Alfred Tarski (1901–1983) described himself as “amathematician (as well as a logician, and perhaps a philosopher of asort)” (1944, p. 369). He is widely considered as one of thegreatest logicians of the twentieth century (often regarded as secondonly to Gödel), and thus as one of the greatest logicians of alltime. Among philosophers he is especially known for his mathematicalcharacterizations of the concepts of truth and logical consequence forsentences of classical formalized languages, and to a lesser extentfor his mathematical characterization of the concept of a logicalconstant for expressions of those same languages. Among logicians andmathematicians he is in addition famous for his work on set theory,model theory and algebra, which includes results and developments suchas the Banach-Tarski paradox, the theorem on the indefinability oftruth (see section 2 below), the completeness and decidability ofelementary algebra and geometry, and the notions of cardinal, ordinal,relation and cylindric algebras. After a biographical sketch, thisentry offers a condensed exposition of the parts of Tarski’swork that are most relevant to philosophy, his theories of truth,logical consequence and logical constants. In this exposition we haveattempted to remain as close as possible to Tarski’s originalpresentations, reducing to a minimum the number of claims that mightbe controversial philosophically or exegetically. The final section onfurther reading refers the reader to other entries and works oncritical and exegetical aspects of Tarski’s work not touchedupon in this entry.

• 1. Biographical sketch
• 2. Truth
• 3. Logical consequence
• 4. Logical constants
• 5. Further reading
• Bibliography
  • Primary Sources: Works by Tarski
  • Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. Biographical sketch

Tarski was born on January 14, 1901 in Warsaw, then a part of theRussian Empire. His family name at birth was Tajtelbaum, changed toTarski in 1923. He studied mathematics and philosophy at theUniversity of Warsaw from 1918 to 1924, taking courses withKotarbiński, Leśniewski, Łukasiewicz, Mazurkiewicz andSierpiński among others. (See Lvov-Warsaw School.) He got adoctoral degree, under Leśniewski’s supervision, in 1924.From then until 1939 he taught mathematics at a high school and heldminor teaching positions at the University of Warsaw. In this periodhe published prolifically on logic and set theory, building a stronginternational reputation for himself. Yet he failed in his attempt toobtain a professorship at the University of Lvov (now Lviv) in 1939.In 1929 he married Maria Witkowska, with whom he soon had twochildren, Ina and Jan.

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

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

2. Truth

In a logic seminar he gave at the University of Warsaw between 1927and 1929, Tarski proved several results that made reference to notionsthat later would be called “semantical”, in particularresults about the notions of definability and of truth in a structure(see Vaught 1974 and 1986). Vaught informs us that Tarski foundcertain difficulties when trying to give a mathematically satisfactoryform to the results presented in the seminar, and that this led him tolook for a precise theory of the semantical notions (cf. Vaught 1974,pp. 160ff. and 1986, pp. 870ff.). Such a theory did not exist at thetime. In particular, there were no definitions of these notions interms of concepts accepted in the foundational systems designed forthe reconstruction of classical mathematics (for example, Russell andWhitehead’s theory of types or Zermelo’s set theory).Because of this, the existing results in which these notions appearedcould not be reconstructed in the accepted foundational systems. Andthere was no rigorous axiomatic theory of the semantical notions inwhich these were taken as primitive, either.

In spite of this, the results about semantical notions were importantand even moderately abundant by 1930. Among these Tarski himself wouldmention Gödel’s completeness theorem and several versionsof the Löwenheim-Skolem theorem (cf. Tarski 1983b, pp.240–1, sp. p. 240 n.1, and p. 241 n. 2). All these results usethe notion of truth in a structure, or functionally equivalentnotions. And Tarski says that

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

Tarski’s aim in a series of works on semantical notions wasprecisely to propose mathematically acceptable definitions of thosenotions, and in particular acceptable in some one of the chieffoundational systems (the system chosen by Tarski will change withtime; see below). With appropriate definitions of those notions, thetheorems about them would be susceptible of being reformulated makinguse of the defined notions, and the uneasiness described by Tarski inthe text just quoted would be alleviated. The alternative of takingthe semantical notions as primitive is also considered by Tarski inseveral places, but he clearly prefers to avoid it if possible. Thereason is that in the axiomatic alternative there is no negligiblerisk that there remain ways of generating the semantical antinomiesfor the semantical primitives of the system (cf. Tarski 1983b, p.255). With the definitional procedure, however, the consistency of thedefinition would depend exclusively on the consistency of the theoryin which it is formulated, and this will be a theory we have reasonsto think is consistent.

The first work containing a mathematical definition of one of thesemantical notions is Tarski (1931) (English version, Tarski 1983d),where Tarski examines a languageL in which it is possible toformalize the arithmetic of the real numbers, and gives a recursivedefinition of the notion “set of real numbers definable inL” (cf. Def. 9 in Tarski 1983d, p. 128). The definitiondepends on others that are a bit cumbersome; today we would define asimilar notion in a simpler way with the help of the definition oftruth that Tarski himself would publish two years later. But theessence of the Tarskian definition of truth is already here. In the1931 paper we find also, as was to be expected, the Tarskian concernwith matters of precision and foundational rigor, and a revealingstatement about the attitude of mathematicians toward the notion ofdefinability, that could be extended to other semantical notions:

The distrust of mathematicians towards the notion in question isreinforced by the current opinion that this notion is outside theproper limits of mathematics altogether. The problems of making itsmeaning more precise, of removing the confusions and misunderstandingsconnected with it, and of establishing its fundamental propertiesbelong to another branch of science—metamathematics (Tarski1983d, p. 110).

In his classic monograph on the concept of truth “The Concept ofTruth in Formalized Languages” (Polish original version, Tarski1933; German translation with an added postscript, Tarski 1935;English version of the German text, Tarski 1983b), Tarski will presenta method for constructing definitions of truth for classicalquantificational formal languages. When one succeeds in applying thismethod to a particular formal language, the end result will be theconstruction of a predicate in a metalanguage for that language whoseessential properties will be that it will be constructed out ofnon-suspicious mathematical vocabulary and that it will be intuitivelysatisfied precisely by the intuitively true sentences of the objectlanguage. At the same time, Tarski shows how, in terms of the definednotion of truth, one can give intuitively adequate definitions of thesemantical notions of definability and denotation, and he indicateshow it is possible to define the notion of truth in a structure in away 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 Löwenheim,Skolem, Gödel and Tarski himself, among others), have beenvindicated (cf. Tarski 1983b, p. 266).

Languages, both object languages and metalanguages, are in themonograph not just interpreted grammars; a language also includes adeductive system. A Tarskian metalanguage always includes its objectlanguage as a part, both its grammar (perhaps under some translation)and its deductive system. And besides, it will always contain a fewmore things if these things are not already in the object language.Specifically, it will always contain a sub-language and sub-theorythat can be used to say a great deal of things about the syntax of theobject language; and it will always contain all the mathematics (bothconcepts and methods of proof) that “may be taken from anysufficiently developed system of mathematical logic” (Tarski1983b, p. 211).

Tarski doesn’t give a general formulation of his method. Such ageneral formulation would not be too illuminating. Instead hehelpfully chooses to illustrate his method as it would work for a fewlanguages. The most basic example he uses, to which about half of hismonograph is devoted, is what he calls “the language of thecalculus of classes” (LCC). Here is a quick description in morecurrent notation (Tarski uses Łukasiewicz’s Polishnotation):

1. primitive signs of LCC: ∀, ∨, ¬, (, ), I (a binarypredicate),x,_′ (a subindexaccent to generate variables by suffixing to “x”;let’s use the notationx_n for thex followed byn accents).
2. grammar of LCC: Atomic formulae are of the formIx_kx_l.Complex formulae are obtained by negation, disjunction and universalquantification.
3. interpretation of LCC: The range of the variables is the class ofall subclasses of individuals (of the universe). I stands for therelation of inclusion among these subclasses. The other signs meanwhat you would expect.

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

Tarski’s most basic idea about how to construct a truthpredicate, a predicate which will be intuitively satisfied exactly bythe intuitively true sentences of LCC, is that such a predicate shouldverify what he calls “conventionT”. Thisconvention imposes a condition on defined truth predicates ((a) in theexample immediately below) that Tarski often talks about as graspingthe intuitions behind the “classical Aristotelian conception oftruth” (cf. Tarski 1983b, p. 155; 1944, pp. 342–3).ConventionT is here spelled out for the case ofLCC:

That a definition is “formally correct” means, roughly,that it will be constructed out of non-suspicious vocabulary followingnon-controversial rules for defining new expressions. But why shouldone think that a predicate verifying conventionTshould be, besides, coextensional with the intuitive predicate oftruth for LCC? The reason can be given by an intuitive argumentrelating both predicates, such as the following (cf. Tarski 1944, pp.353–4):

Tarski defines truth in terms of the notion of the satisfaction of aformula of LCC by an infinite sequence of assignments (of appropriateobjects: subclasses of the universe of individuals in the case ofLCC). He gives first a recursive definition and immediately indicateshow to transform it into a normal or explicit definition. Therecursive definition is this: an infinite sequence of classesf satisfies formulaF if and only iff andF are such that

1. there are natural numbersk andl such thatF =Ix_kx_l andf(k)⊆f(l); or
2. there is a formulaG such thatF=¬Gandf does not satisfyG; or
3. there are formulaeG andH such thatF=(G∨H) andf satisfiesG orf satisfiesH; or finally,
4. there is a natural numberk and a formulaG suchthatF=∀x_kG andevery infinite sequence of subclasses that differs fromf atmost in thekth place satisfiesG (cf. Tarski 1983b,p. 193).

The normal definition is this:

Tarski then defines the truth predicate as follows:

For allx, Tr(x) if and only ifx is asentence of LCC and every infinite sequence of subclasses satisfiesx (cf. Tarski 1983b, p. 195).

Given the way the definition has been constructed, it is intuitivelyclear that the metalanguage will prove all biconditionals of theconventionT for LCC. Tarski does not prove thiscumbersome fact in a metametalanguage, and he helpfully contentshimself with showing how a few of the biconditionals would beestablished in the metalanguage.

The usual reasonings leading to the semantical antinomies cannot bereproduced for the Tarskian defined semantical concepts. Inparticular, the antinomy of the liar cannot be reproduced using thedefined predicate Tr. Consider the version of the antinomy offered byTarski (due to Łukasiewicz). Abbreviate the expression “thefirst underlined sentence of this entry” with the letter“c”. Now consider the sentence:

Given what “c” abbreviates, we get

And the following is intuitively true:

From (a) and (b) follows the contradiction

The reason why this cannot be reproduced for Tr is that this predicateis always a predicate of a language (the metalanguage) different fromthe language of the sentences to which it applies (the objectlanguage). It is not possible to form a sentence of the language forwhich one defines Tr that says of itself that it is (not) Tr, since Tris not a predicate of that language. On the other hand, it iscertainly possible in some cases to form a sentenceS of themetalanguage that says of itself that it is not Tr, but sinceS is a sentence of the metalanguage it is simply true and(presumably) not paradoxical, since there is no reason why abiconditional like (b) should hold for it; biconditionals of this kindhold 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 offinite order”. The idea of order is familiar. Variables forindividuals (non-existent in LCC) are of order 1. Variables forclasses of individuals (all the variables of LCC) are of order 2.Variables for classes of classes of individuals are of order 3, and soon. The order of a language of this series is the greatest positiveintegern with variables of ordern. (Thus, althoughLCC looks grammatically just like a first-order language, semanticallyit can be seen as a fragment of what under present conventions wouldbe called a second-order language.)

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

1. primitive signs of LGTC: ∀, ∨, ¬, (, ), X,_′, ′ (The last two are a subindexaccent and a superindex accent to generate variables by suffixing to“X”; let’s use the notationX_n^m for theX followed byn subindex accents andmsuperindex accents. The superindex must be positive.)
2. grammar of LGTC: Atomic formulae are of the formX_n^m+1X_p^m.Complex formulae are obtained by negation, disjunction and universalquantification with respect to the variables of all orders.
3. interpretation of LGTC: A variable of the formX_n¹ takes as valuesindividuals, a variable of the formX_n² takes as values classes ofindividuals, a variable of the formX_n³ takes as values classes ofclasses of individuals, etc.X_n^m+1X_p^m“means” that the object assigned toX_n^m+1“contains” the object assigned toX_p^m. The other signsmean what you would expect.

Besides all this, Tarski gives a deductive system for LGTC, asrequired of every language. Suffice it to say that it is a version ofa typical simple theory of finite types, with axioms and rules for theconnectives and quantifiers, axioms of comprehension andextensionality for all orders, and an axiom of infinity. Tarskiobserves that LGTC suffices, with the help of some tricks, to developall the mathematics that can be developed in the simple theory offinite types, even though it has formally fewer types ofvariables.

In the 1933 original Polish version of his monograph Tarski says thathis method for constructing truth predicates cannot be applied to theconstruction of a truth predicate for LGTC. The problem is that in1933 Tarski adopts as the mathematical apparatus of his metalanguagesthe simple theory of finite types, or equivalently, LGTC. In thislanguage, supplemented (and even unsupplemented) by a theory of syntaxfor LCC, Tarski has everything he needs to give his definition ofsatisfaction for LCC. In particular, the relation of satisfaction forLCC 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 definesit quantifies only over sequences of classes and over relations amongsequences of classes and formulae (as we may check above), and theseare objects quantifiable over in finite type theory.

But the construction of a Tarskian truth predicate for LGTC could notbe carried out in finite type theory. The relation of satisfaction forLGTC 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 relationis not a relation in the hierarchy of finite types. Much less can onequantify in finite type theory over the objects necessary to apply theTarskian method. One cannot quantify over sequences of: sequences ofindividuals, sequences of classes of individuals, sequences of classesof classes of individuals, etc.; and one cannot quantify overrelations among formulae and sequences of: sequences of individuals,sequences of classes of individuals, sequences of classes of classesof individuals, etc.

It is this situation that leads Tarski to wonder

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

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

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

But

it will be remembered that in the conventionT of§3 the conditions which decide the material correctness of anydefinition of true sentence are exactly stipulated. The constructionof a definition which satisfies these conditions forms in fact theprincipal object of our investigation. From this standpoint theproblem we are now considering takes on a precise form: it is aquestion ofwhether on the basis of the metatheory of the languagewe are considering the construction of a correct definition of truthin the sense of conventionT is in principlepossible. As we shall see, the problem in this form can bedefinitively solved, but in anegative sense (Tarski 1983b,p. 246).

The negative answer is provided by a version of Tarski’s theoremof the indefinability of truth. The weakest Tarskian metalanguage forLGTC will consist in expanding LGTC by means of concepts and a theoryof the syntax of LGTC. But the mathematical part of the metalanguagewill be the same, for that’s the “sufficiently developedsystem 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 themetalanguage (cf. Tarski 1983b, p. 247).

In a word, the metalanguage, which we may call “LGTC+”, isLGTC plus a theory of the syntax of LGTC. The question that Tarskiposes is, then, whether there is a predicate constructible in LGTC+which verifies the conventionT for sentences ofLGTC. Tarski solves it negatively by proving the followingtheorem:

Part (b) of Theorem I is a trivial consequence of part (a), since itis the mark of a consistent system that it doesn’t contain asentence and its negation. The proof of part (a) is an application ofGödel’s technique of diagonalization. The following are thebasic steps described using the anachronistic recourse to a morefamiliar case and to a modern, streamlined notation.

View LGTC as if it were a standard language L for first-orderarithmetic. Suppose we extend it with a theory for talking about thesyntax of L (this would not be a Tarskian metalanguage, since it wouldnot include a “sufficiently developed system of mathematicallogic”, but all this is just for purposes of illustration). Callthe extension L+. Then L+ can be “interpreted” in L usingGödelian techniques, by which Tarski means, roughly, that one cantranslate L+ into L arithmetizing it in such a way that thetranslations of theorems of L+ will be theorems of L. Now suppose thatφ(n) is thenth expression of L (and that thisfunction is definable in L+), thatd(x) is adiagonal function for expressions of L (which is definable in L+; hereI have in mind one involving substitution of numerals for freevariables), and letn be the numeral ofn.Then for every predicateE(x) of expressions of Ldefined in L+ one can prove in L+ a general sentence of the form∀n[¬E(d(φ(n)))↔ψ(n)],where ψ(n) is a formula with one free variable entirelyconstructed in L (through the arithmetization of the L+ predicate“¬E(d(φ(n)))”). Thenone can also prove the sentence¬E(d(φ(k)))↔ψ(k),wherek is the Gödel number of ψ(n).d(φ(k)) is the sentenceψ(k), so ψ(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(φ(k)))↔ψ(k)is already equivalent to the sentence¬[E(d(φ(k)))↔ψ(k)],and part (a) of Theorem I is proved.

In the postscript to the 1935 German translation of the monograph ontruth, Tarski abandons the 1933 requirement that the apparatus of themetatheory be formalizable in finite type theory, and accepts the useof a more powerful theory of transfinite types, where the transfiniteobjects are classes of the objects of lower types, or the use of settheory. Tarski notes that the relation of satisfaction and a truthpredicate for LGTC (and LGTC+) are definable in these more powerfulmetatheories (cf. Tarski 1983b, pp. 271–2). But he also saysthat the proof of the indefinability theorem can be adapted to showthat, in general, one cannot define a truth predicate “if theorder of the metalanguage is at most equal to that of the languageitself” (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). Thisclassic paper begins with some general remarks on the possibility of aprecise definition of the concept of consequence. The essence of theseremarks is that since the common concept is vague, it seems certainlydifficult, and perhaps impossible, to reconcile all features of itsuse in the definition of a corresponding precise concept.Nevertheless, Tarski says, logicians had thought until recently thatthey had managed to define a precise concept that coincided exactly inextension with the intuitive concept of consequence. Tarski mentionsthe extraordinary development of mathematical logic in recent decades,which had shown “how to present mathematical disciplines in theshape of formalized deductive theories” (Tarski 1983c, p. 409).In these theories, consequences are extracted from axioms and theoremsby rules of inference, “such as the rules of substitution anddetachment” (Tarski 1983c, p. 410), of a purely syntactical (or“structural”, in Tarski’s word) nature.“Whenever a sentence follows from others, it can be obtainedfrom them—so it was thought—by means of thetransformations 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 theshape of formalized proofs all the exact reasonings which had everbeen carried out in mathematics” (Tarski 1983c, p. 410).

But Tarski goes on to claim that that belief of the logicians waswrong. There are some non-vague cases in which a certain sentence of ahigher-order language follows in the intuitive sense from a set ofother sentences of that language but cannot be derived from them usingthe accepted axioms and rules. These cases are provided by someω-incomplete theories, theories in which for some predicateP the sentences

and, in general, all sentences of the formA_n can be proved, but the universalsentence

A. Every natural number possesses the given propertyP,

cannot be proved on the basis of the accepted axioms and rules ofinference. (LGTC provides an example of such a theory.) “Yetintuitively it seems certain that the universal sentenceAfollows in the usual sense from the totality of particular sentencesA₀,A₁, …,A_n, …. Provided all thesesentences are true, the sentenceA must also be true”(Tarski 1983c, p. 411).

Tarski considers the possibility of adding an ω-rule to theaccepted rules of inference, that is, a rule which allows us to deducea universal sentence of the form ofA from the set ofsentencesA₀,A₁, etc.However, he says that such a rule would differ in very essentialrespects from the old rules: it is not a finitary rule, while all theaccepted rules in common deductive systems are finitary. Also, Tarskiimmediately takes importance away from the further suggestion ofsupplementing the old system of rules by means of a formalizedfinitary ω-rule. He points out that in view ofGödel’s incompleteness results, no matter how many newfinitary rules or axioms we add to certain higher-order theories, theywill still remain incomplete theories, in fact ω-incompletetheories. This discussion is enough to show that “in order toobtain the proper concept of consequence, which is close in essentialsto the common concept, we must resort to quite different methods andapply quite different conceptual apparatus in defining it”(Tarski 1983c, p. 413). The different methods and the differentconceptual apparatus that Tarski has in mind are going to be“the methods which have been developed in recent years for theestablishment of scientific semantics, and the concepts defined withtheir aid” (Tarski 1983c, p. 414; a footnote refers us toTarski’s monograph on truth).

The best way of appreciating the nature of Tarski’s theory oflogical consequence and how it constitutes an application of themethods of Tarskian semantics is to see how it applies to a particularformal language of a simple structure, of a kind that Tarski seems tohave primarily in mind in his paper of 1936. We will choose a simplelanguage of this kind for a fragment of elementary arithmetic. Thislanguage, LAr, can be given this quick description:

1. primitive signs of LAr: ∀, →, ¬, (, ), x,_′ (a subindex accent to generate variablesby suffixing to “x”; let’s use again thenotationx_n for thex followedbyn accents), 0 (an individual constant), N (a monadicpredicate), M (a dyadic predicate).
2. grammar of LAr: Atomic formulae are of the formsNt₁ andMt₁t₂, wheret₁ andt₂ are either variablesor 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 ofall natural numbers. N stands for the set of natural numbers and M forthe relation of being less than among natural numbers. The other signsmean what you would expect.

In the 1936 paper Tarski seems to be thinking paradigmatically oflanguages in which (as in LAr) there is a predicate (“N”in the case of LAr) that applies exactly to the individuals in thedomain of the intended interpretation of the language. He sayselsewhere that the variables of such a language range exclusively overthe individuals of that set (cf. Tarski 1937, p. 84). (This is not tosay that Tarski adopts these conventions with all the languages heconsiders. For example, he does not adopt them with languages which,like LGTC, are used to talk about arbitrary individuals, as opposed tospecific sets of them).^[2]

In the considerations preliminary to the presentation of his theory,Tarski says that when a sentenceX of a formal language(e.g., LAr) is a logical consequence of a setK of sentencesof that language, the argument with premisesK and conclusionX has the following property, that Tarski calls“condition (F)”:

(F) If, in the sentences of the classK and in the sentenceX, the constants—apart from purely logicalconstants—are replaced by any other constants (like signs beingeverywhere replaced by like signs), and if we denote the class ofsentences thus obtained fromK by“K′”, and the sentence obtained fromX by “X′”, then the sentenceX′ must be true provided only that all sentences of theclassK′ 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 anotherindividual constant, “2”, and another dyadic predicate,“Pd”, whose desired interpretations are the number 2 andthe relation of being the immediate predecessor of, respectively. LetK be the following set of sentences of LAr+:{“∀x(Nx→¬Mx0)”,“N0”} (these sentences are true); and letX bethe sentence “¬M00”. The argument with premisesK and conclusionX is intuitively logically correct;thus, according to Tarski, it must verify condition (F). This meansthatany argument obtained from it by uniform replacement ofnon-logical constants by non-logical constants must be an argumentwhereit is not the case that the premises are true and theconclusion false. Let’s suppose that the non-logicalconstants of LAr+ are “0”, “N”,“M”, “2”, and “Pd”. Replace“0” with “2” and “M” with“Pd” in the argument with premisesK andconclusionX and call the resulting set of premises andconclusion “K′” and“X′”. That is,K′ is{“∀x(Nx→¬Pdx2)”,“N2”} andX′ is “¬Pd22”. Invirtue of condition (F), the argument with premisesK′and conclusionX′ must be an argument where it is notthe case that the premises are true and the conclusion false; andthat’s in fact the case: the first premise is false, and thesecond premise and the conclusion are true.

Tarski wonders if it is possible to offer condition (F) as adefinition of the relation of logical consequence, that is,if we can take (F) not only as a necessary but also a sufficientcondition 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 withdoes not possess a sufficient stock of extra-logical constants. Thecondition (F) could be regarded as sufficient for the sentenceX to follow from the classK only if thedesignations of all possible objects occurred in the language inquestion. This assumption, however, is fictitious and can never berealized (Tarski 1983b, pp. 415–6).

Tarski notes that in order for an argument to be an instance oflogical consequence it need not be sufficient that all arguments ofthe same form be arguments where it is not the case that the premisesare true and the conclusion false. It is conceivable that one mayinterpret the non-logical constants of the argument by means ofcertain objects (individuals, sets, etc.) in such a way that thepremises thus reinterpreted become true and the conclusion becomesfalse, and that nevertheless (some of) those objects not be denoted bynon-logical constants of the language that is being considered; insuch a case we would not say that the argument is an instance oflogical consequence, in spite of the fact that it would satisfycondition (F).

To give an example, suppose that the language we are considering isLAr+. Since both the relation of being less than and the relation ofbeing the immediate predecessor of are irreflexive over the domain ofthe natural numbers, the sentence“∀x(Nx→¬Mxx)”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 thatsentence into a falsehood. But clearly“∀x(Nx→¬Mxx)”is not a logical consequence of, say, “N0”. This can bejustified, 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 lessthan or equal to; under this interpretation,“∀x(Nx→¬Mxx)”is false, although “N0” is true. (Tarski’s remarkthat the supposition that all objects have names in the language cannever be realized can be justified, for example, by observing thatthere are non-denumerably many sets of natural numbers, but in thelanguages he considers there are only denumerably many constants.)

Tarski’s proposal consists in making tighter the requirementexpressed by condition (F), so as to incorporate the idea that alogically correct argument cannot bereinterpreted in such away that the premises become true and the conclusion false; in otherwords, the idea that a sentenceX is a logical consequence ofa set of sentencesK whenevery interpretation on whichall the sentences of K are true is an interpretation on which X istrue (or, to use a common terminology, when every interpretationpreserves the truth of the premises in the conclusion).

As Tarski says, the idea of understanding the notion of logicalconsequence by means of the notion of truth preservation on allinterpretations is not an original idea of his, but one implicit inthe logical and mathematical practice of his time (especially amongmathematicians interested in offering independence proofs). What isnew to Tarski’s proposal is that he makes precise the idea usingthe apparatus he had developed for the mathematical characterizationof satisfaction and truth. He does not give a detailed example, but itseems reasonably clear how he would proceed from the indications hegives in (1936a), (1936b) and (1937).

Tarski uses a certain precise notion of interpretation for a formallanguage. In our example, aninterpretation of LAr is asequence <A,a,R> that assignsappropriate objects to the non-logical constants of LAr: a set ofindividualsA to “N”, an individuala to“0” and a binary relation among individualsR to“M”. Besides, he always requires that the objects assignedby an interpretation to the other non-logical constants of a languagebe drawn from the set assigned to the non-logical predicate thatrestricts the range of quantification to a specific set of individuals(“N” in the case of LAr) (cf. Tarski 1937, §34); soin the case of LAr the individual assigned to “0” mustbelong to the set assigned to “N” and the relationassigned to “M” must be a relation among objects in theset assigned to “N”.

Tarski introduces the notion of asentential function. Asentential functionS′ of a sentenceS is theresult of uniformly replacing the non-logical constants appearing inS with corresponding variables of suitable types (anddifferent from the variables already existing in the language). Forexample, the sentential function determined by the sentence“∀x(Nx→¬Mx0)”is the expression“∀x(Px→¬Yxy)”(in which “P”, “Y” and“y” are new variables). It is equally possible todefine the more general notion of aformula function, in ananalogous way, except that nowS may be an open formula. Thesentential functions of sentences of LAr will not in general besentences, and so will not always be either true or false bythemselves. But they will always be either true or falsewithrespect to interpretations of LAr; or, as Tarski says, they willbesatisfied or not by interpretations of LAr.

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

1. (i)X isPx_n (forsomen); orX isPy; or (ii)X isYx_nx_m(for somem andn) and<f(x_n),f(x_m)>∈R;orX isYyx_n(for somen) and<a,f(x_n)>∈R;orX isYx_ny(for somen) and<f(x_n),a>∈R;orX isYyy and<a,a>∈R; or
2. there is a formula functionY such thatX is¬Y and <A,a,R> doesnot satisfyY with respect to sequencef; or
3. there are formula functionsY andZ such thatX is (Y→Z) and either <A,a,R> does not satisfyY with respect tosequencef or <A,a,R>satisfiesZ with respect to sequencef; or,finally,
4. there is a formula functionZ and a numbernsuch thatX is∀x_n(Px_n→Z)and every sequenceg that assigns values fromA tothe (original) variables of LAr and that differs fromf atmost in what it assigns tox_n is suchthat <A,a,R> satisfiesZwith respect tog.^[3]

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

The notion of satisfaction of a sentential function is easy tocharacterize in terms of the defined notion of satisfaction of aformula function. Say thatan interpretation <A,a,R>satisfies sentential function X ifand only if <A,a,R> satisfiesformula functionX with respect to every sequence. Thisdefinition is analogous to the Tarskian definition of truth.

In terms of the defined notion of satisfaction, Tarski introduces thenotion of amodel of a sentence. A model of a sentenceS is an interpretation that satisfies the sentential functionS′ determined byS; more generally, a model ofa set of sentencesK is an interpretation that satisfies allthe sentential functions determined by sentences ofK. And interms of the defined notion of model Tarski proposes his definednotion of logical consequence. A sentenceX is a (Tarskian)logical consequence of the sentences in setK if and only ifevery model of the setK is also a model of sentenceX (cf. Tarski 1983c, p. 417). Tarski proposes also a definednotion of logical truth (he uses the expression “analytictruth”) using the same apparatus: a sentenceS is a(Tarskian) logical truth if and only if every interpretation ofS is a model ofS. Analogous notions of Tarskianlogical consequence and logical truth can be defined for otherlanguages using the same method we have followed with LAr, just makingthe obvious changes.

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

it can be proved, on the basis of this definition, that everyconsequence of true sentences must be true, and also that theconsequence relation which holds between given sentences is completelyindependent of the sense of the extra-logical constants which occur inthese sentences. In brief, it can be shown that the condition (F)formulated above is necessary if the sentenceX is to followfrom the sentences of the classK (Tarski 1983c, p. 417).

(The proof that Tarski seems to have in mind is this: suppose thatX is a Tarskian logical consequence ofK; then thereis no model ofK which is not a model ofX; so thereis no substitution instance <K′,X′> of <K,X> such that thesentences inK′ are true andX′ isfalse; for if there was one such, it would readily provide aninterpretation—constituted by the extensions of theextra-logical constants of <K′,X′>—which would constitute a model ofK that would not be a model ofX. But whether thisis the proof Tarski had in mind is a disputed exegeticalquestion.)

Hence, as was to be desired, if the defined relation of logicalconsequence holds for a given pair <K,X>,then also condition (F) holds for it. (Although this can be shown, theconverse cannot; that is, it cannot be shown that ifX andK satisfy (F) thenX follows fromKaccording to Tarski’s definition. But this is all right, since,as Tarski has already pointed out, (F) is not a sufficient conditionfor the ordinary notion of consequence.)

4. Logical constants

Tarski did not think that the construction of section 3 completelysolved the problem of offering “a materially adequate definitionof the concept of consequence” (Tarski 1983c, p. 418). Accordingto Tarski, perhaps the most important difficulty that remained towardsolving that problem is created by the fact that “underlying ourwhole construction is the division of all terms of the languagediscussed into logical and extra-logical” (Tarski 1983c, p.418). This situation is tolerable because, as Tarski says in a letterof 1944, “it is clear that for all languages which are familiarto us such definitions [of ‘logical term’ and‘logical truth’] can be given (or rather: have beengiven); moreover, they prove fruitful, and this is really the mostimportant. We can define ‘logical terms’, e.g., byenumeration” (Tarski 1987, p. 29). But since the division is notbased on a previous characterization of logical terms generallyapplicable to arbitrary languages, to that extent the definition oflogical consequence is not fully general, and hence unsatisfactory. Inthe final paragraph of the consequence paper Tarski says that apositive solution to the problem would “enable us to justify thetraditional boundary between logical and extra-logicalexpressions” (Tarski 1983c, p. 420). In fact, this is theboundary which for Tarski is “underlying our wholediscussion”.

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

When he speaks of extending liberally the set of logical terms, Tarskiis 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 presupposedby and form part of every science. But similarly, in developing acertain theory, not only logic but other theories may be taken forgranted; thus Tarski speaks of logic and arithmetic as a convenientlypresupposed basis of theories for the development of geometry. Thiswould account naturally for the fact that Tarski was skeptical aboutthe possibility of finding a sharp distinction between logical andextra-logical terms, for it might very well depend on the context ofinvestigation what terms and what laws are considered as forming partof the “logic” of the investigation.

An example which we know Tarski had in mind is the sign for themembership relation. We find him saying: “sometimes it seems tome convenient to include mathematical terms, like the ∈-relation,in the class of logical ones, and sometimes I prefer to restrictmyself to terms of ‘elementary logic’. Is any probleminvolved here?” (Tarski 1987, p. 29). We may take the sign formembership as a logical sign in some formalizations of the theory oftypes (which can in turn be used to formalize theories with arbitraryuniverses 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 againput forward the view that the notion of logical constant may be of arelative character. In different contexts different terms may be takenas logical and therefore so may vary the extension of the relation oflogical consequence. He says that “the fluctuation in the commonusage of the concept of consequence would—in part atleast—be quite naturally reflected in such a compulsorysituation” of relativity (Tarski 1983c, p. 420).

But this does not mean that no desiderata could or should be imposedon a definition of the concept of a logical constant. Many years afterwriting his paper on logical consequence, Tarski returned to theproblem of the definition of the concept of a logical term, advancingan attempt at a positive solution. Tarski thought that this solutionaccommodated his conviction that the notion of a logical term is notan absolute, but a relative one.

The basis for the proposed solution appears in Tarski’s 1966lecture “What are Logical Notions?” (publishedposthumously as Tarski 1986a). Here Tarski first makes some remarksabout the general nature of his proposal. He says that an answer to aquestion like the one that gives title to the lecture may take severalforms. It may give an account of the prevailing usage of the term“logical notion”, or of the prevailing usage of the termamong people who are qualified to use it. It may be a normativeproposal, a suggestion that the term be used in a certain way,independently of its actual use. As a third possibility, some otheranswers

seem to aim at something very different (…); people speak ofcatching the proper, true meaning of a notion, something independentof actual usage, and independent of any normative proposals, somethinglike the platonic idea behind the notion. This last approach is soforeign and strange to me that I shall simply ignore it for I cannotsay anything intelligent on such matters (Tarski 1986a, p. 145).

Immediately Tarski makes it clear that he is also not concerned withdeveloping a normative proposal, but an account which captures acertain common use of the concept of logical notion: “inanswering the question ‘What are logical notions?’ what Ishall do is make a suggestion or proposal about a possible use of theterm ‘logical notion’. This suggestion seems to me to bein agreement, if not with a prevailing usage of the term‘logical notion’, at least with one usage which actuallyis encountered in practice” (Tarski 1986a, p. 145).

What are notions? Tarski says:

I use the term “notion” in a rather loose and generalsense, to mean, roughly speaking, objects of all possible types insome hierarchy of types like that inPrincipia 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 invariantunder all one-one transformations of the universe of discourse ontoitself:

consider the class ofall one-one transformations of thespace, or universe of discourse, or “world”, onto itself.What will be the science which deals with the notions invariant underthis widest class of transformations? Here we will have very fewnotions, all of a very general character. I suggest that they are thelogical notions, that we call a notion “logical” if it isinvariant under all possible one-one transformations of the world ontoitself (Tarski 1986a, p. 149).

A one-one transformation of a class onto itself, also called apermutation, induces permutations of all the types in the hierarchy oftypes of “notions” determined by the class. Thus, apermutationP of a domain of individualsD induces apermutation of the class ofn-ary relations of elements ofD, a permutation of the class of functions with n argumentswith domainDⁿ and range included inD, a permutation of the class ofn-ary relationsamong relations of elements ofD, etc. A notion or objectO of a certain typet is invariant under allpermutations of the universe of discourse if, for every permutationP of this universe, the permutationP̃ inducedbyP in the class of notions of typet is such thatP̃(O)=O.

There is no straightforward way in which the truth-functions and thequantifiers (over any type of objects) are identifiable with notionsin the sense of Tarski. According to John Corcoran (see footnote 6 inTarski 1986a, p. 150), in one version of the lecture Tarski indicatedthat the truth-functions and classical quantifiers can be constructedas certain objects in the type hierarchy that are invariant under allpermutations, and, in this sense, are logical notions. (Clearly,Tarski took the logicality of these notions almost for granted, andwas concerned with a definition which gave some account of the statusof other more substantive notions in various mathematical languages.)For example, the truth-values “true” and“false” can be identified with the universe of individualsand the null set, respectively, and the truth-functions in turn withfunctions having (tuples of) these classes as arguments and values;and the classical universal and existential quantifiers over a type ofobjectst can be identified with certain functions from theclass of sets of objects of typet into the class oftruth-values—identifying “true” with the universalset of objects of typet and “false” with theempty set of that type. (A universal quantifier will assign“true” to the set of all objects of typet, and“false” to all other subsets oft; and theexistential quantifier will assign “true” to the non-emptysubsets, and “false” to the empty subset.)

In 1966 Tarski does not propose a definition of the concept of logicalconstant. One such definition, based on the 1966 idea, appears in abook Tarski wrote in collaboration with Steven Givant, published in1987, four years after Tarski’s death. The main concern ofTarski and Givant (1987) is to indicate how to develop set theory inseveral different languages and to compare their expressive power.They define the concept of logical constant for constants in thevocabulary of a certain class of languages—all of which are in acertain sense extensions of a basic language in which they areparticularly interested, weaker than the languages in which set theoryis usually developed in means of expression, and yet sufficientlypowerful for many purposes. The vocabulary of the basic language iscomposed of three two-place predicate constants of the second type andfour two-place function constants of the third type; the vocabulary ofthe extensions considered by Tarski and Givant can only includeconstants of those two kinds. (However, as we will see, there does notseem to be any problem in extending the range of applicability oftheir definition to wider classes of languages.) They then proveseveral results about that class of languages, results that involvethe defined concept of logical constant.

Tarski and Givant introduce informally the concept of a derivativeuniverse of a given basic universeU. A derivative universeŨ of a given basic universeU is what we havebeen calling the class of all objects of a certain type, generatedfrom that basic universeU. Thus, the class ofn-aryrelations of elements ofU (for any givenn), theclass ofn-ary relations among relations of elements ofU, etc., are derivative universes ofU. Afterintroducing in the same way as above the concepts of a permutation ofthe universe and of invariance of a member of a derivative universeunder every permutation of the basic universe, Tarski and Givant givethe following definitions:

1. Given a basic universeU, a memberM of anyderivative universeŨ is said to be logical, or alogical object, if it is invariant under every permutationPofU.

   (Strictly speaking, since an objectM can be a member of manyderivative universes, we should use in (i) the phrase “is saidto be logical, or a logical object, as a member ofŨ”.)

2. A symbolS of [a language in the class considered byTarski and Givant] is said to be logical, or a logical constant, if,for every given realizationU of this [language] withthe universeU,S denotes a logical object in somederivative universeŨ (Tarski and Givant 1987, p.57).

(i) and the parenthetical comment that comes after it embody fairlyaccurately the basic idea behind the (relativized) definition of alogical notion offered in Tarski (1986a), and (ii) offers the(absolute) definition of the concept of logical constant. There is noapparent obstacle to applying (ii) to wider classes of languages thanthe class considered for special purposes of their investigation byTarski and Givant. It seems safe to assume that the definition can beapplied to a large class of languages in the classical hierarchy ofquantificational languages. Tarski and Givant themselves say that theusual logical constants of languages not in the class considered bythem, like the symbols for the truth-functional connectives andquantifiers, “can also be subsumed under logical constants inthe sense of (ii)” (Tarski and Givant 1987, p. 57), presumablythrough some artifice in the style of the one discussed above. (As wewill soon see, at the very least Tarski clearly would want hisdefinition to be applicable to some languages whose set of sentencesis that of some formulations of the simple theory of types.)

In Tarski and Lindenbaum (1935) the authors had proved that given abasic universeU, all the notions in derivative universes ofU which can be defined in the language of the simple theoryof types are invariant under all permutations ofU. Ingeneral the Tarski-Lindenbaum theorem guarantees that all mathematicalnotions definable in the logicist fashion in the simple theory oftypes are logical notions no matter what the universe of individualsis taken to be (Tarski says that “we may interpret [the universeof the intended interpretation of the language of the theory of types]as the universe of physical objects, although there is nothing inPrincipia mathematica which compels us to accept such aninterpretation” (Tarski 1986a, p. 152)). Since the theoremapplies to every universeU supplying an interpretation ofthe language of the theory of types, the definition of the concept oflogical constant in Tarski and Givant (1987) implies that allprimitive symbols denoting notions in that language (e.g. quantifiersof all orders) are Tarskian logical constants; also, if the definitionwere applicable to defined symbols, all these symbols would beTarskian logical constants. Such results agree well with a“usage actually encountered in practice” (for example, thepractice of the logicists, but of others as well) according to whichthe constants of the language of the theory of types are logicalconstants.

When the Tarski-Givant definition is applied to interpreted languagesof mathematical theories with undefined mathematical primitives, itwill generally yield the result that the notions denoted by theseprimitives are not Tarskian logical constants. Tarski’s exampleis set theory formalized in first-order with a single primitivepredicate for membership as a relation among elements of the universe(see Tarski 1986a, p. 153). Obviously membership as a relation over adomain of individuals and sets is not invariant under all permutationsof that domain, so it is not declared a logical notion of the languageof set theory by Tarski’s proposed definition. Similarly, theclass of all sets will be declared non-logical provided the class ofindividuals that are not sets is not empty. A predicate whose intendedmeaning is membership (among elements of the universe) is not aTarskian logical constant, simply because there is a universe in whichit denotes a non-logical object (with respect to that universe);similarly, a predicate “S” whose intended meaningis “is a set” (such as is used in some formalizations ofset theory suitable for contemplating individuals other than sets inthe universe) is not a Tarskian logical constant, for there is auniverse in which “S” denotes a non-logicalobject (in such a universe, the class of objects that are not setsmust be non-empty). These results are again in agreement with actualusage, 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 typesand first-order set theory, Tarski extracts a conclusion that hewelcomed. This is the conclusion that, in a certain sense, thedistinction between logical and non-logical constants is relative to acontext of investigation. Recall Tarski’s remarks about the signfor membership in a letter of 1944. In the 1966 lecture Tarski saysthat membership is a logical notion in some formalizations of thetheory of types. He says:

Using this method [ofPrincipia Mathematica], it is clearthat the membership relation is certainly a logical notion. It occursin several types, for individuals are elements of classes ofindividuals, classes of individuals are elements of classes of classesof individuals, and so on. And by the very definition of an inducedtransformation it is invariant under every transformation of the worldonto itself (Tarski 1986a, pp. 152–3).

It is a bit unclear that the “notions” of membership thatTarski talks about are notions in the sense introduced by him earlierin the lecture. They do not “occur” in any type, given thestandard formulations of the theory of types (even in formulations inwhich a sign for membership is used), for they are relations amongobjects of different types, which do not occur in a non-cumulativehierarchy. But Tarski is perhaps thinking of some non-standard,cumulative type hierarchy. Under a suitable expansion of the conceptof a notion, the notions of membership Tarski has in mind will indeedbe invariant under permutations of any universe, and hence logical;and the sign or signs for these notions of membership will be Tarskianlogical constants. On the other hand, as pointed out above, membershipas a relation over the domain of individuals and sets will not be alogical notion in that domain, and hence the sign for this concept ofmembership will not be a Tarskian logical constant. Thus,interestingly, Tarski’s definition of the concept of logicalconstant respects in a sense his early relativistic view in the paperon logical consequence. Tarski describes the situation in thefollowing passage:

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

5. Further reading

Tarski’s papers are reprinted and collected in Tarski 1986b. Thevolume Tarski 1983a contains widely used English translations of hismain papers of the 1920s and 1930s. Givant (1986) gives a completebibliography of Tarski’s publications until 1986. Feferman andFeferman 2004 is a biography of Tarski that also contains a veryvaluable introduction to his logical and mathematical work, includingdescriptions of all the results and developments mentioned in theinitial paragraph of this entry. On Tarski’s logical work seealso Simmons 2009 and Eastaugh 2017. McFarland, McFarland & Smith2014 contains a wealth of biographical and historical informationabout Tarski, as well as updates to Givant’s bibliography andtranslations into English of a number of previously untranslated earlytexts by Tarski. Also of much biographical interest are the lettersfrom Tarski to J. H. Woodger included in Mancosu 2021.

Collections of articles devoted to (parts of) Tarski’s workinclude: vols. 51 (1986) and 53 (1988) of theJournal of SymbolicLogic; Woleński and Köhler (1999); vols. 126 (2001) and142 (2004) ofSynthese; vols. 126 and 127 of theAnnalsof Pure and Applied Logic (2004); Patterson 2008a; and Sagi andWoods 2021. Patterson 2012 is a book-length historical andphilosophical study of many of Tarski’s philosophical ideas,focusing on truth and logical consequence. Gruber 2016 is abook-length commentary of Tarski’s monograph on truth,“The Concept of Truth in Formalized Languages”, whichcompares the Polish original version, the German translation, and theEnglish version of the German text.

There is a considerable bibliography on critical and exegeticalaspects of Tarski’s philosophical work. The bibliography ontruth is especially extensive. It includes the chapters onTarski’s theory of truth in Coffa 1991; Kirkham 1992; Soames1999, 2010, and 2018; Künne 2003; Burgess and Burgess 2011; andBlackburn 2018; many chapters of Woleński 1999 and 2019; thepapers Field 1972; Davidson 1973, 1990; Kripke 1975, 2019; Church1976; Soames 1984; Putnam 1985; Hodges 1985/6, 2004, and 2008; McGee1993; García-Carpintero 1996; Heck 1997; Murawski 1998; DeVidiand Solomon 1999; Ketland 1999; Milne 1999; Sher 1999;Gómez-Torrente 2001, 2004, and 2019; Eklund 2002; Gupta 2002;Rojszczak 2002; Raatikainen 2003, 2008; Ray 2003, 2005, 2018; Sundholm2003; Feferman 2004a; Frost-Arnold 2004; Hintikka 2004; Patterson2004, 2008b; Azzouni 2005, 2008; Horwich 2005; Burgess 2008; David2008; Mancosu 2008; Simons 2008; Asay 2013; Schiemer and Reck 2013;Loeb 2014; Barnard and Ulatowski 2016; Smith 2017; Simchen 2020; andthe entries on theliar paradox,model theory,Tarski: truth definitions, andtruth.

The critical and exegetical bibliography on Tarski on logicalconsequence includes Kneale and Kneale 1962; Etchemendy 1988, 1990,and 2008; McGee 1992, 2004; Gómez-Torrente 1996, 1998/9, 2008,and 2009; Ray 1996;, Sher 1996, 2022; Hanson 1997; Sagüillo 1997;Chihara 1998; Shapiro 1998; Schurz 1999; Blanchette 2000; Bays 2001;Edwards 2003; Jané 2006; Mancosu 2006, 2010; Park 2018; Zinke2018; Griffiths and Paseau 2022; and the entries onlogical consequence andlogical truth.

The Tarskian ideas on logical constants are discussed, among otherplaces, in Simons 1987; Sher 1991, 2008, and 2021; Feferman 1999;Gómez-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 andVäänänen 2021; and the entry onlogical constants.

The literature on general philosophical aspects of Tarski’s workand opinions includes Suppes 1988; Woleński 1993; Sinaceur 2001;Feferman 2004b; Mycielski 2004; Mancosu 2005, 2009; Betti 2008; andFrost-Arnold 2008, 2013.

Bibliography

Primary Sources: Works by Tarski

• Tarski, A., 1931. “Sur les ensembles définissables denombres réels. I.”,Fundamenta Mathematicae, 17:210–239.
• –––, 1933.Pojęcie prawdy wjęzykach nauk dedukcyjnych, Warsaw: NakłademTowarzystwa Naukowego Warszawskiego.
• –––, 1935. “Der Wahrheitsbegriff in denformalisierten Sprachen”,Studia Philosophica, 1:261–405. Translation of Tarski 1933 by L. Blaustein, with apostscript added.
• –––, 1936a. “O pojęciu wynikanialogicznego”,PrzeglądFilozoficzny, 39: 58–68.
• –––, 1936b. “Über den Begriff derlogischen Folgerung”, inActes du CongrèsInternational de Philosophie Scientifique, fasc. 7(Actualités Scientifiques et Industrielles, vol. 394), Paris:Hermann et Cie, pp. 1–11.
• –––, 1937.Einführung in diemathematische Logik und in die Methodologie der Mathematik.Vienna: Julius Springer. (Translated with additions as Tarski1941.)
• –––, 1941.Introduction to Logic and to theMethodology of Deductive Science, translation of Tarski 1937 byO. Helmer, with additions. New York: Oxford University Press.
• –––, 1944. “The Semantic Conception ofTruth: And the Foundations of Semantics”,Philosophy andPhenomenological Research, 4: 341–376.
• –––, 1983a.Logic, Semantics,Metamathematics, second edition, ed. by J. Corcoran.Indianapolis: Hackett.
• –––, 1983b. “The Concept of Truth inFormalized Languages”, translation of Tarski 1935 by J.H.Woodger in Tarski 1983a, pp. 152–278.
• –––, 1983c. “On the Concept of LogicalConsequence”, translation of Tarski 1936b by J.H. Woodger inTarski 1983a, pp. 409–20.
• –––, 1983d. “On Definable Sets of RealNumbers”, translation of Tarski 1931 by J.H. Woodger in Tarski1983a, pp. 110–142.
• –––, 1986a. “What Are LogicalNotions?”, ed. by J. Corcoran.History and Philosophy ofLogic, 7: 143–54.
• –––, 1986b.Collected Papers, fourvolumes, ed. by S. Givant and R. McKenzie. Basel:Birkhäuser.
• –––, 1987. “A Philosophical Letter ofAlfred Tarski”,Journal of Philosophy, 84: 28–32.A 1944 letter of Tarski to Morton White, published with a preface ofthe latter.
• –––, 2002. “On the Concept of FollowingLogically”, translation of Tarski 1936a by M. Stroińska andD. Hitchcock.History and Philosophy of Logic, 23:155–96.
• –––, and S. Givant, 1987.A Formalization ofSet Theory without Variables, Providence, RI: AmericanMathematical Society.
• –––, and A. Lindenbaum, 1935. “Überdie Beschränktheit der Ausdrucksmittel deduktiverTheorien”, inErgebnisse eines mathematischenKolloquiums, fasc. 7, 1934–1935, pp. 15–22.(Translated as Tarski and Lindenbaum 1983.)
• –––, and A. Lindenbaum, 1983. “On theLimitations of the Means of Expression of Deductive Theories”,translation of Tarski and Lindenbaum 1935 by J.H. Woodger in Tarski1983a, pp. 384–392.

Secondary Sources

• Asay, J., 2013. “Tarski and Primitivism about Truth”,Philosophers’ Imprint, 13: 1–18.
• Azzouni, J., 2005. “Tarski, Quine, and the Transcendence ofthe Vernacular ‘True’”,Synthese, 142:273–88.
• –––, 2008. “Alternative Logics and theRole of Truth in the Interpretation of Languages”, in Patterson2008a, pp. 390–429.
• Barnard, R. and J. Ulatowski, 2016. “Tarski’s 1944Polemical Remarks and Naess’ ‘ExperimentalPhilosophy’”,Erkenntnis, 81: 457–477.
• Bays, T., 2001. “On Tarski on Models”,Journal ofSymbolic Logic, 66: 1701–26.
• Bellotti, L., 2003. “Tarski on Logical Notions”,Synthese, 135: 401–13.
• Betti, A., 2008. “Polish Axiomatics and its Truth: OnTarski’s Leśniewskian Background and the AjdukiewiczConnection”, in Patterson 2008a, pp. 44–71.
• Blackburn, S., 2018.Truth, Oxford: Oxford UniversityPress.
• Blanchette, P., 2000. “Models and Modality”,Synthese, 124: 45–72.
• Bonnay, D., 2008. “Logicality and Invariance”,Bulletin of Symbolic Logic, 14: 29–68.
• –––, 2014. “Logical Constants, or How toUse Invariance in Order to Complete the Explication of LogicalConsequence”,Philosophy Compass, 9: 54–65.
• Bonnay, D. and S. Speitel, 2021. “The Ways of Logicality:Invariance and Categoricity”, in Sagi and Woods 2021, pp.55–79.
• Burgess, J.P., 2008. “Tarski’s Tort”, inBurgess,Mathematics, Models, and Modality. Selected PhilosophicalEssays, Cambridge: Cambridge University Press, pp.149–68.
• Burgess, A.G. and J.P. Burgess, 2011.Truth, Princeton,NJ: Princeton University Press.
• Casanovas, E., 2007. “Logical Operations andInvariance”,Journal of Philosophical Logic, 36:33–60.
• Chihara, C., 1998. “Tarski’s Thesis and the Ontologyof Mathematics”, in M. Schirn (ed.),The Philosophy ofMathematics Today, Oxford: Clarendon Press, pp.157–72.
• Church, A., 1976. “Comparison of Russell’s Resolutionof the Semantical Antinomies with That of Tarski”,Journalof Symbolic Logic, 41: 747–60.
• Coffa, J.A., 1991.The Semantic Tradition from Kant toCarnap, Cambridge: Cambridge University Press.
• David, M., 2008. “Tarski’s Convention T and theConcept of Truth”, in Patterson 2008a, pp. 133–56.
• Davidson, D., 1973. “In Defence of Convention T”,reprinted in Davidson,Inquiries into Truth andInterpretation, New York: Oxford University Press, 1984, pp.65–75.
• –––, 1990. “The Structure and Content ofTruth”,Journal of Philosophy, 87: 279–328.
• DeVidi, D. and G. Solomon, 1999. “Tarski on‘Essentially Richer’ Metalanguages”,Journal ofPhilosophical Logic, 28: 1–28.
• Dutilh Novaes, C., 2014. “The Undergeneration of PermutationInvariance as a Criterion for Logicality”,Erkenntnis,79: 81–97.
• Eastaugh, B., 2017. “Tarski”, in A. Malpass and M.Antonutti Marfori (eds.),The History of Philosophical and FormalLogic: From Aristotle to Tarski, London: Bloomsbury, pp.293–313.
• Edwards, J., 2003. “Reduction and Tarski’s Definitionof Logical Consequence”,Notre Dame Journal of FormalLogic, 44: 49–62.
• Eklund, M., 2002. “A Vindication of Tarski’s Claimsabout the Liar Paradox”, in T. Childers (ed.),The LogicaYearbook 2001, Prague: Filosofia, pp. 49–55.
• Etchemendy, J., 1988. “Tarski on Truth and LogicalConsequence”,Journal of Symbolic Logic, 53:51–79.
• –––, 1990.The Concept of LogicalConsequence, Cambridge, MA: Harvard University Press.
• –––, 2008. “Reflections onConsequence”, in Patterson 2008a, pp. 263–99.
• Feferman, S., 1999. “Logic, Logics, and Logicism”,Notre Dame Journal of Formal Logic, 40: 31–54.
• –––, 2004a. “Tarski’s ConceptualAnalysis of Semantical Notions”, in A. Benmakhlouf (ed.),Sémantique et Épistemologie, Casablanca-Paris:Le Fennec-Vrin, pp. 79–108. Reprinted in Patterson 2008a, pp.72–93.
• –––, 2004b. “Tarski’s Conception ofLogic”,Annals of Pure and Applied Logic, 126:5–13.
• Feferman, A.B. and S. Feferman, 2004.Alfred Tarski. Life andLogic, Cambridge: Cambridge University Press.
• Field, H., 1972. “Tarski’s Theory of Truth”,reprinted in Field,Truth and the Absence of Fact, New York:Oxford University Press, 2001, pp. 3–26.
• Frost-Arnold, G., 2004. “Was Tarski’s Theory of TruthMotivated by Physicalism?”,History and Philosophy ofLogic, 25: 265–80.
• –––, 2008. “Tarski’sNominalism”, in Patterson 2008a, pp. 225–46.
• –––, 2013.Carnap, Tarski, and Quine atHarvard. Conversations on Logic, Mathematics, and Science,Chicago: Open Court.
• García-Carpintero, M., 1996. “What Is a TarskianDefinition of Truth?”,Philosophical Studies, 82:113–44.
• Givant, S., 1986. “Bibliography of Alfred Tarski”,Journal of Symbolic Logic, 51: 913–41.
• Gómez-Torrente, M., 1996. “Tarski on LogicalConsequence”,Notre Dame Journal of Formal Logic, 37:125–51.
• –––, 1998/9. “Logical Truth and TarskianLogical Truth”,Synthese, 117: 375–408.
• –––, 2001. “Notas sobre elWahrheitsbegriff”, parts I and II.AnálisisFilosófico, 21: 5–41, 149–85.
• –––, 2002. “The Problem of LogicalConstants”,Bulletin of Symbolic Logic, 8:1–37.
• –––, 2004. “The Indefinability of Truth inthe ‘Wahrheitsbegriff’”,Annals of Pure andApplied Logic, 126: 27–37.
• –––, 2008. “Are There Model-TheoreticLogical Truths that are not Logically True?”, in Patterson2008a, pp. 340–68.
• –––, 2009. “Rereading Tarski on LogicalConsequence”,Review of Symbolic Logic, 2:249–97.
• –––, 2019. “Soames on the LogicalEmpiricists on Truth, Meaning, Convention, and Logical Truth”,Philosophical Studies, 176: 1357–65.
• –––, 2021. “The Problem of LogicalConstants and the Semantic Tradition: From Invariantist Views to aPragmatic Account”, in Sagi and Woods 2021, pp.35–54.
• Griffiths, O. and A.C. Paseau, 2022.One True Logic: A MonistManifesto, Oxford: Oxford University Press.
• Gruber, M., 2016.Alfred Tarski and the “Concept ofTruth in Formalized Languages”. A Running Commentary withConsideration of the Polish Original and the German Translation,Cham: Springer.
• Gupta, A., 2002. “An Argument against Tarski’sConvention T”, in R. Schantz (ed.),What Is Truth?,Berlin: De Gruyter, pp. 225–37.
• Hanson, W., 1997. “The Concept of LogicalConsequence”,Philosophical Review, 106:365–409.
• Heck, R., 1997. “Tarski, Truth, and Semantics”,Philosophical Review, 106: 533–54.
• Hintikka, J., 2004. “On Tarski’s Assumptions”,Synthese, 142: 353–69.
• Hodges, W., 1985/6. “Truth in a Structure”,Proceedings of the Aristotelian Society, 86:135–51.
• –––, 2004. “What Languages Have TarskiTruth Definitions?”,Annals of Pure and Applied Logic,126: 93–113.
• –––, 2008. “Tarski’s Theory ofDefinition”, in Patterson 2008a, pp. 94–132.
• Horwich, P., 2005. “A Minimalist Critique of Tarski onTruth”, in JC Beall (ed.),Deflationism and Paradox,Oxford: Clarendon Press, pp. 75–84.
• Jané, I., 2006. “What Is Tarski’sCommon Concept of Consequence?”,Bulletin ofSymbolic Logic, 12: 1–42.
• Kennedy, J., 2021.Gödel, Tarski and the Lure of NaturalLanguage. Logical Entanglement, Formalism Freeness, Cambridge:Cambridge University Press.
• Kennedy, J. and J. Väänänen, 2021.“Logicality and Model Classes”,Bulletin of SymbolicLogic, 27: 385–414.
• Ketland, J., 1999. “Deflationism and Tarski’sParadise”,Mind, 108: 69–94.
• Kirkham, R., 1992.Theories of Truth, Cambridge, MA: MITPress.
• Kneale, W. and M. Kneale, 1962.The Development of Logic,Oxford: Clarendon Press.
• Kripke, S., 1975. “Outline of a Theory of Truth”,Journal of Philosophy, 72: 690–716.
• –––, 2019. “Ungroundedness in TarskianLanguages”,Journal of Philosophical Logic, 48:603–9.
• Künne, W., 2003.Conceptions of Truth, Oxford:Clarendon Press.
• Loeb, I., 2014. “Towards Transfinite Type Theory: RereadingTarski’s Wahrheitsbegriff”,Synthese, 191:2281–99.
• McFarland, A., J. McFarland, and J. T. Smith (eds.), 2014.Alfred Tarski. Early Work in Poland – Geometry andTeaching, New York: Birkhäuser.
• McGee, V., 1992. “Two Problems with Tarski’s Theory ofConsequence”,Proceedings of the Aristotelian Society,92: 273–92.
• –––, 1993. “A Semantic Conception ofTruth?”,Philosophical Topics, 21: 83–111.
• –––, 2004. “Tarski’s StaggeringExistential Assumptions”,Synthese, 142:371–87.
• Mancosu, P., 2005. “Harvard 1940–1941: Tarski, Carnapand Quine on a Finitistic Language of Mathematics for Science”,History and Philosophy of Logic, 26: 327–57.
• –––, 2006. “Tarski on Models and LogicalConsequence”, in J. Ferreirós and J. J. Gray (eds.),The Architecture of Modern Mathematics, Oxford: OxfordUniversity Press, pp. 209–37.
• –––, 2008. “Tarski, Neurath, andKokoszyńska on the Semantic Conception of Truth”, inPatterson 2008a, pp. 192–224.
• –––, 2009. “Tarski’s Engagement withPhilosophy”, in S. Lapointeet al, (eds.),TheGolden Age of Polish Philosophy, Dordrecht: Springer, pp.131–53.
• –––, 2010. “Fixed- Versus Variable-DomainInterpretations of Tarski’s Account of LogicalConsequence”,Philosophy Compass, 5:745–759.
• –––, 2021. “Logic and Biology: TheCorrespondence Between Alfred Tarski and Joseph H. Woodger”,Journal of Humanistic Mathematics, 11: 18–105.doi:10.5642/jhummath.202102.04
• Milne, P., 1999. “Tarski, Truth and Model Theory”,Proceedings of the Aristotelian Society, 99:141–67.
• Murawski, R., 1998. “Undefinability of Truth. The Problem ofPriority: Tarski vs. Gödel”,History and Philosophy ofLogic, 19: 153–60.
• Mycielski, J., 2004. “On the Tension between Tarski’sNominalism and His Model Theory”,Annals of Pure and AppliedLogic, 126: 215–24.
• Park, W., 2018.Philosophy’s Loss of Logic toMathematics. An Inadequately Understood Take-Over, Cham:Springer.
• Patterson, D., 2004. “The Status of Convention T”, inL. Běhounek (ed.),The Logica Yearbook 2003, Prague:Filosofia, pp. 195–207.
• ––– (ed.), 2008a.New Essays on Tarski andPhilosophy, Oxford: Oxford University Press.
• –––, 2008b. “Tarski’s Conception ofMeaning”, in Patterson 2008a, pp. 157–91.
• –––, 2012.Alfred Tarski: Philosophy ofLanguage and Logic, New York: Palgrave Macmillan.
• Putnam, H., 1985. “A Comparison of Something with SomethingElse”, reprinted in Putnam,Words and Life, Cambridge,MA: Harvard University Press, 1994, pp. 330–50.
• Raatikainen, P., 2003. “More on Putnam and Tarski”,Synthese, 135: 37–47.
• –––, 2008. “Truth, Meaning, andTranslation”, in Patterson 2008a, pp. 247–62.
• Ray, G., 1996. “Logical Consequence: a Defense ofTarski”,Journal of Philosophical Logic, 25:617–77.
• –––, 2003. “Tarski and the MetalinguisticLiar”,Philosophical Studies, 115: 55–80.
• –––, 2005. “On the Matter of EssentialRichness”,Journal of Philosophical Logic, 34:433–57.
• –––, 2018. “Tarski on the Concept ofTruth”, in M. Glanzberg (ed.),The Oxford Handbook ofTruth, Oxford: Oxford University Press, pp. 695–717.
• Rojszczak, A., 2002. “Philosophical Background andPhilosophical Content of the Semantic Definition of Truth”,Erkenntnis, 56: 29–62.
• Sagi, G., 2015. “The Modal and Epistemic Arguments againstthe Invariance Criterion for Logical Terms”,Journal ofPhilosophy, 112: 159–67.
• –––, 2018. “Logicality and Meaning”,Review of Symbolic Logic, 11: 133–59.
• –––, 2021. “Extensionality andLogicality”,Synthese, 198: 1095–1119.
• Sagi, G. and J. Woods (eds.), 2021.The Semantic Conception ofLogic. Essays on Consequence, Invariance, and Meaning, Cambridge:Cambridge University Press.
• Sagüillo, J.M., 1997. “Logical ConsequenceRevisited”,Bulletin of Symbolic Logic, 3:216–241.
• Schiemer, G. and E. Reck, 2013. “Logic in the 1930s: TypeTheory and Model Theory”,Bulletin of Symbolic Logic,19: 433–72.
• Schurz, G., 1999. “Tarski and Carnap on Logical Truth. Or:What is Genuine Logic?”, in Woleński and Köhler 1999,pp. 77–94.
• Shapiro, S., 1998. “Logical Consequence: Models andModality”, in M. Schirn (ed.),The Philosophy of MathematicsToday, Oxford: Clarendon Press, pp. 131–56.
• Sher, G., 1991.The Bounds of Logic, Cambridge, MA: MITPress.
• –––, 1996. “Did Tarski Commit‘Tarski’s Fallacy’?”,Journal of SymbolicLogic, 61: 653–86.
• –––, 1999. “What Is Tarski’s Theoryof Truth?”,Topoi, 59: 149–66.
• –––, 2008. “Tarski’s Thesis”,in Patterson 2008a, pp. 300–39.
• –––, 2021. “Invariance and Logicality inPerspective”, in Sagi and Woods 2021, pp. 13–34.
• –––, 2022.Logical Consequence,Cambridge: Cambridge University Press.
• Simchen, O., 2020. “Modeling Truth for Semantics”,Analytic Philosophy, 61: 28–36.
• Simmons, K., 2009. “Tarski’s Logic”, in D. M.Gabbay and J. Woods (eds.),Handbook of the History of Logic.Volume 5. Logic from Russell to Church, Amsterdam: Elsevier, pp.511–616.
• Simons, P., 1987. “Bolzano, Tarski and the Limits ofLogic”,Philosophia Naturalis, 24: 378–405.
• –––, 2008. “Truth on a Tight Budget:Tarski and Nominalism”, in Patterson 2008a, pp.369–89.
• Sinaceur, H., 2001. “Alfred Tarski: Semantic Shift,Heuristic Shift in Metamathematics”,Synthese, 126:49–65.
• Smith, N. J. J., 2017. “Truth via Satisfaction?”, inP. Arazim and T. Lávička (eds.),The Logica Yearbook2016, London: College Publications, pp. 273–287.
• Soames, S., 1984. “What Is a Theory of Truth?”,Journal of Philosophy, 81: 411–29.
• –––, 1999.Understanding Truth, NewYork: Oxford University Press.
• –––, 2010.Philosophy of Language,Princeton, NJ: Princeton University Press.
• –––, 2018.The Analytic Tradition inPhilosophy. Volume 2. A New Vision, Princeton, NJ: PrincetonUniversity Press.
• Sundholm, G., 2003. “Tarski and Leśniewski on Languageswith Meaning versus Languages without Use”, in J. Hintikkaet al. (eds.),Philosophy and Logic. In Search of thePolish Tradition, Dordrecht: Kluwer, pp. 109–28.
• Suppes, P., 1988. “Philosophical Implications ofTarski’s Work”,Journal of Symbolic Logic, 53:80–91.
• Vaught, R. L., 1974. “Model Theory before 1945”, in L.Henkinet al. (eds.),Proceedings of the TarskiSymposium, Proceedings of Symposia in Pure Mathematics, vol. XXV,Providence, RI: American Mathematical Society, pp. 153–72.
• –––, 1986. “Tarski’s Work in ModelTheory”,Journal of Symbolic Logic, 51:869–82.
• Woleński, J., 1993. “Tarski as a Philosopher”, inF. Coniglioneet al. (eds.),Polish ScientificPhilosophy: The Lvov-Warsaw School, Amsterdam: Rodopi, pp.319–38.
• –––, 1999.Essays in the History of Logicand Logical Philosophy, Cracow: Jagiellonian UniversityPress.
• –––, 2019.Semantics and Truth, Cham:Springer.
• Woleński, J. and E. Köhler (eds.), 1999.AlfredTarski and the Vienna Circle, Dordrecht: Kluwer.
• Zinke, A., 2018.The Metaphysics of Logical Consequence,Frankfurt: Klostermann.

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Acknowledgments

I thank Paolo Mancosu and Richard Zach for helpful comments on anearlier version of this entry.