Jan von Plato

Natural deduction with general elimination rules

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Abstract. The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Through the condition that in a cut-free derivation of the sequent Γ⇒C, no inactive weakening or contraction formulas remain in Γ, a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of Γ become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only.

Cut elimination in the presence of axioms

Away is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic with equality inwhich also cuts on the equality axioms are eliminated. §

The axioms of constructive geometry

Abstract Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions to geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry.

Gentzen's Proof of Normalization for Natural Deduction

Gentzen writes in the published version of his doctoral thesis Untersuchungen uber das logische Schliessen (Investigations into logical reasoning) that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio. We found in February 2005 an early handwritten version of Gentzens thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs . 974: 271. Bernays must have gotten it well before the time of his being expelled from Gottingen on the basis of the racial laws in April 1933.

A proof of Gentzen's Hauptsatz without multicut

Gentzens original proof of the Hauptsatz used a rule of multicut in the case that the right premiss of cut was derived by contraction. Cut elimination is here proved without multicut, by transforming suitably the derivation of the premiss of the contraction.Abstract. Gentzens original proof of the Hauptsatz used a rule of multicut in the case that the right premiss of cut was derived by contraction. Cut elimination is here proved without multicut, by transforming suitably the derivation of the premiss of the contraction.

Ergodic Theory and the Foundations of Probability

In the following, I shall illustrate through examples how useful the notions and results of ergodic theory are for the foundations of probability. Ergodic theory can be discussed from many points of view. It originated with the development of statistical mechanics in classical physics. Later, an abstract formulation of ergodic theory became part of measure theory. The latter treats probabilities as special kinds of measures. Results of ergodic theory can be intuitively explained in probabilistic terms, without special recourse to their physical significance or to the uninterpreted, abstract mathematical formulation. In discussing the relevance of ergodic theory for the foundations of probability, the physical background is significant, however. Most of the notions of the theory are motivated from statistical physics, so that, proceeding in the other direction as it were, an interpretation of the results of ergodic theory can be given in physical terms. It should be noted that there are different possibilities to hand here. Some writers consider the physical application a special case of a priori probabilities.1 Against this view, in which probabilities only relate to ignorance, stand the attempts at justifying probabilistic assumptions from a physical basis.

Proof Theory, History and Philosophical Significance

Preface. Contributing Authors. Introduction. Part 1: Review of Proof Theory. Highlights in Proof Theory S. Feferman. Part 2: The Background of Hilberts Proof Theory. The Empiricist Roots of Hilberts Axiomatic Approach L. Corry. The Calm Before the Storm: Hilberts Early Views on Foundations D. Rowe. Toward Finitist Proof Theory W. Sieg. Part 3: Brouwer and Weyl on Proof Theory and Philosophy of Mathematics. The Development of Brouwers Intuitionism D. van Dalen. Did Brouwers Intuitionistic Analysis Satisfy its own Epistemological Standards? M. Epple. The Significance of Weyls Das Kontinuum S. Feferman. Herman Weyl on the Concept of Continuum E. Scholz. Part 4: Modern Views and Results from Proof Theory. Relationships between Constructive, Predicative and Classical Systems of Analysis S. Feferman. Index.

Sequent calculus in natural deduction style

A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. Therefore it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction. ?

Probability and Determinism

This paper discusses different interpretations of probability in relation to determinism. It is argued that both objective and subjective views on probability can be compatible with deterministic as well as indeterministic situations. The possibility of a conceptual independence between probability and determinism is argued to hold on a general level. The subsequent philosophical analysis of recent advances in classical statistical mechanics (ergodic theory) is of independent interest, but also adds weight to the claim that it is possible to justify an objective interpretation of probabilities in a theory having as a basis the paradigmatically deterministic theory of classical mechanics.This paper discusses different interpretations of probability in relation to determinism. It is argued that both objective and subjective views on probability can be compatible with deterministic as well as indeterministic situations. The possibility of a conceptual independence between probability and determinism is argued to hold on a general level. The subsequent philosophical analysis of recent advances in classical statistical mechanics (ergodic theory) is of independent interest, but also adds weight to the claim that it is possible to justify an objective interpretation of probabilities in a theory having as a basis the paradigmatically deterministic theory of classical mechanics.

A Problem of Normal Form in Natural Deduction

Recently Ekman gave a derivation in natural deduction such that it either contains a substantial redundant part or else is not normal. It is shown that this problem is caused by a non-normality inherent in the usual modus ponens rule.