Wagner de Campos Sanz

Failure of Completeness in Proof-Theoretic Semantics

Several proof-theoretic notions of validity have been proposed in the literature, for which completeness of intuitionistic logic has been conjectured. We define validity for intuitionistic propositional logic in a way which is common to many of these notions, emphasizing that an appropriate notion of validity must be closed under substitution. In this definition we consider atomic systems whose rules are not only production rules, but may include rules that allow one to discharge assumptions. Our central result shows that Harrop’s rule is valid under substitution, which refutes the completeness conjecture for intuitionistic logic.

Constructive semantics, admissibility of rules and the validity of Peirce's law

In his approach to proof-theoretic semantics, Sandqvist claims to provide a justification of classical logic without using the principle of bivalence. Following ideas by Prawitz, his semantics relies on the idea that logical systems extend atomic systems, so-called “bases”, with respect to which the validity of logically complex formulas is defined. We relate this approach to admissibility-based semantics and show that the latter significantly differs from the former. We also relate it to semantics based on the notion of construction, in which case the results obtained are essentially the same as Sandqvist’s. We argue that the form of rules admitted in atomic bases determines which logical rules are validated, as is the fact of whether bases are conceived as information states, which can be monotonely extended, or as non-extensible inductive definitions. This shows that the format of atomic bases is a highly relevant issue in proof-theoretic semantics.

INVERSION BY DEFINITIONAL REFLECTION AND THE ADMISSIBILITY OF LOGICAL RULES

at T¨ ubingen Abstract. The inversion principle for logical rules expresses a relationship between introduction and elimination rules for logical constants. Hallnand Schroeder-Heister (1990/91) proposed the principle of definitional reflection, which embodies basic ideas of inversion in the more general context of clausal definitions. For the context of admissibility statements, this has been further elaborated by Schroeder-Heister (2007). Using the framework of definitional reflection and its admis- sibility interpretation, we show that, in the sequent calculus of minimal propositional logic, the left introduction rules are admissible when the right introduction rules are taken as the definitions of the logical constants and vice versa. This generalises the well-known relationship between introduction and elimination rules in natural deduction to the framework of the sequent calculus.

A New Normalization Strategy for the Implicational Fragment of Classical Propositional Logic

The introduction and elimination rules for material implication in natural deduction are not complete with respect to the implicational fragment of classical logic. A natural way to complete the system is through the addition of a new natural deduction rule corresponding to Peirce’s formula (((A → B) → A) → A). E. Zimmermann [6] has shown how to extend Prawitz’ normalization strategy to Peirce’s rule: applications of Peirce’s rule can be restricted to atomic conclusions. The aim of the present paper is to extend Seldin’s normalization strategy to Peirce’s rule by showing that every derivation Π in the implicational fragment can be transformed into a derivation Π′ such that no application of Peirce’s rule in Π′ occurs above applications of →-introduction and →-elimination. As a corollary of Seldin’s normalization strategy we obtain a form of Glivenko’s theorem for the classical {→}-fragment.

On Dummett’s verificationist justification procedure

We examine the proof-theoretic verificationist justification procedure proposed by Dummett (1991). After some scrutiny, two distinct interpretations with respect to bases are advanced: the independent and the dependent interpretation. We argue that both are unacceptable as a semantics for propositional intuitionistic logic.

A Note on Gentzen's LJ and NJ Systems Isomorphism

In this paper we are going to examine intuitionistic sequent calculus and its negation rules. We state new negation rules defining, in this way, a new sequent system. It will be used to clarify Gentzens NJ and LJ systems isomorphism. These new negation rules are a direct reading of new natural deduction negation rules obtained by a slight modification of NJ rules. We also show that the new system is equivalent to LJ and that the Hauptsatz holds for it.