Harmonising natural deduction

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented.     

The Weak Normalization of the Simply Typed {lambda}se-calculus

In this paper, we show the weak normalization (WN) of the simply-typeds_e-calculus with open terms where abstractions are decoratedwith types, and metavariables, de Bruijn indices and updatingoperators are decorated with environments. We show a proof ofWN using the _e-calculus, a calculus isomorphic to . This proof is strongly influenced by Goubault-Larrecq'sproof of WN for the -calculus but with subtle differences whichshow that the two styles require different attention. Furthermore,we give a new calculus '_e which works like s_e but which iscloser to than _e. For both _e and '_e we prove WN for typedsemi-open terms (i.e. those which allow term variables but nosubstitution variables), unlike the result of Goubault-Larrecqwhich covered all open terms.     

Predicate Logical Extensions of some Subintuitionistic Logics

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier. Presented by Melvin Fitting     

A Deep Inference System for the Modal Logic S5

We present a cut-admissible system for the modal logic S5 in a formalism that makes explicit and intensive use of deep inference. Deep inference is induced by the methods applied so far in conceptually pure systems for this logic. The system enjoys systematicity and modularity, two important properties that should be satisfied by modal systems. Furthermore, it enjoys a simple and direct design: the rules are few and the modal rules are in exact correspondence to the modal axioms. Presented by Heinrich Wansing     

Parsing Pregroup Grammars and Lambek Calculus Using Partial Composition

The paper presents a way to transform pregroup grammars into contextfree grammars using functional composition. The same technique can also be used for the proof-nets of multiplicative cyclic linear logic and for Lambek calculus allowing empty premises.     

The finite model property for BCI and related systems

We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].Presented by H. Ono     

Eine Auch Sich Selbst Missverstehende Kritik: Über das Reflexionsdefizit Formaler Explikationen

The criticism formulated by L. B. Puntel concerning the theory of dialectic proposed by the author is rejected. Puntel's attempt at explicating predication by means of (second order) predicate logic fails: It misjudges predication being already presupposed for the possibility of predicate logic, thus belonging to the transcendental conditions of formal predicate logic, so that predication itself cannot be further explicated by means of such logic. What is in fact criticized by Puntel is something like an artefact of formalization. The unreflected application of formal logic here generates problems instead of solving them.     

Game Logic and its Applications II

This paper provides a Genzten style formulation of the game logic framework GLm (0 m ), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GL used in Part I.     

Sense and Reference of Predicates: Comments on Frege’s Theory of Sense-Reference

Frege argued that a predicate was a functional expression and the reference of it a concept, which as a predicative function had one or more empty places and was thus incomplete. Frege’s view gives rise to what has been known as the paradox of the concept “horse.” In order to resolve this paradox, I argue for an opposite view which retains the point that a predicate is a function, i.e. that a predicative function is complete in a sense. Specifically speaking, a predicate performing the function of a predicate has at least one empty place and has no reference, while a predicate performing the function of a subject does not have any empty place but does have a reference. Frege not only regarded a concept with one or more empty places as the reference of a predicate but also took a set of objects without any empty place to be the extension of a concept with one or more empty places. Thus, it presents a complex relationship between the reference of a predicate and its corresponding extension, leading to disharmony in his theory. I argue that this is because there is a major defect in Frege’s theory of meaning, namely the neglect of common names. What he called extensions of concepts are actually extensions of common names, and the references of predicates and the extensions of common names have a substantial difference despite being closely related.