Intermediate logics with the same disjunctionless fragment as intuitionistic logic

Given an intermediate prepositional logic L, denote by L ^–d its disjuctionless fragment. We introduce an infinite sequence {J _n}_n1 of propositional formulas, and prove:(1)For any L: L ^–d =I ^–d (I=intuitionistic logic) if and only if J _n L for every n 1.Since it turns out that L{J _{n} n}1 = Ø for any L having the disjunction property, we obtain as a corollary that L ^–d = I ^–d for every L with d.p. (cf. open problem 7.19 of [5]). Algebraic semantic is used in the proof of the if part of (1). In the last section of the paper we provide a characterization in Kripke's semantic for the logics J _n =I+ +J _n (n 1).     

Theories of Types and Names with Positive Stratified Comprehension

We introduce a certain extension of -calculus, and show that it has the Church-Rosser property. The associated open-term extensional combinatory algebra is used as a basis to construct models for theories of Explict Mathematics (formulated in the language of "types and names") with positive stratified comprehension. In such models, types are interpreted as collections of solutions (of terms) w.r. to a set of numerals. Exploiting extensionality, we prove some consistency results for special ontological axioms which are refutable under elementary comprehension.     

Completeness theorems for some intermediate predicate calculi

We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].     

On the extension of intuitionistic propositional logic with Kreisel-Putnam's and Scott's schemes

LetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = ((? ?α→α)→α∨ ?α)→ ?α∨ ??α (Scott), andKP = (?α→β∨γ)→(?α→β)∨(?α→γ) (Kreisel-Putnam). Using Kripke's semantics, we prove:

1. SKP has the finite model property;
2. SKP has the disjunction property.

In the last section of the paper we give some results about Scott's logic S = I+S.     

Interactions of in atoms with partial dislocations cores in GaAs: 0.3% In

Abstract

It has been established that dislocation mobilities in GaAs are reduced by In doping. This reduction operates mainly on defects exhibiting at least one α (As core) partial dislocation. We discuss here the different interactions between In and α partials which can occur. We propose interstitial In incorporated on dangling bonds as responsible for the observed reduction in mobility. The role of temperature and stress is also discussed.