It is known (Rapp, 1985) that elementary geometry with an additional quantifier “there exist uncountably many” is decidable. We show that this decidability helps in solving the following problem from combinatorial geometry: does there exist an uncountable family of pair-wise non-congruent tetrahedra that are n−equidecomposable with a cube? DEFINITION 1. By an elementary geometry, following [Tar...

An extension, QLP, of the propositional logic of explicit proofs, LP, is created, allowing quantification over proofs. The resulting logic is given an axiomatization, a Kripke-style semantics, and soundness and completeness are shown. It is shown that S4 embeds into the logic, when we translate the necessity operator using a quantifier: there exists an explicit proof. And it is shown that the p...

We give a representation of distributive lattices with the existential quantifier in terms of spectral spaces, which is an alternative to Cignoli’s representation in terms of Priestley spaces. Then we describe dual spectral spaces of subdirectly irreducible and simple Q-distributive lattices and prove that the variety QDist of Q-distributive lattices does not have the congruence extension prope...

Ordinary modal logic deals with the notion of a proposition being true in at least one possible world. This makes it natural to consider the notion of a proposition being true in n possible worlds for any nonnegative integer n. Such a notion would stand to Tarski’s numerical quantifiers as ordinary possibility stands to the existential quantifier. In this paper I present several logics for nume...

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defin...

Since Horn (1972) the notion of conversational implicature proposed by Grice has been put to use to explain certain interpretive differences between expressions in natural language and their counterparts in formal logic. For example, the sentences in (1) seem to convey more than they would be expected to if the natural language disjunction or had the same meaning as the logical disjunction∨, or...

An extension, QLP, of the propositional logic of explicit proofs, LP, is created, allowing quantification over proofs. The resulting logic is given an axiomatization, a Kripke-style semantics, and soundness and completeness are shown. It is shown that S4 embeds into the logic, when we translate the necessity operator using a quantifier: there exists an explicit proof. And it is shown that the p...

Let Δ be an n-dimensional simplex in Euclidean space E contained in an n-dimensional closed ball B. The following question is considered. Given any point x ∈ Δ, does there exist a reflection r : E → E in one of the facets of Δ such that r(x) ∈ B? Mathematics Subject Classification (2000). 51M04, 51M16, 51B11.

Recall that a commutative domain R in which every finitely generated ideal is principal is called a Bézout domain. By definition, a noetherian Bézout domain is a principal ideal domain. Several examples of non-noetherian Bézout domains are listed in [1], 243-246. Recall also that a commutative domain R is called an Elementary Divisor domain if, given any matrix A with coefficients in R, there e...