The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper

Abstract

Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, collaborator-on his eightieth birthday. This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework-like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as 'the class of all individuals'. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework-like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations-as in post-WWII model theory. In the early 1960s, many logicians-mistakenly, as we show-held the 'contrary alternative' that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege-Russell, fixed-universe paradigm. We further argue that between his Principiainfluenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes.

Fixed‐ versus Variable‐domain Interpretations of Tarski’s Account of Logical Consequence

“…Etchemendy was not the first to have noticed a major difference between Tarski’s (1936) account and the later model‐theoretic account. Corcoran and Sagüillo (2009) retrace Corcoran’s awareness of the issue since 1964. In a previous draft (2003) of the same article (but without Sagüillo as co‐author), Corcoran claimed that Tarski’s commitment to a fixed‐domain notion of logical consequence was ‘evident’ (in the latest version it is ‘obvious’, (6)).…”

“…In addition, fixed‐domain conceptions of consequence were not rare at the time of Tarski’s writing and were in fact upheld by logicians with logicist leanings (Carnap, Ramsey, Russell, Lewis, Langford, etc.) who worked with type theories with a fixed domain of individuals given at the outset (explicit references to such authors are also found, among others, in Gómez‐Torrente 1996 and Corcoran and Sagüillo 2009). Thus, the fixed‐domain conception is not ‘nonstandard’ after all.…”

Philosophical Problems of Foundations of Logic

“…The debate continued in the 21st century (see [14], [111], [28], [88], [5], [108], [63], [77], [78], [31], [47], [89], [25], [101], and [104]). In the last work the author rejects the standard definition of logical consequence and suggests a sufficiently general form of the consequence relation between abstract signs.…”