Whereas the extension of a concept is not
composed
of the objects that belong to it.
Gottlob-Frege-Posthumous-Writings
Inbegri.
ff usually means 'essence' or 'embodiment' (as in 'He is the very embodiment of health').
It is for us impossible to determine from these fragmentary notes to what use Frege was putting the term, and we thought it better to leave it untranslated than to put in a probably false conjecture.
Frege obviously has in mind the different relation of an object to the extension of a concept under which it falls, and to an aggregate of which it is a part; but further than that we leave for the reader to decide (trans.
).
? ? On Schoenflies: Die Logischen Paradoxien der Mengenlehre 177 [Discussion]
The article by S, Ober die logischen Paradoxien der Mengenlehre* induces me to make the following remarks, in which I repeat much that I have already discussed previously, since it does not seem to be widely known. I fail to find in S and also in Korselt** the sharp distinction between concept and object. ? ? ? In the signs, a proper name (nomen proprium) corresponds to an object, a concept-word or concept-sign (nomen appellativum) to a concept. A sentence such as 'Two is a prime' can be analysed into two essentially different component parts: into 'two' and 'is a prime'. The former appears complete, the latter in need of supplementation, unsaturated. 'Two'-at least in this sentence-is a proper name, its meaning is an object, which can also be designated with greater prolixity by 'the number two'. The object, too, appears as a complete whole, whereas the predicative part has something unsaturated in its meaning as well. We count the copula 'is' as belonging to this part of the sentence. But there is usually something combined with it which here must be disregarded: assertoric force. We can of course express a thought, without stating it to be true. The thought is strictly the same, whether we merely express it or whether we also put it forward as true. Thus assertoric force, which is often connected with the copula or else with the grammatical predicate, does not belong to the expression of the thought, and so may be disregarded here.
This predicative component part of our sentence which we have described in this way, is also meaningful. We call it a concept-word or nomen appellativum, even though it is not customary to include the copula in this. Just as it itself appears unsaturated, there is also something unsaturated in the realm of meanings corresponding to it: we call this a concept. This unsaturatedness of one of the components is necessary, since otherwise the parts do not hold together. Of course two complete wholes can stand in a relation to one another; but then this relation is a third element-and one that is doubly unsaturated! In the case of a concept we can also call the unsaturatedness its predicative nature. But in this connection it is necessary to point out an imprecision forced on us by language, which, if we are not conscious of it, will prevent us from recognizing the heart of the matter: i. e. we can scarcely avoid using such expressions as 'the concept prime'. Here there is no trace left of unsaturatedness, of the predicative nature. Rather, the expression is constructed in a way which precisely parallels 'the poet Schiller'. So language brands a concept as an object, since the only way it can fit the designation for a concept into its grammatical structure is as a proper name. But in so doing, strictly speaking it falsifies matters. In the same way, the word 'concept' itself is, taken strictly, already defective, since
? The current Jahresbericht, Vol. XV, p. 19 (Jan. 1906).
? ? The current Jahresbericht, Vol. XV, p. 215 (March-April 1906). ? ? ? Cf. My essay 'Concept and Object'.
? ? 178 On Schoenjltes: Die Logischen Paradoxien der Mengenlehre
the phrase 'is a concept' requires a proper name as grammatical subject; and so, strictly speaking, it requires something contradictory, since no proper name can designate a concept; or perhaps better still, something nonsensical. It is no objection to say that surely the grammatical predicate 'is rectangular' can be combined with the grammatical subject 'every square', which isn't a proper name; for even the sentence 'every square is rectangular' can only make sense in virtue of the fact that you can assert of an object that it is rectangular, either rightly or wrongly, but in either case significantly. By a proper name I understand the sign of an object, independently of the question whether it be a simple word or sign, or a complex one, provided only that it designates the object determinately.
In the sentence 'Two is a prime' we find a relation designated: that of subsumption. We may also say the object falls under the concept prime, but if we do so, we must not forget the imprecision of linguistic expression we have just mentioned. This also creates the impression that the relation of subsumption is a third element supervenient upon the object and the concept. This isn't the case: the unsaturatedness of the concept brings it about that the object, in effecting the saturation, engages immediately with the concept, without need of any special cement. Object and concept are fundamentally made for each other, and in subsumption we have their fundamental union.
We call a concept empty if no object falls under it. The concept-word for an empty concept never yields a true sentence,? no matter what proper name may saturate it, or in other words: no matter what proper name may be attached as a grammatical subject to the concept-word as predicate. A concept under which one and only one object falls must still be distinguished from the latter; its sign is a nomen appellativum, not a nomen proprium.
With the help of the definite article or demonstrative, language forms proper names out of concept-words. So, for instance, the phrase 'this A' on p. 20 of the Schoenfties article is a proper name. If forming a proper name in this way is to be legitimate, the concept whose designation is used in its formation must satisfy two conditions:
1. It may not be empty.
2. Only one object may fall under it.
If the first condition is unsatisfied, there is no object at all to which the proper name would be ascribed. If the second is unsatisfied, there are indeed several such objects, but none of them is determined as the one meant to be designated by the proper name. In science the purpose of a proper name is to designate an object determinately; if this purpose is unfulfilled, the proper name has no justification in science. How things may be in ordinary
? For brevity, I here call a sentence expressing a true thought, a true sentence.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 179
language is no concern of ours here. Our first requirement includes the requirement that the concept be consistent.
But this requirement doesn't mean that every concept ought to be consistent, only that a concept must be consistent if you wish to form a proper name from the concept-word together with the definite article or demonstrative. But for this purpose, this requirement doesn't go far enough: the concept is not to be empty, no matter what the reason for its being empty may be.
But if we ask, under what conditions a concept is admissible in science. the first thing to stress is that consistency is no~ such a condition. The only requirement to be made of a concept is that it should have sharp boundaries; that is, that for every object it holds that it either falls under the concept or does not do so. Essentially this is nothing but the requirement that the principle of non-contradiction should hold. But the admissibility of a concept is entirely independent of the question whether objects fall under it, and if so which, or in other words; whether there be objects, and if so which, of which it can be truly asserted. For, before we can raise such questions, we already need the concept. The requirement that a concept be consistent encounters great difficulties. The only apparent way to show a concept has this property, is to cite an object falling under the concept. But to do that, you already need the concept.
Schoenfties obviously only lays down this requirement as a result of his confounding concepts and objects. It is of course self-evident that an object t:annot have inconsistent properties. That the concept of a right-angled equilateral pentagon contains a contradiction does not make it inadmissible. For we can see no reason why a man should not be able to say of an object that it is not a right-angled equilateral pentagon, or why he should not be permitted to say there are no right-angled equilateral pentagons. And before he arrives at such judgements, he must consider the matter, and to do that he requires this concept. It is completely wrongheaded to imagine that every contradiction is immediately recognizable; frequently the contradiction lies deeply buried and is only discovered by a lengthy chain of inference: throughout which you need the concept. The only thing is, given a concept, you may not presuppose without further question that an object falls under it. But someone does that when, by means of a definite article or demonstrative, he forms out of a concept-word a proper name destined for use in science.
Let us now examine Schoenflies' sentences in the light of what we have said so far. Earlier on p. 20 we read,
'I. The A has the property B, and
2. The A does not have the property B. '
The definite article before the 'A' can only be understood in such a way that 'the A' is being thought of as a proper name. In logic it must be presupposed that every proper name is meaningful; that is, that it serves its purpose of
? 180 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
designating an object. For a sentence containing a meaningless proper name either expresses no thought at all, or it expresses a thought that belongs to myth or fiction. In either case it falls outside the domain governed by the laws of logic. For instance 'the A' would be a name that was in this way inadmissible for scientific use if it were formed by means of the definite article from a name of a concept (nomen appellativum), under which either no object or more than one fell. The mistake would then not lie in the use of
the concept-word in itself, but in forming a proper name from it. Let us now assume 'the A' is a meaningful proper name, and look at the concept or logical predicate 'has the property B'. We must presuppose for this concept that it has sharp boundaries in the sense given above; for only in that case can it be logically recognized as a concept. Then indeed the principle of non- contradiction holds, i. e. that of the two sentences cited always one and only one is right, that is, expresses a true thought; and indeed this holds even if the concept of the predicate is contradictory; for then the second sentence is right, but the first not.
And so the conditions for the validity of the principle of non-contradiction in S's formulation consist in the fact that proper names as well as concept- words satisfy logical requirements. But these do not include the require- ment that a concept-word should designate a consistent concept. Since only concepts with sharp boundaries are to be recognized as meaningful by logic, we may also say: proper names and concept-words must be meaningful.
Now if A is the concept of a right-angled equilateral pentagon, 'this A' is an inadmissible-meaningless-proper name, and nothing whatever can be legitimately inferred either from the sentence 'This A has the property B' or from 'This A does not have the property B'; for, strictly, we do not infer from sentences at all, but from thoughts. And only true thoughts are admissible premises of inferences. Also it isn't strictly sentences, it is thoughts which have contradictory counterparts. If you always bear this in mind, that we cannot legitimately infer from sentences, but only from true thoughts, and that proper names and concept-words must be meaningful, then the case of the failure of the principle of non-contradiction or of indirect proof cannot obtain. The failure of the principle of non- contradiction cannot serve as a criterion for the admissibility of a concept; for then you could never avail yourself of indirect proof. We should always have to reckon with its failing. But before we go on to proofs at all, we must have assured ourselves that the proper names and concept-words we employ are admissible. From the failure of indirect proof we can only conclude that we have made a mistake in this respect, which ought not to have occurred at all. Now whether this mistake consists in the use of a meaningless proper name or a meaningless concept-word, cannot be generally answered. At any rate, an inconsistent concept is not to be rejected of itself. Only if some- one makes a proper name from the corresponding concept-word by means of the definite article or demonstrative, does he fall into error. But whether it is a mistake of this kind or of another kind which is to hlame for the failure of the indirect prm~f. needs special investigation in each case.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 181
Now Schoenfties states that the concept of the set of all sets which do not contain themselves as elements is not consistent. It must first be remarked that 'the set of all sets which do not contain themselves as elements' is not a concept-word, it is a proper name, and it can only be a question of whether this proper name is meaningful. It is evidently not formed from the nomen appellativum is a set ofall sets which do not contain themselves as elements with the definite article; for the use of such a predicate has not preceded it: the expression makes its debut together with the definite article. And so we have here a different case from the previous one with the phrase 'this A'. Hence our proper name requires a special scrutiny. We may distinguish two different ways of using the word 'set', going with two different conceptions, which are probably most plainly identified by the words 'aggregate' and 'extension of a concept'. But frequently these conceptions do not occur in their pure form, but mixed together and this makes for unclarity. The aggregative conception is the first to offer itself, but the requirements of mathematics pull towards the opposite side, and so confusions easily arise.
The word 'aggregate' is connected with 'grex'. An aggregate is something like a herd, a whole whose parts are like one another. But since the agreement is never perfect, and since on the other hand it is surely always possible to find some respect in which the parts do agree, the similarity of the parts is unfitted to serve as a characteristic mark. We can say after all that every object in which we may distinguish parts is an aggregate, for instance the solar system, a heap of sand, a grain of sand, a piece of music, and a corporation. For objects to be recognised as parts of a whole, they have to be held together by relations or interactions of some kind or another. Such relations may be spatial, temporal, physical, psychical, legal, even
intervals of pitch. And strictly we ought really to have a different word for 'part' in each of the cases; for obviously a head is a part of a man in a differ- ent sense from that in which the man himself is part of a corporation. It not infrequently occurs that parts of an aggregate are themselves aggregates. A grain of sand is part of a heap of sand, a silicic acid molecule part of the 11,rain of sand and also part of the heap of sand. What is part of a part is part of the whole, at least if the word 'part' is taken in essentially the same sense. lu this description of an aggregate we perhaps fail to find the precision that is to be expected elsewhere in mathematics. At the same time we have to say that aggregate is not really a mathematical concept. But frequently the word ? ! 11enge' [set) is patently used in the sense which agrees with that given here l'or aggregate. I need only mention 'Menschenmenge' [crowd). But so understood, the word 'Menge' can surely claim no place in mathematics.
I move on to the extension of a concept. The word itself indicates that we ure not here dealing with something spatial and physical, but something lo~~,ical. By means of our logical faculties we lay hold upon the extension of a concept, by starting out from the concept.
Let the letters' C/J' and ? 'I" stand in for concept-words (nomina appellativa). Then we designate subordination in sentences of the form 'If something is a 1/l. then it is 11 'P'. In Sl? ntenccs of the form 'If sorncthin11, is 11 1/1, then it is n 'P
? 182 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
and if something is a 'P then it is a 4J' we designate mutual subordination, a second level relation, which has strong affinities with the first level relation of equality (identity). The properties of equality, that is, which we express in the sentences 'a =a', 'if a~~ b, then b =a', 'if a= band b = c, then a= c', have their analogues for the case of that second level relation. And this compels us almost ineluctably to transform a sentence in which mutual subordination is asserted of concepts into a sentence expressing an equality.
Admittedly, to construe mutual subordination simply as equality is forbidden by the basic difference between first and second level relations. Concepts cannot stand in a first level relation. That wouldn't be false, it would be nonsense. Only in the case of objects can there be any question of equality (identity). And so the said transformation can only occur by concepts being correlated with objects in such a way that concepts which are mutually subordinate are correlated with the same object. It is all, so to speak, moved down a level. The sentence 'Every square root of 1 is a bi- nomial coefficient of the exponent ~I and every binomial coefficient of the exponent --1 is a square root of l' is thus transformed into the sentence 'The extension of the concept square root of 1 is equal to (coincides with) the extension of the concept binomial coefficient ofthe exponent --1'.
And so 'the extension of the concept square root of l' is here to be regarded as a proper name, as is indeed indicated by the definite article. By permitting the transformation, you concede that such proper names have meanings. But by what right does such a transformation take place, in which concepts correspond to extensions of concepts, mutual subordination to equality? 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 ?
? ? On Schoenflies: Die Logischen Paradoxien der Mengenlehre 177 [Discussion]
The article by S, Ober die logischen Paradoxien der Mengenlehre* induces me to make the following remarks, in which I repeat much that I have already discussed previously, since it does not seem to be widely known. I fail to find in S and also in Korselt** the sharp distinction between concept and object. ? ? ? In the signs, a proper name (nomen proprium) corresponds to an object, a concept-word or concept-sign (nomen appellativum) to a concept. A sentence such as 'Two is a prime' can be analysed into two essentially different component parts: into 'two' and 'is a prime'. The former appears complete, the latter in need of supplementation, unsaturated. 'Two'-at least in this sentence-is a proper name, its meaning is an object, which can also be designated with greater prolixity by 'the number two'. The object, too, appears as a complete whole, whereas the predicative part has something unsaturated in its meaning as well. We count the copula 'is' as belonging to this part of the sentence. But there is usually something combined with it which here must be disregarded: assertoric force. We can of course express a thought, without stating it to be true. The thought is strictly the same, whether we merely express it or whether we also put it forward as true. Thus assertoric force, which is often connected with the copula or else with the grammatical predicate, does not belong to the expression of the thought, and so may be disregarded here.
This predicative component part of our sentence which we have described in this way, is also meaningful. We call it a concept-word or nomen appellativum, even though it is not customary to include the copula in this. Just as it itself appears unsaturated, there is also something unsaturated in the realm of meanings corresponding to it: we call this a concept. This unsaturatedness of one of the components is necessary, since otherwise the parts do not hold together. Of course two complete wholes can stand in a relation to one another; but then this relation is a third element-and one that is doubly unsaturated! In the case of a concept we can also call the unsaturatedness its predicative nature. But in this connection it is necessary to point out an imprecision forced on us by language, which, if we are not conscious of it, will prevent us from recognizing the heart of the matter: i. e. we can scarcely avoid using such expressions as 'the concept prime'. Here there is no trace left of unsaturatedness, of the predicative nature. Rather, the expression is constructed in a way which precisely parallels 'the poet Schiller'. So language brands a concept as an object, since the only way it can fit the designation for a concept into its grammatical structure is as a proper name. But in so doing, strictly speaking it falsifies matters. In the same way, the word 'concept' itself is, taken strictly, already defective, since
? The current Jahresbericht, Vol. XV, p. 19 (Jan. 1906).
? ? The current Jahresbericht, Vol. XV, p. 215 (March-April 1906). ? ? ? Cf. My essay 'Concept and Object'.
? ? 178 On Schoenjltes: Die Logischen Paradoxien der Mengenlehre
the phrase 'is a concept' requires a proper name as grammatical subject; and so, strictly speaking, it requires something contradictory, since no proper name can designate a concept; or perhaps better still, something nonsensical. It is no objection to say that surely the grammatical predicate 'is rectangular' can be combined with the grammatical subject 'every square', which isn't a proper name; for even the sentence 'every square is rectangular' can only make sense in virtue of the fact that you can assert of an object that it is rectangular, either rightly or wrongly, but in either case significantly. By a proper name I understand the sign of an object, independently of the question whether it be a simple word or sign, or a complex one, provided only that it designates the object determinately.
In the sentence 'Two is a prime' we find a relation designated: that of subsumption. We may also say the object falls under the concept prime, but if we do so, we must not forget the imprecision of linguistic expression we have just mentioned. This also creates the impression that the relation of subsumption is a third element supervenient upon the object and the concept. This isn't the case: the unsaturatedness of the concept brings it about that the object, in effecting the saturation, engages immediately with the concept, without need of any special cement. Object and concept are fundamentally made for each other, and in subsumption we have their fundamental union.
We call a concept empty if no object falls under it. The concept-word for an empty concept never yields a true sentence,? no matter what proper name may saturate it, or in other words: no matter what proper name may be attached as a grammatical subject to the concept-word as predicate. A concept under which one and only one object falls must still be distinguished from the latter; its sign is a nomen appellativum, not a nomen proprium.
With the help of the definite article or demonstrative, language forms proper names out of concept-words. So, for instance, the phrase 'this A' on p. 20 of the Schoenfties article is a proper name. If forming a proper name in this way is to be legitimate, the concept whose designation is used in its formation must satisfy two conditions:
1. It may not be empty.
2. Only one object may fall under it.
If the first condition is unsatisfied, there is no object at all to which the proper name would be ascribed. If the second is unsatisfied, there are indeed several such objects, but none of them is determined as the one meant to be designated by the proper name. In science the purpose of a proper name is to designate an object determinately; if this purpose is unfulfilled, the proper name has no justification in science. How things may be in ordinary
? For brevity, I here call a sentence expressing a true thought, a true sentence.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 179
language is no concern of ours here. Our first requirement includes the requirement that the concept be consistent.
But this requirement doesn't mean that every concept ought to be consistent, only that a concept must be consistent if you wish to form a proper name from the concept-word together with the definite article or demonstrative. But for this purpose, this requirement doesn't go far enough: the concept is not to be empty, no matter what the reason for its being empty may be.
But if we ask, under what conditions a concept is admissible in science. the first thing to stress is that consistency is no~ such a condition. The only requirement to be made of a concept is that it should have sharp boundaries; that is, that for every object it holds that it either falls under the concept or does not do so. Essentially this is nothing but the requirement that the principle of non-contradiction should hold. But the admissibility of a concept is entirely independent of the question whether objects fall under it, and if so which, or in other words; whether there be objects, and if so which, of which it can be truly asserted. For, before we can raise such questions, we already need the concept. The requirement that a concept be consistent encounters great difficulties. The only apparent way to show a concept has this property, is to cite an object falling under the concept. But to do that, you already need the concept.
Schoenfties obviously only lays down this requirement as a result of his confounding concepts and objects. It is of course self-evident that an object t:annot have inconsistent properties. That the concept of a right-angled equilateral pentagon contains a contradiction does not make it inadmissible. For we can see no reason why a man should not be able to say of an object that it is not a right-angled equilateral pentagon, or why he should not be permitted to say there are no right-angled equilateral pentagons. And before he arrives at such judgements, he must consider the matter, and to do that he requires this concept. It is completely wrongheaded to imagine that every contradiction is immediately recognizable; frequently the contradiction lies deeply buried and is only discovered by a lengthy chain of inference: throughout which you need the concept. The only thing is, given a concept, you may not presuppose without further question that an object falls under it. But someone does that when, by means of a definite article or demonstrative, he forms out of a concept-word a proper name destined for use in science.
Let us now examine Schoenflies' sentences in the light of what we have said so far. Earlier on p. 20 we read,
'I. The A has the property B, and
2. The A does not have the property B. '
The definite article before the 'A' can only be understood in such a way that 'the A' is being thought of as a proper name. In logic it must be presupposed that every proper name is meaningful; that is, that it serves its purpose of
? 180 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
designating an object. For a sentence containing a meaningless proper name either expresses no thought at all, or it expresses a thought that belongs to myth or fiction. In either case it falls outside the domain governed by the laws of logic. For instance 'the A' would be a name that was in this way inadmissible for scientific use if it were formed by means of the definite article from a name of a concept (nomen appellativum), under which either no object or more than one fell. The mistake would then not lie in the use of
the concept-word in itself, but in forming a proper name from it. Let us now assume 'the A' is a meaningful proper name, and look at the concept or logical predicate 'has the property B'. We must presuppose for this concept that it has sharp boundaries in the sense given above; for only in that case can it be logically recognized as a concept. Then indeed the principle of non- contradiction holds, i. e. that of the two sentences cited always one and only one is right, that is, expresses a true thought; and indeed this holds even if the concept of the predicate is contradictory; for then the second sentence is right, but the first not.
And so the conditions for the validity of the principle of non-contradiction in S's formulation consist in the fact that proper names as well as concept- words satisfy logical requirements. But these do not include the require- ment that a concept-word should designate a consistent concept. Since only concepts with sharp boundaries are to be recognized as meaningful by logic, we may also say: proper names and concept-words must be meaningful.
Now if A is the concept of a right-angled equilateral pentagon, 'this A' is an inadmissible-meaningless-proper name, and nothing whatever can be legitimately inferred either from the sentence 'This A has the property B' or from 'This A does not have the property B'; for, strictly, we do not infer from sentences at all, but from thoughts. And only true thoughts are admissible premises of inferences. Also it isn't strictly sentences, it is thoughts which have contradictory counterparts. If you always bear this in mind, that we cannot legitimately infer from sentences, but only from true thoughts, and that proper names and concept-words must be meaningful, then the case of the failure of the principle of non-contradiction or of indirect proof cannot obtain. The failure of the principle of non- contradiction cannot serve as a criterion for the admissibility of a concept; for then you could never avail yourself of indirect proof. We should always have to reckon with its failing. But before we go on to proofs at all, we must have assured ourselves that the proper names and concept-words we employ are admissible. From the failure of indirect proof we can only conclude that we have made a mistake in this respect, which ought not to have occurred at all. Now whether this mistake consists in the use of a meaningless proper name or a meaningless concept-word, cannot be generally answered. At any rate, an inconsistent concept is not to be rejected of itself. Only if some- one makes a proper name from the corresponding concept-word by means of the definite article or demonstrative, does he fall into error. But whether it is a mistake of this kind or of another kind which is to hlame for the failure of the indirect prm~f. needs special investigation in each case.
? On Schoenjlies: Die Logischen Paradoxien der Mengenlehre 181
Now Schoenfties states that the concept of the set of all sets which do not contain themselves as elements is not consistent. It must first be remarked that 'the set of all sets which do not contain themselves as elements' is not a concept-word, it is a proper name, and it can only be a question of whether this proper name is meaningful. It is evidently not formed from the nomen appellativum is a set ofall sets which do not contain themselves as elements with the definite article; for the use of such a predicate has not preceded it: the expression makes its debut together with the definite article. And so we have here a different case from the previous one with the phrase 'this A'. Hence our proper name requires a special scrutiny. We may distinguish two different ways of using the word 'set', going with two different conceptions, which are probably most plainly identified by the words 'aggregate' and 'extension of a concept'. But frequently these conceptions do not occur in their pure form, but mixed together and this makes for unclarity. The aggregative conception is the first to offer itself, but the requirements of mathematics pull towards the opposite side, and so confusions easily arise.
The word 'aggregate' is connected with 'grex'. An aggregate is something like a herd, a whole whose parts are like one another. But since the agreement is never perfect, and since on the other hand it is surely always possible to find some respect in which the parts do agree, the similarity of the parts is unfitted to serve as a characteristic mark. We can say after all that every object in which we may distinguish parts is an aggregate, for instance the solar system, a heap of sand, a grain of sand, a piece of music, and a corporation. For objects to be recognised as parts of a whole, they have to be held together by relations or interactions of some kind or another. Such relations may be spatial, temporal, physical, psychical, legal, even
intervals of pitch. And strictly we ought really to have a different word for 'part' in each of the cases; for obviously a head is a part of a man in a differ- ent sense from that in which the man himself is part of a corporation. It not infrequently occurs that parts of an aggregate are themselves aggregates. A grain of sand is part of a heap of sand, a silicic acid molecule part of the 11,rain of sand and also part of the heap of sand. What is part of a part is part of the whole, at least if the word 'part' is taken in essentially the same sense. lu this description of an aggregate we perhaps fail to find the precision that is to be expected elsewhere in mathematics. At the same time we have to say that aggregate is not really a mathematical concept. But frequently the word ? ! 11enge' [set) is patently used in the sense which agrees with that given here l'or aggregate. I need only mention 'Menschenmenge' [crowd). But so understood, the word 'Menge' can surely claim no place in mathematics.
I move on to the extension of a concept. The word itself indicates that we ure not here dealing with something spatial and physical, but something lo~~,ical. By means of our logical faculties we lay hold upon the extension of a concept, by starting out from the concept.
Let the letters' C/J' and ? 'I" stand in for concept-words (nomina appellativa). Then we designate subordination in sentences of the form 'If something is a 1/l. then it is 11 'P'. In Sl? ntenccs of the form 'If sorncthin11, is 11 1/1, then it is n 'P
? 182 On Schoenjlies: Die Logischen Paradoxien der Mengenlehre
and if something is a 'P then it is a 4J' we designate mutual subordination, a second level relation, which has strong affinities with the first level relation of equality (identity). The properties of equality, that is, which we express in the sentences 'a =a', 'if a~~ b, then b =a', 'if a= band b = c, then a= c', have their analogues for the case of that second level relation. And this compels us almost ineluctably to transform a sentence in which mutual subordination is asserted of concepts into a sentence expressing an equality.
Admittedly, to construe mutual subordination simply as equality is forbidden by the basic difference between first and second level relations. Concepts cannot stand in a first level relation. That wouldn't be false, it would be nonsense. Only in the case of objects can there be any question of equality (identity). And so the said transformation can only occur by concepts being correlated with objects in such a way that concepts which are mutually subordinate are correlated with the same object. It is all, so to speak, moved down a level. The sentence 'Every square root of 1 is a bi- nomial coefficient of the exponent ~I and every binomial coefficient of the exponent --1 is a square root of l' is thus transformed into the sentence 'The extension of the concept square root of 1 is equal to (coincides with) the extension of the concept binomial coefficient ofthe exponent --1'.
And so 'the extension of the concept square root of l' is here to be regarded as a proper name, as is indeed indicated by the definite article. By permitting the transformation, you concede that such proper names have meanings. But by what right does such a transformation take place, in which concepts correspond to extensions of concepts, mutual subordination to equality? 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
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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
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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 ?
