Subtyping, also known as subtype polymorphism, is a concept extensively studied in programming language theory, delineating the substitutability relation among datatypes. This property ensures that programs designed for supertype objects remain compatible with their subtypes. In this paper, we explore the capability of order-sorted logic for utilizing these ideas in the context of Knowledge Representation. We recognize two fundamental limitations: First, the inability of this logic to address the concept rather than the value of non-logical symbols, and second, the lack of language constructs for constraining the type of terms. Consequently, we propose guarded order-sorted intensional logic, where guards are language constructs for annotating typing information and intensional logic provides support for quantification over concepts.

The first phase of developing an intelligent system is the selection of an ontology of symbols representing relevant concepts of the application domain. These symbols are then used to represent the knowledge of the domain. This representation should be \emph{elaboration tolerant}, in the sense that it should be convenient to modify it to take into account new knowledge or requirements. Unfortunately, current formalisms require a significant rewrite of that representation when the new knowledge is about the \emph{concepts} themselves: the developer needs to "\emph{reify}" them. This happens, for example, when the new knowledge is about the number of concepts that satisfy some conditions. The value of expressing knowledge about concepts, or "intensions", has been well-established in \emph{modal logic}. However, the formalism of modal logic cannot represent the quantifications and aggregates over concepts that some applications need. To address this problem, we developed an extension of first order logic that allows referring to the \emph{intension} of a symbol, i.e., to the concept it represents. We implemented this extension in IDP-Z3, a reasoning engine for FO($\cdot$) (aka FO-dot), a logic-based knowledge representation language. This extension makes the formalism more elaboration tolerant, but also introduces the possibility of syntactically incorrect formula. Hence, we developed a guarding mechanism to make formula syntactically correct, and a method to verify correctness. The complexity of this method is linear with the length of the formula. This paper describes these extensions, how their relate to intensions in modal logic and other formalisms, and how they allowed representing the knowledge of four different problem domains in an elaboration tolerant way.

Montague's dUficu/t nulalian and complex model theory have tended to obscure potential insights for

The aim of this paper is to present in a rigorous way the syntax and semantics of a certain fragment of a certain dialect of English. Patrick Suppes claims, in a paper prepared for the present workshop [the 1970 Stanford Workshop on Grammar and Semantics], that at the present time the semantics of natural languages are less satisfactorily formulated than the grammars ¼ [and] a complete grammar for any significant fragment of natural language is yet to be written.'' This claim would of course be accurate if restricted in its application to the attempts emanating from the Massachusetts Institute of Technology, but fails to take into account the syntactic and semantic treatments proposed in Montague (1970a, b). Thus the present paper cannot claim to present the first complete syntax (or grammar, in Suppes' terminology) and semantics for a significant fragment of natural language; and it is perhaps not inappropriate to sketch relations between the earlier proposals and the one given below. Montague (1970b) contains a general theory of languages, their interpretations, and the inducing of interpretations by translation.