Pierluigi Minari

Intermediate Predicate Logics Determined by Ordinals

For each ordinal α>0, L(α) is the intermediate predicate logic characterized by the class of all Kripke frames with the poset α and with constant domain. This paper is devoted to a study of logics of the form L(α). It will be shown that for each uncountable ordinal of the form α+η with a finite or a countable η(>0), there exists a countable ordinal of the form β+η such that L(α+η)=L(β+η). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(α) if α is an uncountable regular ordinal. It will be proved that the mapping L is injective if it is restricted to ordinals less than ω ω

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

AbstractLetSKP be the intermediate prepositional logic obtained by adding toI (intuitionistic p.l.) the axiom schemes:S = ((ℸ ℸα→α)→α∨ ℸα)→ ℸα∨ ℸℸα (Scott), andKP = (ℸα→β∨γ)→(ℸα→β)∨(ℸα→γ) (Kreisel-Putnam). Using Kripkes 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 Scotts logic S = I+S.

Completeness theorems for some intermediate predicate calculi

AbstractWe give completeness results — with respect to Kripkes semantic — for the negation-free intermediate predicate calculi:(1)

A solution to Curry and Hindley’s problem on combinatory strong reduction

\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}

Analytic proof systems for λ-calculus: the elimination of transitivity, and why it matters

(2)

Husserl and Boole

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)