We investigate modal logical aspects of provability predicates PrT(x) satisfying the following condition: M: If T⊢φ→ψ, then T⊢PrT(⌜φ⌝)→PrT(⌜ψ⌝). prove arithmetical completeness theorems for monotonic logics MN, MN4, MNP, MNP4, and MND with respect to condition M. That is, we that each logic L them, there exists a Σ1 predicate M such is exactly L. In particular, formulas P: ¬□⊥ D: ¬(□A∧□¬A) are ...

Well-posedness of constrained minimization problems via saddle-points BIAGIO RICCERI Dedicated to Professor Jean Saint Raymond on his sixtieth birthday, with my greatest admiration and esteem Here and in the sequel, X is a Hausdorff topological space, J, Φ are two real-valued functions defined in X, and a, b are two numbers in [−∞, +∞], with a < b.

The unification problem in a propositional logic is to determine, given formula φ, whether there exists substitution σ such that σ(φ) logic. In case, unifier of φ. When unifiable has minimal complete sets unifiers, it either infinitary, finitary, or unitary, depending on the cardinality its unifiers. Otherwise, nullary. this paper, we prove modal $\mathbf {K}+\square \square \bot $ , formulas a...