We treat the smallest normal modal propositional logic with two modal operators 2 and 2+. While 2 is interpreted in Kripke models by the accessibility relation R, 2+ is interpreted by the transitive closure of R. Intuitively the formula 2+φ means the infinite conjunction 2φ ∧ 22φ ∧ 222φ ∧ · · · . There is a Hilbert style axiomatization of this logic (a characteristic axiom is 2φ ∧ 2+(φ → 2φ) → ...

second-language (l2) acquisition is generally thought to be constrained by maturational factors that circumscribe a critical period for native-like attainment. consistent with the maturational view are age effects among learners who begin l2 acquisition prior to, but not after, closure of the putative critical period. some studies indicate that native-like acquisition of a second language by le...

Unlike knowledge representation in modal logic, human tends to introduce new modalities on ad hoc basis in communication. Similar challenges arise from applications of distributed computing and hybrid systems that involve several time domains. This paper presents a parameterized approach for algebraic manipulation of modalities. Each pair of modalities of possibility and necessity is determined...

In 1981, Paris and Wilkie raised the open question about whether and to what extent the axiom system IΣ0 did satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of IΣ0 appearing in the standard textbooks. However, there was an alter...

Veblen has given a set of axioms for geometry.!. The first eight of these are as follows : % Axiom I. There exist at least two distinct points. Axiom II. If points A, B, C are in the order ABC, they are in the order CBA. Axiom III. If points A, B, C are in the order ABC, they are not in the order BCA. Axiom IV. If points A, B, C are in the order ABC, then A is distinct from C. Axiom V. If A and...

we introduce some new concepts of topological spaces which say α − separable topologicalspace and o-topological group, α − first axiom, α − second axiom, and we find some relations betweenthem with some applications in normed spaces.

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that ...