We prove that a variant of Robinson arithmetic $\mathsf{Q}$ with nontotal operations is interpretable in the theory of concatenation $\mathsf{TC}$ introduced by A. Grzegorczyk. Since $\mathsf{Q}$ is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether $\mathsf{Q}$ is interpretable in $\mathsf{TC}$. An immediate consequence is essential undecidability of $\mathsf{TC}$.

A modal theory Z using the Guaspari witness comparison signs $\leq, is developed. The theory Z is similar to, but weaker than, the theory R of Guaspari and Solovay. Nevertheless, Z proves the independence of the Rosser fixed-point. A Kripke semantics for Z is presented and some arithmetical interpretations of Z are investigated. Then Z is enriched to ZI by adding a new modality sign for interpretability and by axioms expressing some facts about interpretability of theories. Two arithmetical interpretations of (...) ZI are presented. The proofs of the validity of the axioms of ZI in arithmetical interpretations use some strengthening of Solovay's result about interpretability in Godel-Bernays set theory. (shrink)

A Kripke-style semantics developed by de Jongh and Veltman is used to investigate relations between several extensions of interpretability logic, IL.

The decision problem for provability logic remains PSPACE-complete even if the number of propositional atoms is restricted to one.

We present an alternative, purely semantical and relatively simple, proof of the Statman's result that both intuitionistic propositional logic and its implicational fragment are PSPACE-complete.

Axiomatization of Gödel-Dummett predicate logics S2G, S3G, and PG, where PG is the weakest logic in which all prenex operations are sound, and the relationships of these logics to logics known from the literature are discussed. Examples of non-prenexable formulas are given for those logics where some prenex operation is not available. Inter-expressibility of quantifiers is explored for each of the considered logics.

Each closed (i.e. variable free) formula of interpretability logic is equivalent in ILF to a closed formula of the provability logic G, thus to a Boolean combination of formulas of the form n.

Bilder gehören zu den ältesten Zeugnissen der menschlichen Kultur. Sie sind, soweit bekannt, wesentlich älter als jede Form des sprachlich vermittelten diskursiven Denkens. Bereits dieser historische Umstand markiert die Bilder als etwas Vortheoretisches. Philosophische Theorien über Bilder stützen von Beginn an diese geschichtliche Kontingenz. Die Bildtheorie verdankt Platon die bis in unsere Tage überaus wirksame Denkposition, dass sich Bilder primär an die Sinnlichkeit wenden und nicht an den Verstand. Eine alte Feindschaft - die zwischen Sprache und Bild, dem Diskursiven und (...) Nicht-Diskursiven oder dem Propositionalen und Nicht-Propositionalen - wurde im Laufe der Zeit von disparat bestehenden philosophischen Traditionen erhärtet. Die seit Anfang der 90er Jahre des 20. Jh. s immer intensiver betriebene Bildtheorie konzentriert sich ebenfalls auf das, was Bilder von der Sprache trennt. Im folgenden wird dagegen jener Aspekt hervorgehoben, den Bilder mit der Sprache gemeinsam haben. Bilder sind wie sprachliche Äußerungen in erster Linie äußerst flexible und vielseitig einsetzbare Werkzeuge. Die Frage nach ihrer Bedeutung gleicht der Frage nach ihrem Gebrauch durch die Menschen. (shrink)

Steve Fuller has argued that a scientific discovery will not be recognized unless it can be justified within the history of the relevant science. He cites Mendel's work on genetics, which was not recognized until thirty-five years after its publication, as an example. This essay argues that Mendel's work comes out of the tradition of work by both agricultural breeders and academics in nineteenth century Austria. Thus, Fuller is mistaken, and one must look elsewhere for the neglect of Mendel's work. (...) This essay also places Mendel's work within the philosophy of Jan Comenius, whose work was very likely known to Mendel.--------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- -----------------------------------------------. (shrink)

Efforts to bring science into early 19th century breeding practices in Central Europe, organised from Brno, the Hapsburg city in which Mendel would later turn breeding experiments into a body of timeless theory, are here considered as a significant prelude to the great discovery. During those years prior to Mendel's arrival in Brno, enlightened breeders were seeking ways to regulate the process of heredity, which they viewed as a force to be controlled. Many were specialising in sheep breeding for the (...) benefit of the local wool industry while others were showing an interest in commercial plants, especially fruit trees and vines, and later cereals. Breeders explained their problems in regulating heredity in terms of climatic influences disruption due to crossing sports or saltations. Practical experience led them to the concepts of 'inheritance capacity' and the 'mutual elective affinity' of parents. The former was seen to differ among individuals and also among traits; the latter was proposed as a means of adding strength to heredity. The breeders came to recognise that traits might be hidden and yet transmitted as a 'potential' to future generations. They also grew to understand that heredity would be strengthened when a quality was 'fixed' within a lineage by 'pure blood relations.' Continued selection of the desired quality might then lead to 'a higher perfection.' But the ultimate 'physiological' question about breeding, 'what is inherited and how?,' found no answer. Major figures in this development included Abbot Napp, the one who asked this question and who was due to receive Mendel into the monastery in 1843, and Professor Diebl whose lectures on agriculture and natural science at the Brno Philosophical Institute Mendel would attend in 1846. Here we analyse their progress in theorizing about breeding up until about 1840. In discussing this development, we refer to certain international contacts, especially with respect to information transfer and scientific education, within the wider context of the late Enlightenment. (shrink)

We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions of models of arithmetical theories (in the language ) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms. We construct a model and a substructure with e total and (...) (Presburger arithmetic) such that in both and Fermat's last theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples. On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in (even in a weaker theory) and thus holds in and. Finally, we also show that Fermat's last theorem for e is provable (again, under the assumption of ABC in ) in “coprimality for e”. (shrink)

The difference in formulation of the question of heredity on a different level of knowledge in Brno in the 1830s and after 1850 is discussed in this article. In order to solve the problem the most important source is forshown in the new philosophy of plant physiology and in physics. Mendel was pleased to have met excellent teachers of both these fields. This explanation is an example of Mirko Grmek's thesis: 'l'histoire des sciences est le laboratoire de l'épistomologie'.

Die Philosophie wurde von so unterschiedlichen Philosophen wie Wolff und Russell als Moglichkeitswissenschaft bezeichnet. Doch erwiesen sich die modalen Konzepte von Moglichkeit und Notwendigkeit als sperrig und vieldeutig, und ihr Verhaltnis zum Wirklichkeitsbegriff bleibt problematisch. Die vorliegende Sammlung beleuchtet die Metaphysik und Logik von Moglichkeit und Wirklichkeit aufs Neue und betrachtet sie aus unterschiedlichsten Perspektiven jenseits der Dichotomie von analytischer und kontinentaler Philosophie. Die Philosophiegeschichte (von der griechischen Antike bis zu David Lewis) kommt ebenso zu Wort wie die Semantik moglicher (...) Welten; Logik, Mathematik und Computerwissenschaft ebenso wie Literatur und Neue Medien; Formen des wissenschaftlichen ebenso wie des fiktionalen Diskurses. Philosophy has been called the science of the possible by philosophers as diverse as Christian Wolff and Bertrand Russell. The modal concepts of possibility and necessity, however, have proved to be ambiguous and recalcitrant to analysis, and their relation to the concept of reality have remained problematic up to the present day. Transcending the worn-out dichotomy between analytic and continental philosophy, this collection of papers takes a fresh look at the metaphysics and logic of possibility and reality, and illuminates them from a great variety of perspectives. Topics include the history of philosophy (from Greek antiquity to David Lewis) as well as the semantics of possible worlds; logic, mathematics and computer science as well as literature and the new media; forms of scientific as well as fictional discourse.". (shrink)

Efforts to bring science into early 19th century breeding practices in Central Europe, organised from Brno, the Hapsburg city in which Mendel would later turn breeding experiments into a body of timeless theory, are here considered as a significant prelude to the great discovery. During those years prior to Mendel's arrival in Brno, enlightened breeders were seeking ways to regulate the process of heredity, which they viewed as a force to be controlled. Many were specialising in sheep breeding for the (...) benefit of the local wool industry while others were showing an interest in commercial plants, especially fruit trees and vines, and later cereals. Breeders explained their problems in regulating heredity in terms of (1) climatic influences (2) disruption due to crossing (3) sports or saltations. Practical experience led them to the concepts of 'inheritance capacity' and the 'mutual elective affinity' of parents. The former was seen to differ among individuals and also among traits; the latter was proposed as a means of adding strength to heredity. The breeders came to recognise that traits might be hidden and yet transmitted as a 'potential' to future generations. They also grew to understand that heredity would be strengthened when a quality was 'fixed' within a lineage by 'pure blood relations.' Continued selection of the desired quality might then lead to 'a higher perfection.' But the ultimate 'physiological' question about breeding, 'what is inherited and how?,' found no answer. Major figures in this development included Abbot Napp, the one who asked this question and who was due to receive Mendel into the monastery in 1843, and Professor Diebl whose lectures on agriculture and natural science at the Brno Philosophical Institute Mendel would attend in 1846. Here we analyse their progress in theorizing about breeding up until about 1840. In discussing this development, we refer to certain international contacts, especially with respect to information transfer and scientific education, within the wider context of the late Enlightenment. (shrink)

The persisting controversial interpretation of constant hybrids and of the term Entwicklungsgeschichte, mentioned by Mendel in the Pisum paper, is elucidated in the context of his experiments with other plant species and of the growth of knowledge in scientific animal and plant breeding in Moravia.

In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ1, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by Švejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in (...) the degree structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result. (shrink)