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 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 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...

Two main lines have been adopted to prove the cut elimination theorem: the syntactic one, that studies the process of reducing cuts, and the semantic one, that consists in interpreting a sequent in some algebra and extracting from this interpretation a cut-free proof of this very sequent. A link between those two methods was exhibited by studying in a semantic way, syntactical tools that allow ...

In this paper, we introduce the notion of fuzzy obstinate ideals in MV -algebras. Some properties of fuzzy obstinateideals are given. Not only we give some characterizations of fuzzy obstinate ideals, but also bring the extension theorem of fuzzy obstinate ideal of an MV -algebra A. We investigate the relationships between fuzzy obstinate ideals and the other fuzzy ideals of an MV -algebra. We ...