical tools, but also as the privileged way to study properties of programs in the area of computer science.

As the latter represents the conceptual basis of many new technologies, it seems important to address the common background with formal logic and especially the prospects for future research. Most recently, and especially from the community of computer scientists, a novel interest has arose for the mathemat- ical foundations of the semantics of programming languages and in general for the study of formal systems, calculi, proof and type systems that help design- ing models of computation for the most new technologies, such as distributed networks, grid computing, security systems.

This is especially true in view of the connections of the same theoretical problems with other applicational as- pects, such as those induced by modal and epistemic logics.

Trying to address these topics is a crucial aim for both involved communities, the one originating from philosophical logic and that of computer scientists, for their evolution and reciprocal comprehension.

Pfenning and H. Kremer, and L. More precisely, we have a double aim: on the one hand, we want to show some of the latest results in new developping logics; on the other hand, we want to focus on a deep proof- theoretical analysis of more standard logics.

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