The calculus of names -- The legacy of Jan Łukasiewicz
Andrzej Pietruszczak
TL;DR
This work surveys the calculus of names, tracing the Łukasiewicz framework and its set-theoretic semantics, and surveys contemporary extensions including Shepherdson, Słupecki, and Leśniewski’s Ontology. It presents semantic characterizations (set-theoretic, polyreferential, non-monoreferential) and completeness proofs via Shepherdson’s approach and Henkin-style constructions, plus definitional extensions and Ontology fusions. It also discusses extensions with singular sentences and identity (CzM) and analyzes the copula-based ontology within a modern syllogistic setting. The study highlights Łukasiewicz’s lasting legacy and shows how modern logic accommodates emptiness, identity, and ontology with rigorous semantic and proof-theoretic tools.
Abstract
With his research on Aristotle's syllogistic, Jan Łukasiewicz (1934, 1939, 1957, 1963) initiates the branch of logic known as the calculus of names. This field deals with axiomatic systems that analyse various fragments of the logic of names, i.e., that branch of logic that studies various forms of names and functors acting on them, as well as logical relationships between sentences in which these names and functors occur. In this work, we want not only to present the genesis of the calculus of names and its first system created by Łukasiewicz. We also want to deliver systems that extend the first. In this work, we will also show that, from the point of view of modern logic, Łukasiewicz's approach to syllogistic is not the only possible one. However, this does not diminish Łukasiewicz's role in the study of syllogism. We believe that the calculus of names is undoubtedly the legacy of Łukasiewicz.