Abstract We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper.

We solve the minimization problem for finitely axiomatizable theories in Gödel infinite-valued propositional logic. That is, we obtain an algorithm that when input a formula α(X1,…,Xn) outputs a formula β(X1,…,Xm) such that (i) the theories singly axiomatized by {α} and {β} have isomorphic algebraic semantics, and (ii) if β′(X1,…,Xm′) is any formula satisfying (i), then m′≥m.

Stefano Aguzzoli, Brunella Gerla, Vincenzo Marra; Applications of Topological Dualities to Measure Theory in Algebraic Many-valued Logic, Journal of Logic

This paper is divided into two parts. In the present Part I, our main objective is to analyse Mamdani-type fuzzy control systems in logical terms, with special emphasis on the fuzzy inference process. To that end, we provide our own inference procedure, cast in the language of standard many-valued logics. We give an ample discussion of the logical meaning of our procedure. We eventually show how to fully recover Mamdani-type fuzzy inference from the latter. In this sense, then, our proposal may be regarded as a logical interpretation of Mamdani-type fuzzy inference. In Part II of this paper, we report on the results of an experiment on the technical analysis of the financial markets based on fuzzy techniques. The core algorithm implements the inference procedure described in this first part of the paper. In Part II, we will argue that the experimental results support the claim that our present theoretical analysis provides a sound interpretation of Mamdani-type fuzzy inference.

This paper is divided into two parts. In Part I, our main objective was to analyse Mamdani-type fuzzy control systems in logical terms, with special emphasis on the fuzzy inference process. To that end, we provided our own inference procedure, cast in the language of standard many-valued logics. We gave an ample discussion of the logical meaning of our procedure. We eventually showed how to fully recover Mamdani-type fuzzy inference from the latter. In this sense, then, our proposal in Part I may be regarded as a logical interpretation of Mamdani-type fuzzy inference. In the present Part II of this paper, we report on the results of an experiment on the technical analysis of the financial markets based on fuzzy techniques. The core algorithm implements the inference procedure described in the first part of the paper. The experimental results support the claim that our theoretical analysis in Part I provides a sound interpretation of Mamdani-type fuzzy inference.

We address the question, which MV-algebras have finite automorphism group. We prove that finitely presented MV-algebras whose maximal spectral space has topological dimension not exceeding 1 do have finite automorphism group. We give examples to show that finite presentability is an essential hypothesis. Our proof produces as an interesting by-product a complete graph–theoretic isomorphism invariant for the class of MV-algebras involved.

Fuzzy sets featuring in applications to fuzzy control systems are often required to satisfy specific conditions such as, e.g., convexity or normality. In the same connection, a widespread choice is to work with fuzzy sets whose graphs have triangular shape. The purpose of this paper is to show that the former conditions may be regarded as attempts at approximating the latter choice. Specifically, as our main result we prove that a reasonable set of such conditions suffices to characterise families of triangular fuzzy sets. A second result provides an additional characterisation of such families in terms of properties of the curve that they parametrise.

Godel logic is the fuzzy logic of the minimum triangular norm and its residuum. Using the functional representation of the Lindenbaum algebra of Godel logic, we analyze the interaction between the integral operator and the logical connectives. On these grounds, we put forth a notion of finitely additive probability measure for Godel logic. Our first main result shows that such measures precisely correspond to integrating the truth value functions induced by Godel formulas with respect to a Borel probability measure on the real unit cube [0,1]n. Our second main result shows that they also coincide with convex combinations of finitely many [0,1]-valued assignments.

Gödel propositional logic is the logic of the minimum triangular norm, and can be axiomatized as propositional intuitionistic logic augmented by the prelinearity axiom(α → β) V (β → α). Its algebraic counterpart is the subvariety of Heyting algebras satisfying prelinearity, known as Gödel algebras. A Delannoy path is a lattice path in 𝕫2that only uses northward, eastward, and northeastward steps. We establish a representation theorem for free n-generated Gödel algebras in terms of the Boolean 𝓃-cube {0,1}𝓃, enriched by suitably generalized Delannoy paths.

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of propositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum logic several known results about inutitionistic logic with the prelinearity axiom (also called Gödel-Dummett logic), mutatis mutandis. We give several such applications.