But one can also trace the chains of
inference
backwards by asking from what truths each theorem has been inferred.
Gottlob-Frege-Posthumous-Writings
It is an imperative and essential logical requirement that these apparently self-contained sentences should be combined into a single compound sentence; however, if we compl~
? ? Introduction to Logic 195
A sentence proper is a proper name, and its meaning, if it has one, is a truth value: the True or the False. There are many sentences which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part, which means a concept. In the same way there are many proper names, whose meanings are not truth values, which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part. If this latter is to be meaningful, then the result of saturing it with any meaningful proper name whatever must once more be a meaningful proper name. When this happens, we call the meaning of this unsaturated part a function. At this point, however, we need to make a reservation, similar to that we made earlier when the word 'concept' was introduced, about the unavoidable inaccuracy of language. The unsaturated part of a sentence, whose meaning we have called a concept, must have the property of yielding a genuine sentence when saturated by any meaningful proper name; this means that it must yield the proper name of a truth value. This is the requirement that a concept have sharp boundaries. For a given concept, every object must either fall under it or not, tertium non datur. From this it follows that a requirement similar to that we have just laid down is to be made of a function. As an example let us start off from the sentence ' 3 - 2 > 0'. We split this up into the proper name '3 - 2' and the remainder
> 0'. We may say this unsaturated part means the concept of a positive number. This concept must have sharp boundaries. Every object must either fall or not fall under this concept. Let us now go further and split the proper name '3 - 2' up into the proper name '2' and the unsaturated part '3 -
Now we may also split the original sentence '3 - 2 >0' up into the proper name '2' and the unsaturated part '3 - >0'. The meaning ofthis is the concept of something that yields a positive remainder when subtracted from 3. This concept must have sharp boundaries too. Now if there were a meaningful proper name a such that the unsaturated part '3 - ' did not
with these requirements we usually end up with grammatical monstrosities. In the concept-script the judgement-stroke, besides conveying assertoric force, serves to demarcate the scope of the roman letters. In order to be able to narrow the scope over which the generality extends, I make use of gothic letters, and with these the concavity demarcates the scope. Here and there in arithmetic there is also a use of letters which roughly corresponds to that of the gothic letters in my concept-script. But I have found no indication that anyone is aware of this use as a special case. Probably most mathematicians, were they to read this, would have no idea of what I am alluding to. It was not until after some time that I became nware of it myself. We are very dependent on external aids in our thinking, nnd there is no doubt that the language of everyday life-so far, at least, tiS a certain area of discourse is concerned-had first to be replaced by a more sophisticated instrument, before certain distinctions could be noticed. Hut so far the academic world has, for the most part, disdained to master
this instrument.
12. v111. o6
? 196
Introduction to Logic
yield a meaningful proper name when saturated by it, then the unsaturated part '3 - >0', when saturated by a; would not yield a meaningful proper name either; that is to say, we should not be able to say whether the object designated by afell under the concept which is the meaning of'3 - >0'. We can see from this that the usual definitions of the arithmetical signs are inadequate.
? ? A brief Survey of my logical Doctrines 1
[1906]
The thought
When I use the word 'sentence' in what follows I do not mean a sentence that serves to express a wish, a command, or a question, but one that serves to make an assertion. Although a sentence can be perceived by the senses, we use it to communicate a content that cannot be perceived by the senses. We are making a judgement about this content when we accept it as true or reject it as false. When a sentence is uttered the assertion that it is true usually goes hand in hand with the communication of the content. But the hearer does not have to adopt the speaker's stance; not that he has to reject the content either. He can simply refrain from making a judgement. We may now think of the content of a sentence as it is viewed by such a hearer.
Now two sentences A and B can stand in such a relation that anyone who recognizes the content of A as true must thereby also recognize the content of B as true and, conversely, that anyone who accepts the content of B must straightway accept that of A. (Equipollence). It is here being assumed that there is no difficulty in grasping the content of A and B. The sentences need not be equivalent in all respects. For instance, one may have what we may call a poetic aura, and this may be absent from the other. Such a poetic aura will belong to the content of the sentence, but not to that which we accept as true or reject as false. I assume there is nothing in the content ofeither of the two equipollent sentences A and B that would have to be immediately accepted as true by anyone who had grasped it properly. The poetic aura then, or whatever else distinguishes the content of A from that of B, does not belong to what is accepted as true; for if this were the case, then it could not be an immediate consequence of anyone's accepting the content of B that he should accept that of A. For the assumption is that what distinguishes A and 8 is not contained in B at all, nor is it something that anyone must recognize as true straight off.
So one has to separate off from the content of a sentence the part that alone can be accepted as true or rejected as false. I call this part the thought
1 ln places this piece agrees, even in wording, with the Einleitung in die Logik of August 1906 (pp. 185 fT. ). I t may well be the case that it constitutes a revision-probably made shortly afterwards-of part of these diary notes. According to notes of the previous editors on the transcripts that form the basis of this edition, the originals of both pieces were found together in the order in which they have been printed here (ed. ).
? 198 A briefSurvey ofmy logical Doctrines
expressed by the sentence. It is the same in equipollent sentences of the kind given above. It is only with this part of the content that logic is concerned. I call anything else that goes to make up the content of a sentence the colouring of the thought.
Thoughts are not psychological entities and do not consist of ideas in the psychological sense. The thought in Pythagoras's theorem is the same for all men; it confronts everyone in the same way as something objective, whereas each man has his own ideas, sensations, and feelings, which belong only to him. We grasp thoughts but we do not create them.
In myth and fiction thoughts occur that are neither true nor false. Logic has nothing to do with these. In logic it holds good that every thought is either true or false, tertium non datur.
Dissociating assertoric force from the predicate
We can grasp a thought without recognizing it as true. 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 this recognition-make an assertion. We need to be able to express a thought without putting it forward as true. In the Begriffschrift I use a special sign to convey assertoric force: the judgement-stroke. The languages known to me lack such a sign, and assertoric force is closely bound up with the indicative mood of the sentence that forms the main clause. Of course in fiction even such sentences are uttered without assertoric force; but logic has nothing to do with fiction. Fiction apart, it seems that it is only in subordinate clauses that we can express thoughts without asserting them. One should not allow oneself to be misled by this peculiarity of language and confuse grasping a thought and making a judgement.
Negation
Assertoric force is to be dissociated from negation too. To each thought there corresponds an opposite, so that rejecting one of them coincides with accepting the other. To make a judgement is to make a choice between opposite thoughts. Accepting one of them and rejecting the other is one act. So there is no need of a special sign for rejecting a thought. We only need a special sign for negation as such.
The hypothetical mode of sentence composition
If we say that in a hypothetical judgement two judgements are put into a relationship w~th one another, we are using the word 'judgement' so as not to include the recognition of the truth of anything, and thus as I use the word 'thought'. For even if the whole compound sentence is uttered with
? 'If
17 ? 19 211
is greater than 2, then
( 17 . 19) 211
is greater than 2'
A briefSurvey ofmy logical Doctrines 199
assertoric force, one is asserting neither the truth of the antecedent nor that of the consequent. What is being judged to be true is rather a thought that is expressed by the whole compound sentence. On a closer examination one finds that in many cases neither the antecedent nor the consequent expresses a thought. In the compound sentence
'If a is greater than 2, then a2 is greater than 2'
the letter 'a' does not designate an object as does the numeral '2': it only indicates indefinitely. So there is an indefiniteness in both antecedent and consequent, which is why neither of these two clauses expresses a thought. But the whole compound sentence does express a thought, the letter 'a' serving to confer generality of content on the sentence as a whole. It may be remarked that in some of the sentences we utter this indefiniteness is not in evidence at all, or there is but the barest suggestion of it. Nevertheless it is always present when we express a general law. What has the grammatical form of a sentence and yet does not express a thought because it contains something indefinite, I call a quasi-sentence. A sentence proper, on the other hand, expresses a thought. With a general law we are tempted to speak of cases in which the condition is true and of others in which it is false. We must reject this way of speaking. It is only a thought that can be true and a thought is either true or false: it is not true in some cases, false in others. When we should like to distinguish such cases, what we have are quasi- sentences. A sentence proper can be obtained from a quasi-sentence by removing the indefiniteness. Thus from the quasi-sentence 'a > 2' we obtain as a sentence proper '1 >2', as well as '2 >2' and '3 >2'. Some of the sentences thus obtained may express true, and some false, thoughts. However this is only by the way. We shall treat generality more fully later. To begin with; we shall assume that the antecedent and consequent are sentences proper. The following compound sentence will serve as an example:
222
What is being said here? Each of the two clauses
'172 ? 19
- - - is greater than 2'
211
2
and
'(17 ? 19)2 is greater than 2' 211
expresses a thought which is either true or false. Ifwe call the first thought A and the second B, then four cases are possible:
A is true and B is true, A is true and B is false, A is false and 8 is true, A is false and B is false.
? 200 A briefSurvey ofmy logical Doctrines
Obviously it is only the second case that is incompatible with the sentence
'If
172 ? 19 211
is greater than 2, then
(172 ? 19)2 211
is greater than 2'
holding good. So our sentence says that the second case does not obtain; it leaves it open which of the remaining cases holds. This gives us the essence of the hypothetical mode of combining sentences. It is correct to say
'If 3 is greater than 2, then 32 is greater than 2' and 'If 2 is greater than 2, then 22 is greater than 2' and 'If 1 is greater than 2, then 12 is greater than 2'.
If these sentences sound strange, a little absurd even, this is due to the fact that in each of these examples one sees at once which of the four cases holds, whereas in the first example one does not. But this difference is quite inessential.
I will call this combination of the thoughts A and B the hypothetical com- bination of A and B, the first thought being the condition and the second the consequence. If a sentence consists of two sentences combined by 'and' both of which express a thought, then the sense of the whole sentence is to be taken as a thought too, for this sense is either true or false. It is true if each of the constituent thoughts r and L1 is true, and false in every other case, and so if at least one of the two constituent thoughts is false. I will call the thought in this whole sentence the conjunction of r and L1. The conjunction ofrand L1, in common with every thought, has an opposite.
Now the hypothetical combination of A and B is the opposite of the conjunction of A and the opposite of B. But, conversely, the conjunction of r and L1 is the opposite of the hypothetical combination of r with the opposite of L1. By means of negation the hypothetical mode can thus be reduced to conjunction and conjunction to the hypothetical mode. Looked at from a logical standpoint both appear equally primitive. But since the hypothetical mode is more closely connected with drawing inferences, it is best to give it pride of place, and see it as the primitive form, reducing conjunction to it.
Generality
We remarked just now that often in a hypothetical sentence neither the antecedent nor the consequent express thoughts, and that the reason for this is that there is present an indefiniteness, although this does not make the compound sentence as a whole devoid of sense. In the example
'If a is greater than 2, then a2 is greater than 2'
the letter 'a' gives rise to this indefiniteness and it is due to this letter that the thought of the whole compound sentence is general. This is the usual
? A briefSurvey ofmy logical Doctrines 201
employment of letters in arithmetic, even if it is not the only one. Of course natural language has means of accomplishing the same thing (e. g. 'tot-quat' in Latin); but these are less precise. And sentences occur in which we can spot no part at all corresponding to the letter 'a'. Here we may take the above use of letters in arithmetic as basic. It is not only compound hypo- thetical sentences that can be general. We may think of the sentence 'a= a'. In order to make it easier to recognize that the sentence is general, we can add 'no matter what a may be'.
Ifin the sentence
'If a is greater than 2, then a2 is greater than 2'
we substitute for the indefinitely indicating letter 'a' in turn the numerals '1', '2', '3', each of which designates a particular thing, we obtain the sentences
'If 1 is greater than 2, then 12 is greater than 2', 'If 2 is greater than 2, then 22 is greater than 2', 'If 3 is greater than 2, then 32 is greater than 2'.
The thoughts expressed by these sentences are particular cases of the general thought. Likewise the thoughts '1 = 1', '2 = 2', are particular cases of the general thought expressed in the sentence ' a = a'.
Here for the first time we have occasion to split up a sentence into parts that are not themselves sentences. Thus in the general sentence we have a part to which congruent parts in the associated particular sentences correspond and a part-in our case it is the letter 'a'-to which non- congruent parts in the particular sentences-the numerals '1', '2', '3'-correspond. These sentence-parts are different in kind. The part in which the general sentence agrees with the related particular sentences shows gaps at the place where the other part of the sentence, the numeral 'I', say, occurs. If we fill these gaps with the letter ? ~? . we obtain in our cases
'If~is greater than 2, then ~2 is greater than 2', ? ~= ~? .
We obtain specific sentences from these by filling the gaps with, say, the numerals '1', '2', or '3'. These do not show any gaps but fill the gaps in the first parts so that we get a sentence. If we call the parts of the sentence that show gaps unsaturated and the other parts complete, then we can think of a sentence as arising from saturating a saturated part with a complete part. The complete part ofa sentence I call a proper name, the unsaturatedpart a concept-name. To the unsaturated part of the sentence there corresponds an unsaturated part of the thought and to the complete part of the sentence a complete part of the thought, and we can also speak here of saturating the unsaturated part of the thought with a complete part. A thought that is put together in this way is just what traditional logic calls a singular judgement. We must notice, however, that one and the same thought can be
? 202 A briefSurvey ofmy logical Doctrines
split up in different ways and so can be seen as put together out of parts in different ways. The word 'singular' does not apply to the thought in itself but only with respect to a particular way of splitting it up. Each of the sentence- parts
'1 is greater than 2' and ' 12 is greater than 2'
can also be seen as put together out of the proper name '1' and an unsaturated part. The corresponding holds for the related thoughts.
? ? Logic in Mathematics1 [Spring 1914]
Mathematics has closer ties with logic than does any other discipline; for almost the entire activity of the mathematician consists in drawing inferences. In no other discipline does inference play so large a part, although inferences do occur here and there in other disciplines. Part of the mathematician's activity, besides drawing inferences, is to give definitions. Most disciplines are not concerned with the latter at all; only in jurisprudence is it of some importance, for although its subject-matter is quite different, it is in several respects close to mathematics. Jurisprudence takes its materials from history and psychology and for this reason these must claim to have some share in it. And there is nothing resembling this with mathematics.
Inferring and defining are subject to logical laws. From this it follows that logic is of greater importance to mathematics than to any other science. If one counts logic as part of philosophy, there will be a specially close
bond between mathematics and philosophy, and this is confirmed by the history of these sciences (Plato, Descartes, Leibniz, Newton, Kant).
But are there perhaps modes ofinference peculiar to mathematics which, for that very reason, do not belong to logic? Here one may point to the inference by mathematical induction from n to n + 1. Well, even a mode of inference peculiar to mathematics must be subject to a law and this law, if it is not logical in nature, will belong to mathematics, and can be ranked with the theorems or axioms of this science. For instance, mathematical
induction rests on the law that can be expressed as follows:
If the number 1 has the property c[J and if it holds generally for every positive whole number n that if it has the property c[J then n + 1 has the
property C/J, then every positive whole number has the property C/J.
If this law can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. If one
draws inferences by mathematical induction, then one is actually making an application of this theorem or axiom; that is, this truth is taken as a premise of an inference. For example: the proof of the proposition (a + b) + n = a+ (b + n).
So likewise in other cases one can reduce a mode of inference that is peculiar to mathematics to a general law, if not a law of logic, then one of
1 The dating of the previous editors has been adopted (ed. ).
? Tracing the chains of inference backwards
mathematics. And from this law one can then draw consequences m accordance with general logical laws.
Now let us examine somewhat more closely what takes place m mathematics, beginning with inference.
We may distinguish two kinds of inferences: inferences from two premises and inferences from one premise.
Now we make advances in mathematics by choosing as the premises of an inference one or two propositions that have already been recognised as true. The conclusion obtained from these is a new truth of mathematics. And this can in turn be used, alone or together with another truth, in drawing further conclusions. It would be possible to call each truth thus obtained a theorem. But usually a truth is only called a theorem when it has not merely been obtained as the result of an inference, but is itself in turn used as a premise in the development of the science, and that not just for one but for a number of inferences. In this way chains of inference are formed connecting truths; and the further the science develops the longer and more numerous become the chains of inference and the greater the diversity of the theorems.
But one can also trace the chains of inference backwards by asking from what truths each theorem has been inferred. As the diversity of theorems becomes greater as we go forward along the chains of inference, so, as we step backwards, the circle of theorems closes in more and more. Whereas it appears that there is no limit to the number of steps forward we can take, when we go backwards we must eventually come to an end by arriving at truths which cannot themselves be inferred in turn from other truths. Going backwards we come up against the axioms and postulates. We may come up against definitions as well, but we shall take a closer look at that later. If we start from a theorem and trace the chains of inference backwards until we arrive at other theorems or at axioms, postulates or definitions, we discover chains of inference starting from known theorems, axioms, postulates or definitions and terminating with the theorem in question.
The totality of these inference-chains constitutes the proofof the theorem. We may say that a proof starts from propositions that are accepted as true and leads via chains of inferences to the theorem. But it can also happen that a proof consists only of a single chain of inference. In most cases a proof will proceed via truths which are not called theorems for the simple reason that they occur only in this proof, and are not used elsewhere. A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. This is why we already find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one.
Science demands that we prove whatever is suceptible of proof and that we do n,ot rest until we come up against something unprovable. It must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in these primitive truths
Proof
Primitive truths
204 Logic in Mathematics
? Logic in Mathematics 205
as in a kernel. Our only concern is to generate the whole of mathematics from this kernel. The essence of mathematics has to be defined by this kernel of truths, and until we have learnt what these primitive truths are, we cannot be clear about the nature of mathematics. If we assume that we have succeeded in discovering these primitive truths, and that mathematics has been developed from them, then it will appear as a system of truths that are connected with one another by logical inference.
Euclid had an inkling of this idea of a system; but he failed to realize it and it almost seems as if at the present time we were further from this goal than ever. We see mathematicians each pursuing his own work on some fragment of the subject, but these fragments do not fit together into a system; indeed the idea of a system seems almost to have been lost. And yet the striving after a system is a justified one. We cannot long remain content with the fragmentation that prevails at present. Order can only be created by a system. But in order to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved.
When an inference is being drawn, we must know what its premises are. We must not allow the premises to be confused with the laws of inference, which are purely logical; otherwise the logical purity of the inferences will be lost and it would not be possible, in the confusion of premises with laws of inference, clearly to distinguish the former. But if we have no clear recognition of what the premises are, we can have no certainty of arriving at the primitive truths, and failing that we cannot construct a system. For this reason we must avoid such expressions as 'a moment's reflection shows that' or 'as we can easily see'. We must put the moment's reflection into
words so that we can see what inferences it consists of and what premises it makes use of. In mathematics we must never rest content with the fact that something is obvious or that we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction. Only in this way can we discover what the primitive truths are, and only in this way can a system be constructed.
Let us now take a closer look at the axioms, postulates and definitions:
The axioms are truths as are the theorems, but they are truths for which no proof can be given in our system, and for which no proof is needed. It follows from this that there are no false axioms, and that we cannot accept a thought as an axiom if we are in doubt about its truth; for it is either false and hence not an axiom, or it is true but stands in need of proof and hence is not an axiom. Not every truth for which no proof is required is an axiom, for such a truth might still be proved in our system. Whether a truth is an axiom depends therefore on the system, and it is possible for a truth to be an nxiom in one system and not in another. That is to say, it is conceivable that there should be a truth A and a truth B, each of which can be proved from the other in conjunction with truths C, D, E, F, whilst the truths C, D, E, F nre not sufficient on their own to prove either A or B. If now C, D, E, F may serve as axioms, then we have the choice of regarding A, C, D, E, F as
The system
of mathematic! i
The axiom?
? ? Parenthetical remark on sentence and thought
axioms and B as a theorem, orB, C, D, E, F as axioms, and A as a theorem. We can see from this that the possibility of one system does not necessarily rule out the possibility of an alternative system, and that we may have a choice between different systems. So it is really only relative to a particular system that one can speak of something as an axiom.
Here, in passing, I may say something about the expressions 'thought' and 'sentence'. I use the word 'sentence' to refer to a sign that is normally complex, whether it is made up of sounds or written signs. Of course this sign must have a sense. Here I am only considering sentences in which we state or assert something. We can translate sentences into another language. The sentence in the other language is different from the original one, for its constituents (component sounds) are different and are put together differently; but if the translation is correct, it will express the same sense and of course it is really the sense that concerns us. The sentence is of value to us because of the sense that we grasp in it, which is recognizably the same in the translation too. I call this sense a thought. What we prove is not a sen- tence, but a thought. And it is neither here nor there which language is used in giving the proof. It is true that in mathematics we often speak of proofs of a theorem [Lehrsatz], understanding by the word 'sentence' ['Satz'] what I am calling a thought, or perhaps not distinguishing clearly between the expression in words or signs and the thought expressed. 1 A thought cannot be perceived by the senses, but in the sentence it is represented by what can be heard or seen. For this reason I do not use 'Lehrsatz' but 'Theorem', and not 'Grundsatz' but 'Axiom', understanding by theorems and axioms true thoughts. This, however, is to imply that a thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody, and its truth is quite independent of its being thought by so-and-so or indeed by anyone at all. We are not to regard thinking as the act of producing a thought, but as that of grasping a thought.
Postulates seem at first sight to be essentially different from axioms. In Euclid we have the postulate 'Let it be postulated that a straight line may be drawn from any point to any other'.
This is obviously introduced with a view to making constructions. The postulates, so it seems, present the simplest procedures for making every construction, and postulate their possibility. At first we might perhaps think that none of this is of any help in providing proofs, but only for solving problems. But this would be a mistake, for sometimes an auxiliary line is needed for a proof, and sometimes an auxiliary point, an auxiliary number-an auxiliary object of some kind. In the proof of a theorem an auxiliary object is one of which nothing is said in the theorem, but which is
1 The play that Frege is here making on the words 'Lehrsatz' and 'Satz' is lost in translation, so we have enclosed them in square brackets after their English equivalents (trans. ).
Postulates
206 Logic in Mathematics
? Logic in Mathematics 207
required for the proof, so that this would collapse if there were no such object. And if there is no such object, it seems that we must be able to create one and we need a postulate to ensure that this is possible. But what in actual fact is this drawing a line? It is not, at any rate, a line in the geometrical sense that we are creating when we make a stroke with a pencil. And how in this way are we to connect a point in the interior of Sirius with a point in Rigel? Our postulate cannot refer to any such external procedure. It refers rather to something conceptual. But what is here in question is not a subjective, psychological possibility, but an objective one. Surely the truth of a theorem cannot really depend on something we do, when it holds quite independently of us. So the only way of regarding the matter is that by drawing a straight line we merely become ourselves aware of what obtains independently of us. So the content of our postulate is essentially this, that given any two points there is a straight line connecting them. So a postulate is a truth as is an axiom, its only peculiarity being that it asserts the existence of something with certain properties. From this it follows that there is no real need to distinguish axioms and postulates. A postulate can be regarded as a special case of an axiom.
We come to definitions. Definitions proper must be distinguished from illustrative examples. In the first stages of any discipline we cannot avoid the use of ordinary words. But these words are, for the most part, not really appropriate for scientific purposes, because they are not precise enough and fluctuate in their use. Science needs technical terms that have precise and fixed meanings, and in order to come to an understanding about these meanings and exclude possible misunderstandings, we give examples illustrating their use. Of course in so doing we have again to use ordinary words, and these may display defects similar to those which the examples are intended to remove. So it seems that we shall then have to do the same thing over again, providing new examples. Theoretically one will never really achieve one's goal in this way. In practice, however, we do manage to come to an understanding about the meanings of words. Of course we have to be able to count on a meeting of minds, on others guessing what we have in mind. But all this precedes the construction of a system and does not helong within a system. In constructing a system it must be assumed that the words have precise meanings and that we know what they are. Hence we can at this point leave illustrative examples out of account and turn our nttention to the construction of a system.
In constructing a system the same group of signs, whether they are Hounds or combinations of sounds (spoken signs) or written signs, may occur over and over again. This gives us a reason for introducing a simple Nign to replace such a group of signs with the stipulation that this simple sign is always to take the place of that group of signs. As a sentence is generally 11 complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of the thought correspond to parts of the sentence. So as a general rule when a group of signs occurs in a sentence
Definitions
Illustrative examples
Definition
proper
? 208 Logic in Mathematics
it will have a sense which is part of the thought expressed. Now when a simple sign is thus introduced to replace a group of signs, such a stipulation is a definition. The simple sign thereby acquires a sense which is the same as that of the group of signs. Definitions are not absolutely essential to a system. We could make do with the original group o f signs. The introduction of a simple sign adds nothing to the content; it only makes for ease and simplicity of expression. So definition is really only concerned with signs. We shall call the simple sign the definiendum, and the complex group of signs which it replaces the definiens. The definiendum acquires its sense only from the definiens. This sense is built up out of the senses of the parts of the definiens. When we illustrate the use of a sign, we do not build its sense up out of simpler constituents in this way, but treat it as simple. All we do is to guard against misunderstanding where an expression is ambiguous.
A sign has a meaning once one has been bestowed upon it by definition, and the definition goes over into a sentence asserting an identity. Of course the sentence is really only a tautology and does not add to our knowledge. It contains a truth which is so self-evident that it appears devoid of content, and yet in setting up a system it is apparently used as a premise. I say apparently, for what is thus presented in the form of a conclusion makes no addition to our knowledge; all it does in fact is to effect an alteration of expression, and we might dispense with this if the resultant simplification of expression did not strike us as desirable. In fact it is not possible to prove something new from a definition alone that would be unprovable without it. When something that looks like a definition really makes it possible to prove something which could not be proved before, then it is no mere definition but must conceal something which would have either to be proved as a theorem or accepted as an axiom. Of course it may look as if a definition makes it possible to give a new proof. But here we have to distinguish between a sentence and the thought it expresses. If the definiens occurs in a sentence and we replace it by. the definiendum, this does not affect the thought at all. It is true we get a different sentence if we do this, but we do
not get a different thought. Of course we need the definition if, in the proof of this thought, we want it to assume the form of the second sentence. But if the thought can be proved at all, it can also be proved in such a way that it assumes the form of the first sentence, and in that case we have no need of the definition. So if we take the sentence as that which is proved, a definition may be essential, but not if we regard the thought as that which is to be proved.
It appears from this that definition is, after all, quite inessential. In fact considered from a logical point of view it stands out as something wholly inessential and dispensable. Now of course I can see that strong exception will be taken to this. We can imagine someone saying: Surely we are undertaking ~ logical analysis when we give a definition. You might as well say that it doesn't matter whether I carry out a chemical analysis of a body in order to see what elements it is composed of, as say that it is immaterial
? Logic in Mathematics 209
whether I carry out a logical analysis of a logical structure in order to find out what its constituents are or leave it unanalysed as if it were simple, when it is in fact complex. It is surely impossible to make out that the activity of defining something is without any significance when we think of the considerable intellectual effort required to furnish a good definition. -There is certainly something right about this, but before I go into it more closely, I want to stress the following point. To be without logical significance is still by no means to be without psychological significance. When we examine what actually goes on in our mind when we are doing intellectual work, we find that it is by no means always the case that a thought is present to our consciousness which is clear in all its parts. For example, when we use the word 'integral', are we always conscious of everything appertaining to its sense? I believe that this is only very seldom the case. Usually just the word is present to our consciousness, allied no doubt with a more or less dim awareness that this word is a sign which has a sense, and that we can, if we wish, call this sense to mind. But we are usually content with the knowledge that we can do this. If we tried to call to mind everything appertaining to the sense of this word, we should make no headway. Our minds are simply not comprehensive enough. We often need to use a sign with which we associate a very complex sense. Such a sign seems, so to speak, a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we have need of what it contains. It follows from this that a thought, as I understand the word, is in no way to be identified with a content of my consciousness. If therefore we need such signs-signs in which, as it were, we conceal a very complex sense as in a receptacle-we also need definitions so that we can cram this sense into the receptacle and also take it out again. So if from a logical point of view definitions are at hottom quite inessential, they are nevertheless of great importance for thinking as this actually takes place in human beings.
An objection was mentioned above which arose from the consideration that it is by means of definitions that we perform logical analyses. In the development of science it can indeed happen that one has used a word, a sign, an expression, over a long period under the impression that its sense is simple until one succeeds in analysing it into simpler logical constituents. By means of such an analysis, we may hope to reduce the number of axioms; for it may not be possible to prove a truth containing a complex constituent so long as that constituent remains unanalysed; but it may be possible, given nn analysis, to prove it from truths in which the elements of the analysis occur. This is why it seems that a proof may be possible by means of a definition, if it provides an analysis, which would not be possible without this analysis, and this seems to contradict what we said earlier. Thus what seemed to be an axiom before the analysis can appear as a theorem after the analysis.
But how does one judge whether a logical analysis is correct? We cannot prove it to be so. The most one can be certain of is that as far as the form of
? 210 Logic in Mathematics
words goes we have the same sentence after the analysis as before. But that the thought itself also remains the same is problematic. When we think that we have given a logical analysis of a word or sign that has been in use over a long period, what we have is a complex expression the sense of whose parts is known to us. The sense of the complex expression must be yielded by that of its parts. But does it coincide with the sense of the word with the long
established use? I believe that we shall only be able to assert that it does when this is self-evident. And then what we have is an axiom. But that the simple sign that has been in use over a long period coincides in sense with that of the complex expression that we have formed, is just what the definition was meant to stipulate.
We have therefore to distinguish two quite different cases:
(1) We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court.
(2) We have a simple sign with a long established use. We believe that we can give a logical analysis of its sense, obtaining a complex expression which in our opinion has the same sense. We can only allow something as a constituent of a complex expression if it has a sense we recognize. The sense of the complex expression must be yielded by the way in which it is put together. That it agrees with the sense of the long established simple sign is not a matter for arbitrary stipulation, but can only be recognized by an immediate insight. No doubt we speak of a definition in this case too. It might be called an 'analytic definition' to distinguish it from the first case. But it is better to eschew the word 'definition' altogether in this case, because what we should here like to call a definition is really to be regarded as an axiom. In this second case there remains no room for an arbitrary stipulation, because the simple sign already has a sense. Only a sign which as yet has no sense can have a sense arbitrarily assigned to it. So we shall stick to our original way of speaking and call only a constructive definition a definition. According to that a definition is an arbitrary stipulation which confers a sense on a simple sign which previously had none. This sense has, of course, to be expressed by a complex sign whose sense results from the way it is put together.
Now we still have to consider the difficulty we come up against in giving a logical analysis when it is problematic whether this analysis is correct.
Let us assume that A is the long-established sign (expression) whose sense we have attempted to analyse logically by constructing a complex expression that gives the analysis. Since we are not certain whether the analysis is successful, we are not prepared to present the complex expression as one which can be replaced by the simple sign A. If it is our intention to put forward a definition proper, we are not entitled to choose the sign . 4, which already has a sense, but we must choose a fresh sign B, say, which has the sense of the complex expression only in virtue of the definition. The question now is whether A and B have the same sense. But we can bypa11
? Logic in Mathematics 211
this question altogether if we are constructing a new system from the bottom up; in that case we shall make no further use of the sign A-we shall only use B. We have introduced the sign B to take the place of the complex expression in question by arbitrary fiat and in this way we have conferred a sense on it. This is a definition in the proper sense, namely a constructive definition.
If we have managed in this way to construct a system for mathematics without any need for the sign A, we can leave the matter there; there is no need at all to answer the question concerning the sense in which-whatever it may be-this sign had been used earlier. In this way we court no objections. However it may be felt expedient to use sign A instead of sign B. But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition. We must therefore explain that the sense in which this sign was used before the new system was constructed is no longer of any concern to us, that its sense is to be understood purely from the constructive definition that we have given. In constructing the new system we can take no account, logically speaking, of anything in mathematics that existed prior to the new system. Everything has to be made anew from the ground up. Even anything that we may have accomplished by our analytical activities is to be regarded only as preparatory work which does not itself make any appearance in the new system itself.
Perhaps there still remains a certain unclarity. How is it possible, one may ask, that it should be doubtful whether a simple sign has the same sense as a complex expression if we know not only the sense of the simple sign, but can recognize the sense of the complex one from the way it is put together? The fact is that if we really do have a clear grasp of the sense of the simple sign, then it cannot be doubtful whether it agrees with the sense of the complex expression. If this is open to question although we can clearly recognize the sense of the complex expression from the way it is put together, then the reason must lie in the fact that we do not have a clear grasp of the sense of the simple sign, but that its outlines are confused as if we saw it through a mist. The effect of the logical analysis of which we spoke will then be precisely this-to articulate the sense clearly. Work of this kind is very useful; it does not, however, form part of the construction of the system, but must take place beforehand. Before the work of construction is begun, the building stones have to be carefully prepared so ns to be usable; i.
? ? Introduction to Logic 195
A sentence proper is a proper name, and its meaning, if it has one, is a truth value: the True or the False. There are many sentences which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part, which means a concept. In the same way there are many proper names, whose meanings are not truth values, which can be analysed into a complete part, which is in its turn a proper name, and an unsaturated part. If this latter is to be meaningful, then the result of saturing it with any meaningful proper name whatever must once more be a meaningful proper name. When this happens, we call the meaning of this unsaturated part a function. At this point, however, we need to make a reservation, similar to that we made earlier when the word 'concept' was introduced, about the unavoidable inaccuracy of language. The unsaturated part of a sentence, whose meaning we have called a concept, must have the property of yielding a genuine sentence when saturated by any meaningful proper name; this means that it must yield the proper name of a truth value. This is the requirement that a concept have sharp boundaries. For a given concept, every object must either fall under it or not, tertium non datur. From this it follows that a requirement similar to that we have just laid down is to be made of a function. As an example let us start off from the sentence ' 3 - 2 > 0'. We split this up into the proper name '3 - 2' and the remainder
> 0'. We may say this unsaturated part means the concept of a positive number. This concept must have sharp boundaries. Every object must either fall or not fall under this concept. Let us now go further and split the proper name '3 - 2' up into the proper name '2' and the unsaturated part '3 -
Now we may also split the original sentence '3 - 2 >0' up into the proper name '2' and the unsaturated part '3 - >0'. The meaning ofthis is the concept of something that yields a positive remainder when subtracted from 3. This concept must have sharp boundaries too. Now if there were a meaningful proper name a such that the unsaturated part '3 - ' did not
with these requirements we usually end up with grammatical monstrosities. In the concept-script the judgement-stroke, besides conveying assertoric force, serves to demarcate the scope of the roman letters. In order to be able to narrow the scope over which the generality extends, I make use of gothic letters, and with these the concavity demarcates the scope. Here and there in arithmetic there is also a use of letters which roughly corresponds to that of the gothic letters in my concept-script. But I have found no indication that anyone is aware of this use as a special case. Probably most mathematicians, were they to read this, would have no idea of what I am alluding to. It was not until after some time that I became nware of it myself. We are very dependent on external aids in our thinking, nnd there is no doubt that the language of everyday life-so far, at least, tiS a certain area of discourse is concerned-had first to be replaced by a more sophisticated instrument, before certain distinctions could be noticed. Hut so far the academic world has, for the most part, disdained to master
this instrument.
12. v111. o6
? 196
Introduction to Logic
yield a meaningful proper name when saturated by it, then the unsaturated part '3 - >0', when saturated by a; would not yield a meaningful proper name either; that is to say, we should not be able to say whether the object designated by afell under the concept which is the meaning of'3 - >0'. We can see from this that the usual definitions of the arithmetical signs are inadequate.
? ? A brief Survey of my logical Doctrines 1
[1906]
The thought
When I use the word 'sentence' in what follows I do not mean a sentence that serves to express a wish, a command, or a question, but one that serves to make an assertion. Although a sentence can be perceived by the senses, we use it to communicate a content that cannot be perceived by the senses. We are making a judgement about this content when we accept it as true or reject it as false. When a sentence is uttered the assertion that it is true usually goes hand in hand with the communication of the content. But the hearer does not have to adopt the speaker's stance; not that he has to reject the content either. He can simply refrain from making a judgement. We may now think of the content of a sentence as it is viewed by such a hearer.
Now two sentences A and B can stand in such a relation that anyone who recognizes the content of A as true must thereby also recognize the content of B as true and, conversely, that anyone who accepts the content of B must straightway accept that of A. (Equipollence). It is here being assumed that there is no difficulty in grasping the content of A and B. The sentences need not be equivalent in all respects. For instance, one may have what we may call a poetic aura, and this may be absent from the other. Such a poetic aura will belong to the content of the sentence, but not to that which we accept as true or reject as false. I assume there is nothing in the content ofeither of the two equipollent sentences A and B that would have to be immediately accepted as true by anyone who had grasped it properly. The poetic aura then, or whatever else distinguishes the content of A from that of B, does not belong to what is accepted as true; for if this were the case, then it could not be an immediate consequence of anyone's accepting the content of B that he should accept that of A. For the assumption is that what distinguishes A and 8 is not contained in B at all, nor is it something that anyone must recognize as true straight off.
So one has to separate off from the content of a sentence the part that alone can be accepted as true or rejected as false. I call this part the thought
1 ln places this piece agrees, even in wording, with the Einleitung in die Logik of August 1906 (pp. 185 fT. ). I t may well be the case that it constitutes a revision-probably made shortly afterwards-of part of these diary notes. According to notes of the previous editors on the transcripts that form the basis of this edition, the originals of both pieces were found together in the order in which they have been printed here (ed. ).
? 198 A briefSurvey ofmy logical Doctrines
expressed by the sentence. It is the same in equipollent sentences of the kind given above. It is only with this part of the content that logic is concerned. I call anything else that goes to make up the content of a sentence the colouring of the thought.
Thoughts are not psychological entities and do not consist of ideas in the psychological sense. The thought in Pythagoras's theorem is the same for all men; it confronts everyone in the same way as something objective, whereas each man has his own ideas, sensations, and feelings, which belong only to him. We grasp thoughts but we do not create them.
In myth and fiction thoughts occur that are neither true nor false. Logic has nothing to do with these. In logic it holds good that every thought is either true or false, tertium non datur.
Dissociating assertoric force from the predicate
We can grasp a thought without recognizing it as true. 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 this recognition-make an assertion. We need to be able to express a thought without putting it forward as true. In the Begriffschrift I use a special sign to convey assertoric force: the judgement-stroke. The languages known to me lack such a sign, and assertoric force is closely bound up with the indicative mood of the sentence that forms the main clause. Of course in fiction even such sentences are uttered without assertoric force; but logic has nothing to do with fiction. Fiction apart, it seems that it is only in subordinate clauses that we can express thoughts without asserting them. One should not allow oneself to be misled by this peculiarity of language and confuse grasping a thought and making a judgement.
Negation
Assertoric force is to be dissociated from negation too. To each thought there corresponds an opposite, so that rejecting one of them coincides with accepting the other. To make a judgement is to make a choice between opposite thoughts. Accepting one of them and rejecting the other is one act. So there is no need of a special sign for rejecting a thought. We only need a special sign for negation as such.
The hypothetical mode of sentence composition
If we say that in a hypothetical judgement two judgements are put into a relationship w~th one another, we are using the word 'judgement' so as not to include the recognition of the truth of anything, and thus as I use the word 'thought'. For even if the whole compound sentence is uttered with
? 'If
17 ? 19 211
is greater than 2, then
( 17 . 19) 211
is greater than 2'
A briefSurvey ofmy logical Doctrines 199
assertoric force, one is asserting neither the truth of the antecedent nor that of the consequent. What is being judged to be true is rather a thought that is expressed by the whole compound sentence. On a closer examination one finds that in many cases neither the antecedent nor the consequent expresses a thought. In the compound sentence
'If a is greater than 2, then a2 is greater than 2'
the letter 'a' does not designate an object as does the numeral '2': it only indicates indefinitely. So there is an indefiniteness in both antecedent and consequent, which is why neither of these two clauses expresses a thought. But the whole compound sentence does express a thought, the letter 'a' serving to confer generality of content on the sentence as a whole. It may be remarked that in some of the sentences we utter this indefiniteness is not in evidence at all, or there is but the barest suggestion of it. Nevertheless it is always present when we express a general law. What has the grammatical form of a sentence and yet does not express a thought because it contains something indefinite, I call a quasi-sentence. A sentence proper, on the other hand, expresses a thought. With a general law we are tempted to speak of cases in which the condition is true and of others in which it is false. We must reject this way of speaking. It is only a thought that can be true and a thought is either true or false: it is not true in some cases, false in others. When we should like to distinguish such cases, what we have are quasi- sentences. A sentence proper can be obtained from a quasi-sentence by removing the indefiniteness. Thus from the quasi-sentence 'a > 2' we obtain as a sentence proper '1 >2', as well as '2 >2' and '3 >2'. Some of the sentences thus obtained may express true, and some false, thoughts. However this is only by the way. We shall treat generality more fully later. To begin with; we shall assume that the antecedent and consequent are sentences proper. The following compound sentence will serve as an example:
222
What is being said here? Each of the two clauses
'172 ? 19
- - - is greater than 2'
211
2
and
'(17 ? 19)2 is greater than 2' 211
expresses a thought which is either true or false. Ifwe call the first thought A and the second B, then four cases are possible:
A is true and B is true, A is true and B is false, A is false and 8 is true, A is false and B is false.
? 200 A briefSurvey ofmy logical Doctrines
Obviously it is only the second case that is incompatible with the sentence
'If
172 ? 19 211
is greater than 2, then
(172 ? 19)2 211
is greater than 2'
holding good. So our sentence says that the second case does not obtain; it leaves it open which of the remaining cases holds. This gives us the essence of the hypothetical mode of combining sentences. It is correct to say
'If 3 is greater than 2, then 32 is greater than 2' and 'If 2 is greater than 2, then 22 is greater than 2' and 'If 1 is greater than 2, then 12 is greater than 2'.
If these sentences sound strange, a little absurd even, this is due to the fact that in each of these examples one sees at once which of the four cases holds, whereas in the first example one does not. But this difference is quite inessential.
I will call this combination of the thoughts A and B the hypothetical com- bination of A and B, the first thought being the condition and the second the consequence. If a sentence consists of two sentences combined by 'and' both of which express a thought, then the sense of the whole sentence is to be taken as a thought too, for this sense is either true or false. It is true if each of the constituent thoughts r and L1 is true, and false in every other case, and so if at least one of the two constituent thoughts is false. I will call the thought in this whole sentence the conjunction of r and L1. The conjunction ofrand L1, in common with every thought, has an opposite.
Now the hypothetical combination of A and B is the opposite of the conjunction of A and the opposite of B. But, conversely, the conjunction of r and L1 is the opposite of the hypothetical combination of r with the opposite of L1. By means of negation the hypothetical mode can thus be reduced to conjunction and conjunction to the hypothetical mode. Looked at from a logical standpoint both appear equally primitive. But since the hypothetical mode is more closely connected with drawing inferences, it is best to give it pride of place, and see it as the primitive form, reducing conjunction to it.
Generality
We remarked just now that often in a hypothetical sentence neither the antecedent nor the consequent express thoughts, and that the reason for this is that there is present an indefiniteness, although this does not make the compound sentence as a whole devoid of sense. In the example
'If a is greater than 2, then a2 is greater than 2'
the letter 'a' gives rise to this indefiniteness and it is due to this letter that the thought of the whole compound sentence is general. This is the usual
? A briefSurvey ofmy logical Doctrines 201
employment of letters in arithmetic, even if it is not the only one. Of course natural language has means of accomplishing the same thing (e. g. 'tot-quat' in Latin); but these are less precise. And sentences occur in which we can spot no part at all corresponding to the letter 'a'. Here we may take the above use of letters in arithmetic as basic. It is not only compound hypo- thetical sentences that can be general. We may think of the sentence 'a= a'. In order to make it easier to recognize that the sentence is general, we can add 'no matter what a may be'.
Ifin the sentence
'If a is greater than 2, then a2 is greater than 2'
we substitute for the indefinitely indicating letter 'a' in turn the numerals '1', '2', '3', each of which designates a particular thing, we obtain the sentences
'If 1 is greater than 2, then 12 is greater than 2', 'If 2 is greater than 2, then 22 is greater than 2', 'If 3 is greater than 2, then 32 is greater than 2'.
The thoughts expressed by these sentences are particular cases of the general thought. Likewise the thoughts '1 = 1', '2 = 2', are particular cases of the general thought expressed in the sentence ' a = a'.
Here for the first time we have occasion to split up a sentence into parts that are not themselves sentences. Thus in the general sentence we have a part to which congruent parts in the associated particular sentences correspond and a part-in our case it is the letter 'a'-to which non- congruent parts in the particular sentences-the numerals '1', '2', '3'-correspond. These sentence-parts are different in kind. The part in which the general sentence agrees with the related particular sentences shows gaps at the place where the other part of the sentence, the numeral 'I', say, occurs. If we fill these gaps with the letter ? ~? . we obtain in our cases
'If~is greater than 2, then ~2 is greater than 2', ? ~= ~? .
We obtain specific sentences from these by filling the gaps with, say, the numerals '1', '2', or '3'. These do not show any gaps but fill the gaps in the first parts so that we get a sentence. If we call the parts of the sentence that show gaps unsaturated and the other parts complete, then we can think of a sentence as arising from saturating a saturated part with a complete part. The complete part ofa sentence I call a proper name, the unsaturatedpart a concept-name. To the unsaturated part of the sentence there corresponds an unsaturated part of the thought and to the complete part of the sentence a complete part of the thought, and we can also speak here of saturating the unsaturated part of the thought with a complete part. A thought that is put together in this way is just what traditional logic calls a singular judgement. We must notice, however, that one and the same thought can be
? 202 A briefSurvey ofmy logical Doctrines
split up in different ways and so can be seen as put together out of parts in different ways. The word 'singular' does not apply to the thought in itself but only with respect to a particular way of splitting it up. Each of the sentence- parts
'1 is greater than 2' and ' 12 is greater than 2'
can also be seen as put together out of the proper name '1' and an unsaturated part. The corresponding holds for the related thoughts.
? ? Logic in Mathematics1 [Spring 1914]
Mathematics has closer ties with logic than does any other discipline; for almost the entire activity of the mathematician consists in drawing inferences. In no other discipline does inference play so large a part, although inferences do occur here and there in other disciplines. Part of the mathematician's activity, besides drawing inferences, is to give definitions. Most disciplines are not concerned with the latter at all; only in jurisprudence is it of some importance, for although its subject-matter is quite different, it is in several respects close to mathematics. Jurisprudence takes its materials from history and psychology and for this reason these must claim to have some share in it. And there is nothing resembling this with mathematics.
Inferring and defining are subject to logical laws. From this it follows that logic is of greater importance to mathematics than to any other science. If one counts logic as part of philosophy, there will be a specially close
bond between mathematics and philosophy, and this is confirmed by the history of these sciences (Plato, Descartes, Leibniz, Newton, Kant).
But are there perhaps modes ofinference peculiar to mathematics which, for that very reason, do not belong to logic? Here one may point to the inference by mathematical induction from n to n + 1. Well, even a mode of inference peculiar to mathematics must be subject to a law and this law, if it is not logical in nature, will belong to mathematics, and can be ranked with the theorems or axioms of this science. For instance, mathematical
induction rests on the law that can be expressed as follows:
If the number 1 has the property c[J and if it holds generally for every positive whole number n that if it has the property c[J then n + 1 has the
property C/J, then every positive whole number has the property C/J.
If this law can be proved, it will be included amongst the theorems of mathematics; if it cannot, it will be included amongst the axioms. If one
draws inferences by mathematical induction, then one is actually making an application of this theorem or axiom; that is, this truth is taken as a premise of an inference. For example: the proof of the proposition (a + b) + n = a+ (b + n).
So likewise in other cases one can reduce a mode of inference that is peculiar to mathematics to a general law, if not a law of logic, then one of
1 The dating of the previous editors has been adopted (ed. ).
? Tracing the chains of inference backwards
mathematics. And from this law one can then draw consequences m accordance with general logical laws.
Now let us examine somewhat more closely what takes place m mathematics, beginning with inference.
We may distinguish two kinds of inferences: inferences from two premises and inferences from one premise.
Now we make advances in mathematics by choosing as the premises of an inference one or two propositions that have already been recognised as true. The conclusion obtained from these is a new truth of mathematics. And this can in turn be used, alone or together with another truth, in drawing further conclusions. It would be possible to call each truth thus obtained a theorem. But usually a truth is only called a theorem when it has not merely been obtained as the result of an inference, but is itself in turn used as a premise in the development of the science, and that not just for one but for a number of inferences. In this way chains of inference are formed connecting truths; and the further the science develops the longer and more numerous become the chains of inference and the greater the diversity of the theorems.
But one can also trace the chains of inference backwards by asking from what truths each theorem has been inferred. As the diversity of theorems becomes greater as we go forward along the chains of inference, so, as we step backwards, the circle of theorems closes in more and more. Whereas it appears that there is no limit to the number of steps forward we can take, when we go backwards we must eventually come to an end by arriving at truths which cannot themselves be inferred in turn from other truths. Going backwards we come up against the axioms and postulates. We may come up against definitions as well, but we shall take a closer look at that later. If we start from a theorem and trace the chains of inference backwards until we arrive at other theorems or at axioms, postulates or definitions, we discover chains of inference starting from known theorems, axioms, postulates or definitions and terminating with the theorem in question.
The totality of these inference-chains constitutes the proofof the theorem. We may say that a proof starts from propositions that are accepted as true and leads via chains of inferences to the theorem. But it can also happen that a proof consists only of a single chain of inference. In most cases a proof will proceed via truths which are not called theorems for the simple reason that they occur only in this proof, and are not used elsewhere. A proof does not only serve to convince us of the truth of what is proved: it also serves to reveal logical relations between truths. This is why we already find in Euclid proofs of truths that appear to stand in no need of proof because they are obvious without one.
Science demands that we prove whatever is suceptible of proof and that we do n,ot rest until we come up against something unprovable. It must endeavour to make the circle of unprovable primitive truths as small as possible, for the whole of mathematics is contained in these primitive truths
Proof
Primitive truths
204 Logic in Mathematics
? Logic in Mathematics 205
as in a kernel. Our only concern is to generate the whole of mathematics from this kernel. The essence of mathematics has to be defined by this kernel of truths, and until we have learnt what these primitive truths are, we cannot be clear about the nature of mathematics. If we assume that we have succeeded in discovering these primitive truths, and that mathematics has been developed from them, then it will appear as a system of truths that are connected with one another by logical inference.
Euclid had an inkling of this idea of a system; but he failed to realize it and it almost seems as if at the present time we were further from this goal than ever. We see mathematicians each pursuing his own work on some fragment of the subject, but these fragments do not fit together into a system; indeed the idea of a system seems almost to have been lost. And yet the striving after a system is a justified one. We cannot long remain content with the fragmentation that prevails at present. Order can only be created by a system. But in order to construct a system it is necessary that in any step forward we take we should be aware of the logical inferences involved.
When an inference is being drawn, we must know what its premises are. We must not allow the premises to be confused with the laws of inference, which are purely logical; otherwise the logical purity of the inferences will be lost and it would not be possible, in the confusion of premises with laws of inference, clearly to distinguish the former. But if we have no clear recognition of what the premises are, we can have no certainty of arriving at the primitive truths, and failing that we cannot construct a system. For this reason we must avoid such expressions as 'a moment's reflection shows that' or 'as we can easily see'. We must put the moment's reflection into
words so that we can see what inferences it consists of and what premises it makes use of. In mathematics we must never rest content with the fact that something is obvious or that we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction. Only in this way can we discover what the primitive truths are, and only in this way can a system be constructed.
Let us now take a closer look at the axioms, postulates and definitions:
The axioms are truths as are the theorems, but they are truths for which no proof can be given in our system, and for which no proof is needed. It follows from this that there are no false axioms, and that we cannot accept a thought as an axiom if we are in doubt about its truth; for it is either false and hence not an axiom, or it is true but stands in need of proof and hence is not an axiom. Not every truth for which no proof is required is an axiom, for such a truth might still be proved in our system. Whether a truth is an axiom depends therefore on the system, and it is possible for a truth to be an nxiom in one system and not in another. That is to say, it is conceivable that there should be a truth A and a truth B, each of which can be proved from the other in conjunction with truths C, D, E, F, whilst the truths C, D, E, F nre not sufficient on their own to prove either A or B. If now C, D, E, F may serve as axioms, then we have the choice of regarding A, C, D, E, F as
The system
of mathematic! i
The axiom?
? ? Parenthetical remark on sentence and thought
axioms and B as a theorem, orB, C, D, E, F as axioms, and A as a theorem. We can see from this that the possibility of one system does not necessarily rule out the possibility of an alternative system, and that we may have a choice between different systems. So it is really only relative to a particular system that one can speak of something as an axiom.
Here, in passing, I may say something about the expressions 'thought' and 'sentence'. I use the word 'sentence' to refer to a sign that is normally complex, whether it is made up of sounds or written signs. Of course this sign must have a sense. Here I am only considering sentences in which we state or assert something. We can translate sentences into another language. The sentence in the other language is different from the original one, for its constituents (component sounds) are different and are put together differently; but if the translation is correct, it will express the same sense and of course it is really the sense that concerns us. The sentence is of value to us because of the sense that we grasp in it, which is recognizably the same in the translation too. I call this sense a thought. What we prove is not a sen- tence, but a thought. And it is neither here nor there which language is used in giving the proof. It is true that in mathematics we often speak of proofs of a theorem [Lehrsatz], understanding by the word 'sentence' ['Satz'] what I am calling a thought, or perhaps not distinguishing clearly between the expression in words or signs and the thought expressed. 1 A thought cannot be perceived by the senses, but in the sentence it is represented by what can be heard or seen. For this reason I do not use 'Lehrsatz' but 'Theorem', and not 'Grundsatz' but 'Axiom', understanding by theorems and axioms true thoughts. This, however, is to imply that a thought is not something subjective, is not the product of any form of mental activity; for the thought that we have in Pythagoras's theorem is the same for everybody, and its truth is quite independent of its being thought by so-and-so or indeed by anyone at all. We are not to regard thinking as the act of producing a thought, but as that of grasping a thought.
Postulates seem at first sight to be essentially different from axioms. In Euclid we have the postulate 'Let it be postulated that a straight line may be drawn from any point to any other'.
This is obviously introduced with a view to making constructions. The postulates, so it seems, present the simplest procedures for making every construction, and postulate their possibility. At first we might perhaps think that none of this is of any help in providing proofs, but only for solving problems. But this would be a mistake, for sometimes an auxiliary line is needed for a proof, and sometimes an auxiliary point, an auxiliary number-an auxiliary object of some kind. In the proof of a theorem an auxiliary object is one of which nothing is said in the theorem, but which is
1 The play that Frege is here making on the words 'Lehrsatz' and 'Satz' is lost in translation, so we have enclosed them in square brackets after their English equivalents (trans. ).
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required for the proof, so that this would collapse if there were no such object. And if there is no such object, it seems that we must be able to create one and we need a postulate to ensure that this is possible. But what in actual fact is this drawing a line? It is not, at any rate, a line in the geometrical sense that we are creating when we make a stroke with a pencil. And how in this way are we to connect a point in the interior of Sirius with a point in Rigel? Our postulate cannot refer to any such external procedure. It refers rather to something conceptual. But what is here in question is not a subjective, psychological possibility, but an objective one. Surely the truth of a theorem cannot really depend on something we do, when it holds quite independently of us. So the only way of regarding the matter is that by drawing a straight line we merely become ourselves aware of what obtains independently of us. So the content of our postulate is essentially this, that given any two points there is a straight line connecting them. So a postulate is a truth as is an axiom, its only peculiarity being that it asserts the existence of something with certain properties. From this it follows that there is no real need to distinguish axioms and postulates. A postulate can be regarded as a special case of an axiom.
We come to definitions. Definitions proper must be distinguished from illustrative examples. In the first stages of any discipline we cannot avoid the use of ordinary words. But these words are, for the most part, not really appropriate for scientific purposes, because they are not precise enough and fluctuate in their use. Science needs technical terms that have precise and fixed meanings, and in order to come to an understanding about these meanings and exclude possible misunderstandings, we give examples illustrating their use. Of course in so doing we have again to use ordinary words, and these may display defects similar to those which the examples are intended to remove. So it seems that we shall then have to do the same thing over again, providing new examples. Theoretically one will never really achieve one's goal in this way. In practice, however, we do manage to come to an understanding about the meanings of words. Of course we have to be able to count on a meeting of minds, on others guessing what we have in mind. But all this precedes the construction of a system and does not helong within a system. In constructing a system it must be assumed that the words have precise meanings and that we know what they are. Hence we can at this point leave illustrative examples out of account and turn our nttention to the construction of a system.
In constructing a system the same group of signs, whether they are Hounds or combinations of sounds (spoken signs) or written signs, may occur over and over again. This gives us a reason for introducing a simple Nign to replace such a group of signs with the stipulation that this simple sign is always to take the place of that group of signs. As a sentence is generally 11 complex sign, so the thought expressed by it is complex too: in fact it is put together in such a way that parts of the thought correspond to parts of the sentence. So as a general rule when a group of signs occurs in a sentence
Definitions
Illustrative examples
Definition
proper
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it will have a sense which is part of the thought expressed. Now when a simple sign is thus introduced to replace a group of signs, such a stipulation is a definition. The simple sign thereby acquires a sense which is the same as that of the group of signs. Definitions are not absolutely essential to a system. We could make do with the original group o f signs. The introduction of a simple sign adds nothing to the content; it only makes for ease and simplicity of expression. So definition is really only concerned with signs. We shall call the simple sign the definiendum, and the complex group of signs which it replaces the definiens. The definiendum acquires its sense only from the definiens. This sense is built up out of the senses of the parts of the definiens. When we illustrate the use of a sign, we do not build its sense up out of simpler constituents in this way, but treat it as simple. All we do is to guard against misunderstanding where an expression is ambiguous.
A sign has a meaning once one has been bestowed upon it by definition, and the definition goes over into a sentence asserting an identity. Of course the sentence is really only a tautology and does not add to our knowledge. It contains a truth which is so self-evident that it appears devoid of content, and yet in setting up a system it is apparently used as a premise. I say apparently, for what is thus presented in the form of a conclusion makes no addition to our knowledge; all it does in fact is to effect an alteration of expression, and we might dispense with this if the resultant simplification of expression did not strike us as desirable. In fact it is not possible to prove something new from a definition alone that would be unprovable without it. When something that looks like a definition really makes it possible to prove something which could not be proved before, then it is no mere definition but must conceal something which would have either to be proved as a theorem or accepted as an axiom. Of course it may look as if a definition makes it possible to give a new proof. But here we have to distinguish between a sentence and the thought it expresses. If the definiens occurs in a sentence and we replace it by. the definiendum, this does not affect the thought at all. It is true we get a different sentence if we do this, but we do
not get a different thought. Of course we need the definition if, in the proof of this thought, we want it to assume the form of the second sentence. But if the thought can be proved at all, it can also be proved in such a way that it assumes the form of the first sentence, and in that case we have no need of the definition. So if we take the sentence as that which is proved, a definition may be essential, but not if we regard the thought as that which is to be proved.
It appears from this that definition is, after all, quite inessential. In fact considered from a logical point of view it stands out as something wholly inessential and dispensable. Now of course I can see that strong exception will be taken to this. We can imagine someone saying: Surely we are undertaking ~ logical analysis when we give a definition. You might as well say that it doesn't matter whether I carry out a chemical analysis of a body in order to see what elements it is composed of, as say that it is immaterial
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whether I carry out a logical analysis of a logical structure in order to find out what its constituents are or leave it unanalysed as if it were simple, when it is in fact complex. It is surely impossible to make out that the activity of defining something is without any significance when we think of the considerable intellectual effort required to furnish a good definition. -There is certainly something right about this, but before I go into it more closely, I want to stress the following point. To be without logical significance is still by no means to be without psychological significance. When we examine what actually goes on in our mind when we are doing intellectual work, we find that it is by no means always the case that a thought is present to our consciousness which is clear in all its parts. For example, when we use the word 'integral', are we always conscious of everything appertaining to its sense? I believe that this is only very seldom the case. Usually just the word is present to our consciousness, allied no doubt with a more or less dim awareness that this word is a sign which has a sense, and that we can, if we wish, call this sense to mind. But we are usually content with the knowledge that we can do this. If we tried to call to mind everything appertaining to the sense of this word, we should make no headway. Our minds are simply not comprehensive enough. We often need to use a sign with which we associate a very complex sense. Such a sign seems, so to speak, a receptacle for the sense, so that we can carry it with us, while being always aware that we can open this receptacle should we have need of what it contains. It follows from this that a thought, as I understand the word, is in no way to be identified with a content of my consciousness. If therefore we need such signs-signs in which, as it were, we conceal a very complex sense as in a receptacle-we also need definitions so that we can cram this sense into the receptacle and also take it out again. So if from a logical point of view definitions are at hottom quite inessential, they are nevertheless of great importance for thinking as this actually takes place in human beings.
An objection was mentioned above which arose from the consideration that it is by means of definitions that we perform logical analyses. In the development of science it can indeed happen that one has used a word, a sign, an expression, over a long period under the impression that its sense is simple until one succeeds in analysing it into simpler logical constituents. By means of such an analysis, we may hope to reduce the number of axioms; for it may not be possible to prove a truth containing a complex constituent so long as that constituent remains unanalysed; but it may be possible, given nn analysis, to prove it from truths in which the elements of the analysis occur. This is why it seems that a proof may be possible by means of a definition, if it provides an analysis, which would not be possible without this analysis, and this seems to contradict what we said earlier. Thus what seemed to be an axiom before the analysis can appear as a theorem after the analysis.
But how does one judge whether a logical analysis is correct? We cannot prove it to be so. The most one can be certain of is that as far as the form of
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words goes we have the same sentence after the analysis as before. But that the thought itself also remains the same is problematic. When we think that we have given a logical analysis of a word or sign that has been in use over a long period, what we have is a complex expression the sense of whose parts is known to us. The sense of the complex expression must be yielded by that of its parts. But does it coincide with the sense of the word with the long
established use? I believe that we shall only be able to assert that it does when this is self-evident. And then what we have is an axiom. But that the simple sign that has been in use over a long period coincides in sense with that of the complex expression that we have formed, is just what the definition was meant to stipulate.
We have therefore to distinguish two quite different cases:
(1) We construct a sense out of its constituents and introduce an entirely new sign to express this sense. This may be called a 'constructive definition', but we prefer to call it a 'definition' tout court.
(2) We have a simple sign with a long established use. We believe that we can give a logical analysis of its sense, obtaining a complex expression which in our opinion has the same sense. We can only allow something as a constituent of a complex expression if it has a sense we recognize. The sense of the complex expression must be yielded by the way in which it is put together. That it agrees with the sense of the long established simple sign is not a matter for arbitrary stipulation, but can only be recognized by an immediate insight. No doubt we speak of a definition in this case too. It might be called an 'analytic definition' to distinguish it from the first case. But it is better to eschew the word 'definition' altogether in this case, because what we should here like to call a definition is really to be regarded as an axiom. In this second case there remains no room for an arbitrary stipulation, because the simple sign already has a sense. Only a sign which as yet has no sense can have a sense arbitrarily assigned to it. So we shall stick to our original way of speaking and call only a constructive definition a definition. According to that a definition is an arbitrary stipulation which confers a sense on a simple sign which previously had none. This sense has, of course, to be expressed by a complex sign whose sense results from the way it is put together.
Now we still have to consider the difficulty we come up against in giving a logical analysis when it is problematic whether this analysis is correct.
Let us assume that A is the long-established sign (expression) whose sense we have attempted to analyse logically by constructing a complex expression that gives the analysis. Since we are not certain whether the analysis is successful, we are not prepared to present the complex expression as one which can be replaced by the simple sign A. If it is our intention to put forward a definition proper, we are not entitled to choose the sign . 4, which already has a sense, but we must choose a fresh sign B, say, which has the sense of the complex expression only in virtue of the definition. The question now is whether A and B have the same sense. But we can bypa11
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this question altogether if we are constructing a new system from the bottom up; in that case we shall make no further use of the sign A-we shall only use B. We have introduced the sign B to take the place of the complex expression in question by arbitrary fiat and in this way we have conferred a sense on it. This is a definition in the proper sense, namely a constructive definition.
If we have managed in this way to construct a system for mathematics without any need for the sign A, we can leave the matter there; there is no need at all to answer the question concerning the sense in which-whatever it may be-this sign had been used earlier. In this way we court no objections. However it may be felt expedient to use sign A instead of sign B. But if we do this, we must treat it as an entirely new sign which had no sense prior to the definition. We must therefore explain that the sense in which this sign was used before the new system was constructed is no longer of any concern to us, that its sense is to be understood purely from the constructive definition that we have given. In constructing the new system we can take no account, logically speaking, of anything in mathematics that existed prior to the new system. Everything has to be made anew from the ground up. Even anything that we may have accomplished by our analytical activities is to be regarded only as preparatory work which does not itself make any appearance in the new system itself.
Perhaps there still remains a certain unclarity. How is it possible, one may ask, that it should be doubtful whether a simple sign has the same sense as a complex expression if we know not only the sense of the simple sign, but can recognize the sense of the complex one from the way it is put together? The fact is that if we really do have a clear grasp of the sense of the simple sign, then it cannot be doubtful whether it agrees with the sense of the complex expression. If this is open to question although we can clearly recognize the sense of the complex expression from the way it is put together, then the reason must lie in the fact that we do not have a clear grasp of the sense of the simple sign, but that its outlines are confused as if we saw it through a mist. The effect of the logical analysis of which we spoke will then be precisely this-to articulate the sense clearly. Work of this kind is very useful; it does not, however, form part of the construction of the system, but must take place beforehand. Before the work of construction is begun, the building stones have to be carefully prepared so ns to be usable; i.
