This defect can be removed by stipulating that by the square of an
object we are to understand the object itself if this object is not a number, hut that 'the square of a number' is to be understood in its arithmetical sense.
object we are to understand the object itself if this object is not a number, hut that 'the square of a number' is to be understood in its arithmetical sense.
Gottlob-Frege-Posthumous-Writings
An actual proof can scarcely be furnished.
We will have to assume an unprovable law here.
Of course it isn't as self-evident as one would wish for a law of logic.
And if it was possible for there to be doubts previously, these doubts have been reinforced by the shock the law has sustained from Russell's paradox.
Yet let us set these doubts on one side for the moment. When we undertake the transformation as above, we acknowledge that there is one and only one object which we designate by the proper name 'the extension of the concept square root o f I', and that we also designate the same object by the proper name 'the extension of the concept binomial coefficient ofthe exponent of-1'. Perhaps you suspect that this object is something which we have just called an aggregate; but we will see that an extension of a concept is essentially different from an aggregate.
In the first instance we have in the case of a concept the fundamental case of subsumption. which we express in such a sentence as 'A is a 1/>'; where 'A' stands for a proper name, '1/>' for a concept-word. Now if the extension of the concept 1/>, coincides with the extension of the concept 'P, it follows from 'A is a 1/>' that A is a 'P too. Therefore we then also have a relation of the object A to th~ extension of the concept cfJ, which I will call B. And the fact that this relation holds I will express thus: 'A belongs to IJ'. And so this is meant to be tantamount to 'A is a 1/>, if B is the extension of the concept C/J'.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 183
On superficial reftexion you might compare this relation with that of part to whole; but here we have nothing corresponding to the sentence: What is part of a part, is part of the whole. To be sure, A itself can be the extension of a concept; but ifL1 belong to A, and A to B, L1 need not belong to B. In the sentence 'The extension of the concept prime is an extension of a concept to which 3 belongs' the phrase 'extension of a concept to which 3 belongs' is to be regarded as a nomen appellativum. Now let B be the extension of the concept hereby designated, and A the extension of the concept prime. Then A belongs to B and 2 belongs to A; but 2 does not belong to B; for 2 isn't an extension of a concept to which 3 belongs. From this alone it follows that an extension of a concept is at bottom completely different from an aggregate. The aggregate is composed of its parts. Whereas the extension of a concept is not composed of the objects that belong to it. For the case is conceivable that no objects belong to it. The extension of a concept simply has its being in the concept, not in the objects which belong to it; these are not its parts.
! 'here cannot be an aggregate which has no parts.
Now of course it can happen that all objects which belong to the
extension of a concept are at the same time parts of an aggregate and what is more in such a way that the whole being of the aggregate is completely exhausted by them. In this way it may look as though in such a case the aggregate coincides with the extension of the concept; but it isn't necessary that every part of the aggregate also belongs to the extension of the concept; for it can be that a part of the aggregate, without itself belonging to the extension of the concept, has parts which belong to the extension of the concept, or that a part of the aggregate, without itself belonging to the extension of the concept, is part of an object which belongs to the extension of the concept. So the relation of a part to the aggregate must still always be distinguished from that of an object to the extension of a concept to which it belongs. The extension of the concept is not determined by the aggregate even in this case, where they apparently coincide. A grain of sand is an aggregate. And it can be that the extension of the concept silicic acid molecule contained in this grain of sand apparently coincides with the aggregate which we call this grain of sand. But we could just as well let the extension of the concept atom contained in this grain ofsand coincide with our aggregate. But in that case the two extensions of concepts would coincide, which is impossible. From which it follows that neither of the two extensions of concepts coincides with the aggregate, for if one of them were to do so, then the other could with equal right be said to do so.
? ? What may I regard as the Result ofmy Work? 1 [Aug. 1906]
It is almost all tied up with the concept-script. a concept construed as a function. a relation as a function of two arguments. the extension of a concept or class is not the primary thing for me. unsaturatedness both in the case of concepts and functions. the true nature of concept and function recognized.
strictly I should have begun by mentioning the judgement-stroke, the dissociation of assertoric force from the predicate . . .
Hypothetical mode of sentence composition . . .
Generality . . .
Sense and meaning . . .
1 According to a note of the previous editors these jottings bore the date '5. VIII. 06'. They are no doubt prefatory to the piece 'Einleitung in die Logik', which is divided into sections according to the captions given here (ed. ).
? Introduction to Logic1 [August 1906]
Dissociating assertoric force from the predicate
We can express a thought without asserting it. But there is no word or sign in language whose function is simply to assert something. This is why, apparently even in logical works, predicating is confused with judging. As a result one is never quite sure whether what logicians call a judgement is meant to be a thought alone or one accompanied by the judgement that it is true. To go by the word, one would think it meant a thought accompanied hy a judgement; but often in common usage the word does not include the actual passing of judgement, the recognition of the truth of something. I use the word 'thought' in roughly the same way as logicians use 'judgement'. To think is to grasp a thought. Once we have grasped a thought, we can recognize it as true (make a judgement) and give expression to our recognition of its truth (make an assertion). Assertoric force is to be dissociated from negation too. To each thought there corresponds an opposite, so that rejecting one of them is accepting the other. One can say that to make a judgement is to make a choice between opposites. Rejecting the one and accepting the other is one and the same act. Therefore there is no need of a special name, or special sign, for rejecting a thought. We may speak of the negation of a thought before we have made any distinction of parts within it. To argue whether negation belongs to the whole thought or to the predicative part is every bit as unfruitful as to argue whether a coat clothes a man who is already clothed or whether it belongs together with the rest of his clothing. Since a coat covers a man who is already clothed, it automatically becomes part and parcel with the rest of his apparel. We may, metaphorically speaking, regard the predicative component of a thought as a covering for the subject-component. If further coverings are added, these automatically become one with those already there.
The hypothetical mode of sentence composition
If someone says that in a hypothetical judgement two judgements are set in relation to one another, he is using the word 'judgement' so as not to include the recognition of the truth of anything. For even if the whole compound
5. VIll. 06
1 Parts of these uiary notes were later revised hy Frege; cf. pp. I97 IT. (ed. ).
? 186 Introduction to Logic
sentences is uttered with assertoric force, one is still asserting neither the truth of the thought in the antecedent nor that of the thought in the consequent. The recognition of truth extends rather over a thought that is expressed in the whole compound sentence. And on closer examination we find that in many cases the antecedent on its own does not express a thought, and nor does the consequent either (quasi-sentences). What we generally have in these cases is the relation of subordination between concepts. Here it is quite common to conflate two different kinds of things, which I was probably the first to distinguish: the relation I designate by the conditional stroke, and generality. The former corresponds roughly to what logicians intend by 'a relation between judgements'. That is, the sign for the relation (the conditional stroke) connects sentences with one another: these are sentences proper, and so each of them expresses a thought.
Now leaving myth and fiction on one side, and considering only those cases in which truth in the scientific sense is in question, we can say that every thought is either true or false, tertium non datur. It is nonsense to speak of cases in which a thought is true and cases in which it is false. The same thought cannot be true at one time, false at another. On the contrary, the cases people have in mind in speaking in this way always involve different thoughts, and the reason they believe the thought to be the same is that the form of words is the same; this form of words will then be a quasi- sentence. We do not always adequately distinguish the sign from what it expresses.
If there are two thoughts, only four cases are possible:
1. the first is true, and likewise the second; 2. the first is true, the second false;
3. the first is false, the second true;
4. both are false.
Now if the third of these cases does not hold, then the relation I have designated by the conditional stroke obtains. The sentence expressing the first thought is the consequent, the sentence expressing the second the antecedent. It is now almost 28 years since I gave this definition. I believed at the time that I had only to mention it and everyone else would immediately know more about it than I did. And now, after more than a quarter of a century has elapsed, the great majority of mathematicians have no inkling of the matter, and the same goes for the logicians. What pig? headedness! This way academics have of behaving reminds me of nothing so much as that of an ox confronted by a new gate: it gapes, it bellows, it tries to squeeze by sideways, but going through it-that might be dangerous. I can readily believe that it looks strange at first sight, but if it didn't, it would have been discovered long ago. Do we really have to go by our first cursOFy impression of the matter? Have we no time at all to reflect upon it? No, for what good could come of that! People probably feel the lack of an inner connection between the thoughts: we find it hard to accept
? Introduction to Logic 187
that it is only the truth or falsity of the thoughts that is to be taken into account, that their content doesn't really come into it all. This is connect;;d with what I realized about sense and meaning. Well, just let someone try to give an account in which the thought itself plays a bigger role and it will probably turn out that what has been added from the thought is at bottom quite superfluous, and that one has only succeeded in complicating the issue, or that the antecedent and consequent are not sentences proper, neilht:r being such as to express a thought, so that a relation has not been established between two thoughts as one intended, but rather betwecll concepts or relations. Is the relation I designate by the conditional ~troh. c i11 fact such as can obtain between thoug;,ts? Strictly speaking, no! Tlw most we can say here is that the sign for this relation (i. e. the conditional ~trokc) connects sentences. Later the definition will be filled out in such a way as to allow also names of objects to be connected hy the conditione1l strok'? . Al lirst this will be even harder to swallow. We need first to take a cln~er k1uk :rt generality if we are to find it acceptable.
Generality
It is only at this point that the need arises to ana(vse a thought inftJ purts none of which are thoughts. The simplest case is that of splitting a thought 111to two parts. The parts are different in kind, one being unsaturnted. the <lther saturated (complete). The thoughts we have to consider here are those designated in traditional logic as singular judgements. In such a thought something is asserted of an object. The sentence expressing such a 1bought 1s composed of a proper name-and this corresponds to the complete pan o f t h e t h o u g h t - - a n d a p r e d i c a t i v e p a r t , w h i c h corre~ponds t o t ! J , : unsaturated part of the thought. We should mention that, strictly spenkiuf. it is not in itself that a thought is singular, but only with respect to a po~sil>k way of analysing it. It is possible for the same thought, with respect tt' rJ. different analy~is, to appear as particular (Christ converted some men to his teaching). Proper names designate objects, and a singular thought i~ about nhjccts. But we can't say that an object is part of a thought as a proper rrame is part of the corresponding sentence. Mont Blanc with its masst~s nf snow and ice is not part of the thought that Mont Blanc is more than 4000 m. high; all we can say is that to the object there corresponds. in a certain \'ii'. Y that has yet to be considered, a part of the thought. By analysing a singulM 1hought we obtain components of the complete and of the unsaturated t~intL which of course cannot occur in isolation; but any component of the t? ne kind together with any component of the other kind will form n thuught. If we now keep the unsaturated part constant but vary the complete part, w? ? should expect some of the thoughts so formed to he true, and some fal>t'. But it can also happen that the whole lot arc true. F. g. let tire unsatur:1tcd romponent be expressed in the words 'is idcnticnl With itsdf'. This is then :1
? 188 Introduction to Logic
particular property of the unsaturated part. We thus obtain a new thought (everything is identical with itself), which compared to the singular thoughts (two is identical with itself, the moon is identical with itself) is general. However the word 'everything', which here takes the place of a proper name ('the moon'), is not itself a proper name, doesn't designate an object, but serves to confer generality of content on the sentence. In logic we can often be too influenced by language and it is in this way that the concept-script is of value: it helps to emancipate us from the forms of language. Instead of saying 'the moon is identical with itself' we can also say 'the moon is identical with the moon' without changing the thought. But in language it is impossible, in making the transition to the general statement, to allow the word 'everything' also to occur in two places. The sentence 'everything is identical with everything' would not have the desired sense. We may, taking a leaf from mathematics, employ a letter and say 'a is identical with a'. This letter then occupies the place (or places) of a proper name, but it is not itself a proper name; it does not have a meaning, but only serves to confer generality of content on a sentence. This use of letters, being simpler and, from a logical point of view, more appropriate, is to be preferred to the means which language provides for this purpose.
If a whole is composed of two sentences connected by 'and', each of which expresses a thought, then the sense of the whole is also to be construed as a thought, for this sense is either true or false; it is true if each component thought is true, and false in every other case-hence when at least one of the two component thoughts is false. If we call the thought of the whole the conjunction of the two component thoughts, then the conjunction too has its opposite thought, as does every thought. Now it is clear what the opposite of a conjunction of the opposite of a thought A with a thought B is. It is what I express by means of the conditional stroke. The seJ\tence expressing thought A is the consequent, that expressing thought B the antecedent. But the whole sentence expressing the opposite of the conjunction of the opposite of A with B may be called the hypothetical sentence whose consequent expresses A and whose antecedent expresses B. The thought expressed by the hypothetical sentence we shall call the
s. vnLo6 hypothetical thought whose consequence is A and whose condition is B. Now if the same proper name occurs in both consequent and antecedent, we may regard the hypothetical thought as singular if we think of it as bein& analysed into the complete part that corresponds to the proper name and the unsaturated part left over. If we now keep the unsaturated part fixed, and vary the complete part, it may turn out that we always obtain a true thought, no matter what we choose for the complete part. In saying this, we are assuming, as we are throughout this enquiry, that we are operating not in the realm of myth and fiction, but in that of truth (in the scientific sense); consequently every proper name really does achieve its goal of designatinl an object, and so is not empty. The complete parts of the thoughts that are here in question are of course not themselves the objects designated by the
? Introduction to Logic 189
proper names, but are connected with them, and it is essential that there should be such objects if everything is not to fall within the realm of fiction. Otherwise we cannot speak of the truth of thoughts at all. So we are assuming that we have a hypothetical thought-one which can at the same time be construed as a singular thought-from which we, as was said above, always obtain a true thought by keeping the unsaturated part fixed, whatever complete part we saturate it with. In this way we arrive at a general thought, and the singular hypothetical thought from which we started is seen to be a special case of it. For instance:
Thought A: that 3 squared is greater than 2.
Thought B: that 3 is greater than 2.
Opposite of Thought A: that 3 squared is not greater than 2.
Conjunction o f the opposite o f thought A with thought B: that 3 squared is not greater than 2, and that 3 is greater than 2.
Opposite ofthe conjunction ofthe opposite ofthought A with thought B: that it is false both that 3 squared is not greater than 2 and that 3 is greater than 2.
This is the hypothetical thought with thought A as its consequence and thought B as its condition. There is something unnatural about the form of words 'If 3 is greater than 2, then 3 squared is greater than 2' and perhaps ~:ven more so about what we get when we replace '3' by '2': 'If 2 is greater than 2, then 2 squared is greater than 2'. But the thought that it is false both 1hat 2 squared is not greater than 2 and that 2 is greater than 2 is a true one. And whatever number we take instead of 3, we always obtain a true thought. But what if we take an object that is not a number? Any sentence obtained from 'a is greater than 2' by putting the proper name of an object ror 'a' expresses a thought, and this thought is of course false if the object is not a number. It is different with the first sentence, because the expression which results when the proper name of an object is put for 'a' in 'a squared' only designates an object in ordinary discourse if this object is a number. The incompleteness of the usual definition of 'squared' is to be blamed for 1his.
This defect can be removed by stipulating that by the square of an
object we are to understand the object itself if this object is not a number, hut that 'the square of a number' is to be understood in its arithmetical sense. We shall then always obtain from the schema 'that a squared is Rreater than 2' a sentence expressing a false thought if 'a' is replaced by the JllllllC of an object that is not a number. Once this stipulation has been made, we can replace the numeral '3' in our hypothetical by the name of any object whatever, and we shall always obtain a sentence expressing a true thought. The general thought at which we thus arrive is therefore also true. We could express it as follows: 'If something is greater than 2, then its square is 11rcatcr than 2' or better 'if a is greater than 2, then a squared is greater than 2'. In this context the construction with 'if' seems the most idiomatic. But now we no longer have two thoughts combined. If we replace 'a' by the
? ? ? ! 1
' ' ' ' "
prop. :r name of an object, then the sentence we obtain expresses a thought which is seen to be a particular case of the general thought; in such a partit:ular case we have two thoughts present in the condition and consequence, besides the thought which is present in the whole sentence. We can grasp these in isolation. But we cannot proceed in this way to split up 1 h e sent~. :nce e x p r e s s i n g t h e g e n e r a l t h o u g h t w i t h o u t m a k i n g t h e p a r t s sen~;elcs~. For the letter 'a' is meant to confer generality of content upon the whok sentence, not on its clauses. With 'a is greater than 2' we no longer have a part expressing a thought: it neither expresses a thought that is true nor one that is false, because 'a' is neither meant to designate an object as Joes a proper name, nor to confer generality of content upon this part. It has no function at all in relation to this part: it has no contribution to make to it, as it would have if it conferred a sense on it. The same holds of the
ultH:r part ? a squared is greater than 2'. The 'a' in the one clause refers to the ? ? a' in the other, and for this very reason we cannot separate the clauses; for if we did, the contribution that 'a' is meant to make to the sense of the whole would be utterly destroyed and its function lost. Just so, in Latin a . compound sentence whose clauses are introduced by tot and quot cannot be split up into these clauses without rendering each of them senseless. I call ~nmdhing a quasi sentence if it has the grammatical form of a sentence and yd i~ not an expression of a thought, although it may be part of a sentence l that d1h. :s express a thought, and thus part of a sentence proper. Hence in the? ea~,;of a general sentence we cannot draw the distinction we drew earlier . l between a condition and a consequence. The antecedent and consequent are ? now quasi-sentences, no longer expressing thoughts. Now we do indeed ; ~r:. . :ak as if the condition were satisfied in some cases and not in others. This makes it clear that what we are here calling the condition is not a thought, for a thought ---leaving as always myth and fiction on one side---is only <:;llh. ;r true or false. There cannot be a case of the same thought being now , n 11e, now false. What we have in such a case is simply a quasi-sentence from
which ~emew. :es proper can be derived, some of which express true thoughts dlHi some false. But then these thoughts are just different. Letters which, like the ? ci in our example, serve to confer generality of content upon a sentence ~r(\ iu virtu~: uf this role, essentially different from proper names. I say that ? a proper name designates (or means) an object; 'a' indicates an object, it dPes not ha"e a meaniug, it designates or means nothing. In ordinary hmt;uagc words like 'something' and 'it' often take over the role of letters; in ' ~;,,,11e cases even there seems to be nothing at all to take over this role. In thi,; n~gartL as in others, language is defective. For discerning logical ' ~utll:t ! Ire i1 is better to use letters than to rely on the vernacular. Let us now J lotlk at the component quasi? sentences of our general sentence. Each of ' tliese C1H1tains a letter. If we replace these letters by the proper name of an \>bJcct, we nhtain a sentence proper, which is now manifestly composed of tlti~ prupcr name and the remainder. This remainder corresponds to the ] lllbalmatcd part of the thought and is also part of the quasi sentence. So
190 Introduction to Logic
? Introduction to Logic 191
t'ach of the component quasi-sentences contains, besides the letter. a constituent which corresponds to the unsaturated part of a thought. These unsaturated parts of a thought are now in turn parts of our general thought, hut they need a cement to hold them together; in the same way tv. o complete parts of a thought cannot hold together without a cement. If we t'Xpress our example of a general thought as follows: 'If a is greattf tkw 2, 11len a is something whose square is greater than 2', then tht? words 'is -;omething whose square is greater than 2' and 'is greater than 2' correspond lo the two unsaturated parts of a thought that we were speaking about. Bm 1he 'is' here must be taken throughout as being devoid of asserturk force. What correspond to the cement are the words 'if' and 'then', the letter ? a' aud 11te occurrence of the word 'is', first immediately after the 'a' and s? :condly irnrnediately after the 'then'. But, as we know, the truth of the matter is that 1ili~ particular mode of composition is effected by negating, forming a un~junction, negating again, and generalizing ~sit venia verbo).
Sense and Meaning
l';oper names are meant to designate objects, and we call the object tksignated by a proper name its meaning. On the other hand, a pmper name rs a constituent of a sentence, which expresses a thought. Now what has tilt. ~ object got to do with the thought? We have seen from the sentence 'Mont lllanc is over 4000 m high' that it is not part of the thought. Is then the object necessary at all for the sentence to express a thought? People l'l:rtainly say that Odysseus is not an histo,"ical person, and mean by this contradictory expression that the name 'Odysseus' designates nothing, ha-; no meaning. But if we accept this, we do not on that account deny a thought-content to all sentences of the Odyssey in which the namt:: 't>dy&seus' occurs. Let us just imagine that we have convinced ourselves. l? ontrary to our former opinion, that the name 'Odysseus', as it occur:, in the Od_t? ssey, does designate a man after all. Would this mean that the sentences c1? ntaining the name 'Odysseus' expressed different thoughts? 1 think not. 1'11e thoughts would strictly remain the same; they would only be trausposcd frnm the realm of fiction to that of truth. So the object designated by a proper name seems to be quite inessential to the thought-content of a ~,? ntence which contains it. To the thought-content! For the rest, it goc;. ; without saying that it is by no means a matter of indifference tu us whethe;? we are operating in the realm of fiction or of truth. But we can immediately infer from what we have just said that something further must be as:;ocialt:d with the proper name, something which is different from the ub. it:et designated and which is essential to the thought of the sentence in which the proper name occurs. I call it the sense of the proper na111e. As the p w p n name is part of the scntenc~. :, so its sense is part of the thuu! '. ht.
The same point ctul be approached in other ways. 11 i~ not uncnnlllloll lo1
? to. v111. o6
the same object to have different proper names; but for all that they are not simply interchangeable. This is only to be explained by the fact that proper names of the same object can have different senses. The sentence 'Mont Blanc is over 4000 m high' does not express the same thought as the sentence 'The highest mountain in Europe is over 4000 m high', although the proper name 'Mont Blanc' designates the same mountain as the expression 'the highest mountain in Europe'. The two sentences 'The Evening Star is the same as the Evening Star' and 'The Morning Star is the same as the Evening Star' differ only by a single name having the same meaning in each. Nevertheless they express different thoughts. So the sense of the proper name 'the Evening Star' must be different from that of the proper name 'the Morning Star'. The upshot is that there is something associated with a proper name, different from its meaning, which can be different as between proper names with the same meaning, and which is essential to the thought-content of the sentence containing it. A sentence proper, in which a proper name occurs, expresses a singular thought, and in this we distinguished a complete part and an unsaturated one. The former corresponds to the proper name, but it is not the meaning of the proper name, but its sense. The unsaturated part of the thought we take to be a sense too: it is the sense of the part of the sentence over and above the proper name. And it is in line with these stipulations to take the thought itself as a sense, namely the sense of the sentence. As the thought is the sense of the whole sentence, so a part of the thought is the sense of part of the sentence. Thus the thought appears the same in kind as the sense of a proper name, but quite different from its meaning.
Now the question arises whether to the unsaturated part of the thought, which is to be regarded as the sense of the corresponding part of the sentence, there does not also correspond something which is to be construed as the meaning of this part. As far as the mere thought-content is concerned it is indeed a matter of indifference whether a proper name has a meaning, but in any other regard it is of the greatest importance; at least it is so if we are concerned with the acquisition of knowledge. It is this which determines whether we are in the realm of fiction or truth. Now it is surely unlikely that a proper name should behave so differently from the rest of a singular sentence that it is only in its case that the existence of a meaning should be of importance. If the thought as a whole is to belong to the realm of truth, we must rather assume that something in the realm of meaning must correspond to the rest of the sentence, which has the unsaturated part of the thought for its sense. We may add to this the fact that in this part of the sentence too there may occur proper names, where it does matter that they should have a meaning. If several proper names occur in a sentence, the corresponding thought can be analysed into a complete and unsaturated part in differeqt ways. The sense of each of these proper names can be set up as the complete part over against the rest of the thought as the unsaturated part. We know that even in speech the same thought can be expressed in
192 Introduction to Logic
diffcrenl ways, by making now this proper name, now that one, the
? Introduction to Logic 193
grammatical subject. No doubt we shall say that these different phrasings are not equivalent. This is true. But we must not forget that language does not simply express thoughts; it also imparts a certain tone or colouring to them. And this can be different even where the thought is the same. It is inconceivable that it is only for the proper names that there can be a question of meaning and not for the other parts of the sentence which connect them. If we say 'Jupiter is larger than Mars', what are we talking about? About the heavenly bodies themselves, the meanings of the proper names 'Jupiter' and 'Mars'. We are saying that they stand in a certain relation to one another, and this we do by means of the words 'is larger than'. This relation holds between the meanings of the proper names, and so must itself belong to the realm of meanings. It follows that we have to acknowledge that the part of the sentence 'is larger than Mars' is meaningful, and not merely possessed of a sense. If we split up a sentence into a proper name and the remainder, then this remainder has for its sense an unsaturated part of a thought. But we call its meaning a concept. In doing so we are of course making a mistake, a mistake which language forces upon us. By the very fact of introducing the word 'concept', we countenance the possibility of sentences of the form 'A is a concept', where A is a proper name. We have thereby stamped as an object what-as being completely different in kind-is the precise opposite of an object. For the same reason the definite article at the beginning of 'the meaning of the remaining part of the sentence' is a mistake too. But language forces us into such inaccuracies, and so nothing remains for us but to bear them constantly in mind, if we are not to fall into error and thus blur the sharp distinction between concept and object. We can, metaphorically speaking, call the concept unsaturated too; alternatively we can say that it is predicative in character.
We have considered the case of a compound sentence consisting of a 4uasi-antecedent and -consequent, where these quasi-sentences contain a letter ('a', say). When the letter is subtracted from each of these quasi- sentences the remainder corresponds to an unsaturated part of a thought, and we may now say that such a part of a thought is the sense of the part of a sentence referred to as the remainder. Now such a part also has a meaning, and this we have called a concept. So we have one concept occurring as the meaning of what is left over from the quasi-antecedent, and one concept occurring as the meaning of what is left over from the quasi- consequent. These concepts are here brought into a special connection with one another (we could also say 'relation') and this we call subordination: that is to say, the concept in the quasi-antecedent is made subordinate to the concept in the quasi-consequent. If we regard a singular sentence as composed of a proper name and the remainder, then to a proper name there corresponds an object as its meaning and to the remainder a concept. Here the concept and object present themselves as connected or related in a
special way, which we call subsumption. The object is subsumed under the concept. ll is clear the subsumption is totally different from subordination.
We have seen that it is true of parts of sentences thut they have meunings.
? 194 1ntroduction to Logic
\VhaL of a whole sentence, does this have a meaning too? If we an ''UJiCct ned with truth, if we are aiming at knowledge, then we demand ol each proper name occurring in a sentence that it should have a meaning. Or the other hand, we know that as far as the sense of a sentence, the thought is concerned, it doesn't matter whether the parts of the sentence havt nwaning~ or not. It follows that there must be something associated with~ ~:entenc. : which is difl'erent from the thought, something to which it i! essemial that the parts of the sentence should have meanings. This is to be ,? alled the meaning of a sentence. But the only thing to which this is essential is what I call the truth-value-whether the thought is true or false. Thought! in myth and tlction do not need to have truth-values. A sentence containin~ a weaning,less proper name is neither true nor false; if it expresses a though1 at alL then that thought belongs to fiction. In that case the sentence has ne <1lt:a! 11! 16. We have two truth-values, the True and the False. If s ~;entt;nc. : has a meaning at all, this is either the True or the False. If a sentence can be split up into parts, each of which is meaningful, then the sentence also has a meaning. The True and the False are to be regarded a5 objects, for both the sentence and its sense, the thought, are complete in dmra. ;:t. . ::r, not unsaturated. If, instead of the True and the False, I had discovered two chemical elements, this would have created a greater stir in the a( adcmic world. If we say 'the thought is true', we seem to be ascribins truth to the thought as a property. If that were so, we should have a case ol ~ub~umption. The thought as an object would be subsumed under the concept of the true. But here we are misled by language. We don't have the rdation of an object to a property, but that of the sense of a sign to its meaning. In fact at bottom the sentence 'it is true that 2 is prime' says no more than the sentence '2 is prime'. If in the first case we express a judge? ment, this is not because of the word 'true', but because of the assertoric: force we give the word 'is'. But we can do that equally well in the second sentence, and an actor on the stage, for example, would be able to utter the first sentence without assertoric force just as easily as the second. *
*Remarks on the use of letters in arithmetic (12. VIII. 06): people use letters in arithmetic without, as a rule, voicing any opinion on how, for what purpose, or with what justification, they are used, and presumably they are not wholly clear about these matters themselves. Their use in algebra al signs for unknowns (to make it plain what I am referring to, this expression may be allowed here although objections can be brought against it) is not all that different from the use that is standard in arithmetic, despite appearances to the contrary. There is no doubt that, by and large, letters in arithmetic arc meant to confer generality of content. But on what? Not, in most cases, on a single sentence or a compound sentence in the grammatical sense, but on a group of apparently self-contained sentences, where it is not always easy to make out where these begin or break off. It is an imperative and essential logical requirement that these apparently self-contained sentences should be combined into a single compound sentence; however, if we compl~
? ? Introduction to Logic 195
A sentence proper is a proper name, and its meaning, if it has one, is a truth value: the True or the False. There are many sentences which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part, which means a concept. In the same way there are many proper names, whose meanings are not truth values, which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part. If this latter is to be meaningful, then the result of saturing it with any meaningful proper name whatever must once more be a meaningful proper name. When this happens, we call the meaning of this unsaturated part a function. At this point, however, we need to make a reservation, similar to that we made earlier when the word 'concept' was introduced, about the unavoidable inaccuracy of language. The unsaturated part of a sentence, whose meaning we have called a concept, must have the property of yielding a genuine sentence when saturated by any meaningful proper name; this means that it must yield the proper name of a truth value.
Yet let us set these doubts on one side for the moment. When we undertake the transformation as above, we acknowledge that there is one and only one object which we designate by the proper name 'the extension of the concept square root o f I', and that we also designate the same object by the proper name 'the extension of the concept binomial coefficient ofthe exponent of-1'. Perhaps you suspect that this object is something which we have just called an aggregate; but we will see that an extension of a concept is essentially different from an aggregate.
In the first instance we have in the case of a concept the fundamental case of subsumption. which we express in such a sentence as 'A is a 1/>'; where 'A' stands for a proper name, '1/>' for a concept-word. Now if the extension of the concept 1/>, coincides with the extension of the concept 'P, it follows from 'A is a 1/>' that A is a 'P too. Therefore we then also have a relation of the object A to th~ extension of the concept cfJ, which I will call B. And the fact that this relation holds I will express thus: 'A belongs to IJ'. And so this is meant to be tantamount to 'A is a 1/>, if B is the extension of the concept C/J'.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 183
On superficial reftexion you might compare this relation with that of part to whole; but here we have nothing corresponding to the sentence: What is part of a part, is part of the whole. To be sure, A itself can be the extension of a concept; but ifL1 belong to A, and A to B, L1 need not belong to B. In the sentence 'The extension of the concept prime is an extension of a concept to which 3 belongs' the phrase 'extension of a concept to which 3 belongs' is to be regarded as a nomen appellativum. Now let B be the extension of the concept hereby designated, and A the extension of the concept prime. Then A belongs to B and 2 belongs to A; but 2 does not belong to B; for 2 isn't an extension of a concept to which 3 belongs. From this alone it follows that an extension of a concept is at bottom completely different from an aggregate. The aggregate is composed of its parts. Whereas the extension of a concept is not composed of the objects that belong to it. For the case is conceivable that no objects belong to it. The extension of a concept simply has its being in the concept, not in the objects which belong to it; these are not its parts.
! 'here cannot be an aggregate which has no parts.
Now of course it can happen that all objects which belong to the
extension of a concept are at the same time parts of an aggregate and what is more in such a way that the whole being of the aggregate is completely exhausted by them. In this way it may look as though in such a case the aggregate coincides with the extension of the concept; but it isn't necessary that every part of the aggregate also belongs to the extension of the concept; for it can be that a part of the aggregate, without itself belonging to the extension of the concept, has parts which belong to the extension of the concept, or that a part of the aggregate, without itself belonging to the extension of the concept, is part of an object which belongs to the extension of the concept. So the relation of a part to the aggregate must still always be distinguished from that of an object to the extension of a concept to which it belongs. The extension of the concept is not determined by the aggregate even in this case, where they apparently coincide. A grain of sand is an aggregate. And it can be that the extension of the concept silicic acid molecule contained in this grain of sand apparently coincides with the aggregate which we call this grain of sand. But we could just as well let the extension of the concept atom contained in this grain ofsand coincide with our aggregate. But in that case the two extensions of concepts would coincide, which is impossible. From which it follows that neither of the two extensions of concepts coincides with the aggregate, for if one of them were to do so, then the other could with equal right be said to do so.
? ? What may I regard as the Result ofmy Work? 1 [Aug. 1906]
It is almost all tied up with the concept-script. a concept construed as a function. a relation as a function of two arguments. the extension of a concept or class is not the primary thing for me. unsaturatedness both in the case of concepts and functions. the true nature of concept and function recognized.
strictly I should have begun by mentioning the judgement-stroke, the dissociation of assertoric force from the predicate . . .
Hypothetical mode of sentence composition . . .
Generality . . .
Sense and meaning . . .
1 According to a note of the previous editors these jottings bore the date '5. VIII. 06'. They are no doubt prefatory to the piece 'Einleitung in die Logik', which is divided into sections according to the captions given here (ed. ).
? Introduction to Logic1 [August 1906]
Dissociating assertoric force from the predicate
We can express a thought without asserting it. But there is no word or sign in language whose function is simply to assert something. This is why, apparently even in logical works, predicating is confused with judging. As a result one is never quite sure whether what logicians call a judgement is meant to be a thought alone or one accompanied by the judgement that it is true. To go by the word, one would think it meant a thought accompanied hy a judgement; but often in common usage the word does not include the actual passing of judgement, the recognition of the truth of something. I use the word 'thought' in roughly the same way as logicians use 'judgement'. To think is to grasp a thought. Once we have grasped a thought, we can recognize it as true (make a judgement) and give expression to our recognition of its truth (make an assertion). Assertoric force is to be dissociated from negation too. To each thought there corresponds an opposite, so that rejecting one of them is accepting the other. One can say that to make a judgement is to make a choice between opposites. Rejecting the one and accepting the other is one and the same act. Therefore there is no need of a special name, or special sign, for rejecting a thought. We may speak of the negation of a thought before we have made any distinction of parts within it. To argue whether negation belongs to the whole thought or to the predicative part is every bit as unfruitful as to argue whether a coat clothes a man who is already clothed or whether it belongs together with the rest of his clothing. Since a coat covers a man who is already clothed, it automatically becomes part and parcel with the rest of his apparel. We may, metaphorically speaking, regard the predicative component of a thought as a covering for the subject-component. If further coverings are added, these automatically become one with those already there.
The hypothetical mode of sentence composition
If someone says that in a hypothetical judgement two judgements are set in relation to one another, he is using the word 'judgement' so as not to include the recognition of the truth of anything. For even if the whole compound
5. VIll. 06
1 Parts of these uiary notes were later revised hy Frege; cf. pp. I97 IT. (ed. ).
? 186 Introduction to Logic
sentences is uttered with assertoric force, one is still asserting neither the truth of the thought in the antecedent nor that of the thought in the consequent. The recognition of truth extends rather over a thought that is expressed in the whole compound sentence. And on closer examination we find that in many cases the antecedent on its own does not express a thought, and nor does the consequent either (quasi-sentences). What we generally have in these cases is the relation of subordination between concepts. Here it is quite common to conflate two different kinds of things, which I was probably the first to distinguish: the relation I designate by the conditional stroke, and generality. The former corresponds roughly to what logicians intend by 'a relation between judgements'. That is, the sign for the relation (the conditional stroke) connects sentences with one another: these are sentences proper, and so each of them expresses a thought.
Now leaving myth and fiction on one side, and considering only those cases in which truth in the scientific sense is in question, we can say that every thought is either true or false, tertium non datur. It is nonsense to speak of cases in which a thought is true and cases in which it is false. The same thought cannot be true at one time, false at another. On the contrary, the cases people have in mind in speaking in this way always involve different thoughts, and the reason they believe the thought to be the same is that the form of words is the same; this form of words will then be a quasi- sentence. We do not always adequately distinguish the sign from what it expresses.
If there are two thoughts, only four cases are possible:
1. the first is true, and likewise the second; 2. the first is true, the second false;
3. the first is false, the second true;
4. both are false.
Now if the third of these cases does not hold, then the relation I have designated by the conditional stroke obtains. The sentence expressing the first thought is the consequent, the sentence expressing the second the antecedent. It is now almost 28 years since I gave this definition. I believed at the time that I had only to mention it and everyone else would immediately know more about it than I did. And now, after more than a quarter of a century has elapsed, the great majority of mathematicians have no inkling of the matter, and the same goes for the logicians. What pig? headedness! This way academics have of behaving reminds me of nothing so much as that of an ox confronted by a new gate: it gapes, it bellows, it tries to squeeze by sideways, but going through it-that might be dangerous. I can readily believe that it looks strange at first sight, but if it didn't, it would have been discovered long ago. Do we really have to go by our first cursOFy impression of the matter? Have we no time at all to reflect upon it? No, for what good could come of that! People probably feel the lack of an inner connection between the thoughts: we find it hard to accept
? Introduction to Logic 187
that it is only the truth or falsity of the thoughts that is to be taken into account, that their content doesn't really come into it all. This is connect;;d with what I realized about sense and meaning. Well, just let someone try to give an account in which the thought itself plays a bigger role and it will probably turn out that what has been added from the thought is at bottom quite superfluous, and that one has only succeeded in complicating the issue, or that the antecedent and consequent are not sentences proper, neilht:r being such as to express a thought, so that a relation has not been established between two thoughts as one intended, but rather betwecll concepts or relations. Is the relation I designate by the conditional ~troh. c i11 fact such as can obtain between thoug;,ts? Strictly speaking, no! Tlw most we can say here is that the sign for this relation (i. e. the conditional ~trokc) connects sentences. Later the definition will be filled out in such a way as to allow also names of objects to be connected hy the conditione1l strok'? . Al lirst this will be even harder to swallow. We need first to take a cln~er k1uk :rt generality if we are to find it acceptable.
Generality
It is only at this point that the need arises to ana(vse a thought inftJ purts none of which are thoughts. The simplest case is that of splitting a thought 111to two parts. The parts are different in kind, one being unsaturnted. the <lther saturated (complete). The thoughts we have to consider here are those designated in traditional logic as singular judgements. In such a thought something is asserted of an object. The sentence expressing such a 1bought 1s composed of a proper name-and this corresponds to the complete pan o f t h e t h o u g h t - - a n d a p r e d i c a t i v e p a r t , w h i c h corre~ponds t o t ! J , : unsaturated part of the thought. We should mention that, strictly spenkiuf. it is not in itself that a thought is singular, but only with respect to a po~sil>k way of analysing it. It is possible for the same thought, with respect tt' rJ. different analy~is, to appear as particular (Christ converted some men to his teaching). Proper names designate objects, and a singular thought i~ about nhjccts. But we can't say that an object is part of a thought as a proper rrame is part of the corresponding sentence. Mont Blanc with its masst~s nf snow and ice is not part of the thought that Mont Blanc is more than 4000 m. high; all we can say is that to the object there corresponds. in a certain \'ii'. Y that has yet to be considered, a part of the thought. By analysing a singulM 1hought we obtain components of the complete and of the unsaturated t~intL which of course cannot occur in isolation; but any component of the t? ne kind together with any component of the other kind will form n thuught. If we now keep the unsaturated part constant but vary the complete part, w? ? should expect some of the thoughts so formed to he true, and some fal>t'. But it can also happen that the whole lot arc true. F. g. let tire unsatur:1tcd romponent be expressed in the words 'is idcnticnl With itsdf'. This is then :1
? 188 Introduction to Logic
particular property of the unsaturated part. We thus obtain a new thought (everything is identical with itself), which compared to the singular thoughts (two is identical with itself, the moon is identical with itself) is general. However the word 'everything', which here takes the place of a proper name ('the moon'), is not itself a proper name, doesn't designate an object, but serves to confer generality of content on the sentence. In logic we can often be too influenced by language and it is in this way that the concept-script is of value: it helps to emancipate us from the forms of language. Instead of saying 'the moon is identical with itself' we can also say 'the moon is identical with the moon' without changing the thought. But in language it is impossible, in making the transition to the general statement, to allow the word 'everything' also to occur in two places. The sentence 'everything is identical with everything' would not have the desired sense. We may, taking a leaf from mathematics, employ a letter and say 'a is identical with a'. This letter then occupies the place (or places) of a proper name, but it is not itself a proper name; it does not have a meaning, but only serves to confer generality of content on a sentence. This use of letters, being simpler and, from a logical point of view, more appropriate, is to be preferred to the means which language provides for this purpose.
If a whole is composed of two sentences connected by 'and', each of which expresses a thought, then the sense of the whole is also to be construed as a thought, for this sense is either true or false; it is true if each component thought is true, and false in every other case-hence when at least one of the two component thoughts is false. If we call the thought of the whole the conjunction of the two component thoughts, then the conjunction too has its opposite thought, as does every thought. Now it is clear what the opposite of a conjunction of the opposite of a thought A with a thought B is. It is what I express by means of the conditional stroke. The seJ\tence expressing thought A is the consequent, that expressing thought B the antecedent. But the whole sentence expressing the opposite of the conjunction of the opposite of A with B may be called the hypothetical sentence whose consequent expresses A and whose antecedent expresses B. The thought expressed by the hypothetical sentence we shall call the
s. vnLo6 hypothetical thought whose consequence is A and whose condition is B. Now if the same proper name occurs in both consequent and antecedent, we may regard the hypothetical thought as singular if we think of it as bein& analysed into the complete part that corresponds to the proper name and the unsaturated part left over. If we now keep the unsaturated part fixed, and vary the complete part, it may turn out that we always obtain a true thought, no matter what we choose for the complete part. In saying this, we are assuming, as we are throughout this enquiry, that we are operating not in the realm of myth and fiction, but in that of truth (in the scientific sense); consequently every proper name really does achieve its goal of designatinl an object, and so is not empty. The complete parts of the thoughts that are here in question are of course not themselves the objects designated by the
? Introduction to Logic 189
proper names, but are connected with them, and it is essential that there should be such objects if everything is not to fall within the realm of fiction. Otherwise we cannot speak of the truth of thoughts at all. So we are assuming that we have a hypothetical thought-one which can at the same time be construed as a singular thought-from which we, as was said above, always obtain a true thought by keeping the unsaturated part fixed, whatever complete part we saturate it with. In this way we arrive at a general thought, and the singular hypothetical thought from which we started is seen to be a special case of it. For instance:
Thought A: that 3 squared is greater than 2.
Thought B: that 3 is greater than 2.
Opposite of Thought A: that 3 squared is not greater than 2.
Conjunction o f the opposite o f thought A with thought B: that 3 squared is not greater than 2, and that 3 is greater than 2.
Opposite ofthe conjunction ofthe opposite ofthought A with thought B: that it is false both that 3 squared is not greater than 2 and that 3 is greater than 2.
This is the hypothetical thought with thought A as its consequence and thought B as its condition. There is something unnatural about the form of words 'If 3 is greater than 2, then 3 squared is greater than 2' and perhaps ~:ven more so about what we get when we replace '3' by '2': 'If 2 is greater than 2, then 2 squared is greater than 2'. But the thought that it is false both 1hat 2 squared is not greater than 2 and that 2 is greater than 2 is a true one. And whatever number we take instead of 3, we always obtain a true thought. But what if we take an object that is not a number? Any sentence obtained from 'a is greater than 2' by putting the proper name of an object ror 'a' expresses a thought, and this thought is of course false if the object is not a number. It is different with the first sentence, because the expression which results when the proper name of an object is put for 'a' in 'a squared' only designates an object in ordinary discourse if this object is a number. The incompleteness of the usual definition of 'squared' is to be blamed for 1his.
This defect can be removed by stipulating that by the square of an
object we are to understand the object itself if this object is not a number, hut that 'the square of a number' is to be understood in its arithmetical sense. We shall then always obtain from the schema 'that a squared is Rreater than 2' a sentence expressing a false thought if 'a' is replaced by the JllllllC of an object that is not a number. Once this stipulation has been made, we can replace the numeral '3' in our hypothetical by the name of any object whatever, and we shall always obtain a sentence expressing a true thought. The general thought at which we thus arrive is therefore also true. We could express it as follows: 'If something is greater than 2, then its square is 11rcatcr than 2' or better 'if a is greater than 2, then a squared is greater than 2'. In this context the construction with 'if' seems the most idiomatic. But now we no longer have two thoughts combined. If we replace 'a' by the
? ? ? ! 1
' ' ' ' "
prop. :r name of an object, then the sentence we obtain expresses a thought which is seen to be a particular case of the general thought; in such a partit:ular case we have two thoughts present in the condition and consequence, besides the thought which is present in the whole sentence. We can grasp these in isolation. But we cannot proceed in this way to split up 1 h e sent~. :nce e x p r e s s i n g t h e g e n e r a l t h o u g h t w i t h o u t m a k i n g t h e p a r t s sen~;elcs~. For the letter 'a' is meant to confer generality of content upon the whok sentence, not on its clauses. With 'a is greater than 2' we no longer have a part expressing a thought: it neither expresses a thought that is true nor one that is false, because 'a' is neither meant to designate an object as Joes a proper name, nor to confer generality of content upon this part. It has no function at all in relation to this part: it has no contribution to make to it, as it would have if it conferred a sense on it. The same holds of the
ultH:r part ? a squared is greater than 2'. The 'a' in the one clause refers to the ? ? a' in the other, and for this very reason we cannot separate the clauses; for if we did, the contribution that 'a' is meant to make to the sense of the whole would be utterly destroyed and its function lost. Just so, in Latin a . compound sentence whose clauses are introduced by tot and quot cannot be split up into these clauses without rendering each of them senseless. I call ~nmdhing a quasi sentence if it has the grammatical form of a sentence and yd i~ not an expression of a thought, although it may be part of a sentence l that d1h. :s express a thought, and thus part of a sentence proper. Hence in the? ea~,;of a general sentence we cannot draw the distinction we drew earlier . l between a condition and a consequence. The antecedent and consequent are ? now quasi-sentences, no longer expressing thoughts. Now we do indeed ; ~r:. . :ak as if the condition were satisfied in some cases and not in others. This makes it clear that what we are here calling the condition is not a thought, for a thought ---leaving as always myth and fiction on one side---is only <:;llh. ;r true or false. There cannot be a case of the same thought being now , n 11e, now false. What we have in such a case is simply a quasi-sentence from
which ~emew. :es proper can be derived, some of which express true thoughts dlHi some false. But then these thoughts are just different. Letters which, like the ? ci in our example, serve to confer generality of content upon a sentence ~r(\ iu virtu~: uf this role, essentially different from proper names. I say that ? a proper name designates (or means) an object; 'a' indicates an object, it dPes not ha"e a meaniug, it designates or means nothing. In ordinary hmt;uagc words like 'something' and 'it' often take over the role of letters; in ' ~;,,,11e cases even there seems to be nothing at all to take over this role. In thi,; n~gartL as in others, language is defective. For discerning logical ' ~utll:t ! Ire i1 is better to use letters than to rely on the vernacular. Let us now J lotlk at the component quasi? sentences of our general sentence. Each of ' tliese C1H1tains a letter. If we replace these letters by the proper name of an \>bJcct, we nhtain a sentence proper, which is now manifestly composed of tlti~ prupcr name and the remainder. This remainder corresponds to the ] lllbalmatcd part of the thought and is also part of the quasi sentence. So
190 Introduction to Logic
? Introduction to Logic 191
t'ach of the component quasi-sentences contains, besides the letter. a constituent which corresponds to the unsaturated part of a thought. These unsaturated parts of a thought are now in turn parts of our general thought, hut they need a cement to hold them together; in the same way tv. o complete parts of a thought cannot hold together without a cement. If we t'Xpress our example of a general thought as follows: 'If a is greattf tkw 2, 11len a is something whose square is greater than 2', then tht? words 'is -;omething whose square is greater than 2' and 'is greater than 2' correspond lo the two unsaturated parts of a thought that we were speaking about. Bm 1he 'is' here must be taken throughout as being devoid of asserturk force. What correspond to the cement are the words 'if' and 'then', the letter ? a' aud 11te occurrence of the word 'is', first immediately after the 'a' and s? :condly irnrnediately after the 'then'. But, as we know, the truth of the matter is that 1ili~ particular mode of composition is effected by negating, forming a un~junction, negating again, and generalizing ~sit venia verbo).
Sense and Meaning
l';oper names are meant to designate objects, and we call the object tksignated by a proper name its meaning. On the other hand, a pmper name rs a constituent of a sentence, which expresses a thought. Now what has tilt. ~ object got to do with the thought? We have seen from the sentence 'Mont lllanc is over 4000 m high' that it is not part of the thought. Is then the object necessary at all for the sentence to express a thought? People l'l:rtainly say that Odysseus is not an histo,"ical person, and mean by this contradictory expression that the name 'Odysseus' designates nothing, ha-; no meaning. But if we accept this, we do not on that account deny a thought-content to all sentences of the Odyssey in which the namt:: 't>dy&seus' occurs. Let us just imagine that we have convinced ourselves. l? ontrary to our former opinion, that the name 'Odysseus', as it occur:, in the Od_t? ssey, does designate a man after all. Would this mean that the sentences c1? ntaining the name 'Odysseus' expressed different thoughts? 1 think not. 1'11e thoughts would strictly remain the same; they would only be trausposcd frnm the realm of fiction to that of truth. So the object designated by a proper name seems to be quite inessential to the thought-content of a ~,? ntence which contains it. To the thought-content! For the rest, it goc;. ; without saying that it is by no means a matter of indifference tu us whethe;? we are operating in the realm of fiction or of truth. But we can immediately infer from what we have just said that something further must be as:;ocialt:d with the proper name, something which is different from the ub. it:et designated and which is essential to the thought of the sentence in which the proper name occurs. I call it the sense of the proper na111e. As the p w p n name is part of the scntenc~. :, so its sense is part of the thuu! '. ht.
The same point ctul be approached in other ways. 11 i~ not uncnnlllloll lo1
? to. v111. o6
the same object to have different proper names; but for all that they are not simply interchangeable. This is only to be explained by the fact that proper names of the same object can have different senses. The sentence 'Mont Blanc is over 4000 m high' does not express the same thought as the sentence 'The highest mountain in Europe is over 4000 m high', although the proper name 'Mont Blanc' designates the same mountain as the expression 'the highest mountain in Europe'. The two sentences 'The Evening Star is the same as the Evening Star' and 'The Morning Star is the same as the Evening Star' differ only by a single name having the same meaning in each. Nevertheless they express different thoughts. So the sense of the proper name 'the Evening Star' must be different from that of the proper name 'the Morning Star'. The upshot is that there is something associated with a proper name, different from its meaning, which can be different as between proper names with the same meaning, and which is essential to the thought-content of the sentence containing it. A sentence proper, in which a proper name occurs, expresses a singular thought, and in this we distinguished a complete part and an unsaturated one. The former corresponds to the proper name, but it is not the meaning of the proper name, but its sense. The unsaturated part of the thought we take to be a sense too: it is the sense of the part of the sentence over and above the proper name. And it is in line with these stipulations to take the thought itself as a sense, namely the sense of the sentence. As the thought is the sense of the whole sentence, so a part of the thought is the sense of part of the sentence. Thus the thought appears the same in kind as the sense of a proper name, but quite different from its meaning.
Now the question arises whether to the unsaturated part of the thought, which is to be regarded as the sense of the corresponding part of the sentence, there does not also correspond something which is to be construed as the meaning of this part. As far as the mere thought-content is concerned it is indeed a matter of indifference whether a proper name has a meaning, but in any other regard it is of the greatest importance; at least it is so if we are concerned with the acquisition of knowledge. It is this which determines whether we are in the realm of fiction or truth. Now it is surely unlikely that a proper name should behave so differently from the rest of a singular sentence that it is only in its case that the existence of a meaning should be of importance. If the thought as a whole is to belong to the realm of truth, we must rather assume that something in the realm of meaning must correspond to the rest of the sentence, which has the unsaturated part of the thought for its sense. We may add to this the fact that in this part of the sentence too there may occur proper names, where it does matter that they should have a meaning. If several proper names occur in a sentence, the corresponding thought can be analysed into a complete and unsaturated part in differeqt ways. The sense of each of these proper names can be set up as the complete part over against the rest of the thought as the unsaturated part. We know that even in speech the same thought can be expressed in
192 Introduction to Logic
diffcrenl ways, by making now this proper name, now that one, the
? Introduction to Logic 193
grammatical subject. No doubt we shall say that these different phrasings are not equivalent. This is true. But we must not forget that language does not simply express thoughts; it also imparts a certain tone or colouring to them. And this can be different even where the thought is the same. It is inconceivable that it is only for the proper names that there can be a question of meaning and not for the other parts of the sentence which connect them. If we say 'Jupiter is larger than Mars', what are we talking about? About the heavenly bodies themselves, the meanings of the proper names 'Jupiter' and 'Mars'. We are saying that they stand in a certain relation to one another, and this we do by means of the words 'is larger than'. This relation holds between the meanings of the proper names, and so must itself belong to the realm of meanings. It follows that we have to acknowledge that the part of the sentence 'is larger than Mars' is meaningful, and not merely possessed of a sense. If we split up a sentence into a proper name and the remainder, then this remainder has for its sense an unsaturated part of a thought. But we call its meaning a concept. In doing so we are of course making a mistake, a mistake which language forces upon us. By the very fact of introducing the word 'concept', we countenance the possibility of sentences of the form 'A is a concept', where A is a proper name. We have thereby stamped as an object what-as being completely different in kind-is the precise opposite of an object. For the same reason the definite article at the beginning of 'the meaning of the remaining part of the sentence' is a mistake too. But language forces us into such inaccuracies, and so nothing remains for us but to bear them constantly in mind, if we are not to fall into error and thus blur the sharp distinction between concept and object. We can, metaphorically speaking, call the concept unsaturated too; alternatively we can say that it is predicative in character.
We have considered the case of a compound sentence consisting of a 4uasi-antecedent and -consequent, where these quasi-sentences contain a letter ('a', say). When the letter is subtracted from each of these quasi- sentences the remainder corresponds to an unsaturated part of a thought, and we may now say that such a part of a thought is the sense of the part of a sentence referred to as the remainder. Now such a part also has a meaning, and this we have called a concept. So we have one concept occurring as the meaning of what is left over from the quasi-antecedent, and one concept occurring as the meaning of what is left over from the quasi- consequent. These concepts are here brought into a special connection with one another (we could also say 'relation') and this we call subordination: that is to say, the concept in the quasi-antecedent is made subordinate to the concept in the quasi-consequent. If we regard a singular sentence as composed of a proper name and the remainder, then to a proper name there corresponds an object as its meaning and to the remainder a concept. Here the concept and object present themselves as connected or related in a
special way, which we call subsumption. The object is subsumed under the concept. ll is clear the subsumption is totally different from subordination.
We have seen that it is true of parts of sentences thut they have meunings.
? 194 1ntroduction to Logic
\VhaL of a whole sentence, does this have a meaning too? If we an ''UJiCct ned with truth, if we are aiming at knowledge, then we demand ol each proper name occurring in a sentence that it should have a meaning. Or the other hand, we know that as far as the sense of a sentence, the thought is concerned, it doesn't matter whether the parts of the sentence havt nwaning~ or not. It follows that there must be something associated with~ ~:entenc. : which is difl'erent from the thought, something to which it i! essemial that the parts of the sentence should have meanings. This is to be ,? alled the meaning of a sentence. But the only thing to which this is essential is what I call the truth-value-whether the thought is true or false. Thought! in myth and tlction do not need to have truth-values. A sentence containin~ a weaning,less proper name is neither true nor false; if it expresses a though1 at alL then that thought belongs to fiction. In that case the sentence has ne <1lt:a! 11! 16. We have two truth-values, the True and the False. If s ~;entt;nc. : has a meaning at all, this is either the True or the False. If a sentence can be split up into parts, each of which is meaningful, then the sentence also has a meaning. The True and the False are to be regarded a5 objects, for both the sentence and its sense, the thought, are complete in dmra. ;:t. . ::r, not unsaturated. If, instead of the True and the False, I had discovered two chemical elements, this would have created a greater stir in the a( adcmic world. If we say 'the thought is true', we seem to be ascribins truth to the thought as a property. If that were so, we should have a case ol ~ub~umption. The thought as an object would be subsumed under the concept of the true. But here we are misled by language. We don't have the rdation of an object to a property, but that of the sense of a sign to its meaning. In fact at bottom the sentence 'it is true that 2 is prime' says no more than the sentence '2 is prime'. If in the first case we express a judge? ment, this is not because of the word 'true', but because of the assertoric: force we give the word 'is'. But we can do that equally well in the second sentence, and an actor on the stage, for example, would be able to utter the first sentence without assertoric force just as easily as the second. *
*Remarks on the use of letters in arithmetic (12. VIII. 06): people use letters in arithmetic without, as a rule, voicing any opinion on how, for what purpose, or with what justification, they are used, and presumably they are not wholly clear about these matters themselves. Their use in algebra al signs for unknowns (to make it plain what I am referring to, this expression may be allowed here although objections can be brought against it) is not all that different from the use that is standard in arithmetic, despite appearances to the contrary. There is no doubt that, by and large, letters in arithmetic arc meant to confer generality of content. But on what? Not, in most cases, on a single sentence or a compound sentence in the grammatical sense, but on a group of apparently self-contained sentences, where it is not always easy to make out where these begin or break off. It is an imperative and essential logical requirement that these apparently self-contained sentences should be combined into a single compound sentence; however, if we compl~
? ? Introduction to Logic 195
A sentence proper is a proper name, and its meaning, if it has one, is a truth value: the True or the False. There are many sentences which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part, which means a concept. In the same way there are many proper names, whose meanings are not truth values, which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part. If this latter is to be meaningful, then the result of saturing it with any meaningful proper name whatever must once more be a meaningful proper name. When this happens, we call the meaning of this unsaturated part a function. At this point, however, we need to make a reservation, similar to that we made earlier when the word 'concept' was introduced, about the unavoidable inaccuracy of language. The unsaturated part of a sentence, whose meaning we have called a concept, must have the property of yielding a genuine sentence when saturated by any meaningful proper name; this means that it must yield the proper name of a truth value.
