In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.

We investigate modal logics of high probability having two unary modal operators: an operator K expressing probabilistic certainty and an operator B expressing probability exceeding a fixed rational threshold c ≥ 1 2 . Identifying knowledge with the former and belief with the latter, we may think of c as the agent’s betting threshold, which leads to the motto “belief is willingness to bet.” The...

A system is classified as multimodal if its language has more than one modal operator as primitive, and such operators are not interdefinable. We extend the anodic and cathodic modal systems, introduced in [BS09a] and [BS10], to a class of the so-called basilar multimodal systems generating, in this way, the classes of anodic and cathodic multimodal logics. The cathodic multimodal systems are d...

We extend propositional Gödel logic by a unary modal operator, which we interpret as 5 Gödel homomorphisms, i.e. functions [0, 1] → [0, 1] that distribute over the interpretations of the 6 binary connectives of Gödel logic. We show that validity in the propositional fragment has a simple 7 superintuitionistic Hilbert-type proof system, which is not structurally complete, and that validity 8 doe...

In a fascinating exposition, Dazeley and Gombocz (1979) study the logical analysis of the verb facere given by St. Anselm of Canterbury in Lambeth Manuscript 59.1 According to St. Anselm, someone slaying a man with a sword is said to facere that the man is dead. Other cases elaborated by St. Anselm include facere non esse, non facere esse, and non facere non esse. From these cases, St. Anselm s...

The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues relate...

The first paper published on Abstract Stone Duality showed that the overt discrete objects (those admitting ∃ and = internally) form a pretopos, i.e. a category with finite limits, stable disjoint coproducts and stable effective quotients of equivalence relations. Using an N-indexed least fixed point axiom, here we show that this full subcategory is an arithmetic universe, having a free semilat...

This paper formalizes the communication of agents with modal operators in arrow logic. A communication between agents consists of an agent's utterance and the other agent's perception, thus, both of the utterance and the perception are regarded as parts of a communication channel between agents. Information is regarded as a propositional content of a sentence. An information channel where infor...

In this paper we introduce DML: Default Modal Logic. DML is a logic endowed with a two-place modal connective that has the intended meaning of \If , then normally ". On top of providing a well-de ned tool for analyzing common default reasoning, DML allows nesting of the default operator. We present a semantic framework in which many of the known default proof systems can be naturally characteri...