250 Logic in Mathematics
'the concept positive number' were a proper name designating an object and as if the intention were to assert of this object that it is satisfied.
'the concept positive number' were a proper name designating an object and as if the intention were to assert of this object that it is satisfied.
Gottlob-Frege-Posthumous-Writings
E.
g.
we may use the sign 'I' when what is in question is just the addition of positive integers, but define it completely so that the value of the function ~ I Cis determined whatever is taken as the ~- and the C-argument.
E.
g.
we may stipulate that the value of this function is to be the False when one of the two arguments is something other than a positive integer.
So piecemeal definition and what is referred to as the extension of con- cepts by stages must be rejected. Definitions must be given once and for all; for whilst the definition of a concept remains incomplete, the concept itself does not have sharp boundaries and cannot be acknowledged as such.
Let us take one more look at the ground we have just covered
A sentence has a sense and we call the sense of an assertoric sentence a thought. A sentence is uttered either with assertoric force or without. It is not enough for science that a sentence should only have a sense; it must have a truth-value too and this we call the meaning of the sentence. If a sentence only has a sense, but no meaning, it belongs to fiction, and not to science.
Language has the power to express, with comparatively few means such a profusion of thoughts that no one could possibly command a view of them all. What makes this possible is that a thought has parts out of which it is constructed and that these parts correspond to parts of sentences, by which they are expressed. The simplest case is that of a thought which consists of a complete part and an unsaturated one. The latter we may also call the predicative part. Each of these parts must equally have a meaning, if the whole sentence is to have a meaning, a truth-value. We call the meaning of the complete part an object, that of the part which is in need of supplementation, which is unsaturated or predicative, we call a concept. We may call the way in which object and concept are combined in a sentence the subsumption of the object under the concept. Objects and concepts are fundamentally different. We call the complete part of a sentence the proper name of the object it designates. The part of a sentence that is in need of supplementation we call a concept-word or concept-sign. It is a necessary requirement for concepts that they have sharp boundaries. Both parts of a sentence, the proper name and the concept-word, may in turn be complex. The proper name may itself consist of a complete part and a part in need of supplementation. The former is again a proper name and designates an object; the latter we call a function-sign. As a result of completing a
concept-sign with a proper name we obtain a sentence, whose meaning is a
? ? 244
Logic in Mathematics
truth-value. As a result of supplementing a function-sign with a proper name we obtain a proper name, whose meaning is an object. We obtain the same perspective on both if we count a concept as a function, namely a function whose value is always a truth-value, and if we count a truth-value as an object. Then a concept is a function whose value is always a truth-value. '
But a function-sign may be complex too: it may be composed of a complete part which is again a proper name, and a part that is doubly in need of supplementation-what is a name or sign of a function of two arguments. A function of two arguments whose value is always a truth- value, we call a relation. The requirement that a concept have sharp boundaries corresponds to the more general requirement that the name of a function of one argument, when supplemented with a meaningful proper name, must in turn yield a meaningful proper name. And the same holds mutatis mutandis for functions of two arguments.
Let us take a look at something that came still earlier. We realized the necessity of constructing mathematics as a system, which is not to rule out the possibility of there being different systems. It turned out that the foundations of a system are
1. the axioms and
2. thedefinitions.
The axioms of a system serve as premises for the inferences by means of
which the system is built up, but they do not figure as inferred truths. Since they are intended as premises, they have to be true. An axiom that is not true is a contradiction in terms. An axiom must not contain any term with which we are unfamiliar.
The definitions are something quite different. Their role is to bestow a meaning on a sign or word that hitherto had none. So a definition has to contain a new sign. Once a meaning has been given to this sign by the definition, the definition is transformed into an independent sentence which can be used in the development of the system as a premise for inferences. How are inferences carried out within the system?
Let us assume that we have a sentence of the form 'If A holds, so does B'. If we add to this the further sentence 'A holds', then from both premises we can infer 'B holds'. But for the conclusion to be possible, both premises have to be true. And this is why the axioms also have to be true, if they are to serve as premises. For we can draw no conclusion from something false. But it might perhaps be asked, can we not, all the same, draw consequences from a sentence which may be false, in order to see what we should get if it were true? Yes, in a certain sense this is possible. From the premises
If Fholds, so does A If A holds, so does E
1 This is how the sentence reads in the German. Since it merely repeats the first part of the preceding sentence, the editors suggest that we read 'object' in place of 'truth value', so that the sentence marks a natural inference from the one precedina (trans. ).
? we can infer
If Fholds, so does E From this and the further premise
we can infer
If E holds, so does Z
IfFholds, so does Z.
Logic in Mathematics 245
And so we can go on drawing consequences without knowing whether r is true or false. But we must notice the difference. In the earlier example the premise 'A holds' dropped out of the conclusion altogether. In this example the condition 'If Fholds' is retained throughout. We can only detach it when we have seen that it is fulfilled. In the present case 'r holds' cannot be regarded as a premise at all: what we have as a premise is
If Fholds, so does A,
and thus something of which 'r holds' is only a part. Of course this whole premise must be true; but this is possible without the condition being fulfilled, without r holding. So, strictly speaking, we simply cannot say that consequences are here being drawn from a thought that is false or doubtful; for this does not occur independently as a premise, but is only part of a premise which as such has indeed to be true, but which can be true without that part of the thought-the part which it contains as a condition-being true.
In indirect proofs it looks as if consequences are being drawn from something false. As an example, suppose we have to prove that in a triangle
the longer side subtends the greater angle. To prove:
1
A
c
If LB > LA, then AC >BC. We take as given:
I IfBC>AC,thenLA>LB. II IfBC=AC,thenLA= LB.
III If not AC >BC, and if not BC> AC, then BC=AC.
IV IfLA=LB,thennotLB>LA. V If LA> LB, then not LB >LA.
From Il and Ill there follows:
IfnotAC >BC and if not BC> AC, then LA= LB.
From this and IV we have: IfnotAC>BCandifnotBC>AC,thennot LB>L. A.
? 246 Logic in Mathematics From I and V:
IfBC> AC, then not LB >LA. From the last two sentences there follows:
If not AC >BC, then not LB > LA. And then by contraposition:
If LB >LA, then AC >BC.
To simplify matters, I shall assume that we are not speaking of triangles in general, but of a particular triangle. LA and LB may be understood as numbers, arrived at by measuring the angles by some unit-measure, as e. g. a right-angle. AC and BC may likewise be understood as numbers, arrived at by using some unit-measure for the sides, as e. g. a metre. Then the signs 'LA',' LB', 'AC', and 'BC' are to be taken as proper names of numbers.
We see that 'not AC > BC' does not occur here as a premise, but that it is contained in Ill as a part-as a condition. So strictly speaking, we cannot say that consequences are being drawn from the false thought (not AC >BC). Therefore, we ought not really to say 'suppose that not AC >BC', because this makes it look as though 'not AC >BC' was meant to serve as a premise for inference, whereas it is only a condition.
We make far too much of the peculiarity of indirect proof vis-d-vis direct proof. The truth is that the difference between them is not at all important.
The proof can also be set out in the following way. We now take as given:
I' Ifnot LA >LB then not BC> AC. II' IfnotLA= LB,thennotBC=AC.
Ill' IfnotBC>ACandifnotBC=AC,thenAC>BC. IV' If LB >LA then not LA= LB.
V' If LB >LA then not LA> LB.
From V' and I' there follows:
If LB > LA, then not BC> AC.
From this and Ill' we have:
If LB >LA and if not BC= AC, then AC >BC.
From IV' and II' there follows:
If LB >LA, then not BC= AC.
From the last two sentences we have:
If LB >LA, then AC >BC.
At no point in this proof have we entertained 'not AC >BC' even as a mere hypothesis.
? Logic in Mathematics 247
In an investigation of the foundations of geometry it may also look as if consequences are being drawn from something false or at least doubtful. Can we not put to ourselves the question: How would it be if the axiom of parallels didn't hold? Now there are two possibilities here: either no use at all is made of the axiom of parallels, but we are simply asking how far we can get with the other axioms, or we are straightforwardly supposing something which contradicts the axiom of parallels. It can only be a question of the latter case here. But it must constantly be borne in mind that what is false cannot be an axiom, at least if the word 'axiom' is being used in the traditional sense. What are we to say then? Can the axiom of parallels be acknowledged as an axiom in this sense? When a straight line intersects one of two parallel lines, does it always intersect the other? This question, strictly speaking, is one that each person can only answer for himself. I can only say: so long as I understand the words 'straight line', 'parallel' and 'intersect' as I do, I cannot but accept the parallels axiom. If someone else does not accept it, I can only assume that he understands these words differently. Their sense is indissolubly bound up with the axiom of parallels. Hence a thought which contradicts the axiom of parallels cannot be taken as a premise of an inference. But a true hypothetical thought, whose condition contradicted the axiom, could be used as a premise. This condition would then be retained in all judgements arrived at by means of our chains of inference. If at some point we arrived at a hypothetical judgement whose consequence contradicted known axioms, then we could conclude that the condition contradicting the axiom of parallels was false, and we should thereby have proved the axiom of parallels with the help of other axioms. But because it had been proved, it would lose its status as an axiom. In such a case we should really have given an indirect proof.
If, however, we went on drawing inference after inference and still did not come up against a contradiction anywhere, we should certainly become more and more inclined to regard the axiom as incapable of proof. Nevertheless we should still, strictly speaking, not have proved this to be so.
Now in his Grundlagen der Geometrie Hilbert is preoccupied with such questions as the consistency and independence of axioms. But here the sense of the word 'axiom' has shifted. For if an axiom must of necessity be true, it is impossible for axioms to be inconsistent with one another. So any discussion here would be a waste of words. But obvious though it is, it seems just not to have entered Hilbert's mind that he is not speaking of axioms in Euclid's sense at all when he discusses their consistency and independence. We could say that the word 'axiom', as he uses it, fluctuates from one sense to another without his noticing it. It is true that if we concentrate on the words of one of his axioms, the immediate impression is that we are dealing with an axiom of the Euclidean variety; but the words mislead us, because all the words have a different use from what they have in Euclid. At? 31 we
1 The quotation here is from the first edition of the Grundlagen der Geometric. Later editions give a different uxiom in plnce of thnt which Frege cites under ((. 4 (cd. ).
? 248 Logic in Mathematics
read 'Definition. The points of a straight line stand in certain relations to one another, for the description of which we appropriate the word "between". ' Now this definition is only complete once we are given the four axioms
11. 1 If A, B, C are points on a straight line, and B lies between A and C, then B lies between C and A.
11. 2 If A and C are two points on a straight line, then there is at least one point B lying between A and C, and at least one point D such that C lies between A and D.
11. 3 Given any three points on a straight line, there is one and only one which lies between the other two.
11. 4 Any four points A, B, C, D on a straight line can be so ordered that B lies between A and C and between A and D, and so that C lies between A and D and between B and D.
These axioms, then, are meant to form parts of a definition. Consequently these sentences must contain a sign which hitherto had no meaning, but which is given a meaning by all these sentences taken together. This sign is apparently the word 'between'. But a sentence that is meant to express an axiom may not contain a new sign. All the terms in it must be known to us. As long as the word 'between' remains without a sense, the sentence 'If A, B, C, are points on a straight line and B lies between A and C, then B lies between C and A' fails to express a thought.
An axiom, however, is always a true thought. Therefore, what does not express a thought, cannot express an axiom either. And yet one has the impression, on reading the first of these sentences, that it might be an axiom. But the reason for this is only that we are already accustomed to associate a sense with the word 'between'. In fact if in place of
we say
'B lies between A and C' 'B pat A nam C',
then we associate no sense with it. Instead of the so-called axiom 11. 1 we should have
'If B pat A nam C, then B pat C nam A'.
No one to whom these syllables 'pat' and 'nam' are unfamiliar will associate a sense with this apparent sentence. The same holds of the other three pseudo-axioms.
The question now arises whether, if we understand by A, B, C points on a straight line, an expression of the form
'B patA nam C'
will not at least ~ome to acquire a sense through the totality of these pseudo-sentences. I think not. We may perhaps hazard the guess that it will
come to the same thing as
? Logic in Mathematics 249 'B lies between A and C',
but it would be a guess and no more. What is to say that this matrix could not have several solutions?
But does, then, a definition have to be unambiguous? Are there not circumstances in which a certain give and take is a good thing?
Of course a2 = 4 does not determine unequivocally what a is to mean, but is there any harm in that? Well, if a is to be a proper name whose meaning is meant to be fixed, this goal will obviously not be achieved. On the contrary one can see that in this formula there is designated a concept under which
the numbers 2 and -2 fall. Once we see this, the ambiguity is harmless, but it means that we don't have a definition of an object.
If we want to compare this case with our pseudo-axiom, we have to compare the letter 'a' with 'between' or with 'pat-nam'. We must distinguish signs that designate from those which merely indicate. In fact the word 'between' or 'pat-nam' no more designates anything than does the letter 'a'. So we have here to disregard the fact that we usually associate a sense with the word 'between'. In this context it no more has a sense than does 'pat- nam'. Now to say that a sign which only indicates neither designates anything nor has a sense is not yet to say it could not contribute to the expression of a thought. It can do this by conferring generality of content on a simple sentence or on one made up of sentences.
Now there is, to be sure, a difference between our two cases; for whilst 'a' stands in for a proper name, 'pat-nam' stands in for the designation of a relation with three terms. As we call a function of one argument, whose value is always a truth-value, a concept, and a function of two arguments, whose value is always a truth-value, a relation, we can go yet a step further and call a function of three arguments whose value is always a truth-value, a relation with three terms. Then whilst the 'between-and' or the 'pat-nam' do not designate such a relation with three terms, they do indicate it, as 'a' indicates an object. Still there remains a distinction. We were able to find a concept designated in 'a2 = 4'.
What would correspond to this in the case of our pseudo-axioms? It is what I call a second level concept. In order to see more clearly what I understand by this, consider the following sentences:
'There is a positive number' 'There is a cube root of 1'.
We can see that these have something in common. A statement is being made, not about an object, however, but about a concept. In the first sentence it is the concept positive number, in the second it is the concept cube root of 1. And in each case it is asserted of the concept that it is not empty, but satisfied. It is indeed strictly a mistake to say 'The concept positive number is satisfied', for by saying this I seem to make the concept into an object, as the definite article 'the concept' shows. lt now looks as if
?
250 Logic in Mathematics
'the concept positive number' were a proper name designating an object and as if the intention were to assert of this object that it is satisfied. But the truth is that we do not have an object here at all. By a necessity oflanguage, we have to use an expression which puts things in the wrong perspective; even so there is an analogy. What we designate by 'a positive number' is related to what we designate by 'there is', as an object (e. g. the earth) is related to a concept (e. g. planet).
I distinguish concepts under which objects fall as concepts of first level from concepts of second level within which, as I put it, concepts of first level fall. Of course it goes without Saying that all these expressions are only to be understood metaphorically; for taken literally, they would put things in the wrong perspective. We can also admit second level concepts within which relations fall. E. g. in the sentence:
'It is to hold for all A, B, C that if A stands in the p-relation to B and if A stands in the p-relation to C, then B = C',
we have a designation of a second-level concept within which relations fall and 'stands in the p-relation to . . . ' here stands in for the argument- sign-for, that is, the designation of the relation presented as argument. If we substitute e. g. the equals sign, we get
'It holds for all A, B, C that ifA= Band ifA= C, then B = C'.
If this is true, then it follows that the relation of equality falls within this second level concept.
As we call a function of one argument, whose value is always a truth value, a concept, and as we call a function of two arguments, whose value is always a truth value, a relation, so we could introduce a special name for a function o f three arguments, whose value is always a truth value. Provisionally, such a function may be called a relation with three terms. That designated by the words 'lies between . . . and . . . ' would be of this kind, if these words were understood as we should ordinarily understand them when used in speaking of Euclidean points on a Euclidean straight line. However, they are not used in our pseudo-axioms as a sign which designates, but only as one which indicates, as are letters in arithmetic. So in this context they do not designate such a three-termed relation: they only indicate such a relation. Even if we wish, for the time being, to understand the actual words 'point' and 'straight line' in the Euclidean sense, still the words 'lies between . . . and . . . ' are not to be regarded, strictly speaking, as words with a sense, but only as a stand-in for an argument, as is the letter 'a' in 'a2'. But the function, whose arguments they stand in for, is a second level function within which only relations with three terms can fall.
? ? My basic logical Insights1 [1915]
The following may be of some use as a key to the understanding of my results.
Whenever anyone recognizes something to be true, he makes a judgement. What he recognizes to be true is a thought. It is impossible to recognize a thought as true before it has been grasped. A true thought was true before it was grasped by anyone. A thought does not have to be owned by anyone. The same thought can be grasped by several people. Making a judgement does not alter the thought that is recognized to be true. When something is judged to be the case, we can always cull out the thought that is recognized as true; the act of judgement forms no part of this. The word 'true' is not an adjective in the ordinary sense. If I attach the word 'salt' to the word 'sea- water' as a predicate, I form a sentence that expresses a thought. To make it clearer that we have only the expression of a thought, but that nothing is meant to be asserted, I put the sentence in the dependent form 'that sea- water is salt'. Instead of doing this I could have it spoken by an actor on the stage as part of his role, for we know that in playing a part an actor only seems to speak with assertoric force. Knowledge of the sense of the word 'salt' is required for an understanding of the sentence, since it makes an essential contribution to the thought-in the mere word 'sea-water' we should of course not have a sentence at all, nor an expression for a thought. With the word 'true' the matter is quite different. If I attach this to the words 'that sea-water is salt' as a predicate, I likewise form a sentence that expresses a thought. For the same reason as before I put this also in the dependent form 'that it is true that sea-water is salt'. The thought expressed in these words coincides with the sense of the sentence 'that sea-water is salt'. So the sense of the word 'true' is such that it does not make any essential contribution to the thought. If I assert 'it is true that sea-water is salt', I assert the same thing as if I assert 'sea-water is salt'. This enables us to recognize that the assertion is not to be found in the word 'true', but in the assertoric force with which the sentence is uttered. This may lead us to think that the word 'true' has no sense at all. But in that case a sentence in which 'true' occurred as a predicate would have no sense either. All one can say
1
is based, this piece is to be dated around 1915 (ed. ).
According to a note by Heinrich Scholz on the transcripts on which this edition
? ? 252 My basic logical Insights
is: the word 'true' has a sense that contributes nothing to the sense of the whole sentence in which it occurs as a predicate.
But it is precisely for this reason that this word seems fitted to indicate the essence of logic. Because of the particular sense that it carried any other adjective would be less suitable for this purpose. So the word 'true' seems to make1 the impossible possible: it allows what corresponds to the assertoric force to assume the form of a contribution to the thought. And although this attempt miscarries, or rather through the very fact that it miscarries, it indicates what is characteristic of logic. And this, from what we have said, seems something essentially different from what is characteristic of aesthetics and ethics. For there is no doubt that the word 'beautiful' actually does indicate the essence of aesthetics, as does 'good' that of ethics, whereas 'true' only makes an abortive attempt to indicate the essence of logic, since what logic is really concerned with is not contained in the word 'true' at all but in the assertoric force with which a sentence is uttered.
Many things that belong with the thought, such as negation or generality, seem to be more closely connected with the assertoric force of the sentence or the truth of the thought. 2 One has only to see that such thoughts occur in e. g. conditional sentences or as spoken by an actor as part of his role for this illusion to vanish.
How is it then that this word 'true', though it seems devoid of content, cannot be dispensed with? Would it not be possible, at least in laying the foundations of logic, to avoid this word altogether, when it can only create confusion? That we cannot do so is due to the imperfection of language. If our language were logically more perfect, we would perhaps have no further need of logic, or we might read it off from the language. But we are far from being in such a position. Work in logic just is, to a large extent, a struggle with the logical defects of language, and yet language remains for us an in- dispensable tool. Only after our logical work has been completed shall we possess a more perfect instrument.
Now the thing that indicates most clearly the essence of logic is the assertoric force with which a sentence is uttered. But no word, or part of a sentence, corresponds to this; the same series of words may be uttered with assertoric force at one time, and not at another. In language assertoric force is bound up with the predicate.
1 A different version of the manuscript has 'to be trying to make' in place of 'to make' (ed. ).
2 This sentence and the one following are crossed out in the manuscript (ed. ).
? ? [Notes for Ludwig DarmstaedterP [July 1919]
I started out from mathematics. The most pressing need, it seemed to me, was to provide this science with a better foundation. I soon realized that number is not a heap, a series of things, nor a property of a heap either, but that in stating a number which we have arrived at as the result of counting we are making a statement about a concept. (Plato, The Greater Hippias. )
The logical imperfections of language stood in the way of such investigations. I tried to overcome these obstacles with my concept-script. In this way I was led from mathematics to logic.
What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word 'true', and then immediately go on to introduce a thought as that to which the question 'Is it true? ' is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgement; I come by the parts of a thought by analysing the thought. This marks off my concept-script from the similar inventions of Leibniz and his successors, despite what the name suggests; perhaps it was not a very happy choice on my part.
Truth is not part of a thought. We can grasp a thought without at the same time recognizing it as true-without making a judgement. Both grasping a thought and making a judgement are acts of a knowing subject, and are to be assigned to psychology. But both acts involve something that does not belong to psychology, namely the thought.
False thoughts must be recognized too, not of course as true, but as indispensable aids to knowledge, for we sometimes arrive at the truth by way of false thoughts and doubts. There can be no questions if it is essential to the content of any question that that content should be true.
Negation does not belong to the act of judging, but is a constituent of a thought. The division of thoughts (judgements) into affirmative and negative is of no use to logic, and I doubt if it can be carried through.
Where we have a compound sentence consisting of an antecedent and a consequent, there are two main cases to distinguish. The antecedent and consequent may each have a complete thought as its sense. Then, over and
1 This piece is dated at the end by Frege himself. It is one of the few manuscripts of Frege which have come down to us in the original and is in the possession of the Staatsbibliothek der Stiftung Prezij)ischer Ku/turbesitz where it forms part of the collection of the historian of science Ludwig Darmstaedter, ref. number 1919-95 (ed. ).
? ? 254
[Notes for Ludwig Darmstaedter]
above these, we have the thought expressed by the whole compound sentence. By recognizing this thought as true, we recognize neither the thought in the antecedent as true, nor that in the consequent as true. A second case is where neither antecedent nor consequent has a sense in itself, but where nevertheless the whole compound sentence does express a thought-a thought which is general in character. In such a case we have a relation, not between judgements or thoughts, but between concepts, the relation, namely, of subordination. The antecedent and consequent are here sentences only in the grammatical, not in the logical, sense. The first thing that strikes us here is that a thought is made up out of parts that are not themselves thoughts. The simplest case of this kind is where one of the two parts is in need of supplementation and is completed by the other part, which is saturated: that is to say, it is not in need of supplementation. The former part then corresponds to a concept, the latter to an object (subsumption of an object under a concept). However, the object and concept are not constituents of the thought expressed. The constituents of the thought do refer to the object and concept, but in a special way. There is also the case where a part doubly in need of supplementation is completed by two saturated parts. The former part corresponds to a relation. -An object stands in a relation to an object. -Where logic is concerned, it seems that every combination of parts results from completing something that is in need of supplementation; where logic is concerned, no whole can consist of saturated parts alone. The sharp separation of what is in need of supplementation from what is saturated is very important. When all is said and done, people have long been familiar with the former in mathematics (+, :, . . ;-, sin, =, >). In this connection they speak of functions, and yet it would seem that in most cases they have only a vague notion of what is at stake.
A general statement can be negated. In this way we arrive at what logicians call existential judgements and particular judgements. The existential thoughts I have in mind here are such as are expressed in German by 'es gibt'. 1 This phrase is never followed immediately by a proper name in the singular, or by a word accompanied by the definite article, but always by a concept-word (nomen appellativum) without a definite article. In existential sentences of this kind we are making a statement about a concept. Here we have an instance of how a concept can be related to a second level concept in a way analogous to that in which an object is related to a concept under which it falls. Closely akin to these existential thoughts are thoughts that are particular: indeed they may be included among them. But we can also say that what is expressed by a sentence of the particular form is that a concept stands in a certain second level relation to a concept. The distinction between first and second level concepts can only be grasped clearly by one who has clearly grasped the distinction between what is in
1 i. e. judgements that are expressed in English by 'there is' or 'there are' (trans. ).
? [Notes for Ludwig Darmstaedter] 255
need of supplementation and what is saturated. A saturated part obtained by analysing a thought can sometimes itself be split up in the same way into a part in need of supplementation and a saturated part. The sentence 'The capital of Sweden is situated at the mouth of Lake Malar' can be split up into a part in need of supplementation and the saturated part 'the capital of Sweden'. This can further be split up into the part 'the capital of, which stands in need of supplementation, and the saturated part 'Sweden'. Splitting up the thought expressed by a sentence corresponds to such a splitting up of the sentence. The functions of Analysis correspond to parts of thoughts
that are thus in need of supplementation, without however being such.
A distinction has to be drawn between the sense and meaning of a sign (word, expression). If an astronomer makes a statement about the moon, the moon itself is not part of the thought expressed. The moon itself is the meaning of the expression 'the moon'. Therefore in addition to its meaning this expression must also have a sense, which can be a constituent of a thought. We can regard a sentence as a mapping of a thought: corresponding to the whole-part relation of a thought and its parts we have, by and large, the same relation for the sentence and its parts. Things are different in the domain of meaning. We cannot say that Sweden is a part of the capital of Sweden. The same object can be the meaning of different expressions, and any one of them can have a sense different from any other. Identity of meaning can go hand in hand with difference of sense. This is what makes it possible for a sentence o f the form ' A = B ' to express a thought with more content than one which merely exemplifies the law of identity. A statement in which something is recognized as the same again can be of far greater cognitive value than a particular case of the law of
identity.
Even a part of a thought, or a part of a part of a thought, that is in need of
supplementation, has something corresponding to it in the realm of meaning. Yet it is wrong to call this a concept, say, or a relation, or a function, although we can hardly avoid doing so. Grammatically, 'the concept of God' has the form of a saturated expression. Accordingly its sense cannot be anything in need of supplementation. When we use the words 'concept', 'relation', 'function' (as this is understood in Analysis), our words fail of their intended target. In this case even the expression 'the meaning', with the
definite article, should really be avoided.
It is not, however, only parts of sentences that have meaning; even a
whole sentence, whose sense is a thought, has one. All sentences that express a true thought have the same meaning, and all sentences that express a false thought have the same meaning (the True and the False). Sentences and parts of sentences with different meanings also have different senses. If in a sentence or part of a sentence one constituent is replaced by another with a different meaning, the different sentence or part that results does not have to have a different meaning from the original; on the other hand, it always has a different sense. If in a sentence or part of a sentence
? ? 256 [Notes for Ludwig Darmstaedter]
one constituent is replaced by another with the same meaning but not with the same sense, the different sentence or part that results has the same meaning as the original, but not the same sense. All this holds for direct, not for indirect, speech.
A thought can also be the meaning of a sentence (indirect speech, the subjunctive mood)1? The sentence does not then express this thought, but can be regarded as its proper name. Where we have a clause in indirect speech
occurring within direct speech, and we replace a constituent of this clause by another which has the same meaning in direct speech, then the whole which results from this transformation does not necessarily have the same meaning as the original.
The miracle of number. The adjectival use of number-words is misleading. In arithmetic a number-word makes its appearance in the singular as a proper name of an object of this science; it is not accompanied by the indefinite article, but is saturated. Subsumption: 'Two is a prime,' not subordination. The combinations 'each two', 'all twos' do not occur.
Yet amongst mathematicians we find a great lack of clarity and little agreement. Is number an object that arithmetic investigates or is it a counter in a game? Is arithmetic a game or a science? According to one man, we are to understand by 'number' a series of objects of the same kind; according to another, a spatial, material structure produced by writing. A third denies that number is spatial at all. Perhaps there are times when arithmeticians merely delude themselves into thinking that they understand by 'number' what they say they do. If this is not so, then they are attaching different senses to sentences with same wording; and if they still believe that they are working within one and the same discipline, then they are just deluding themselves.
So piecemeal definition and what is referred to as the extension of con- cepts by stages must be rejected. Definitions must be given once and for all; for whilst the definition of a concept remains incomplete, the concept itself does not have sharp boundaries and cannot be acknowledged as such.
Let us take one more look at the ground we have just covered
A sentence has a sense and we call the sense of an assertoric sentence a thought. A sentence is uttered either with assertoric force or without. It is not enough for science that a sentence should only have a sense; it must have a truth-value too and this we call the meaning of the sentence. If a sentence only has a sense, but no meaning, it belongs to fiction, and not to science.
Language has the power to express, with comparatively few means such a profusion of thoughts that no one could possibly command a view of them all. What makes this possible is that a thought has parts out of which it is constructed and that these parts correspond to parts of sentences, by which they are expressed. The simplest case is that of a thought which consists of a complete part and an unsaturated one. The latter we may also call the predicative part. Each of these parts must equally have a meaning, if the whole sentence is to have a meaning, a truth-value. We call the meaning of the complete part an object, that of the part which is in need of supplementation, which is unsaturated or predicative, we call a concept. We may call the way in which object and concept are combined in a sentence the subsumption of the object under the concept. Objects and concepts are fundamentally different. We call the complete part of a sentence the proper name of the object it designates. The part of a sentence that is in need of supplementation we call a concept-word or concept-sign. It is a necessary requirement for concepts that they have sharp boundaries. Both parts of a sentence, the proper name and the concept-word, may in turn be complex. The proper name may itself consist of a complete part and a part in need of supplementation. The former is again a proper name and designates an object; the latter we call a function-sign. As a result of completing a
concept-sign with a proper name we obtain a sentence, whose meaning is a
? ? 244
Logic in Mathematics
truth-value. As a result of supplementing a function-sign with a proper name we obtain a proper name, whose meaning is an object. We obtain the same perspective on both if we count a concept as a function, namely a function whose value is always a truth-value, and if we count a truth-value as an object. Then a concept is a function whose value is always a truth-value. '
But a function-sign may be complex too: it may be composed of a complete part which is again a proper name, and a part that is doubly in need of supplementation-what is a name or sign of a function of two arguments. A function of two arguments whose value is always a truth- value, we call a relation. The requirement that a concept have sharp boundaries corresponds to the more general requirement that the name of a function of one argument, when supplemented with a meaningful proper name, must in turn yield a meaningful proper name. And the same holds mutatis mutandis for functions of two arguments.
Let us take a look at something that came still earlier. We realized the necessity of constructing mathematics as a system, which is not to rule out the possibility of there being different systems. It turned out that the foundations of a system are
1. the axioms and
2. thedefinitions.
The axioms of a system serve as premises for the inferences by means of
which the system is built up, but they do not figure as inferred truths. Since they are intended as premises, they have to be true. An axiom that is not true is a contradiction in terms. An axiom must not contain any term with which we are unfamiliar.
The definitions are something quite different. Their role is to bestow a meaning on a sign or word that hitherto had none. So a definition has to contain a new sign. Once a meaning has been given to this sign by the definition, the definition is transformed into an independent sentence which can be used in the development of the system as a premise for inferences. How are inferences carried out within the system?
Let us assume that we have a sentence of the form 'If A holds, so does B'. If we add to this the further sentence 'A holds', then from both premises we can infer 'B holds'. But for the conclusion to be possible, both premises have to be true. And this is why the axioms also have to be true, if they are to serve as premises. For we can draw no conclusion from something false. But it might perhaps be asked, can we not, all the same, draw consequences from a sentence which may be false, in order to see what we should get if it were true? Yes, in a certain sense this is possible. From the premises
If Fholds, so does A If A holds, so does E
1 This is how the sentence reads in the German. Since it merely repeats the first part of the preceding sentence, the editors suggest that we read 'object' in place of 'truth value', so that the sentence marks a natural inference from the one precedina (trans. ).
? we can infer
If Fholds, so does E From this and the further premise
we can infer
If E holds, so does Z
IfFholds, so does Z.
Logic in Mathematics 245
And so we can go on drawing consequences without knowing whether r is true or false. But we must notice the difference. In the earlier example the premise 'A holds' dropped out of the conclusion altogether. In this example the condition 'If Fholds' is retained throughout. We can only detach it when we have seen that it is fulfilled. In the present case 'r holds' cannot be regarded as a premise at all: what we have as a premise is
If Fholds, so does A,
and thus something of which 'r holds' is only a part. Of course this whole premise must be true; but this is possible without the condition being fulfilled, without r holding. So, strictly speaking, we simply cannot say that consequences are here being drawn from a thought that is false or doubtful; for this does not occur independently as a premise, but is only part of a premise which as such has indeed to be true, but which can be true without that part of the thought-the part which it contains as a condition-being true.
In indirect proofs it looks as if consequences are being drawn from something false. As an example, suppose we have to prove that in a triangle
the longer side subtends the greater angle. To prove:
1
A
c
If LB > LA, then AC >BC. We take as given:
I IfBC>AC,thenLA>LB. II IfBC=AC,thenLA= LB.
III If not AC >BC, and if not BC> AC, then BC=AC.
IV IfLA=LB,thennotLB>LA. V If LA> LB, then not LB >LA.
From Il and Ill there follows:
IfnotAC >BC and if not BC> AC, then LA= LB.
From this and IV we have: IfnotAC>BCandifnotBC>AC,thennot LB>L. A.
? 246 Logic in Mathematics From I and V:
IfBC> AC, then not LB >LA. From the last two sentences there follows:
If not AC >BC, then not LB > LA. And then by contraposition:
If LB >LA, then AC >BC.
To simplify matters, I shall assume that we are not speaking of triangles in general, but of a particular triangle. LA and LB may be understood as numbers, arrived at by measuring the angles by some unit-measure, as e. g. a right-angle. AC and BC may likewise be understood as numbers, arrived at by using some unit-measure for the sides, as e. g. a metre. Then the signs 'LA',' LB', 'AC', and 'BC' are to be taken as proper names of numbers.
We see that 'not AC > BC' does not occur here as a premise, but that it is contained in Ill as a part-as a condition. So strictly speaking, we cannot say that consequences are being drawn from the false thought (not AC >BC). Therefore, we ought not really to say 'suppose that not AC >BC', because this makes it look as though 'not AC >BC' was meant to serve as a premise for inference, whereas it is only a condition.
We make far too much of the peculiarity of indirect proof vis-d-vis direct proof. The truth is that the difference between them is not at all important.
The proof can also be set out in the following way. We now take as given:
I' Ifnot LA >LB then not BC> AC. II' IfnotLA= LB,thennotBC=AC.
Ill' IfnotBC>ACandifnotBC=AC,thenAC>BC. IV' If LB >LA then not LA= LB.
V' If LB >LA then not LA> LB.
From V' and I' there follows:
If LB > LA, then not BC> AC.
From this and Ill' we have:
If LB >LA and if not BC= AC, then AC >BC.
From IV' and II' there follows:
If LB >LA, then not BC= AC.
From the last two sentences we have:
If LB >LA, then AC >BC.
At no point in this proof have we entertained 'not AC >BC' even as a mere hypothesis.
? Logic in Mathematics 247
In an investigation of the foundations of geometry it may also look as if consequences are being drawn from something false or at least doubtful. Can we not put to ourselves the question: How would it be if the axiom of parallels didn't hold? Now there are two possibilities here: either no use at all is made of the axiom of parallels, but we are simply asking how far we can get with the other axioms, or we are straightforwardly supposing something which contradicts the axiom of parallels. It can only be a question of the latter case here. But it must constantly be borne in mind that what is false cannot be an axiom, at least if the word 'axiom' is being used in the traditional sense. What are we to say then? Can the axiom of parallels be acknowledged as an axiom in this sense? When a straight line intersects one of two parallel lines, does it always intersect the other? This question, strictly speaking, is one that each person can only answer for himself. I can only say: so long as I understand the words 'straight line', 'parallel' and 'intersect' as I do, I cannot but accept the parallels axiom. If someone else does not accept it, I can only assume that he understands these words differently. Their sense is indissolubly bound up with the axiom of parallels. Hence a thought which contradicts the axiom of parallels cannot be taken as a premise of an inference. But a true hypothetical thought, whose condition contradicted the axiom, could be used as a premise. This condition would then be retained in all judgements arrived at by means of our chains of inference. If at some point we arrived at a hypothetical judgement whose consequence contradicted known axioms, then we could conclude that the condition contradicting the axiom of parallels was false, and we should thereby have proved the axiom of parallels with the help of other axioms. But because it had been proved, it would lose its status as an axiom. In such a case we should really have given an indirect proof.
If, however, we went on drawing inference after inference and still did not come up against a contradiction anywhere, we should certainly become more and more inclined to regard the axiom as incapable of proof. Nevertheless we should still, strictly speaking, not have proved this to be so.
Now in his Grundlagen der Geometrie Hilbert is preoccupied with such questions as the consistency and independence of axioms. But here the sense of the word 'axiom' has shifted. For if an axiom must of necessity be true, it is impossible for axioms to be inconsistent with one another. So any discussion here would be a waste of words. But obvious though it is, it seems just not to have entered Hilbert's mind that he is not speaking of axioms in Euclid's sense at all when he discusses their consistency and independence. We could say that the word 'axiom', as he uses it, fluctuates from one sense to another without his noticing it. It is true that if we concentrate on the words of one of his axioms, the immediate impression is that we are dealing with an axiom of the Euclidean variety; but the words mislead us, because all the words have a different use from what they have in Euclid. At? 31 we
1 The quotation here is from the first edition of the Grundlagen der Geometric. Later editions give a different uxiom in plnce of thnt which Frege cites under ((. 4 (cd. ).
? 248 Logic in Mathematics
read 'Definition. The points of a straight line stand in certain relations to one another, for the description of which we appropriate the word "between". ' Now this definition is only complete once we are given the four axioms
11. 1 If A, B, C are points on a straight line, and B lies between A and C, then B lies between C and A.
11. 2 If A and C are two points on a straight line, then there is at least one point B lying between A and C, and at least one point D such that C lies between A and D.
11. 3 Given any three points on a straight line, there is one and only one which lies between the other two.
11. 4 Any four points A, B, C, D on a straight line can be so ordered that B lies between A and C and between A and D, and so that C lies between A and D and between B and D.
These axioms, then, are meant to form parts of a definition. Consequently these sentences must contain a sign which hitherto had no meaning, but which is given a meaning by all these sentences taken together. This sign is apparently the word 'between'. But a sentence that is meant to express an axiom may not contain a new sign. All the terms in it must be known to us. As long as the word 'between' remains without a sense, the sentence 'If A, B, C, are points on a straight line and B lies between A and C, then B lies between C and A' fails to express a thought.
An axiom, however, is always a true thought. Therefore, what does not express a thought, cannot express an axiom either. And yet one has the impression, on reading the first of these sentences, that it might be an axiom. But the reason for this is only that we are already accustomed to associate a sense with the word 'between'. In fact if in place of
we say
'B lies between A and C' 'B pat A nam C',
then we associate no sense with it. Instead of the so-called axiom 11. 1 we should have
'If B pat A nam C, then B pat C nam A'.
No one to whom these syllables 'pat' and 'nam' are unfamiliar will associate a sense with this apparent sentence. The same holds of the other three pseudo-axioms.
The question now arises whether, if we understand by A, B, C points on a straight line, an expression of the form
'B patA nam C'
will not at least ~ome to acquire a sense through the totality of these pseudo-sentences. I think not. We may perhaps hazard the guess that it will
come to the same thing as
? Logic in Mathematics 249 'B lies between A and C',
but it would be a guess and no more. What is to say that this matrix could not have several solutions?
But does, then, a definition have to be unambiguous? Are there not circumstances in which a certain give and take is a good thing?
Of course a2 = 4 does not determine unequivocally what a is to mean, but is there any harm in that? Well, if a is to be a proper name whose meaning is meant to be fixed, this goal will obviously not be achieved. On the contrary one can see that in this formula there is designated a concept under which
the numbers 2 and -2 fall. Once we see this, the ambiguity is harmless, but it means that we don't have a definition of an object.
If we want to compare this case with our pseudo-axiom, we have to compare the letter 'a' with 'between' or with 'pat-nam'. We must distinguish signs that designate from those which merely indicate. In fact the word 'between' or 'pat-nam' no more designates anything than does the letter 'a'. So we have here to disregard the fact that we usually associate a sense with the word 'between'. In this context it no more has a sense than does 'pat- nam'. Now to say that a sign which only indicates neither designates anything nor has a sense is not yet to say it could not contribute to the expression of a thought. It can do this by conferring generality of content on a simple sentence or on one made up of sentences.
Now there is, to be sure, a difference between our two cases; for whilst 'a' stands in for a proper name, 'pat-nam' stands in for the designation of a relation with three terms. As we call a function of one argument, whose value is always a truth-value, a concept, and a function of two arguments, whose value is always a truth-value, a relation, we can go yet a step further and call a function of three arguments whose value is always a truth-value, a relation with three terms. Then whilst the 'between-and' or the 'pat-nam' do not designate such a relation with three terms, they do indicate it, as 'a' indicates an object. Still there remains a distinction. We were able to find a concept designated in 'a2 = 4'.
What would correspond to this in the case of our pseudo-axioms? It is what I call a second level concept. In order to see more clearly what I understand by this, consider the following sentences:
'There is a positive number' 'There is a cube root of 1'.
We can see that these have something in common. A statement is being made, not about an object, however, but about a concept. In the first sentence it is the concept positive number, in the second it is the concept cube root of 1. And in each case it is asserted of the concept that it is not empty, but satisfied. It is indeed strictly a mistake to say 'The concept positive number is satisfied', for by saying this I seem to make the concept into an object, as the definite article 'the concept' shows. lt now looks as if
?
250 Logic in Mathematics
'the concept positive number' were a proper name designating an object and as if the intention were to assert of this object that it is satisfied. But the truth is that we do not have an object here at all. By a necessity oflanguage, we have to use an expression which puts things in the wrong perspective; even so there is an analogy. What we designate by 'a positive number' is related to what we designate by 'there is', as an object (e. g. the earth) is related to a concept (e. g. planet).
I distinguish concepts under which objects fall as concepts of first level from concepts of second level within which, as I put it, concepts of first level fall. Of course it goes without Saying that all these expressions are only to be understood metaphorically; for taken literally, they would put things in the wrong perspective. We can also admit second level concepts within which relations fall. E. g. in the sentence:
'It is to hold for all A, B, C that if A stands in the p-relation to B and if A stands in the p-relation to C, then B = C',
we have a designation of a second-level concept within which relations fall and 'stands in the p-relation to . . . ' here stands in for the argument- sign-for, that is, the designation of the relation presented as argument. If we substitute e. g. the equals sign, we get
'It holds for all A, B, C that ifA= Band ifA= C, then B = C'.
If this is true, then it follows that the relation of equality falls within this second level concept.
As we call a function of one argument, whose value is always a truth value, a concept, and as we call a function of two arguments, whose value is always a truth value, a relation, so we could introduce a special name for a function o f three arguments, whose value is always a truth value. Provisionally, such a function may be called a relation with three terms. That designated by the words 'lies between . . . and . . . ' would be of this kind, if these words were understood as we should ordinarily understand them when used in speaking of Euclidean points on a Euclidean straight line. However, they are not used in our pseudo-axioms as a sign which designates, but only as one which indicates, as are letters in arithmetic. So in this context they do not designate such a three-termed relation: they only indicate such a relation. Even if we wish, for the time being, to understand the actual words 'point' and 'straight line' in the Euclidean sense, still the words 'lies between . . . and . . . ' are not to be regarded, strictly speaking, as words with a sense, but only as a stand-in for an argument, as is the letter 'a' in 'a2'. But the function, whose arguments they stand in for, is a second level function within which only relations with three terms can fall.
? ? My basic logical Insights1 [1915]
The following may be of some use as a key to the understanding of my results.
Whenever anyone recognizes something to be true, he makes a judgement. What he recognizes to be true is a thought. It is impossible to recognize a thought as true before it has been grasped. A true thought was true before it was grasped by anyone. A thought does not have to be owned by anyone. The same thought can be grasped by several people. Making a judgement does not alter the thought that is recognized to be true. When something is judged to be the case, we can always cull out the thought that is recognized as true; the act of judgement forms no part of this. The word 'true' is not an adjective in the ordinary sense. If I attach the word 'salt' to the word 'sea- water' as a predicate, I form a sentence that expresses a thought. To make it clearer that we have only the expression of a thought, but that nothing is meant to be asserted, I put the sentence in the dependent form 'that sea- water is salt'. Instead of doing this I could have it spoken by an actor on the stage as part of his role, for we know that in playing a part an actor only seems to speak with assertoric force. Knowledge of the sense of the word 'salt' is required for an understanding of the sentence, since it makes an essential contribution to the thought-in the mere word 'sea-water' we should of course not have a sentence at all, nor an expression for a thought. With the word 'true' the matter is quite different. If I attach this to the words 'that sea-water is salt' as a predicate, I likewise form a sentence that expresses a thought. For the same reason as before I put this also in the dependent form 'that it is true that sea-water is salt'. The thought expressed in these words coincides with the sense of the sentence 'that sea-water is salt'. So the sense of the word 'true' is such that it does not make any essential contribution to the thought. If I assert 'it is true that sea-water is salt', I assert the same thing as if I assert 'sea-water is salt'. This enables us to recognize that the assertion is not to be found in the word 'true', but in the assertoric force with which the sentence is uttered. This may lead us to think that the word 'true' has no sense at all. But in that case a sentence in which 'true' occurred as a predicate would have no sense either. All one can say
1
is based, this piece is to be dated around 1915 (ed. ).
According to a note by Heinrich Scholz on the transcripts on which this edition
? ? 252 My basic logical Insights
is: the word 'true' has a sense that contributes nothing to the sense of the whole sentence in which it occurs as a predicate.
But it is precisely for this reason that this word seems fitted to indicate the essence of logic. Because of the particular sense that it carried any other adjective would be less suitable for this purpose. So the word 'true' seems to make1 the impossible possible: it allows what corresponds to the assertoric force to assume the form of a contribution to the thought. And although this attempt miscarries, or rather through the very fact that it miscarries, it indicates what is characteristic of logic. And this, from what we have said, seems something essentially different from what is characteristic of aesthetics and ethics. For there is no doubt that the word 'beautiful' actually does indicate the essence of aesthetics, as does 'good' that of ethics, whereas 'true' only makes an abortive attempt to indicate the essence of logic, since what logic is really concerned with is not contained in the word 'true' at all but in the assertoric force with which a sentence is uttered.
Many things that belong with the thought, such as negation or generality, seem to be more closely connected with the assertoric force of the sentence or the truth of the thought. 2 One has only to see that such thoughts occur in e. g. conditional sentences or as spoken by an actor as part of his role for this illusion to vanish.
How is it then that this word 'true', though it seems devoid of content, cannot be dispensed with? Would it not be possible, at least in laying the foundations of logic, to avoid this word altogether, when it can only create confusion? That we cannot do so is due to the imperfection of language. If our language were logically more perfect, we would perhaps have no further need of logic, or we might read it off from the language. But we are far from being in such a position. Work in logic just is, to a large extent, a struggle with the logical defects of language, and yet language remains for us an in- dispensable tool. Only after our logical work has been completed shall we possess a more perfect instrument.
Now the thing that indicates most clearly the essence of logic is the assertoric force with which a sentence is uttered. But no word, or part of a sentence, corresponds to this; the same series of words may be uttered with assertoric force at one time, and not at another. In language assertoric force is bound up with the predicate.
1 A different version of the manuscript has 'to be trying to make' in place of 'to make' (ed. ).
2 This sentence and the one following are crossed out in the manuscript (ed. ).
? ? [Notes for Ludwig DarmstaedterP [July 1919]
I started out from mathematics. The most pressing need, it seemed to me, was to provide this science with a better foundation. I soon realized that number is not a heap, a series of things, nor a property of a heap either, but that in stating a number which we have arrived at as the result of counting we are making a statement about a concept. (Plato, The Greater Hippias. )
The logical imperfections of language stood in the way of such investigations. I tried to overcome these obstacles with my concept-script. In this way I was led from mathematics to logic.
What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word 'true', and then immediately go on to introduce a thought as that to which the question 'Is it true? ' is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgement; I come by the parts of a thought by analysing the thought. This marks off my concept-script from the similar inventions of Leibniz and his successors, despite what the name suggests; perhaps it was not a very happy choice on my part.
Truth is not part of a thought. We can grasp a thought without at the same time recognizing it as true-without making a judgement. Both grasping a thought and making a judgement are acts of a knowing subject, and are to be assigned to psychology. But both acts involve something that does not belong to psychology, namely the thought.
False thoughts must be recognized too, not of course as true, but as indispensable aids to knowledge, for we sometimes arrive at the truth by way of false thoughts and doubts. There can be no questions if it is essential to the content of any question that that content should be true.
Negation does not belong to the act of judging, but is a constituent of a thought. The division of thoughts (judgements) into affirmative and negative is of no use to logic, and I doubt if it can be carried through.
Where we have a compound sentence consisting of an antecedent and a consequent, there are two main cases to distinguish. The antecedent and consequent may each have a complete thought as its sense. Then, over and
1 This piece is dated at the end by Frege himself. It is one of the few manuscripts of Frege which have come down to us in the original and is in the possession of the Staatsbibliothek der Stiftung Prezij)ischer Ku/turbesitz where it forms part of the collection of the historian of science Ludwig Darmstaedter, ref. number 1919-95 (ed. ).
? ? 254
[Notes for Ludwig Darmstaedter]
above these, we have the thought expressed by the whole compound sentence. By recognizing this thought as true, we recognize neither the thought in the antecedent as true, nor that in the consequent as true. A second case is where neither antecedent nor consequent has a sense in itself, but where nevertheless the whole compound sentence does express a thought-a thought which is general in character. In such a case we have a relation, not between judgements or thoughts, but between concepts, the relation, namely, of subordination. The antecedent and consequent are here sentences only in the grammatical, not in the logical, sense. The first thing that strikes us here is that a thought is made up out of parts that are not themselves thoughts. The simplest case of this kind is where one of the two parts is in need of supplementation and is completed by the other part, which is saturated: that is to say, it is not in need of supplementation. The former part then corresponds to a concept, the latter to an object (subsumption of an object under a concept). However, the object and concept are not constituents of the thought expressed. The constituents of the thought do refer to the object and concept, but in a special way. There is also the case where a part doubly in need of supplementation is completed by two saturated parts. The former part corresponds to a relation. -An object stands in a relation to an object. -Where logic is concerned, it seems that every combination of parts results from completing something that is in need of supplementation; where logic is concerned, no whole can consist of saturated parts alone. The sharp separation of what is in need of supplementation from what is saturated is very important. When all is said and done, people have long been familiar with the former in mathematics (+, :, . . ;-, sin, =, >). In this connection they speak of functions, and yet it would seem that in most cases they have only a vague notion of what is at stake.
A general statement can be negated. In this way we arrive at what logicians call existential judgements and particular judgements. The existential thoughts I have in mind here are such as are expressed in German by 'es gibt'. 1 This phrase is never followed immediately by a proper name in the singular, or by a word accompanied by the definite article, but always by a concept-word (nomen appellativum) without a definite article. In existential sentences of this kind we are making a statement about a concept. Here we have an instance of how a concept can be related to a second level concept in a way analogous to that in which an object is related to a concept under which it falls. Closely akin to these existential thoughts are thoughts that are particular: indeed they may be included among them. But we can also say that what is expressed by a sentence of the particular form is that a concept stands in a certain second level relation to a concept. The distinction between first and second level concepts can only be grasped clearly by one who has clearly grasped the distinction between what is in
1 i. e. judgements that are expressed in English by 'there is' or 'there are' (trans. ).
? [Notes for Ludwig Darmstaedter] 255
need of supplementation and what is saturated. A saturated part obtained by analysing a thought can sometimes itself be split up in the same way into a part in need of supplementation and a saturated part. The sentence 'The capital of Sweden is situated at the mouth of Lake Malar' can be split up into a part in need of supplementation and the saturated part 'the capital of Sweden'. This can further be split up into the part 'the capital of, which stands in need of supplementation, and the saturated part 'Sweden'. Splitting up the thought expressed by a sentence corresponds to such a splitting up of the sentence. The functions of Analysis correspond to parts of thoughts
that are thus in need of supplementation, without however being such.
A distinction has to be drawn between the sense and meaning of a sign (word, expression). If an astronomer makes a statement about the moon, the moon itself is not part of the thought expressed. The moon itself is the meaning of the expression 'the moon'. Therefore in addition to its meaning this expression must also have a sense, which can be a constituent of a thought. We can regard a sentence as a mapping of a thought: corresponding to the whole-part relation of a thought and its parts we have, by and large, the same relation for the sentence and its parts. Things are different in the domain of meaning. We cannot say that Sweden is a part of the capital of Sweden. The same object can be the meaning of different expressions, and any one of them can have a sense different from any other. Identity of meaning can go hand in hand with difference of sense. This is what makes it possible for a sentence o f the form ' A = B ' to express a thought with more content than one which merely exemplifies the law of identity. A statement in which something is recognized as the same again can be of far greater cognitive value than a particular case of the law of
identity.
Even a part of a thought, or a part of a part of a thought, that is in need of
supplementation, has something corresponding to it in the realm of meaning. Yet it is wrong to call this a concept, say, or a relation, or a function, although we can hardly avoid doing so. Grammatically, 'the concept of God' has the form of a saturated expression. Accordingly its sense cannot be anything in need of supplementation. When we use the words 'concept', 'relation', 'function' (as this is understood in Analysis), our words fail of their intended target. In this case even the expression 'the meaning', with the
definite article, should really be avoided.
It is not, however, only parts of sentences that have meaning; even a
whole sentence, whose sense is a thought, has one. All sentences that express a true thought have the same meaning, and all sentences that express a false thought have the same meaning (the True and the False). Sentences and parts of sentences with different meanings also have different senses. If in a sentence or part of a sentence one constituent is replaced by another with a different meaning, the different sentence or part that results does not have to have a different meaning from the original; on the other hand, it always has a different sense. If in a sentence or part of a sentence
? ? 256 [Notes for Ludwig Darmstaedter]
one constituent is replaced by another with the same meaning but not with the same sense, the different sentence or part that results has the same meaning as the original, but not the same sense. All this holds for direct, not for indirect, speech.
A thought can also be the meaning of a sentence (indirect speech, the subjunctive mood)1? The sentence does not then express this thought, but can be regarded as its proper name. Where we have a clause in indirect speech
occurring within direct speech, and we replace a constituent of this clause by another which has the same meaning in direct speech, then the whole which results from this transformation does not necessarily have the same meaning as the original.
The miracle of number. The adjectival use of number-words is misleading. In arithmetic a number-word makes its appearance in the singular as a proper name of an object of this science; it is not accompanied by the indefinite article, but is saturated. Subsumption: 'Two is a prime,' not subordination. The combinations 'each two', 'all twos' do not occur.
Yet amongst mathematicians we find a great lack of clarity and little agreement. Is number an object that arithmetic investigates or is it a counter in a game? Is arithmetic a game or a science? According to one man, we are to understand by 'number' a series of objects of the same kind; according to another, a spatial, material structure produced by writing. A third denies that number is spatial at all. Perhaps there are times when arithmeticians merely delude themselves into thinking that they understand by 'number' what they say they do. If this is not so, then they are attaching different senses to sentences with same wording; and if they still believe that they are working within one and the same discipline, then they are just deluding themselves.
