Lesniewski Stanislaw (1886–1939), Polish philosopher-logician, cofounder, with Lukasiewicz and Kotarbigski, of the Warsaw Center of Logical Research. He perfected the logical reconstruction of classical mathematics by Frege, Schröder, Whitehead, and Russell in his synthesis of mathematical with modernized Aristotelian logic. A pioneer in scientific semantics whose insights inspired Tarski, Les’niewski distinguished genuine antinomies of belief, in theories intended as true mathematical sciences, from mere formal inconsistencies in uninterpreted calculi. Like Frege an acute critic of formalism, he sought to perfect one comprehensive, logically true instrument of scientific investigation. Demonstrably consistent, relative to classical elementary logic, and distinguished by its philosophical motivation and logical economy, his system integrates his central achievements. Other contributions include his ideographic notation, his method of natural deduction from suppositions and his demonstrations of inconsistency of other systems, even Frege’s revised foundations of arithmetic. Fundamental were (1) his 1913 refutation of Twardowski’s Platonistic theory of abstraction, which motivated his ‘constructive nominalism’; and (2) his deep analyses of Russell’s paradox, which led him to distinguish distributive from collective predication and (as generalized to subsume Grelling and Nelson’s paradox of self-reference) logical from semantic paradoxes, and so (years before Ramsey and Gödel) to differentiate, not just the correlatives object language and metalanguage, but any such correlative linguistic stages, and thus to relativize semantic concepts to successive hierarchical strata in metalinguistic stratification.
His system of logic and foundations of mathematics comprise a hierarchy of three axiomatic deductive theories: protothetic, ontology, and mereology. Each can be variously based on just one axiom introducing a single undefined term. His prototheses are basic to any further theory. Ontology, applying them, complements protothetic to form his logic. Les’niewski’s ontology develops his logic of predication, beginning (e.g.) with singular predication characterizing the individual so-and-so as being one (of the one or more) such-and-such, without needing classabstraction operators, dispensable here as in Russell’s ‘no-class theory of classes.’ But this, his logic of nouns, nominal or predicational functions, etc., synthesizing formulations by Aristotle, Leibniz, Boole, Schröder, and Whitehead, also represents a universal theory of being and beings, beginning with related individuals and their characteristics, kinds, or classes distributively understood to include individuals as singletons or ‘one-member classes.’ Les’niewski’s directives of definition and logical grammar for his systems of protothetic and ontology provide for the unbounded hierarchies of ‘open,’ functional expressions. Systematic conventions of contextual determinacy, exploiting dependence of meaning on context, permit unequivocal use of the same forms of expression to bring out systematic analogies between homonyms as analogues in Aristotle’s and Russell’s sense, systematically ambiguous, differing in semantic category and hence significance. Simple distinctions of semantic category within the object language of the system itself, together with the metalinguistic stratification to relativize semantic concepts, prevent logical and semantic paradoxes as effectively as Russell’s ramified theory of types. Lesniewski’s system of logic, though expressively rich enough to permit Platonist interpretation in terms of universals, is yet ‘metaphysically neutral’ in being free from ontic commitments. It neither postulates, presupposes, nor implies existence of either individuals or abstractions, but relies instead on equivalences without existential import that merely introduce and explicate new terms. In his ‘nominalist’ construction of the endless Platonic ladder of abstraction, logical principles can be elevated step by step, from any level to the next, by definitions making abstractions eliminable, translatable by definition into generalizations characterizing related individuals. In this sense it is ‘constructively nominalist,’ as a developing language always open to introduction of new terms and categories, without appeal to ‘convenient fictions.’ Les’niewski’s system, completely designed by 1922, was logically and chronologically in advance of Russell’s 1925 revision of Principia Mathematica to accommodate Ramsey’s simplification of Russell’s theory of types. Yet Les’niewski’s premature death, the ensuing disruption of war, which destroyed his manuscripts and dispersed survivors such as Sobocigski and Lejewski, and the relative inaccessibility of publications delayed by Les’niewski’s own perfectionism have retarded understanding of his work.
See also POLISH LOGIC. E.C.L.