From Closure-Operatic Deductive Methodology to Non-Standard Alternatives

Abstract

The best known abstract setting for the traditional deductive methodology that has in the course of time become widely accepted as the most essential ingredient of methodological orthodoxy is due to the fundamental work of A. Tarski. In a series of papers of the thirties, especially in papers (1930a), (1930b), (1933) and (1936), Tarski developed a general conceptual set-up which he based on the idea of the consequence operation, or closure operator as it is called in the context of lattice theory today. He took a closure operator Cn to be a mapping from P(S) to P(S) where P(S) is the power set of the set S of all sentences of a fixed language and he characterised the operator as reflexive (i.e. $X \subseteq Cn(X))$; monotonic (i.e. $Cn(X) \subseteq Cn(Y)$ whenever $X \subseteq Y$) and idempotent (i.e. $Cn(Cn(X)) \subseteq Cn(X))$ where X, Y are arbitrary subsets of S. In his (1935), Tarski re-modelled the Cn-based set-up by employing, instead of Cn, the idea of what he called a deductive system or the idea of a closure system, as we say today. A closure system Th is a subset of P(S) closed under arbitrary intersections i.e. a subset of P(S) such that $\cap K \subseteq Th$ for any $K \subseteq Th$. Various aspects of this Tarskian methodological set-up have been subjected to an extensive investigation in the subsequent years both in Poland and abroad. For more details see Blok and Pigozzi (1989), Cohn (1965), Grzegorczyk (1961), łoś and Suszko (1958), Pogorzelski (1969) and (1994), Schmidt (1952), Thiele (1956), Weaver (1992) and Wójcicki (1988). In (1952) J. Schmidt proved that, loosely speaking,

$${\rm{Cn(Th(Cn)) = Cn and Th(Cn}}$$

for any Cn and Th.