Cover semantics for quantified lax logic

Abstract

Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of Lawvere-Tierney-Grothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the Beth-Kripke-Joyal cover semantics for first-order intuitionistic logic with the classical relational semantics for a "diamond" modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of "propositions" of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4.An alternative completeness proof is given for (non-modal) first-order intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkin-style construction of a characteristic model whose points are principal theories rather than prime saturated ones.The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.

Grishin Algebras and Cover Systems for Classical Bilinear Logic

“…Proof. This is shown in [10,Theorem 5] and [11,Lemma 3.3]. Briefly: the Refinement axiom ensures that if X is an up-set, then so is jX; and the Existence axiom ensures that j is an inflationary operator on up-sets, i.e.…”

“…A poset is order-complete, or give a new semantics for relevant logic, a logic that has distribution of conjunction over disjunction. For further background and discussion of the cover semantics methodology see also [11], which develops interpretations of intuitionistic modal logic giving the diamond modality ♦ its standard Kripkean semantics, without validating distribution of ♦ over disjunction.…”

“…So is a relation of refinement in the sense of increase of information. Interpret covering as a relation expressing commonality of information content, so that if C x then the information content of x consists of that which is common to all the states in C. Further discussion of these and other motivations can be found in [11] and [12,Chapter 6].…”

“…But even that weaker condition is sufficient to make Prop(S) into a complete Heyting algebra, as is shown in [11], and so it too must be abandoned when interpreting non-distributive logics.…”

A semantic hierarchy for intuitionistic logic

“…37 It is straightforward to show that PLL is sound and complete with respect to its algebraic semantics based on nuclear algebras. A number of other, more concrete semantics for PLL have also been proposed [Goldblatt, 1981, Fairtlough and Mendler, 1997, Benton et al, 1998, Alechina et al, 2001, Goldblatt, 2011].…”