Truth and Proof

TRUTH AND PROOF.

The antinomy of the liar, a basic obstacle to an adequate definition of truth in natural languages, reappears In formalized languages as a constructive argument showing not all true sentences can be proved

by Alfred Tarski

he subject of this article is an old sentence.) Moreover, when speaking of Here and in the subsequent discussion one. It has been frequently dis sentences, we shall always have in mind the word "false" means the same as the Tcussed in modern logical and phil what are called in grammar declarative expression "not true" and can be re osophical literature, and it would not be sentences, and not interrogative or im placed by the latter. easy to contribute anything original to perative sentences. The intuitive content of Aristotle's the discussion. To many readers, I am Whenever one explains the meaning formulation appears to be rather clear. afraid, none of the ideas put forward in of any term drawn from everyday lan Nevertheless, the formulation leaves. the article will appear essentially novel; guage, he should bear in mind that the much to be desired from the point of nonetheless, I hope they may findso me goal and the logical status of such an ex view of precision and formal correctness. interest in the way the material has been planation may vary from one case to an For one thing, it is not general enough; arranged and knitted together. other. For instance, the explanation may it refers only to sentences that "say" As the title indicates, I wish to discuss be intended as an account of the actual about something "that it is" or "that it is here two differentthough related no use of the term involved, and is thus sub not"; in most cases it would hardly be tions: the notion of truth and the notion ject to questioning whether the account possible to cast a sentence in this mold of proof. Actually the article is divided is indeed correct. At some other time an without slanting the sense of the sen into three sections. The first section is explanation may be of a normative na tence and forcing the spirit of the lan concerned exclusively with the notion of ture, that is, it may be offered as a sug guage. This is perhaps one of the rea truth, the second deals primarily with the gestion that the term be used in some sons why in modern philosophy various notion of proof, and the third is a dis definite way, without claiming that the substitutes for the Aristotelian formula cussion of the relationship between these suggestion conforms to the way in which tion have been offered. As examples we . two notions. the term is actuall y used; such an ex quote the following: planation can be evaluated, for instance, The Notion of Truth from the point of view of its usefulness A sentence is true if it denotes the but not of its correctness. Some further existing state of affairs. The task of explaining the meaning of alternatives could also be listed. The truth of a sentence consists the term "true" will be interpreted here The explanation we wish to give in the in its conformity with (or corre in a restric ted way. The notion of truth present case is, to an extent, of mixed spondence to) the reality. occurs in many different contexts, and character. What will be offered can be there are several distinct categories of treated in principle as a suggestion for Due to the use of technical philo objects to which the term "true" is ap a definite way of using the term "true", sophical terms these fo rmulations have plied. In a psychological discussion one but the offering will be accompanied by undoubtedly a very "scholarly" sound. might speak of true emotions as well as the belief that it is in agreement with the Nonetheless, it is my feeling that the true beliefs; in a discourse from the do prevailing usage of this term in everyday new formulations, when analyzed more main of esthetics the inner truth of an language. closely, prove to be less clear and un object of art might be analyzed. In this Our understanding of the notion of equivocal than the one put forward by article, however, we are interested only truth seems to agree es sentially with Aristotle. in what might be called the logical no various explanations of this notion that The conception of truth that found its tion of truth. More specifically, we con have been given in philosophical litera expression in the Aristotelian form ula cern ourselves exclusively with the mean ture. What may be the earliest explana tion (and in related formulations of more ing of the term "true" when this term is tion can be found in Aristotle's Meta recent origin) is usually referred to as used to refer to sentences. Presumably physics: the classical, or semantic conception of this was the original use of the term truth. By semantics we mean the part of logic that, loosely speaking, discusses "true" in human language. Sentences To say of what is that it is not, or are treated here as linguistic objects, as of what is not that it is, is false, the relations between linguistic objects certain strings of sounds or written signs. while to say of what is that it is, or (such as sentences) and what is ex (Of course, not every such strin g is a of what is not that it is not, is true. pressed by these obj ects. The semantic

© 1969 SCIENTIFIC AMERICAN, INC character of the term "true" is clearly tence. Notice that (1), as well as (1'), has move these difficulties let us try another revealed by the explanation offered by the form prescribed for definitions by the method of forming names of expressions, Aristotle and by some formulations that rules of logiC, namely the form of logical in fact a method that can be character will be given later in this article. One equivalence. It consists of two parts, the ized as a letter-by-letter description of speaks sometimes of the correspondence left and the right side of the equivalence, an expression. Using this method we ob theory of truth as the theory based on combined by the connective "if and only tain instead of (1) the following lengthy the classical conception. if". The left side is the definiendum, the formulation: (In modern philosophical literature phrase whose meaning is explained by some other .conceptions and theories of the definition; the right side is the defin (2) The string of three words, the truth are also discussed, such as the prag iens, the phrase that provides the expla first of which is the string of the letters Es, En, and Double-U, matic conception and the coherence nation. In the present case the definien 0 the second is the string of letters theory. These conceptions seem to be of dum is the following expression: I and Es, and the third is the an excluSively nOlmative character and string of the letters Double-U, "snow is white" is true; have little connection with the actual Aitch, I, Te, and E, is a true sen usage of the term "true"; none of them the definiens has the form: tence if and only if snow is white. has been formulated so far with any de gree of clarity and precision. They will snow is white. Formulation (2) does not differ from not be discussed in the present article.) (1) in its meaning; (1) can sim ply be re We shall attempt to obtain here a It might seem at first sight that (1), garded as an abbreviated form of (2). more precise explanation of the classical when regarded as a definition, exhibits The new formulation is certainly much conception of truth, one that could an essential flawwidely discussed in less perspicuous than the old one, but it supersede the Aristotelian formulation traditional logic as a vicious circle. The has the ad vantage that it creates no ap while preserving its basic intentions. To reason is that certain words, for example pearance of a vicious circle. this end we shall have to resort to some "snow", occur in both the definiens and Partial definitions of truth analogous techniques of contemporary logic. We the definiendum. Actually, however, to (1) (or (2)) can be constructed for oth shall also have to specify the language these occurrences have an entirely differ er sentences as well. Each of these defini whose sentences we are concerned with; ent character. The word "snow" is a syn tions has the form: this is necessary if only for the reason tactical, or organic, part of the definiens; " " that a string of sounds or signs, which is in fact the definiens is a sentence, and ( 3 ) p is true if and only if p, a true or a false sentence but at any rate the word "snow" is its subject. The de a meaningful sentence in one language, finiendum is also a sentence; it expresses where "p" is to be replaced on both sides may be a meaningless expression in an the fact that the definiens is a true sen of (3) by the sentence for which the defi other. For the time being let us assume tence. Its subject is a name of the defin nition is cons tructed. Special attention that the language with which we are iens formed by putting the definiens in should be paid, however, to those situa concerned is the common English lan quotes. (When saying something of an tions in which the sentence put in place guage. object, one alw ays uses a name of this of "p" happens to contain the word We begin with a simple problem. object and not the ob ject itself, even "true" as a syntactical part. The corre Consider a sentence in Engli sh whose when dealing with linguistic objects.) sponding equivalence (3) cannot then be meaning does not raise any doubts, say For several reasons an expression en viewed as a partial defini tion of truth the sentence "snow is white". For brev closed in quotes must be treated gram since, when treated as such, it would ob ity we denote this sentence by "S", so matically as a Single word haVing no syn viously exhibit a vicious circle. Even in that "S" be comes the name of the sen tactical parts. Hence the word "snow", this case, however, (3) is a meaningful tence. We ask ourselves the questi on: which undoubtedly occurs in the defin sentence, and it is actu ally a true sen What do we mean by saying that S is iendum as a part, does not occur there tence from the point of view of the clas true or that it is false? The answer to as a syntactical part. A medieval logician sical conception of truth. For illustra this question is Simple: in the spirit of would say that "snow" occurs in the de tion, imagine that in a review of a book Aristotelian explanation, by saying that finiens in suppositione formalis and in one finds the following sentence: S is true we mean simply that snow is the definiendum in suppositione materi white, and by saying that S is false we aliso However , words which are not syn ( 4) Not every sentence in this book is mean that snow is not white. By elimi tactical parts of the definiendum cannot true. nating the symbol "S" we arrive at the create a vicious circle, and the danger of following formulations: a vicious circle vanishes. By applying to (4) the Aristotelian cri The preceding remarks touch on some terion, we see that the sentence (4) is ( 1) "snow is white" is true if and questions which are rat her subtle and true if, in fact, not every sentence in the only if snow is white. not quite simple from the logical point of book concerned is true, and that (4) is ( 1') "snow is white" is false if and view. Instead of elaborating on them, only if snow is not white. false otherwise; in other words, we can I shall indicate another manner in which assert the equivalence obtained from Thus (1) and (1 ') provide satisfactory any fears of a vicious circle can be dis (3) by taking (4) for "p". Of course, this explanations of the meaning of the terms pelled. In formulating (1) we have ap equivalence states merely the conditions "true" and "false" when these terms are plied a common method of forming a under which the sentence (4) is true or referred to the sentence "snow is white". name of a sentence, or of any other ex is not true, but by itself the equivalence We can regard (1) and (1') as partial pression, which consists in putting the does not enable us to decide which is definitions of the terms "true" and expression in quotes. The method has actually the case. To verify the judgment "false", in fact, as definitions of these many virtues, but it is also the source of expressed in (4) one would have to read terms with respect to a particular sen- the difficulties discussed above. To re- attentively the book reviewed and ana-

© 1969 SCIENTIFIC AMERICAN, INC lyze the truth of the sentences contained "and" between any two consecutive par obtained in some other way, pOSSibly by in it. tial definitions. The only thing that re using a different idea. There is, however, In the light of the preceding discus mains to be done is to give the result a more serious and fundamental reason sion we can now reformulate our main ing conjunction a different, but logically that seems to preclude this possibility. problem. We stipulate that the use of the equivalent, form, so as to satisfy formal More than that, the mere supposition term "true" in its reference to sentences requirements imposed on definitions by that an adequate use of the term "true" in English then and only then conforms rules of logic: (in its reference to arbitrary sentences in with the classical conception of truth English) has been secured by any meth if it enables us to ascertain every equiv (5) For every sentence x (in the lan od whatsoever appears to lead to a con alence of the form (3) in which "p" is guage L), x is true if and only if tradiction. The simplest argument that replaced on both sides by an arbitrary either provides such a contradiction is known and is identical to English sentence. If this condition is sat S1' x "S1", as the antinomy of the liar; it will be or isfied, we shall say simply that the use of carried through in the next few lines. and is identical to the term "true" is adequate. Thus our S�, x " s:,!" , Consider the following sentence: main problem is: can we establish an ( 6 ) The sentence printed in red on adequate use of the term "true" for sen or finally, page 65 of the June 1969 issue of tences in English and, if so, then by and is identical to s1.000' x Scientific American is false. what methods? We can, of course, raise "81,00 0". " " an analogous question for sentences in Let us agree to use s as an abbrevia any other language. vVe have thus arrived at a statement tion for this sentence. Loo king at the The problem will be solved complete which can indeed be accepted as the de date of this magazine, and the number " " ly if we manage to construct a general sired general definition of tru th: it is of this page, we easily check that s is definition of truth that will be adequate formally correct and is adequate in the just the only sentence printed in red on in the sense tha t it will carry with it sense that it implies all the equivalences page 65 of the June 1969 issue of Scien " " as logical consequences all the equ iva of the form (3) in which p has been re tific Ame1'ican.Hence it follows, in par lences of form (3). If such a definition is placed by any sentence of the language ticular, that accepted by English-speaking people, it L. We notice in passing that (5) is a sen will obviously establish an adequate use tence in English but obviously not in the (7) "s" is false if and only if the sen tence printed in red on page 65 of of the term "true". language L; since (5) contains all sen the June 1969 issue of Scientific Under certain special assumptions the tences in L as proper parts, it cannot co American is false. construction of a general definition of incide with any of them. Further discus " " truth is easy. Assume, in fact, that we sion will throw more light on this point. On the other hand, s is undoubtedly are interested, not in the whole common For obvious reasons the procedure a sentence in English. Therefore, as English language, but only in a frag just outlined cannot be followed if we suming that our use of the term "true" is ment of it, and that we wish to define are interested in the whole of the English adequate, we can assert the equivalence " " " " the term "true" exclusively in reference language and not merely in a fragment (3) in which p is replaced by s . Thus to sentences of the fragmentary lan of it. When trying to prepare a complete we can state: guage; we shall refer to this fragmentary list of English sentences, we meet from language as the language L. As sume the start the difficulty that the rules of (8) "s" is true if and only if s. further that L is pro vided with precise English grammar do not determine pre " " syntactical rules which enable us, in each cisely the form of expressions (strings of vVe now recall that s stands for the particular case, to distinguish a sentence words) which should be regarded as sen whole sentence (6). Hence we can re " " from an expression which is not a sen tences: a particular expression, sayan place s by (6) on the rig ht side of (8); tence, and that the number of all sen exclamation, may function as a sentence we then obtain tences in the language L is finite (though in some given context, whereas an ex pOSSibly very large). Assume, finally, that pression of the same form will not func (9) "s" is true if and only if the sen ' tence printed in red on page 65 the word "true" does not occur in L and tion so in some other context. Further of the June 1969 issue of Scientific that the meaning of all words in L is more, the set of all sentences in English American is false. sufficiently clear, so that we have no ob is, potentially at least, infinite. Although jection to using them in definingtruth. it is certainly true that only a finite num By now comparing (8) and (9), we Under these assumptions proceed as fol ber of sentences have been formulated conclude: lows. First, prepare a complete list of all in speech and writing by human beings sentences in L; suppose, for example, up to the present moment, probably no ( 10) "s" is false if and only if " s" is that there are exactly 1,000 sentences in body would agree that the list of all these true. " L, and agree to use the symbols st, sentences comprehends all sentences in "S2", ... , "S1.000" as abbreviations for English. On the contrary, it seems likely This leads to an obvious contradicti on: " " consecutive sentences on the list . Next, that on seeing such a list each of us could s proves to be both true and fals e. " for each of the sentences st", "S2", "', easily produce an English sentence Thus we are confronted with an antino "S1.000" con struct a partial defini tion of which is not on the list. Finally, the fact my. The above formulation of the an truth by substituting successively these that the word "true" does occur in En tinomy of the liar is due to the Polish " " sentences for p on both sides of the glish prevents by itself an application of logician Jan Lukasiewicz. schema (3). Finally, form the logical con the procedure previously described. Some more involved formulations of junction of all these partial definitions; From these remarks it does not follow this antinomy are also known. Imagine, in other words, combine them in one that the desired definition of truth for for instance, a book of 100 pages, with statement by putting the co nnective arbitrary sentences in English cannot be just one sentence printed on each page.

© 1969 SCIENTIFIC AMERICAN, INC On page 1 we read: origin. It is usually ascribed to the Greek formulated so as to apply to ot her nat logician Eubulides; it tormented many ural languages. We are confronted with The sentence printed on page 2 ancient logicians and caused the pre a serious problem: how can we avoid of this book is true. mature death of at least one of them, the contradictions induced by this an Philetas of Cos. A number of other an tinomy? A radical solution of the prob On page 2 we read: tinomies and paradoxes were found in lem which may readily occur to us would The sentence printed on page 3 antiquity, in the Middle Ages, and in be simply to remove the word "true" of this book is true. modern times. Although many of them from the English vocabulary or at least are now entirely forgotten, the antinomy to abstain from using it in any serious And so it goes on up to page 99. How of the liar is still analyzed and discussed discussion. ever, on page 100, the last page of the in contemporary writings. Together with Those people to whom such an ampu book, we find: some recent antinomies discovered tation of English seems highly unsatis around the turn of the century (in par factory and illegitimate may be inclined The sentence printed on page 1 ticular, the antinomy of Russell), it has of this book is false. to accept a somewhat more compromis had a great impact on the development ing solution, which consists in adopting Assume that the sentence printed on of modern logic. what could be called (following the con page 1 is indeed false. By means of an Two diametrically opposed approach temporary Polish philosopher Tadeusz argument which is not difficult but is es to antinomies can be found in the lit Kotarbi{lski) "the nihilistic approach to very long and requires leafing through erature of the subject. One approach is the theory of truth". According to this the entire book, we conclude that our to disregard them, to treat them as approach, the word "true" has no inde assumption is wrong. Consequently we sophistries, as jokes that are not serious pendent meaning but can be used as a assume now that the sentence printed on but malicious, and that aim mainly at component of the two meaningful ex page 1 is true-and, by an argument showing the cleverness of the man who pressions "it is true that" and "it is not which is as easy and as long as the orig formulates them. The opposite approach true that". These expressions are thus inal one, we convince ourselves that the is characteristic of certain thinkers of the treated as if they were single words with new assumption is wrong as well. Thus 19th century and is still represented, or no organic parts. The meaning ascribed we are again confronted with an an was so a short while ago, in certain parts to them is such that they can be immedi tinomy. of our globe. According to this approach ately eliminated from any sentence in It turns out to be an easy matter to antinomies cons titute a very essential which they occur. For instance, instead compose many other "antinomial books" element of human tho ught; they must of saying that are variants of the one just de appear again and again in intellectual scribed. Each of them has 100 pages. activities, and their presence is the basic it is true that all cats are black Every page contains just one sentence, source of real progress. As often hap and in fact a sentence of the form: pens, the truth is probably som ewhere we can simply say in between. Personally, as a logician, I The sentence printed on page 00 all cats are black, could not reconcile myself with antino of this book is XX. mies as a permanent element of our sys and instead of In each particular case "XX" is replaced tem of knowledge. However, I am not by one of the words "true"or "false", the least inclined to treat ant inomies it is not true that all cats are black while "00" is replaced by one of the nu lightly. The appearance of an antinomy merals "1", "2", ... , "100"; the same nu is for me a symptom of disease. Starting we can say meral may occur on many pages. Not with premises that seem intui tively ob every variant of the original book com vious, using forms of reasoning that not all cats are black. posed according to these rules actually seem intuitively certain, an antinomy yields an antinomy. The reader who is leads us to nonsense, a contradiction. In other contexts the word "true" is fond of logical puzzles will hardly find Whenever this happens, we have to sub meaningless. In particular, it cannot be it difficult to describe all those variants mit our ways of thinking to a thorough used as a real predicate qualifying names that do the job. The following warning revision, to reject some premises in of sentences. Employing the terminology may prove useful in this connection. Im which we believed or to improve some of medieval logic, we can say that the agine that somewhere in the book, say forms of argument which we used. We word "true" can be used syncategore on page 1, it is said that the sentence on do this with the hope not only that the matically in some special situations, but page 3 is true, while somewhere else, old antinomy will be disposed of but also it cannot ever be used categorematically. say on page 2, it is claimed that the same that no new one will appear. To this end To realize the implications of this ap sentence is false. From this information we test our reformed system of thinking proach, consider the sentence which was it does not follow at all that our book is by all available means, and, firs t of all, the starting point for the antinomy of "antinomial"; we can only draw the con we try to reconstruct the old antinomy in the liar; that is, the sentence printed in clusion that either the sentence on page the new setting; this testing is a very red on page 65 in this magazine. From 1 or the sentence on page 2 must be important activity in the realm of specu the "nihilistic" point of view it is not a false. An antinomy does arise, however, lative thought, akin to carrying out cru meaningful sentence, and the an tinomy whenever we are able to show that one cial experiments in empirical science. simply vanishes. Unfortunately, many of the sentences in the book is both true From this point of view consider now uses of the word "true", which otherwise and false, indep endent of any assump specifically the antinomy of the liar. The seem quite legitimate and reasonable, tions concerning the truth or falsity of antinomy involves the notion of truth are similarly affected by this approach. the remaining sentences. in reference to arbitrary sentences of Imagine, for instance, that a certain The antinomy of the liar is of very old common English; it could eas ily be re- term occurring repeatedly in the works

© 1969 SCIENTIFIC AMERICAN, INC of an anci ent mathematician admits of several interpretations. A historian of sci ence who studies the works arrives at the conclusion that under one of these interpretations all the theorems stated by the mathematician prove to be true; this leads him naturally to the conjecture that the same will apply to any work of this mathematician that is not known at pres ent but may be discovered in the future. If, however, the historian of science shares the "nihilistic" approach to the notion of truth, he lacks the possi bility of expressing his conjecture in words. One could say that truth-theoretical "ni hilism" pays lip service to some popular forms of human speech, while actually removing the notion of truth from the conceptual stock of the human mind. We shall look, therefore, for another way out of our predicament. We shall try to find a solution that will keep the classical concept of truth essentially in tact. The applicability of the notion of truth will have to undergo some restric tions, but the notion will remain avail able at least for the purpose of scholarly discourse. To this end we have to analyze those features of the common language that are the real source of the antinomy of the liar. When carrying through this analysis, we notice at once an outstand ing feature of this language-its all-com prehensive, universal character. The common language is universal and is in QUESTAR'S SEVEN-INCH IS VERY 81G WITH R&D tended to be so. It is su pposed to pro vide adequate facilities for expressing -yet its scant 20-pound weight is so everything that can be expressed at all, easily portable in this 27" aluminum case that in any language whatsoever; it is con tinually expanding to satisfy this re you can take it with you quirement. In particular, it is semanti Questar's commitment to quality, which cally universal in the following sense. has built its world-wide reputation, now Together with the linguistic objects, such gives you the Seven-inch, an instrument as sentences and terms, which are com with the finest possible resolution for ponents of this language, names of these every optical need. Those who use it for objects are also included in the language laser sending or receiving, for rocket borne instrumentation, for closed-circuit (as we know, names of expressions can television, or just for taking pictures of be obtained by putting the expressions nature, marvel at the performance which in quotes); in addition, the language easily doubles that of its 31j2-inch parent. contains semantic terms such as "truth", Its Cer-Vit mirror has essentially no ther "name", "designation", which directly or mal expansion, thereby increasing the instrument's usefulness for all the special indirectly refer to the relationship be optical problems encountered in scientific tween linguistic objects and what is research and engineering. And it's on the expressed by them. Consequently, for shelf, of course, ready for delivery. every sentence formulated in the com Here, at last, is the perfect telescope for manned spacecraft. mon language, we can form in the same language another sentence to the effect QUESTAR, THE WORLD'S FINEST, MOST VERSATILE SMALL TELESCOPE, PRICED FROM $795, IS that the first sentence is true or that it is DESCRIBED IN OUR NEWEST BOOKLET WHICH CONTAINS MORE THAN 100 PHOTOGRAPHS BY false. Using an additional "trick" we can QUESTAR OWNERS. SEND $1 FOR MAILING ANYWHERE IN NORTH AMERICA. BY AIR TO REST OF WESTERN HEMISPHERE, EUROPE AND NORTH AFRICA, ELSEWHERE, even construct in the language what is $2.50; $3.00; $3.50. sometimes called a self-referential sen tence, that is, a sentence S which asserts the fact that S itself is true or that it is QUIESrA� false. In case S asserts its own falsity we BOX 120, NEW HOPE, PENN, 18938

© 1969 SCIENTIFIC AMERICAN, INC can show by means of a simple argument syntactical rules should be purely formal, as the metalanguage and the former as that S is both true and false-and we are that is, they should refer exclusively to the object-language. The metalanguage confronted again with the antinomy of the form (the shape) of expressions; the must be sufficiently rich; in particular, it the liar. function and the meaning of an expres must include the object-language as a There is, however, no need to use uni sion should depend exclUSively on its part. In fact, according to our stipula versal languages in all possible situa form. In particular, looking at an expres tions, an adequate definition of truth tions. In particular, su ch languages are sion, one should be able in each case to will imply as consequences all partial in general not needed for the purposes decide whether or not the expression is definitions of this notion, that is, all of scienc e (and by science I mean here a sen tence. It should never happen that equivalences of form (3): the whole realm of intellectual inquiry). an expression functions as a sentence at " " In a particular branch of science, say in one place while an expression of the p is true if and only if p, chemistry, one discusses certain special same form does not function so at some objects, such as elements, molecules, and other place, or that a sentence can be as where "p" is to be replaced (on both so on, but not for instance linguistic ob serted in one context while a sentence sides of the equivalence) by an arbitrary jects such as sentences or terms. The of the same form can be denied in an sentence of the obje ct-language. Since language that is well adapted to this dis other. (Hence it foll ows, in particular, all these consequences are formulated cussion is a restricted language with that demonstrative pronouns and ad in the metalanguage, we conclude that a limited vocabulary; it must contain verbs such as "this" and "here" should every sentence of the object-language names of chemical obj ects, terms such not occur in the vocabulary of the lan must also be a sentence of the metalan as "element" and "molecule", but not guage.) Languages that satisfy these guage. Furthermore, the metalanguage names of linguistic objects; hence it does conditions are referred to as formalized must contain names for sentences (and not have to be semantically universal. languages. When discussing a formal other expressions) of the object-lan The same applies to most of the other ized language there is no need to distin guage, since these names occur on the branches of science. The situation be guish between expressions of the same left sides of the above equivalences. It comes somewhat confused when we turn form which hav� been written or uttered must also contain some further terms to linguistics. This is a science in which in different places; one often speaks of that are needed for the discussion of the we study languages; thus the languag e them as if they were one and the same object-language, in fact terms denoting of linguistics must certainly be provided expression. The reader may have noticed certain special sets of expressions, rela with names of linguistic objects. How we sometimes use this way of speaking tions between expressions, and opera ever, we do not have to identify the lan even when discussing a natural lan tions on expressions; for instance, we guage of linguistics with the universal guage, that is, one which is not formal must be able to speak of the set of all language or any of the languages that are ized; we do so for the sake of simplicity, sentences or of the operation of juxta objects of linguistic discussion, and we and only in those cases in which there position, by means of which, putting one are not bound to assume that we use in seems to be no danger of confusion. of two given expressions immediately linguistics one and the same language Formalized languages are fully ade after the other, we form a new expres for all discussions. The language of lin quate for the presentation of logical and sion. Finally, by defining truth, we show guistics has to contain the names of lin mathematical theories; I see no essential that semantic terms (expressing relations guistic components of the languages reasons why they cannot be adapted for between sentences of the object-lan discussed but not the names of its own use in other scientific diSciplines and in guage and objects referred to by these components; thus, again, it does not have particular to the development of theo sentences) can be introduced in the met to be semantically universal. The same retical parts of empirical sciences. I alanguage by means of definitions. applies to the language of logic, or rath should like to emphaSize that, when Hence we conclude that the metalan er of that part of logic known as meta using the term "formalized languages", . guage which provides sufficient means logic and metamathematics; here we I do not refer exclUSively to linguistic for defining truth must be essentially again concernourselves with certain systems that are formulated entirely in richer than the obj ect-language; it can languages, primarily with languages of symbols, and I do not have in mind any not coincide with or be translatable into logical and mathematical th eories (al thing essentially opposed to natural lan the latter, since otherwise both lan though we discuss these languages from guages. On the contrary, the only formal guages would turn out to be semantical a different point of view than in the case ized languages that seem to be of real ly universal, and the antinomy of the liar of linguistics). interest are those which are fragments could be reconstructed in both of them. The question now arises whether the of natural languages (fragments provid \lVe shall return to this question in the notion of truth can be precisely defined, ed with complete vocabularies and pre last section of this article. and thus a consistent and adequ ate cise syntactical rules) or those which can If all the above conditions are satis usage of this notion can be established at least be adequately translated. into fied, the construction of the desired def at least for the semantically restricted natural languages. inition of truth presents no essential languages of scientific discourse. Under There are some further conditions on difficulties. Technically, however, it is certain conditions the answer to this which the realization of our program too involved to be explained here in de question proves to be affirmative. The depends. We should ma ke a strict dis tail. For any given sentence of the ob main conditions imposed on the lan tinction between the lan guage which is ject-language one can easily formulate guage are that its full vocabulary should the object of our discussion and for the corresponding partial definition of be available and its syntactical rules which in particular we intend to con form (3). Since, however, the set of all concerning the formation of sentences struct the definition of truth, and the sentences in the object-language is as a and other mea ningful expressions from language in which the definitionis to rule infinite, whereas every sentence of words listed in the vocabulary should be be formulated and its implicatio ns are the metalanguage is a finite string of precisely formulated. Furthermore, the to be studied. The latter is referred to signs, we cannot arrive at a general defi-

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© 1969 SCIENTIFIC AMERICAN, INC decide whether or not any such sentence a certain stock of terms, and were accept pline. Two analogous principles concern is true is a task of the science itself, and ed as true. This aggregate of sentences the use of terms in constructing the dis not of logic or the theory of truth. lacked any structural order. A sentence cipline. By the firstof them we list at Some philosophers and methodolo was accepted as true either because it the beginning a few terms, called un gists of science are inclined ·to reject seemed intuitively evident, or else be defined or primitive terms, which ap every definition that does not provide a cause it was proved on the basis of some pear to be directly understandable and criterion for deciding whether any given intuitively evident sentences, and thus which we decide to use (in formulating particular object falls under the notion was shown, by means of an intuitively and proving theorems) without explain defined or not. In the methodology of certain argument, to be a consequence ing their meanings; by the second prin empirical sciences such a tendency is of these other sentences. The criterion of ciple we agree not to use any further represented by the doctrine of opera intuitive evidence (and intuitive certain term unless we are able to explain its tionalism; philosophers of mathematics ty of argum ents) was applied without meaning by definingit with the help of who belong to the constructivist school any restrictions; every sentence recog undefined terms and terms previously seem to exhibit a similar tendency. In nized as true by means of this criterion defined. These four principles are cor both cases, however, the people who \-vas automatically included in the dis nerstones of the axiomatic method; theo hold this opinion appear to be in a small cipline. This description seems to fit, ries developed in accordance with these minority. A consistent attempt to carry for instance, the science of geometry as principles are called axiomatic theories. out the program in pract ice (that is, to it was known to ancient Egyptians As is well known, the axiomatic meth develop a science without using undesir and Greeks in its early, pre-Euclidean od was applied to the development of able definitions) has hardly ever been stage. geometry in the Elements of Euclid made. It seems clear that under this pro It was 'realized rather soon, however, about 300 B.C. Thereafter it was used for gram much of contemporary mathemat that the criterion of intuitive evidence over 2,000 years with practically no ics would disappear, and theoretical is far from being infallible, has no ob change in its main principles (which, by parts of physics, chemistry, biology, and jective character, and often leads to seri the way, were not even explicitly forn1U other empirical sc iences would be se ous errors. The entire subseq uent de lated for a long period of time) nor in the verely mutilated. The definitions of such velopment of the axiomatic method can general approach to the subject. How notions as atom or gene as well as most be viewed as an expression of the tend ever, in the 19th and 20th centuries the definitions in mathematics do not carry ency to restrict the recourse to intuitive concept of the axiomatic method did with them any criteria for deciding evidence. undergo a profound evolution . Those whether or not an object falls under the This tendency first revealed itself in features of the evolution which concern term that has been defined. the effort to prove as many sentences the notion of proof are particularly sig Since the definition of truth does not as possible, and hence to restrict as much nificant for our discussion. provide us with any such criterion and as possible the number of sentences ac Until the last years of the 19th cen at the same time the search for truth is cepted as true merely on the basis of in tury the notion of proof was primarily rightly considered the essence of scien tuitive evidence. The ideal from this of a psychological character. A proof tific activities, it appears as an important point of view would be to prove every was an intellectual activity that aimed problem to find at least partial criteria of sentence that is to be accepted as true. at convincing oneself and others of the truth and to develop procedures that For obvious reasons this ideal cannot be truth of a sentence discussed; more spe may enable us to ascertain or negate the realized. Indeed, we prove each sentence cifically, in developing a mathematical truth (or at least the likelihood of truth) on the basis of other sentences, we prove theory proofs were used to convince our of as many sentences as possible. Such these other sentences on the basis of selves and others that a sent ence dis procedures are known indeed; some of some further sentences, and so on: if we cussed had to be accepted as true once them are used exclusively in empirical are to avoid both a vicious circle and an some other sentences had been previous science and some primarily in deductive infinite regress, the procedure must be ly accepted as such. No restrictions were science. The notion of proof-the second discontinued somewhere. As a compro put on arguments used in proofs, except notion to be discussed in this paper-re mise between that unattainable ideal that they had to be intuitively convinc fers just to a procedure of ascertaining and the realizable possibilities, two prin ing. At a certain period, however, a need the truth of sentences which is employed ciples emerged and were su bsequently began to be felt for submitting the no primarily in deductive science. This pro applied in constructing mathematical tion of proof to a deeper analysis that cedure is an essential element of what is diSciplines. By the first of these princi would result in restricting the recourse known as the axiomatic method, the only ples every diScipline begins with a list of to intuitive evidence in this context as method now used to develop mathemati a small number of sentences, called ax well. This was probably related to some cal diSciplines. ioms or primitive sentences, which seem specific developments in mathematics, The axiomatic method and the notion to be intuitively evident and which are in particular to the discovery of non of proof within its framework are prod recognized as true without any further Euclidean geometries. The analYSis was ucts of a long historical development. justification. According to the second carried out by logicians, beginning with Some rough knowledge of this develop principle, no other sentence is accepted the German logician Gottlob Frege; it ment is probably essential for the under in the diScipline as true unless we are led to the introduction of a new notion, standing of the contemporary notion of able to prove it with the exclusive help that of a formal proof, which turned out proof. of axioms and those sentences that were to be an adequate substitute and an es Originally a mathematical discipline previously proved. All the sentences that sential improvement over the old psy was an aggregate of sentences that con can be recognized as true by virtue of chological notion. cerned a certain class of objects or phe these two principles are called theorems, The first step toward supplying a nomena, were formulated by means of or provable sentences, of the given disci- mathematical theory with the notion of

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© 1969 SCIENTIFIC AMERICAN, INC a formal proof is the formalization of the any finite sequence of sentences with the quate only if all sentences acquired with language of the theory, in the sense dis three properties just listed. its help prove to be true and all true sen cussed previously in connection with the An axiomatic theory whose language tences can be acquired with its help. definition of truth. Thus formal syntac has been formalized and for which the Hence the problem naturally arises: is tical rules are provided which in particu notion of a formal proof has been sup the formal proof actually an adequate lar enable us simply by looking at shapes plied is called a formalized theory. We procedure for acquiring truth? In other of expressions, to distinguish a sentence stipulate that the only proofs which can words: does the set of all (formally) prov from an expression that is not a sentence. be used in a formalized theory are formal able sentences coincide with the set of The next step consists in formulating a proofs; no sentence can be accepted as all true sentences? few rules of a different nature, the so a theorem unless it appears on the list To be specific,we refer this problem called rules of proof (or of inference). of axioms or a formal proof can be found to a particular, very elementary mathe By these rules a sentence is regarded as for it. The method of presenting a for matical discipline, namely to the arith directly derivable from given sentences malized theory at each st age of its metic of natural numbers (the ele if, generally speaking, its shape is related development is in principle very ele mentary number theory). We assume in a prescribed manner to the shapes of mentary. vVe list firstthe axioms and that this discipline has been presented given sentences. The number of rules of then all the known theorems in such an as a formalized theory. The vocabulary proof is small, and their content is order that every sentence on the list of the theory is meager. It consists, in " " " " simple. Just like the syntactical rules, which is not an axiom can be directly fact, of variables such as m , "n", p , they all have a formal character, that recognized as a theorem, simply by com ... representing arbitrary natural num is, they refer exclusively to shapes of paring its shape with the shapes of sen bers; of numerals "0", "1", "2", ... de sentences involved. Intuitively all the tences that precede it on the list; no com noting particular numbers; of symbols rules of derivation appear to be in plex processes of reasoning and convinc denoting some familiar relations be fallible, in the sense that a sentence ing are involved. (I am not speaking here tween numbers and operations on num " which is directly derivable from true sen of psychological processes by means of bers sllch as "=", "<", "+", "- ; tences by any of these rules must be true which the theorems have actually been and, finally, of certain logical terms, itself. Actually the infallibility of the discovered.) The recourse to intuitive namely sentential connectives ("and", rules of proof can be established on the evidence has been indeed conSiderably "or", "if", "not") and quantifiers (expres " basis of an adequate definition of truth. restricted; doubts concerning the truth sions of the form "for every number m " The best-known and most important ex of theorems have not been entirely and "for some number m ). The syn ample of a rule of proof is the rule of de eliminated but have been reduced to tactical rules and the rules of proof are tachment known also as modus ]Jonens. possible doubts concerning the truth of simple. When speaking of sentences in By this rule (which in some theories the few sentences listed as axioms and the subsequent discussion, we always serves as the only rule of proof) a sen the infallibility of the few simple rules have in mind sentences of the formalized tence "q" is directly derivable from two of proof. It may be added that the proc language of arithmetic. given sentences if one of them is the con ess of introducing new terms in the lan We know from the discussion of truth " ditional sentence "if ]J, then q while the guage of a theory can also be formalized in the first section that, taking this lan " " " " other is ]1 ; here ]1 and "q" are, as by supplying special formal rules of guage as the object-language, we can usual, abbreviations of any two sentences definitions. construct an appropriate metalanguage of our formalized language. vV e can now It is now known that all the existing and formulate in it an adequate defini explain in what a formal proof of a given mathematical disciplines can be pre tion of truth. It proves convenient in this sentence consists. First, we apply the sented as formalized theories. Formal context to say that what we have thus rules of proof to axioms and obtain new proofs can be provided for the deep defined is the set of true sentences; in sentences that are directly derivable est and most complicated mathematical fact, the definition of truth states that a from axioms; next, we apply the same theorems, which were originally estab certain condition formulated in the meta rules to new sentences, or jointly to new lished by intuitive arguments. language is satisfied by all elements of sentences and axioms, and obtain further this set (that is, all true sentences) and sentences; and we continue this process. The Relationship of Truth and Proof only by these elements. Even more read If after a finite number of steps we arrive ily we can define in the metalanguage at a given sentence, we say that the sen It was undoubtedly a great achieve the set of provable sentences; the defini tence has been formally proved. This can ment of modern logiC to have replaced tion conforms entirely with the explana also be expressed more precisely in the the old psychological notion of proof, tion of the notion of formal proof that following way: a formal proof of a given which could hardly ever be made clear was given in the second section. Strictly sentence consists in constructing a finite and precise, by a new simple notion of a speaking, the definitions of both truth sequence of sentences such that (1) the purely formal character. But the triumph and provability belong to a new theory first sentence in the sequence is an ax of the formal method carried with it the formulated in the metalanguage and iom, (2) each of the following sentences germ of a futur e setback. As we shall specifically designed for the study of our either is an axiom or is directly derivable see, the very simplicity of the new notion formalized arithmetic and its language. from some of the sentences that precede turned out to be its Achilles heel. The new theory is called the metatheory it in the sequence, by virtue of one of the To assess the notion of formal proof or, more specifically, the meta-arithme rules of proof, and (3) the last sentence we have to clarify its relation to the no tic. vVe shall not elaborate here on the in the sequence is the sentence to be tion of truth. After all, the formal proof, way in which the metatheory is con proved. Changing somewhat the use of just like the old intuitive proof, is a pro structed-on its axioms, undefined terms, the term "proof", we can even say that cedure aimed at acquiring new true sen and so on. We only point out that it is a formal proof of a sentence is simply tences. Such a procedure will be ade- within the framework of this metatheory

© 1969 SCIENTIFIC AMERICAN, INC that we formulate and solve the problem of these notions is explained essentially arithmetical theory, and that the final of whether the set of provable sentences in terms of certain simple relations outcome of the discussion could be dif coincides with that of true sentences. among sentences prescribed by a few ferent if we appropriately enriched the The solution of the problem proves to rules of proof; the reader may recall theory by adjoining new axioms or new be negative. We shall give here a very here the rule of modus ponens. The rules of inference. A closer analysis rough account of the method by which corresponding relations among numbers shows, however, that the argument de the solution has been reached .. The main of sentences are equally simple; it turns pends very little on specific properties idea is closely related to the one used by out that they can be characterized in of the theory discussed, and that it the contemporary American logician (of terms of the simplest arithmetical opera actually extends to most other formalized Austrian origin) Kurt Godel in his fa tions and relations, such as addition, theories. Assuming that a theory includes mous paper on the incompleteness of multiplication, and equality-thus in the arithmetic of natural numbers as a arithmetic. terms occurring in our arithmetical theo part (or that, at least, arithmetic can be It was pOinted out in the first section ry. As a consequence the set of provable" reconstructed in it), we can repeat the that the metalanguage which enables us numbers can also be characterized in essential portion of our argument in a to define and discuss the notion of truth such terms. One can describe briefly practically unchanged form; we thus must be rich. It contains the entire "vhat has been achieved by saying that conclude again that the set of provable object-language as a part, and therefore the definition of provability has been sentences of the theory is different from we can speak in it of natural numbers, translated from the metalanguage into the set of its true sentences. If, more sets of numbers, relations among num the object-language. over, we can show (as is frequently the bers, and so for th. But it also contains On the other hand, the discussion of case) that all the axioms of the theory terms needed for the discussion of the the notion of truth in common languages are true and all the rules of inference are object-language and its components ; strongly suggests the conjecture that no infallible, we further conclude that there consequently we can speak in the meta such translation can be obtained for the are true sentences of the theory which language of expressions and in particular definition of truth; otherwise the object are not provable. Apart from some frag of sentences, of sets of sentences, of language would prove to be in a sense mentary theories with restricted means relations among sentences, and so forth. semantically universal, and a reappear of expression, the assumption concerning Hence in the metatheory we can study ance of the antinomy of the liar would be the relation of the theory to the arith properties of these various kinds of imminent. We confirm this conjecture by metic of natural numbers is generally objects and establish connections be showing that, if the set of true" numbers satisfied, and hence our conclusions have tween them. could be definedin the language of a nearly universal character. (Regarding In particular, using the description of arithmetic, the antinomy of the liar could those fragmentary theories which do not sentences provided by the syntactical actually be reconstructed in this lan include the arithmetic of natural num rules of the object-language, it is easy to guage. Since, however, we are dealing bers, their languages may not be provid arrange all sentences (from the simplest now with a restricted formalized lan ed with sufficient means for defining the ones through the more and more com guage, the antinomy would assume a notion of provability, and their provable plex) in an infinite sequence and to more involved and sophisticated form. sentences may in fact coincide with their number them consecutively. \Ve thus In particular, no expressions with an true sentences. Elementary geometry correlate with every sentence a natural empirical content such as "the sentence and elementary algebra of real numbers number in such a way that two numbers printed in such-and-such place", which are the best known, and perhaps most correlated with two different sentences played an essential part in the original important, examples of theories in which are always different; in other words, we formulation of the antinomy, would these notions coincide.) establish a one-to-one correspondence occur in the new formulation. 'Ve shall The dominant part played in the between sentences and numbers. This in not go into any further details here. whole argument by the antinomy of the turn leads to a similar correspondence Thus the set of provable" numbers liar throws some interesting light on our between sets of sentences and sets of does not coincide with the. set of true" earlier remarks concerning the role of numbers, or relations among sentences numbers, since the former is definable antinomies in the history of human and relations among numbers. In par in the language of arithmetic while the thought. The antinomy of the liar first ticular, we can consider numbers of latter is not. Consequently the sets of appeared in our discussion as a kind of provable sentences and numbers of true provable sentences and true sen tences evil force with a great destructive power. sentences; we call them briefly provable" do not coincide either. On the other It compelled us to abandon all attempts numbers and true" numbers. Our main hand, using the definition of truth we at clarifying the notion of truth for problem is reduced then to the ques easily show that all the axioms of arith natural languages. We had to restrict tion: are the set of provable" numbers metic are true and all the rules of proof our endeavors to formalized languages and the set of true" numbers identical? are infallible. Hence all the provable of scientific discourse. As a safeguard To answer this question negatively, sentences are true; therefore the con against a possible reappearance of the it suffices, of course, to indicate a single verse cannot hold. Thus our final con antinomy, we had to complicate con property that applies to one set but not clusion is: there are sentences formubt siderably the discussion by distinguish to the other. The property we shall ed in the language of arithmetic that are ing between a language and its meta actually exhibit may seem rather un true but cannot be proved on the basis language. Subsequently, however, in the expected, a kind of deus ex machina. of the axioms and rules of proof accept new, restricted setting, we have man The intrinsic simplicity of the notions ed in arithmetic. aged to tame the destructive energy and of formal proof and formal provabilit y One might think that the conclusion harness it to peaceful, constructive will play a basic role here. We have seen essentially depends on specificaxioms purposes. The antinomy has not reap in the second section that the meaning and rules of inference, chosen for our peared, but its basic idea has been used

© 1969 SCIENTIFIC AMERICAN, INC to establish a significant metalogical re sult with far-reaching implications. Nothing is detracted from the si gnifi Component cance of this result by the fact that its philosophical implications are essentially Propulsion negative in character. The result shows indeed that in no domain of mathematics Engineer: is the notion of provability a perfect substitute for the no tion of truth. The belief that formal proof can serve as an adequate instrument for establishing truth of all mathematical statements has proved to be unfounded . The original �\ IL triumph of formal methods has been ...... followed by a serious setback. Whatever can be said to conclude this /,\ discussion is bound to be an anticlim ax. The notion of truth for formalized theo ries can now be introduced by means of Yo u'll get a bo ng a precise and adequate definition. It can therefore be used without any restric out of working at JPL. tions and reservations in metalogical dis We're looking for a rare man -who will be responsible for the design and cussion. It has actually become a basic development of liquid propulsion elements and controls for planetary explora metalogical notion involved in important tion spacecraft applications. If you have a degree in Mechanical Engineer problems and results. On the other hand, ing and five years' applicable experience, send your resume in confidence the notion of proof has not lost its sig to Mr. Wallace Peterson, JPL's supervisor of Employment. nificance either. Proof is still the only method used to ascertain the truth of sentences within any specific mathe • ��I !kRG��,�a�d��i��!�l�O� matical theory. We are now aware of the Attention: Professional Staffing Department � 6 fact, however, that there are sentences "An equal opportunity employer." Jet Propulsion laboratory is operated by the Ca lif. formulated in the language of the theory ornie Institute of Technol ogy for the National Aeronautics and Space Administration. which are true but not provable, and we cannot discount the possibility that some The Minolta SR -T 101 35mmsingle lens reflex compose a picture, set aperture and shutter such sentences occur among those in isn't a regular camera kind of camera. It's a speed without taking your eye from the view- which we are in terested and which we system with a full range of accessories finder. The SR-T 101 with a standard and interchangeable lenses, from wide Rokkor f/1 .7 lens and a through the attEmpt to prove. Hence in some situa angle to telephoto. (The telephoto lens lens meter is under $245 plus case. The tions we may wish to explore the pos shown is optional at under $200.) The SR-T 101: it can make the dif ference camera part of the system le ts you between pho tography and photography. sibility of widening the set of provable  sentences. To this end we enrich the given theory by including new sentences in its axiom syst em or by providing it with new rules of proof. In doing so we use the notion of truth as a guide ; for we do not wish to add a new axiom or a new rule of proof if we have reason to believe that the new axiom is not a true sentence, or that the new rule of proof The difference when applied to true sentences may between picture taking yield a false sentence. The process of and photography. extending a theory may of course be repeated arbitrarily many ti mes. The notion of a true sentence functions thus as an ideal limit which can never be reached but whic.:h we try to approximate by gradually widening the set of prov able sentences. (It seems likely, although for different reasons, that the notion of truth plays an analogous role in the realm of empirical kn owledge.) There is no conRict between the notions of truth and proof in the development of mathematics; the two notions are not at war but live in peaceful coexistence. South, N.Y., N.Y. 1 0003.

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