It is true of course that we speak of men's
thoughts
as being liable to change.
Gottlob-Frege-Posthumous-Writings
There is, however, the difference that these sciences have to take into account other properties besides these we have mentioned, and that there is no one property by which their nature is so completely characterized as logic is by the word 'true'.
Like ethics, logic can also be called a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject-matter. On the. contrary, the task we assign logic is only that of saying what holds with the utmost generality for all thinking, whatever its subject-matter. We must assume that the rules for our thinking and for our holding something to be true are prescribed by the laws of truth. The former are given along with the latter. Consequently we can also say: logic is the science of the most general laws of truth. The reader may find that he can form no very precise impression from this description of what is meant. The author's inadequacy and the awkwardness of language are probably to blame for this. But it il only a question of giving a rough indication of the goal of logic. What is still lacking in the account will have to be made good as we go on.
Now it would be futile to employ a definition in order to make it clearer what is to be understood by 'true'. If, for example, we wished to say 'an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form 'A is true if and only if it has such-and-such properties or stands in
1 Frege only took the table of contents as far as p. 141 (ed. ).
? Logic 129
such-and-such a relation to such-and-such a thing'. In each case in hand it would always come back to the question whether it is true that A has such- and-such properties, or stands in such-and-such a relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever.
If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, ifl assert that it agrees with reality. Therefore it is really by using the form of an assertoric sentence that we assert truth, and to do this we do not need the word 'true'. Indeed we can say that even where we use the form of expression 'it is true that . . . 'the essential thing is really the assertoric form of the sentence.
We now ask: what can the predicate 'true' be applied to? The issue here is to delimit the range of application of the word. Whatever else may be the case, the word cannot be applied to anything that is material. If there is any doubt about this, it could arise only for works of art. But if we speak of truth in connection with these, then we are surely using the word with a different meaning from the one that is meant here. In any case it is only as a work of art that a thing is called true. If a thing had come into existence through the
blind play of natural forces, our predicate would be clearly inappropriate. Por the same reason we are excluding from consideration the use that is made by, say, an art critic when he calls feelings and experiences true.
No one would deny that our predicate is, for the most part, ascribed to sentences. We are not, however, concerned with sentences expressing wishes, questions, requests and commands, but only with assertoric sentences, sentences that is to say, in which we communicate facts and propound mathematical laws or laws of nature.
Further, it is clear that we do not, properly speaking, ascribe truth to the series of sounds which constitute a sentence, but to its sense; for, on the one hand, the truth of a sentence is preserved when it is correctly translated into another language, and, on the other hand, it is at least conceivable that the same series of sounds should have a true sense in one language and a false sense in another.
We are here including under the word 'sentence' the main clause of a sentence and clauses that are subordinate to it.
In the cases which alone concern logic the sense of an assertoric sentence is either true or false, and then we have what we call a thought proper. But there remains a third case of which at least some mention must be made here.
The sentence 'Scylla has six heads' is not true, but the sentence 'Scylla does not have six heads' is not true either; for it to be true the proper name 'Scylla' would have to designate something. Perhaps we think that the name
? ? 130 Logic
'Scylla' does designate something, namely an idea. In that case the first question to ask is 'Whose idea? ' We often speak as if one and the same idea occurred to different men, but that is false, at least if the word 'idea' is used in the psychological sense: each man has his own idea. But then an idea does not have heads, and so one cannot cut heads off an idea either. The name 'Scylla' does not therefore designate an idea. Names that fail to fulfil the usual role of a proper name, which is to name something, may be called mock proper names. Although the tale of Willian Tell is a legend and not history and the name 'William Tell' is a mock proper name, we cannot deny it a sense. But the sense of the sentence 'William Tell shot an apple off his son's head' is no more true than is that of the sentence 'William Tell did not shoot an apple off his son's head'. I do not say, however, that this sense is false either, but I characterize it as fictitious. This may elucidate the sense in which I am using the word 'false', which is as little susceptible of a definition proper as is the word 'true'.
If the idealist theory of knowledge is correct then all the sciences would belong to the realm of fiction. Indeed one might try to reinterpret all sentences in such a way that they were about ideas. By doing this, however, their sense would be completely changed and we should obtain quite a different science; this new science would be a branch of psychology.
Instead of speaking of 'fiction', we could speak of 'mock thoughts'. Thus if the sense of an assertoric sentence is not true, it is either false or fictitious, and it will generally be the latter if it contains a mock proper name. * The writer, in common with, for example, the painter, has his eye on appearances. Assertions in fiction are not to be taken seriously: they are only mock assertions. Even the thoughts are not to be taken seriously as in the sciences: they are only mock thoughts. If Schiller's Don Car/os were to be regarded as a piece of history, then to a large extent the drama would be false. But a work of fiction is not meant to be taken seriously in this way ati all: it's all play. Even the proper names in the drama, though they: correspond to names of historical personages, are mock proper names; they are not meant to be taken seriously in the work. We have a similar thing in the case of an historical painting. As a work of art it simply does not claim to give a visual representation of things that actually happened. A picture that was intended to portray some significant moment in history with photographic accuracy would not be a work of art in the higher sense of the word, but would be comparable rather to an anatomical drawing in a scientific work.
The logician does not have to bother with mock thoughts, just as 1 physicist, who sets out to investigate thunder, will not pay any attention to stage-thunder. When we speak of thoughts in what follows we mean thoughts proper, thoughts that are either true or false.
*We have an exception where a mock proper name occurs within a clause in indirect speech.
? ? Logic 131
The sense of an assertoric sentence I call a thought. Examples of thoughts are laws of nature, mathematical laws, historical facts: all these find expression in assertoric sentences. I can now be more precise and say: The predicate 'true' applies to thoughts.
Of course we speak of true ideas1 as well. By an idea we understand a picture that is called up by the imagination: unlike a perception it does not consist of present impressions, but of the reactivated traces of past impressions or actions. Like any other picture, an idea is not true in itself, hut only in relation to something to which it is meant to correspond. If it is said that a picture is meant to represent Cologne Cathedral, fair enough; it can then be asked whether this intention is realized; if there is no reference to an intention to depict something, there can be no question of the truth of a picture. It can be seen from this that the predicate true is not really conferred on the idea itself, but on the thought that the idea depicts a certain object. And this thought is not an idea, nor is it made up of ideas in any way. Thoughts are fundamentally different from ideas (in the psychological sense). The idea of a red rose is something different from the thought that this rose is red. Associate ideas or run them together as we may, we shall still finish up with an idea and never with something that could be true. This difference also comes out in the modes we have of communicating. The proper means of expression for a thought is a sentence. But a sentence is hardly an appropriate vehicle for conveying an idea. I have only to remind you how inadequate any description is by comparison with a pictorial representation. Things are not so bad where it is a matter of representing sounds, since we have the resources of onomatopoeia; but onomatopoeia has nothing whatever to do with the expression of thoughts, and whilst in translation the play of sounds is easily lost, the thought must be preserved if we are to speak of a translation at all. Conversely, pictures and musical
compositions without accompanying words are hardly suited for expressing thoughts. It is true that we may associate all kinds of thoughts with some work of art or other but there is no necessary connection between the two, and we are not surprised when someone else associates different thoughts with it.
In order to shed a clearer light on the peculiarity of the predicate true, let us compare it with the predicate beautiful. We can see, to begin with, that what is beautiful admits of degree, but what is true does not. We can think two objects beautiful, and yet think one more beautiful than the other. On the other hand, if two thoughts are true, one is not more true than the other. And here there emerges the essential difference that what is true is true
1 In the German, 'Vorstellungen'. Throughout this essay the difficult word ? Vorstellung' has been generally rendered by 'idea'. Admittedly this makes certain passages read unnaturally, but the gist of what Frege is saying should be clear if the reader bears in mind the explanation he gives here of how the term 'VorstellunJ. :' is hcing used (trans. ).
? 132 Logic
independently of our recognizing it as such, but what is beautiful is beautiful only for him who experiences it as such. What is beautiful for one person is not necessarily beautiful for another. There is no disputing tastes. Where truth is concerned, there is the possibility of error, but not where beauty is concerned. By the very fact that I consider something beautiful it is beautiful for me. But something does not have to be true because I consider it to be true, and if it is not true in itself, it is not true for me either. Nothing is beautiful in itself: it is only beautiful for some being experiencing it and this is necessarily implicit in any aesthetic judgement. Now it is true that we also make judgements of this kind which seem to lay claim to being objective. Whether we are aware of it or not the assumption of a normal human being always underlies such judgements, and each one of us cannot help but think that he himself is so close to the normal human being that he believes he can speak in his name. What, then, we mean by 'This rose is beautiful' is 'This rose is beautiful for a normal human being'. But what is normal? That depends on the circle of human beings one has in mind. If there is some remote mountain valley where nearly all the people have goitres, then having a goitre will be looked on as normal there, and those who lack such an adornment will be considered ugly. How is a negro from the heart of Africa to be weaned from the view that the narrow nose of the European is ugly, whereas the broad nose of the negro is beautiful? And cannot a negro qua negro be just as normal as a white man qua white man? Cannot a child be
just as normal as a grown-up? The ideas that are awakened in us by the power of association have a great influence on the judgements a man forms of what is beautiful, and these ideas depend upon what he has absorbed in earlier life. But this varies from person to person. And even if we managed to define a normal human being and so 'beautiful' in an objective sense, it would still be only possible to do this on the basis of the subjective sense. Far from having rid ourselves of this, we would have recognized it as the root sense. We could not alter the situation by trying to substitute an ideal human being for a normal one. In the absence of experiences and ideas there would be no instance of anything subjectively beautiful and therefore no instance of anything objectively beautiful either. There is therefore much to be said for the view that the real work of art is a structure of ideas within us and that the external thing-the painting, the statue-is only a means for producing the real work of art in us. On this view, anyone who enjoys a work of art has his own work of art, with the consequence that there is no contradiction whatever between varying aesthetic judgements. Hence: dl gustibus non disputandum.
If anyone tried to contradict the statement that what is true is true independently of our recognizing it as such, he would by his very assertion contradict what he had asserted; he would be in a similar position to the Cretan who said that all Cretans are liars.
To elaborate: if something were true only for him who held it to be true, there would be no contradiction between the opinions of different people. So
? Logic 133
to be consistent, any person holding this view would have no right whatever to contradict the opposite view; he would have to espouse the principle: non disputandum est. He would not be able to assert anything at all in the normal sense, and even if his utterances had the form of assertions, they would only have the status of interjections--of expressions of mental states or processes, between which and such states or processes in another person there could be no contradiction. And in that case his assertion that something was true only for us and through being recognized by us as such would have this status too. If this view were true, it would be impossible to daim that any of his own opinions was more justified in the eyes of others than the opposite opinion. A view that made such a claim would be unjustified; this would mean, however, that every opinion would be unjustified in the usual sense of the word, and so also those opinions to which we were opposed. There would be no science, no error and no ~:orrection of error; properly speaking, there would be nothing true in the uormal sense of the word. For this is so closely bound up with that independence of being recognized as true, which we are emphasizing here, that it cannot be separated from it. If anyone seriously and sincerely defended the view we are here attacking, we should have no recourse but to ussume that he was attaching a different sense to the word 'true'.
We can go a step further. In order to be true, thoughts--e. g. laws of nature-not only do not need to be recognized by us as true: they do not have to have been thought by us at all. A law of nature is not invented by us, hut discovered, and just as a desolate island in the Arctic Ocean was there long before anyone had set eyes on it, so the laws of nature, and likewise those of mathematics, have held good at all times and not just since they were discovered. This shows us that these thoughts, if true, are not only true independently of our recognizing them to be so, but that they are indepen- dent of our thinking as such. A thought does not belong specially to the person who thinks it, as does an idea to the person who has it: everyone who grasps it encounters it in the same way, as the same thought. Otherwise two people would never attach the same thought to the same sentence, but cuch would have his own thought; and if, say, one man put 2 ? 2 = 4 forward ns true whilst another denied it, there would be no contradiction, because what was asserted by one would be different from what was rejected by the other. It would be quite impossible for the assertions of different people to contradict one another, for a contradiction occurs only when it is the very Nume thought that one person is asserting to be true and another to be false. So a dispute about the truth of something would be futile. There would Nimply be no common ground to fight on; each thought would be enclosed in its own private world and a contradiction between the thoughts of ditTerent people would be like a war between ourselves and the inhabitants of Mars. Nor must we say that one person might communicate his thought to another and a conflict would then flare up in the latter's private world. It would be quite impossible for a thought to be so communicated that it
? 134 Logic
should pass out of the private world of one person into that of another. The thought that entered the latter's mind as a result of the communication would be different from the thought in the former's mind; and the slightest alteration can transform a truth into a falsehood. If we wanted to regard a thought as something psychological, as a structure of ideas, without, however, adopting a wholly subjective standpoint, we should have to explain the assertion that 2 + 3 = 5 on something like the following lines 'It has been observed that with many people certain ideas form themselves in association with the sentence "2 + 3 = 5". We call a formation of this kind the sense of the sentence "2 + 3 = 5". So far as we have observed hitherto these formations are always true; we may therefore make the provisional statement "Going by the observations made hitherto, the sense of the sentence '2 + 3 = 5' is true". ' But it is obvious that this explanation would not work at all. And it would leave us where we were, for the sense of the sentence 'It has been observed that with many people certain ideas form themselves etc. ' would of course be a formation of ideas too and the whole thing would begin over again. A soup that tastes pleasant to one person, may be nauseous to another. In such a case each person is really making a judgement about his own sensation of taste, and this is different from the other's. The same would hold for thoughts if a thought were related to a sentence in the same kind of way as sensations of taste are related to the chemical stimuli that excite them.
If a thought, like an idea, were something private and mental, then the truth of a thought could surely only consist in a relation to something that was not private or mental. So if we wanted to know whether a thought was true, we should have to ask whether the relation in question obtained and thus whether the thought that this relation obtained was true. And so we should be in the position of a man on a treadmill who makes a step forwards and upwards, but the step he treads on keeps giving way and he falls back to where he was before.
A thought is something impersonal. If we see the sentence '2 + 3 = 5' written on a wall, we have no difficulty at all in recognizing the thought expressed by it, and we do not need to know who has written it there in order to understand it.
A sentence like 'I am cold' may seem to be a counter-example to our thesis that a thought is independent of the person thinking it, in so far as it can be true for one person and false for another, and thus not true in itself. The reason for this is that the sentence expresses a different thought in the mouth of one person from what it expresses in the mouth of another. In this case the mere words do not contain the entire sense: we have in addition to take into account who utters it. There are many cases like this in which the spoken word has to be supplemented by the speaker's gesture and expression, and the accompanying circumstances. The word 'I' simply designates a different person in the mouths of different people. It is not necessary that the person who feels cold should himself give utterance to the
? Logic 135
thought that he feels cold. Another person can do this by using a name to designate the one who feels cold.
In this way a thought can be clothed in a sentence that is more in keeping with its being independent of the person thinking it. The possibility of doing this distinguishes it from a mental state expressed by an interjection. Words like 'here' and 'now' only acquire their full sense through the circumstances in which they are used. If someone says 'it is raining' the time and place of utterance has to be supplied. If such a sentence is written down it often no longer has a complete sense because there is nothing to indicate who uttered it, and where and when. As regards a sentence containing a judgement of taste like 'This rose is beautiful', the identity of the speaker is essential to the sense, even though the word 'I' does not occur in it. So the explanation for all these apparent exceptions is that the same sentence does not always express the same thought, because the words need to be supplemented in order to get a complete sense, and how this is done can vary according to the circumstances.
Whereas ideas (in the psychological sense of the word) have no fixed boundaries, but are constantly changing and, Proteus-like, assume different forms, thoughts always remain the same. It is of the essence of a thought to he non-temporal and non-spatial. In the case of the thought that 3 + 4 = 7 and the laws of nature there is hardly any need to support this statement. If it should turn out that the law of gravitation ceased to be true from a certain moment onwards, we should conclude that it was not true at all, and put ourselves out to discover a new law: the new one would differ in containing a condition which would be satisfied at one time but not at another. It is the same with place. If it should transpire that the law of gravitation was not valid in the neighbourhood of Sirius, we should search for another law which contained a condition that was satisfied in our solar system but not in the neighbourhood of Sirius. If someone wished to cite, say, 'The total number of inhabitants of the German Empire is 52 000 000', as a counter-example to the timelessness of thoughts, I should reply: This sentence is not a com- plete expression of a thought at all, since it lacks a time-determination. If we add such a determination, for example, 'at noon on l January 1897 by central European time', then the thought is either true, in which case it is always, or better, timelessly, true, or it is false and in that case it is false without qualification. This holds of any particular historical fact: if it is true, it is true independently of the time at which it is judged to be true. It is no objection that a sentence may acquire a different sense in the course of time; for what changes in such a case is of course the language, not the thought. In another language this shift need not take place.
It is true of course that we speak of men's thoughts as being liable to change. However it is not the thoughts which are true at one time and false at another: it is only that they arc held to be true at one time and false at another.
What if it is objected that I am attaching to the word 'thought' a sense that it does not ordinarily have, and that other people understand by it an
? 136 Logic
act of thinking, which is obviously private and mental? Well, the important thing is that I remain true to my way of using it; whether this agrees with the ordinary use is of less importance. It may well be the case that people sometimes understand by the word 'thought' an act of thinking-in any case this is not always so*-and such an act cannot be true.
In logic, as in other sciences, it is open to us to coin technical terms,
? Dedekind, in theor. 66 of his book Was sind und was sol/en die Zahlen? , uses this word as I do. For he is attempting there to prove that the totality of things that can be objects of his thinking is infinite. Let s be such an object; then Dedekind calls rp(s) the thought that s can be an object of his thinking. And this thought can now itself be an object of his thinking. Thus rp(rp(s)) is the thought that the thought that s can be an object of his thinking can be an object of his thinking. We can see from this what 'rp(rp(rp(s)))', 'rp(rp(rp(rp(s))))' and so on, are supposed to mean. It is essential to the proof that the sentence 's can be an object of Dedekind's thinking' always expresses a thought when the letter 's' designates such an object. Now if, as Dedekind wishes to prove, there are infinitely many such objects s, there must also be
infinitely many such thoughts rp(s). Now presumably we shall not hurt Dedekind's feelings if we assume that he has not thought infinitely many thoughts. Equally he should not assume that others have already thought infinitely many thoughts which could be the objects of his thinking; for this would be to assume what was to be proved. Now if infinitely many thoughts have not yet been thought, the infinitely many thoughts rp(s) must comprise infinitely many thoughts that are not thought, in which case it cannot be essential to a thought that it should be thought. And this is precisely what I am maintaining. If there were only thoughts that are thought, the sign 'rp(s)' would not always have a meaning; to ensure that it did have, it is not sufficient for 's' to mean something that could be an object of Dedekind's thinking: it would also have to have been thought by someone in order to be a possible object of Dedekind's thinking. If this were not the case, then the sign 'rp(s)' would have no meaning for the given s. The sun (0) can be an object of Dedekind's thinking; hence the first two members and perhaps a few successive members of the series '0, 'rp(0)', rp(rp(0))' . . . have a meaning. But as we progressed along the series we would be bound eventually to reach a member that was meaningless, because the thought which it was meant to designate had not been thought, and so was not to hand. In that case 'rp(s)' would resemble a power series which did not converge for every value of the argument. The fact that the series diverged would correspond to the sign 'rp(s)' becoming meaningless. If we assume a power series with radius of convergence 4 and if we assume, further, that the series has the value 2 for l as argument and the value 5 for 2 as argument, then the corresponding series of numbers l, 2, 5 comes to an end at this point and does not go on to infinity. In the same way the series 0, rp(0), rp(rp(0)), does not go on to infinity if there are only thoughts that are thought. So the validity of Dedekind's proofs rests on the assumption that thoughts obtain independently of our thinking. We can see how this use is one to which the word 'thought' naturally lends itself.
? Logic 137
regardless of whether the words are always used in precisely that way in l'vcryday life. It doesn't matter if the meaning we fix on is not altogether in line with everyday use or doesn't accord with the word's etymology; what does matter is to make it as appropriate a vehicle as possible for use in npressing laws. Provided there is no loss of rigour, the more compendious the formulation of the complete system of laws, the more felicitous is the apparatus of technical terms.
Now we cannot regard thinking as a process which generates thoughts. It would be just as wrong to identify a thought with an act of thinking, so that a thought is related to thinking as a leap is to leaping. This view is in harmony with many of our ways of talking. For do we not say that the same thought ~~grasped by this person and by that person? And that each person has the same thought over and over again? Now if thoughts only came into txistence as a result of thinking or if they were constituted by thinking, then the same thought could come into existence, cease to exist, a! )d then come 111to existence again, which is absurd. As I do not create a tree by looking at 1t or cause a pencil to come into existence by taking hold of it, neither do I 1/,Cnerate a thought by thinking. And still less does the brain secrete thoughts, as the liver does gall.
The metaphors that underlie the expressions we use when we speak of 11. rasping a thought, of conceiving, laying hold of, seizing, understanding, of mpere, percipere, comprehendere, intelligere, put the matter in essentially the right perspective. What is grasped, taken hold of, is already there and all we do is take possession of it. Likewise, what we see into or single out from nmongst other things is already there and does not come into existence as a result of these activities. Of course all metaphors go lame at some point. We nrc inclined to regard what is independent of our mental processes as Nornething spatial or material, and the words that we have just listed make it look as if this is what a thought actually is. But this is not where the point of lhc comparison lies. What is independent of our mental processes, what is ohjcctive, does not have to be spatial or material or actual. If we were to disregard this we should easily slip into a kind of mythology. To say 'The lnws of gravitation, of inertia, of the parallelogram of forces cause the earth lo move as it does move', might make it look as if these laws, so to speak, look the earth by the ears and kept it on the path they prescribe. Such a use llfthe words 'affect', 'cause', would be misleading. On the other hand, it is nil right to say that the sun and planets act on one another in accordance with the laws of gravitation.
So even if physical bodies and thoughts resemble one another in being Independent of my inner life, we are not entitled to conclude from this that thoughts can be moved as bodies can, or can be smelled or tasted, and it would be invalid to seek somehow to draw from the absurdity of such Inference an objection to our views. Although a law of nature obtains quite Independently of whether we think of it or not, it does not emit light or Mound waves by which our visual or auditory nerves could be affected. But
? 138 Logic
do I not then see that this flower has five petals? We can say this, but if we do, the word 'see' is not being used in the sense of having a mere visual experience: what we mean by it is bound up with thinking and judging. Newton did not discover the law of gravitation because his senses were especially acute.
If we wish to speak of a thought as being actual, we can do so only in the sense that the knowledge that a man has of e. g. a law of nature has an influence on the decisions he makes, which in their turn may affect the course of history. We should then be thinking of the recognition of a law as a case of a law's acting upon us, and it is perhaps possible to do this, just as we can regard, say, the seeing of a flower as the flower's indirectly acting on us.
We can disregard thoughts and we can take possession of them. We might conceive of the latter as a case of our acting on thoughts, which seems to speak against their being timeless. But the thought is not changed in itself by being thus acted on, just as the moon is apparently unaffected whether we take any notice of it or not. So even though it may be possible to speak of thoughts as acting on us, we cannot speak of ourselves as acting on thoughts. We might cite, as an instance of thoughts being subject to change, the fact that they are not always immediately clear. But what is called the clarity of a thought in our sense of this word is really a matter of how thoroughly it has been assimilated or grasped, and is not a property of a thought.
It would be wrong to think that it is only true thoughts that obtained independently of our mental life, and that false ones, on the other hand, belonged, as ideas do, to our inner life. Almost everything that we have said about the predicate true holds for the predicate false as well. In the strict sense it applies only to thoughts. Where it looks to be predicated of sentences or ideas, still at bottom it is being predicated of thoughts. What is false is false in itself and independently of our opinions. A dispute over the falsity of something is at the same time a dispute over the truth of some- thing. Therefore the thing whose falsity can be a matter for dispute does not belong to some mind or other.
Separating a Thought from its Trappings
In an assertoric sentence two different kinds of thing are usually intimately bound up with one another: the thought expressed and the assertion of its truth. And this is why these are often not clearly distinguished. However, one can express a thought without at the same time putting it forward as true. A scientist who makes a scientific discovery usually begins by grasping just a though,t, and then he asks himself whether it is to be recognized as true; it is not until his investigation has turned out in favour of the hypothesis, that he ventures to put it forward as true. We express the same
? Logic 139
thought in the question 'Is oxygen condensable? ' and in the sentence 'Oxygen is condensable', joining it in the one case with a request and in the other with an assertion.
When we inwardly recognize that a thought is true, we are making a judgement: when we communicate this recognition, we are making an a. 1? sertion.
We can think without making a judgement.
We have seen that the series of sounds that compose a sentence is often not sufficient for the complete expression of a thought. If we wish to bring the essence of a thought into as sharp a focus as possible, we ought not to overlook the fact that the converse case is not uncommon, the case where a sentence does more than express a thought and assert its truth. In many cases a sentence is meant to have an effect on the ideas and feelings of the hearer as well; and the more closely it approximates to the language of poetry, the greater this effect is meant to be. We have indeed stressed the lnct that language is but poorly suited for calling up at will an idea in the 111ind of a hearer with any precision. Who would ever rely on words to evoke as precise a mental picture of an Apollo as can be produced without difficulty by looking at a piece of sculpture? Even so, we do say that the poet paints pictures. And in fact it cannot be denied that the spoken word affects the ideas we have just because it enters consciousness as a complex of auditory sensations. Right from the start we experience the series of sounds themselves, the tone of the voice, the intonation and rhythm with kclings of pleasure or displeasure. These sensations of sound are linked to nuditory ideas that resemble them and these latter are linked in turn with further ideas reactivated by them. This is the domain of onomatopoeia. Here we may cite the Homeric verse (Odyssey IX, 71): rpzx{}d Kaz' rerpax{}a l~ti:oxzoev le; av'ep,ow.
This is quite independent of the aim of words to express thoughts. Here the sounds are acting only as a sensory stimulus. But because sequences of such sounds are meant to have a sense they act upon the imagination in yet n different way. Anyone who hears the word 'horse' and understands it will probably have straightaway a picture of a horse in his mind. This picture, however, is not to be confused with the sense of the word 'horse'; for the word 'horse' gives no clue to the colour of the horse, or to its carriage when Ntnnding still or in motion, or to the side from which it is seen and the like. If different men were able, say, immediately to project onto a canvas the ideas that sprung up in their minds on hearing the word 'horse', then we should be presented with quite different pictures. And even with the same man the word 'horse' does not always conjure up the same idea. Here a great deal depends on the context. We may compare e. g. the sentences 'With what joy he rides his gallant horse' and 'I just saw a horse stumble on the wet nsphalt'.
So there can be no question of the same idea always being associated with the word 'horse'. Thus in virtue of its sense such a word will excite a certain
? 140 Logic
idea in us, but by itself it is far from determining this idea completely. Generally speaking the most we are entitled to assume is that the ideas of the speaker and hearer are very roughly in agreement. If several artists produce, independently of one another, illustrations of the same poem, they will diverge considerably from one another in the portrayal they give. Thus the poet does not really depict anything: he only provides the impetus for others to do so, furnishing hints to this end, and leaving it to the hearer to give his words body and shape. And in this connection it is useful to the poet to have at his disposal a number of different words that can be substituted for one another without altering the thought, but which can act in different ways on the feelings and imagination of the hearer. We may think e. g. of the words 'walk', 'stroll', 'saunter'. These means are also used to the same end in everyday language. If we compare the sentences 'This dog howled the whole night' and 'This cur howled the whole night', we find that the thought is the same. The first sentence tells us neither more nor less than does the second. But whilst the word 'dog' is neutral as between having pleasant or unpleasant associations, the word 'cur' certainly has unpleasant rather than pleasant associations and puts us rather in mind of a dog with a somewhat unkempt appearance. Even if it is grossly unfair to the dog to think of it in this way, we cannot say that this makes the second sentence false. True, anyone who utters this sentence speaks pejoratively, but this is
not part of the thought expressed. What distinguishes the second sentence from the first is of the nature of an interjection. It might be thought that the second sentence does nevertheless tell us more than the first, namely that the speaker has a poor opinion of the dog. In that case, the word 'cur' would contain an entire thought. We can put this to the test in the following way.
We assume that the first sentence is true and the second sentence is spoken by someone who does not actually feel the contempt which the word 'cur' seems to imply. If the objection were correct, the second sentence would now contain two thoughts, one of which was false; so it would assert something false as a whole, whilst the first sentence would be true. We shall hardly go along with this; rather the use of the word 'cur' does not prevent us from holding that the second sentence is true as well. For we have to make a distinction between the thoughts that are expressed and those which the speaker leads others to take as true although he does not express them. If a commander conceals his weakness from the enemy by making his troops keep changing their uniforms, he is not telling a lie; for he is not expressing any thoughts, although his actions are calculated to induce thoughts in others. And we find the same thing in the case of speech itself, as when one gives a special tone to the voice or chooses special words. If someone announces the news of a death in a sad tone of voice without actually being sad, the thought expressed is still true even if the sad tone is assumed in oq:ler to create a false impression. And we can substitute words like 'ah', and 'unfortunately' for such a tone of voice without altering the thought. Naturally things are different if certain actions are specifically
? Logic 141
agreed on as a means of communicating something. In language common usage takes the place of such agreements. Of course borderline cases can arise because language changes. Something that was not originally employed as a means of expressing a thought may eventually come to do this because it has constantly been used in cases of the same kind. A thought which to hegin with was only suggested by an expression may come to be explicitly asserted by it. And in the period in between different interpretations will he possible. But the distinction itself is not obliterated by such fluctuations in language. In the present context the only essential thing for us is that a different thought does not correspond to every difference in the words used, and that we have a means of deciding what is and what is not part of the thought, even though, with language constantly developing, it may at times he difficult to apply.
The distinction between the active and passive voice belongs here too. The sentences 'M gave document A to N', 'Document A was given by N by M', 'N received document A from M' express exactly the same thought; we learn not a whit more or less from any one of these sentences than we do from the others. Hence it is impossible that one of them should be true whilst another is false. It is the very same thing that is here capable of being true or false. For all this we are not in a position to say that it is a matter of complete indifference which of these sentences we use. As a rule stylistic and uesthetic reasons will give the preference to one of them. If someone asks 'Why has A been arrested? ' it would be unnatural to reply 'B has been rnurdered by him', because it would require a needless switch of the uttention from A to B. Although in actual speech it can certainly be very important where the attention is directed and where the stress falls, it is of no concern to logic.
In translating from one language to another it is sometimes necessary to dispense with the original grammatical construction altogether. Nevertheless, this need not affect the thought and it must not do so, if the translation is to be correct. But it is sometimes necessary to sacrifice the feeling and colour of the original.
Again in the two sentences 'Frederick the Great won the battle of Rossbach' and 'It is true that Frederick the Great won the battle of Rossbach', we have, as we said earlier, the same thought in a different verbal form. In affirming the thought in the first sentence we thereby affirm the thought in the second, and conversely. There are not two different acts of judgement, but only one.
(From all this we can see that the grammatical categories of subject and predicate can have no significance for logic. )
The distinction between what is part of the thought expressed in a Ncntence and what only gets attached to the thought is of the greatest Importance for logic. The purity of the object of one's investigation is not of Importance only to the chemist. How would the chemist be able to recognize, hcyond any doubt, that he has arrived at the same results by different
? 142 Logic
means, if the apparent difference of means could be traced back to impurities in the substances used? There is no doubt that the first and most important discoveries in a science are often a matter of recognizing something as the same again. However self-evident it may seen to us that it is the same sun which went down yesterday and rose today, and however insignificant therefore this discovery may seem to us, it has certainly been one of the most important in astronomy and perhaps the one that really laid the foundations of the science. It was also important to recognize that the morning star is the same as the evening star, that three times five is the same as five times three. It is just as important not to distinguish what is the same as it is to be alive to differences when they don't hit one in the eye. So it is quite wrong to think that one can never make too many distinctions. It does nothing but harm to insist on distinctions where they are not relevant. Thus in general mechanics we shall take care not to speak of the chemical differences between substances and not to state the law of inertia in a special form for, say, each chemical element. We shall only take those differences into account that are essential to the formulation of the laws with which we are actually concerned. Above all, we must not let ourselves be seduced by the presence of extraneous factors into seeing distinctions where there are none.
In logic we must reject all distinctions that are made from a purely psychological point of view. What is referred to as a deepening of logic by psychology is nothing but a falsification of it by psychology.
In human beings it is natural for thinking to be intermingled with having images and feeling. Logic has the task of isolating what is logical, not, to be sure, so that we should think without having images, which is no doubt impossible, but so that we should consciously distinguish the logical from what is attached to it in the way of ideas and feelings. There is a difficulty here in that we think in some language or other and that grammar, which has a significance for language analogous to that which logic has for judgement, is a mixture of the logical and the psychological. If this were not so, all languages would necessarily have the same grammar. It is true that we can express the same thought in different languages; but the psychological trappings, the clothing of the thought, will often be different. This is why the learning of foreign languages is useful for one's logical education. Seeing that the same thought can be worded in different ways, we learn better to distinguish the verbal husk from the kernel with which, in any given language, it appears to be organically bound up. This is how the differences between languages can facilitate our grasp of what is logical. But still the difficulties are not wholly removed in this way, and our logic books still keep dragging in a number of things-subject and predicate, for example-which do not, strictly speaking, belong to logic. For this reason it is useful to be acquainted also with a means of expressing thoughts that is of a radically different nature, such as we have in the formula-language of arithmetic or in my concept-script.
? Logic 143
The first and most important task is to set out clearly what the objects to be investigated are. Only if we do this shall we be able to recognize the same as the same: in logic too, such acts of recognition probably constitute the fundamental discoveries. Therefore let us never forget that two different sentences can express the same thought, that we are concerned with only that part of a sentence's content which can be true or false.
Even if there were only a jot more to the thought contained in the passive form than in the active, it would be conceivable that this jot should be false whilst the thought contained in the active form was true, and that we should not be entitled without more ado to go over from the active to the passive form. Likewise if there were only a jot more to the thought contained in the active form than in the passive, we should not be able to go over from the passive form to the active without examining the particular case in hand. But if both transitions can always be made sa/va veritate, then this con- firms that what is true here, namely the thought, is not affected by this change of form. This serves as a warning not to attach too much weight to linguistic distinctions, as logicians are prone to: a case in point being the assumption that every thought-or judgement as it is usually called-has a subject and a predicate, so that the subject and predicate of a thought are determined by the thought, as the subject and predicate of a sentence are unambiguously given along with the sentence.
Like ethics, logic can also be called a normative science. How must I think in order to reach the goal, truth? We expect logic to give us the answer to this question, but we do not demand of it that it should go into what is peculiar to each branch of knowledge and its subject-matter. On the. contrary, the task we assign logic is only that of saying what holds with the utmost generality for all thinking, whatever its subject-matter. We must assume that the rules for our thinking and for our holding something to be true are prescribed by the laws of truth. The former are given along with the latter. Consequently we can also say: logic is the science of the most general laws of truth. The reader may find that he can form no very precise impression from this description of what is meant. The author's inadequacy and the awkwardness of language are probably to blame for this. But it il only a question of giving a rough indication of the goal of logic. What is still lacking in the account will have to be made good as we go on.
Now it would be futile to employ a definition in order to make it clearer what is to be understood by 'true'. If, for example, we wished to say 'an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form 'A is true if and only if it has such-and-such properties or stands in
1 Frege only took the table of contents as far as p. 141 (ed. ).
? Logic 129
such-and-such a relation to such-and-such a thing'. In each case in hand it would always come back to the question whether it is true that A has such- and-such properties, or stands in such-and-such a relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever.
If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and 3 make 5. So I assert that it is true that my idea of Cologne Cathedral agrees with reality, ifl assert that it agrees with reality. Therefore it is really by using the form of an assertoric sentence that we assert truth, and to do this we do not need the word 'true'. Indeed we can say that even where we use the form of expression 'it is true that . . . 'the essential thing is really the assertoric form of the sentence.
We now ask: what can the predicate 'true' be applied to? The issue here is to delimit the range of application of the word. Whatever else may be the case, the word cannot be applied to anything that is material. If there is any doubt about this, it could arise only for works of art. But if we speak of truth in connection with these, then we are surely using the word with a different meaning from the one that is meant here. In any case it is only as a work of art that a thing is called true. If a thing had come into existence through the
blind play of natural forces, our predicate would be clearly inappropriate. Por the same reason we are excluding from consideration the use that is made by, say, an art critic when he calls feelings and experiences true.
No one would deny that our predicate is, for the most part, ascribed to sentences. We are not, however, concerned with sentences expressing wishes, questions, requests and commands, but only with assertoric sentences, sentences that is to say, in which we communicate facts and propound mathematical laws or laws of nature.
Further, it is clear that we do not, properly speaking, ascribe truth to the series of sounds which constitute a sentence, but to its sense; for, on the one hand, the truth of a sentence is preserved when it is correctly translated into another language, and, on the other hand, it is at least conceivable that the same series of sounds should have a true sense in one language and a false sense in another.
We are here including under the word 'sentence' the main clause of a sentence and clauses that are subordinate to it.
In the cases which alone concern logic the sense of an assertoric sentence is either true or false, and then we have what we call a thought proper. But there remains a third case of which at least some mention must be made here.
The sentence 'Scylla has six heads' is not true, but the sentence 'Scylla does not have six heads' is not true either; for it to be true the proper name 'Scylla' would have to designate something. Perhaps we think that the name
? ? 130 Logic
'Scylla' does designate something, namely an idea. In that case the first question to ask is 'Whose idea? ' We often speak as if one and the same idea occurred to different men, but that is false, at least if the word 'idea' is used in the psychological sense: each man has his own idea. But then an idea does not have heads, and so one cannot cut heads off an idea either. The name 'Scylla' does not therefore designate an idea. Names that fail to fulfil the usual role of a proper name, which is to name something, may be called mock proper names. Although the tale of Willian Tell is a legend and not history and the name 'William Tell' is a mock proper name, we cannot deny it a sense. But the sense of the sentence 'William Tell shot an apple off his son's head' is no more true than is that of the sentence 'William Tell did not shoot an apple off his son's head'. I do not say, however, that this sense is false either, but I characterize it as fictitious. This may elucidate the sense in which I am using the word 'false', which is as little susceptible of a definition proper as is the word 'true'.
If the idealist theory of knowledge is correct then all the sciences would belong to the realm of fiction. Indeed one might try to reinterpret all sentences in such a way that they were about ideas. By doing this, however, their sense would be completely changed and we should obtain quite a different science; this new science would be a branch of psychology.
Instead of speaking of 'fiction', we could speak of 'mock thoughts'. Thus if the sense of an assertoric sentence is not true, it is either false or fictitious, and it will generally be the latter if it contains a mock proper name. * The writer, in common with, for example, the painter, has his eye on appearances. Assertions in fiction are not to be taken seriously: they are only mock assertions. Even the thoughts are not to be taken seriously as in the sciences: they are only mock thoughts. If Schiller's Don Car/os were to be regarded as a piece of history, then to a large extent the drama would be false. But a work of fiction is not meant to be taken seriously in this way ati all: it's all play. Even the proper names in the drama, though they: correspond to names of historical personages, are mock proper names; they are not meant to be taken seriously in the work. We have a similar thing in the case of an historical painting. As a work of art it simply does not claim to give a visual representation of things that actually happened. A picture that was intended to portray some significant moment in history with photographic accuracy would not be a work of art in the higher sense of the word, but would be comparable rather to an anatomical drawing in a scientific work.
The logician does not have to bother with mock thoughts, just as 1 physicist, who sets out to investigate thunder, will not pay any attention to stage-thunder. When we speak of thoughts in what follows we mean thoughts proper, thoughts that are either true or false.
*We have an exception where a mock proper name occurs within a clause in indirect speech.
? ? Logic 131
The sense of an assertoric sentence I call a thought. Examples of thoughts are laws of nature, mathematical laws, historical facts: all these find expression in assertoric sentences. I can now be more precise and say: The predicate 'true' applies to thoughts.
Of course we speak of true ideas1 as well. By an idea we understand a picture that is called up by the imagination: unlike a perception it does not consist of present impressions, but of the reactivated traces of past impressions or actions. Like any other picture, an idea is not true in itself, hut only in relation to something to which it is meant to correspond. If it is said that a picture is meant to represent Cologne Cathedral, fair enough; it can then be asked whether this intention is realized; if there is no reference to an intention to depict something, there can be no question of the truth of a picture. It can be seen from this that the predicate true is not really conferred on the idea itself, but on the thought that the idea depicts a certain object. And this thought is not an idea, nor is it made up of ideas in any way. Thoughts are fundamentally different from ideas (in the psychological sense). The idea of a red rose is something different from the thought that this rose is red. Associate ideas or run them together as we may, we shall still finish up with an idea and never with something that could be true. This difference also comes out in the modes we have of communicating. The proper means of expression for a thought is a sentence. But a sentence is hardly an appropriate vehicle for conveying an idea. I have only to remind you how inadequate any description is by comparison with a pictorial representation. Things are not so bad where it is a matter of representing sounds, since we have the resources of onomatopoeia; but onomatopoeia has nothing whatever to do with the expression of thoughts, and whilst in translation the play of sounds is easily lost, the thought must be preserved if we are to speak of a translation at all. Conversely, pictures and musical
compositions without accompanying words are hardly suited for expressing thoughts. It is true that we may associate all kinds of thoughts with some work of art or other but there is no necessary connection between the two, and we are not surprised when someone else associates different thoughts with it.
In order to shed a clearer light on the peculiarity of the predicate true, let us compare it with the predicate beautiful. We can see, to begin with, that what is beautiful admits of degree, but what is true does not. We can think two objects beautiful, and yet think one more beautiful than the other. On the other hand, if two thoughts are true, one is not more true than the other. And here there emerges the essential difference that what is true is true
1 In the German, 'Vorstellungen'. Throughout this essay the difficult word ? Vorstellung' has been generally rendered by 'idea'. Admittedly this makes certain passages read unnaturally, but the gist of what Frege is saying should be clear if the reader bears in mind the explanation he gives here of how the term 'VorstellunJ. :' is hcing used (trans. ).
? 132 Logic
independently of our recognizing it as such, but what is beautiful is beautiful only for him who experiences it as such. What is beautiful for one person is not necessarily beautiful for another. There is no disputing tastes. Where truth is concerned, there is the possibility of error, but not where beauty is concerned. By the very fact that I consider something beautiful it is beautiful for me. But something does not have to be true because I consider it to be true, and if it is not true in itself, it is not true for me either. Nothing is beautiful in itself: it is only beautiful for some being experiencing it and this is necessarily implicit in any aesthetic judgement. Now it is true that we also make judgements of this kind which seem to lay claim to being objective. Whether we are aware of it or not the assumption of a normal human being always underlies such judgements, and each one of us cannot help but think that he himself is so close to the normal human being that he believes he can speak in his name. What, then, we mean by 'This rose is beautiful' is 'This rose is beautiful for a normal human being'. But what is normal? That depends on the circle of human beings one has in mind. If there is some remote mountain valley where nearly all the people have goitres, then having a goitre will be looked on as normal there, and those who lack such an adornment will be considered ugly. How is a negro from the heart of Africa to be weaned from the view that the narrow nose of the European is ugly, whereas the broad nose of the negro is beautiful? And cannot a negro qua negro be just as normal as a white man qua white man? Cannot a child be
just as normal as a grown-up? The ideas that are awakened in us by the power of association have a great influence on the judgements a man forms of what is beautiful, and these ideas depend upon what he has absorbed in earlier life. But this varies from person to person. And even if we managed to define a normal human being and so 'beautiful' in an objective sense, it would still be only possible to do this on the basis of the subjective sense. Far from having rid ourselves of this, we would have recognized it as the root sense. We could not alter the situation by trying to substitute an ideal human being for a normal one. In the absence of experiences and ideas there would be no instance of anything subjectively beautiful and therefore no instance of anything objectively beautiful either. There is therefore much to be said for the view that the real work of art is a structure of ideas within us and that the external thing-the painting, the statue-is only a means for producing the real work of art in us. On this view, anyone who enjoys a work of art has his own work of art, with the consequence that there is no contradiction whatever between varying aesthetic judgements. Hence: dl gustibus non disputandum.
If anyone tried to contradict the statement that what is true is true independently of our recognizing it as such, he would by his very assertion contradict what he had asserted; he would be in a similar position to the Cretan who said that all Cretans are liars.
To elaborate: if something were true only for him who held it to be true, there would be no contradiction between the opinions of different people. So
? Logic 133
to be consistent, any person holding this view would have no right whatever to contradict the opposite view; he would have to espouse the principle: non disputandum est. He would not be able to assert anything at all in the normal sense, and even if his utterances had the form of assertions, they would only have the status of interjections--of expressions of mental states or processes, between which and such states or processes in another person there could be no contradiction. And in that case his assertion that something was true only for us and through being recognized by us as such would have this status too. If this view were true, it would be impossible to daim that any of his own opinions was more justified in the eyes of others than the opposite opinion. A view that made such a claim would be unjustified; this would mean, however, that every opinion would be unjustified in the usual sense of the word, and so also those opinions to which we were opposed. There would be no science, no error and no ~:orrection of error; properly speaking, there would be nothing true in the uormal sense of the word. For this is so closely bound up with that independence of being recognized as true, which we are emphasizing here, that it cannot be separated from it. If anyone seriously and sincerely defended the view we are here attacking, we should have no recourse but to ussume that he was attaching a different sense to the word 'true'.
We can go a step further. In order to be true, thoughts--e. g. laws of nature-not only do not need to be recognized by us as true: they do not have to have been thought by us at all. A law of nature is not invented by us, hut discovered, and just as a desolate island in the Arctic Ocean was there long before anyone had set eyes on it, so the laws of nature, and likewise those of mathematics, have held good at all times and not just since they were discovered. This shows us that these thoughts, if true, are not only true independently of our recognizing them to be so, but that they are indepen- dent of our thinking as such. A thought does not belong specially to the person who thinks it, as does an idea to the person who has it: everyone who grasps it encounters it in the same way, as the same thought. Otherwise two people would never attach the same thought to the same sentence, but cuch would have his own thought; and if, say, one man put 2 ? 2 = 4 forward ns true whilst another denied it, there would be no contradiction, because what was asserted by one would be different from what was rejected by the other. It would be quite impossible for the assertions of different people to contradict one another, for a contradiction occurs only when it is the very Nume thought that one person is asserting to be true and another to be false. So a dispute about the truth of something would be futile. There would Nimply be no common ground to fight on; each thought would be enclosed in its own private world and a contradiction between the thoughts of ditTerent people would be like a war between ourselves and the inhabitants of Mars. Nor must we say that one person might communicate his thought to another and a conflict would then flare up in the latter's private world. It would be quite impossible for a thought to be so communicated that it
? 134 Logic
should pass out of the private world of one person into that of another. The thought that entered the latter's mind as a result of the communication would be different from the thought in the former's mind; and the slightest alteration can transform a truth into a falsehood. If we wanted to regard a thought as something psychological, as a structure of ideas, without, however, adopting a wholly subjective standpoint, we should have to explain the assertion that 2 + 3 = 5 on something like the following lines 'It has been observed that with many people certain ideas form themselves in association with the sentence "2 + 3 = 5". We call a formation of this kind the sense of the sentence "2 + 3 = 5". So far as we have observed hitherto these formations are always true; we may therefore make the provisional statement "Going by the observations made hitherto, the sense of the sentence '2 + 3 = 5' is true". ' But it is obvious that this explanation would not work at all. And it would leave us where we were, for the sense of the sentence 'It has been observed that with many people certain ideas form themselves etc. ' would of course be a formation of ideas too and the whole thing would begin over again. A soup that tastes pleasant to one person, may be nauseous to another. In such a case each person is really making a judgement about his own sensation of taste, and this is different from the other's. The same would hold for thoughts if a thought were related to a sentence in the same kind of way as sensations of taste are related to the chemical stimuli that excite them.
If a thought, like an idea, were something private and mental, then the truth of a thought could surely only consist in a relation to something that was not private or mental. So if we wanted to know whether a thought was true, we should have to ask whether the relation in question obtained and thus whether the thought that this relation obtained was true. And so we should be in the position of a man on a treadmill who makes a step forwards and upwards, but the step he treads on keeps giving way and he falls back to where he was before.
A thought is something impersonal. If we see the sentence '2 + 3 = 5' written on a wall, we have no difficulty at all in recognizing the thought expressed by it, and we do not need to know who has written it there in order to understand it.
A sentence like 'I am cold' may seem to be a counter-example to our thesis that a thought is independent of the person thinking it, in so far as it can be true for one person and false for another, and thus not true in itself. The reason for this is that the sentence expresses a different thought in the mouth of one person from what it expresses in the mouth of another. In this case the mere words do not contain the entire sense: we have in addition to take into account who utters it. There are many cases like this in which the spoken word has to be supplemented by the speaker's gesture and expression, and the accompanying circumstances. The word 'I' simply designates a different person in the mouths of different people. It is not necessary that the person who feels cold should himself give utterance to the
? Logic 135
thought that he feels cold. Another person can do this by using a name to designate the one who feels cold.
In this way a thought can be clothed in a sentence that is more in keeping with its being independent of the person thinking it. The possibility of doing this distinguishes it from a mental state expressed by an interjection. Words like 'here' and 'now' only acquire their full sense through the circumstances in which they are used. If someone says 'it is raining' the time and place of utterance has to be supplied. If such a sentence is written down it often no longer has a complete sense because there is nothing to indicate who uttered it, and where and when. As regards a sentence containing a judgement of taste like 'This rose is beautiful', the identity of the speaker is essential to the sense, even though the word 'I' does not occur in it. So the explanation for all these apparent exceptions is that the same sentence does not always express the same thought, because the words need to be supplemented in order to get a complete sense, and how this is done can vary according to the circumstances.
Whereas ideas (in the psychological sense of the word) have no fixed boundaries, but are constantly changing and, Proteus-like, assume different forms, thoughts always remain the same. It is of the essence of a thought to he non-temporal and non-spatial. In the case of the thought that 3 + 4 = 7 and the laws of nature there is hardly any need to support this statement. If it should turn out that the law of gravitation ceased to be true from a certain moment onwards, we should conclude that it was not true at all, and put ourselves out to discover a new law: the new one would differ in containing a condition which would be satisfied at one time but not at another. It is the same with place. If it should transpire that the law of gravitation was not valid in the neighbourhood of Sirius, we should search for another law which contained a condition that was satisfied in our solar system but not in the neighbourhood of Sirius. If someone wished to cite, say, 'The total number of inhabitants of the German Empire is 52 000 000', as a counter-example to the timelessness of thoughts, I should reply: This sentence is not a com- plete expression of a thought at all, since it lacks a time-determination. If we add such a determination, for example, 'at noon on l January 1897 by central European time', then the thought is either true, in which case it is always, or better, timelessly, true, or it is false and in that case it is false without qualification. This holds of any particular historical fact: if it is true, it is true independently of the time at which it is judged to be true. It is no objection that a sentence may acquire a different sense in the course of time; for what changes in such a case is of course the language, not the thought. In another language this shift need not take place.
It is true of course that we speak of men's thoughts as being liable to change. However it is not the thoughts which are true at one time and false at another: it is only that they arc held to be true at one time and false at another.
What if it is objected that I am attaching to the word 'thought' a sense that it does not ordinarily have, and that other people understand by it an
? 136 Logic
act of thinking, which is obviously private and mental? Well, the important thing is that I remain true to my way of using it; whether this agrees with the ordinary use is of less importance. It may well be the case that people sometimes understand by the word 'thought' an act of thinking-in any case this is not always so*-and such an act cannot be true.
In logic, as in other sciences, it is open to us to coin technical terms,
? Dedekind, in theor. 66 of his book Was sind und was sol/en die Zahlen? , uses this word as I do. For he is attempting there to prove that the totality of things that can be objects of his thinking is infinite. Let s be such an object; then Dedekind calls rp(s) the thought that s can be an object of his thinking. And this thought can now itself be an object of his thinking. Thus rp(rp(s)) is the thought that the thought that s can be an object of his thinking can be an object of his thinking. We can see from this what 'rp(rp(rp(s)))', 'rp(rp(rp(rp(s))))' and so on, are supposed to mean. It is essential to the proof that the sentence 's can be an object of Dedekind's thinking' always expresses a thought when the letter 's' designates such an object. Now if, as Dedekind wishes to prove, there are infinitely many such objects s, there must also be
infinitely many such thoughts rp(s). Now presumably we shall not hurt Dedekind's feelings if we assume that he has not thought infinitely many thoughts. Equally he should not assume that others have already thought infinitely many thoughts which could be the objects of his thinking; for this would be to assume what was to be proved. Now if infinitely many thoughts have not yet been thought, the infinitely many thoughts rp(s) must comprise infinitely many thoughts that are not thought, in which case it cannot be essential to a thought that it should be thought. And this is precisely what I am maintaining. If there were only thoughts that are thought, the sign 'rp(s)' would not always have a meaning; to ensure that it did have, it is not sufficient for 's' to mean something that could be an object of Dedekind's thinking: it would also have to have been thought by someone in order to be a possible object of Dedekind's thinking. If this were not the case, then the sign 'rp(s)' would have no meaning for the given s. The sun (0) can be an object of Dedekind's thinking; hence the first two members and perhaps a few successive members of the series '0, 'rp(0)', rp(rp(0))' . . . have a meaning. But as we progressed along the series we would be bound eventually to reach a member that was meaningless, because the thought which it was meant to designate had not been thought, and so was not to hand. In that case 'rp(s)' would resemble a power series which did not converge for every value of the argument. The fact that the series diverged would correspond to the sign 'rp(s)' becoming meaningless. If we assume a power series with radius of convergence 4 and if we assume, further, that the series has the value 2 for l as argument and the value 5 for 2 as argument, then the corresponding series of numbers l, 2, 5 comes to an end at this point and does not go on to infinity. In the same way the series 0, rp(0), rp(rp(0)), does not go on to infinity if there are only thoughts that are thought. So the validity of Dedekind's proofs rests on the assumption that thoughts obtain independently of our thinking. We can see how this use is one to which the word 'thought' naturally lends itself.
? Logic 137
regardless of whether the words are always used in precisely that way in l'vcryday life. It doesn't matter if the meaning we fix on is not altogether in line with everyday use or doesn't accord with the word's etymology; what does matter is to make it as appropriate a vehicle as possible for use in npressing laws. Provided there is no loss of rigour, the more compendious the formulation of the complete system of laws, the more felicitous is the apparatus of technical terms.
Now we cannot regard thinking as a process which generates thoughts. It would be just as wrong to identify a thought with an act of thinking, so that a thought is related to thinking as a leap is to leaping. This view is in harmony with many of our ways of talking. For do we not say that the same thought ~~grasped by this person and by that person? And that each person has the same thought over and over again? Now if thoughts only came into txistence as a result of thinking or if they were constituted by thinking, then the same thought could come into existence, cease to exist, a! )d then come 111to existence again, which is absurd. As I do not create a tree by looking at 1t or cause a pencil to come into existence by taking hold of it, neither do I 1/,Cnerate a thought by thinking. And still less does the brain secrete thoughts, as the liver does gall.
The metaphors that underlie the expressions we use when we speak of 11. rasping a thought, of conceiving, laying hold of, seizing, understanding, of mpere, percipere, comprehendere, intelligere, put the matter in essentially the right perspective. What is grasped, taken hold of, is already there and all we do is take possession of it. Likewise, what we see into or single out from nmongst other things is already there and does not come into existence as a result of these activities. Of course all metaphors go lame at some point. We nrc inclined to regard what is independent of our mental processes as Nornething spatial or material, and the words that we have just listed make it look as if this is what a thought actually is. But this is not where the point of lhc comparison lies. What is independent of our mental processes, what is ohjcctive, does not have to be spatial or material or actual. If we were to disregard this we should easily slip into a kind of mythology. To say 'The lnws of gravitation, of inertia, of the parallelogram of forces cause the earth lo move as it does move', might make it look as if these laws, so to speak, look the earth by the ears and kept it on the path they prescribe. Such a use llfthe words 'affect', 'cause', would be misleading. On the other hand, it is nil right to say that the sun and planets act on one another in accordance with the laws of gravitation.
So even if physical bodies and thoughts resemble one another in being Independent of my inner life, we are not entitled to conclude from this that thoughts can be moved as bodies can, or can be smelled or tasted, and it would be invalid to seek somehow to draw from the absurdity of such Inference an objection to our views. Although a law of nature obtains quite Independently of whether we think of it or not, it does not emit light or Mound waves by which our visual or auditory nerves could be affected. But
? 138 Logic
do I not then see that this flower has five petals? We can say this, but if we do, the word 'see' is not being used in the sense of having a mere visual experience: what we mean by it is bound up with thinking and judging. Newton did not discover the law of gravitation because his senses were especially acute.
If we wish to speak of a thought as being actual, we can do so only in the sense that the knowledge that a man has of e. g. a law of nature has an influence on the decisions he makes, which in their turn may affect the course of history. We should then be thinking of the recognition of a law as a case of a law's acting upon us, and it is perhaps possible to do this, just as we can regard, say, the seeing of a flower as the flower's indirectly acting on us.
We can disregard thoughts and we can take possession of them. We might conceive of the latter as a case of our acting on thoughts, which seems to speak against their being timeless. But the thought is not changed in itself by being thus acted on, just as the moon is apparently unaffected whether we take any notice of it or not. So even though it may be possible to speak of thoughts as acting on us, we cannot speak of ourselves as acting on thoughts. We might cite, as an instance of thoughts being subject to change, the fact that they are not always immediately clear. But what is called the clarity of a thought in our sense of this word is really a matter of how thoroughly it has been assimilated or grasped, and is not a property of a thought.
It would be wrong to think that it is only true thoughts that obtained independently of our mental life, and that false ones, on the other hand, belonged, as ideas do, to our inner life. Almost everything that we have said about the predicate true holds for the predicate false as well. In the strict sense it applies only to thoughts. Where it looks to be predicated of sentences or ideas, still at bottom it is being predicated of thoughts. What is false is false in itself and independently of our opinions. A dispute over the falsity of something is at the same time a dispute over the truth of some- thing. Therefore the thing whose falsity can be a matter for dispute does not belong to some mind or other.
Separating a Thought from its Trappings
In an assertoric sentence two different kinds of thing are usually intimately bound up with one another: the thought expressed and the assertion of its truth. And this is why these are often not clearly distinguished. However, one can express a thought without at the same time putting it forward as true. A scientist who makes a scientific discovery usually begins by grasping just a though,t, and then he asks himself whether it is to be recognized as true; it is not until his investigation has turned out in favour of the hypothesis, that he ventures to put it forward as true. We express the same
? Logic 139
thought in the question 'Is oxygen condensable? ' and in the sentence 'Oxygen is condensable', joining it in the one case with a request and in the other with an assertion.
When we inwardly recognize that a thought is true, we are making a judgement: when we communicate this recognition, we are making an a. 1? sertion.
We can think without making a judgement.
We have seen that the series of sounds that compose a sentence is often not sufficient for the complete expression of a thought. If we wish to bring the essence of a thought into as sharp a focus as possible, we ought not to overlook the fact that the converse case is not uncommon, the case where a sentence does more than express a thought and assert its truth. In many cases a sentence is meant to have an effect on the ideas and feelings of the hearer as well; and the more closely it approximates to the language of poetry, the greater this effect is meant to be. We have indeed stressed the lnct that language is but poorly suited for calling up at will an idea in the 111ind of a hearer with any precision. Who would ever rely on words to evoke as precise a mental picture of an Apollo as can be produced without difficulty by looking at a piece of sculpture? Even so, we do say that the poet paints pictures. And in fact it cannot be denied that the spoken word affects the ideas we have just because it enters consciousness as a complex of auditory sensations. Right from the start we experience the series of sounds themselves, the tone of the voice, the intonation and rhythm with kclings of pleasure or displeasure. These sensations of sound are linked to nuditory ideas that resemble them and these latter are linked in turn with further ideas reactivated by them. This is the domain of onomatopoeia. Here we may cite the Homeric verse (Odyssey IX, 71): rpzx{}d Kaz' rerpax{}a l~ti:oxzoev le; av'ep,ow.
This is quite independent of the aim of words to express thoughts. Here the sounds are acting only as a sensory stimulus. But because sequences of such sounds are meant to have a sense they act upon the imagination in yet n different way. Anyone who hears the word 'horse' and understands it will probably have straightaway a picture of a horse in his mind. This picture, however, is not to be confused with the sense of the word 'horse'; for the word 'horse' gives no clue to the colour of the horse, or to its carriage when Ntnnding still or in motion, or to the side from which it is seen and the like. If different men were able, say, immediately to project onto a canvas the ideas that sprung up in their minds on hearing the word 'horse', then we should be presented with quite different pictures. And even with the same man the word 'horse' does not always conjure up the same idea. Here a great deal depends on the context. We may compare e. g. the sentences 'With what joy he rides his gallant horse' and 'I just saw a horse stumble on the wet nsphalt'.
So there can be no question of the same idea always being associated with the word 'horse'. Thus in virtue of its sense such a word will excite a certain
? 140 Logic
idea in us, but by itself it is far from determining this idea completely. Generally speaking the most we are entitled to assume is that the ideas of the speaker and hearer are very roughly in agreement. If several artists produce, independently of one another, illustrations of the same poem, they will diverge considerably from one another in the portrayal they give. Thus the poet does not really depict anything: he only provides the impetus for others to do so, furnishing hints to this end, and leaving it to the hearer to give his words body and shape. And in this connection it is useful to the poet to have at his disposal a number of different words that can be substituted for one another without altering the thought, but which can act in different ways on the feelings and imagination of the hearer. We may think e. g. of the words 'walk', 'stroll', 'saunter'. These means are also used to the same end in everyday language. If we compare the sentences 'This dog howled the whole night' and 'This cur howled the whole night', we find that the thought is the same. The first sentence tells us neither more nor less than does the second. But whilst the word 'dog' is neutral as between having pleasant or unpleasant associations, the word 'cur' certainly has unpleasant rather than pleasant associations and puts us rather in mind of a dog with a somewhat unkempt appearance. Even if it is grossly unfair to the dog to think of it in this way, we cannot say that this makes the second sentence false. True, anyone who utters this sentence speaks pejoratively, but this is
not part of the thought expressed. What distinguishes the second sentence from the first is of the nature of an interjection. It might be thought that the second sentence does nevertheless tell us more than the first, namely that the speaker has a poor opinion of the dog. In that case, the word 'cur' would contain an entire thought. We can put this to the test in the following way.
We assume that the first sentence is true and the second sentence is spoken by someone who does not actually feel the contempt which the word 'cur' seems to imply. If the objection were correct, the second sentence would now contain two thoughts, one of which was false; so it would assert something false as a whole, whilst the first sentence would be true. We shall hardly go along with this; rather the use of the word 'cur' does not prevent us from holding that the second sentence is true as well. For we have to make a distinction between the thoughts that are expressed and those which the speaker leads others to take as true although he does not express them. If a commander conceals his weakness from the enemy by making his troops keep changing their uniforms, he is not telling a lie; for he is not expressing any thoughts, although his actions are calculated to induce thoughts in others. And we find the same thing in the case of speech itself, as when one gives a special tone to the voice or chooses special words. If someone announces the news of a death in a sad tone of voice without actually being sad, the thought expressed is still true even if the sad tone is assumed in oq:ler to create a false impression. And we can substitute words like 'ah', and 'unfortunately' for such a tone of voice without altering the thought. Naturally things are different if certain actions are specifically
? Logic 141
agreed on as a means of communicating something. In language common usage takes the place of such agreements. Of course borderline cases can arise because language changes. Something that was not originally employed as a means of expressing a thought may eventually come to do this because it has constantly been used in cases of the same kind. A thought which to hegin with was only suggested by an expression may come to be explicitly asserted by it. And in the period in between different interpretations will he possible. But the distinction itself is not obliterated by such fluctuations in language. In the present context the only essential thing for us is that a different thought does not correspond to every difference in the words used, and that we have a means of deciding what is and what is not part of the thought, even though, with language constantly developing, it may at times he difficult to apply.
The distinction between the active and passive voice belongs here too. The sentences 'M gave document A to N', 'Document A was given by N by M', 'N received document A from M' express exactly the same thought; we learn not a whit more or less from any one of these sentences than we do from the others. Hence it is impossible that one of them should be true whilst another is false. It is the very same thing that is here capable of being true or false. For all this we are not in a position to say that it is a matter of complete indifference which of these sentences we use. As a rule stylistic and uesthetic reasons will give the preference to one of them. If someone asks 'Why has A been arrested? ' it would be unnatural to reply 'B has been rnurdered by him', because it would require a needless switch of the uttention from A to B. Although in actual speech it can certainly be very important where the attention is directed and where the stress falls, it is of no concern to logic.
In translating from one language to another it is sometimes necessary to dispense with the original grammatical construction altogether. Nevertheless, this need not affect the thought and it must not do so, if the translation is to be correct. But it is sometimes necessary to sacrifice the feeling and colour of the original.
Again in the two sentences 'Frederick the Great won the battle of Rossbach' and 'It is true that Frederick the Great won the battle of Rossbach', we have, as we said earlier, the same thought in a different verbal form. In affirming the thought in the first sentence we thereby affirm the thought in the second, and conversely. There are not two different acts of judgement, but only one.
(From all this we can see that the grammatical categories of subject and predicate can have no significance for logic. )
The distinction between what is part of the thought expressed in a Ncntence and what only gets attached to the thought is of the greatest Importance for logic. The purity of the object of one's investigation is not of Importance only to the chemist. How would the chemist be able to recognize, hcyond any doubt, that he has arrived at the same results by different
? 142 Logic
means, if the apparent difference of means could be traced back to impurities in the substances used? There is no doubt that the first and most important discoveries in a science are often a matter of recognizing something as the same again. However self-evident it may seen to us that it is the same sun which went down yesterday and rose today, and however insignificant therefore this discovery may seem to us, it has certainly been one of the most important in astronomy and perhaps the one that really laid the foundations of the science. It was also important to recognize that the morning star is the same as the evening star, that three times five is the same as five times three. It is just as important not to distinguish what is the same as it is to be alive to differences when they don't hit one in the eye. So it is quite wrong to think that one can never make too many distinctions. It does nothing but harm to insist on distinctions where they are not relevant. Thus in general mechanics we shall take care not to speak of the chemical differences between substances and not to state the law of inertia in a special form for, say, each chemical element. We shall only take those differences into account that are essential to the formulation of the laws with which we are actually concerned. Above all, we must not let ourselves be seduced by the presence of extraneous factors into seeing distinctions where there are none.
In logic we must reject all distinctions that are made from a purely psychological point of view. What is referred to as a deepening of logic by psychology is nothing but a falsification of it by psychology.
In human beings it is natural for thinking to be intermingled with having images and feeling. Logic has the task of isolating what is logical, not, to be sure, so that we should think without having images, which is no doubt impossible, but so that we should consciously distinguish the logical from what is attached to it in the way of ideas and feelings. There is a difficulty here in that we think in some language or other and that grammar, which has a significance for language analogous to that which logic has for judgement, is a mixture of the logical and the psychological. If this were not so, all languages would necessarily have the same grammar. It is true that we can express the same thought in different languages; but the psychological trappings, the clothing of the thought, will often be different. This is why the learning of foreign languages is useful for one's logical education. Seeing that the same thought can be worded in different ways, we learn better to distinguish the verbal husk from the kernel with which, in any given language, it appears to be organically bound up. This is how the differences between languages can facilitate our grasp of what is logical. But still the difficulties are not wholly removed in this way, and our logic books still keep dragging in a number of things-subject and predicate, for example-which do not, strictly speaking, belong to logic. For this reason it is useful to be acquainted also with a means of expressing thoughts that is of a radically different nature, such as we have in the formula-language of arithmetic or in my concept-script.
? Logic 143
The first and most important task is to set out clearly what the objects to be investigated are. Only if we do this shall we be able to recognize the same as the same: in logic too, such acts of recognition probably constitute the fundamental discoveries. Therefore let us never forget that two different sentences can express the same thought, that we are concerned with only that part of a sentence's content which can be true or false.
Even if there were only a jot more to the thought contained in the passive form than in the active, it would be conceivable that this jot should be false whilst the thought contained in the active form was true, and that we should not be entitled without more ado to go over from the active to the passive form. Likewise if there were only a jot more to the thought contained in the active form than in the passive, we should not be able to go over from the passive form to the active without examining the particular case in hand. But if both transitions can always be made sa/va veritate, then this con- firms that what is true here, namely the thought, is not affected by this change of form. This serves as a warning not to attach too much weight to linguistic distinctions, as logicians are prone to: a case in point being the assumption that every thought-or judgement as it is usually called-has a subject and a predicate, so that the subject and predicate of a thought are determined by the thought, as the subject and predicate of a sentence are unambiguously given along with the sentence.
