Should Tarski’s Idea of Consequence Operation be Revised?

Abstract

Tarski’s papers, in which he examines the idea of a consequence operation Cn,divide into two groups. One of them is formed by the papers (see especially Tarski (1936) that offer an analysis of the general idea of the consequence operation. Resorting to fundamental ideas of logical semantics, Tarski explains what, in his view, it means to say that a formula a of a language L is a logical consequence Cn(X) of a set of formulas X of that language. Under the definition he proposed Cn is the strongest operation on sets of formulas of L that preserves truth under all possible interpretations of the language. In the papers belonging to the second group (Tarski 1930, 1930a, 1935–36), the consequence operation Cn is treated as an object of abstract studies. More specifically, it is treated as an abstract unary operation defined on the power set of a set S such that the following familiar conditions are satisfied:

$$X \subseteq Cn\left( {Cn\left( X \right)} \right) \subseteq Cn\left( X \right)$$

(T1.)

and

$$If\,X\, \subset \,Ythen\,Cn\left( X \right) \subset \,Cn\left( Y \right)$$

(T2.)

Actually, Tarski imposed on Cn one more requirement which in more recent investigations (a large survey of them is Wójcicki 1988, see also 1984) is often omitted. It is as follows:

$$If\,\alpha \, \in \,Cn\left( X \right),then\,for\,a\,finite\,subset\,X'\, \subseteq \,X,\alpha \, \in \,Cn\left( {X'} \right)$$

(T3.)

(T3) is known as the finitaryness (or, finiteness or compactness — all these terms are applied) condition.

The tide under which this paper was presented at the International Symposium Alfred Tarski and Vienna Circle; Austro-Polish Connections in Logical Empiricism was “Is the notion of truth an indispensable element of Tarski’s conception of Logical Consequence?” In its present version the title may suggest that I am going to discuss issues similar to those one may found in Etchemedy (1990). In fact, the central issue of this essay is the relevance of some ideas derived from cognitive science to Tarski’s way of approaching both the question of how the notion of a logical consequence operation should be defined and that of how it should be studied. In the version delivered at the Vienna meeting I concluded my paper by showing that the approach based on the idea of a belief system and its transformations with time yields the notion of consequence operation alternative to that familiar from Tarski’s investigations. The former is not monotonic, while monotonicity is one of the basic properties of the latter. In the present written version of the paper I am going a step further and argue that the problems that have resulted in developing the idea of non-monotonic logic can be handled within a Tarski-style framework. I neither deny that the phenomenon of non-monotonic reasoning is real nor that it deserves to be studied. I express my doubt, however, whether its logical aspects justify developing the idea of logical inference alternative to the traditional one.

My Vienna talk was delivered in so general terms that I was reluctant to prepare this paper for publication. Now, when the work has been done, I am sincerely grateful Professors Friedrich Stadler and Eckehart Köhler the organizers of the Symposium who was urging me to write down the ideas presented in my Vienna talk. I hope that the paper will contribute to clarifying some, I believe, worth being examined points. Of course, the responsibility for both the contents of this paper and its rather informal style is fully mine.