Alfred Tarski (January 14, 1901 - October 26, 1983) was a Polish logician considered to be one of the greatest logicians of all time in a manner after Aristotle, Gottlob Frege, and Kurt Gödel.
Tarski made contributions to algebra, measure theory, mathematical logic, set theory and metamathematics. See Truth for a brief description of the "Convention T" standard in his "inductive definition of truth". This was an important contribution to symbolic logic and the philosophy of language.
Tarski studied logic and philosophy in Warsaw with Łukasiewicz, Leśniewski, and Kotarbiński.
He joined the University of California, Berkeley in 1942, and became professor of mathematics there in 1949.
(Givant 1986) provides a comprehensive bibliography of Tarski's writings.
The concept of truth in formalized languages
This paper is a long (over 100 pages) presentation of a mathematical definition of truth for logical languages. It first appeared in 1933 in Polish and then in 1935 in German, under the title "Der Wahrheitsbegriff in den Sprachen der deduktiven Disziplinen." It is thus sometimes referred to as the "Wahrheitsbegriff." Its first appearance in full in English was in 1956 in the first edition of Logic, Semantics, Metamathematics.
Some fairly recent philosophical debate has examined to what extent Tarski's theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centres on how to read Tarski's condition of adequacy for a truth definition. That condition requires that the truth theory have the following as theorems for all sentences P of the language for which truth is being defined:
'P' is True if and only if p.
(where p is the proposition expressed by "P")
The debate amounts to whether to read sentences of this form, such as
"Snow is white" is true if and only if snow is white
as expressing merely a deflationary theory of truth or as embodying truth as a more substantial property.
On the concept of logical consequence
In 1935, Tarski gave a lecture to the International Congress of Scientific Philosophy in Paris. It appeared in 1936 in a Polish and then a German version. (First appearance in English?) In it he gave either the modern model-theoretic definition of (semantic) logical consequence, or the basis for that modern notion. The question of whether Tarski's notion was the modern one turns on the question of whether he intended to admit models with varying domains (and in particular, models with domains of different cardinalities). This question is currently being debated in the philosophical literature.
Tarski ends his paper by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. Thus, the talk "What are Logical Notions?" can be viewed as continuing the work of "On the Concept of Logical Consequence."
Much of the recent discussion over the treatment of varying domains in this paper was stimulated by (Etchemendy 1999).
A new translation of this paper has recently been produced (Tarski 2002). It gives extensive details of the differences between the German and Polish versions of the paper and corrects a number of mistranslations in the previous translation.
What are logical notions?
A theory of Tarski's that has been attracting attention in the recent philosophical literature is outlined in his "What are Logical Notions?" (Tarski 1986). This is an edited version of a talk that Tarski first gave in 1966, and was edited without his direct involvement.
In the talk, Tarski proposed a demarcation of the logical operations (which he calls "notions") from the non-logical. The suggested criteria was derived from the Erlangen programme of the German 19th century Mathematician, Felix Klein. (Tarski was preceded in applying the Erlanger Program to logic by (Mautner 1946) as well as possibly by an article by the Italian mathematician Silva.)
That program classified the various types of geometry (Euclidean geometry, affine geometry, topology, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.
As the range or permissible transformations becomes broader the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (equilateral triangles from non-equilateral triangles, for instance). Continuous transformations (which can intuitively be though of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or gluing) allow us to distinguish a polygon from an annulus (ring with a whole in the centre), but does not allow us to distinguish two polygons from each other.
Tarski's proposal was to demarcate the logical notions by considering all possible one-one transformations of a domain onto itself. (By domain here is meant the universe of discourse of a model for the semantic theory of a logic. A one-one transformation of a set onto itself is also known as an automorphism.) If one identifies the truth-value True with the domain set and the truth-value False with the empty set, then the following sorts of operations are counted as logical under the proposal: