We extend to the exponential connectives of linear logic the study initiated in Bucciarelli and Ehrhard 247). We define an indexed version of propositional linear logic and provide a sequent calculus for this system. To a formula A of indexed linear logic, we associate an underlying formula of linear logic, and a family A of elements of , the interpretation of in the category of sets and relations. Then A is provable in indexed linear logic iff the family A (...) is contained in the interpretation of some proof of . We extend to this setting the product phase semantics of indexed multiplicative additive linear logic introduced in Bucciarelli and Ehrhard , defining the symmetric product phase spaces. We prove a soundness result for this truth-value semantics and show how a denotational model of linear logic can be associated to any symmetric product phase space. Considering a particular symmetric product phase space, we obtain a new coherence space model of linear logic, which is non-uniform in the sense that the interpretation of a proof of !A−B contains informations about the behavior of this proof when applied to “chimeric” arguments of type A . In this coherence semantics, an element of a web can be strictly coherent with itself, or two distinct elements can be “neutral”. (shrink)

We relate two sequential models of PCF: the sequential algorithm model due to Berry and Curien and the strongly stable model due to Bucciarelli and the author. More precisely, we show that all the morphisms araising in the strongly stable model of PCF are sequential in the sense that they are the “extensional projections” of some sequential algorithms. We define a model of PCF where morphisms are “extensional” sequential algorithms and prove that any equation between PCF terms which holds in (...) this model also holds in the strongly stable model. (shrink)

Linear Logic is a branch of proof theory which provides refined tools for the study of the computational aspects of proofs. These tools include a duality-based categorical semantics, an intrinsic graphical representation of proofs, the introduction of well-behaved non-commutative logical connectives, and the concepts of polarity and focalisation. These various aspects are illustrated here through introductory tutorials as well as more specialised contributions, with a particular emphasis on applications to computer science: denotational semantics, lambda-calculus, logic programming and concurrency theory. The (...) volume is rounded-off by two invited contributions on new topics rooted in recent developments of linear logic. The book derives from a summer school that was the climax of the EU Training and Mobility of Researchers project 'Linear Logic in Computer Science'. It is an excellent introduction to some of the most active research topics in the area. (shrink)

In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to define new denotational models of linear logic using symmetric product phase models of indexed linear logic. We present here a strict extension of indexed linear logic for which symmetric product phase spaces provide a complete semantics. We study the connection between this new system and indexed linear logic.

Presents for the first time in English, this 1967 text which came to be known as the 'bible' of a new movement in German philosophy of language, the 'Erlanger School.' This school of linguistic philosophy's treatment of language is rooted in the tradition of transcendentalism, and bases its system on Kant and his Continental successors. For the Erlanger School, 'language is not just a fact we discover...but a human cultural accomplishment whose construction reason can and should be controlled.' An influential (...) work which will be a valuable resource for students of philosophy, particularly philosophy of language and transcendentalism. (shrink)

We study the notion of logical relation in the coherence space semantics of multiplicative-additive linear logic . We show that, when the ground-type logical relation is “closed under restrictions”, the logical relation associated to any type can be seen as a map associating facts of a phase space to families of points of the web of the corresponding coherence space. We introduce a sequent calculus extension of whose formulae denote these families of points. This logic admits a truth-value semantics in (...) the previously mentioned phase space, and this truth-value semantics faithfully describes the logical relation model we started from. Then we generalize this notion of phase space, we prove a truth-value completeness result for and we derive from any phase model of a denotational model for . Using the truth-value completeness result, we obtain a weak denotational completeness result based on this new denotational semantics. (shrink)

Modern Philosophy of Science has not yet recognized the significance of physical invariance principles for science and daily life. In this paper we investigate as a simple example, how time independence or time translational invariance of natural laws determines the time scale. We start with informal definitions of the invariance of concepts and laws. We then ask if time independence is an essential feature of natural laws or if time dependent laws are plausible without loss of predictive relevance. In section (...) 3 we investigate the question wether time independence of natural laws is a matter of convention. In sections 5-7 the determination of the time scale by time translational invariance is studied. In section 8 we study the conditions under which this determination is unique. (shrink)

Summary In the last decade analytical philosophy of science has been considered by many people as a descriptive activity. In part I of this paper we show that philosophy of science has been designed as normative logical analysis by Reichenbach and Carnap before world war II. Thus the identification analytical = descriptive is historically unjustified. In part II we discuss three tasks of analytical philosophy of science, the logical reconstruction of concepts, theories, and methods. While the first is mainly descriptive, (...) the third mainly normative, the second may be both, depending on the intended goal. (shrink)

The concept of necessity plays a central role in Kant's philosophy, but seems to lead to severe paradoxes. On the one hand he states: ‘Notwendigkeit und strenge Allgemeinheit sind sichere Kennzeichen einer Erkenntnis a priori’. On the other hand he talks also about ‘notwendig (d. i. nach einer Regel)’, which means ‘necessary according to the empirical natural laws’. However, he never states explicitly the distinction between these two different concepts of necessity. Either Kant's philosophy is inconsistent or we have to (...) assume that he indeed interchanges two different concepts, even if he does not tell us that there is a difference between synthetic apriori necessity and natural necessity. This paper defends the hypothesis of two different necessities and then shows how this strategy makes possible a rational reconstruction of Kant's epistemology. (shrink)

These arguments are fairly well known today. It is interesting to note that v. Kries already knew them, and that they have been ignored by Reichenbach and v. Mises in their original account of probability.2This observation leads to the interesting question why the frequency theory of probability has been adopted by many people in our century in spite of severe counterarguments. One may think of a change in scientific attitude, of a scientific revolution put forward by Feyerabendarian propaganda- and who (...) would deny that Reichenbach was an excellent propagandist!My suggested explanation is the following:J. v. Kries is still a mentalist in the tradition of Descartes and Locke. This can be shown very well from his logic, which he wrote even much later (1916). A mentalist does not pose semantical questions in a proper sense: he rather asks which ideas in the human mind make up a concept. And studying probability in this way, he will find in any case that what is in the human mind is first of all an expectation. Therefore, his analysis of the concept of probability will be centered around the concept of expectation.People tended to approach the problem in quite a different way after the scientific revolution which led from mentalism to lingualism. They tended in many cases to use operationalism and to ask how probabilities are determined in science, if they wanted to clear up the meaning of probability. This new aspect of probability, which was caused by looking at it from a new position or standpoint, suggested it to be the limit of relative frequencies in the first instance. This suggestion was so strong that all objections were ignored, and all counterarguments encountered deaf ears.Of course I have used here an oversimplification. Neither was J. v. Kries untouched by Poincaré's conventionalism, which is essentially a movement leading to the breakdown of mentalism; nor was Reichenbach, when he wrote his thesis in 1915, already a lingualist in any respect.Nevertheless, I think that the lingualist revolution, which took place particularly in mathematics and physics, has also changed the attitude towards probability. It would be interesting to corroborate this thesis by extensive study of the history of science in the late 19th century.Besides this historical aspect of v. Kries' theory, there is a systematic one which should be noticed. Though his theory still has a mentalist outlook, many of v. Kries' statements can be translated into the present lingualist idiom and then become most interesting contributions to the present discussion. We shall then state that for v. Kries probabilities appear first of all in probabilistic hypotheses which may or may not be confirmed by empirical data. Therefore, they have a role similar to theoretical concepts.3But that is not the only predominant feature of v. Kries' theory. He is approaching objective probability in a way which is quite unfamiliar to present day philosophers or mathematicians. Objective probabilities are connected with other concepts in three different ways. First of all, probabilistic hypotheses are confirmed by empirically found relative frequencies in a finite series of events. Secondly, they serve as an aid for decisions. And finally, there is a third relationship. Probabilities have to be explained by theories which are in many cases not formulated in probabilistic terms themselves. We can try to understand probability in one of the first two ways via their connection to other concepts. That is what is usually done by interpretations in terms of frequencies or decisions. A third approach, however, is to understand the nature of probability by the way probabilities are explained, and that is what v. Kries does. Such an approach does not seem unusual at all. In many cases the nature of a kind of objects is characterized by giving an explanation. If somebody asks what an eclipse of the moon is, we shall answer - as Aristotle already proposed - by giving an explanation of it. Thus it is quite natural to answer a question as to the nature of probability by describing the general type of explanation of probability distributions in natural or social science. The answer given by v. Kries is - as we know today — of limited validity. Quantum mechanical probabilities are not explainable by Spielräume. V. Kries solution, however, is remarkable in one important respect. It accounts for the time direction of objective probabilities, which are always predictive or forward probabilities, while retrodictive or backward probabilities are always subjective. The relation of objective probabilities to time direction is surely of utmost importance for natural philosophy. I do not know of any other philosophical foundation of probability which takes into account this deep-rooted relation. Therefore, v. Kries' interpretation may help us to understand one of the most intricate puzzles of philosophy. It may be nearly one hundred years old, but is by no means out of date. (shrink)

Hans Reichenbach's 1928 thesis of the relativity of geometry has been misunderstood as the statement that the geometrical structure of space can be described in different languages. In this interpretation the thesis becomes an instance of trivial semantical conventionalism, as Grünbaum calls it. To understand Reichenbach correctly, we have to interpret it in the light of the linguistic turn, the transition from thought oriented philosophy to language oriented philosophy, which mainly took place in the first decades of our century. Reichenbach (...) — as Poincaré before him — is undermining the prejudice of thought oriented philosophy, that two propositions have different factual content, if they are associated with different ideas in our mind. Thus Reichenbach prepared the change to language oriented philosophy, which he also accepted later. (shrink)

Summary Some leading ideas of the constructivist protophysics are discussed on the basis of P. Janich's Protophysik der Zeit. After having reviewed the contents of the second edition Janich's claim that analytical philosophy of science is purely affirmative and not critical towards science in its historical appearence is refuted. In the next section the principles of constructivist methodology of physics are criticised, and the claim is refuted that prescriptions for measurement cannot without circularity be shown to be invalid by experimental (...) results. In the last section we discuss Janich's foundation of time measurement in a mathematical frame similar to that for extensive systems. Thus the relation between Janich's and the ordinary account is cleared up. (shrink)