Jaakko Hintikka on Truth
Hans Sluga
- In the Spring of 1967, Jaakko Hintikka published two contributions to the journal Synthese, of which he was then the editor, that have proved to be of singular importance to the further development of analytic philosophy. The first was Donald Davidson’s well known essay on “Meaning and Truth,” the second Jean van Heijenoort’s no less influential note on “Logic as Calculus and Logic as Language.” In publishing these two programmatic statements side by side Hintikka as editor of Synthese helped to propel analytic philosophy into an entirely phase of its evolution. He did not, of course, foresee this at the time nor did he anticipate that the two pieces would eventually also become crucial to his own philosophical thinking. But so great is their bearing on Hintikka’s concerns with questions of language, meaning, and truth that no serious examination of his work can bypass an assessment of Davidson’s and van Heijenoort’s essays. These influences are particularly evident in the eight papers collected now in Hintikka’s Lingua Universalis vs. Calculus Ratiocinator on which I will focus my attention in this essay.[1] In its theoretical parts, Hintikka’s volume examines once more the Davidsonian question whether and how we can construct a theory of meaning for ordinary language and, like Davidson, it takes such a theory of meaning to require a definition and theory of truth. In the other sections, the volume follows van Heijenoort in trying to situate such theoretical concerns in a larger historical context. To appreciate Hintikka’s contribution to the ongoing philosophical debate means then to triangulate between his writings and Davidson’s and van Heijenoort’s essays and since the latter can help us to locate the two others in a broader context it is plausible to consider it first.
- Jean van Heijenoort’s essay – in fact a five-page note – presents itself modestly as providing “a useful insight into the history of logic.” (H, p. 324)[2] Its stated purpose is to alert us to the absence of systematic work in formal semantics in the early phases of the development of modern logic (the period, roughly, from 1879 to 1913). The essay depicts logicians like Peano, Frege, and Russell as altogether unconcerned with metatheoretical investigations and argues that this branch of logic was first explored by Leopold Löwenheim in his essay on “Möglichkeiten im Relativkalkül” and subsequently expanded by Skolem, Herbrand, and Gödel. Van Heijenoort concludes with the simple but provocative observation that Frege’s “Begriffsschrift (1879) Löwenheim’s paper (1915), and Chapter 5 of Herbrand’s thesis (1929) are the three cornerstones of modern logic.” (H, p. 329) This leaves Tarski and his work on the concept of truth out of consideration which is unfortunate given that philosophical interest in formal semantics is certainly more deeply indebted to Tarski than to either Löwenheim or Herbrand.[3] Carnap, in particular, to whom we can trace back much of the contemporary philosophical concern with questions of semantics, came to these matters precisely through Tarski’s intervention.[4]
“Logic as Calculus and Logic as Language” begins with a review of the logical achievements of Frege’s Begriffsschrift. Frege is given credit for formulating “with all the necessary accuracy, a cardinal notion of modern thought, that of a formal system” and for developing classical propositional and predicate logic. At the same time van Heijenoort charges Frege with advancing a conception of logic that, in effect, helped to stop the development of metamathematical reasoning. He traces this obstacle to metatheory back to Frege’s belief in “the universality of logic.” This belief, he writes, is “perhaps not discussed explicitly but nevertheless constantly guides Frege.” (H, p. 324) But to what extent was Frege actually guided by a belief in the universality of logic? What we find in his texts is, in fact, something significantly weaker though in some ways related to that doctrine. It is a characterization of the logical laws as “laws of thought that transcend all particulars.” (B, p. iv; 48)[5] Frege writes similarly also in his Foundations of Arithmetic that logic concerns “general (allgemeine) logical laws” (F, p. 4)[6] and that its truths are “of a general logical nature.” (Ibid.) He observes, moreover, that “the truths of arithmetic govern all that is numerable. This is the widest domain of all; for to it belongs not only the actual, but everything thinkable.” And he asks, in consequence: “Should not the laws of number, then be connected very intimately with the laws of thought.” (F, p. 21) This generality of logic expresses itself in Frege’s notation most immediately in the fact that the quantifiers binding individual variables range over all objects. Van Heijenoort comments: “For Frege it cannot be a question of changing universes. One could not even say that he restricts himself to one universe. His universe is the universe.” (H., p. 325)
While all this is correct, van Heijenoort’s conclusion that in Frege’s logic “nothing can be, or has to be, said outside the system” and that for this reason Frege “never raises any metasystematic question” (H, p. 326) is problematic. For Frege emphasizes in the preface to the Begriffsschrift that his logical symbolism is by no means meant to be a universal language in such a strong, Leibnizian sense. He considers it rather to be a tool for specialized investigations that is not intended to replace ordinary language at all. Comparing his logic to a microscope and ordinary language to the human eye, Frege writes: “[A]s soon as scientific purposes place great demands on sharpness of resolution, the eye turns out to be inadequate. The microscope, on the other hand, is perfectly suited for just such purposes, but precisely because of this is useless for all others.” (B, p. v; 49) Frege’s notation is thus to be taken as “an aid for particular scientific purposes,” and nothing else. (Ibid.)[7] If we follow this metaphor through, we ought to say that the logical notation is for Frege a technical aid to ordinary language; it can, therefore, not replace ordinary language which retains its own original and distinctive function. There is also no reason to think that the structure (or deep structure) of ordinary language is identical with that of the logical notation. On the contrary, while the sentences of ordinary language have a subject-predicate structure, those expressed in the logical notation do not.[8] The formal notation is a constructed device and this means, presumably, that we first have to design a set of symbols and then attach appropriate meanings to them, meanings that are always specified with the help of the resources of ordinary language. By contrast, ordinary language itself has arisen in an organic fashion from sense-perceptions and ideas and from groups of memory-images that have gathered around our perceptions. The symbolic devices of ordinary language allow us to pursue these images further and prevent them from sinking into darkness as new perceptions arise. “But if we produce the symbol of an idea which a perception has called to mind, we create in this way a firm, new focus about which ideas gather.”[9] That is why language is not governed by logical laws in the way our artificial notation can be.
It is true that, in order to describe his logical notation, Frege adopts the Leibnizian term “lingua characterica” but that does not mean he also adopts Leibniz’s ambitions for such a language. The preface to the Begriffsschrift, in fact, dismisses Leibniz’s view that it would be easy or useful to construct a universal language. “The enthusiasm that seized its originator in considering what an immense increase in the mental power of mankind would arise from a symbolism suited to things themselves let him underestimate the difficulties that such an enterprise faces.” (B, p. V; 50) Frege admittedly conceives of extensions of his symbolism that would make it applicable to physics, chemistry, and possibly other sciences, but even then he does not commit himself to the Leibnizian conception of a universal language. His notation is universal only in the restricted sense that the individual variables range over all objects. In contrast to Leibniz, he is not committed to the idea that all possible predicates should be expressible in that notation. Thus, semantic predicates (like “means,” “refers,” “is true”), in particular, find no representation in Frege’s logical language. He is, in fact, quite clear about what the notation must express and what not. Logic concerns for him “the laws of correct inference,” as he puts it occasionally.[10] The Begriffsschrift, he writes in the preface to his book, “is thus intended to serve primarily to test in the most reliable way the validity of a chain of inference.” (B, p. iv; 48) And it follows that “[e]verything that is necessary for a valid inference is fully expressed; but what is not necessary is mostly not even indicated.” (B, p. 3; 54)
The decisive point in van Heijenoort’s argument is, in fact, not that Frege conceives of his notation as a universal language, but that he thinks of it as a language. This is to say, that Frege takes his notation to be, like ordinary language, a system of signs with fixed meanings. By contrast, Löwenheim treats the logical notation as a calculus, that is, as a system of signs that can (a) be studied as a pure formalism and (b) be given various interpretations. He, thus, “takes the liberty to change the universe of discourse at will and to base considerations on such changes.” (H, p. 328) This makes it possible for him to establish that a well-formed formula is valid in every domain, if it is valid in a denumerable domain (Löwenheim-Skolem theorem). Van Heijenoort argues, moreover, that the calculus conception of the logical symbolism derives ultimately from Boole. Frege himself, he points out, had characterized Boole’s symbolism as a mere calculus for the purpose of making deductions (a calculus ratiocinator), whereas he had called his own symbolism a characteristic language. With Löwenheim’s paper we have, thus, “a return to, or at least a connection with pre-Fregean or non-Fregean logic.” (H, p. 328)[11] Here again it is possible to grant van Heijenoort’s historical observations but to question their philosophical interpretation. He is certainly right in saying that not only Frege but also Russell treated the logical symbolism as a language with a fixed interpretation. But from this does not follow that “[s]emantic notions are unknown” to them. (H, p. 326) This is a travesty of the truth, at least if we take van Heijenoort’s words literally. For semantic considerations were, in fact, of prime importance to both Frege and Russell, as any serious reader of their writings will know. But van Heijenoort means, presumably, that neither Frege nor Russell worked in model-theoretical semantics and this is also correct and in this respect Löwenheim’s 1915 paper does, indeed, mark “a sharp break with the Frege-Russell approach to the foundations of logic.” (H, p. 328) However, to this we must add that there is nothing in Frege’s and Russell’s conception of the symbolism as a language that bars them from considering questions of meaning and from examining notions like reference and truth. And that should come as no surprise, for if the symbolism is a language, as they believe, then a semantics of the symbolism is no more problematic than the semantics of ordinary language.
We may grant van Heijenoort that there are semantic problem that can be attacked only with model-theoretical methods, but from this it follows in no way that Frege’s understanding of his logic prohibited him in principle from raising metasystematic questions and that it was a matter of principle that “[q]uestions about the system are as absent from Principia Mathematica as they are from Frege’s work.” (H, p. 326) In contrast to what van Heijenoort suggests, Frege does, in fact, occasionally engage in metasystematic arguments. One such argument can be found in his Basic Laws of Arithmetic, another in his debate with Hilbert over the foundations of geometry.[12] In his introduction to the Tractatus, Russell, in turn, takes up Wittgenstein’s challenge that it is impossible to describe the syntactic and semantics characteristics of a language within that language and suggests that we might solve the problem by constructing a hierarchy of languages. Thus, it may turn out that “every language has, as Mr. Wittgenstein says, a structure concerning which, in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit.” (TLP, p. xxii)[13] Russell’s remark draws our attention to what may well be the real source of van Heijenoort’s claims, i.e., Wittgenstein’s Tractatus. In that work, the anti-metatheoretical view is spelled out boldly in the programmatic statement that “[l]ogic must take look after itself.” (TLP, 5.473) It follows for Wittgenstein that “[l]ogic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental.” (TLP, 6.13) It follows, in particular, that “propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it.” (TLP, 4.12) However, no corresponding statements can be found in the writings of either Frege or Russell. Van Heijenoort’s claim that both Frege and Russell are committed to the belief in “the universality of logic” and can thus not encounter the possibility of metatheoretical reasoning derives from an illegitimate (or, at least, questionable) imposition of Tractarian ideas on their work.
What is more, in 1939 Löwenheim wrote to Paul Bernays that the “precise formulation of a strict formalism” which he had developed was ultimately due to Frege and that he had corresponded extensively with Frege on this matter. That correspondence, consisting of ten letters each by Frege and Löwenheim written between 1908 and 1910, was unfortunately lost in the Second World War. But Heinrich Scholz, who had collected it, reports that its authors considered it so important that they had planned to publish it.[14] Its significance, as Scholz tells us in another place, lay in the fact that starting from Basic Laws, vol. II, § 90, Löwenheim had succeeded in convincing Frege of the possibility of constructing formal arithmetic in an unobjectionable manner.”[15] Though Frege seems to have been initially skeptical about the possibility of the model-theoretical reasoning that Löwenheim had developed, it appears that he eventually became convinced of its possibility. Scholz’s short statement does not allow us to state with full confidence what Frege thought that admission came to. There is, however, no reason to think that it changed his general views on the concepts of meaning and truth as one can see from his post-1910 writings.
One must then admit that Frege never pursued metamathematics systematically and that Russell never pursued the proposal he had sketched in his introduction to the Tractatus. However, it is not implausible to conjecture that this was due to a natural division of labor in the development of modern logic. Frege and Russell had first to create their systems of symbolic logic before the project of a scientific examination of such systems could be formulated (plausible-development theory). Van Heijenoort quotes Kurt Gödel as saying that metamathematical questions arise “at once” when we adopt the methods of symbolic logic and he wonders why Frege and Russell did not see this point. He explains: “If the question of the semantic completeness of quantification theory did not ‘at once’ arise, it is because of the universality… of Frege’s and Russell’s logic.” (H, p. 327) Is it not apparent that the plausible-development theory is sufficient to explain this fact? It is surely obvious that formal systems had first to be constructed before meta-theoretical questions could meaningfully be asked about them. Van Heijenoort thinks, however, that much more is at stake. According to him, Frege and Russell failed to develop metamathematics (including formal semantics) because they conceived of their logical notation as a universal language. By contrast, the metamathematicians treated the logical notation as an interpretable calculus. In other words, van Heijenoort adheres to what might be called an ultimate-presuppositions theory of the history of modern logic.
This latter view has implications for the whole of twentieth-century analytic philosophy. While van Heijenoort essay is meant to give us insight, in the first instance, into the history of modern logic, it proves just as relevant to an understanding of twentieth century analytic philosophy because of the great bearing that the new logic has had on this branch of philosophy. Van Heijenoort speaks, in fact, of Frege not only as the founder of a new logic but also as the beginning of the analytic tradition in modern philosophy. “Frege’s philosophy is analytic,” he writes, “in the sense that logic has a constant control over his philosophical investigations; this marked a sharp break with the past, especially in Germany, and Frege influenced philosophers as different as Russell, Wittgenstein, and Austin.” (H, p. 324) This point was first fully appreciated by Burton Dreben who in turn communicated it to his students at Harvard; but because Dreben was such a reluctant writer, it took some while before his appropriation of van Heijenoort’s scheme became visible.[16] It revealed itself first in Warren Goldfarb’s article on “Logic in the Twenties: The Nature of the Quantifier” twelve years after the original essay which, however, like its source, still concentrated on the development of modern logic. It was only in the eighties and nineties that some of Dreben’s students and a few other scholars began to weigh the broader implications of van Heijenoort’s claims for the development of analytic philosophy and among these scholars was Jaakko Hintikka.
- Throughout the essays that make up Lingua Universalis vs. Calculus Ratiocinator Hintikka emphasizes that van Heijenoort’s essay has unearthed “an ultimate presupposition” of twentieth-century philosophy, not only of twentieth-century logic. This presupposition presents itself not as a shared belief but as a choice which all contemporary philosophy is said to face – “a choice between two competing overall views concerning our relationship to our language. I shall call them (i) the view of language as a universal medium of communication, (in brief, language as the universal medium) or otherwise expressed, the view of the universality of language, and (ii) the view of language as a calculus.” (HI, p. 21) Hintikka is aware at the same time that “the universalist position,” understood in this way. “should perhaps be described as a syndrome of different ideas.” (HI, p. 193) What really matters to him is one single element in this syndrome: the belief in what he calls “the ineffability” or “inexpressibility” of semantics. In this respect, his concern appears narrower than van Heijenoort’s who is, after all, trying to explain more generally why Frege, Russell, and their followers raised no “metasystematic” questions. Among those would, of course, be semantic ones; but they would also include questions of consistency and independence that can be cast in purely syntactic terms. According to van Heijenoort, none of these questions were asked by Frege and Russell and this, he thinks, needs explanation. The great division in the history of logic is for him between the systematic and the metasystematic thinkers and he finds that division marked by two different conceptions of the logical symbolism. For Hintikka, on the other hand, the critical issue matter is that of the inexpressibility of semantics. Thus, Frege is for him a paradigmatic universalist because he thinks of language that “[i]ts semantics cannot be defined without circularity in that language itself without circularity, for this semantics is assumed in all its uses, and it cannot be defined in a metalanguage, because there is no such language beyond our actual working language. In brief, the semantics of our one and only actual language is inexpressible in it.” (HI, p. x) Opposed to this view is for Hintikka the conception of language as a calculus or, rather “the model-theoretical tradition in logic and philosophy of language” of which he writes: “According to this tradition, we are not prisoners of our own language in the same way as [we are] according to the universalist tradition. We can speak in a suitable language of its own semantics; we can vary its interpretation; we can construct a model theory for it; we can theorize about its semantics…” (HI, p. xi) But even this is not yet the crux of the matter for Hintikka. The ultimate and decisive question is rather whether a philosopher considers the notion of truth to be definable. He assumes “that the notion of truth is the single most important semantic relation” which is for him the “relation between language and the world.” (HI, p. 7) Thus, the question whether language is universal turns out to be “to a considerable extent equivalent to the question” whether truth is “ineffable” or “indefinable.” (HI, p. 23)
In speaking of the problem of “the universality of language” or, more specifically, the problem of “the inexpressibility of semantics” or, even more specifically the problem of “the definability of truth” as an ultimate presupposition of twentieth century philosophy Hintikka is borrowing consciously from R.G. Collingwood. On this account, the distinction between the language universalists and the model-theorists is so basic to analytic philosophy that it determines much of what goes on in the field. Those who proceed from one or the other of these two opposing conceptions may often not be aware of this or only partially so. In any case, they generally fail to reason about their respective presuppositions and instead take their own positions for granted. On one side of this dispute stand, as far as Hintikka is concerned, the early authoritative figures in analytic philosophy: Frege, Russell, and Wittgenstein, but also the members of the Vienna Circle as well as more recent authors like Quine and Church; on the other side stand the later Carnap but also, ultimately, Hintikka himself. Tarski and Gödel are said to have played an “ambivalent role… in the unfolding of the model-theoretical vision.” (HI, p. xii) For Hintikka the division is also responsible outside the analytic tradition for “one of the most consequential and most intriguing love-hate relationships in twentieth century philosophy, the relationship between Edmund Hussar and Martin Heidegger.” (HI, p. xiii) Because of Heidegger’s and Wittgenstein’s wide influence, the universalist conception of language is, according to Hintikka, reflected furthermore in philosophers as diverse as Richard Rorty and Jacques Derrida. That the front between the universalist and the model-theoretical view should be so long and cut across the boundary of contemporary philosophical schools will not surprise us, if we remember that, as Hintikka puts it, “an important part of the ineffability view is Kant’s transcendental philosophy.” (p. xviii) Hintikka can, of course, not argue that the universalist conception is entirely under the influence of Kant since such stoutly anti-Kantian philosophers as Russell and Quine are, according to him, also committed to it. But he draws support for his reading from Wittgenstein’s characterization of “[t]he impossibility of expressing in language the conditions of agreement between meaningful propositions – a thought – and reality” as “the Kantian solution to the problem of philosophy.” (HI, p. xviii) The ultimate presupposition of twentieth century philosophy is then, in a sense, the Kantian heritage in philosophy and the question to what extent it is necessary to take over and maintain that heritage.
Hintikka’s goal is, however, not simply to diagnose the ultimate presuppositions of twentieth century philosophy and of analytic philosophy more specifically. He wants rather to resolve the dispute between the two fundamentally different viewpoints. He asks accordingly in Lingua Character vs. Calculus Ratiocinator whether “the believers in language as a universal medium or the philosophers who see the wave of the future in the model-theoretic approach” are right and answers forthrightly: “The results reported here constitute a powerful argument for the conception of language as a calculus and against the thesis of the indefinability of semantics.” (HI, p. xvii) I quote these words to show that he is not speaking of a decisive argument against the universalist view but only of a “powerful” one. For if it is the case that the opposing positions represent ultimate presuppositions in analytic philosophy and more generally in contemporary philosophy as a whole, we should not expect to come up with decisive arguments for or against them. It is in the nature of ultimate presuppositions that they are ultimate and, hence, cannot be dislodged by refutations. Arguments can be relevant to an ultimate presupposition only by showing that further adherence to it requires accommodations in one’s theory which one may or may not find attractive.
- Before turning to Hintikka’s theoretical claims about language, meaning, and truth, we must look more closely at his historical observations. The first thing to emphasize here is that the classing together of a large number of twentieth century philosophers into two great, opposing camps is, at best, a hazardous undertaking. Philosophers are notorious individualists and philosophical schools or movements are, for that reason, characterized as often by their internal tensions and divisions as by their agreements. I have doubts, in any case, that the defining issues of twentieth century philosophy can be described in the way Hintikka assumes. He is certainly right in thinking that language, meaning, and truth are of pervasive concern in twentieth century thought – and that across the boundaries of individual schools – and that differences in how these concepts are understood may well be definitive of the different philosophical schools. We can also note that much of twentieth century philosophy shows a particular interest in the concept of truth, that this is due to large-scale dissatisfaction with the classical conception of truth as correspondence, and that different philosophers seek to grasp the concept of truth in radically different ways. A short list of what philosophers and groups of philosophers have said about truth will make this evident:
pragmatists: truth is usefulness.
Nietzsche: truth is interpretation.
Frege: truth is simple and indefinable and unlike any other predicate.[17]
Early Moore and Russell: truth is simple, indefinable, and exactly like any other predicate.
Logical positivists: replace the concept of truth by that of verification/falsification.
Heidegger: truth is unhiddenness (a-letheia).
Early Wittgenstein: truth is mirroring but to say that is really meaningless.
Later Wittgenstein: attempts to characterize truth are empty.
Foucault: truth is a system of procedures linked in a circular relation to power.
These variations show not only how intensively the concept of truth is discussed; they also make clear that we cannot easily reduce this multiplicity to an opposition between those who consider truth definable and those who do not. The definability or indefinability of truth is only one of the issues that divides philosophers. Equally fundamental divisions arise from the question whether truth should be considered a semantic, a pragmatic, or an ontological concept. Another fundamental division seems to exist between those who want to adhere to some notion of truth and those, like the logical positivists, who seek to it aside as a residue of traditional metaphysics.
The variety and complexity of philosophical thinking about the concept of truth becomes more evident when we look closely at those philosophers who want to retain the notion of truth but also consider it indefinable. We then discover that this thesis allows for a number of very different readings and that those who agree on the formula “truth is indefinable” may be motivated by different and irreconcilable concerns. We will also be able to see that of all the thinkers who adhere to the indefinability thesis Frege takes the most worked out and most complex view which seeks to incorporate a number of justifications for that thesis and which is, for that reason, multiply overdetermined. According to the first and weakest reading the thesis comes to saying:
(1) The concept of the truth of a sentence is a semantic notion but it cannot be explained in
terms of the semantic properties of subsentential components such as the reference of
proper names and the meanings of predicates (however the latter is to be specified).
In this form the indefinability thesis opposes itself to the traditional correspondence theory of truth according to which a sentence is true if and only if it corresponds to the facts (if and only if it says how things are) and in which this is, in turn, meant to say that an elementary sentence is true if the thing named by the subject term has the property indicated by the predicate. Such a characterization would, of course, need to be expanded, if elementary sentences can be of other than subject-predicate structure but even with this modification, proposition (1) may seem to be compelling.
On this reading of the indefinability thesis, it opposes, in effect, only certain kinds of definitions and does not, in principle, rule out the possibility of defining truth in some other semantic terms. Thus, an adherent of (1) could, perhaps trivially, write:
(1a) p is true =Df p is not false.
Or he could proceed disjunctively:
(1b) p is true =Df p is either analytic or synthetic a priori or empirically true.
The two definitions are certainly formally unobjectionable, even if they seem to be of little explanatory value. It might, however, be argued that (1a) and (1b) are, philosophically speaking, not proper definitions because they employ the notions of falsehood, analyticity, the synthetic a priori, and empirical truth. And these, in turn, it might then be said, implicitly presuppose the notion of truth, as is certainly suggested by the term “empirical truth.” But how is one to determine whether one concept implicitly presupposes another one? The question has, unfortunately, no technical answer. We have to rely here, instead, on our intuitions and on philosophical argumentation and these also allow someone like Frege to maintain that the notion of the meanings of sub-sentential components implicitly presupposes that of the truth and falsity of sentences. This is, indeed, what Frege implies when he writes in his 1918 summary of his philosophical achievements that his logic begins with the notions of a thought and proceeds from there directly to that of the truth of the thought and that he comes to the parts of a thought through a process of decomposition.[18] When we translate this into semantic terms, Frege seems to be saying that the notions of the sense and the reference of proper names and predicates have to be explained in terms of the more primitive notion of truth. Here truth is considered a semantic notion but one which cannot be explicated in terms of the semantic properties of sub-sentential components. Already in his Begriffsschrift Frege shows us how we are to conceive of this possibility. He treats the judgment-sign there as his first and most fundamental logical symbol and with it comes immediately the notion of a “judgeable content.” He then argues that we can recognize relations of inference between such contents and that we must assign just enough structure to judgeable contents to account for these relations. It is for this reason that he rejects a subject-predicate analysis since it ascribes, on his view, more structure to judgeable contents than is needed to explain their inference relations. This account prefigures Frege’s famous context principle which he made explicit in The Foundations of Arithmetic. It is often objected to such an account that it conflicts with the “principle of compositionality” according to which the meaning of a sentence is a function of the meanings of its parts. This principle must be true, it is said, if we are to explain how we are able to form and understand an indefinite number of new sentences. It is also said that Frege himself actually gave expression to the principle of compositionality in his essay “On Sense and Reference.” But such objections overlook that the question is not whether the principle of compositionality is true or not but whether it is basic or derivative. On the interpretation of Frege put forward, the principle can be said to follow from more basic assumptions, for once we have decomposed a sentence in the manner suggested, we can, of course, also describe its meaning as a function of the elements into which we have decomposed it. But because the principle is derivative, the attempt to define truth in terms of a sentence in terms of the meanings of its components would be philosophically speaking circular though, perhaps, technically feasible. In other words, such an attempted definition would have the same shortcomings as the suggested definitions (1a) and (1b)
Proposition (1) was meant to spell out one interpretation of the thesis that truth is indefinable. There is, however, another and more radical reading of that thesis which is also advanced by Frege. That this is so has, however, been mostly overlooked by the interpreters. We can summarize this second reading of the indefinability thesis in the following proposition.
(2) The concept of truth cannot be defined because it is presupposed in the use of any
attempted definition.
We can describe the difference between (1) and (2) as follows: according to (1) the concept of truth is semantically prior to other semantic concepts and according to (2) the concept is pragmatically prior to other semantic concepts. These two claims are, of course, not incompatible and so Frege is able to subscribe to both of them. Frege accepts (2) because of his belief that judgment (or, linguistically speaking, assertion) is the most basic logical notion. The concept of truth is, in turn, contained in our practice of judgment or assertion. To judge (or assert) that p is to judge (or assert) that p is true. So, consider any attempted definition of truth, that is consider any formula of the form
(2a) P is true =Df X.
Now, before we can judge in a particular case that P is true, we must ask ourselves whether X. If we can assert X, then we are also entitled to assert that P is true. But in asserting X, we are already presupposing that we have a concept of truth and so the concept of truth is presupposed in our attempt to explain what it means to say that P is true. But does Frege’s argument not founder on the distinction between object- and meta-language? All he seems to be proving is that in order to define a notion of truth in our object language we must already be able to employ it in our meta-language? And that is surely not incompatible with advancing a meta-linguistic truth-definition for sentences in the object language. That is, precisely, what Tarski has shown. Frege knows, of course, no such distinction as that between object- and meta-language and for him this objection is, in consequence, hardly decisive. But Frege’s argument must, in any case, be of concern to anyone who holds that it is unnecessary to maintain a separation of object- and meta-language and that the concept of truth must be definable within the language in which it is being used (and Hintikka, we will see, is one of these.)
Frege’s considerations in support of (2) come close here to a third interpretation of the thesis that truth is indefinable. For we may also mean by that claim:
(3) The concept of truth is not a semantic notion at all and can therefore not be defined
in semantic terms.
Frege’s views come close to this, if we take proposition (2) to imply that truth is really a pragmatic notion, or rather that truth is a notion that needs to be explained in terms of and, perhaps, reduced to pragmatic notions. There are passages in which Frege suggests that conclusion, such as the intriguing note “My Logical Insights” in which he argues that we have a concept of truth only because our language is imperfect and that we would need no such concept in a logically adequate language. His reason is that the concept of truth is merely a device for trying to make the assertive function of language explicit. In a perfect language we would be able to do this by just making judgments.[19] It is sometimes supposed that Frege was here advancing a redundancy theory of truth according to which the concept of truth is defined as:
(3a) P is true =Df P.
But the redundancy theory is best considered in conjunction with another a forth interpretation of the thesis that truth is indefinable. Frege’s assertion in “My Logical Insights” that the concept of truth is without (semantic) content is rather to be understood as akin to the pragmatist assertion that truth is to be understood as usefulness and to the logical positivists’ desire to replace the notion altogether by that of verification/falsification.
There is, in addition, a second and quite different take on proposition (3) according to which truth is neither a semantic nor a pragmatic but an ontological notion and that the concept is for this reason not definable in semantic terms. This is a view first put forward in the writings of the early Moore and the early Russell at the turn of last century when they were breaking with their own idealist past. It is a view, moreover, which was later taken up for different reasons by Martin Heidegger. Moore and Russell argued in their revolt against idealism that the semantic conception of truth – and more specifically the correspondence conception of truth – leads eventually and inevitably to the monistic idealism proposed by Bradley. We need not concern ourselves here with the question how compelling these objections are, but their consequence is that Moore1 and Russell1 take judgments (or propositions) to be constituents of reality not things that are about or correspond to reality and thus stand apart from it. Their judgments (or propositions) are, in other and more familiar words, states of affairs. Truth and falsity are accordingly for them simple and semantically indefinable properties of such states of affairs. To say that a state of affairs is true means as much as that it is there in the world and that it can be apprehended and known. Russell makes the point most brutally by saying that some propositions are true and others false just as some roses are red and others white. Logically, he argues, there is nothing more to be said. This conception of truth is as close as it can get to Heidegger’s concept of truth as unhiddenness, a notion much derided by analytic philosophers. For Heidegger, as for the early Moore and Russell, propositional truth is an uninteresting and derivative concept. It presupposes that there be something (which Heidegger calls “Being itself”) that is presented to us or reveals itself to us. And this unhiddenness, this truth of Being, is the fundamental meaning of truth. What distinguishes Heidegger from Moore1 and Russell1 is simply that for the former the truth of Being is historical in that different things reveal or fail to reveal themselves over time. For Moore1 and Russell1, on the other hand, truth and falsity are timeless characteristics of equally timeless judgments or propositions. On both understandings, however, truth turns out to be an ontological characteristic rather than a semantic one and the concept of truth can, for that reason, not be explicated in semantic terms. It should be added that Frege, too, comes close to such unorthodox ways of speaking about truth. This manifests itself in the doctrine of truth-values according to which the truth and falsity of a sentences consists in it referring to “the True” or “the False.” These truth-values are clearly not semantic in character and the semantic notion of truth is thus also for Frege derivative; primary is for him rather an ontology of value-objects that has affinities to Neo-Kantian value-theory.
There is still a forth, “Kantian” reading of the claim that truth is indefinable and it may well be that this is the reading that Frege was also groping for:
(4) The concept of truth may or may not be formally definable but such a definition can tell
us in principle nothing about the relation of our language to the world.
This view was first expressed by Kant who writes in The Critique of Pure Reason that we may take “the nominal definition of truth, that it is the agreement of knowledge with its object” for granted. (A58; my emphasis) But this is admission is for Kant of limited philosophical significance since truth substantively concerns the content of the knowledge and “it is impossible, and indeed absurd, to ask for a general test of the truth of such content.” (A59) It follows that “a sufficient and at the same time general criterion of truth cannot possibly be given” (Ibid.) The purely logical criterion of truth is only formal or negative in character. “[F]urther than this logic cannot go.” (A60) Hence, “no one can venture with the help of logic alone to judge regarding objects, or to make any assertion.” (Ibid.) The gist of this is for Kant that we may be justified in saying that the sentence “S is P” is true if and only if S has the property P, but this can tell us nothing about whether there (metaphysically) exists an object S and a property P. We can speak, in other words, of the correspondence of a sentence to appearance; but this reveals nothing about the constitution of the thing in itself. This may, indeed, be close to the Wittgenstein’s views, as Hintikka seems to recognize. For in the same period in which Wittgenstein expresses the “Kantian” view that it is impossible to describe a fact corresponding to a sentence without repeating that sentence, he also asserts in the Tractatus that “[a] proposition can be true or false only in virtue of being a picture of reality.” (TLP, 4.06). I suspect, moreover, that Frege’s assertion that the concept of truth is unique and indefinable really points to the same conclusion. To state that conclusion more generally: while it may or may not be possible to give a satisfactory formal definition of the concept of truth, such a definition can in no way add to our philosophical understanding of reality. Or even more generally: a characterization of the semantics of a sentence P will not tell us anything more about the world than the original sentence P. There is, to put it a third way, no semantic super-knowledge which can take us beyond the knowledge we communicate in our non-semantic language. No metaphysical or ontological benefits are to be derived from a definition of truth. This has been understood even by some of those philosophers who have sought to construct a theory of truth for ordinary language. Thus, Davidson writes about his proposed theory of meaning and truth: “The theory reveals nothing new about the conditions under which an individual sentence is true; it does not make those conditions any clearer than the sentence itself does.” (TI, p. 25)
All in all, we are thus left with a series of doubts about Hintikka’s thesis that we can divide contemporary philosophers into two camps according to whether they believe truth to be definable or not. Such a classification may be too broad to capture the multiple and varying views that twentieth century philosophers have expressed concerning the concept of truth. Our examination also raises doubts about how useful it is to divide philosophers into those who engage in semantic theorizing and those who do not as well as doubts about distinguishing between thinkers who engage in metatheoretical reasoning and those who do not. It is, in any case, clear that the indefinability thesis does not imply that there can be no (other) semantic theorizing; even less does it imply that there can be no metatheoretical reasoning. It is the latter view we should identify with the doctrine of the universality of language and we can see now that this doctrine follows in no way from doubts about truth-definitions or about semantics. Van Heijenoort’s reconstruction of the ultimate presuppositions of twentieth-century logic is thus doubtful and so is Hintikka’s story about the ultimate presuppositions of twentieth-century philosophy.
- 5. Hintikka’s concern with the question of the definability of truth is (as I have said) both historically and theoretically motivated. On the theoretical side he proceeds from the assumption that it is possible to construct a theory of meaning as a theory of truth and that the latter will have to include a definition of the concept of truth. That conviction is indubitably shaped by Donald Davidson’s programmatic statement of these claims in his essay on “Meaning and Truth.” In retrospect we can see that this essay has had a formative influence on the course of analytic philosophy since 1967. For one thing it initiated a research program on which its author would work for decades to come. In elaborating that program in ever new ways Davidson also motivated a whole generation of other philosophers to expand, revise, or contest it. Hintikka is, no doubt, one of those who have taken up the challenge posed by that essay.
When we set aside the philosophical concerns that have subsequently accreted around “Meaning and Truth,” we will find that two fundamental ideas characterize Davidson’s essay. The first is that the meaning of a sentence is to be understood in terms of the conditions of its truth, the second that the notion of truth is to be explicated by means of application and extension of Tarski’s theory of truth for formalized languages. These were not obvious claims to make in 1967 nor was it obvious how they could be made together. For the first claim is, in effect, the outcome of a set of deliberations that began with a critique of the correspondence theory of truth whereas Tarski’s work was generally regarded as a modern reformulation of that theory. It was (as I have shown) precisely the critics of the correspondence theory of truth who insisted that the concept of the truth of a sentence is the fundamental semantic notion and that this notion cannot therefore be explicated in terms of the reference of the components of the sentence. Tarski’s model-theoretical definition appears therefore, at least at first sight, incompatible with this belief in the priority of the concept of truth. The complexity of Davidson’s undertaking is due to the fact that he sets out to maintain the priority claim while at the same time subscribing to a Tarski-style theory of truth. On the one hand he is committed to treating a theory of meaning as a theory of truth, on the other he also wants to agree with those philosophers who hold that “a satisfactory theory of meaning must give an account of how the meanings of sentences depend upon the meanings of words.” (TI, p. 17) Much of Davidson’s later work on the concept of truth results from the need to resolve the resulting tensions.
It is easy to forget today how provocative Davidson’s two theses must have originally sounded, for by now they have become almost a common place among analytic philosophers. They are, nevertheless, by no means obvious. In “Meaning and Truth” Davidson himself admits that there exists “a staggering list of difficulties and conundrums” for his theory. (TI, p. 35) For one thing, such a theory presupposes an understanding of the formal structure of the sentences to which it is applied. But Davidson has to admit that “we do not know the logical form of counterfactuals and subjunctive sentences; nor of sentences about probabilities and about causal relations; we have no good idea what the logical role of adverbs is, nor the role of attributive adjectives; we have no theory of mass terms like ‘fire’, ‘water’, and ‘snow’, nor for sentences about belief, perception, and intention, nor for verbs of action that imply purpose. And finally there are all the sentences that seem not to have truth-values at all.” (TI, p. 35f.) Davidson himself and others have since then sought to fill these gaps. For all that, it remains true that it is still an open issue whether a theory of meaning for ordinary language can be successfully executed as a theory of truth. For one thing, such a theory is not likely to tell us what the connections are between the assumed semantic properties of our sentences and their use. It may be easy, moreover, to explain the truth-conditions and the meaning of simple sentences like “snow is white,” but how about all those sentences we find in scientific, technical, and other kinds of theoretical writing? How about the propositions and formulas that make up sophisticated physical theories? How about meaning in the sense in which it is the concern of interpreters and translators of literary or philosophical texts? Could a theory of meaning conceived as a theory of truth make us understand Kant’s transcendental deduction any better? Do the dark lines of an Ezra Pound become more transparent by means of a theory of truth? Davidson’s hope in “Meaning of Truth” was, of course, that all these questions could be answered satisfactorily, but such answers are still not forthcoming.
There is another challenge that Davidson takes up in “Meaning and Truth.” For Tarski himself had always denied the possibility of extending his results to ordinary language. Davidson argues by contrast that this caution is unwarranted and that one can conceive of the possibility of a truth-theoretical account of the meaning for the sentences of ordinary language. He concludes his essay accordingly with the admission that “I have taken an optimistic and programmatic view of the possibilities for a formal characterization of a truth predicate for natural languages.” (TI, p. 35) Such an optimistic view would not have seemed plausible without Noam Chomsky’s work in transformational grammar. Commenting on the slightly later paper “Semantics for Natural Languages,” Davidson wrote, in fact, in 1984, that it urged, “that truth theories could provide a formal semantics for natural languages to match the sort of formal syntax linguists from Chomsky on have favored.” (TI, p. xv) It was the publication of Chomsky’s Syntactic Structures in 1963 that evidently prepared the way for Davidson’s undertaking. One forgets today the challenge that Chomsky’s work in linguistics together with Davidson’s programmatic statement in “Truth and Meaning” presented to the analytic philosophers of the nineteen sixties. The shock was not, however, due to a sudden disillusionment with the doctrine that language is a universal medium, as Hintikka’s account might suggest. It was due, rather, to the fact that most analytic philosophers had been convinced that ordinary language was too fluid a medium to allow for a formal characterization of its syntactic and semantic properties. The target of Davidson’s pointed remarks were, indeed, the ordinary language philosophers at Oxford and all those philosophers whose picture of language was shaped by the thought of Moore and, perhaps, even more importantly, by the work of the later Wittgenstein. It is not too much to say that Chomsky and Davidson together brought the dominance of ordinary language philosophy to an end and helped to make Wittgenstein a more contested figure in the field. This development had certainly been prepared by a number of factors. One of them was the shift of the center of analytic philosophy from Great Britain to the United States with its accompanying shift towards a more scientistic conception of philosophy. Another significant factor were changes in the philosophical curriculum both in England and in the United States. The younger generation of philosophers emerging in the nineteen sixties were more highly trained in symbolic logic than their teachers had been. They were familiar not only with the rudiments of the propositional and predicate calculus, but also learned set theory and metamathematics. They read Hilbert, Gödel, and Tarski. They were, in other words, catching up with the developments in symbolic logic in the first third of the century that van Heijenoort discusses in “Logic as Calculus and Logic as Language.” The ground was thus well prepared for the acceptance of Chomsky’s and Davidson’s program.
These observations help us to reflect more broadly on the development of philosophy in the twentieth century. They get us to understand that the most significant division in philosophy in this period is not, as Hintikka will have it, between those who consider language a universal medium and those who conceive it as a calculus; the most radical division is rather between those who think of philosophy on the model of the formal and natural sciences and those who do not; those who consider the objective of philosophy to be the generation of theories and those who do not, those who think that human reason can be fully formalized and those who do not; those who conceive of language as a formal system and those who do not. In sum, the fundamental division in twentieth-century philosophy is between, as we might say, the formalizers and the anti-formalizers. While the formalizers think of philosophy as a science, as aiming at the construction of theories, as a spelling out of the formal rules of human reason, as committed to the picture of language as a formal structure which can be analyzed by means of metatheoretical tools, the anti-formalizers speak of philosophy as a questioning, as describing, as phenomenology, as a concern with difference, or as deconstruction, and proceeding in this way they see philosophy as distinct from the sciences. These philosophers aim not at the generality of theories, but at the grasp of particularity, of the historically unique, of the distinctive. They understand that some parts of human reason may follow formal rules, but they are wary of generalizing that claim, they are wary of the very distinction between the formal and the substantive, they are wary most of all of supposing that rules govern everything. These anti-formalist philosophers are, finally, attuned to the loose play of language, observant of the ever-changing surfaces of that language and skeptical about postulating a fixed, underlying, logical deep structures. They are, in a word, wary of depth. They are, for that reason, interested not only in the regulated uses of language in science or for making statements, but just as much in informal, metaphorical, literary, poetical, and political uses of language.
The division of twentieth century philosophy along these two broad fronts is due to the peculiar historical condition of contemporary philosophizing. For all modern philosophy takes place in the shadow of the sciences. These sciences have absorbed some of the old philosophical concerns (about the constitution of matter, the origin of the world, about space and time, etc.) and their methods have generally proved effective and progressive. Hence, the inevitable question: what if anything is left for philosophical labor. If we are to look for “ultimate presuppositions” in our philosophizing, it is presumably to be found in this peculiar and historically unique condition of twentieth-century thought. In the face of these worries there arises the possibility of two very different understandings of the philosophical enterprise: one is the conviction that philosophy, too, can be given a scientific character and the other that philosophy lies in principle outside the scope of the scientific enterprise. Both conceptions of philosophy permit a number of variations. Philosophy can be seen as a science in its own right (logical analysis) but also as something continuous with scientific investigation (positivism, naturalism) or, again, as laying the conceptual groundwork of the sciences (Neo-Kantianism, foundationalism). Philosophy, understood in the second way, thinks of itself as a radical questioning (skepticism, Heideggerian “piety of thought”), as a therapeutic undertaking (later Wittgenstein) as a practical pursuit, or as the recovery of a way of life. This conception can be accompanied by a radical critique of reason, science, and technology or it can involve the decision to bypass the results of science and to consider them as having no bearing on philosophy (Heidegger, Wittgenstein). It is in these multiple forms that the distinction between philosophical formalizers and the philosophical anti-formalists expresses itself. As such it cuts across the classification that Hintikka considers decisive and is quite different in character from the popular distinction between analytic and continental philosophy (which divides philosophical thought into two classes by pitting a methodological term against a geographic one). On the distinction here suggested, Frege and Russell belong into the same camp as Husserl, the French structuralists, Carnap, Chomsky, Davidson, and Hintikka. On the other side of the division we have thinkers as diverse as Moore and (the later) Wittgenstein, the Oxford ordinary language philosophers of the nineteen fifties, Heidegger, Foucault, Feyerabend, Rorty, and Derrida.
Admittedly, if this marks a fundamental division in twentieth-century philosophy, it still remains to write its history and until this is done, the classification has at best a suggestive and polemical function. A history of twentieth-century philosophy written in terms of division between formalizers and anti-formalizing thinkers will, of course, have plenty of reasons for dissolving the apparent dualism into a scheme involving multiple and intersecting divisions. Everything I have said in criticism of Hintikka’s dualistic scheme will no doubt apply. But in whatever way such a history of philosophy will be written, it will not be based on the divisions suggested by Hintikka and van Heijenoort.
- It should be evident by now that Davidson’s essay on “Meaning and Truth” and van Heijenoort’s note on “Logic as Calculus and Logic as Calculus” were symptoms, expressions, and motivating forces of a reorientation in philosophy which began around the time of the publication of these two pieces and that Hintikka, as editor of Synthese, played a seminal role is helping to bring about that reorientation. His own theoretical concern with questions of language, meaning, and truth has to be considered, thus, part of that sea change.
It would certainly be difficult to imagine Hintikka’s writings on these topics without that assumption. His guiding conviction that a theory of meaning must be conceived as a theory of truth and his resulting pre-occupation with the definition of truth reflects directly the programmatic statements of Davidson’s essay. But this does not mean that Hintikka ever intended to adopt Davidson’s program in its entirety. Where Davidson sees himself as applying Tarski-style truth-theory to ordinary language, Hintikka remains largely critical of such an approach. At first sight, Hintikka writes, “Tarski might seem to have aided and abetted theorists of language as calculus.” But since he thought that a truth-definition for one language can be given only in a stronger metalanguage and since he considered the actual working language as our highest metalanguage, it followed for him that “in the case that really matters philosophically, truth-definitions are impossible. In this sense, truth is literally ineffable, and the universalists have won.” (HI, p. 13) Davidson, it turns out, had also been somewhat ambiguous on this matter. In characterizing his project to of a truth-theoretical account of meaning, he had written that “it is assumed that the language for which truth is being characterized is part of the language used and understood by the characterizer.” (TI, p. 25) This suggests that, like Hintikka, he wanted to lift Tarski’s restriction that the concept of truth is definable for a language only within another language. But Davidson had continued immediately: “Under these circumstances, the framer of a theory will as a matter of course avail himself when he can of the built-in convenience of a metalanguage with a sentence guaranteed equivalent to each sentence in the object language.” (Ibid.) This leaves open the question whether the metalanguage has to be, in Tarski’s sense, “essentially richer” than the object language. Davidson notes that there were two reasons for Tarski’s pessimism about the possibility of giving a truth-definition for our actual working language. The first was that “the universal character of natural languages leads to contradiction and the second that these languages are “too confused and amorphous to permit the direct application of formal methods.” (TI, p. 28) Though Davidson grants that he has no “serious answer” to the first point, he is confident that the claim that natural languages are truly universal is suspect, now we know such universality leads to paradox. (TI, p. 29) His argument is unfortunately vitiated here by a confusion, for Tarski’s claim was, of course, not that natural languages lead to paradoxes, but that the attempt to define truth for such language in that very language leads to paradoxes and nothing that Davidson says contradicts that possibility. Davidson counters Tarski’s second objection, that natural language is too amorphous for the application of formal methods to it by arguing that we can always construct a formal language that approximates to our ordinary idiom. “Philosophers have long been at work of applying theory to ordinary language by the device of matching sentences in the vernacular with sentences for which they have a theory.” (TI, p. 29) In making that point Davidson refers once again to “recent work by Chomsky and others.” (TI, p. 30)
In contrast top Davidson, Hintikka holds out no hope for a Tarski-style truth-definition for ordinary language. He is convinced, rather, that “[r]esults like Gödel’s and Tarski’s in fact constitute the hard core of any rational basis of the overall ineffability thesis.” (HI, p. 17) He proposes, therefore, to rethink the problem of a definition and theory of truth by departing from Tarski’s model. His first step in this direction is to reconstruct the theory of quantification we have inherited from Frege. For Hintikka the classical, Fregean account is problematic because in any formula containing more than one quantifier, a quantifier R within the scope of another quantifier Q will always be considered dependent on Q. As Hintikka says of the classical theory: “In that notation, each quantifier is associated with a segment of the formula as its scope. It is then required that these scopes are ordered, that is, that the scopes of the two different quantifiers must either be exclusive or else that the scope of one is included in the scope of the other.” (HI, p. 49) This makes it sounds as if the classical theory was based on a tautology according to which two quantifiers are always either exclusive or inclusive of each other’s scope. Hintikka adds therefore in explanation that he means to say that the scopes of two quantifiers may overlap only partially. Even that explanation is, however, open to misinterpretation. What Hintikka really intends is, however, made clear by the notational variation he introduces into quantification theory. Thus, in the standard formula:
(5) (Vx)(Vz)(Ey)(Eu) S[x,y,z,u]
the two existential quantifiers are both dependent on the two universal quantifiers preceding them.
Hintikka adds to the standard quantifier notation a slash operator notation that exempts the quantifier immediately preceding the slash from dependence on the quantifier that immediately follows the slash. Thus, in:
(6) (Vx) (Vz) (Ey/Vz) (Eu/Vx) S[x,y,z,u]
the first existential quantifier depends on (Vx) but not on (Vz) while the second depends on (Vz) but not on (Vx). Hintikka calls a logic including such a slash operator “an independence friendly logic” and argues that the exclusion of such a device from classical quantification theory is arbitrary and that the use of generality in ordinary language corresponds, in fact, more closely to IF logic than to classical quantification theory.
He also writes: “Interpretationally, IF logic does not mark a single step beyond ordinary first-order logic, and notationally it can be considered merely a liberated variant of the same logic.” (HI, p. 54) That is, however, a modest understatement of the differences. For one of the characteristics of IF first-order logic is that its valid formulas are not recursively enumerable – a fact that may well explain logicians’ preference for classical quantification theory. Another is that IF logic violates the principle of compositionality, that is, we can no longer assume that the semantic interpretation of a complex expression can be determined from the semantic interpretation of its component expressions and the order of their composition. It follows from this that we will also find it impossible to construct a Tarski-style truth-definition for such a logic. None of this worries Hintikka, for he is convinced that IF logic possesses certain decisive advantages over classical quantificational logic. The most important of these is that there exists “a kinship of IF languages and natural languages.” (HI, p. 86) There is, as Hintikka also writes,” important similarity between extended IF first-order logic and our Sprachlogik.” (HI, p. 87) I will have to come back to this claim, but it is in any case obvious what role it plays in Hintikka’s reasoning. For if we are to show that a theory of meaning for ordinary language can be conceived as a theory of truth, we can make that claim plausible only by constructing a theory of truth for a logical notation that possesses “important similarity” to ordinary language.
That a Tarski-style truth theory cannot be constructed for IF logic may, at first, seem an obstacle to this line of reasoning. For the question must then be asked what other procedures there might be for defining the concept of truth. Hintikka argues that while Tarski-style truth theory is unavailable, we can apply a game-theoretical semantics of the sort that he first developed in The Game of Language and that we can do so without any changes in IF logic.[20] What is more, in IF logic the game-theoretical truth-conditions of any first-order sentence S can be expressed by a second-order sentence S* in that logic. In order to turn this observation into an actual truth-definition, we must first apply the technique of Gödel numbering to the sentences of our logic (which requires that IF contains, at least, elementary arithmetic); we can then formulate a truth-predicate Tr in IF such that
(7) Tr(┌S┐) ↔ S
where ┌S┐is the Gödel number of S. Hintikka’s goal is then to find a finite conjunction of sentences containing Tr(x) that together can be taken to provide an implicit characterization of the concept of truth for IF first-order sentences. He proceeds to show how this is to be done and concludes that he has, thus, established that the concept of truth can be defined in IF itself (and not just in a metalanguage of IF). With this, he thinks, he is in a position to show that at least one of the requirements for an ideal semantic theory for a language L can be satisfied, namely that “[t]he truth-definition should be formulated in L itself, not in some metalanguage with a stronger and hence presumably more problematic semantics.” (HI, p. 46) Since Hintikka’s ultimate goal is to convince us that it he can provide a truth-definition for ordinary language and since ordinary language will, in a way, be the highest metalanguage, we can see how important it is for Hintikka to show that one can define truth for a language L in L itself. But this argumentation exposes him, as he is aware, of the danger that a form of the liar paradox can be reconstructed as well. Thus, we can show that there is a numeral ┌g┐which is the Gödel number of the sentence
(8) ~T(┌g┐).
This sentence says therefore roughly of itself that it is false. But here a decisive aspect of IF logic comes in. It is that the law of excluded middle holds only in the fragment of such a logic that consists of the corresponding ordinary first-order language. A contradiction follows from (8) only by assuming that it must be either true or false. But in IF logic it need not be either and thus no contradiction can be derived. While there are other logics in which truth-value gaps or third truth-values lead to violations of the law of excluded middle, Hintikka considers IF logic distinctive in that the failure of the law of excluded middle in it “is a consequence of the eminently natural basic assumptions of the entire theory.” (HI, p. 61)
The failure of the principle of excluded middle in IF logic is, in particular, reflected in the way that negation works in such a logic. Here, Hintikka sees once again affinities between IF logic and natural usages. Extended first-order IF languages, he argues, offer “an interesting novel framework for analyzing the behavior of negation in natural languages.” (HI, p. 88) He thinks, in particular, that they can explain the difference between verbal and sentential negation in English. But he is forced to admit, in the end, that the analogy is only loose (and therefore, presumably, also inconclusive). For he is forced to admit that “the overt facts of the grammar of negation in English are too complex to be subject to any correlation with what happens in logical languages.” (HI, p. 89) Still, he is confident he has shown that “the main theoretical underpinning of the idea of the ineffability of semantics is eliminated if truth is defined in a suitable language, approximating the logical power of natural languages.” (HI, p. 93)
Hintikka is thus confident that he has shown us how truth can be technically defined in a formal symbolism which approximates ordinary language and he concludes that his work bears thus directly on the philosophical problem of truth. “What we have here is an interesting example of how apparently technical results can have truly striking consequences for the fundamental assumptions of entire philosophical traditions.” (HI, p. 7) The remark leaves us, however, with two questions: How strong are the technical results? And how significant are the philosophical implications? As far as the first question is concerned, we must note that Hintikka’s Gödelized truth-theory cannot entirely avoid making a distinction between object- and meta-language for the assignment of Gödel numbers to expressions in the symbolism must itself be carried through outside the symbolism. And in this outside language we must have such metalinguistic expressions as “sentence,” “quantifier,” “variable,” etc., at our command. We must, moreover, be able to say in this metalanguage that the Gödel number of sentence S is n and such sentences will have, in turn, truth-conditions which, however, are not captured by the proposed definition. The Gödelization of our symbolic notation is, moreover, a mere thought experiment that proves too complicated to be carried out in actual practice. On practical grounds then we find ourselves forced to employ the metalinguistic idiom that Hintikka’s theory is meant to circumvent. It is in this in this metalanguage that we will actually express the truth-theory of our symbolism and this is evident from Hintikka’s own words. For when he sets out to explain to us his theory of truth for IF logic he has to fall back on ordinary English to do so. But once we have granted that much, we are back to Tarski’s observation that our truth-theory will then be a theory only for the notation which is the object of our study and not a theory of truth for the language in which we are speaking. Hintikka will, presumably respond to this by arguing that ordinary language, the language in which we are speaking can itself be conceived as an IF logic. But this he has, of course, not yet shown. On the contrary, he has been forced to admit that the similarity between ordinary language and IF logic is only an approximate one. And even that result holds only for a carefully chosen fragment of ordinary language. Hintikka has, in fact, made no step beyond Davidson’s expression of hope that it will eventually become possible to describe the logical structure of all the formations of ordinary language. What is more, he has also given us no further reasons beyond those advanced by Davidson to think that a comprehensive theory of meaning for ordinary language can be developed as a theory of truth. Hintikka has certainly not shown to us how ordinary language could be Gödelized and how a truth-predicate could be defined for that language.
All these concerns do, however, not yet touch the fundamental philosophical issue at stake in this discussion. Even if Hintikka succeeded in ironing out the technical difficulties of a comprehensive theory of truth for ordinary language, he would in no way be settling the broader questions concerning the notion of truth that have motivated philosophers since Kant. For the ultimate philosophical concern is, as we have seen, with the relation of language to the world. This relation, the philosophers have concluded, reveals itself in the way we speak. In other words, in speaking we speak about the world. No theory of truth can tell us more about the world than our ordinary speaking does. The sentence “snow is white” tells us no less about the world than the sentence that “the sentence ‘snow is white’ is true.” No additional understanding of the world is achieved when we consider the “semantics” of that sentence. In other words, if we want to understand the physical universe, we should consult the words of the physicists not the analyses of the semanticists. There is nothing the semanticists can add to physicists’ story. A theory of truth will certainly not help us in any way with understanding the universe any better. This may appear to be a trivial point, but philosophers from Russell onwards have thought that we can determine what there is by considering the logical structure of our sentences. In opposition to such armchair science we should be firm in concluding that what we mean is contained in our words that we do not, in general, require saying also what we mean with our words. That would, in any case, be futile for in trying to say what we mean we would, once again, have to rely on the meaning of what we are saying to be contained in our words. It is this discovery that Kant sought to embody in the conclusion that truth may be nominally definable but that for all that we lack a general and sure criterion of truth. It is this which lies at the heart of the thesis that “truth is indefinable” which again and again has motivated philosophers since Kant though not always in a manner that is recognizably Kantian. Such a philosophical concern with truth does, however, not stand on its own; it is not itself the ultimate presupposition of twentieth century philosophy. It arises rather from the broader question whether we can step away from or outside the forms through which we understand and describe the world and describe and analyze these structures. The dream of the formalizers in philosophy has always been that of a science behind all science; but that may be, just as the anti-formalizers have always said, merely a dream.
Notes
[1] Kluwer, Dordrecht 1997. Herafter cited as HI.
[2] Jean van Heijenoort, “Logic as Calculus and Logic as Language,” Synthese, 17, 1967, pp. 324-330. Here abbreviated as “H.”
[3] The omission is hardly accidental as one can see from van Heijenoort’s From Frege to Gödel, Harvard University Press, Cambridge, Mass 1967 where Tarski also given short shrift. It is unclear from this text whether van Heijenoort intended to downplay Tarski’s significance or whether he considered him to belong to a later phase of the history of logic.
[4] Hans Sluga, “Truth before Tarski,” in Afred Tarski and the Vienna Circle. Austro-Polish Connections in Logical Empiricism, ed. by Jan Woleński and Eckehart Köhler, Kluwer, Dordrecht 1999.
[5] Gottlob Frege, Begriffsschrift, transl. by Michael Beaney, in Michael Beaney, The Frege Reader, Oxford 1997 hereaftter cited as “B.” In quoting this text I will each time give the original page number of the German original followed by the page number of the Beaney collection.
[6] Gottlob Frege, The Foundations of Arithmetic, transl. by J.L. Austin, hereafter cited as “F.”
[7] In the somewhat later essay “On the Scientific Justification of a Conceptual Notation” Frege compares ordinary language to the human hand which despite its “adaptability to the most diverse tasks” proves inadequate for some purposes. We then “build for ourselves artificial hands, tools for particular purposes.” These prove useful thro\ugh their “stiffness and inflexibility of parts” but lack the dexterity of the human hand.” (Gottlob Frege, Conceptual Notation and related articles, translated and edited by Terrell Ward Bynum, Oxford, The Clarendon Press 1972, p. 86) He also speaks of the invention of his notation as “an advance in technology” that makes possible “the construction of new instruments.” (p. 89)
[8] “In my first draft of a formula language I was misled by the example of ordinary language into constructing judgments out of subject and predicate. But I soon convinced myself that this was an obstacle to my particular goal and only led to useless prolixity.” (B, p. 4; 54)
[9] Gottlob Frege, “On the Scientific Justification of a Conceptual Notation,” in Gottlob Frege, Conceptual Notation, loc. cit., p. 83f.
[10] Gottlob Frege, “Logik,” Nachgelassene Schriften, ed. by Hans Hermes, et al, Felix Meiner, Hamburg 1969, p. 3.
[11] Readers have generally overlooked the Hegelian, dialectic tone of van Heijenoort’s account in which the Frege-Russell conception of logic as language represents the thesis and the calculus view the appropriate antithesis which together generate the inevitable synthesis: “During the ‘twenties,” so van Heijenoort concludes his story of the two conceptions of logic,” the work of Skolem, Herbrand, and Gödel produced an amalgamation and also a dé passement of these two trends.” (H, p. 328)
[12] Jamie Tappenden, “Metatheory and Mathematical Practice in Frege,” Philosophical Topics, vol. 29, 1997, pp. 213-264; also Tappenden, “Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?,” in Notre Dame Journal of Formal Logic, vol. 41, 2000.
[13] Ludwig Wittgenstein, Tractatus Logico-Philosophicus, transl. By D.F. Pears and B.F. McGuinness, with an introduction by Bertrand Russell, Routledge, London 1961. The text will be cited as “T”; references to Wittgenstein’s own words will indicate the numbered propositions.
[14] Gottlob, Frege, Wissenschaftlicher Briefwechsel, ed. by Gottfried Gabriel, et al., Felix Meiner, Hamburg 1976, p. 161.
[15] Heinrich Scholz and Friedrich Bachmann, “Der wissenschaftliche Nachlaß von Gottlob Frege,” Actes du Congrès International de Philosophie Scientific, Paris 1936, p. 29.
[16] See van Heijenoort’s and Dreben’s joint “Introductory note to 1929a, 1930, and 1930a,” in Kurt Gödel, Collected Works, vol. 1, ed. by Solomon Feferman et al, Oxford University Press, Oxford 1986, p. 44.
[17] I discuss the Fregean version of the doctrine that truth is indefinable and its place in Frege’s overall reflections on the concept of truth in “Frege and the Indefinability of Truth,” in From Frege to Wittgenstein, ed. by Erich Reck, Oxford University Press, Oxford 2000. A revised version of my argument can be found in “Freges These von der Undefinierbarkeit der Wahrheit,” in ed. by Dirk Greimann, 2003.
[18] Gottlob Frege, “Notes for Ludwig Darmstaedter,” in Beaney, loc. cit., p. 362.
[19] The word ‘true’ “allows what corresponds to the assertoric force to assume the form of a contribution to the thought.” But this attempt fails. That we cannot do without the notion of truth “is due to the imperfections of language.” (Gottlob Frege, “My Basic Logical Insights,” Beaney, loc. cit., p. 323)
[20] The Game of Language, Reidel, Dordrecht 1983.
