## 1. Biographical sketch



Tarski was born on January 14, 1901 in Warsaw, then a part of the
Russian Empire. His family name at birth was Tajtelbaum, changed to
Tarski in 1923. He studied mathematics and philosophy at the
University of Warsaw from 1918 to 1924, taking courses with
Kotarbinski, Lesniewski, Lukasiewicz, Mazurkiewicz and
Sierpinski among others. (See Lvov-Warsaw School.) He got a
doctoral degree, under Lesniewski's supervision, in 1924.
From then until 1939 he taught mathematics at a high school and held
minor teaching positions at the University of Warsaw. In this period
he published prolifically on logic and set theory, building a strong
international reputation for himself. Yet he failed in his attempt to
obtain a professorship at the University of Lvov (now Lviv) in 1939.
In 1929 he married Maria Witkowska, with whom he soon had two
children, Ina and Jan.



In August 1939 Tarski traveled to the United States to attend a
congress of the Unity of Science movement (see Vienna Circle). World
War II broke out soon after that, leaving Tarski no option but to stay
in the States. He spent the war years separated from his family,
forced to remain in Poland. In this period he held several temporary
university positions, at Harvard University, the City College of New
York, the Institute for Advanced Study at Princeton, and the
University of California at Berkeley, where he was eventually given
tenure in 1945 and a professorship in mathematics in 1948. Maria, Ina
and Jan were able to join him in Berkeley in 1946.



In Berkeley Tarski built a prominent school of research in logic and
the foundations of mathematics and science, centered around the
prestigious graduate program in logic and methodology of science,
which he was also instrumental in creating. He received many academic
honors, such as the membership in the National Academy of Sciences of
the USA and the election as Corresponding Fellow of the British
Academy. Tarski remained affiliated to Berkeley until his death, on
October 27, 1983.


## 2. Truth



In a logic seminar he gave at the University of Warsaw between 1927
and 1929, Tarski proved several results that made reference to notions
that later would be called "semantical", in particular
results about the notions of definability and of truth in a structure
(see Vaught 1974 and 1986). Vaught informs us that Tarski found
certain difficulties when trying to give a mathematically satisfactory
form to the results presented in the seminar, and that this led him to
look for a precise theory of the semantical notions (cf. Vaught 1974,
pp. 160ff. and 1986, pp. 870ff.). Such a theory did not exist at the
time. In particular, there were no definitions of these notions in
terms of concepts accepted in the foundational systems designed for
the reconstruction of classical mathematics (for example, Russell and
Whitehead's theory of types or Zermelo's set theory).
Because of this, the existing results in which these notions appeared
could not be reconstructed in the accepted foundational systems. And
there was no rigorous axiomatic theory of the semantical notions in
which these were taken as primitive, either.



In spite of this, the results about semantical notions were important
and even moderately abundant by 1930. Among these Tarski himself would
mention Godel's completeness theorem and several versions
of the Lowenheim-Skolem theorem (cf. Tarski 1983b, pp.
240-1, sp. p. 240 n.1, and p. 241 n. 2). All these results use
the notion of truth in a structure, or functionally equivalent
notions. And Tarski says that



> 
> it is evident that all these results only receive a clear content and
> can only then be exactly proved, if a concrete and precisely
> formulated definition of [true] sentence is accepted as a basis for
> the investigation (Tarski 1983b, p. 241).
> 



Tarski's aim in a series of works on semantical notions was
precisely to propose mathematically acceptable definitions of those
notions, and in particular acceptable in some one of the chief
foundational systems (the system chosen by Tarski will change with
time; see below). With appropriate definitions of those notions, the
theorems about them would be susceptible of being reformulated making
use of the defined notions, and the uneasiness described by Tarski in
the text just quoted would be alleviated. The alternative of taking
the semantical notions as primitive is also considered by Tarski in
several places, but he clearly prefers to avoid it if possible. The
reason is that in the axiomatic alternative there is no negligible
risk that there remain ways of generating the semantical antinomies
for the semantical primitives of the system (cf. Tarski 1983b, p.
255). With the definitional procedure, however, the consistency of the
definition would depend exclusively on the consistency of the theory
in which it is formulated, and this will be a theory we have reasons
to think is consistent.



The first work containing a mathematical definition of one of the
semantical notions is Tarski (1931) (English version, Tarski 1983d),
where Tarski examines a language *L* in which it is possible to
formalize the arithmetic of the real numbers, and gives a recursive
definition of the notion "set of real numbers definable in
*L*" (cf. Def. 9 in Tarski 1983d, p. 128). The definition
depends on others that are a bit cumbersome; today we would define a
similar notion in a simpler way with the help of the definition of
truth that Tarski himself would publish two years later. But the
essence of the Tarskian definition of truth is already here. In the
1931 paper we find also, as was to be expected, the Tarskian concern
with matters of precision and foundational rigor, and a revealing
statement about the attitude of mathematicians toward the notion of
definability, that could be extended to other semantical notions:



> 
> The distrust of mathematicians towards the notion in question is
> reinforced by the current opinion that this notion is outside the
> proper limits of mathematics altogether. The problems of making its
> meaning more precise, of removing the confusions and misunderstandings
> connected with it, and of establishing its fundamental properties
> belong to another branch of science--metamathematics (Tarski
> 1983d, p. 110).
> 



In his classic monograph on the concept of truth "The Concept of
Truth in Formalized Languages" (Polish original version, Tarski
1933; German translation with an added postscript, Tarski 1935;
English version of the German text, Tarski 1983b), Tarski will present
a method for constructing definitions of truth for classical
quantificational formal languages. When one succeeds in applying this
method to a particular formal language, the end result will be the
construction of a predicate in a metalanguage for that language whose
essential properties will be that it will be constructed out of
non-suspicious mathematical vocabulary and that it will be intuitively
satisfied precisely by the intuitively true sentences of the object
language. At the same time, Tarski shows how, in terms of the defined
notion of truth, one can give intuitively adequate definitions of the
semantical notions of definability and denotation, and he indicates
how it is possible to define the notion of truth in a structure in a
way analogous to the one used to define
 truth.[1]
 That allows him to conclude that the acceptability and rigor of the
"recent methodological studies" (of Lowenheim,
Skolem, Godel and Tarski himself, among others), have been
vindicated (cf. Tarski 1983b, p. 266).



Languages, both object languages and metalanguages, are in the
monograph not just interpreted grammars; a language also includes a
deductive system. A Tarskian metalanguage always includes its object
language as a part, both its grammar (perhaps under some translation)
and its deductive system. And besides, it will always contain a few
more things if these things are not already in the object language.
Specifically, it will always contain a sub-language and sub-theory
that can be used to say a great deal of things about the syntax of the
object language; and it will always contain all the mathematics (both
concepts and methods of proof) that "may be taken from any
sufficiently developed system of mathematical logic" (Tarski
1983b, p. 211).



Tarski doesn't give a general formulation of his method. Such a
general formulation would not be too illuminating. Instead he
helpfully chooses to illustrate his method as it would work for a few
languages. The most basic example he uses, to which about half of his
monograph is devoted, is what he calls "the language of the
calculus of classes" (LCC). Here is a quick description in more
current notation (Tarski uses Lukasiewicz's Polish
notation):


1. primitive signs of LCC: [?], [?], !, (, ), I (a binary
predicate), *x*, ' (a subindex
accent to generate variables by suffixing to "*x*";
let's use the notation *x**n* for the
*x* followed by *n* accents).
2. grammar of LCC: Atomic formulae are of the form
I*x**k**x**l*.
Complex formulae are obtained by negation, disjunction and universal
quantification.
3. interpretation of LCC: The range of the variables is the class of
all subclasses of individuals (of the universe). I stands for the
relation of inclusion among these subclasses. The other signs mean
what you would expect.



Besides all this, Tarski gives a deductive system for LCC, as required
of every language.



Tarski's most basic idea about how to construct a truth
predicate, a predicate which will be intuitively satisfied exactly by
the intuitively true sentences of LCC, is that such a predicate should
verify what he calls "convention **T**". This
convention imposes a condition on defined truth predicates ((a) in the
example immediately below) that Tarski often talks about as grasping
the intuitions behind the "classical Aristotelian conception of
truth" (cf. Tarski 1983b, p. 155; 1944, pp. 342-3).
Convention **T** is here spelled out for the case of
LCC:




**Convention T.** A formally correct definition of the
symbol "Tr", formulated in the metalanguage, will be
called an *adequate definition of truth* if the deductive
system of the metatheory proves the following:


1. all sentences which are obtained from the expression
"Tr(*x*) if and only if *p*" by substituting
for the symbol "*x*" a structural-descriptive name
of any sentence of the language in question and for the symbol
"*p*" the expression which forms the translation of
this sentence into the metalanguage;
2. the sentence "for any *x*, if Tr(*x*) then
*x* is a sentence of LCC" (cf. Tarski 1983b, pp.
187-8).




That a definition is "formally correct" means, roughly,
that it will be constructed out of non-suspicious vocabulary following
non-controversial rules for defining new expressions. But why should
one think that a predicate verifying convention **T**
should be, besides, coextensional with the intuitive predicate of
truth for LCC? The reason can be given by an intuitive argument
relating both predicates, such as the following (cf. Tarski 1944, pp.
353-4):




(=) Suppose that "*p*" is a sentence of LCC
such that Tr("*p*"). By assumption,
"Tr("*p*") if and only if *p*"
is provable in the metalanguage. So Tr("*p*") if
and only if *p* (the metalanguage must only prove truths). So
*p*. So "*p*" is true.



(=) Suppose that "*p*" is true. Then
*p*. And then, as before, Tr("*p*").




Tarski defines truth in terms of the notion of the satisfaction of a
formula of LCC by an infinite sequence of assignments (of appropriate
objects: subclasses of the universe of individuals in the case of
LCC). He gives first a recursive definition and immediately indicates
how to transform it into a normal or explicit definition. The
recursive definition is this: an infinite sequence of classes
*f* satisfies formula *F* if and only if *f* and
*F* are such that


1. there are natural numbers *k* and *l* such that
*F* =
I*x**k**x**l* and
*f*(*k*)[?]*f*(*l*); or
2. there is a formula *G* such that *F*=!*G*
and *f* does not satisfy *G*; or
3. there are formulae *G* and *H* such that
*F*=(*G*[?]*H*) and *f* satisfies
*G* or *f* satisfies *H*; or finally,
4. there is a natural number *k* and a formula *G* such
that *F*=[?]*x**k**G* and
every infinite sequence of subclasses that differs from *f* at
most in the *k*th place satisfies *G* (cf. Tarski 1983b,
p. 193).



The normal definition is this:




An infinite sequence of subclasses *f* satisfies a formula
*F* if and only if *f* and *F* stand in every
relation *R* among sequences and formulae such that, for any
*g* and *G*, *g* and *G* stand in
*R* if and only if *g* is an infinite sequence of
subclasses, *F* is a formula and either


1. there are natural numbers *k* and *l* such that
*G* =
I*x**k**x**l* and
*g*(*k*)[?]*g*(*l*); or
2. there is a formula *H* such that *G* is
!*H* and *g* and *H* don't stand in
*R*; or
3. there are formulae *H* and *I* such that *G*
is (*H*[?]*I*) and *g* and *H* stand in
*R* or *g* and *I* stand in *R*; or
4. there is a natural number *k* and a formula *H* such
that *G* is [?]*x**k**H*
and, for every infinite sequence of subclasses *h* that differs
from *g* at most in the *k*th place, *h* and
*H* stand in *R* (cf. Tarski 1983b, p. 193, n. 1).




Tarski then defines the truth predicate as follows:



> 
> For all *x*, Tr(*x*) if and only if *x* is a
> sentence of LCC and every infinite sequence of subclasses satisfies
> *x* (cf. Tarski 1983b, p. 195).
> 



Given the way the definition has been constructed, it is intuitively
clear that the metalanguage will prove all biconditionals of the
convention **T** for LCC. Tarski does not prove this
cumbersome fact in a metametalanguage, and he helpfully contents
himself with showing how a few of the biconditionals would be
established in the metalanguage.



The usual reasonings leading to the semantical antinomies cannot be
reproduced for the Tarskian defined semantical concepts. In
particular, the antinomy of the liar cannot be reproduced using the
defined predicate Tr. Consider the version of the antinomy offered by
Tarski (due to Lukasiewicz). Abbreviate the expression "the
first underlined sentence of this entry" with the letter
"*c*". Now consider the sentence:



*c* is not a true sentence.


Given what "*c*" abbreviates, we get



(a) "*c* is not a true sentence" = *c*


And the following is intuitively true:



(b) "*c* is not a true sentence" is a true sentence
if and only if *c* is not a true sentence.


From (a) and (b) follows the contradiction



*c* is a true sentence if and only if *c* is not a true
sentence.


The reason why this cannot be reproduced for Tr is that this predicate
is always a predicate of a language (the metalanguage) different from
the language of the sentences to which it applies (the object
language). It is not possible to form a sentence of the language for
which one defines Tr that says of itself that it is (not) Tr, since Tr
is not a predicate of that language. On the other hand, it is
certainly possible in some cases to form a sentence *S* of the
metalanguage that says of itself that it is not Tr, but since
*S* is a sentence of the metalanguage it is simply true and
(presumably) not paradoxical, since there is no reason why a
biconditional like (b) should hold for it; biconditionals of this kind
hold only for sentences of the language for which Tr has been defined,
not for sentences of the metalanguage.



LCC has the property of being what Tarski calls a "language of
finite order". The idea of order is familiar. Variables for
individuals (non-existent in LCC) are of order 1. Variables for
classes of individuals (all the variables of LCC) are of order 2.
Variables for classes of classes of individuals are of order 3, and so
on. The order of a language of this series is the greatest positive
integer *n* with variables of order *n*. (Thus, although
LCC looks grammatically just like a first-order language, semantically
it can be seen as a fragment of what under present conventions would
be called a second-order language.)



But there are also languages of infinite order. In these the order of
the available variables is not bounded above. Tarski gives an example,
which he calls "the language of the general theory of
classes" (LGTC). Here again is a quick description in more
current notation:


1. primitive signs of LGTC: [?], [?], !, (, ), X,
', ' (The last two are a subindex
accent and a superindex accent to generate variables by suffixing to
"*X*"; let's use the notation
*X**n**m* for the
*X* followed by *n* subindex accents and *m*
superindex accents. The superindex must be positive.)
2. grammar of LGTC: Atomic formulae are of the form
*X**n**m*+1*X**p**m*.
Complex formulae are obtained by negation, disjunction and universal
quantification with respect to the variables of all orders.
3. interpretation of LGTC: A variable of the form
*X**n*1 takes as values
individuals, a variable of the form
*X**n*2 takes as values classes of
individuals, a variable of the form
*X**n*3 takes as values classes of
classes of individuals, etc.
*X**n**m*+1*X**p**m*
"means" that the object assigned to
*X**n**m*+1
"contains" the object assigned to
*X**p**m*. The other signs
mean what you would expect.



Besides all this, Tarski gives a deductive system for LGTC, as
required of every language. Suffice it to say that it is a version of
a typical simple theory of finite types, with axioms and rules for the
connectives and quantifiers, axioms of comprehension and
extensionality for all orders, and an axiom of infinity. Tarski
observes that LGTC suffices, with the help of some tricks, to develop
all the mathematics that can be developed in the simple theory of
finite types, even though it has formally fewer types of
variables.



In the 1933 original Polish version of his monograph Tarski says that
his method for constructing truth predicates cannot be applied to the
construction of a truth predicate for LGTC. The problem is that in
1933 Tarski adopts as the mathematical apparatus of his metalanguages
the simple theory of finite types, or equivalently, LGTC. In this
language, supplemented (and even unsupplemented) by a theory of syntax
for LCC, Tarski has everything he needs to give his definition of
satisfaction for LCC. In particular, the relation of satisfaction for
LCC is a relation that can be found in the hierarchy of finite types.
(Full set theory is not needed.) And the form in which Tarski defines
it quantifies only over sequences of classes and over relations among
sequences of classes and formulae (as we may check above), and these
are objects quantifiable over in finite type theory.



But the construction of a Tarskian truth predicate for LGTC could not
be carried out in finite type theory. The relation of satisfaction for
LGTC is intuitively a relation among formulae and sequences of:
sequences of individuals, sequences of classes of individuals,
sequences of classes of classes of individuals, etc. And this relation
is not a relation in the hierarchy of finite types. Much less can one
quantify in finite type theory over the objects necessary to apply the
Tarskian method. One cannot quantify over sequences of: sequences of
individuals, sequences of classes of individuals, sequences of classes
of classes of individuals, etc.; and one cannot quantify over
relations among formulae and sequences of: sequences of individuals,
sequences of classes of individuals, sequences of classes of classes
of individuals, etc.



It is this situation that leads Tarski to wonder



> 
> whether our failure is accidental and in some way connected with
> defects in the methods actually used, or whether obstacles of a
> fundamental kind play a part which are connected with the nature of
> the concepts we wish to define, or of those with the help of which we
> have tried to construct the required definitions (Tarski 1983b, p.
> 246).
> 



There is a problem even in formulating the question precisely, for, as
Tarski notes,



> 
> if the second supposition [of the preceding quotation] is the correct
> one all efforts intended to improve the methods of construction would
> clearly be fruitless (Tarski 1983b, p. 246).
> 



But



> 
> it will be remembered that in the convention **T** of
> SS3 the conditions which decide the material correctness of any
> definition of true sentence are exactly stipulated. The construction
> of a definition which satisfies these conditions forms in fact the
> principal object of our investigation. From this standpoint the
> problem we are now considering takes on a precise form: it is a
> question of *whether on the basis of the metatheory of the language
> we are considering the construction of a correct definition of truth
> in the sense of convention* **T** *is in principle
> possible*. As we shall see, the problem in this form can be
> definitively solved, but in a *negative sense* (Tarski 1983b,
> p. 246).
> 



The negative answer is provided by a version of Tarski's theorem
of the indefinability of truth. The weakest Tarskian metalanguage for
LGTC will consist in expanding LGTC by means of concepts and a theory
of the syntax of LGTC. But the mathematical part of the metalanguage
will be the same, for that's the "sufficiently developed
system of mathematical logic" of which Tarski spoke. Further,
the metalanguage will obviously contain its object language as part,
both its grammar (under no translation) and its deductive system,
because LGTC will appear both in the object language and in the
metalanguage (cf. Tarski 1983b, p. 247).



In a word, the metalanguage, which we may call "LGTC+", is
LGTC plus a theory of the syntax of LGTC. The question that Tarski
poses is, then, whether there is a predicate constructible in LGTC+
which verifies the convention **T** for sentences of
LGTC. Tarski solves it negatively by proving the following
theorem:




**Theorem I.** (a) In whatever way the symbol
"Tr", denoting a class of expressions, is defined in
LGTC+, it will be possible to derive from it the negation of one of
the sentences which were described in the condition (a) of the
convention **T**;



(b) assuming that the class of all provable sentences of the
metatheory is consistent, it is impossible to construct an adequate
definition of truth in the sense of convention **T** on
the basis of LGTC+ (cf. Tarski 1983b, p. 247).




Part (b) of Theorem I is a trivial consequence of part (a), since it
is the mark of a consistent system that it doesn't contain a
sentence and its negation. The proof of part (a) is an application of
Godel's technique of diagonalization. The following are the
basic steps described using the anachronistic recourse to a more
familiar case and to a modern, streamlined notation.



View LGTC as if it were a standard language L for first-order
arithmetic. Suppose we extend it with a theory for talking about the
syntax of L (this would not be a Tarskian metalanguage, since it would
not include a "sufficiently developed system of mathematical
logic", but all this is just for purposes of illustration). Call
the extension L+. Then L+ can be "interpreted" in L using
Godelian techniques, by which Tarski means, roughly, that one can
translate L+ into L arithmetizing it in such a way that the
translations of theorems of L+ will be theorems of L. Now suppose that
ph(*n*) is the *n*th expression of L (and that this
function is definable in L+), that *d*(*x*) is a
diagonal function for expressions of L (which is definable in L+; here
I have in mind one involving substitution of numerals for free
variables), and let **n** be the numeral of *n*.
Then for every predicate *E*(*x*) of expressions of L
defined in L+ one can prove in L+ a general sentence of the form
[?]*n*[!*E*(*d*(ph(*n*)))-ps(*n*)],
where ps(*n*) is a formula with one free variable entirely
constructed in L (through the arithmetization of the L+ predicate
"!*E*(*d*(ph(*n*)))"). Then
one can also prove the sentence
!*E*(*d*(ph(**k**)))-ps(**k**),
where *k* is the Godel number of ps(*n*).
*d*(ph(**k**)) is the sentence
ps(**k**), so ps(**k**) is a
"fixed point" of the predicate !*E*(x): it
"says of itself" that it has the property expressed by
!*E*(x). The sentence
!E(*d*(ph(**k**)))-ps(**k**)
is already equivalent to the sentence
![*E*(*d*(ph(**k**)))-ps(**k**)],
and part (a) of Theorem I is proved.



In the postscript to the 1935 German translation of the monograph on
truth, Tarski abandons the 1933 requirement that the apparatus of the
metatheory be formalizable in finite type theory, and accepts the use
of a more powerful theory of transfinite types, where the transfinite
objects are classes of the objects of lower types, or the use of set
theory. Tarski notes that the relation of satisfaction and a truth
predicate for LGTC (and LGTC+) are definable in these more powerful
metatheories (cf. Tarski 1983b, pp. 271-2). But he also says
that the proof of the indefinability theorem can be adapted to show
that, in general, one cannot define a truth predicate "if the
order of the metalanguage is at most equal to that of the language
itself" (Tarski 1983b, p. 273).


## 3. Logical consequence



Tarski presents his theory of logical consequence in Tarski (1936a)
(German version, Tarski 1936b; English translation of the Polish,
Tarski 2002; English translation of the German, Tarski 1983c). This
classic paper begins with some general remarks on the possibility of a
precise definition of the concept of consequence. The essence of these
remarks is that since the common concept is vague, it seems certainly
difficult, and perhaps impossible, to reconcile all features of its
use in the definition of a corresponding precise concept.
Nevertheless, Tarski says, logicians had thought until recently that
they had managed to define a precise concept that coincided exactly in
extension with the intuitive concept of consequence. Tarski mentions
the extraordinary development of mathematical logic in recent decades,
which had shown "how to present mathematical disciplines in the
shape of formalized deductive theories" (Tarski 1983c, p. 409).
In these theories, consequences are extracted from axioms and theorems
by rules of inference, "such as the rules of substitution and
detachment" (Tarski 1983c, p. 410), of a purely syntactical (or
"structural", in Tarski's word) nature.
"Whenever a sentence follows from others, it can be obtained
from them--so it was thought--by means of the
transformations prescribed by the rules" (Tarski 1983c, p. 410).
According to Tarski, this belief of the logicians was justified by
"the fact that they had actually succeeded in reproducing in the
shape of formalized proofs all the exact reasonings which had ever
been carried out in mathematics" (Tarski 1983c, p. 410).



But Tarski goes on to claim that that belief of the logicians was
wrong. There are some non-vague cases in which a certain sentence of a
higher-order language follows in the intuitive sense from a set of
other sentences of that language but cannot be derived from them using
the accepted axioms and rules. These cases are provided by some
o-incomplete theories, theories in which for some predicate
*P* the sentences 




*A*0. 0 possesses the given property *P*,



*A*1. 1 possesses the given property *P*,




and, in general, all sentences of the form
*A**n* can be proved, but the universal
sentence



*A*. Every natural number possesses the given property
*P*,



cannot be proved on the basis of the accepted axioms and rules of
inference. (LGTC provides an example of such a theory.) "Yet
intuitively it seems certain that the universal sentence *A*
follows in the usual sense from the totality of particular sentences
*A*0, *A*1, ...,
*A**n*, .... Provided all these
sentences are true, the sentence *A* must also be true"
(Tarski 1983c, p. 411).



Tarski considers the possibility of adding an o-rule to the
accepted rules of inference, that is, a rule which allows us to deduce
a universal sentence of the form of *A* from the set of
sentences *A*0, *A*1, etc.
However, he says that such a rule would differ in very essential
respects from the old rules: it is not a finitary rule, while all the
accepted rules in common deductive systems are finitary. Also, Tarski
immediately takes importance away from the further suggestion of
supplementing the old system of rules by means of a formalized
finitary o-rule. He points out that in view of
Godel's incompleteness results, no matter how many new
finitary rules or axioms we add to certain higher-order theories, they
will still remain incomplete theories, in fact o-incomplete
theories. This discussion is enough to show that "in order to
obtain the proper concept of consequence, which is close in essentials
to the common concept, we must resort to quite different methods and
apply quite different conceptual apparatus in defining it"
(Tarski 1983c, p. 413). The different methods and the different
conceptual apparatus that Tarski has in mind are going to be
"the methods which have been developed in recent years for the
establishment of scientific semantics, and the concepts defined with
their aid" (Tarski 1983c, p. 414; a footnote refers us to
Tarski's monograph on truth).



The best way of appreciating the nature of Tarski's theory of
logical consequence and how it constitutes an application of the
methods of Tarskian semantics is to see how it applies to a particular
formal language of a simple structure, of a kind that Tarski seems to
have primarily in mind in his paper of 1936. We will choose a simple
language of this kind for a fragment of elementary arithmetic. This
language, LAr, can be given this quick description:


1. primitive signs of LAr: [?], -, !, (, ), x,
' (a subindex accent to generate variables
by suffixing to "*x*"; let's use again the
notation *x**n* for the *x* followed
by *n* accents), 0 (an individual constant), N (a monadic
predicate), M (a dyadic predicate).
2. grammar of LAr: Atomic formulae are of the forms
N*t*1 and
M*t*1*t*2, where
*t*1 and *t*2 are either variables
or the constant 0. Complex formulae are obtained by negation,
conditionalization and universal quantification relativized to
"N".
3. interpretation of LAr: The range of the variables is the set of
all natural numbers. N stands for the set of natural numbers and M for
the relation of being less than among natural numbers. The other signs
mean what you would expect.



In the 1936 paper Tarski seems to be thinking paradigmatically of
languages in which (as in LAr) there is a predicate ("N"
in the case of LAr) that applies exactly to the individuals in the
domain of the intended interpretation of the language. He says
elsewhere that the variables of such a language range exclusively over
the individuals of that set (cf. Tarski 1937, p. 84). (This is not to
say that Tarski adopts these conventions with all the languages he
considers. For example, he does not adopt them with languages which,
like LGTC, are used to talk about arbitrary individuals, as opposed to
specific sets of
 them).[2]



In the considerations preliminary to the presentation of his theory,
Tarski says that when a sentence *X* of a formal language
(e.g., LAr) is a logical consequence of a set *K* of sentences
of that language, the argument with premises *K* and conclusion
*X* has the following property, that Tarski calls
"condition (F)":



> 
> (F) If, in the sentences of the class *K* and in the sentence
> *X*, the constants--apart from purely logical
> constants--are replaced by any other constants (like signs being
> everywhere replaced by like signs), and if we denote the class of
> sentences thus obtained from *K* by
> "*K*'", and the sentence obtained from
> *X* by "*X*'", then the sentence
> *X*' must be true provided only that all sentences of the
> class *K*' are true (Tarski 1983c, p. 415).
> 



Let's clarify the sense of condition (F) with an example.
Consider a language LAr+ which is like LAr but has besides another
individual constant, "2", and another dyadic predicate,
"Pd", whose desired interpretations are the number 2 and
the relation of being the immediate predecessor of, respectively. Let
*K* be the following set of sentences of LAr+:
{"[?]*x*(N*x*-!M*x*0)",
"N0"} (these sentences are true); and let *X* be
the sentence "!M00". The argument with premises
*K* and conclusion *X* is intuitively logically correct;
thus, according to Tarski, it must verify condition (F). This means
that *any* argument obtained from it by uniform replacement of
non-logical constants by non-logical constants must be an argument
where *it is not the case that the premises are true and the
conclusion false*. Let's suppose that the non-logical
constants of LAr+ are "0", "N",
"M", "2", and "Pd". Replace
"0" with "2" and "M" with
"Pd" in the argument with premises *K* and
conclusion *X* and call the resulting set of premises and
conclusion "*K*'" and
"*X*'". That is, *K*' is
{"[?]*x*(N*x*-!Pd*x*2)",
"N2"} and *X*' is "!Pd22". In
virtue of condition (F), the argument with premises *K*'
and conclusion *X*' must be an argument where it is not
the case that the premises are true and the conclusion false; and
that's in fact the case: the first premise is false, and the
second premise and the conclusion are true.



Tarski wonders if it is possible to offer condition (F) as a
*definition* of the relation of logical consequence, that is,
if we can take (F) not only as a necessary but also a sufficient
condition for an argument to be an instance of logical consequence.
His answer is that we cannot. The reason is that condition (F)



> 
> may in fact be satisfied only because the language we are dealing with
> does not possess a sufficient stock of extra-logical constants. The
> condition (F) could be regarded as sufficient for the sentence
> *X* to follow from the class *K* only if the
> designations of all possible objects occurred in the language in
> question. This assumption, however, is fictitious and can never be
> realized (Tarski 1983b, pp. 415-6).
> 



Tarski notes that in order for an argument to be an instance of
logical consequence it need not be sufficient that all arguments of
the same form be arguments where it is not the case that the premises
are true and the conclusion false. It is conceivable that one may
interpret the non-logical constants of the argument by means of
certain objects (individuals, sets, etc.) in such a way that the
premises thus reinterpreted become true and the conclusion becomes
false, and that nevertheless (some of) those objects not be denoted by
non-logical constants of the language that is being considered; in
such a case we would not say that the argument is an instance of
logical consequence, in spite of the fact that it would satisfy
condition (F).



To give an example, suppose that the language we are considering is
LAr+. Since both the relation of being less than and the relation of
being the immediate predecessor of are irreflexive over the domain of
the natural numbers, the sentence
"[?]x(N*x*-!M*x**x*)"
would be a logical consequence of every set of premises on criterion
(F): no replacement of the non-logical constants "N" and
"M" by other non-logical constants of LAr+ turns that
sentence into a falsehood. But clearly
"[?]*x*(N*x*-!M*x**x*)"
is not a logical consequence of, say, "N0". This can be
justified, e.g., keeping fixed the usual interpretation of
"0" and "N" but observing that "M"
can be interpreted by means of the reflexive relation of being less
than or equal to; under this interpretation,
"[?]*x*(N*x*-!M*x**x*)"
is false, although "N0" is true. (Tarski's remark
that the supposition that all objects have names in the language can
never be realized can be justified, for example, by observing that
there are non-denumerably many sets of natural numbers, but in the
languages he considers there are only denumerably many constants.)



Tarski's proposal consists in making tighter the requirement
expressed by condition (F), so as to incorporate the idea that a
logically correct argument cannot be *reinterpreted* in such a
way that the premises become true and the conclusion false; in other
words, the idea that a sentence *X* is a logical consequence of
a set of sentences *K* when *every interpretation on which
all the sentences of K are true is an interpretation on which X is
true* (or, to use a common terminology, when every interpretation
*preserves* the truth of the premises in the conclusion). 



As Tarski says, the idea of understanding the notion of logical
consequence by means of the notion of truth preservation on all
interpretations is not an original idea of his, but one implicit in
the logical and mathematical practice of his time (especially among
mathematicians interested in offering independence proofs). What is
new to Tarski's proposal is that he makes precise the idea using
the apparatus he had developed for the mathematical characterization
of satisfaction and truth. He does not give a detailed example, but it
seems reasonably clear how he would proceed from the indications he
gives in (1936a), (1936b) and (1937).



Tarski uses a certain precise notion of interpretation for a formal
language. In our example, an *interpretation* of LAr is a
sequence <*A*, *a*, *R*> that assigns
appropriate objects to the non-logical constants of LAr: a set of
individuals *A* to "N", an individual *a* to
"0" and a binary relation among individuals *R* to
"M". Besides, he always requires that the objects assigned
by an interpretation to the other non-logical constants of a language
be drawn from the set assigned to the non-logical predicate that
restricts the range of quantification to a specific set of individuals
("N" in the case of LAr) (cf. Tarski 1937, SS34); so
in the case of LAr the individual assigned to "0" must
belong to the set assigned to "N" and the relation
assigned to "M" must be a relation among objects in the
set assigned to "N".



Tarski introduces the notion of a *sentential function*. A
sentential function *S*' of a sentence *S* is the
result of uniformly replacing the non-logical constants appearing in
*S* with corresponding variables of suitable types (and
different from the variables already existing in the language). For
example, the sentential function determined by the sentence
"[?]*x*(N*x*-!M*x*0)"
is the expression
"[?]*x*(*P**x*-!*Y**x**y*)"
(in which "*P*", "*Y*" and
"*y*" are new variables). It is equally possible to
define the more general notion of a *formula function*, in an
analogous way, except that now *S* may be an open formula. The
sentential functions of sentences of LAr will not in general be
sentences, and so will not always be either true or false by
themselves. But they will always be either true or false *with
respect to interpretations of LAr*; or, as Tarski says, they will
be *satisfied* or not by interpretations of LAr.



The concept of satisfaction of a sentential function by an
interpretation can be defined using the Tarskian method for defining
satisfaction. Say that *an interpretation <A, a, R> of LAr
satisfies the formula function X with respect to a sequence f*
(that assigns values from *A* to the original variables of LAr)
if and only if: 


1. (i) *X* is *P**x**n* (for
some *n*); or *X* is *P**y*; or (ii)
*X* is
*Y**x**n**x**m*
(for some *m* and *n*) and
<*f*(*x**n*),*f*(*x**m*)>[?]*R*;
or *X* is *Y**y**x**n*
(for some *n*) and
<*a*,*f*(*x**n*)>[?]*R*;
or *X* is *Y**x**n**y*
(for some *n*) and
<*f*(*x**n*),*a*>[?]*R*;
or *X* is *Y**y**y* and
<*a*,*a*>[?]*R*; or
2. there is a formula function *Y* such that *X* is
!*Y* and <*A*, *a*, *R*> does
not satisfy *Y* with respect to sequence *f*; or
3. there are formula functions *Y* and *Z* such that
*X* is (*Y*-*Z*) and either <*A*,
*a*, *R*> does not satisfy *Y* with respect to
sequence *f* or <*A*, *a*, *R*>
satisfies *Z* with respect to sequence *f*; or,
finally,
4. there is a formula function *Z* and a number *n*
such that *X* is
[?]*x**n*(*P**x**n*-*Z*)
and every sequence *g* that assigns values from *A* to
the (original) variables of LAr and that differs from *f* at
most in what it assigns to *x**n* is such
that <*A*, *a*, *R*> satisfies *Z*
with respect to
 *g*.[3]



This is a recursive definition, entirely parallel to the definition of
satisfaction of formulae by sequences in the definition of
satisfaction for LCC in Tarski's monograph on truth, that we saw
in section 1 above. In the same way that that definition, the one just
given can be turned into an explicit one by the same method.



The notion of satisfaction of a sentential function is easy to
characterize in terms of the defined notion of satisfaction of a
formula function. Say that *an interpretation* <*A*,
*a*, *R*> *satisfies sentential function X* if
and only if <*A*, *a*, *R*> satisfies
formula function *X* with respect to every sequence. This
definition is analogous to the Tarskian definition of truth.



In terms of the defined notion of satisfaction, Tarski introduces the
notion of a *model* of a sentence. A model of a sentence
*S* is an interpretation that satisfies the sentential function
*S*' determined by *S*; more generally, a model of
a set of sentences *K* is an interpretation that satisfies all
the sentential functions determined by sentences of *K*. And in
terms of the defined notion of model Tarski proposes his defined
notion of logical consequence. A sentence *X* is a (Tarskian)
logical consequence of the sentences in set *K* if and only if
every model of the set *K* is also a model of sentence
*X* (cf. Tarski 1983c, p. 417). Tarski proposes also a defined
notion of logical truth (he uses the expression "analytic
truth") using the same apparatus: a sentence *S* is a
(Tarskian) logical truth if and only if every interpretation of
*S* is a model of *S*. Analogous notions of Tarskian
logical consequence and logical truth can be defined for other
languages using the same method we have followed with LAr, just making
the obvious changes.



After presenting his definition of logical consequence, Tarski
immediately adds that condition (F) can be shown to hold of arguments
falling under the defined notion: 



> 
> it can be proved, on the basis of this definition, that every
> consequence of true sentences must be true, and also that the
> consequence relation which holds between given sentences is completely
> independent of the sense of the extra-logical constants which occur in
> these sentences. In brief, it can be shown that the condition (F)
> formulated above is necessary if the sentence *X* is to follow
> from the sentences of the class *K* (Tarski 1983c, p. 417).
> 



(The proof that Tarski seems to have in mind is this: suppose that
*X* is a Tarskian logical consequence of *K*; then there
is no model of *K* which is not a model of *X*; so there
is no substitution instance <*K*',
*X*'> of <*K*, *X*> such that the
sentences in *K*' are true and *X*' is
false; for if there was one such, it would readily provide an
interpretation--constituted by the extensions of the
extra-logical constants of <*K*',
*X*'>--which would constitute a model of
*K* that would not be a model of *X*. But whether this
is the proof Tarski had in mind is a disputed exegetical
question.)



Hence, as was to be desired, if the defined relation of logical
consequence holds for a given pair <*K*, *X*>,
then also condition (F) holds for it. (Although this can be shown, the
converse cannot; that is, it cannot be shown that if *X* and
*K* satisfy (F) then *X* follows from *K*
according to Tarski's definition. But this is all right, since,
as Tarski has already pointed out, (F) is not a sufficient condition
for the ordinary notion of consequence.)


## 4. Logical constants



Tarski did not think that the construction of section 3 completely
solved the problem of offering "a materially adequate definition
of the concept of consequence" (Tarski 1983c, p. 418). According
to Tarski, perhaps the most important difficulty that remained toward
solving that problem is created by the fact that "underlying our
whole construction is the division of all terms of the language
discussed into logical and extra-logical" (Tarski 1983c, p.
418). This situation is tolerable because, as Tarski says in a letter
of 1944, "it is clear that for all languages which are familiar
to us such definitions [of 'logical term' and
'logical truth'] can be given (or rather: have been
given); moreover, they prove fruitful, and this is really the most
important. We can define 'logical terms', e.g., by
enumeration" (Tarski 1987, p. 29). But since the division is not
based on a previous characterization of logical terms generally
applicable to arbitrary languages, to that extent the definition of
logical consequence is not fully general, and hence unsatisfactory. In
the final paragraph of the consequence paper Tarski says that a
positive solution to the problem would "enable us to justify the
traditional boundary between logical and extra-logical
expressions" (Tarski 1983c, p. 420). In fact, this is the
boundary which for Tarski is "underlying our whole
discussion".



Tarski says that the distinction between logical and extra-logical
terms is not entirely arbitrary, because if we were to include signs
like the implication sign or the universal quantifier among the
extra-logical terms, "our definition of the concept of
consequence would lead to results which obviously contradict ordinary
usage" (Tarski 1983c, p. 418). This is so because in this case
the definition would not declare logical consequences many instances
of the relation consecrated by the common usage of logicians. However,
Tarski does not seem to worry that the opposite problem may arise. The
possibility exists, according to him, of extending the set of logical
terms without making the definition of logical consequence useless.
Even if all terms of the language are considered logical, the
definition results in a characterization of a special concept of
consequence, that of material consequence.



When he speaks of extending liberally the set of logical terms, Tarski
is perhaps thinking of the phenomenon of what he calls the
"disciplines preceding a given discipline" in Tarski
(1937) (see p. 80). He speaks of logic as preceding every discipline,
in the sense that logical constants and logical laws are presupposed
by and form part of every science. But similarly, in developing a
certain theory, not only logic but other theories may be taken for
granted; thus Tarski speaks of logic and arithmetic as a conveniently
presupposed basis of theories for the development of geometry. This
would account naturally for the fact that Tarski was skeptical about
the possibility of finding a sharp distinction between logical and
extra-logical terms, for it might very well depend on the context of
investigation what terms and what laws are considered as forming part
of the "logic" of the investigation. 



An example which we know Tarski had in mind is the sign for the
membership relation. We find him saying: "sometimes it seems to
me convenient to include mathematical terms, like the [?]-relation,
in the class of logical ones, and sometimes I prefer to restrict
myself to terms of 'elementary logic'. Is any problem
involved here?" (Tarski 1987, p. 29). We may take the sign for
membership as a logical sign in some formalizations of the theory of
types (which can in turn be used to formalize theories with arbitrary
universes of individuals). But when doing set theory for its own sake,
the appropriate thing is not to take membership as a logical notion,
and not to assume principles about membership as forming part of the
"logic" of the theory, but as postulates thereof.
Tarski's final remarks in the paper on logical consequence again
put forward the view that the notion of logical constant may be of a
relative character. In different contexts different terms may be taken
as logical and therefore so may vary the extension of the relation of
logical consequence. He says that "the fluctuation in the common
usage of the concept of consequence would--in part at
least--be quite naturally reflected in such a compulsory
situation" of relativity (Tarski 1983c, p. 420).



But this does not mean that no desiderata could or should be imposed
on a definition of the concept of a logical constant. Many years after
writing his paper on logical consequence, Tarski returned to the
problem of the definition of the concept of a logical term, advancing
an attempt at a positive solution. Tarski thought that this solution
accommodated his conviction that the notion of a logical term is not
an absolute, but a relative one.



The basis for the proposed solution appears in Tarski's 1966
lecture "What are Logical Notions?" (published
posthumously as Tarski 1986a). Here Tarski first makes some remarks
about the general nature of his proposal. He says that an answer to a
question like the one that gives title to the lecture may take several
forms. It may give an account of the prevailing usage of the term
"logical notion", or of the prevailing usage of the term
among people who are qualified to use it. It may be a normative
proposal, a suggestion that the term be used in a certain way,
independently of its actual use. As a third possibility, some other
answers 



> 
> seem to aim at something very different (...); people speak of
> catching the proper, true meaning of a notion, something independent
> of actual usage, and independent of any normative proposals, something
> like the platonic idea behind the notion. This last approach is so
> foreign and strange to me that I shall simply ignore it for I cannot
> say anything intelligent on such matters (Tarski 1986a, p. 145).
> 



Immediately Tarski makes it clear that he is also not concerned with
developing a normative proposal, but an account which captures a
certain common use of the concept of logical notion: "in
answering the question 'What are logical notions?' what I
shall do is make a suggestion or proposal about a possible use of the
term 'logical notion'. This suggestion seems to me to be
in agreement, if not with a prevailing usage of the term
'logical notion', at least with one usage which actually
is encountered in practice" (Tarski 1986a, p. 145). 



What are notions? Tarski says:



> 
> I use the term "notion" in a rather loose and general
> sense, to mean, roughly speaking, objects of all possible types in
> some hierarchy of types like that in *Principia mathematica*.
> Thus notions include individuals (...), classes of individuals,
> relations of individuals, classes of classes of individuals, and so on
> (Tarski 1986a, p. 147).
> 



He then proposes to define logical notions as those notions invariant
under all one-one transformations of the universe of discourse onto
itself:



> 
> consider the class of *all* one-one transformations of the
> space, or universe of discourse, or "world", onto itself.
> What will be the science which deals with the notions invariant under
> this widest class of transformations? Here we will have very few
> notions, all of a very general character. I suggest that they are the
> logical notions, that we call a notion "logical" if it is
> invariant under all possible one-one transformations of the world onto
> itself (Tarski 1986a, p. 149).
> 



A one-one transformation of a class onto itself, also called a
permutation, induces permutations of all the types in the hierarchy of
types of "notions" determined by the class. Thus, a
permutation *P* of a domain of individuals *D* induces a
permutation of the class of *n*-ary relations of elements of
*D*, a permutation of the class of functions with n arguments
with domain *D**n* and range included in
*D*, a permutation of the class of *n*-ary relations
among relations of elements of *D*, etc. A notion or object
*O* of a certain type *t* is invariant under all
permutations of the universe of discourse if, for every permutation
*P* of this universe, the permutation *P* induced
by *P* in the class of notions of type *t* is such that
*P*(*O*)=*O*.



There is no straightforward way in which the truth-functions and the
quantifiers (over any type of objects) are identifiable with notions
in the sense of Tarski. According to John Corcoran (see footnote 6 in
Tarski 1986a, p. 150), in one version of the lecture Tarski indicated
that the truth-functions and classical quantifiers can be constructed
as certain objects in the type hierarchy that are invariant under all
permutations, and, in this sense, are logical notions. (Clearly,
Tarski took the logicality of these notions almost for granted, and
was concerned with a definition which gave some account of the status
of other more substantive notions in various mathematical languages.)
For example, the truth-values "true" and
"false" can be identified with the universe of individuals
and the null set, respectively, and the truth-functions in turn with
functions having (tuples of) these classes as arguments and values;
and the classical universal and existential quantifiers over a type of
objects *t* can be identified with certain functions from the
class of sets of objects of type *t* into the class of
truth-values--identifying "true" with the universal
set of objects of type *t* and "false" with the
empty set of that type. (A universal quantifier will assign
"true" to the set of all objects of type *t*, and
"false" to all other subsets of *t*; and the
existential quantifier will assign "true" to the non-empty
subsets, and "false" to the empty subset.)



In 1966 Tarski does not propose a definition of the concept of logical
constant. One such definition, based on the 1966 idea, appears in a
book Tarski wrote in collaboration with Steven Givant, published in
1987, four years after Tarski's death. The main concern of
Tarski and Givant (1987) is to indicate how to develop set theory in
several different languages and to compare their expressive power.
They define the concept of logical constant for constants in the
vocabulary of a certain class of languages--all of which are in a
certain sense extensions of a basic language in which they are
particularly interested, weaker than the languages in which set theory
is usually developed in means of expression, and yet sufficiently
powerful for many purposes. The vocabulary of the basic language is
composed of three two-place predicate constants of the second type and
four two-place function constants of the third type; the vocabulary of
the extensions considered by Tarski and Givant can only include
constants of those two kinds. (However, as we will see, there does not
seem to be any problem in extending the range of applicability of
their definition to wider classes of languages.) They then prove
several results about that class of languages, results that involve
the defined concept of logical constant.



Tarski and Givant introduce informally the concept of a derivative
universe of a given basic universe *U*. A derivative universe
*U* of a given basic universe *U* is what we have
been calling the class of all objects of a certain type, generated
from that basic universe *U*. Thus, the class of *n*-ary
relations of elements of *U* (for any given *n*), the
class of *n*-ary relations among relations of elements of
*U*, etc., are derivative universes of *U*. After
introducing in the same way as above the concepts of a permutation of
the universe and of invariance of a member of a derivative universe
under every permutation of the basic universe, Tarski and Givant give
the following definitions:


1. Given a basic universe *U*, a member *M* of any
derivative universe *U* is said to be logical, or a
logical object, if it is invariant under every permutation *P*
of *U*.



(Strictly speaking, since an object *M* can be a member of many
derivative universes, we should use in (i) the phrase "is said
to be logical, or a logical object, as a member of
*U*".)
2. A symbol *S* of [a language in the class considered by
Tarski and Givant] is said to be logical, or a logical constant, if,
for every given realization **U** of this [language] with
the universe *U*, *S* denotes a logical object in some
derivative universe *U* (Tarski and Givant 1987, p.
57).



(i) and the parenthetical comment that comes after it embody fairly
accurately the basic idea behind the (relativized) definition of a
logical notion offered in Tarski (1986a), and (ii) offers the
(absolute) definition of the concept of logical constant. There is no
apparent obstacle to applying (ii) to wider classes of languages than
the class considered for special purposes of their investigation by
Tarski and Givant. It seems safe to assume that the definition can be
applied to a large class of languages in the classical hierarchy of
quantificational languages. Tarski and Givant themselves say that the
usual logical constants of languages not in the class considered by
them, like the symbols for the truth-functional connectives and
quantifiers, "can also be subsumed under logical constants in
the sense of (ii)" (Tarski and Givant 1987, p. 57), presumably
through some artifice in the style of the one discussed above. (As we
will soon see, at the very least Tarski clearly would want his
definition to be applicable to some languages whose set of sentences
is that of some formulations of the simple theory of types.)



In Tarski and Lindenbaum (1935) the authors had proved that given a
basic universe *U*, all the notions in derivative universes of
*U* which can be defined in the language of the simple theory
of types are invariant under all permutations of *U*. In
general the Tarski-Lindenbaum theorem guarantees that all mathematical
notions definable in the logicist fashion in the simple theory of
types are logical notions no matter what the universe of individuals
is taken to be (Tarski says that "we may interpret [the universe
of the intended interpretation of the language of the theory of types]
as the universe of physical objects, although there is nothing in
*Principia mathematica* which compels us to accept such an
interpretation" (Tarski 1986a, p. 152)). Since the theorem
applies to every universe *U* supplying an interpretation of
the language of the theory of types, the definition of the concept of
logical constant in Tarski and Givant (1987) implies that all
primitive symbols denoting notions in that language (e.g. quantifiers
of all orders) are Tarskian logical constants; also, if the definition
were applicable to defined symbols, all these symbols would be
Tarskian logical constants. Such results agree well with a
"usage actually encountered in practice" (for example, the
practice of the logicists, but of others as well) according to which
the constants of the language of the theory of types are logical
constants.



When the Tarski-Givant definition is applied to interpreted languages
of mathematical theories with undefined mathematical primitives, it
will generally yield the result that the notions denoted by these
primitives are not Tarskian logical constants. Tarski's example
is set theory formalized in first-order with a single primitive
predicate for membership as a relation among elements of the universe
(see Tarski 1986a, p. 153). Obviously membership as a relation over a
domain of individuals and sets is not invariant under all permutations
of that domain, so it is not declared a logical notion of the language
of set theory by Tarski's proposed definition. Similarly, the
class of all sets will be declared non-logical provided the class of
individuals that are not sets is not empty. A predicate whose intended
meaning is membership (among elements of the universe) is not a
Tarskian logical constant, simply because there is a universe in which
it denotes a non-logical object (with respect to that universe);
similarly, a predicate "*S*" whose intended meaning
is "is a set" (such as is used in some formalizations of
set theory suitable for contemplating individuals other than sets in
the universe) is not a Tarskian logical constant, for there is a
universe in which "*S*" denotes a non-logical
object (in such a universe, the class of objects that are not sets
must be non-empty). These results are again in agreement with actual
usage, for example in the model theory of first-order set theory,
where the membership predicate is taken as a non-logical constant.



From the consideration of the examples provided by the theory of types
and first-order set theory, Tarski extracts a conclusion that he
welcomed. This is the conclusion that, in a certain sense, the
distinction between logical and non-logical constants is relative to a
context of investigation. Recall Tarski's remarks about the sign
for membership in a letter of 1944. In the 1966 lecture Tarski says
that membership is a logical notion in some formalizations of the
theory of types. He says:



> 
> Using this method [of *Principia Mathematica*], it is clear
> that the membership relation is certainly a logical notion. It occurs
> in several types, for individuals are elements of classes of
> individuals, classes of individuals are elements of classes of classes
> of individuals, and so on. And by the very definition of an induced
> transformation it is invariant under every transformation of the world
> onto itself (Tarski 1986a, pp. 152-3).
> 



It is a bit unclear that the "notions" of membership that
Tarski talks about are notions in the sense introduced by him earlier
in the lecture. They do not "occur" in any type, given the
standard formulations of the theory of types (even in formulations in
which a sign for membership is used), for they are relations among
objects of different types, which do not occur in a non-cumulative
hierarchy. But Tarski is perhaps thinking of some non-standard,
cumulative type hierarchy. Under a suitable expansion of the concept
of a notion, the notions of membership Tarski has in mind will indeed
be invariant under permutations of any universe, and hence logical;
and the sign or signs for these notions of membership will be Tarskian
logical constants. On the other hand, as pointed out above, membership
as a relation over the domain of individuals and sets will not be a
logical notion in that domain, and hence the sign for this concept of
membership will not be a Tarskian logical constant. Thus,
interestingly, Tarski's definition of the concept of logical
constant respects in a sense his early relativistic view in the paper
on logical consequence. Tarski describes the situation in the
following passage:



> 
> Since it is now well known that the whole of mathematics can be
> constructed within set theory, or the theory of classes, the problem
> [of whether mathematical notions are logical notions] reduces to the
> following one: Are set-theoretical notions logical notions or not?
> Again, since it is known that all usual set-theoretical notions can be
> defined in terms of one, the notion of belonging, or the membership
> relation, the final form of our question is whether the membership
> relation is a logical one in the sense of my suggestion. The answer
> will seem disappointing. For we can develop set theory, the theory of
> the membership relation, in such a way that the answer to this
> question is affirmative, or we can proceed in such a way that the
> answer is negative (Tarski 1986a, pp. 151-2).
> 


## 5. Further reading



Tarski's papers are reprinted and collected in Tarski 1986b. The
volume Tarski 1983a contains widely used English translations of his
main papers of the 1920s and 1930s. Givant (1986) gives a complete
bibliography of Tarski's publications until 1986. Feferman and
Feferman 2004 is a biography of Tarski that also contains a very
valuable introduction to his logical and mathematical work, including
descriptions of all the results and developments mentioned in the
initial paragraph of this entry. On Tarski's logical work see
also Simmons 2009 and Eastaugh 2017. McFarland, McFarland & Smith
2014 contains a wealth of biographical and historical information
about Tarski, as well as updates to Givant's bibliography and
translations into English of a number of previously untranslated early
texts by Tarski. Also of much biographical interest are the letters
from Tarski to J. H. Woodger included in Mancosu 2021.



Collections of articles devoted to (parts of) Tarski's work
include: vols. 51 (1986) and 53 (1988) of the *Journal of Symbolic
Logic*; Wolenski and Kohler (1999); vols. 126 (2001) and
142 (2004) of *Synthese*; vols. 126 and 127 of the *Annals
of Pure and Applied Logic* (2004); Patterson 2008a; and Sagi and
Woods 2021. Patterson 2012 is a book-length historical and
philosophical study of many of Tarski's philosophical ideas,
focusing on truth and logical consequence. Gruber 2016 is a
book-length commentary of Tarski's monograph on truth,
"The Concept of Truth in Formalized Languages", which
compares the Polish original version, the German translation, and the
English version of the German text.



There is a considerable bibliography on critical and exegetical
aspects of Tarski's philosophical work. The bibliography on
truth is especially extensive. It includes the chapters on
Tarski's theory of truth in Coffa 1991; Kirkham 1992; Soames
1999, 2010, and 2018; Kunne 2003; Burgess and Burgess 2011; and
Blackburn 2018; many chapters of Wolenski 1999 and 2019; the
papers Field 1972; Davidson 1973, 1990; Kripke 1975, 2019; Church
1976; Soames 1984; Putnam 1985; Hodges 1985/6, 2004, and 2008; McGee
1993; Garcia-Carpintero 1996; Heck 1997; Murawski 1998; DeVidi
and Solomon 1999; Ketland 1999; Milne 1999; Sher 1999;
Gomez-Torrente 2001, 2004, and 2019; Eklund 2002; Gupta 2002;
Rojszczak 2002; Raatikainen 2003, 2008; Ray 2003, 2005, 2018; Sundholm
2003; Feferman 2004a; Frost-Arnold 2004; Hintikka 2004; Patterson
2004, 2008b; Azzouni 2005, 2008; Horwich 2005; Burgess 2008; David
2008; Mancosu 2008; Simons 2008; Asay 2013; Schiemer and Reck 2013;
Loeb 2014; Barnard and Ulatowski 2016; Smith 2017; Simchen 2020; and
the entries on the
 liar paradox,
 model theory,
 Tarski: truth definitions,
 and
 truth.



The critical and exegetical bibliography on Tarski on logical
consequence includes Kneale and Kneale 1962; Etchemendy 1988, 1990,
and 2008; McGee 1992, 2004; Gomez-Torrente 1996, 1998/9, 2008,
and 2009; Ray 1996;, Sher 1996, 2022; Hanson 1997; Saguillo 1997;
Chihara 1998; Shapiro 1998; Schurz 1999; Blanchette 2000; Bays 2001;
Edwards 2003; Jane 2006; Mancosu 2006, 2010; Park 2018; Zinke
2018; Griffiths and Paseau 2022; and the entries on
 logical consequence
 and
 logical truth.



The Tarskian ideas on logical constants are discussed, among other
places, in Simons 1987; Sher 1991, 2008, and 2021; Feferman 1999;
Gomez-Torrente 2002, 2021; Bellotti 2003; Casanovas 2007;
Bonnay 2008, 2014; Dutilh Novaes 2014; Sagi 2015, 2018, and 2021;
Bonnay and Speitel 2021; Kennedy 2021; Kennedy and
Vaananen 2021; and the entry on
 logical constants.



The literature on general philosophical aspects of Tarski's work
and opinions includes Suppes 1988; Wolenski 1993; Sinaceur 2001;
Feferman 2004b; Mycielski 2004; Mancosu 2005, 2009; Betti 2008; and
Frost-Arnold 2008, 2013.