JAAKKO HINTIKKA

A REVOLUTION IN THE FOUNDATIONS OF MATHEMATICS?

1. FIRST-ORDER LOGIC AND ITS SHORTCOMINGS

The standard contemporary view of the foundations of mathematics andof the role of logic in it is well known and firmly entrenched. The groundfloor of the edifice of mathematics is on this view our basic logic, thatis to say first-order logic, a.k.a. quantification theory or lower predicatecalculus. It has various desirable features, such as completeness, com-pactness, Lowenheim–Skolem property and the validity of the separationtheorem. Indeed, it is the strongest possible logic in the sense of abstract(model-theoretical) logic that has the pleasant properties of compactnessand Lowenheim–Skolem property, assuming only the usual behavior ofpropositional connectives plus a few plausible structural properties. Thisis what the famous theorem of Lindstrom’s shows.1 This theorem seems toassign a special position to ordinary first-order logic. At the same time, it isa kind of impossibility theorem, showing that we cannot hope to strengthenfirst-order logic without losing some of its desirable properties.

First-order logic was first formulated explicitly by Frege as a part ofhis more comprehensive Begriffsschrift. Frege had to go further, however.His logic is not first-order, but higher-order.2 In other words, Frege’s logicallows quantification not only over individuals (particulars), but also overhigher-order entities, such as functions and other concepts applying toindividuals. The problems caused by this transgression beyond first-orderconcepts will be discussed later in this paper.

It is sometimes said that the idea of a freestanding first-order logic wasnot known to Frege and that it crystallized only later, for the first timeapparently in Hilbert’s and Ackermann’s 1928 textbook (Moore 1988).Maybe so. But even if Frege should have entertained the idea of purefirst-order logic, he would have had plenty of good prima facie reasons togo beyond what is in our days known as first-order logic. The most basicreason is that ordinary first-order logic does not suffice for mathematics.Another reason is that it is not even self-sufficient. The first of these twofailures is illustrated by the fact that several of the most basic concepts of

Synthese 111: 155–170, 1997.c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

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all mathematics cannot be expressed by means of ordinary first-order logic.Perhaps the most fundamental concept in the foundations of mathematics,the concept of equicardinality, cannot be expressed in first-order terms.When do the extensions of two conceptsA andBhave the same cardinality?If and only if there are functions f and (its inverse) g such that

(8x)(8z)((A(x) � B(f(x))&(1)

(B(z) � A(g(z)))&((z = f(x))$ (x = g(z))))

Here we cannot dispense with the quantification over the functions f andg.

Other concepts that likewise cannot be expressed in first-order termsinclude infinity, continuity in the sense of general topology, mathematicalinduction, etc. No wonder that Frege resorted to higher-order logic in hisattempted reduction of mathematics to logic. For one thing, he needed theconcept of equicardinality for his definition of number.

For another thing, first-order languages are not self-sufficient in thesense that the model theory of a first-order language or first-order axiomsystem cannot be formulated in first-order terms. The most central con-cept of all model theory (logical semantics), the concept of truth, can bedefined for first-order languages along the lines Tarski staked out (Tarski1956). Alas, these lines lead us away from a first-order language to thecorresponding second-order language. A Tarski-type truth predicate for afirst-order language is a second-order predicate asserting the existence ofa suitable kind of valuation, that is, of a function from the expressions ofthe language to their potential values in the given model.

2. TRADITIONAL PICTURE OF THE FOUNDATIONS OF MATHEMATICS

Hence we apparently need higher-order logic in our mathematical theoriz-ing. But higher-order logic is not only inevitably incomplete. It is entangledwith all the problems with the existence of higher-order entities like setsand functions. Hence it might seem only fair to fess up and admit thatQuine is right in calling higher-order logic set theory in sheep’s clothing.The proper framework for mathematical theorizing therefore appears tobe set theory – a view which is currently shared by many, perhaps most,mathematicians. Because in set theory we need axioms over and abovethose of first-order logic, it is usually considered a mathematical ratherthan logical theory.

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The study of set theory is usually conducted in the same way as that ofany other mathematical theory, that is to say, in the form of an axiomatictheory using first-order logic. Higher-order logic is not used, for the wholeidea of axiomatic set theory is to dispense with it. Instead, the only logicused to regulate the consequences of set-theoretical axioms is the ordinaryfirst-order logic, once again reflecting the dogma that this logic is ournatural basic logic.

First-order axiomatizations of nontrivial mathematical theories are nev-ertheless inevitably incomplete, no matter whether set theory is beingresorted to or not.3 For one thing, first-order set theories cannot do thesame job as higher-order logics in one important respect. No explicit first-order axiomatization can capture the intended standard interpretation ofhigher-order quantifiers. In the (ordinary first-order) axiomatization of settheory, there will inevitably be models in which e.g. functions variables(like the f , g of (1)) do not range over literally all extensionally possiblefunctions, but only some specified subset of the set of all such functions.But this means that formulas like (1) cannot quite do the job they weredrafted to do in all models of the underlying language. For instance, theextensions ofA(x) andB(x) might be equinumerous, and yet there mightnot exist any functions f , g (in the sense of the first-order theory) doingthe one-to-one correlation expressed by (1).

Now the first-order character of the usual axiomatizations of set theorymeans that set theory cannot capture the standard interpretation of higher-order quantifiers.4 Model-theoretically speaking, any first-order axiomati-zation of set theory admits of nonstandard models which violate the stan-dard interpretation of set theory thought of as a substitute for higher-orderlogic.

Moreover, since first-order logic cannot even capture the principle ofmathematical induction, even first-order theories such elementary mathe-matical theories as arithmetic are bound to be incomplete, as Godel showedin 1931 (Godel 1986a). This holds of course also when elementary numbertheory is reconstructed within first-order axiomatic set theory.

Thus the usual line of thought results in a two-tier model of the founda-tions of mathematics. The ground floor according to this view is first-orderlogic, which is semantically complete but not strong enough to cope withactual mathematical conceptualizations. The next floor is set theory and/orhigher-order logic. On this level, we can express the basic mathemati-cal concepts, but we cannot hope to achieve completeness. Nevertheless,axiomatic set theory is in this view the best general framework for mathe-matical theorizing that we have available to us.

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On this view, the main foundational problems can be located Theypertain to the higher levels of the structure I have described. In otherwords, practically all the difficulties in the foundations of mathematicsseem to lie in formulating complete mathematical theories. The task ofstudying the logical consequences of axiomatic theories can apparentlytake place by means of a simple logic that can be mastered once and forall in the sense of admitting a semantically complete axiomatization, thatis, a recursive enumeration of all valid formulas.

Even though on this view logic is unproblematic, its scope is severelylimited. One might even say that the picture I have painted is overshad-owed by several impossibility results, including Godel’s incompletenesstheorem, Tarski’s result that truth can be defined for a first-order theoryonly in a stronger theory, Lindstrom’s theorem, and the impossibility ofexpressing notions like equicardinality on the first-order level.

This generally accepted picture has one thing against it. It is wrong. Itrests on a woefully inadequate analysis of the entire problem situation inthe foundations of logic and mathematics. It is based on an unnecessarilyrestrictive idea of what our basic logic is like. It relies on an ambiguous andconsequently misleading picture of what completeness means. Moreover,and perhaps most fundamentally, it relies on an unclear idea what the endsof mathematical theorizing are in the first place.

I will explain these four points in the rest of this paper.

3. INDEPENDENCE-FRIENDLY FIRST-ORDER LOGIC

To take the first point first, whether we credit the ordinary first-order logicto Frege or to Hilbert and Ackermann, it does not do the job fully adequatelythat it was calculated to do.5 For what is first-order logic supposed to do? Itis supposed to be the logic of quantifiers. Now the source of the expressivepower of quantifiers is their interplay. Quantifiers cannot be adequatelycharacterized as higher-order predicates as Frege thought, or as “rangingover” a class of values, as most philosophers seem to think in these days.Quantification theory is essentially a study of the interplay of differentquantifiers, their dependencies and independencies. And the construal ofquantifiers as higher-order predicates cannot do justice to this interplay.

It has turned out that Frege’s approach to logic, including the forma-tion rules for the first-order fragment of his total logic, rules out certaincombinatorially possible and easily interpretable patterns of dependenceand independence between quantifiers. The simplest instances of a patternnot expressible in ordinary first-order logic are the so-called Henkin quan-

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tifier formulas expressible as the following two-dimensional “branchingquantifier” formula:

(8x)(9y)

(8z)(9u)>S[x; y; z; u](2)

Here the truth making value of y depends only on x and the similar value ofu only on z. Hence (3) is equivalent to the following second-order formula

(9f)(9g)(8x)(8z)S[x; f(x); z; g(z)](3)

which is easily seen to be irreducible to any ordinary first-order formula.In practice, it is simplest to introduce a special symbol to indicate theindependence of a quantifier, say (9y), of another one, say (8z), of which itotherwise depends on, by writing it (9y=8z). Then (2) = (3) can be writtenon the first-order level linearly as

(8x)(8z)(9y=8z)(9u=8x)S[x; y; z; u](4)

When this slash notation is extended also to propositional connectives, weobtain what has been called independence-friendly (IF) first-order logic(Hintikka 1995a; Hintikka 1996, Chaps. 3, 7).

This logic has a better claim to be our true basic logic than ordinaryfirst-order logic, in that it is free from the needless and arbitrary restrictionsthat beset ordinary first-order logic. IF first-order logic has several of thesame desirable metalogical properties as its received counterpart. It iscompact and has the Lowenheim–Skolem property. Separation theoremholds in it in an even stronger form than before. The reason it does notviolate Lindstrom’s Theorem is that the law of excluded middle fails inIF first-order logic, which it is one of the apparently minor premises ofLindstrom’s Theorem but which turns out to be crucially important for itsapplicability.

Both ordinary and IF first-order logic admit of a simple translation to thecorresponding second-order language. The translation t(S) of a sentenceS simply asserts that its Skolem functions exist. In other words, if S isof the form S[(9x)S1[x]], where (9x)S1[x] occurs within the scope of theuniversal quantifiers (8y1), (8y2) : : : , it is replaced by

(9f)S[S1[f(y1; y2; : : :)]]

where f is a new function symbol expressing one of the Skolem functionsof S. (It is of course assumed that S is in the negation normal form, i.e.,

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that all its negation signs are prefixed to atomic sentences and that theonly propositional connectives in S are &, _.) In order to reach t(S), thiselimination of first-order existential quantifiers must be applied to all ofthem. likewise, whenever S is of the form

S[(S1 _ S2)]

where (S1_S2) occurs within the scope of (8y1), (8y2), : : : , it is replacedby

S[((S1 & g(y1; y2; : : :) = 0) _ (S2&g(y1; y2; : : :) 6= 0)]:

In other words we must also associate a “Skolem function” to each dis-junction dependent on universal quantifiers. It is immediately seen thatthe translation t(S) belongs to what is known as the �1

1 fragment of thesecond-order language in question, that is, its only second-order ingre-dient is an initial sequence of second-order existential quantifiers. Theremarkable thing here is that this fragment can be translated back into thecorresponding IF first-order language.

The failure of the law of excluded middle is an inevitable consequenceof the most obvious and most natural semantical rules for IF logic (Hin-tikka 1996, Chap. 7). Admittedly, I can extend the IF first-order logicby introducing contradictory negation by fiat. But then it turns out thatthis contradictory negation : cannot be given any semantical rules exceptfor saying that it is the contradictory negation. As a consequence, it canoccur only sentence-initially (or fronting an atomic formula). In the result-ing extended IF first-order logic, most of the “nice” metatheorems listedabove fail.

One of the most remarkable things about IF first-order logic is that itextends what can be done in the foundations of mathematics by purelylogical means. Consider for instance, the following statement

(8x)(8z)(9y=8z)(9u=8x)((Ax) � B(y))&(5)

(B(z) � A(u))&((y = z) ! (u = x))

The second-order translation of (5) is

(9f)(9g)(8x)(8z)((A(x) � B(f(x)))&(6)

(B(z) � A(g(z)))&((z = f(x)) ! (x = g(z))))

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A comparison with (1) shows that (6) in fact expresses the equicardinalityof the extensions of A(x);and B(x). At the same time, (5) is perfectlyfirst-order. Its quantifiers “range over” individuals, not over higher-orderentities. It expresses a combinatorial fact about the model (“world”) inquestion. It could even be expressed by merely revising the received scopeconventions of first-order logic. Indeed, we could reach the entire IF first-order language in this way, merely by allowing that the scope associatedwith a quantifier need not be a continuous segment of the formula inquestion. For instance, the Henkin quantifier formula (2) (or (4)) couldthen be written as

(8x)f(9y)(8z)g(9u)fS[x; y; z; y]g(7)

where the brackets fg indicate the scope of (8x).Consider next the statement

(8x)(8z)(9y=8z)(9u=8x)((y 6= x)&(u 6= z)&(8)

((x = z) ! (y = u)))

This is easily seen to be equivalent to

(9f)(8x)(8z)((f(x) 6= x)&((x = z) ! (f(x) = f(z))))(9)

which clearly is true if and only if the universe of discourse is infinite (orempty).

Likewise, the topological notion of continuity can be expressed bymeans of IF first-order logic, as can be several other basic mathematicalconcepts that could not be expressed in terms of ordinary first-order logic.In extended IF first-order logic, even more concepts become expressible.They include mathematical induction, well-ordering, power set, etc.

One might nevertheless doubt the power of IF first-order logic as a toolin mathematics. It corresponds after all only to a small fragment of second-order logic, viz. to its �1

1 fragment. Even extended IF first-order logic doesnot extend any further than to the �1

1 [�11 fragment of second-order logic.

Neither one goes beyond second-order logic in contradistinction from afull higher-order logic. I will return later to this apparent restriction.

4. DIFFERENT KINDS OF COMPLETENESS AND INCOMPLETENESS

In any case, IF first-order logic extends significantly the range of what logiccan do in the foundations of mathematics. But it seems to have an important

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shortcoming: It is incomplete. It is impossible to give a list of axioms fromwhich all the valid formulas of IF first-order logic can be derived by purelyformal rules. Or, to speak the language of mathematical logicians, the setof all valid formulas of IF first-order logic is not recursively enumerable.

Far from being a defect, however, this feature of IF first-order logiccan be turned into a forceful reminder of the fact that the notion of com-pleteness traditionally used in the foundations of mathematics is a mess,albeit perhaps not a hopeless mess. If the reader has found it hard to graspwhat is meant by different references to completeness and incompletenessearlier in this paper, he or she has had a good reason to feel at sea. AsI have pointed out on an earlier occasion (Hintikka; 1989, 1996, Chap.5), there are at least three entirely different notions of completeness andincompleteness. They even apply to different kinds of theories:

(i) Descriptive completeness applies to nonlogical axiom systems. Itmeans that the models of the system include all and only intended models.

This is a model-theoretical notion in the sense that only the notions oftruth and validity are involved in it, but not any axiomatization of logic.

(ii) Semantic completeness is a property of a so-called axiomatizationof some part of logic. It says that the axiomatization in question effects arecursive enumeration of all valid formulas.

(iii) Deductive completeness is a property of a nonlogical axiom systemtogether with an axiomatization of the underlying logic. It says that fromthe axioms of this system one can logically prove either S or� S for eachsentence S of the language in question.

The drastic differences between these notions can be illustrated byrelaxing our distinctions to Godel’s first incompleteness theorem. Thetheorem establishes the incompleteness of elementary (first-order) arith-metic, but in what sense? The answer is clear. Godel showed the deductiveincompleteness of elementary arithmetic. But this is not the only incom-pleteness in town, and it concerns more the computational manipulabilityof proofs in first-order logic than its power to capture the right structuresas its models. This latter power is at issue in descriptive completeness andincompleteness, not in deductive ones. So the sixty-four-thousand-dollarquestion becomes: Does Godel’s result entail the descriptive incomplete-ness of arithmetic? A closer examination shows that it does so only if theunderlying logic is semantically complete. Of course, the logic Godel wasrelying on, ordinary first-order logic, had just been proved complete byGodel himself. But there is nothing in Godel’s result that precludes thepossibility that by using some other kind of logic, which would have tobe semantically incomplete, we could formulate a descriptively completeaxiomatization of elementary arithmetic.

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This is one of the many directions in which IF logic opens new pos-sibilities. Its semantical incompleteness turns out to be a hidden asset, inthat it might open new possibilities of formulating descriptively completetheories. Actually, it turns out that although unextended IF first-order logicdoes not allow a complete axiomatization of elementary number theo-ry, extended IF first-order logic does so. Indeed, one way of obtaining adescriptively complete axiomatization is as follows:

Let ' be the sentence

(9z)[(8x)(8y)('11(_=8y)'12)(_=8x)('21(_=8y)'22)]

where

'11 : :(x = y) '12 : :(x = 0)

'21 : :(y = z) �22 : :(x = y)&:(y = f(x))

Let be the conjunction of the sentence s

(8x):(f(x) = 0); (8x)(8y)(f(x) = f(y) � x = y);

(8x)(x 6= 0 � (9y)(f(y) = x)):

It has been shown by Sandu and Vaananen that M j= (' & ) iff M isa non-standard model of arithmetic (Sandu and Vaananen 1992). HenceM j= :(' & ) (where : is contradictory negation) iff M is the standardmodel. (I owe this observation to Gabriel Sandu.)

5. AXIOMATIZING MATHEMATICAL THEORIES

This does not yet show anything about the prospects of formulating descrip-tively complete axiomatic theories by means of IF logic in general. Thismatter can be put into a clear perspective by noting that practically all usualmathematical theories admit of a descriptively complete axiomatization inhigher-order logic. There are, for instance, scarcely any major unsolvedproblems in mathematics that cannot be expressed faithfully in terms ofsecond-order or higher-order logic (cf. Shapiro 1991). This is possiblebecause higher-order logics are semantically incomplete.

This descriptive completeness is nevertheless bought at a very highprice. This price is not the loss of semantical completeness but the vexingproblems concerning set existence and more generally the existence of

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higher-order entities that are inevitable in higher-order logic. These diffi-culties are the justification of branding higher-order logic “set theory insheeps’ clothing”. For it is in set theory that the questions of set existencecome to a head. These problems are often thought of as being mainly abouthow to avoid paradoxes in making existence assumptions in set theory. Inreality, the really puzzling difficulties in set theory concern the formulationof sufficiently strong assumptions of set existence, as will be emphasizedbelow in connection with the requirement of standard interpretation.

As a consequence of the haunting difficulties, it becomes important toinquire whether descriptively complete axiomatizations for mathematicaltheories can be formulated by means of a suitable first-order logic, whichof course has to be semantically incomplete.6

Here I can turn an old confusion into a successful strategy. In the lightof hindsight it can be said that Frege and Russell dealt – or tried to deal– with higher-order logics as if they were many-sorted first-order logics,with each type constituting a separate “sort”. The interrelations of differentsorts can be handled in first-order terms.

So what did Frege and Russell miss? What they missed was the intendedstandard interpretation of higher-order quantifiers. Standardness is hereunderstood in Henkin’s sense, as a requirement that bindable variables ofany given type range over all the extensionally possible sets of entities ofthe appropriate lower type (Henkin 1950; Hintikka 1995a). For instance,set variables range standardly over the entire power set of the domain, thatis, all extensionally possible sets, and not only over sets that e.g. exist in thesense of some set theory. Likewise, function variables range over what aresometimes called all arbitrary functions, relation variables over “relationsin extension”, as Russell called them, and so on. Standardness cannot beenforced by first-order means, for instance through first-order axiomaticset theory, as is illustrated by the so-called Skolem paradox. Becauseof this failure of first-order logic to capture the standard interpretation,dealing with higher-order logic as a many-sorted first-order logic cannotever succeed completely. As was spelled out by Ramsey (1931), many ofthe shortcomings of Russell’s and Whitehead’s Principia can be tracedto their failure to countenance the standard interpretation of higher-orderquantifiers.

What is remarkable here is that the many-sorted first-order treatmentalmost succeeds. For the only thing in higher-order theories which cannotbe captured by means of ordinary (many-sorted) first-order logic is pre-cisely the requirement that for each extensionally possible collection oflower-type entities there corresponds an entity of the appropriate highertype. This point can be illustrated by reference to what happens in general

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topology (Kelley 1955). There all of the most elementary concepts can beformulated in terms of ordinary first-order quantification over points on theone hand and sets on the other hand. This includes such concepts as base,closure, openness, etc. But when we proceed further, at some point webegin to need the assumption that quantification over, say, the subsets ofa given set really means quantification over all the extensionally possiblesubsets. One place where this is likely to happen is in connection with theconcept of connectedness.

Obviously, the missing standardness requirements can be formulatedquite simply (cf. Hintikka 1996, Chap. 9). For each type (higher than that ofindividuals andn-tuples of individuals), it suffices to require that it containsrepresentatives of all the sets of entities of the appropriate lower type. Andsuch a requirement can be expressed by �1

1 second-order statements. Wehave no reason to resort to any higher-type conceptualizations for thepurposes of the sortal reconstruction of higher-order logic and higher-order theories. We do not need any more complex second-order formulas,either. With a modicum of stage-setting, all such �1

1 statements can evenbe integrated into one single �1

1 statement.These simple observations have striking consequences. They imply

among other things that each mathematical theory that can be formulatedby means of higher-order logic can be construed as a �1

1 theory. The setof structures that the given higher-order theory can capture as the set of itsmodels can be captured by a �1

1 theory, albeit imbedded in the additionalstructure which the many-sortal reconstruction introduces. In this sense,�1

1 theories are all that is needed in ordinary mathematics. Indeed, there is asense in which the basic form of mathematical reasoning viz. mathematicalinduction, formulated generally as the closure of a certain set under somemathematical operations belongs to �1

1 logic and hence to extended IFfirst-order logic. For it was shown by Moschovakis (1974) that any suchinductive theory is equivalent to a �1

1 sentence.But each such �1

1 theory can be translated so as to become a theoryexpressed in extended IF first-order logic. Hence there is a sense in whichall ordinary mathematics can be reduced to extended IF first-order logic.

Moreover, in the many-sorted first-order reconstruction of a higher-order theory, all the putative theorems can obviously be expressed as ordi-nary first-order statements. The question of the status of such a putativetheorem T in an axiom systemX thus becomes a question concerning thevalidity of

(X � T )(10)

whereX is�11 andT first-order. But that means that (10) is itself�1

1. Henceit has a translation in IF first-order logic. Hence there is a sense in which

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every problem of ordinary mathematics is equivalent with the problem ofvalidity for a sentence of IF first-order logic.

In the sense that appears from these remarks, IF first-order logic is theonly logic that is in principle needed in ordinary mathematics.

I am not suggesting for a moment that IF first-order logic is a prac-tical framework for doing mathematics. For such purposes, a judicioususe of second-order logic, typically restricted to �1

1 and �11 conceptual-

izations, seems to be the best bet. In practice, the upshot would probablybe something rather like general topology (Kelley 1955). The main advan-tages of my reduction of mathematical theories to IF first-order level arephilosophical and theoretical. The main payoff is a complete liberationof mathematical theorizing from all problems of set existence and of allproblems concerning the existence of higher-order entities in general. Thequestion of the validity of a sentence of IF first-order logic is a purelycombinatorial one: It concerns the possibility or impossibility of differentstructures of individuals (particulars). It is nominalistic in Quine’s sense.It is to such combinatorial problems that I am reducing all questions oftheoremhood in mathematical theories. This is a tremendous advantagein principle, for the problems of set existence are notoriously the mostconfused and confusing ones in the foundations of mathematics.

6. IF FIRST-ORDER LOGIC IS SELF-APPLICABLE MODEL-THEORETICALLY

So far, I have not said anything of the second massive reason why ordinaryfirst-order logic has not been thought of as a self-sufficient foundation formathematical reasoning. This reason is that the model theory of ordinaryfirst-order logic cannot be done by means of itself. The crucial conceptof all model theory is the concept of truth (truth in a model), and it wasnoted earlier that for Tarski-type truth definitions we need second-orderconcepts.

In this respect, too, IF first-order logic puts things in a new perspective.The result has been investigated in some detail elsewhere (cf. Hintikka1995, Chap. 6). Suffice it here to indicate only the most general features ofthe situation. Both Tarski-type truth-predicates for a given finite first-orderlanguage and the game-theoretical ones that are in many ways superior tothe Tarskian ones can be expressed as �1

1 statements in the correspondingsecond-order language. This holds also for truth-definitions for an (unex-tended) IF first-order language. But such �1

1 statements can be translatedback into the correlated IF first-order language.

In such a language we therefore can formulate a truth-predicate foritself. Paradoxes are avoided because of the inevitable failure of the law

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of excluded middle in IF first-order logic (Hintikka 1996, Chap. 7). Thisfailure creates the truth-value gaps that are needed to avoid a contradiction.

In extended IF first-order logic we have a contradictory negation present.However, it can occur only sentence-initially. This makes it impossible toapply the diagonal lemma so as to create a liar-type sentence that wouldgive rise to a contradiction.

Truth-predicates are not all that there is to the model theory of a languageor a theory. However, their possibility is an eloquent indication that we donot need set theory or higher-order logic for the main ingredients of amodel theory of a given first-order theory. Furthermore, it is easily seenthat many other familiar notions of model theory can be formulated interms of a (possibly extended) IF first-order language.

7. A NEW PERSPECTIVE ON THE FOUNDATIONS OF MATHEMATICS

The total picture of mathematical thinking that is suggested by these resultsdiffers sharply from the usual one. The core of mathematical thinking is notset theoretical. We can in principle dispense with set-theoretical assump-tions and concepts altogether. Instead, mathematical thinking can – andperhaps should – be thought of as combinatorial, dealing with the structuresof particular objects, especially with questions as to which configurationsof individuals are possible and impossible. These are essentially the samekinds of questions or are dealt with in received first-order logic.

These results might in one respect seem too good, or perhaps too simple,to be true. If all of ordinary mathematics can be done in extended IF first-order logic or, equivalently, in the �1

1[�11 fragment of second-order logic,

what use and what interest does the rest of higher-order logic have? Surelyit has some content and use beyond its simplest special case.

I am not denying that higher-order logic has uses beyond the �11 [ �1

1case. But the crucial question is: What kind of use? By and large, logic hastwo kinds of uses in its applications, especially in mathematics (Hintikka1996, Chap. 1). They can be characterized as follows:

(a) Descriptive use. This is the contribution of logic to the specificationof the kind of structure or structures that a mathematician or a scientistwants to study. One’s aim in this direction can for instance be to formulatea descriptively complete axiom system. An example might be offered e.g.by the use of quantifiers and other logical constants in the axioms of amathematical or scientific theory. The descriptive function of logic couldand perhaps should be called its theoretical function.

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(b) Deductive use. In this employment, logical concepts are used to studyvalid inferential relationships between propositions.

This function might also be called the computational or explicativefunction of logic in axiomatic theories. What has been seen is that the task(a) can typically be performed by logic already on the first-order level. Butfrom this it does not follow that the deductive task can be so performed.Indeed, it can be seen that the deductive task is awkward to fulfil as soonas we go beyond ordinary first-order logic. For what a logician would liketo do is to express in his or her language the proposition that says that if S1

is true, then S2 is true. In ordinary propositional logic, this is expressed by

(S1 � S2)(11)

But in IF first-order logic (11) does not semantically speaking say that S2

is true if S1 is. What it says is that eitherS1 is false or S2 is true. Since in IFfirst-order logic there are propositions that are neither true nor false, this isan unnecessarily strong requirement. What is needed in the study of logicalinference is a statement whose truth authorizes us to infer the truth of S2

from the truth of S1. Such a statement can be formulated in higher-orderterms. Expressed in game-theoretical jargon it will say that there is aneffective (recursive) functional � which from a winning verifier’s strategy' in the game S1[', ] associated with S1 yields a winning verifier’sstrategy �(') in the game S2[', ] associated with S2 (Hintikka 1993).Winning strategies are in each case characterized by the fact that they resultin a win even when one’s opponent is aware of them. The possibility ofinfering the truth of S2 from the truth of S1 will thus be expressible by

(9�)((9')(8 )S1['; ] � (8�)S2[�('); �])(12)

However, it is natural to require also that the knowledge of a strategyfalsifying S2 should enable us to find a strategy falsifying S1. This iscaptured by changing (12) into

(9�)(9 )(8')(8�)(S1['; ('; �] � S2[�('); �])(13)

which is Godel’s functional interpretation for conditionals (Godel 1986b).It can be seen that (12) is of a higher order (type) than S1 or S2. Whenconditionals are nested in a sentence, as we might very well want to nestthem for purposes of inference, we are pushed to even higher types.

What this means is that for inferential and deductive purposes, therecan be plenty of reasons to use higher-order logics. The overall picture ofthe foundations of mathematics which we thus arrive at is almost diamet-rically opposite to the traditional one, explained above. The descriptive,

THE FOUNDATIONS OF MATHEMATICS 169

that is, model-specifying tasks in mathematics can be accomplished bymeans of relatively elementary (first-order) logic. These descriptive tasksare paramount on the level of theorizing, be this theorizing mathematical,scientific or philosophical. It is the inferential, deductive and computationalaspects of the task of a mathematician that force us to consider increasinglyhigher-order logics and draft them to our service. What these computation-al aspects amount to is to elicit consequences of a theory which has alreadybeen formulated concerning a class of structures (models) which alreadyhave been captured by the theory, assuming of course descriptive com-pleteness. Hence the use of higher-order logics belongs to the technologyof logic rather than its theory.

Obviously neither the descriptive nor the deductive task of logic shouldbe neglected. In our day and age, which is overwhelmingly the age of com-puters and computing, equally obviously it is the descriptive and theoreticalfunction that is in danger of being neglected.7

NOTES

1 See Per Lindstrom, (1969), pp. 1–11. The most accessible formulation of his result isfound in H. D. Ebbinghaus, J. Flum and W. Thomas, 1984, chapter 12.2 An important qualification needed here is that Frege adopted a nonstandard interpretation(in Henkin’s sense, see Note 4 below and Henkin 1950), which means that his higher-orderlogic could be dealt with like a many-sorted first-order logic. See Jaakko Hintikka andGabriel Sandu.3 Indeed, if a first-order theory is deductively incomplete, then it is also descriptively incom-plete as pointed out. On the other side, if a theory is descriptively complete, in the senseof e.g. characterizing up to isomorphism the standard model of arithmetic, then the theoryis semantically incomplete, hence it cannot be a first-order theory. This follows from factsproved in Jon Barwise and S. Feferman (1985).4 For the standard vs. nonstandard distinction, see Jaakko Hintikka (1995a), pp. 21–44.5 With the following, cf. Jaakko Hintikka (1996), chapters 3–4 as well as “A revolution inlogic?” (forthcoming).6 By the results mentioned at the end of Section 4 above, if we have a descriptively com-plete axiomatization of a non-trivial mathematical theory, the theory must be semanticallyincomplete, hence it cannot be an ordinary first-order theory. (Of course it can be an IFfirst-order theory.)7 In working on this paper, I have greatly profited from the advice of Dr. Gabriel Sandu.

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Department of PhilosophyBoston UniversityBoston, MAU.S.A.