Online research in philosophy

From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a crucial (...) advantage over natural deduction, where substitution is built into the general framework. (shrink)

. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a principle for reasoning (...) on the basis of rule-based atomic definitions, proposed by Hallnäs and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω-principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself. (shrink)

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used in normalization theory. It (...) is argued that these two sorts of concepts must be kept strictly apart. (shrink)

any other one with the False, without contradicting any stipulations previously introduced (we shall call this claim the identiability thesis, following Schroeder-Heister [13]). As far as we are aware, there is no consensus in the literature as to (i) the proper interpretation of the permutation argument and the identiability thesis, (ii) the validity of the permutation argument, and (iii) the truth of the identiability thesis.1 In this paper, we undertake a detailed technical study of the two main lines of interpretation, (...) and gather some evidence for favoring one interpretation over the other. (shrink)

In the 1920s, Paul Hertz (1881-1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is regarded as a part of structural proof theory. Furthermore, it is shown that Gentzen, in his first paper of 1933, which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of (...) prepositional SLD-resolution in logic programming. (shrink)

We reconstruct Frege?s treatment of certain deducibility problems posed by Boole. It turns out that in his formalization and solution of Boole?s problems Frege anticipates the idea of propositional resolution.

Theories in the usual sense, as characterized by a language and a set of theorems in that language ("statement view"), are related to theories in the structuralist sense, in turn characterized by a set of potential models and a subset thereof as models ("non-statement view", J. Sneed, W. Stegmüller). It is shown that reductions of theories in the structuralist sense (that is, functions on structures) give rise to so-called "representations" of theories in the statement sense and vice versa, where representations (...) are understood as functions that map sentences of one theory into another theory. It is argued that commensurability between theories should be based on functions on open formulas and open terms so that reducibility does not necessarily imply commensurability. This is in accordance with a central claim by Stegmüller on the compatibility of reducibility and incommensurability that has recently been challenged by D. Pearce. (shrink)

This paper deals with Popper's little-known work on deductive logic, published between 1947 and 1949. According to his theory of deductive inference, the meaning of logical signs is determined by certain rules derived from ?inferential definitions? of those signs. Although strong arguments have been presented against Popper's claims (e.g. by Curry, Kleene, Lejewski and McKinsey), his theory can be reconstructed when it is viewed primarily as an attempt to demarcate logical from non-logical constants rather than as a semantic foundation for (...) logic. A criterion of logicality is obtained which is based on conjunction, implication and universal quantification as fundamental logical operations. (shrink)

MICHAEL D. RESNIK, Frege and the philosophy of mathematics. Ithaca and London: Cornell University Press, 1980. 244 pp. $16.50. HANS D. SLUGA, Gottlob Frege. London, Boston and Henley: Routledge & Kegan Paul, 1980. xi + 203 pp. £ 12.95.