The question of truth in early twentieth century philosophy and logic
Hans Sluga
- Heidegger on truth
In an essay on Plato’s doctrine of truth Martin Heidegger argued in 1929 that the Republic marks a decisive point in the evolution of our concept of truth.[1] According to him, that text jettisons an earlier Greek understanding of truth as “the unhiddenness of being,” and conceives of it, instead, as correctness. According to Heidegger, this is made evident in the allegory of the cave which depicts things in the temporal world as reflections of eternal forms and thereby speaks of truth as the similarity of something in the world to something else.
This observation is of interest in what it tells us about both Plato and Heidegger. It is common today to ascribe the first statement of the correspondence notion of truth to Aristotle who writes: “A truth is a statement of that which is that it is or of that which is not that it is not.”[2] What is determines, on this account, the correctness and thereby the truth of the statement. Statement and fact must be adequate to each other. We speak accordingly now of Aristotle’s correspondence theory of truth – even though the notion of correspondence itself appears only in medieval reworkings of the original formula. Heidegger reminds us in his essay that the Aristotelian idea is already anticipated by Plato. And in this observation we may agree with him though it seems more obvious to refer to Plato’s Sophist rather than the Republic as support for it.
Heidegger’s observation is, however, not made in the spirit of detached scholarship, it is meant to lead rather to a critique of the Platonic-Aristotelian tradition and to a resurrection of the concept that preceded it. According to this earlier understanding, truth belongs to the object of knowledge itself, to “being” in Heidegger’s terminology, and not to a representing idea or proposition. The original concept of truth is, moreover, non-relational in character: that something is true does not mean that it stands in a relation, is correlated, or corresponds to something else. Truth in the original sense is, rather, intrinsic to whatever is called true. Thirdly, truth in this sense is simple and indefinable. It shows itself for what it is and is recognized as such. Its characterization as unhiddenness is therefore not to be taken as definitory but as an explication of the literal meaning of aletheia, the Greek word for truth.
On Heidegger’s account, the Platonic-Aristotelian notion of truth has determined large parts of the history of philosophy since then. Critical reflection on that concept must therefore lead to a radical rethinking of major rethinking of Western philosophy.
- Where things themselves had previously been called true, such talk comes from Plato onwards to be considered secondary and and metaphorical. The statement, the thought, or the idea (in the psychological sense of the world) will from now on be considered the primary locus of truth. With this the notion o f representation proves crucial to philosophy and the concern with language as the medium of conceptual representation.
- Questions arise now about how statements, thoughts, or ideas convey meaning, how they succeed or fail in corresponding to reality, how they are to be analyzed, what parts they have and how these hang together. These are from Aristotle onwards the concerns of a new undertaking that goes under the name of “logic.” The new discipline is closely related to the enterprise of metaphysics which explores reality by way of an investigation of the forms and constituents of statements, thoughts, and ideas. As a third discipline, there comes about the examination of the subject, soul, or self (philosophical psychology) since it is the subject which is said to represent the world by means of language and thought. Finally, questions are generated about the subject’s capacity for representing reality and its limitations. The fields of logic, metaphysics, philosophical psychology, and epistemology all have their roots in Plato’s conception of truth as correctness.
- The most pressing issue turns out to be, from now on, the distinction between the
objective and the subjective. Truth had initially been conceived as belonging to
being itself. But when it comes to be understood as correctness or
correspondence, when it is described as representation of what is, we are forced to
distinguish sharply between the object and the subject of knowledge. All
representation requires a medium: a “language” of words, thoughts, or ideas. But
the representational function of any such medium is (so it seems) always dependent
on a human subject who knows how to employ and interpret it correctly. It seems
that from now truth must always be thought of as relative to a language and relative
to the users of such a language. But how can truth be objective if it depend in this
manner on a human subject?
Heidegger’s observations, though admittedly speculative, deserve the attention of the historian of the concept of truth, though perhaps for reasons other than the ones that he had in mind. Heidegger claims that our whole philosophical understanding, how we conceive of logic, metaphysics, psychology, and epistemology depends on our concept of truth. Of all the philosophical disciplines logic is, of course, most intimately connected with the concept of truth. If Heidegger is right, we must expect radical changes in philosophy and, most specifically, radical changes in logic to reflect and be based on changes in our concept of truth. One such change and, indeed, the most radical change in the history of logic occurred in the second half of the nineteenth century. For the emergence, in this period, of a completely formalized (mathematical or symbolic)logic marked a complete break with the Platonic-Aristotelian conception of logic which had dominated till then. The question for the historian is why this break occurred at all and what conceptual shifts made it possible. The question is what intellectual resources the new logicians had that their predecessors lacked. Or alternatively, what philosophical commitments held these predecessors within the logical tradition but were no longer maintained by the new logicians. It is, of course, possible that the decisive difference lies only in the technical and mathematical skills of the new logicians. But Heidegger offers us a deeper and more interesting explanation: that classical logic was held in place by the Platonic-Aristotelian conception of truth and that the appearance of formalized logic must therefore be due to a break with that conception of truth and the emergence of new modes of thinking about the concept of truth.
This claim can, indeed, be made plausible and to this effect it is is helpful to compare and contrast the thought of Bernard Bolzano with that of Gottlob Frege: two outstanding logicians of their time, both experienced mathematicians and committed, so it seems, to a number of similar philosophical assumptions, but standing on different sides of the great divide in the history of logic. That they stand so close to each other yet end up with radically different conceptions of logic, may allow us to pinpoint the decisive difference that separates a sophisticated exponent of the classical tradition from a revolutionary innovator in logic. Since both were accomplished mathematicians we can see, for instance, that the difference between the old and the new logicians was probably not just one of technical or mathematical skill. That they both were determined “antipsychologists” and both believed in the objectivity of logic and truth, makes it equally unlikely that the decisive difference between the old and the new logic can be defined in these philosophical terms. Bolzano and Frege differed, however, in one important respect: while the former adhered to the classical notion of truth, Frege declared truth to be something non-relational, simple, and indefinable. Given this crucial difference, there is then reason to think Heidegger right when he declared logic held in place by its underlying notion of truth.
- Bolzano on truth
The question of truth is, certainly, crucial to Bolzano. He writes in his Wissenschaftslehre that there exists “no textbook of logic in which ‘the concept of truth’ is not more or less talked about.” (Sect. 29, p. 125)[3] He understands his logic as a theory of science because a science is a “totality of truths of a certain kind” and his logic is meant to be concerned with “the rules by which we must proceed in this business of dividing the whole area of truth into separate sciences and in the writing of textbooks for each of them.” (Sect. 1, pp. 4 and 6)Surveying the different traditional uses of the term “true” he concludes:
The first and most distinctive is without doubt the one according to which one
understands by truth a certain characteristic propositions… a characteristic by
means of which they state something as it is. (Sect. 24, p. 108)
And he declares accordingly that “by a truth in itself I will understand any particular proposition which states something as it is, whether this proposition is actually thought or uttered by anyone or not.” A proposition is a truth in itself “whenever the object with which it deals really has the properties that it ascribes to it.” (Sect. 25, p. 112) With this we are within a traditional, propositional, relational, “correspondence” account of truth.
Bolzano is aware that among the varying conceptions of truth is the one that Heidegger has characterized as pre-Platonic. He acknowledges that the Greeks disagreed about the original meaning of aletheia and that writers such as Sextus Empiricus explain it “not as truth” (in Bolzano’s own sense) but as the unhidden, “to me lethon.” (Sect. 24, p. 111) He mentions the related scholastic formula that “verum et ens convertuntur,” but seeks to show that it agrees with his own conception of truth. The scholastics, he argues, chose their formula only to avoid the threat of subjectivism for they did not want to speak of the truth of sentences or assertions afraid that this would necessitate reference to the subject that utters such sentences or makes such assertions. (Sect. 27, p. 118) Bolzano draws finally also attention to the principle omne ens est verum asserted by some modern writers (Locke, Wolff, and Baumgarten among them) who sought to make “metaphysical truth” into a property of beings. But in agreement with Leibniz he dismisses such a conception, however, as “utterly useless and devoid of sense.” (Sect. 29, p. 143) He is, in any case, convinced, that such a use of the word “true” is merely figurative (uneigentlich) and “a mere abbreviation.” (Sect. 24, pp. 110 and 111)
He agrees rather with Malebranche according to whom “veritas
nil aliud est, quam relatio realis sive aequalitatis sive inaequalitatis.” (Sect. 27, p. 119) He balks, however, at calling this a correspondence theory of truth, writing:
I cannot omit the demand that one should indicate precisely what is meant to
be understood by the correspondence (Übereinstimmung) which is supposed to
obtain between ideas or propositions and their correlated objects. One can certainly
not imagine here an absolute identity or sameness. For propositions or ideas are not
absolutely the same as the objects to which they refer; nor are the properties of the
former also properties of the latter. (Sect. 29, p. 128)
It is not, for instance, the case that “the ideas of a house and a garden are related to each other as those objects, i.e., the house and the garden themselves.” (p. 129) Bolzano concludes that all such talk reduces substantively to his own preferred formula that a proposition is true if it states something as it is. More precisely, he declares, that “in every sentence there must be an object with which it deals (the subject) and also a certain something that is said of this object (the predicate). In a true sentence, moreover, that which is said of the object must really belong to it.” (Sect. 28, p. 122) This is, indeed, he says, also the opinion of Aristotle according to whom the expression “this belongs to that” means the same as “this can be truly said of that.” (p. 124)
Having adopted the classical relational and representational notion of truth Bolzano finds himself at once embroiled in the metaphysical and epistemic questions to which, according to Heidegger, this notion gives rise. The possible threat of subjectivism emanating from the classical notion constitutes, indeed, the major philosophical concern of his whole Wissenschaftslehre. The essential thing is for him to “separate the logical from all admixture of the psychological,” he writes in agreement with Herbart, to reveal “the judgment as no appearance in the mind, but as something objective.” (Sect. 21, p. 85) While he speaks of the proposition (Satz) as true or false, he insists that the terms must not be taken in its ordinary sense as meaning something we “propose” (setzen). If we were to take propositions in this sense, they would be dependent on human action and hence truth and falsity would be so dependent. Bolzano argues:
Through its derivation from the verb “to propose” the term “proposition” used
here suggests admittedly an action, a something proposed by someone (in other
words, something that is produced or altered in some way). But in the case of truths
in themselves one must ignore this. (Sect. 25, p. 114f.)
Similarly, when we declare that a proposition “says” or “states” something this must be taken as figurative (uneigentlich). When we speak of an assertion (Aussage) as true or false, that too must be considered a figurative expression. For in reality truth or falsity are not asserted. The peculiar outcome of this reasoning is that Bolzano can explain the supposedly non-figurative meaning of the word “true” only by means of figuratively used terms. “It is true that in some parts of logic it will be necessary to speak also of sentences in the until now ordinary sense; but in such places we will be in no difficult to make ourselves understood; for in most cases the context will be sufficient to show of which kind of proposition we are talking.” (Sect. 20, p. 82)
Since a proposition is not to be something produced by human action it cannot be identified with a sentence or a thought. Hobbes may believe that only verbal assertions can be considered true and false, that truth belongs to words and not to things, that therefore only beings capable of language can possess truths. But this is either “merely love for the absurd” or rests on a confusion between representations (Vorstellungen) and the words we use to indicate them. (Sect. 29, p. 144) A proposition is, in Bolzano’s competing view, rather “only the sense which a certain combination of words can express.” (Sect. 28, p. 121) Propositions, understood in this way, “have no real existence, i.e., they are not something that is in any particular place, at any particular time, or is in any other way something real.” (Sect. 25, p. 112) He writes accordingly that “the number of blossoms that were on a certain tree last spring is a statable, if unknown figure. Thus, the proposition which states this figure I call an objective truth, even if nobody knows it.” (Ibid.) In order to obviate the threat of subjectivism inherent in the classical conception of truth, Bolzano postulates thus a realm of objective entities that can remain true independent of any human cognitive acts, independent of the changing circumstances of the temporal world.
One might argue at this point that Bolzano could have avoided the resort to propositions in themselves, had he taken Hobbesian realism more seriously. While we do, indeed, say that a proposition remains true, even if no one entertains it, this need not lead us to postulate propositions in themselves. We might argue, rather, that the locution has only the force of a subjunctive conditional claim, that it says only “If someone were to assert proposition p at time t, he would be making a true assertion.” This interpretation is made more attractive by Bolzano’s own assumption that a true proposition says what there is and that what there is must be independent of the corresponding proposition. For this suggests the possibility of the existence of what there is (a fact) even when there is no corresponding proposition. Thus, we may hold that the blossoming tree last spring had, indeed, a specific number of flowers, but this does not require us to postulate an unthought true proposition corresponding to that fact. Bolzano seems here caught in an unnecessary duplication of entities. Bolzano would probably respond that propositions in themselves are still needed because a true proposition may speak of something that does not exist, as in “There are no unicorns,” or, to his Bolzano’s own words:
Objects that have no reality may still have characteristics which may be stated
in true propositions. The truth that there is no magnitude whose square = -1 would be
of this kind. (Sect. 29, 126)
This is, however, hardly a decisive objection. But to answer it would involve a solution to the problem what makes negative propositions and logical or mathematical propositions. One such solution can be found in Wittgenstein’s Tractatus. Whether or not we find it compelling, it is sufficient to show that one can address the problem without having to follow Bolzano into postulating both objective facts and propositions in themselves.
- Frege on truth
Those who are acquainted with both Bolzano’s and Frege’s writings have often been struck by their apparent affinities. These are, of course, not evident in the technical parts of Frege’s work. For Bolzano’s logic remains within the boundaries of the classical theory of the syllogism and contains neither a systematically developed, truth-functional, axiomatized theory of propositional logic, nor an adequate theory of quantification and a formally constructed predicate logic. But there are echoes of Bolzano’s views in Frege’s more philosophical writings and these are particularly apparent in his late essay “The Thought.” Frege published that article in 1919 when he was seventy years old and intended it obviously as a definitive final statement of his views. It summarizes, in fact, many of Frege’s earlier philosophical ideas but does so in words that appear particularly close to Bolzano’s formulations.
Like Bolzano’s Wissenschaftslehre, the essay begins with declaring the concept of truth fundamental to logic: “As the word ‘beautiful’ points the direction for aesthetics and ‘good’ that for ethics, so ‘true’ points the direction for logic… To discover truths is the task of all sciences; logic is concerned with recognizing the laws of truth.”[4] Frege goes on to denounce the idea that logic might be concerned with the psychological process of thinking. He insists that there is a boundary between psychology and logic which must not be breached. Like Bolzano in the early sections of the Wissenschaftslehre, he proceeds from this denunciation of psychologism to the question of the meaning of truth. “To begin with I want to draw the rough outlines of what I want to call true in this context.” He points out that truth is commonly asserted of “pictures, representations, propositions, and thoughts.” And he considers the possibility that in propositions as in pictures we speak of truth “insofar as there exists a correspondence between the picture and what it depicts.” But a correspondence, he insists, “can only be perfect if the corresponding things coincide and are, therefore, not distinct things at all.” (Ibid.) This is not what we expect in the case of propositions. Here we assume, at most, that idea and reality correspond only in certain respects. But, in this case, “we should have to inquire whether it were true that an idea and reality, perhaps, corresponded in the laid down respect. And then we should be confronted by a question of the same kind and the game would begin again.” (p. 344)
The affinity between Bolzano and Frege on all these points is, indeed, blatant. And there are still others that draw our attention. For both of them insist that we must distinguish between a sentence and its sense which Bolzano calls a proposition in itself and Frege a thought. The proposition in itself as well as the Fregean thought are objective and timeless. Frege writes, accordingly that thoughts are neither things of the external, physical world nor are they psychological representations: “The thought which we state in the Pythagorean theorem is atemporal, eternal, unchangeable.” (p. 361) When we are concerned with truth, we are according to him really concerned with the sense of sentences, not with verbal sounds. And when we speak of such a sense of a sentence as true this “cannot consist in the correspondence of this sense to something else; for otherwise the question of truth would repeat itself to infinity.” (p. 344)
There is still uncertainty among scholars over the question whether Frege was actually acquainted with Bolzano’s writings. Even though his debates with Benno Kerry and Alwin Korselt, two of Bolzano’s admirers, might have led him to read the Wissenschaftslehre, he makes no reference to it or its author. What sound like ideas akin to Bolzano’s in his work, may easily have been derived from Herbart (with whom Bolzano was also familiar) or from Lotze. The latter is certainly responsible for the Fregean doctrine of objective thoughts as well as for numerous other assumptions that Frege makes. It is, however, in any case a mistake to associate Bolzano and Frege too closely. Whatever ideas they shared must not obscure the fact that on at least one crucial point they take diametrically opposed stands.
This point is their actual conception of truth. That Bolzano and Frege are here in discord becomes apparent when we turn once more to the beginning of “The Thought.” Having criticized the notion of correspondence there in terms reminiscent of Bolzano, Frege proceeds in his essay to a general attack on all attempts to define the notion of truth, arguing that for the same reasons as the correspondence theory of truth “every other attempt to define truth collapses too.” Any definition whatever of the concept of truth would have to say that a sentence is true, if and only if it has certain characteristics. But in this case “the question would always arise whether it were true that the characteristics were present. So one goes round in a circle.” The ultimate conclusion is then that “the content of the word ‘true’ is unique and undefinable.” (p. 344) It is obvious then that Frege would reject Bolzano’s characterization of truth just as much as any other.
Frege’s reasoning at this point has been subjected to repeated criticism. But what matters here for us is not initially whether his argument is compelling, it is rather what conclusions he reaches by it, what concept of truth he ends up with. There is certainly no doubt that the Fregean notion of truth is radically different from Bolzano’s. We have seen already that Bolzano’s view leads him to a peculiar duplication of entities such that for every temporal fact there must be a correlated atemporal true proposition in itself. Since Frege rejects any representational account of truth he is not drawn into a similar duplication of entities. For he does not assume that besides objectively real thoughts there exist also ontologically independent facts to which these thoughts may correspond. In “The Thought” he declares, instead: “A fact is a true thought.” (p. 359) And this
remark takes up an earlier statement from 1897 where he had said: “Thoughts are, for instance, natural laws, mathematical laws, historical facts.”[5] Facts are, on this view, not what thoughts are about but are themselves thoughts. While we tend to speak of thoughts as correlates of possible facts such correlates would, in Frege’s terms, be at best “representations” (Vorstellungen). But such representations are for Frege unsuitable as truth-bearers, not only because they are subjective but because they are strictly speaking incommunicable. Fregean thoughts, on the other hand, are not only truth-bearers they also constitute the world instead of being its representations. This idea is not without its attractions. For it is plausible to argue that the identity criteria of facts must be intensional. The fact that Venus is the morning star is surely different from the fact that Venus is the evening star. Facts can therefore, on Frege’s scheme, not be located at the level of reference where identity criteria are unfailingly extensional and the truth and falsity of thought cannot be explained by reference to a supposed correlation or lack of correlation between thoughts and facts. Frege’s claim that there are objective true thoughts which no one has grasped comes thus to the assertion that there are unknown facts. It follows that, unlike Bolzano, he does not engage in an unnecessary duplication of facts and correlated propositions in themselves. His doctrine of objectivity is, for that reason, more difficult to dislodge and requires something like Dummett’s antirealism as a counterposition.
Frege’s claim that the notion of truth is simple and indefinable makes its appearance at only one point in his publications: i.e., in the initial passage to the essay “The Thought” just discussed. For that reason it has often been ignored by his interpreters and where they have taken notice of it they have often dismissed Frege’s argument as insufficient or confused and, in the light of Tarski’s formal theory of truth, as dated or as a regrettable slip.
For all that the claim is of key importance to Frege’s whole conception of logic. Admittedly, Frege came to the doctrine only in 1897, after all the other elements of his thinking about truth and meaning were in place. The doctrine is however, in fact, the capstone of his whole construction. It is fully anticipated in his treatment of logic in the Begriffsschrift of 1879, in the context principle of 1884, in the claim that the distinction between objects and functions is categorial in nature, as well as in the distinction of sense and reference and the idea that the True and the False are objects referred to by declarative sentences.
Frege first formulated it explicitly in a lengthy but incomplete and hence unpublished piece of writing from 1897, which he simply entitled “Logic.” From the early 1880’s onwards he had repeatedly tried to describe his conception of logic in informal terms. The 1897 manuscript constituted one such effort and as such is the predecessor of his final attempt to make the philosophical implications of his ideas clear. Just like “The Thought,” the 1897 manuscript begins with a comparison between the terms good, beautiful, and true. While the term first characterizes the “goal” of ethics and the second that of aesthetics, “true” characterizes the goal of science. Even though all science is concerned with truth, logic is concerned with it in a unique manner. Like ethics, logic can be called a normative science. Logic is “the science of the most general laws of being true.” Frege continues:
It would now be in vain to try to make clearer by a definition what is to be
understood by “true.” If one were to say: “An idea is true if it corresponds to
reality”, nothing would be gained, for in order to apply it we would have to ask
in each case whether an idea corresponds to reality, in other words, whether it is
true that the idea corresponds to reality. One would have to presuppose what is to
be defined. The same would hold of any explanation of this form: “A is true if it
has such and such a property or if it stands in this or that relation to this or that.”…
Truth is obviously something so primordial and simple that a reduction to
something even simpler is impossible. We are therefore forced to illuminate what
is unique in our predicate through a comparison with hers.[6]
When Rudolf Carnap attended Frege’s lectures at Jena in the Winter of 1910, he records that Frege spoke of these same ideas in his class. While much of the course was devoted to an exposition of the formal machinery of the Begriffsschrift logic, Frege ever so often allowed himself more general observations of a philosophical kind. It was on one of these occasions when he said, according to Carnap:
Truth cannot be defined as “correspondence of an idea with reality”; for
something objective cannot be compared to something subjective. Truth cannot
be defined, analyzed, or reduced [to anything else]. It is something simple,
primordial.[7]
This is an intriguing argument but certainly also an incomplete one. It is intriguing, because Frege attacks here a traditional conception of truth that characterizes it as a correspondence between idea and reality and applies to it considerations concerning the relation between the subjective and the objective he had first formulated in The Foundations of Arithmetic. He had argued there that what is subjective cannot be communicated. In 1910 he seems to be saying that we would be able to communicate and compare ideas, if we could say of two of them entertained by two different persons that they both correspond to the same reality. This does surely not prove that truth is simple and primordial. It may show that we cannot conceive truth in terms of a relation between idea and reality, but it fails to establish the impossibility of any other definition. Why could we not, for instance, think of truth as a relation between an objective thought (in the Fregean sense) and reality? That argument is blocked, however, in the assumption of the 1897 manuscript that facts are true thoughts.
The significance Frege attached to the doctrine that truth is simple and indefinable comes out only in two posthumously published texts. In notes written for the historian of science Ludwig Darmstaedter, at about the time of the publication of “The Thought,” Frege summarized one of his points in the following words:
What is distinctive about my conception of logic is that I give primacy to the
content of the word ‘true’, and then immediately go on to introduce a thought
as that to which the question ‘Is it true?’ is in principle applicable. So I do not
begin with concepts and put them together to form a judgment; I come to the
parts of a thought by analyzing the thought.[8].
In consequence, truth must be considered a semantically primitive term. Other semantic notions may be explained by means of it, but since they all presuppose the notion of truth, they cannot, in turn, be used to explain it. The notes for Darmstaedter seek to show, for that reason, how by starting with the notion of truth as basic one can come to the notions of sense and reference, to the distinction of functions and objects, and to the other
fundamental notions of Fregean logic.
Frege’s second conclusion from the observation that truth is indefinable is even more radical. He draws it in a short note from 1915, entitled “My basic logical insights.” He observes there that “the ocean is salty” and “it is true that the ocean is Šsalty” assert the same thing and concludes that “the word ‘true’ has a sense that contributes nothing to the sense of a whole sentence in which it occurs as a predicate.” What the word seeks to make explicit is the force with which we assert propositions. “Thus, the word ‘true’ seems to make the impossible possible, namely, to make that which corresponds to the force a contribution to the thought itself.” The claim to truth is implicitly contained in all our assertions. To assert something means to assert that it is true. Frege can therefore write:
How is it then that this word “true”, though it seems devoid of content, cannot
be dispensed with? Would it not be possible, at least in laying the foundations of
logic, to avoid this word altogether? That we cannot do so is due to the imperfection
of language. If our language were logically more perfect we would perhaps have no
further need of logic, or we might read it off from the language.[9]
That does not mean that there is nothing at all to be said about the concept of truth. Frege’s conclusion is rather that nothing can be said about the concept in semantic terms, if that means reducing it to other, more primitive semantic notions. But there are still lots of other ways to speak about truth. We can speak about truth, above all, in epistemic terms. We can ask under what conditions we are justified in asserting something as true. This may lead us to say, for instance, that we take certain propositions as self-evidently true or that we regard them as true because we have made this or that empirical observation or because they are derivable in this or that manner from other established truths. We can, thus, speak of the ways in which the truth and falsity of one proposition depends on the truth or falsity of certain other propositions. In other words, we can explore inference relations between propositions. Frege suggests that the laws of logic, that is, the laws of our object-language propositional and quantificational logic, should be considered “laws of truth.” He does not mean by this that these laws contain the word “true” and say something about it. He means rather that those laws explicate our understanding of what “truth” means by showing us truth-relations between propositions.
Such considerations were the guiding principles of Frege’s construction of a radically new logic. We can see this clearly in his Begriffsschrift, the monograph in which he first laid out his new system of logic. The Begriffsschrift begins with an explicit reference to the concept of truth. In the first sentence of its preface Frege writes: “In apprehending a scientific truth we pass, as a rule, through various degrees of certainty.” And the notion of truth occurs, altogether, two more times in that same first paragraph. For Frege goes on to argue that over time propositions come to be more securely established “by being connected with other truths through chains of inferences” and he says that we can divide “all truths that require justification into two kinds,” those that can be proved “purely by means of logic” and those whose proof requires appeal to “facts of experience.”
Such remarks might lead one to expect a detailed examination of the concept of truth in the body of the text. But no such thing is provided. On the contrary, truth is hardly mentioned anymore in the rest of the Begriffsschrift and plays no role in the exposition of the new logic. When we consider the beginning of the preface once more with this in mind, we realize that even there Frege was really concerned not with truth but with the apprehension, justification, and interconnection of truths. As he leaves the concept of truth behind Frege can circumvent the logical, metaphysical, psychological, and epistemological discussions that preoccupy traditional logic books. The difference between the Fregean approach to logic and the traditional one becomes visible when one compares the _Begriffsschrift_ with such standard works as Christoph Sigwart’s or Wilhelm Wundt’s treatises on logic. Frege’s work is a short monograph of less than a hundred pages. Standard textbooks have hundreds of pages and consist of several volumes. The reason for Frege’s conciseness is that he has succeeded in excluding most of the topics that concern their authors. Beginning from the classical notion of truth they are driven into a labyrinth of logical, metaphysical, psychological, and epistemological questions from which there is no exit. Frege, by contrast, has set the classical notion of truth aside and can therefore construct an elegantly sparse account of his logic. It should be obvious that in this comparison, Bolzano’s Wissenschaftslehre with its four heavy volumes belongs on the side of the tradition.
The effects of Frege’s exclusionary approach to logic can be traced very precisely in his development of a propositional calculus. By treating truth as a simple, undefined notion he can isolate a logic in which the truth connections between whole propositions are the exclusive concern. Traditional questions about what makes the elementary propositions true are set aside as immaterial. This permits the construction of a simple truth-functional propositional calculus as a primary logic. In this logic only the relations between whole propositions are at stake, only the relations of inference and justification. By setting aside the question of how language might relate to the world, Frege’s attention is thus focused on a concern with how propositions relate to each other and freed from the elaborate scaffolding of traditional logic.
- Truth after Frege
Thirty years ago Jean van Heijenoort argued in a seminal article, entitled “Logic as Calculus and Logic as Language,” that none of the original authorities in the rise of mathematical logic had been interested in questions of semantics whose formal development was left to later writers such as Alfred Tarski. The “logicist” tradition, in which van Heijenoort included Frege, Russell, and the early Wittgenstein, had taken logic instead to be a universal language such that “nothing can be, or has to be, said outside the system.” For that reason “Frege never raises any metasystematic question” and “questions about the system are as absent from Principia Mathematica as they are from Frege’s work. Semantic notions are unknown.”[10]
Given Frege’s, Russell’s, and the early Wittgenstein’s extensive discussions of the notions of meaning and truth, such claims may strike us as problematic. But it is certainly true that none of the three constructed a formal theory of meaning or truth. What we have in their writings are at most incidental metatheoretical arguments or metasystematic claims that are disowned as soon as they are made. Van Heijenoort was therefore right in concluding that the logicist tradition shunned systematic work in syntactical and semantic theory. By setting aside such concerns and by setting aside at the same time all the traditional questions about truth and meaning Frege, Russell, and the early Wittgenstein succeeded in clearing the deck for an altogether new approach to the logic. Still, van Heijenoort overstates his case. It is not that semantic notions were unknown
to the logicist tradition but that it sought to rethink them from the ground up.
This required above all new ways of thinking about the concept of truth. Far from the history of truth coming to a halt in this period, the logicist tradition accelerated that history and transformed logic by transforming the ways we think about truth. Russell’s interest in formal logic developed at a time when he and G.E. Moore were rebelling against their own youthful adherence to Bradleyan idealism and that rebellion took the form of a break with the classical notion of truth. Bradley had argued for his idealism on the basis the correspondence theory. Moore’s and Russell’s revolt issued, therefore, from a challenge to that theory of truth. Against Bradley, they maintained that the concept of truth is a simple and undefinable term. Moore’s revolutionary essay on “The Nature of Judgment” begins by recalling that for Bradley “truth and falsehood depend on the relation of our ideas to reality.” He grants that “it is at first sight tempting to say that the truth of a proposition depends on its relation to reality,” but concludes that “truth cannot be defined by a reference to existence, but existence only by a reference to truth.” Moore concludes that “what kind of relation makes a proposition true, what false, cannot be further defined, but must be immediately recognized.”[11]
It took Moore and Russell several years to extract themselves from the complexities and absurdities of this doctrine. By 1910 they had both come back to some form of correspondence theory of truth. Russell concluded reluctantly at that time that “we are driven back to correspondence with fact as constituting the nature of truth.” But he added: “It remains to define what we mean by ‘fact’, and what is the nature of the correspondence which must subsist between belief and fact, in order that belief may be true.”[12] But his attempts to work out these problems led him to views far away from the traditional notion of truth. “Judging or believing is a certain complex unity of which a mind is a constituent,” he concluded at one point. “If the remaining constituents, taken in the order in which they occur in the belief, form a complex unity, then the belief is true; if not it is false.” (p. 128f.)
Wittgenstein became acquainted with these ideas when he arrived in Cambridge in 1911. His insistent probing forced Russell eventually to abandon much of the line of reasoning he had followed in attempting to define truth as correspondence. Wittgenstein convinced him, instead, of the viability of his own so-called “picture-theory of meaning.” Where Russell had spoken ore generally of a correspondence as fundamental to the concept of truth, Wittgenstein conceived of a picturing-relation, i.e., of a strict mapping relation. On this view, a sentence is itself a fact. Both the sentence and the fact in the world it depicts are composed of objects. The sentence and the depicted fact must have the same logical multiplicity and the same logical form. The sentence “does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects.” (T, 5.542) But Wittgenstein also held that ultimately no formal semantic theory could be constructed. All attempts to speak about logic were bound to fail. Having spelled out his conception of logic and truth, Wittgenstein concluded the Tractatus therefore by announcing that “my propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical… He must transcend these propositions, and then he will see the world aright.” (6.54)[13] Wittgenstein’s declaration that “logic is not a theory but a reflexion of the world,” and that “logic is transcendental.” (6.13 and 6.2) Hence also his memorable phrase that “logic must take care of itself.” (5.4731)
These sketchy remarks must suffice to show that the concept of truth became a lively topic for logical and philosophical reflection in the period between Frege and Tarski. Frege’s views about truth were obviously not the same as Russell’s and Wittgenstein’s. But van Heijenoort was certainly right in thinking that all three were most specifically concerned with the symbolism, the notation, the language and that this had to do with their doubts about the traditional manner in which philosophers and logicians had talked about meaning and truth. They all sought to bracket that tradition out and to begin anew and it was this rejection of the past that opened their eyes to the possibilities of the new logic.
All this may strike us who have grown up in the Tarskian radition as peculiar. For in Tarski’s truth theory and truth definition we seem to have achieved what the logicists either neglected or claimed to be impossible. Tarski’s theory seems to take us back, in fact, to the correspondence notion of truth and Tarski himself has alerted us to affinities between his definition of truth and the classical, Aristotelian notion. Insofar as Bolzano can be seen as the last and most mature expression of the classical tradition in logic, he, too, seems to find confirmation in the triumph of formal semantics.
This view rests, however, on philosophical confusions about the nature of Tarski’s achievement. Tarski’s theory resurrects in no way the classical theory of truth as correspondence and the philosophical tradition cannot be seen as confirmed by Tarski’s results. Tarski’s truth-definition has, indeed, little to do with the traditional conception of truth as correspondence to the world. It offers to us, instead, an account of how we can correlate sentences of one language, the object-language, with those of another, the metalanguage. Tarski’s famous criterion of adequacy says in terms of a metalanguage that we can call an object-language sentence true when we are justified in asserting its correlate metalanguage sentence. There is here, strictly speaking, no talk of correspondence between language and world but only of a correspondence between two languages.
By adopting the view that Tarski has explicated for us the philosophical problem of truth we are, moreover, likely to lose certain insights that Frege, the Russell, and Wittgenstein appear to have had about the concept of truth. They had understood, on the one hand, that our talking is in some way or other responsible to the world and that this is what philosophical attempts to define the concept of truth have always been about. But they also understood that any attempt to talk about the relation between language and world is problematic since it must, of necessity, remain within the bounds of language. Their insight can, of course, be conjoined to Tarski’s formal semantics and is, in fact, needed to make sense of its formal constructions. Post-Tarski philosophizing has, however, for the most part failed to grasp this point and has therefore come to believe, quite falsely, that Tarski connects us back to the classical theory of truth.
The true story is that modern logic marks the end of a thinking about the notion of truth that, as Heidegger has argued, began with Plato and Aristotle and that was responsible not only for the rise of classical logic but also for much else in philosophy, not least, the growth of both classical metaphysics and of classical epistemology. Heidegger has often worried over the question where the end of this tradition might be found and who should be considered the last metaphysician. In the history of logic, it seems clear, that Bolzano is one of these last men and that with Frege begins a new turn – a turn away from the classical notion of truth and, made possible by this, the turn to a new kind of logic.[14]
NOTES
[1] Martin Heidegger, “Platons Lehre von der Wahrheit,” Wegmarken, Vittorio Klostermann, Frankfurt 1967.
[2] Aristotle, Metaphysics, 1011b28.
[3] These references are to section and page numbers of Bernard Bolzano, Wissenschaftslehre, 2nd ed., ed. by Wolfgang Schultz, Felix Meiner, Leipzig 1929, vol. 1.
[4] Gottlob Frege, “Der Gedanke,” Kleine Schriften, Georg Olms, Hildesheim 1967, p. 342. The following page references are to the same text.
[5] Gottlob Frege, “Logik”, _Nachgelassene Schriften_, Felix Meiner, Hamburg 1969 (herafter “NS”), p. 140.
[6] “Logik,” NS, pp. 139 and 140.
[7] Gottlob Frege, “Vorlesungen über Begriffsschrift“, History and Philosophy of Logic, vol. 17, 1996, p. 15.
[8] Gottlob Frege, “Aufzeichnungen für Ludwig Darmstaedter, NS, p. 273.
[9] Gottlob Frege, “Meine grundlegenden logischen Einsichten,” NS, p. 272.
[10] Jean van Heijenoort, “Logic as Calculus and Logic as Language”, in H. Sluga, ed., The Philosophy of Frege, Garland, New York 1993, vol. 1, p. 326.
[11] G.E. Moore, “The Nature of Judgment”, _Mind_, vol. 8, 1899, pp. 176, 179, 180, respectively.
[12] Bertrand Russell, The Problems of Philosophy, Oxford University Press, Oxford 1959, p. 123. The following page references are to the same text.
[13] Ludwig Wittgenstein, Tractatus Logico-Philosophicus. I follow here the traditional way of citing the text by means of Wittgenstein’s own numeration.
[14] I am particularly grateful to Wolfgang Künne, Goran Sundholm, and Mark Textor for having forced me to clarify my thoughts on this topic.
