A strict implication calculus for compact Hausdorff spaces

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.

Outstanding Contributions to Logic, 2014

In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, T1-spaces, dense-in-themselves spaces, a zerodimensional dense-in-itself separable metric space, R n (n ≥ 2). We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.

Studia Logica, 1977

Post Mgebras were introduced for the s time by l~osenbloom [6] and investigated after that by many autors (for the full bibliography and historical remarks see and ). These Mgcbras play such a role for the m-vMued logics of E. Post as Boolean algebras for the classical logic. The first standaxd systems (in the sense of YCasiowa [4]) of m-valued pro-positionM calculi, complete with respect to the class of Post algebras, were constructed by Rousseau [7], . The logic con'esponding to these calculi was called by Rousseau ch~ssical many-vMued logic. In [7] and [8] l~ousseau introduced also the notions of intuitionistic many-vMued logic and pseudo-Post algebras as a semantic basis of this logic.

2010

Order, 2011

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axiomatization of the intermediate logic of each metrizable Stone space and of each extremally disconnected Stone space. In particular, we obtain that the intuitionistic logic is the logic of the Pelczynski compactification of the natural numbers and the logic of weak excluded middle is the logic of the Gleason cover of the Cantor space.

Lecture Notes in Computer Science, 2007

In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by and . We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra.

Advances in Modal Logic, 2022

De Vries duality generalizes Stone duality between Boolean algebras and Stone spaces to a duality between de Vries algebras (complete Boolean algebras equipped with a subordination relation satisfying some axioms) and compact Hausdorff spaces. This duality allows for an algebraic approach to region-based theories of space that differs from point-free topology. Building on the recent choice-free version of Stone duality developed by Bezhanishvili and Holliday, this paper establishes a choice-free duality between de Vries algebras and a category of de Vries spaces. We also investigate connections with the Vietoris functor on the category of compact Hausdorff spaces and with the category of compact regular frames in point-free topology, and we provide an alternative, choice-free topological semantics for the Symmetric Strict Implication Calculus of Bezhanishvili et al.

2006

Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an epistemic reading of this formula: "t is a possible justification of φ". Motivated, in part, by this epistemic reading, Fitting introduced a Kripke style semantics for LP in . In this note, we prove soundness and completeness of some axiom systems which are not covered in [8].

Motivated by questions like: which spatial structures may be characterized by means of modal logic, what is the logic of space, how to encode in modal logic different geometric relations, topological logic provides a framework for studying the confluence of the topological semantics for $\sf S4$ modalities, based on topological spaces rather than Kripke frames. Following research initiated by Sgro, and further pursued algebraically by Georgescu, we prove an interpolation theorem and an omitting types theorem for various extensions of predicate topological logic and Chang's modal logic. Our proof is algebraic addressing expansions of cylindric algebras using interior operators and boxes, respectively. Then we proceed like is done in abstract algebraic logic by studing algebraisable extensions of both logics; obtaining a plethora of results on the amalgamation property for various subclasses of their algebraic counterparts, which are varieties. Notions like atom-canonicity and com...

2007

Implication algebras, originally introduced in order to study algebraic properties of the implication operation in Boolean algebras, are generalized and it is shown that these more general algebras are in one-to-one correspondence to semilattices with 1 the principal filters of which are posets with an antitone involution, respectively to commutative directoids with 1 the principal filters of which are posets with a switching involution.