Tarski on Semantical Completeness, Categoricity and Logical Consequence

1. Tarski on Semantical Completeness, Categoricity and Logical Consequence Paolo Mancosu Department of Philosophy U.C. Berkeley mancosu@socrates.berkeley.edu

2. Alfred Tarski, On the Completeness and Categoricity of Deductive Systems, Unpublished typescript, Alfred Tarski Papers, Carton 15, Bancroft Library, U.C. Berkeley To appear in: P. Mancosu, The Adventure of Reason. Interplay between mathematical logic and philosophy of mathematics, 1900-1940, Oxford University Press, 2010.

3. Dear Van, The precise relationship between the completeness and the categoricity (both relative to logic) is the following one. Every categorical axiom-system is complete; the problem whether the converse holds remains open; if, however, a system which is complete possesses an interpretation in logic, it is categorical. I wonder when and where my paper concerned (“On completeness and categoricity of deductive theories”) will be published. You can quote the paper by Lindenbaum and myself “Über die Beschränkheit der Ausdrucksmitteln…” in Mengers “Ergebnisse Math. Coll.” (7 or 8?) since it contains essentially the same results. In this case you would have to add that the concept of “Nicht-gabelbarkeit” which is discussed there is equivalent to the relative completeness. But I would be glad if you could mention the forthcoming paper on completeness and categoricity or refer to my lecture at Harvard. Tarski to Quine, July 1, 1940. Quine archive, MS Storage 299, box 8, folder Tarski

4. The latter notion (synthetically complete), under the name ‘completeness relative to logic’, is due to Tarski. It is easier to formulate than the older concept of categoricity, and is related to the latter as follows: systems which are categorical (with respect to a given logic) are synthetically complete, and synthetically complete systems possessed of logical models are categorical. These matters were set forth by Tarski at the Harvard Logic Club in January, 1940 and will appear in a paper “On completeness and categoricity of deductive theories” Quine, Goodman, Elimination of Extra-logical Postulates JSL 5, 1940, footnote 3, 109

5. A system of sentences of a deductive theory is called absolutely complete or simply complete if every sentence which can be formulated in the language of this theory is decidable, that is, either derivable or refutable in this system. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p.1

6. In general the domain of application of this result is very extensive, and is not essentially limited by the premises which condition the incompleteness of the system. For it is well known that the arithmetic of whole numbers can be formalized within any deductive theory with a sufficiently rich logical structure, even if the concepts of arithmetic themselves do not occur explicitly in this theory. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 3

7. If the logical basis of our theory is rich enough, we can formalize within its boundaries the arithmetic of natural numbers and for this reason its logical basis is incomplete: i.e. there are logical sentences which are not logically valid and whose negations are likewise not logically valid. In other words, there are problems belonging entirely to the logical part of our theory which cannot be solved either affirmatively or negatively with the purely logical devices at our disposal. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 4

8. Definition. Two sentences A and B are equivalent with respect to a theory T iff T  {A}|-B and T  {B}|-A. Definition. A theory T is relatively complete (or complete with respect to its logical basis) iff for every B in L(T) there is an A expressed in the logical vocabulary of T such that A and B are equivalent with respect to T. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940.

9. The most important role is played here by the concept of model or realization. Let us consider a system of non-logical sentences and let, for instance “C1”, “C2”…“Cn” be all the non-logical constants which occur. If we replace these constants by variables “X1”, “X2”…“Xn” our sentences are transformed into sentential functions with n free variables and we can say that these functions express certain relations between n objects or certain conditions to be fulfilled by n objects. Now we call a system of n objects O1, O2,…,On a model of the considered system of sentences if these objects really fulfill all conditions expressed in the obtained sentential functions. It is of course possible that the whole system reduces to one sentence; in this case we speak simply of the model of this sentence. We now say that a given sentence is a logical consequence of the system of sentences if every model of the system is likewise a model of this sentence. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 5

10. An axiomatic theory is called semantically complete (relative to a given semantics) if any of the following four equivalent conditions holds: (1) For all formulas j and all models M, N of T, if M|=j then N|= j. (2) For all formulas j, either T |=j or T |= ¬j. (3) For all formulas j, either T |=j or T  {j}is not satisfiable (4) There is no formula j such that both T {j} and T {¬j} are satisfiable. S. Awodey, E. Reck, Completeness and Categoricity, HPL, 23, 2002.

11. Thus a system of sentences of a given deductive theory is called semantically complete if every sentence which can be formulated in the given theory is such that either it or its negation is a logical consequence of the considered set of sentences. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 5

12. It should be noted that the condition just mentioned is satisfied by any logical sentence: hence we can deduce without difficulty that the concept of semantical completeness is a generalization of the concept of relative completeness: every system that is relatively complete is likewise semantically complete (but it can be shown by an example that the converse is not true). Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 5

13. P. Mancosu, Fixed versus variable domain interpretations of Tarski’s account of logical consequence, Philosophy Compass, 4/1, 2010, 1-15 Consequences for logical consequence; the debate on the fixed domain interpretation

14. We shall distinguish here two variants of the concept of categoricity: semantical categoricity, and categoricity with respect to the logical basis or relative categoricity, which parallel respectively semantical and relative completeness. But we do not know and therefore do not introduce any concept of categoricity which would parallel the absolute completeness. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 5

15. We shall call a system of sentences semantically categorical if any two models of this system are isomorphic. The concept of isomorphism is a well known logical concept, not a methodological or semantical one, and thus I could take it that it is understood; but in any case I should like to have it noted that the precise definition of this concept depends on the logical foundation of our methodological investigations. Roughly speaking [and adapting ourselves to the language of Principia Mathematica, but disregarding the theory of types] we can say that two system of objects [classes, relations, etc.] O1, O2, …, On and P1, P2, …, Pn are isomorphic if there is a one-one correspondence which maps the class of all individuals onto itself and simultaneously the objects O1, O2, …, On onto P1, P2, …, Pn respectively. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, pp. 5-6

16. In order to obtain the second variant of the concept of categoricity we shall confine ourselves for the sake of simplicity, to such a case in which the considered system of sentences is finite and contains only one non-logical constant, say “C”. Let “P(C)” represent the logical product of all these sentences. “C” can denote, for instance, a class of individuals or a relation between individuals, or a class of such classes or relations etc. We assume further that the logical structure of the deductive theory is rich enough to express the fact that two objects X and Y of the same logical type as C are isomorphic and we assume that this fact is expressed by the formula “X~Y”. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 6

17. We can now correlate with the semantical sentence stating that our system is semantically categorical an equivalent sentence formulated in the language of the deductive theory itself. This is the following sentence: For every X and Y, if P(X) and P(Y) then X~Y Or in symbols (X)(Y)[P(X)&P(Y)X~Y] Now we say that the considered system of sentences is categorical with respect to its logical basis, or relatively categorical, if the sentence formulated above is logically valid. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 6

18. Theorem I. Every system of sentences which is categorical with respect to its logical basis is also complete with respect to this basis. Theorem II. Every semantically categorical system of sentences is also semantically complete Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 6

19. We use the word ‘categorical’ in a different, somewhat stronger sense than is customary: usually it is required of the relation R […] only that it maps x’, y’, z’, …, onto x’’, y’’, z’’,… respectively, but not that it maps the class of all individuals onto itself. The sets of sentences which are categorical in the usual (Veblen’s) sense can be called intrinsically categorical, those in the new sense absolutely categorical. The axiom systems of various deductive theories are for the most part intrinsically but not absolutely categorical. It is, however, easy to make them absolutely categorical. It suffices, for example, to add a single sentence to the axiom system of geometry which asserts that every individual is a point (or more generally one which determines the number of individuals which are not points). Tarski, Einige methodologische Untersuchungen über die Definierbarkeit der Begriffe, 1934/5

20. There are some problems concerning the mutual relations of the concepts of completeness and categoricity which still remain open; for example, the problem as to whether the converses of theorems I and II are true.We know many systems of sentences that are categorical; we know, for instance, categorical systems of axioms for the arithmetic of natural, integral, rational, real, and complex numbers, for the metric, affine, projective geometry of any number of dimensions etc. All these systems are semantically categorical, but if we base them on a sufficiently rich logic they become also categorical with respect to the logical basis. From theorems I and II we see that all mentioned systems are at the same time semantically or relatively complete. Thus, in opposition to absolute completeness, relative or semantical completeness occurs as a common phenomenon. Tarski, On the Completeness and Categoricity of Deductive Systems, 1940, p. 7

21. Is there a theory in second order logic with finitely many axioms (and thus finitely many non-logical constants) that is semantically complete but not categorical? Question

22. 1. (Solovay, see FOM, 5/16/06): Let A be a second order theory. ZFC+ V=L proves “If A is finitely axiomatizable and complete then A is categorical”. 2. (Solovay, see FOM, 5/16/06): there is a model of ZFC+V≠L where there is a finitely axiomatizable and complete second order theory which is not categorical. Answer

23. The argument can be spelled out as follows. Suppose S is relatively complete. As Tarski does let us assume that from S all logically valid sentences can be derived. Let f be an arbitrary sentence in L(S) (L(S) includes also all the logical symbols). We want to show that either f or ¬ f is a logical consequence of S. By definition of relative completeness for any sentence f there is a logical sentence f * such that S  { f } |- f* and S  { f*} |- f. Every system that is relatively complete is semantically complete.

24. If f is a logical sentence then, by claim C, it or its negation is a logical consequence of S, so there is nothing to prove. If f is not logical then let f* be a logical sentence satisfying the condition given in the definition of relative completeness for S. We consider two cases. Because f* is a logical sentence either all models of S are models of f* or all models of S are models of ¬f*. First assume all models of S are models of f*. Then any model M of S is also a model of S  { f*}. Since S  { f*}|- f and the logical system is assumed sound, M is a model of f. So f is a logical consequence of S. Every system that is relatively complete is semantically complete.

25. Now assume that every model of S is a model of ¬ f*. Furthermore, by way of contradiction, assume there is a model M’ of S such that M’ is not a model of ¬f. Thus M’ is a model of f. Because S  { f } |- f* hence by the soundness of the logical system, M’ is a model of f*. This contradicts all models of S being models of ¬f*. So ¬f is a logical consequence of S. Consequently, S is semantically complete. Every system that is relatively complete is semantically complete.