Modeling Nonintersective Adjectives Using Operator Logics

Our topic is one that involves the interface between natural language and mathematical logic. First-order predicate language/logic does a good job approximating many parts of (English) speech, i.e., nouns, verbs and prepositions, but fails decidedly when it comes to, say, adjectives. In particular, it cannot account for the quite different ways in which the adjectives green and big modify a noun such as chair. In the former case, we can easily view a world in which the class of green chairs is the intersection of the class of green things with the class of chair-things. By contrast, the way big modifies a noun depends on the noun itself: a big chair is microscopic when compared to the smallest of galaxies. We investigate logical languages inspired by this phenomenon; particularly those with variables ranging over individuals and with variable-binding operators akin to generalized quantifiers

A Galois connection between classical and intuitionistic logics. II: Semantics

Three classes of models of QHC, the joint logic of problems and propositions, are constructed, including a class of subset/sheaf-valued models that is related to solutions of some actual problems (such as solutions of algebraic equations) and combines the familiar Leibniz-Euler-Venn semantics of classical logic with a BHK-type semantics of intuitionistic logic. To test the models, we consider a number of principles and rules, which empirically appear to cover all "sufficiently simple" natural conjectures about the behaviour of the operators ! and ?, and include two hypotheses put forward by Hilbert and Kolmogorov, as formalized in the language of QHC. Each of these turns out to be either derivable in QHC or equivalent to one of only 13 principles and 1 rule, of which 10 principles and 1 rule are conservative over classical and intuitionistic logics. The three classes of models together suffice to confirm the independence of these 10 principles and 1 rule, and to determine the full lattice of implications between them, apart from one potential implication.Comment: 35 pages. v4: Section 4.6 "Summary" is added at the end of the paper. v3: Major revision of a half of v2. The results are improved and rewritten in terms of the meta-logic. The other half of v2 (Euclid's Elements as a theory over QHC) is expected to make part III after a revisio

Categories for Dynamic Epistemic Logic

The primary goal of this paper is to recast the semantics of modal logic, and dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We first review the category of relations and categories of Kripke frames, with particular emphasis on the duality between relations and adjoint homomorphisms. Using these categories, we then reformulate the semantics of DEL in a more categorical and algebraic form. Several virtues of the new formulation will be demonstrated: The DEL idea of updating a model into another is captured naturally by the categorical perspective -- which emphasizes a family of objects and structural relationships among them, as opposed to a single object and structure on it. Also, the categorical semantics of DEL can be merged straightforwardly with a standard categorical semantics for first-order logic, providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Kripke Semantics for Martin-L\"of's Extensional Type Theory

It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of interpretations of dependent type theory, i.e., to show that the interpretations of equal expressions are indeed equal. Several classes of models have been used to remedy this problem. We contribute to this investigation by giving a semantics that is standard, coherent, and sufficiently general for completeness while remaining relatively easy to compute with. Our models interpret types of Martin-L\"of's extensional dependent type theory as sets indexed over posets or, equivalently, as fibrations over posets. This semantics can be seen as a generalization to dependent type theory of the interpretation of intuitionistic first-order logic in Kripke models. This yields a simple coherent model theory, with respect to which simple and dependent type theory are sound and complete