We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a s...

We prove that the lattice of normal forms over an arbitrary set, introduced in [12], is an implicative lattice. The relative pseudo-complement α ⇒ β is defined as α 1 ∪α 2 =α −α 1 α 2 β, where −α is the pseudo-complement of α and α β is a rather strong implication introduced in this paper. [1] provide the notation and terminology for this paper. The following proposition is true (1) Let A, B, C...

In the paper we introduce logical calculi for representation of propositions with modal operators indexed by fuzzy values. There calculi are called Heyting-valued modal logics. We introduce the concept of a Heyting-valued Kripke model and consider a semantics of Heyting-valued modal logics at the class of Heyting-valued Kripke models.

A review is given of recent work aimed at constructing a quantum theory of cosmology in which all observables refer to information measurable by observers inside the universe. At the classical level the algebra of observables should be modified to take into account the fact that observers can only give truth values to observables that have to do with their backwards light cone. The resulting al...

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a s...

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice D(e,R), which is a sublattice of the lattice L of all...

We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : : :) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula there exists ...

We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : : :) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula there exists ...

We consider the families L of propositional superintuitionistic logics (s.i.l.) andNE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig’s int...