The Logic of Partial Terms LPT is a strict negative free logic that provides an economical framework for developing many traditional mathematical theories having partial functions. In these traditional theories, all functions and predicates are strict. For example, if a unary function (predicate) is applied to an undefined argument, the result is undefined (respectively, false). On the other hand, every practical programming language incorporates at least one nonstrict or lazy construct, such as the if-then-else, but nonstrict functions cannot be either (...) primitive or introduced in definitional extensions in LPT. Consequently, lazy programming language constructs do not fit the traditional mathematical mold inherent in LPT. A nonstrict (positive free) logic is required to handle nonstrict functions and predicates. Previously developed nonstrict logics are not fully satisfactory because they are verbose in describing strict functions (which predominate in many programming languages), and some logicians find their semantics philosophically unpalatable. The newly developed Lazy Logic of Partial Terms LL is as concise as LPT in describing strict functions and predicates, and strict and nonstrict functions and predicates can be introduced in definitional extensions of traditional mathematical theories. LL is "built on top of" LPT, and, like LPT, admits only one domain in the semantics. In the semantics, for the case of a nonstrict unary function h in an LL theory T, we have $\models_T h(\perp) = y \longleftrightarrow \forall x(h(x) = y)$ , where $\perp$ is a canonical undefined term. Correspondingly, in the axiomatization, the "indifference" (to the value of the argument) axiom $h(\perp) = y \longleftrightarrow \forall x(h(x) = y)$ guarantees a proper fit with the semantics. The price paid for LL's naturalness is that it is tailored for describing a specific area of computer science, program specification and verification, possibly limiting its role in explicating classical mathematical and philosophical subjects. (shrink)

This paper embeds a theory of proper names in a general approach to singular reference based on type‐free property theory. It is proposed that a proper name “N” is a sortal common noun whose meaning is essentially tied to the linguistic type “N”. Moreover, “N” can be singularly referring insofar as it is elliptical for a definite description of the form the “N” Following Montague, the meaning of a definite description is taken to be a property of properties. The proposed (...) theory fulfils the major desiderata stemming from Kripke's works on proper names. (shrink)

Church's simple theory of types is a system of higher-order logic in which functions are assumed to be total. We present in this paper a version of Church's system called PF in which functions may be partial. The semantics of PF, which is based on Henkin's general-models semantics, allows terms to be nondenoting but requires formulas to always denote a standard truth value. We prove that PF is complete with respect to its semantics. The reasoning mechanism in PF for partial (...) functions corresponds closely to mathematical practice, and the formulation of PF adheres tightly to the framework of Church's system. (shrink)