Avenarius for the
Vierteljahrsschrift
fii.
Gottlob-Frege-Posthumous-Writings
Frege's footnote, p.
6) we clearly have a fragment of what was Intended as a textbook on logic.
! 'he footnote on p. 6 refers to the Begriffschrift {1879). In section B of the table 111' (on tents Frege uses the expression 'content of possible judgement'. From a letter 111 llusscrl dated 24. 5. 1891 it is clear that he had given up using this designation by lhr first half of 1891 at the latest. Therefore the present piece should be dated ! Jetween 1879 and 1891 (ed. ).
? ? 2 [2f. ] [3]
[4] [5]
[5f. ]
Logic
Superstitions about the weather have a basis in experience. Furnishing grounds of this kind is no proof.
Grounds that aflord a justification are often found in other truths. Inference. To establish laws of inference is the task of logic. Logic, like psychology, has for its subject-matter things that cannot be perceived by the senses. There is a sharp divide, however, marked by 'true'. Logic considers its objects in so far as they are true. What is true is true independently of the person who recognizes it to be true, and so is not a product of an inner process.
Comparison with ethics.
Comment on technical terms of logic. Rejection of psychological distinctions.
Isolating what is psychological, by consciously marking it off. Warning against confusing points of view and switching from one question to another. Danger lies in language. Translation possible? Yes, so far as the logical kernel is concerned. Value of learning languages for one's logical education.
[6] The formula-language of algebra: analysis of the logically complex. Reducing the laws of logic to one another.
The goal of scientific endeavour is truth. Inwardly to recognize something as true is to make a judgement, and to give expression to this judgement is to make an assertion.
What is true is true independently of our recognizing it as such. We can make mistakes. The grounds on which we make a judgement may justify our recognizing it as true; they may, however, merely give rise to our making a judgement, or make up our minds for us, without containing a justification for our judgement. Although each judgement we make is causally conditioned, it is nevertheless not the case that all these causes are grounds that afford a justification. There is an empirical tendency in philosophy which does not take sufficient heed of this distinction, and so, because our thinking takes its rise from experience, philosophy ends up by declaring all our knowledge to be empirical. The causes which merely give rise to acts of judgement do so in accordance with psychological laws; they are just as capable of leading to error as of leading to truth; theY. have no inherent relation to truth whatsoever; they know nothing of the opposition of true and false. The farmer whose fortunes are, for good or ill, bound up with the weather, seeks for means of determining what it will be like in advance. Little wonder that he attempts to link phases of the moon with variations in the weather and asks himself whether a full moon does not herald a change in the weather. If this appears to be confirmed-as may well be the case, since by and large the weather does not change abruptly and since it is not altogether easy to say whether the weather has changed-from that moment on he believes the weather is connected with
? Logic 3
1ht? moon, and this belief takes root because the cases that speak in its lnvour make a greater impression than those that do not and imprint themselves more firmly on his memory; and he thinks he now knows this 1mm experience. This is exactly what experience is in the case of the said t'mpirical tendency amongst philosophers. And so it is with every ~uperstition. Usually it is possible to make out the psychological causes. <'learly such an account of how men have come to hold something to be 1rue is no proof; and in science, too, the history of how a mathematical law wns discovered cannot take the place of the grounds that justify it. These will always be ahistorical; in other words, it will never depend on who first ~tnve them, what provided him with the incentive to follow up such a fruitful line of thought, when and where this occurred, and so forth.
Now the grounds which justify the recognition of a truth often reside in ol her truths which have already been recognized. But if there are any truths 1ccognized by us at all, this cannot be the only form that justification takes.
l'here must be judgements whose justification rests on something else, if 1hey stand in need ofjustification at all.
And this is where epistemology comes in. Logic is concerned only with 1hose grounds ofjudgement which are truths. To make a judgement because we are cognisant of other truths as providing a justification for it is known us inferring. There are laws governing this kind ofjustification, and to set up lhese laws of valid inference is the goal of logic.
The subject-matter of logic is therefore such as cannot be perceived by 1he senses and in this respect it compares with that of psychology and contrasts with that of the natural sciences. Instincts, ideas etc. are also neither visible nor tangible. All the same there is a sharp divide between 1hese disciplines, and it is marked by the word 'true'. Psychology is only concerned with truth in the way every other science is, in that its goal is to extend the domain of truths; but in the field it investigates it does not study the property 'true' as, in its field, physics focuses on the properties 'heavy', 'warm', etc. This is what logic does. It would not perhaps be beside the mark tu say that the laws of logic are nothing other than an unfolding of the content of the word 'true'. Anyone who has failed to grasp the meaning of this word-what marks it ofT from others-cannot attain to any clear idea uf what the task of logic is.
For psychology it is neither here nor there whether the products of the mental processes it studies may be called true. What is true is true independently of the person who recognizes it as true. What is true is therefore not a product of a mental process or inner act; for the product of one person's mind is not that of another's, however similar they may seem 10 be, just as the hunger of one person is not that of another or the eye of one person is not that of another, however close the resemblance may be. We do not directly observe the processes in the mind of another, only the
effects they have in the physical world. Strictly speaking, therefore, we can only form a superficial judgement of the similarity between mental
? 4 Logic
processes, since we are unable to unite the inner states experienced by different people in one consciousness and so compare them. If the content of the sentence 2 + 3 = 5 is exactly the same, in the strictest sense, for all those who recognize it to be true, this means that it is not a product of the mind of this person and a product of the mind of that person, but that it is grasped and recognized as true by both equally. Even if subjective elements are a necessary part and parcel of this grasping of a content, we shall not include them in what we call 'true'.
Logic has a closer affinity with ethics. The property 'good' has a significance for the latter analogous to that which the property 'true' has for the former. Although our actions and endeavours are all causally conditioned and explicable in psychological terms, they do not all deserve to be called good. Here, too, we can talk of justification, and here, too, this is not simply a matter of relating what actually took place or of showing that things had to happen as they did and not in any other way. Certainly we say 'tout comprendre, c'est tout pardonner', but we can only pardon what we consider not to be good.
What mak10s us so prone to embrace such erroneous views is that we define the task of logic as the investigation of the laws of thought, whilst understanding by this expression something on the same footing as the laws of nature: we understand them as laws in accordance with which thinking actually takes place and by whose means we could explain a single thought process in a particular person in a way analogous to that in which we explain, say, the movement of a planet by means of the law of gravity. The laws in accordance with which we actually draw inferences are not to be identified with laws of valid inference; otherwise we could never draw a wrong inference.
In these times when the theory of evolution is marching triumphantly through the sciences and the method of interpreting everything historically threatens to exceed its proper bounds, we must be prepared to face some strange and disconcerting questions. If man, like all other living creatures, has undergone a continuous process of evolution, have the laws of his thinking always been valid and will they always retain their validity? Will an inference that is valid now still be valid after thousands of years and was it already valid thousands of years ago? Clearly, the laws of how men do in fact think are being confounded here with the laws of valid inference. Let us take a somewhat closer look at this question. In the sense in which we speak of natural laws, psychological, mathematical or logical laws, it is, strictly speaking, impossible for laws to change at all. For such a law, expressed in full, must include mention of all relevant conditions, in which case it will hold independently of time and place. The law of inertia, for instance, claims to be valid for all times and regions of space. If it appeared not to be valid in, say, the neighbourhood of Sirius, we should assume that it had not been fully expressed, a condition having been overlooked which is satisfied here but not in the neighbourhood of Sirius. A genuine condition always contains
? Logic 5
something indefinite, and so, according to how this something is determined, 11 can assume the form of a true or false proposition. Thus if after some time 1he law of inertia no longer seemed to hold, this would be an indication that n further condition needed adding, a condition which had been satisfied up lo a certain date but not subsequently. The supposed change in the laws of
1hought would have to be interpreted in this way too; this could be no more thnn an apparent change and would be an indication that our knowledge of 1hese laws was incomplete. Now if by the laws of thought we understand the luws of logic, it is easy to see the absurdity of a condition relating, say, to the phosphorus content of our brains or to something else in human beings which is subject to change. In that case it would be quite possible that such a dtunge should have taken place in some people, but not in others, so that for Nomc people there would follow from certain truths the opposite to what would follow for others. This is utterly contrary to the nature of a law of In~tic, since it is contrary to the sense of the word 'true', which excludes any tciCrence to a knowing subject.
If, on the other hand, by the laws of thought we understand psychological luws, then we cannot rule out in advance the possibility that they should l'onlain mention of something that varies with time and place and, nl'wrdingly, that the process of thinking is different nowadays from what it wus 3000 years ago.
I,ogic, in common with every science, has its technical terms, words some uf which are also used outside the sciences, though not in quite the same ? ruse. It doesn't matter in the least if the meaning we fix on is not altogether In line with the everyday use of the word or it doesn't accord with its olymology; what does matter is that the word should be as appropriate a vehicle as possible for use in expressing the laws. Provided there is no loss of rl1uur, the more compendious the formulation of the whole system of laws I? . I he more felicitous is the apparatus of technical terms.
The task of logic being what it is, it follows that we must turn our backs 1111 unything that is not necessary for setting up the laws of inference. In pltrticular we must reject all distinctions in logic that are made from a purely p? ychological standpoint and have no bearing on inference. Similarly, in pure mechanics we don't distinguish substances according to their chemical properties, but speaks only of 'mass' and physical bodies, so that we don't h~tve, say, to establish a special law for each chemical substance in place of the one law of inertia. Therefore let us only distinguish where it serves our purpose. The so-called deepening of logic by psychology is nothing but a raiNiflcation of logic by psychology. In the form in which thinking naturally llovelops the logical and the psychological are bound up together. The task In hund is precisely that of isolating what is logical. This does not mean that
we want to banish any trace of what is psychological from thinking as it nMiurally takes place, which would be impossible; we only want to become ~twure of the logical justification for what we think. So the required 1epuration of the logical from the psychological is only a matter of
? ? 6 Logic
distinguishing in our minds between them. That is why we cannot give too many warnings against the danger of confusing points of view and switching from one question to another, a danger to which we are particularly exposed because we are accustomed to thinking in some language or other and because grammar, which has for speech a significance analogous to that which logic has for thought, is a mixture of the logical and the psychological. Otherwise all languages would necessarily have the same grammar. Can the same thought be expressed in different languages? Without a doubt, so far as the logical kernel is concerned; for otherwise it would not be possible for human beings to share a common intellectual life. But if we think of the kernel with the psychological husk added, a precise translation is impossible. Indeed we may go so far as to doubt whether the outer covering is the same for any two men. From this we can see the value of learning foreign languages for one's logical education; when we see that the same thought can be worded in different ways, our mind separates off the husk from the kernel, though, in any given language, it appears as a natural and integral part of it. This is how the differences between languages can facilitate our grasp of what is logical. But still the difficulties are not wholly removed in this way and our logicians still keep dragging in a number of things which are really of no logical concern, though they belong to the grammar of languages akin to our own, if not to others. For this reason it is useful to be acquainted also with a means of expression of a quite different kind, such as we have, for instance, in the formula-language of algebra. *
But even when we have completely isolated what is logical in some form or phrase from the vernacular or in some combination of words, our task is still not complete. What we obtain will generally turn out to be complex; we have to analyse this, for here as elsewhere we only attain full insight by pressing forwards until we arrive at what is absolutely simple. In this respect, too, logic, because of its attachment to language and grammar, has fallen short in a number of ways. The laws of logic are themselves truths and here again there arises the question how a judgement is justified. If it is not justified in terms of other truths, then logic doesn't need to bother itself with it any further. If, on the other hand, a law of logic can be reduced to other laws by a process of inference, then it is evidently the task of logic to carry out this reduction; for it is only by doing this that we can reach a vantage point from which we can take a conspectus of the laws of logic, and not count as many a law that is one and the same.
To sum up briefly, it is the business of the logician to conduct an unceasing struggle against psychology and those parts of language and
*In this connection mention might also be made of my concept-script. I would not be in a position to write this work on logic without benefit of my earlier endeavours to devise a concept-script.
? Logic 7
l(rammar which fail to give untrammelled expression to what is logical. He does not have to answer the question: how does thinking normally take place in human beings? What course does it naturally follow in the human 111ind? What is natural to one person may well be unnatural to another. To . ~ee this we need look no further than to the great difference between l(rammars. There is no reproach the logician need fear less than the 1eproach that his way of formulating things is unnatural, that the actual 11rocess of thinking follows a different course. If you aim to teach a beginner the rudiments of mathematics in as logically rigorous a form as possible, he will, unless he is quite exceptional, find this very unnatural, and for the very 1enson that such a degree of rigour is employed. As a result, he will understand what he is taught either not at all or only imperfectly. Therefore, oue needs to temper the rigour of one's approach in the early stages, and o11ly as one advances try by degrees to arouse a need for it. Even in utathematics we find that the most rigorous work always belongs to the lntcst stages in the history of the subject. If we were to heed those who ? ? h. iect that logic is unnatural, we would run the risk of becoming embroiled 111 interminable disputes about what is natural, disputes which are quite 1111:apable of being resolved within the province of logic, and which therefore lluvc no place in logic at all. Perhaps definitive answers are simply not to be hud, or cannot be given until we have observed primitive peoples and made u scientific study of languages.
[B. ] Content of possible judgement
Inwardly to recognize something as true is to make a judgement. Thus an instance of a content of possible judgement is the content of the equation 2 1 3 ~ 5. As we have seen, such a content is not the result of an inner pro- cess or the product of a mental act which men perform, but something objective: that is to say, it is something that is exactly the same for all mtional beings, for all who are capable of grasping it, just as the sun, say, is 11ornething objective. But isn't the sun perhaps for some people a beneficent or maleficent deity, for others a shining disk hurled into the heavens from the oust and rolling back down again towards the west, and for yet others an Immense, spherical white-hot body enveloped by a cloud of incandescent I&IINcs'! No. To some it may seem one thing, to others another; it is what it is.
A judgement is often preceded by questions. A mathematician will formulate a theorem to himself before he can prove it. A physicist will 11. a:cpt a law as an hypothesis in order to test it by experience. We grasp the l)unlcnt of a truth before we recognize it as true, but we grasp not only this; we grasp the opposite as well. In asking a question we are poised between opposite sentences. Although it is usually only one side that is expressed when we speak,? the other is still always implied; for the sense of the
? Here, of course, we are referring only to sentence-questions, not to wurd questions.
? 8 Logic
question remains the same if we add 'or not? '. It is this very fact which makes possible such verbal economy. Now whatever can thus be posed in a question, we wish to call a content of possible judgement. Therefore the content of any truth is 'a content of possible judgement', but so too is the opposite content. This opposition or conflict is such that we automatically reject one limb as false when we accept the other as true, and conversely. The rejection of the one and the acceptance of the other are one and the same.
? Boole's logical Calculus and the Concept-script1 [1880/81]
In his writings, Leibniz threw out such a profusion of seeds of ideas that in IIus respect he is virtually in a class of his own. A number of these seeds wrrc developed and brought to fruition within his own lifetime and with his 11tllaboration, yet more were forgotten, then later rediscovered and drvcloped further. This justifies the expectation that a great deal in his work lhul is now to all appearance dead and buried will one day enjoy a rr,urrcction. As part of this, I count an idea which Leibniz clung to lluuughout his life with the utmost tenacity, the idea of a lingua l'haracterica, an idea which in his mind had the closest possible links with lhnl of a calculus ratiocinatur. That it made it possible to perform a type of ru111putation, it was precisely this fact that Leibniz saw as a principal ndvanlage of a script which compounded a concept out of its constituents rnl her than a word out of its sounds, and of all hopes he cherished in this muller, we can even today share this one with complete confidence. I will IJlrulc just the following from the relevant passages:
'Si daretur vel lingua quaedam exacta, vel genus scripturae vere philosophiae, . . . omnia quae ex datis ratione assequi, inveniri possent l. fUodam genere calculi, perinde ac resolvuntur problemata arithmetica aut jcomctrica. '*
? I>c Scientia universali seu calculo philosophico.
1 In 188 I, this article was submitted by Frege in turn to the Zeitschrift fiir Matlrematik und Physik, the Mathematischen Annalen and the Zeitschrift fiir l'lr/lu. wphie und philosophisehe Kritik, but was in every case rejected by the editors. lllluully remained unpublished.
l'rom the report of H. Scholz and F. Bachmann: Der wissenschaftliche Nachlass r? mt {,'o/1/ob Frege (Paris 1936) we learn that the lost original was 'a manuscript rrrJlnrcd for publication of 103 closely written sides of quarto'. Scholz and lt11chmann mention besides that Frege also submitted the manuscript to R.
Avenarius for the Vierteljahrsschrift fii. r wissenschaftliche Philosophie. However it 11uuld be that what Frege submitted to Avenarius was the essay published in this v11lumc on pp. 47 IT. 'Boole's logical Formula-language and my Concept-Script', llnce Avenarius in his letter to Frege of 20/4/1882 cites the title of the manuscript rtlurncd by him as 'Boole's logical Formula-Language'.
The article can scarcely have been composed before 1880, the year in which the rtview by Schroder mentioned on p. 11 appeared. A great number of the reflections
? ? ? 10 Boole's logical Calculus and the Concept-script
Among the various sorties Leibniz made upon his goal, the beginnings of a symbolic logic come closest to what seems to be indicated by the phrase 'calculus ratiocinatur'. They are to be found in the essays:
Non inelegans specimen demonstrandi in abstractis and Addenda ad specimen calculi universalis
In these Leibniz stuck very close to language. Just as the words we use for the attributes of a thing follow one another, so he simply juxtaposes the letters corresponding to properties in order to express the formation of a concept. If, for instance, A means right-angled, B isosceles, C triangle, then Leibniz represents right-angled isosceles triangle by ABC. He uses a sign for identity, oo, and the sign + with the definition:
'A +Boo L significat A inesse ipsi L'.
This seems to coincide with the meaning recent logicians have given this sign, according to which A + B represents the class of individuals which belong to A or to B or to both. Since I am passing over less important details, the only other fact I will mention is that Leibniz allows the words 'non' and 'ens' to occur in his formulae. In this project he surely has the lingua characterica in mind, even though he made no express connection with the attempts he made to represent a content.
This way of setting up a formal logic seems to suggest itself naturally. At any rate recent German and English logicians have arrived at the same conception quite independently of Leibniz, though, as far as I know, in doing so they do not have a general characteristic in mind. However much Boolean logic* may stand out as a systematic working out of the fragmen- tary hints in Leibniz, it only goes beyond him in one point of fundamental importance-in the way it reduces hypothetical and disjunctive judgements to categorical. ? ? The Leibnizian 'ens' is left out.
In a short monograph,? ? ? I have now attempted a fresh approach to the Leibnizian idea of a lingua characterica. In so doing, I had to treat in part the same subject-matter as Boole, even if in a different way. This has
? Boole's main work is An Investigation ofthe Laws ofThought on which are founded the mathematical theories of logic and probabilities. London 1854.
** On one point indeed Boole has taken a retrograde step away from Leibniz, in adding to the Leibnizian meaning of A + B the condition that the classes A and B should have no common element. W. Stanley Jevons, E. Schroder and others have quite rightly not followed him in this.
*** Begriffschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle aS. 1879. (Note continued, pp. 11-12. )
put forward here by Frege are along the same lines as the contents of the essay published in 1882 Uber der Zweck der Begriffsschrift.
The last two sides of the lost original are preserved on a photostat, which was presented at the Paris Congress in 1935 as a specimen of the Frege NachlqfJ.
? ? BooZe's logical Calculus and the Concept-script 11
prompted many critics to draw comparisons between the two, of which the most detailed are those set out by E. Schroder in the Zeitschrift fiir Math. 11. Phys. Vol. XXV. He will not regard it as detracting from the gratitude which I here express for his thorough study of my monograph and its friendly review, if in what follows I attempt to supplement and correct that comparison.
A few of the signs I introduced there should be explained briefly here:
I deviate from usual practice in drawing a distinction between judgement und content of possible judgement. For me the relation in a hypothetical is uot one between judgements but between contents of possible judgement. 11ut ifl affirm that this relation holds, I then express a judgement.
( I) The content-stroke is horizontal, it is always prefixed to the expression of a content of possible judgement, serving to connect it to the judgement uml negation-strokes, and also to combine it with other contents of possible 111dgement by means of the conditional-stroke(? 2): e. g.
- ( 2 + 3 = 5).
(2) The judgement-stroke is placed vertically at the left hand end of the
content-stroke, it converts the content of possible judgement into a Judgement (? 2): e. g.
1--(2 +3 = 5).
(3) the negation-stroke is placed vertically under the content-stroke dividing it into two parts. The part to the right of the negation-stroke is the content-stroke of the original content, the part to the left that of its negation (~ 7): e. g.
. . . . . -(2 + 3 = 6).
(4) The conditional stroke connects two content-strokes by running from
the under side of the upper one to the left hand end of the lower. Like the negation-stroke it splits the upper stroke in two, of which the part to the right remains the content-stroke of the upper content, while the part to the left becomes the content-stroke of the whole, which means the negation of Ihe case that the upper content is false, and the lower true(? 5): e. g.
tx=7
X+ 3 = 10: means:thecasethatx+3= 10withoutx= 7doesnotoccur. Wecansay
in this case: ifx + 3 = 10, then x = 7. The roman letters such as the x here, confer generality on the content of the judgement by presenting it as true, whatever one may substitute for the roman letter (? 11, p. 21). If, for instance, we replace x by 2 we obtain
t,-2=7
L 2 + 3 = 10.
This is a true judgement, although the upper content, as well as the lower, is 1'11lsc. Here, the rendering 'if would jar with normal usage. Yet here too, the CIUIC is denied that 2 + 3 = 10 is true and 2 = 7 false.
If a negation-stroke stands to the left of the conditional stroke, as in
~2 = 7
2 + 3 = 5,
? ? 12 Boole's logical Calculus and the Concept-script
Above all, if we are not to go astray it is necessary that we should always bear in mind the purpose that governed Boole in his symbolic logic and the one that governed me in my Begriffsschrift.
If I understand him aright, Boole wanted to construct a technique for resolving logical problems systematically, similar to the technique of elimination and working out the unknown that algebra teaches. To this end, he represents judgements in the form of equations that he constructs out of letters and arithmetical signs such as +, 0 and 1. Logical laws then assume the form of algorithms, although these only coincide in part with those that hold in arithmetic for the same signs. In all this there is no concern about content whatsoever. In the main these means fulfil their purpose, at least as far as the range of problems that Boole has in mind are concerned. But one may think of logical problems lying outside this range. The use of arithmetical signs for logical purposes has the advantage that one is spared the necessity of learning a completely new algorithm. A large number of the transformations already familiar to our eyes still remain valid. ? Higher demands, which however Boole does not set himself, are naturally not met by this enterprise. Anyone demanding the closest possible agreement between the relations of the signs and the relations of the things themselves will always feel it to be back to front when logic, whose concern is correct thinking and which is also the foundation of arithmetic, borrows its signs
from arithmetic. To such a person it will seem more appropriate to develop for logic its own signs, derived from the nature of logic itself; we can then go on to use them throughout the other sciences wherever it is a question of preserving the formal validity of a chain of inference.
In contrast we may now set out the aim of my concept-script. Right from the start I had in mind the expression ofa content. What I am striving after is a lingua characterica in the first instance for mathematics, not a calculus restricted to pure logic. But the content is to be rendered more exactly than is done by verbal language. For that leaves a great deal to guesswork, even if only of the most elementary kind. There is only an imperfect correspondence between the way words are concatenated and the structure
the negation of the case that 2 + 3 = 5 without 2 = 7 is thereby converted into its affirmation we may render: 2 + 3 = 5 and it is not the case that 2 = 7 (? 7, p. 13). In
fTrr-2= l+l L2+3=5
-. - 2 = 1 + 1takes the place ofthe 2 = 7 ofthe preceding example. In rendering this, two denials come together and yield an affirmation:
2+3=5,and2=1+1(? 7. p. 12).
? I should mention that the deviations from arithmetic are for all that so fundamental ti}at solving logical equations is not at all like solving algebraic ones. And this greatly diminishes the value of the agreement in the algorithm.
? Boole's logical Calculus and the Concept-script 13
of the concepts. The words 'lifeboat' and 'deathbed' are similarly constructed though the logical relations of the constituents are different. So the latter isn't expressed at all, but is left to guesswork. Speech often only indicates by inessential marks or by imagery what a concept-script should spell out in full. At a more external level, the latter is distinguished from verbal language in being laid out for the eye rather than for the ear. Verbal script is of course also laid out for the eye, but since it simply reproduces verbal speech, it scarcely comes closer to a concept-script than speech: in fact it is at an even greater remove from it, since it consists in signs for signs, not of signs for the things themselves. A lingua characterica ought, as Leibniz says, peindre non pas les paroles, mais les pensees. The formula- languages of mathematics come much closer to this goal, indeed in part they nrrive at it. But that of geometry is still completely undeveloped and that of urithmetic itself is inadequate for its own domain; for at precisely the most important points, when new concepts are to be introduced, new foundations laid, it has to abandon the field to verbal language, since it only forms numbers out of numbers and can only express those judgements which treat 111' the equality of numbers which have been generated in different ways. But arithmetic in the broadest sense also forms concepts-and concepts of such richness and fineness in their internal structure that in perhaps no other science are they to be found combined with the same logical perfection. And there are other judgements which arithmetic makes, besides mere equations und inequalities. The reason for this inability to form concepts in a scientific manner lies? in the lack of one of the two components of which every highly developed language must consist. That is, we may distinguish the formal part which in verbal language comprises endings, prefixes, suffixes and auxiliary words, from the material part proper. The signs of arithmetic correspond to the latter. What we still lack is the logical cement that will bind these building stones firmly together. Up till now verbal language took over this role, and hence it was impossible to avoid using it in the proof itself, and not merely in parts that can be omitted without affecting the cogency of the patterns of inference, whose only purpose is to make it easier lo grasp connections. In contrast, Boole's symbolic logic only represents the formal part of language, and even that incompletely. The result is that
Boole's formula-language and the formula-language of arithmetic each solve only one part of the problem of a concept-script. What we have to do now, in order to produce a more adequate solution, is to supplement the signs of 111athematics with a formal element, since it would be inappropriate to leave the signs we already have unused. But on this score alone Boole's logic is nlready completely unsuited to the task of making this supplementation, since it employs the signs +, 0 and 1 in a sense which diverges from their nrithrnetical ones. It would lead to great inconvenience if the same signs were to occur in one formula with different meanings. This is not an
? In the case of the formula-language of arithmetic.
? 14 Boole's logical Calculus and the Concept-script
objection to Boole, since such an application of his formulae obviously lay completely outside his intentions. Thus, the problem arises of devising signs for logical relations that are suitable for incorporation into the formula- language of mathematics, and in this way of forming-at least for a certain domain-a complete concept-script. This is where my booklet comes in.
Despite all differences in our further aims, it is evident from what has been said already that the first problem for Boole and me was the same: the perspicuous representation of logical relations by means of written signs. This implies the possibility of comparing the two. If I now turn to this, it cannot be done in the sense of adjudicating between the two formula- languages, which is to be preferred. To raise such a question would mean referring back to their ultimate aims, which are more ambitious in my case than in Boole's. It would indeed be more than possible that each set of signs was the more appropriate for its own ends. Nevertheless, it seems to me worthwhile to work out the comparison in detail, since in that way many of the peculiarities of my concept-script are thrown into sharper focus.
To begin with, as far as the extent of these sign languages is concerned, one is entitled to expect that these would have to coincide, once everything had been sifted from my concept-script which lies beyond the confines of pure logic. Yet even then, not everything that I express can be translated into Boolean notation, whereas the converse transformation is possible. E. Schroder indeed thinks that my concept-script would only correspond to the second part of Boole's. This is refuted by the fact that I too can easily express such a judgement as 'Every square root of 4 is a fourth root of 16' by
~a4 =16 La2 = 4
although Boole would count it as one of the primary propositions. I represent all judgements in the first part of Boole in a similar way though in so doing I admittedly construe them quite differently. The real difference is that I avoid such a division into two parts, the first dedicated to the relation of concepts (primary propositions), the second to the relation of judgements (secondary propositions) and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it.
On the other hand, Boole had only an inadequate expression for particular judgements such as 'some 4th roots of 16 are square roots of 4'
~a2 =4
La4 = 16,
and for existential judgements such as 'there is at least one 4th root of 16'
IT&ra4 = 16,
So it transpires that even when we restrict ourselves to pure logic my
apparently no expression at all.
? Boole's logical Calculus and the Concept-script 15
concept-script commands a somewhat wider domain than Boole's formula- lunguage. This is a result of my having departed further from Aristotelian logic. For in Aristotle, as in Boole, the logically primitive activity is the formation of concepts by abstraction, and judgement and inference enter in through an immediate or indirect comparison of concepts via their extensions. ? The only difference is that Aristotle places in the foreground 1he case where the extension of one concept completely includes that of unother-i. e. that of subordination-whereas Boole reduces other cases to lhe case of equality of extensions. ? ? Here hypothetical and disjunctive 111dgements are for the moment left on one side. Boole now construes the hypothetical judgement 'If B, then A' as a case of subordination of con- cepts, by saying 'the class of time instants at which B is included in the class ,,rtime instants at which A'.
The full incongruity of the introduction here of the idea of time instants slunds out most clearly if you think of eternal truths such as those of mathematics. Schroder seems to want to avoid the artificiality this involves, siuce, in company with Hugh McColl, he explains expressions like A = 0, A t 8 = 1 etc. -whose sense, on the Boolean conception, is self-explanatory when taken in conjunction with the stipulations of the first part-all over U! (ain without referring back. But in this way the last weak link between the 1wo parts is also snapped, and the signs 0, 1, = receive yet a third meaning 111 addition to their Boolean and arithmetical ones. According to Boole 0 111cans the extension of a concept under which nothing falls, as for example lhc extension of the concept 'whole number whose square is 2'. By 1, Boole uuderstands the extension of his universe ofdiscourse. These meanings hold for the first as much as the second part. I f one now ruptures this connection, then strictly speaking 0 has no longer an independent meaning in the Nccond part; combined with the identity sign it means a denial expressed as a
Judgement, while '= 1' designates an affirmation, which I express by the judgement-stroke. Then, besides this, the identity sign still has an iudependent meaning in formulae like A = B. But that even Boole on his npproach does not establish an organic connection between the two parts cun be seen from the fact that he does not use the equations of the first part us constituents of equations of the second part, and, if you hold strictly to Iheir meanings, cannot so use them. For in the first part A = B is a judge- ment. whereas if it were made a constituent of an equation of the second purl, as say in
( A = B )C = D
? This is not meant to imply that concept formation takes up a great deal uf space in their presentations. Rather, that their logics are essentially doctrines of inference, in which the formation of concepts is presupposed AN something that has already been completed.
** The relation of identity has greater content and so is less extensive nru. l general than that of subordination.
? ? ? 16 Boole's logical Calculus and the Concept-script
A = B would mean the class of time instants at which the content of the judgement 'A = B' was to be affirmed. On the other hand, if, to avoid this, you do not resort to the expedient of the idea of a class of time instants, then you get in return the double sense of the 0, the 1 etc. in the same formula, which would then, to say the least, become unperspicuous.
As opposed to this, I start out from judgements and their contents, and not from concepts. The precisely defined hypothetical relation between contents of possible judgement has a similar significance for the foundation of my concept-script to that which identity of extensions has for Boolean logic. I only allow the formation of concepts to proceed from judgements. If, that is, you imagine the 2 in the content of possible judgement
24 = 16
to be replaceable by something else, by (-2) or by 3 say, which may be
indicated by putting an x in place of the 2: x4 = 16,
the content of possible judgement is thus split into a constant and a variable part. The former, regarded in its own right but holding a place open for the latter, gives the concept '4th root of 16'.
! 'he footnote on p. 6 refers to the Begriffschrift {1879). In section B of the table 111' (on tents Frege uses the expression 'content of possible judgement'. From a letter 111 llusscrl dated 24. 5. 1891 it is clear that he had given up using this designation by lhr first half of 1891 at the latest. Therefore the present piece should be dated ! Jetween 1879 and 1891 (ed. ).
? ? 2 [2f. ] [3]
[4] [5]
[5f. ]
Logic
Superstitions about the weather have a basis in experience. Furnishing grounds of this kind is no proof.
Grounds that aflord a justification are often found in other truths. Inference. To establish laws of inference is the task of logic. Logic, like psychology, has for its subject-matter things that cannot be perceived by the senses. There is a sharp divide, however, marked by 'true'. Logic considers its objects in so far as they are true. What is true is true independently of the person who recognizes it to be true, and so is not a product of an inner process.
Comparison with ethics.
Comment on technical terms of logic. Rejection of psychological distinctions.
Isolating what is psychological, by consciously marking it off. Warning against confusing points of view and switching from one question to another. Danger lies in language. Translation possible? Yes, so far as the logical kernel is concerned. Value of learning languages for one's logical education.
[6] The formula-language of algebra: analysis of the logically complex. Reducing the laws of logic to one another.
The goal of scientific endeavour is truth. Inwardly to recognize something as true is to make a judgement, and to give expression to this judgement is to make an assertion.
What is true is true independently of our recognizing it as such. We can make mistakes. The grounds on which we make a judgement may justify our recognizing it as true; they may, however, merely give rise to our making a judgement, or make up our minds for us, without containing a justification for our judgement. Although each judgement we make is causally conditioned, it is nevertheless not the case that all these causes are grounds that afford a justification. There is an empirical tendency in philosophy which does not take sufficient heed of this distinction, and so, because our thinking takes its rise from experience, philosophy ends up by declaring all our knowledge to be empirical. The causes which merely give rise to acts of judgement do so in accordance with psychological laws; they are just as capable of leading to error as of leading to truth; theY. have no inherent relation to truth whatsoever; they know nothing of the opposition of true and false. The farmer whose fortunes are, for good or ill, bound up with the weather, seeks for means of determining what it will be like in advance. Little wonder that he attempts to link phases of the moon with variations in the weather and asks himself whether a full moon does not herald a change in the weather. If this appears to be confirmed-as may well be the case, since by and large the weather does not change abruptly and since it is not altogether easy to say whether the weather has changed-from that moment on he believes the weather is connected with
? Logic 3
1ht? moon, and this belief takes root because the cases that speak in its lnvour make a greater impression than those that do not and imprint themselves more firmly on his memory; and he thinks he now knows this 1mm experience. This is exactly what experience is in the case of the said t'mpirical tendency amongst philosophers. And so it is with every ~uperstition. Usually it is possible to make out the psychological causes. <'learly such an account of how men have come to hold something to be 1rue is no proof; and in science, too, the history of how a mathematical law wns discovered cannot take the place of the grounds that justify it. These will always be ahistorical; in other words, it will never depend on who first ~tnve them, what provided him with the incentive to follow up such a fruitful line of thought, when and where this occurred, and so forth.
Now the grounds which justify the recognition of a truth often reside in ol her truths which have already been recognized. But if there are any truths 1ccognized by us at all, this cannot be the only form that justification takes.
l'here must be judgements whose justification rests on something else, if 1hey stand in need ofjustification at all.
And this is where epistemology comes in. Logic is concerned only with 1hose grounds ofjudgement which are truths. To make a judgement because we are cognisant of other truths as providing a justification for it is known us inferring. There are laws governing this kind ofjustification, and to set up lhese laws of valid inference is the goal of logic.
The subject-matter of logic is therefore such as cannot be perceived by 1he senses and in this respect it compares with that of psychology and contrasts with that of the natural sciences. Instincts, ideas etc. are also neither visible nor tangible. All the same there is a sharp divide between 1hese disciplines, and it is marked by the word 'true'. Psychology is only concerned with truth in the way every other science is, in that its goal is to extend the domain of truths; but in the field it investigates it does not study the property 'true' as, in its field, physics focuses on the properties 'heavy', 'warm', etc. This is what logic does. It would not perhaps be beside the mark tu say that the laws of logic are nothing other than an unfolding of the content of the word 'true'. Anyone who has failed to grasp the meaning of this word-what marks it ofT from others-cannot attain to any clear idea uf what the task of logic is.
For psychology it is neither here nor there whether the products of the mental processes it studies may be called true. What is true is true independently of the person who recognizes it as true. What is true is therefore not a product of a mental process or inner act; for the product of one person's mind is not that of another's, however similar they may seem 10 be, just as the hunger of one person is not that of another or the eye of one person is not that of another, however close the resemblance may be. We do not directly observe the processes in the mind of another, only the
effects they have in the physical world. Strictly speaking, therefore, we can only form a superficial judgement of the similarity between mental
? 4 Logic
processes, since we are unable to unite the inner states experienced by different people in one consciousness and so compare them. If the content of the sentence 2 + 3 = 5 is exactly the same, in the strictest sense, for all those who recognize it to be true, this means that it is not a product of the mind of this person and a product of the mind of that person, but that it is grasped and recognized as true by both equally. Even if subjective elements are a necessary part and parcel of this grasping of a content, we shall not include them in what we call 'true'.
Logic has a closer affinity with ethics. The property 'good' has a significance for the latter analogous to that which the property 'true' has for the former. Although our actions and endeavours are all causally conditioned and explicable in psychological terms, they do not all deserve to be called good. Here, too, we can talk of justification, and here, too, this is not simply a matter of relating what actually took place or of showing that things had to happen as they did and not in any other way. Certainly we say 'tout comprendre, c'est tout pardonner', but we can only pardon what we consider not to be good.
What mak10s us so prone to embrace such erroneous views is that we define the task of logic as the investigation of the laws of thought, whilst understanding by this expression something on the same footing as the laws of nature: we understand them as laws in accordance with which thinking actually takes place and by whose means we could explain a single thought process in a particular person in a way analogous to that in which we explain, say, the movement of a planet by means of the law of gravity. The laws in accordance with which we actually draw inferences are not to be identified with laws of valid inference; otherwise we could never draw a wrong inference.
In these times when the theory of evolution is marching triumphantly through the sciences and the method of interpreting everything historically threatens to exceed its proper bounds, we must be prepared to face some strange and disconcerting questions. If man, like all other living creatures, has undergone a continuous process of evolution, have the laws of his thinking always been valid and will they always retain their validity? Will an inference that is valid now still be valid after thousands of years and was it already valid thousands of years ago? Clearly, the laws of how men do in fact think are being confounded here with the laws of valid inference. Let us take a somewhat closer look at this question. In the sense in which we speak of natural laws, psychological, mathematical or logical laws, it is, strictly speaking, impossible for laws to change at all. For such a law, expressed in full, must include mention of all relevant conditions, in which case it will hold independently of time and place. The law of inertia, for instance, claims to be valid for all times and regions of space. If it appeared not to be valid in, say, the neighbourhood of Sirius, we should assume that it had not been fully expressed, a condition having been overlooked which is satisfied here but not in the neighbourhood of Sirius. A genuine condition always contains
? Logic 5
something indefinite, and so, according to how this something is determined, 11 can assume the form of a true or false proposition. Thus if after some time 1he law of inertia no longer seemed to hold, this would be an indication that n further condition needed adding, a condition which had been satisfied up lo a certain date but not subsequently. The supposed change in the laws of
1hought would have to be interpreted in this way too; this could be no more thnn an apparent change and would be an indication that our knowledge of 1hese laws was incomplete. Now if by the laws of thought we understand the luws of logic, it is easy to see the absurdity of a condition relating, say, to the phosphorus content of our brains or to something else in human beings which is subject to change. In that case it would be quite possible that such a dtunge should have taken place in some people, but not in others, so that for Nomc people there would follow from certain truths the opposite to what would follow for others. This is utterly contrary to the nature of a law of In~tic, since it is contrary to the sense of the word 'true', which excludes any tciCrence to a knowing subject.
If, on the other hand, by the laws of thought we understand psychological luws, then we cannot rule out in advance the possibility that they should l'onlain mention of something that varies with time and place and, nl'wrdingly, that the process of thinking is different nowadays from what it wus 3000 years ago.
I,ogic, in common with every science, has its technical terms, words some uf which are also used outside the sciences, though not in quite the same ? ruse. It doesn't matter in the least if the meaning we fix on is not altogether In line with the everyday use of the word or it doesn't accord with its olymology; what does matter is that the word should be as appropriate a vehicle as possible for use in expressing the laws. Provided there is no loss of rl1uur, the more compendious the formulation of the whole system of laws I? . I he more felicitous is the apparatus of technical terms.
The task of logic being what it is, it follows that we must turn our backs 1111 unything that is not necessary for setting up the laws of inference. In pltrticular we must reject all distinctions in logic that are made from a purely p? ychological standpoint and have no bearing on inference. Similarly, in pure mechanics we don't distinguish substances according to their chemical properties, but speaks only of 'mass' and physical bodies, so that we don't h~tve, say, to establish a special law for each chemical substance in place of the one law of inertia. Therefore let us only distinguish where it serves our purpose. The so-called deepening of logic by psychology is nothing but a raiNiflcation of logic by psychology. In the form in which thinking naturally llovelops the logical and the psychological are bound up together. The task In hund is precisely that of isolating what is logical. This does not mean that
we want to banish any trace of what is psychological from thinking as it nMiurally takes place, which would be impossible; we only want to become ~twure of the logical justification for what we think. So the required 1epuration of the logical from the psychological is only a matter of
? ? 6 Logic
distinguishing in our minds between them. That is why we cannot give too many warnings against the danger of confusing points of view and switching from one question to another, a danger to which we are particularly exposed because we are accustomed to thinking in some language or other and because grammar, which has for speech a significance analogous to that which logic has for thought, is a mixture of the logical and the psychological. Otherwise all languages would necessarily have the same grammar. Can the same thought be expressed in different languages? Without a doubt, so far as the logical kernel is concerned; for otherwise it would not be possible for human beings to share a common intellectual life. But if we think of the kernel with the psychological husk added, a precise translation is impossible. Indeed we may go so far as to doubt whether the outer covering is the same for any two men. From this we can see the value of learning foreign languages for one's logical education; when we see that the same thought can be worded in different ways, our mind separates off the husk from the kernel, though, in any given language, it appears as a natural and integral part of it. This is how the differences between languages can facilitate our grasp of what is logical. But still the difficulties are not wholly removed in this way and our logicians still keep dragging in a number of things which are really of no logical concern, though they belong to the grammar of languages akin to our own, if not to others. For this reason it is useful to be acquainted also with a means of expression of a quite different kind, such as we have, for instance, in the formula-language of algebra. *
But even when we have completely isolated what is logical in some form or phrase from the vernacular or in some combination of words, our task is still not complete. What we obtain will generally turn out to be complex; we have to analyse this, for here as elsewhere we only attain full insight by pressing forwards until we arrive at what is absolutely simple. In this respect, too, logic, because of its attachment to language and grammar, has fallen short in a number of ways. The laws of logic are themselves truths and here again there arises the question how a judgement is justified. If it is not justified in terms of other truths, then logic doesn't need to bother itself with it any further. If, on the other hand, a law of logic can be reduced to other laws by a process of inference, then it is evidently the task of logic to carry out this reduction; for it is only by doing this that we can reach a vantage point from which we can take a conspectus of the laws of logic, and not count as many a law that is one and the same.
To sum up briefly, it is the business of the logician to conduct an unceasing struggle against psychology and those parts of language and
*In this connection mention might also be made of my concept-script. I would not be in a position to write this work on logic without benefit of my earlier endeavours to devise a concept-script.
? Logic 7
l(rammar which fail to give untrammelled expression to what is logical. He does not have to answer the question: how does thinking normally take place in human beings? What course does it naturally follow in the human 111ind? What is natural to one person may well be unnatural to another. To . ~ee this we need look no further than to the great difference between l(rammars. There is no reproach the logician need fear less than the 1eproach that his way of formulating things is unnatural, that the actual 11rocess of thinking follows a different course. If you aim to teach a beginner the rudiments of mathematics in as logically rigorous a form as possible, he will, unless he is quite exceptional, find this very unnatural, and for the very 1enson that such a degree of rigour is employed. As a result, he will understand what he is taught either not at all or only imperfectly. Therefore, oue needs to temper the rigour of one's approach in the early stages, and o11ly as one advances try by degrees to arouse a need for it. Even in utathematics we find that the most rigorous work always belongs to the lntcst stages in the history of the subject. If we were to heed those who ? ? h. iect that logic is unnatural, we would run the risk of becoming embroiled 111 interminable disputes about what is natural, disputes which are quite 1111:apable of being resolved within the province of logic, and which therefore lluvc no place in logic at all. Perhaps definitive answers are simply not to be hud, or cannot be given until we have observed primitive peoples and made u scientific study of languages.
[B. ] Content of possible judgement
Inwardly to recognize something as true is to make a judgement. Thus an instance of a content of possible judgement is the content of the equation 2 1 3 ~ 5. As we have seen, such a content is not the result of an inner pro- cess or the product of a mental act which men perform, but something objective: that is to say, it is something that is exactly the same for all mtional beings, for all who are capable of grasping it, just as the sun, say, is 11ornething objective. But isn't the sun perhaps for some people a beneficent or maleficent deity, for others a shining disk hurled into the heavens from the oust and rolling back down again towards the west, and for yet others an Immense, spherical white-hot body enveloped by a cloud of incandescent I&IINcs'! No. To some it may seem one thing, to others another; it is what it is.
A judgement is often preceded by questions. A mathematician will formulate a theorem to himself before he can prove it. A physicist will 11. a:cpt a law as an hypothesis in order to test it by experience. We grasp the l)unlcnt of a truth before we recognize it as true, but we grasp not only this; we grasp the opposite as well. In asking a question we are poised between opposite sentences. Although it is usually only one side that is expressed when we speak,? the other is still always implied; for the sense of the
? Here, of course, we are referring only to sentence-questions, not to wurd questions.
? 8 Logic
question remains the same if we add 'or not? '. It is this very fact which makes possible such verbal economy. Now whatever can thus be posed in a question, we wish to call a content of possible judgement. Therefore the content of any truth is 'a content of possible judgement', but so too is the opposite content. This opposition or conflict is such that we automatically reject one limb as false when we accept the other as true, and conversely. The rejection of the one and the acceptance of the other are one and the same.
? Boole's logical Calculus and the Concept-script1 [1880/81]
In his writings, Leibniz threw out such a profusion of seeds of ideas that in IIus respect he is virtually in a class of his own. A number of these seeds wrrc developed and brought to fruition within his own lifetime and with his 11tllaboration, yet more were forgotten, then later rediscovered and drvcloped further. This justifies the expectation that a great deal in his work lhul is now to all appearance dead and buried will one day enjoy a rr,urrcction. As part of this, I count an idea which Leibniz clung to lluuughout his life with the utmost tenacity, the idea of a lingua l'haracterica, an idea which in his mind had the closest possible links with lhnl of a calculus ratiocinatur. That it made it possible to perform a type of ru111putation, it was precisely this fact that Leibniz saw as a principal ndvanlage of a script which compounded a concept out of its constituents rnl her than a word out of its sounds, and of all hopes he cherished in this muller, we can even today share this one with complete confidence. I will IJlrulc just the following from the relevant passages:
'Si daretur vel lingua quaedam exacta, vel genus scripturae vere philosophiae, . . . omnia quae ex datis ratione assequi, inveniri possent l. fUodam genere calculi, perinde ac resolvuntur problemata arithmetica aut jcomctrica. '*
? I>c Scientia universali seu calculo philosophico.
1 In 188 I, this article was submitted by Frege in turn to the Zeitschrift fiir Matlrematik und Physik, the Mathematischen Annalen and the Zeitschrift fiir l'lr/lu. wphie und philosophisehe Kritik, but was in every case rejected by the editors. lllluully remained unpublished.
l'rom the report of H. Scholz and F. Bachmann: Der wissenschaftliche Nachlass r? mt {,'o/1/ob Frege (Paris 1936) we learn that the lost original was 'a manuscript rrrJlnrcd for publication of 103 closely written sides of quarto'. Scholz and lt11chmann mention besides that Frege also submitted the manuscript to R.
Avenarius for the Vierteljahrsschrift fii. r wissenschaftliche Philosophie. However it 11uuld be that what Frege submitted to Avenarius was the essay published in this v11lumc on pp. 47 IT. 'Boole's logical Formula-language and my Concept-Script', llnce Avenarius in his letter to Frege of 20/4/1882 cites the title of the manuscript rtlurncd by him as 'Boole's logical Formula-Language'.
The article can scarcely have been composed before 1880, the year in which the rtview by Schroder mentioned on p. 11 appeared. A great number of the reflections
? ? ? 10 Boole's logical Calculus and the Concept-script
Among the various sorties Leibniz made upon his goal, the beginnings of a symbolic logic come closest to what seems to be indicated by the phrase 'calculus ratiocinatur'. They are to be found in the essays:
Non inelegans specimen demonstrandi in abstractis and Addenda ad specimen calculi universalis
In these Leibniz stuck very close to language. Just as the words we use for the attributes of a thing follow one another, so he simply juxtaposes the letters corresponding to properties in order to express the formation of a concept. If, for instance, A means right-angled, B isosceles, C triangle, then Leibniz represents right-angled isosceles triangle by ABC. He uses a sign for identity, oo, and the sign + with the definition:
'A +Boo L significat A inesse ipsi L'.
This seems to coincide with the meaning recent logicians have given this sign, according to which A + B represents the class of individuals which belong to A or to B or to both. Since I am passing over less important details, the only other fact I will mention is that Leibniz allows the words 'non' and 'ens' to occur in his formulae. In this project he surely has the lingua characterica in mind, even though he made no express connection with the attempts he made to represent a content.
This way of setting up a formal logic seems to suggest itself naturally. At any rate recent German and English logicians have arrived at the same conception quite independently of Leibniz, though, as far as I know, in doing so they do not have a general characteristic in mind. However much Boolean logic* may stand out as a systematic working out of the fragmen- tary hints in Leibniz, it only goes beyond him in one point of fundamental importance-in the way it reduces hypothetical and disjunctive judgements to categorical. ? ? The Leibnizian 'ens' is left out.
In a short monograph,? ? ? I have now attempted a fresh approach to the Leibnizian idea of a lingua characterica. In so doing, I had to treat in part the same subject-matter as Boole, even if in a different way. This has
? Boole's main work is An Investigation ofthe Laws ofThought on which are founded the mathematical theories of logic and probabilities. London 1854.
** On one point indeed Boole has taken a retrograde step away from Leibniz, in adding to the Leibnizian meaning of A + B the condition that the classes A and B should have no common element. W. Stanley Jevons, E. Schroder and others have quite rightly not followed him in this.
*** Begriffschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle aS. 1879. (Note continued, pp. 11-12. )
put forward here by Frege are along the same lines as the contents of the essay published in 1882 Uber der Zweck der Begriffsschrift.
The last two sides of the lost original are preserved on a photostat, which was presented at the Paris Congress in 1935 as a specimen of the Frege NachlqfJ.
? ? BooZe's logical Calculus and the Concept-script 11
prompted many critics to draw comparisons between the two, of which the most detailed are those set out by E. Schroder in the Zeitschrift fiir Math. 11. Phys. Vol. XXV. He will not regard it as detracting from the gratitude which I here express for his thorough study of my monograph and its friendly review, if in what follows I attempt to supplement and correct that comparison.
A few of the signs I introduced there should be explained briefly here:
I deviate from usual practice in drawing a distinction between judgement und content of possible judgement. For me the relation in a hypothetical is uot one between judgements but between contents of possible judgement. 11ut ifl affirm that this relation holds, I then express a judgement.
( I) The content-stroke is horizontal, it is always prefixed to the expression of a content of possible judgement, serving to connect it to the judgement uml negation-strokes, and also to combine it with other contents of possible 111dgement by means of the conditional-stroke(? 2): e. g.
- ( 2 + 3 = 5).
(2) The judgement-stroke is placed vertically at the left hand end of the
content-stroke, it converts the content of possible judgement into a Judgement (? 2): e. g.
1--(2 +3 = 5).
(3) the negation-stroke is placed vertically under the content-stroke dividing it into two parts. The part to the right of the negation-stroke is the content-stroke of the original content, the part to the left that of its negation (~ 7): e. g.
. . . . . -(2 + 3 = 6).
(4) The conditional stroke connects two content-strokes by running from
the under side of the upper one to the left hand end of the lower. Like the negation-stroke it splits the upper stroke in two, of which the part to the right remains the content-stroke of the upper content, while the part to the left becomes the content-stroke of the whole, which means the negation of Ihe case that the upper content is false, and the lower true(? 5): e. g.
tx=7
X+ 3 = 10: means:thecasethatx+3= 10withoutx= 7doesnotoccur. Wecansay
in this case: ifx + 3 = 10, then x = 7. The roman letters such as the x here, confer generality on the content of the judgement by presenting it as true, whatever one may substitute for the roman letter (? 11, p. 21). If, for instance, we replace x by 2 we obtain
t,-2=7
L 2 + 3 = 10.
This is a true judgement, although the upper content, as well as the lower, is 1'11lsc. Here, the rendering 'if would jar with normal usage. Yet here too, the CIUIC is denied that 2 + 3 = 10 is true and 2 = 7 false.
If a negation-stroke stands to the left of the conditional stroke, as in
~2 = 7
2 + 3 = 5,
? ? 12 Boole's logical Calculus and the Concept-script
Above all, if we are not to go astray it is necessary that we should always bear in mind the purpose that governed Boole in his symbolic logic and the one that governed me in my Begriffsschrift.
If I understand him aright, Boole wanted to construct a technique for resolving logical problems systematically, similar to the technique of elimination and working out the unknown that algebra teaches. To this end, he represents judgements in the form of equations that he constructs out of letters and arithmetical signs such as +, 0 and 1. Logical laws then assume the form of algorithms, although these only coincide in part with those that hold in arithmetic for the same signs. In all this there is no concern about content whatsoever. In the main these means fulfil their purpose, at least as far as the range of problems that Boole has in mind are concerned. But one may think of logical problems lying outside this range. The use of arithmetical signs for logical purposes has the advantage that one is spared the necessity of learning a completely new algorithm. A large number of the transformations already familiar to our eyes still remain valid. ? Higher demands, which however Boole does not set himself, are naturally not met by this enterprise. Anyone demanding the closest possible agreement between the relations of the signs and the relations of the things themselves will always feel it to be back to front when logic, whose concern is correct thinking and which is also the foundation of arithmetic, borrows its signs
from arithmetic. To such a person it will seem more appropriate to develop for logic its own signs, derived from the nature of logic itself; we can then go on to use them throughout the other sciences wherever it is a question of preserving the formal validity of a chain of inference.
In contrast we may now set out the aim of my concept-script. Right from the start I had in mind the expression ofa content. What I am striving after is a lingua characterica in the first instance for mathematics, not a calculus restricted to pure logic. But the content is to be rendered more exactly than is done by verbal language. For that leaves a great deal to guesswork, even if only of the most elementary kind. There is only an imperfect correspondence between the way words are concatenated and the structure
the negation of the case that 2 + 3 = 5 without 2 = 7 is thereby converted into its affirmation we may render: 2 + 3 = 5 and it is not the case that 2 = 7 (? 7, p. 13). In
fTrr-2= l+l L2+3=5
-. - 2 = 1 + 1takes the place ofthe 2 = 7 ofthe preceding example. In rendering this, two denials come together and yield an affirmation:
2+3=5,and2=1+1(? 7. p. 12).
? I should mention that the deviations from arithmetic are for all that so fundamental ti}at solving logical equations is not at all like solving algebraic ones. And this greatly diminishes the value of the agreement in the algorithm.
? Boole's logical Calculus and the Concept-script 13
of the concepts. The words 'lifeboat' and 'deathbed' are similarly constructed though the logical relations of the constituents are different. So the latter isn't expressed at all, but is left to guesswork. Speech often only indicates by inessential marks or by imagery what a concept-script should spell out in full. At a more external level, the latter is distinguished from verbal language in being laid out for the eye rather than for the ear. Verbal script is of course also laid out for the eye, but since it simply reproduces verbal speech, it scarcely comes closer to a concept-script than speech: in fact it is at an even greater remove from it, since it consists in signs for signs, not of signs for the things themselves. A lingua characterica ought, as Leibniz says, peindre non pas les paroles, mais les pensees. The formula- languages of mathematics come much closer to this goal, indeed in part they nrrive at it. But that of geometry is still completely undeveloped and that of urithmetic itself is inadequate for its own domain; for at precisely the most important points, when new concepts are to be introduced, new foundations laid, it has to abandon the field to verbal language, since it only forms numbers out of numbers and can only express those judgements which treat 111' the equality of numbers which have been generated in different ways. But arithmetic in the broadest sense also forms concepts-and concepts of such richness and fineness in their internal structure that in perhaps no other science are they to be found combined with the same logical perfection. And there are other judgements which arithmetic makes, besides mere equations und inequalities. The reason for this inability to form concepts in a scientific manner lies? in the lack of one of the two components of which every highly developed language must consist. That is, we may distinguish the formal part which in verbal language comprises endings, prefixes, suffixes and auxiliary words, from the material part proper. The signs of arithmetic correspond to the latter. What we still lack is the logical cement that will bind these building stones firmly together. Up till now verbal language took over this role, and hence it was impossible to avoid using it in the proof itself, and not merely in parts that can be omitted without affecting the cogency of the patterns of inference, whose only purpose is to make it easier lo grasp connections. In contrast, Boole's symbolic logic only represents the formal part of language, and even that incompletely. The result is that
Boole's formula-language and the formula-language of arithmetic each solve only one part of the problem of a concept-script. What we have to do now, in order to produce a more adequate solution, is to supplement the signs of 111athematics with a formal element, since it would be inappropriate to leave the signs we already have unused. But on this score alone Boole's logic is nlready completely unsuited to the task of making this supplementation, since it employs the signs +, 0 and 1 in a sense which diverges from their nrithrnetical ones. It would lead to great inconvenience if the same signs were to occur in one formula with different meanings. This is not an
? In the case of the formula-language of arithmetic.
? 14 Boole's logical Calculus and the Concept-script
objection to Boole, since such an application of his formulae obviously lay completely outside his intentions. Thus, the problem arises of devising signs for logical relations that are suitable for incorporation into the formula- language of mathematics, and in this way of forming-at least for a certain domain-a complete concept-script. This is where my booklet comes in.
Despite all differences in our further aims, it is evident from what has been said already that the first problem for Boole and me was the same: the perspicuous representation of logical relations by means of written signs. This implies the possibility of comparing the two. If I now turn to this, it cannot be done in the sense of adjudicating between the two formula- languages, which is to be preferred. To raise such a question would mean referring back to their ultimate aims, which are more ambitious in my case than in Boole's. It would indeed be more than possible that each set of signs was the more appropriate for its own ends. Nevertheless, it seems to me worthwhile to work out the comparison in detail, since in that way many of the peculiarities of my concept-script are thrown into sharper focus.
To begin with, as far as the extent of these sign languages is concerned, one is entitled to expect that these would have to coincide, once everything had been sifted from my concept-script which lies beyond the confines of pure logic. Yet even then, not everything that I express can be translated into Boolean notation, whereas the converse transformation is possible. E. Schroder indeed thinks that my concept-script would only correspond to the second part of Boole's. This is refuted by the fact that I too can easily express such a judgement as 'Every square root of 4 is a fourth root of 16' by
~a4 =16 La2 = 4
although Boole would count it as one of the primary propositions. I represent all judgements in the first part of Boole in a similar way though in so doing I admittedly construe them quite differently. The real difference is that I avoid such a division into two parts, the first dedicated to the relation of concepts (primary propositions), the second to the relation of judgements (secondary propositions) and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it.
On the other hand, Boole had only an inadequate expression for particular judgements such as 'some 4th roots of 16 are square roots of 4'
~a2 =4
La4 = 16,
and for existential judgements such as 'there is at least one 4th root of 16'
IT&ra4 = 16,
So it transpires that even when we restrict ourselves to pure logic my
apparently no expression at all.
? Boole's logical Calculus and the Concept-script 15
concept-script commands a somewhat wider domain than Boole's formula- lunguage. This is a result of my having departed further from Aristotelian logic. For in Aristotle, as in Boole, the logically primitive activity is the formation of concepts by abstraction, and judgement and inference enter in through an immediate or indirect comparison of concepts via their extensions. ? The only difference is that Aristotle places in the foreground 1he case where the extension of one concept completely includes that of unother-i. e. that of subordination-whereas Boole reduces other cases to lhe case of equality of extensions. ? ? Here hypothetical and disjunctive 111dgements are for the moment left on one side. Boole now construes the hypothetical judgement 'If B, then A' as a case of subordination of con- cepts, by saying 'the class of time instants at which B is included in the class ,,rtime instants at which A'.
The full incongruity of the introduction here of the idea of time instants slunds out most clearly if you think of eternal truths such as those of mathematics. Schroder seems to want to avoid the artificiality this involves, siuce, in company with Hugh McColl, he explains expressions like A = 0, A t 8 = 1 etc. -whose sense, on the Boolean conception, is self-explanatory when taken in conjunction with the stipulations of the first part-all over U! (ain without referring back. But in this way the last weak link between the 1wo parts is also snapped, and the signs 0, 1, = receive yet a third meaning 111 addition to their Boolean and arithmetical ones. According to Boole 0 111cans the extension of a concept under which nothing falls, as for example lhc extension of the concept 'whole number whose square is 2'. By 1, Boole uuderstands the extension of his universe ofdiscourse. These meanings hold for the first as much as the second part. I f one now ruptures this connection, then strictly speaking 0 has no longer an independent meaning in the Nccond part; combined with the identity sign it means a denial expressed as a
Judgement, while '= 1' designates an affirmation, which I express by the judgement-stroke. Then, besides this, the identity sign still has an iudependent meaning in formulae like A = B. But that even Boole on his npproach does not establish an organic connection between the two parts cun be seen from the fact that he does not use the equations of the first part us constituents of equations of the second part, and, if you hold strictly to Iheir meanings, cannot so use them. For in the first part A = B is a judge- ment. whereas if it were made a constituent of an equation of the second purl, as say in
( A = B )C = D
? This is not meant to imply that concept formation takes up a great deal uf space in their presentations. Rather, that their logics are essentially doctrines of inference, in which the formation of concepts is presupposed AN something that has already been completed.
** The relation of identity has greater content and so is less extensive nru. l general than that of subordination.
? ? ? 16 Boole's logical Calculus and the Concept-script
A = B would mean the class of time instants at which the content of the judgement 'A = B' was to be affirmed. On the other hand, if, to avoid this, you do not resort to the expedient of the idea of a class of time instants, then you get in return the double sense of the 0, the 1 etc. in the same formula, which would then, to say the least, become unperspicuous.
As opposed to this, I start out from judgements and their contents, and not from concepts. The precisely defined hypothetical relation between contents of possible judgement has a similar significance for the foundation of my concept-script to that which identity of extensions has for Boolean logic. I only allow the formation of concepts to proceed from judgements. If, that is, you imagine the 2 in the content of possible judgement
24 = 16
to be replaceable by something else, by (-2) or by 3 say, which may be
indicated by putting an x in place of the 2: x4 = 16,
the content of possible judgement is thus split into a constant and a variable part. The former, regarded in its own right but holding a place open for the latter, gives the concept '4th root of 16'.
