Pretopologies were introduced in [S], and there shown to give a complete semantics for a propositional sequent calculus BL, here called basic linear logic, as well as for its extensions by structural rules,ex falso quodlibetor double negation. Immediately after Logic Colloquium '88, a conversation with Per Martin-Löf helped me to see how the pretopology semantics should be extended to predicate logic; the result now is a simple and fully constructive completeness proof for first order BL and virtually all its (...) extensions, including the usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger set-theoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as the usual two-valued models.To make the paper self-contained, I briefly review in §1 the definition of pretopologies; §2 deals with syntax and §3 with semantics. The completeness proof in §4, though similar in structure, is sensibly simpler than that in [S], and this is why it is given in detail. In §5 it is shown how little is needed to obtain completeness for extensions of BL in the same language. Finally, in §6 connections with proofs with respect to more traditional semantics are briefly investigated, and some open problems are put forward. (shrink)

We introduce the notion of infinitary preorder and use it to obtain a predicative presentation of sup-lattices by generators and relations. The method is uniform in that it extends in a modular way to obtain a presentation of quantales, as “sup-lattices on monoids”, by using the notion of pretopology.Our presentation is then applied to frames, the link with Johnstone’s presentation of frames is spelled out, and his theorem on freely generated frames becomes a special case of our results on (...) quantales.The main motivation of this paper is to contribute to the development of formal topology. That is why all our definitions and proofs can be expressed within an intuitionistic and predicative foundation, like constructive type theory. (shrink)

We present a semantic study of a family of modal intuitionistic linear systems, providing various logics with both an algebraic semantics and a relational semantics, to obtain completeness results. We call modality a unary operator on formulas which satisfies only one rale (regularity), and we consider any subsetW of a list of axioms which defines the exponential of course of linear logic. We define an algebraic semantics by interpreting the modality as a unary operation on an IL-algebra. Then we introduce (...) a relational semantics based on pretopologies with an additional binary relationr between information states. The interpretation of is defined in a suitable way, which differs from the traditional one in classical modal logic. We prove that such models provide a complete semantics for our minimal modal system, as well as, by requiring the suitable conditions onr (in the spirit of correspondence theory), for any of its extensions axiomatized by any subsetW as above. We also prove an embedding theorem for modal IL-algebras into complete ones and, after introducing the notion of general frame, we apply it to obtain a duality between general frames and modal IL-algebras. (shrink)

Formal spaces have become commonplace conceptual and computational tools in a large array of scientific disciplines, including both the natural and the social sciences. Morphological spaces are spaces describing and relating organismal phenotypes. They play a central role in morphometrics, the statistical description of biological forms, but also underlie the notion of adaptive landscapes that drives many theoretical considerations in evolutionary biology. We briefly review the topological and geometrical properties of the most common morphospaces in the biological literature. In contemporary (...) geometric morphometrics, the notion of a morphospace is based on the Euclidean tangent space to Kendall’s shape space, which is a Riemannian manifold. Many more classical morphospaces, such as Raup’s space of coiled shells, lack these metric properties, e.g., due to incommensurably scaled variables, so that these morphospaces typically are affine vector spaces. Other notions of a morphospace, like Thomas and Reif’s skeleton space, may not give rise to a quantitative measure of similarity at all. Such spaces can often be characterized in terms of topological or pretopological spaces. (shrink)

It is assumed that a Kripke–Joyal semantics \({\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}\) has been defined for a first-order language \({\mathcal{L}}\) . To transform \({\mathbb{C}}\) into a Heyting algebra \({\overline{\mathbb{C}}}\) on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of \({\mathbb{C}}\) . A pretopology \({\overline{{\rm Cov}}}\) is defined on \({\overline{\mathbb{C}}}\) using the pretopology on \({\mathbb{C}}\) . A sheaf \({\overline{{\it F}}}\) is made up of (...) sections of F that obey functoriality. A forcing relation \({\overline{\Vdash}}\) is defined and it is shown that \({\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }\) is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of \({\mathcal{A}}\) . That is to say, an object a of \({\mathbb{C}Ob}\) forces a sentence with respect to \({\mathcal{A}}\) if and only if the maximal a-crible forces it with respect to \({\overline{\mathcal{A}}}\) . This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra. (shrink)