There are, in general, two kinds of logical foundations of mathematics, namely set theory and higher-order logic (aka. type theory). Fuzzy set theory and class theory have been studied extensively for a long time. Studies on higher-order fuzzy logic, by contrast, just started more recently and there is much yet to be done. Here we introduce higher-order fuzzy logics over MTL (monoidal t-norm logic; uniform foundations of fuzzy logics such as Hájek's basic logic, Łukasiewicz logic, and Gödel logic); higher-order MTL boils down to the standard higherorder intuitionistic logic (i.e., the internal logic of topos) with the pre-linearity axiom when equipped with the contraction rule. We give uniform categorical semantics for all higher-order fuzzy logics over MTL in terms of tripos theory. We prove the linear completeness of tripos semantics for higher-order fuzzy logics, and a tripos-theoretical Baaz translation theorem, which allows us to simulate higher-order classical logic within fuzzy logics. The relationships between topos theory and fuzzy set theory have been pursued for a long time; yet no complete topos semantics of fuzzy set theory has been found. Here we give complete tripos semantics of higher-order fuzzy logic (or fuzzy type theory).

Knowledge does not necessarily exist only within a single agent; it may exist collectively within a number of agents combined together (cf. the wisdom of the crowd), and this sort of knowledge is called distributed knowledge. In light of bounded rationality and uncertainty, distributed knowledge can be incomplete, and there may be gradations in the certainty of distributed knowledge, that is, knowledge may be distributed within a system of agents up to some degree of certainty only. Fagin, Halpern, Moses, and Vardi gave an axiomatic system for reasoning about distributed knowledge, and developed a special technique to prove its fundamental properties such as completeness. Here we extend their classic results so as to incorporate fuzzy distributed knowledge; in addition we prove several other properties of fuzzy modal systems such as the finite model property and Gödel-style translation theorems.

Abstract Lawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural (or Lambekian) algebraic logic and hyperdoctrinal (or Lawverian) categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics.

It is known that some fuzzy predicate logics, such as Łukasiewicz predicate logic, are not complete with respect to the standard real-valued semantics. In the present paper we focus upon a typed version of first-order MTL (Monoidal T-norm Logic), which gives a unified framework for different fuzzy logics including, inter alia, Hajek's basic logic, Łukasiewicz logic, and Gödel logic. And we show that any extension of first-order typed MTL, including Łukasiewicz predicate logic, is sound and complete with respect to the corresponding categorical semantics in the style of Lawvere's hyperdoctrine, and that the so-called Baaz delta translation can be given in the first-order setting in terms of Lawvere's hyperdoctrine. A hyperdoctrine may be seen as a fibred algebra, and the first-order completeness, then, is a fibred extension of the algebraic completeness of propositional logic. While the standard real-valued semantics for Łukasiewicz predicate logic is not complete, the hyperdoctrine, or fibred algebraic, semantics is complete because it encompasses a broader class of models that is sufficient to prove completeness; in this context, incompleteness may be understood as telling that completeness does not hold when the class of models is restricted to the standard class of real-valued hyperdoctrine models. We expect that this finally leads to a unified categorical understanding of Takeuti-Titani's fuzzy models of set theory.

We propose the concept of topological dualizability as the condition of possibility of Stone duality, and thereby give a non-Hausdorff extension of the primal duality theorem in natural duality theory in universal algebra. The primal duality theorem is a vast generalization of the classic Stone duality for Boolean algebras, telling that any varieties generated by functionally complete algebras, such as the algebras of Emil Post's finite-valued logics, are categorically equivalent to zero-dimensional compact Hausdorff spaces. Here we show a non-Hausdorff extension of primal duality: any varieties generated by certain weakly functionally complete or topologically dualizable algebras are categorically dually equivalent to coherent spaces, a special class of compact sober spaces. This generalizes the Stone duality for distributive lattices and Heyting algebras (as a subclass of distributive lattices) in the spirit of primal duality theory. And we give applications of the general theorem to algebras of Łukasiewicz many-valued logics. The concept of topological dualizability is arguably the key to the universal algebraic unification of Stone-type dualities; in the present paper, we take the first steps in demonstrating this thesis.

Utilising and expanding concepts from categorical topology and algebra, we contrive a moderately general theory of dualities between algebraic, point-free spaces and set-theoretical, point-set spaces, which encompasses infinitary Stone dualities, such as the well-known duality between frames (aka. locales) and topological spaces, and a duality between \sigma-complete Boolean algebras and measurable spaces, as well as the classic finitary Stone, Gelfand, and Pontryagin dualities. Among different applications of our theory, we focus upon domain-convexity duality in particular: from the theory we derive a duality between Scott's continuous lattices and convexity spaces, and exploit the resulting insights to identify intrinsically the dual equivalence part of a dual adjunction for algebras of the distribution monad; the dual adjunction was uncovered by Bart Jacobs, but with no characterisation of the induced equivalence, which we do give here. In the Appendix, we place categorical duality in a wider context, and elucidate philosophical underpinnings of duality.

We formalize reasoning about fuzzy belief and fuzzy common belief, especially incomparable beliefs, in multi-agent systems by using a logical system based on Fitting's many-valued modal logic, where incomparable beliefs mean beliefs whose degrees are not totally ordered. Completeness and decidability results for logic of fuzzy belief and common belief are established while implicitly exploiting duality-theoretic perspective on Fitting's logic that builds upon author's previous work. A conceptually novel feature is that incomparable beliefs and qualitative fuzziness can be formalized in developed system, whereas they cannot be formalized in previously proposed systems for reasoning about fuzzy belief. We believe that belief degrees can ultimately be reduced to truth and we call this the reduction thesis about belief degrees, which is assumed in present paper and motivates an axiom of our system. We finally argue that fuzzy reasoning sheds new light on old epistemic issues such as coordinated attack problem.

Stone-type duality connects logic, algebra, and topology in both conceptual and technical senses. This paper is intended to be a demonstration of this slogan. In this paper we focus on some versions of Fitting's L-valued logic and L-valued modal logi