Alfred Tarski - What Are Logical Notions?

What Are Logical Notions?

Another theory of Tarski's attracting attention in the recent philosophical literature is that outlined in his "What are Logical Notions?" (Tarski 1986). This is the published version of a talk that he gave in 1966; it 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 were derived from the Erlangen programme of the German 19th century Mathematician, Felix Klein. (Mautner 1946, and possibly an article by the Portuguese mathematician Sebastiao e Silva, anticipated Tarski in applying the Erlangen Program to logic.)

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-to-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 of 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 (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other.

Tarski's proposal was to demarcate the logical notions by considering all possible one-to-one transformations (automorphisms) of a domain onto itself. By domain is meant the universe of discourse of a model for the semantic theory of a logic. If one identifies the truth value True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal:

1. Truth-functions: All truth-functions are admitted by the proposal. This includes, but is not limited to, all n-ary truth-functions for finite n. (It also admits of truth-functions with any infinite number of places.)
2. Individuals: No individuals, provided the domain has at least two members.
3. Predicates:
   • the one-place total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension
   • two-place total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension
   • the two-place identity predicate, with the set of all order-pairs <a,a> in its extension, where a is a member of the domain
   • the two-place diversity predicate, with the set of all order pairs <a,b> where a and b are distinct members of the domain
   • n-ary predicates in general: all predicates definable from the identity predicate together with conjunction, disjunction and negation (up to any ordinality, finite or infinite)
4. Quantifiers: Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates Fx and Gy, "More(x, y)", which says "More things have F than have G."
5. Set-Theoretic relations: Relations such as inclusion, intersection and union applied to subsets of the domain are logical in the present sense.
6. Set membership: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo–Fraenkel set theory.
7. Logical notions of higher order: While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well.

In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Russell and Whitehead's Principia Mathematica are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987).

Solomon Feferman and Vann McGee further discussed Tarski's proposal in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary homomorphisms. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard first-order logic without identity.

McGee (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends first-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.