We need only glance at the
formulae
to
(14)
?
(14)
?
Gottlob-Frege-Posthumous-Writings
Of the four possibilities
A andB
11 AandnotB Ill notAandB
IV not A and not B,
my T~ denies the third, Boole's identity sign the middle two; for Boole
himself the addition sign denies the first and the last, for Leibniz and Stanley Jevons only the last; and finally the multiplication sign affirms the first possibility, and so denies the other three. The first thing one notices is that Buole uses a greater number of signs. Indeed I too have an identity sign, but I use it between contents of possible judgement almost exclusively to
? ? 36
Boole's logical Calculus and the Concept-script
stipulate the sense of a new designation. Furthermore I now no longer regard it as a primitive sign but would define it by means of others. In that case there would be one sign of mine to three of Boole's. I follow the basic principle of introducing as few primitives as possible, not from any aversion to new signs-in that case, I would, like Boole, have endowed old ones with new meanings-but because it makes it difficult to survey the state of a science if the same thing is dressed up in different garbs. That seems to me the only reasonable ground for resisting new designations. This does not prevent the subsequent introduction of a simple sign for a very complicated
combination of signs that occurs frequently. But then you don't lay down as primitive the sentences which hold for such signs, you derive them from their meanings. The more primitive signs you introduce, the more axioms you need. But it is a basic principle of science to reduce the number of axioms to the fewest possible. Indeed the essence of explanation lies precisely in the fact that a wide, possibly unsurveyable, manifold is governed by one or a few sentences. The value of an explanation can be directly measured by this condensation and simplification: it is zero if the number of assumptions is as great as the number of facts to be explained. Now, to arrive at the fewest possible primitive signs, I must choose those with the simplest possible meanings, just as in chemistry the only hope of decreasing the number of elements is further analysis. But the simpler a content is, the less it says. For instance, my conditional stroke, which only denies the third of the four cases, says less than the Boolean identity sign which denies the second as well. The multiplication sign says even more, because it denies the fourth possibility as well, eliminating all choice. Only the addition sign, like my conditional stroke, excludes only one case, if you adopt Stanley Jevons' improvement,* and it only is an improvement because it diminishes the content of the sign. Of course, in some cases, the result is more cumbersome formulae. The exclusive 'a orb', which Boole can simply express by a + b, has to be written by Schroder in the form ab, + a, b. But this only concerns particular cases. In general it is always the sign with the simplest content which is the most widely applicable and leads to the clearest way of putting things. A content which is a component part of another, as that of my conditional stroke is of Boole's identity sign (if we simply ignore the idea of the class of time instants), as that of the inclusive 'or' is ofthe exclusive, will probably occur in several other contexts beside this one; indeed it will probably occur more often in other contexts. Even if two contents of possible judgement do in fact stand in the exclusive 'or' relation, in many inferences what matters is that one of the two contents holds; for others it's only essential that they don't both hold; finally there will be a few inferences, but probably the smallest number, in which both facts are needed. And this still doesn't take any account of the fact that for the most part contents of possible judgement only stand in one of the two relations
*See note above on p. 10 [**I.
? Boole's logical Calculus and the Concept-script 37
anyway. Boole's identity sign does the work of two of my conditional strokes: TA a n d T B Here, as above, it is true that in many cases use will
B A.
only be made ofT~ or only o f T ! , in a few use will be made of both, and
the fact is that the combination of precisely these two assumptions may not occur appreciably more frequently than any others. It might be held that we have to choose primitive signs with a simple content if only because we cannot express a content by means of signs with more content. But in fact it 1sn't impossible; it is only that a frequent construction will then be expressed hy a more complex formula than one that is relatively uncommon. E. g. Boole for his part has to use a more cumbersome expression for Schroder's 11 t b, the inclusive 'a orb'. But the exclusive 'or' perhaps only occurs once l11r every ten occurrences of the inclusive. So in chemistry everyone will regard it as more appropriate to represent the elements hydrogen and oxygen by single letters H and 0, and to form OH from them, than to designate the hydroxyl complex OH by a single letter, while using a L"ombination of signs to designate hydrogen as de-oxidized hydroxyl.
Now to obtain a sign joining two contents of possible judgement whose 111caning was as simple as possible, I had four choices open, all from this point of view equally justified: I could have adopted as the meaning of such n sign the denial of any one of the four cases mentioned above. But it Mulliced to choose one, since the four cases can be converted into one unother by replacing A and B by their denials. To use a chemical metaphor, Ihey are only allotropes of the same element. I chose the denial of the third ense, because of the ease with which it can be used in inference, and because IIN content has a close affinity with the important relation of ground and ~onsequent.
The fundamental principle of reducing the number of primitive laws as far IN possible wouldn't be fully satisfied without a demonstration that the few lcl\ are also sufficient. It is this consideration which determined the form of lhc second and third sections of my book. Here too it would be wrong to 1uppose that a direct comparison with Boole's work is possible. In his case lhcre is nothing remarkable in the attempt to manage everything with the fcwcst possible primitive laws. His only object is to find a brief and practical Wily to solve his problems. I sought as far as possible to translate into furrnulae everything that could also be expressed verbally as a rule of lnrcrence, so as not to make use of the same thing in different forms. ltccuuse modes of inference must be expressed verbally, I only used a single unc by giving as formulae what could otherwise have also been introduced ? ? modes of inference. This admittedly gave rise to a longwindedness which mliht appear pedantic. Not that it would not have been a simple matter for me tu give the transitions a briefer form, as I have done in the examples alvcn here and already indicated in the preface to my book. But it wasn't my Intention to provide a sample of how to carry out such derivations in a brief
? 38 Boole's logical Calculus and the Concept-script
and practical way: it was to show that I can manage throughout with my basic laws. Of course the fact that I managed with them in several cases could not render this more than probable. But it wasn't a matter of indifference which example I chose for my demonstration. So as not perhaps to overlook precisely those transformations which are of value in scientific use, I chose the step by step derivation of a sentence which, it seems to me, is indispensable to arithmetic, although it is one that commands little attention, being regarded as self-evident. The sentence in question is the following:
If a series is formed by first applying a many-one operation to an object (which need not belong to arithmetic), and then applying it successively to its own results, and if in this series two objects follow one and the same object, then the first follows the second in the series, or vice versa, or the two objects are identical.
I proved this sentence from the definitions of the concepts of following in a series, and of many-oneness by means of my primitive laws. In the process I derived the sentence that if in a series one member follows a second, and the second a third, then the first follows the third. Apart from a few formulae introduced to cater for Aristotelian modes of inference, I only assumed such as appeared necessary for the proof in question.
These were the principles which guided me in setting up my axioms and in the choice and derivation of other sentences. It was a matter of complete indifference to me whether a formula seemed interesting or to say nothing. That my sentences have enough content, in so far as you can talk of the content of sentences of pure logic at all, follows from the fact that they were adequate for the task. Sentences that were indispensable links in a chain of inference had to be assumed even if they contained superfluous conditions. We have a similar situation in Boolean computation. If there you multiply an equation through by a letter, you introduce into it something which is superfluous for its validity, and so reduce its content, just as when you add an unnecessary condition to a judgement. But there are times when such a diminution in content, far from being a loss, is a necessary point of transition in the development.
In accordance with my guiding principles, I also had to assume formulae which merely express the different ways in which you may alter the order of a number of conditions. Instead of giving a general rule that conditions may be ordered at random, I only introduced a much weaker axiom that two conditions may be interchanged, and then derived from this the permissi- bility of other transpositions. We have something similar in Boolean computation where it is a matter of changing the order of factors or summands. Schroder lays down the commutative and associative laws of multiplication and addition as axioms in his 'Operationskreise des Logik- kalkuls', but d,oesn't derive from it for the case of more than three factors or summands that the order and grouping is arbitrary. But such proofs would be necessary, if you wished to prove in Boole's formal logic, as far as
? Boole's logical Calculus and the Concept-script 39
this is possible, the sentences derived by me, with an equally complete chain of inference. This wouldn't be afforded by 'mental multiplying out'. You also need the sentence that you may interchange two sides of an equation, and that equals may always be substituted for equals. Schroder does not include these among his thirteen axioms, although there is no justification for leaving them out, even if you regard them as self-evident truths of logic. And so he really uses fifteen axioms. In my Begrif. fsschrift I laid down nine axioms, to which we must add the rules set out in words, than in essentials are determined by the modes of designation adopted. They are as follows:
(I) What follows the content-stroke must be a content of possible judgement (p. 2).
(2) Theruleofinference.
(3) Different gothic letters are to be chosen when one occurs within the
scope of another* (p. 21 ).
(4) A rule for replacing roman letters by gothic (p. 21).
(5) A rule for exporting a condition outside the scope of a gothic letter
(p. 21).
We may ignore here what I have to say about the use of Greek small
letters, since it lies outside the domain within which we may compare the mncept-script and Boole's formula-language. So with 14 primitive sentences I command a somewhat wider domain than does Schroder with 15. But I have since seen that the two basic laws for identity are completely dispensable, and that we may reduce the three basic laws for negation to two. After this simplification I need only 11 basic sentences. I see in this the success of my endeavour to have simple primitive constituents and proofs free from gaps. And so I replace the logical forms which in prose proliferate indefinitely by a few. This seems to me essential if our trains of thought are lo he relied on; for only what is finite and determinate can be taken in at once, and the fewer the number of primitive sentences, the more perfect a
mastery can we have of them.
Since, then, Boolean computations cannot be compared with the
llerivations I gave in the Begrif. fsschrift, it may not be out of place to introduce here an example where there can be a comparison. It would not be Nurprising and I could happily concede the point, if Boolean logic were hetter suited than my concept-script to solve the kind of problems it was Npecifically designed for, or for which it was specifically invented. But 11111yhe not even this is the case. Since the question involved is for me one of Nli~ht importance, I will confine myself to using the concept-script to solve a prohlem that has been treated by Boole,** then by Schroder,*** and then Wundt,**** while very briefly indicating how it differs from Boole's method.
? Strictly, this rule is implicit in the first.
? ? Op. cif. pp. 146 f.
? ? ? Der Operationskreis des Logikkalkuls, pp. 25 f. ? ? ? ? Logik l, p. 356.
? 40 Boole's logical Calculus and the Concept-script
In Schroder's formulation, the problem is as follows. Suppose we observe a class of phenomena (natural kinds or artefacts, e. g. substances) and arrive at the following general results:
(a) If the characteristics or properties A and C are simultaneously absent from any of the phenomena, the property E is found together with either the property B or the property D but not both.
(/J) Wherever A and D are found together in the absence of E, B and C are either both present or both absent.
(y) Wherever A is found together with BorE or both, either C or D is to be found but not both. And conversely wherever one of C, D is found without the other, then A is to be found together with BorE or both.
We now have to find:
(1) What in general can be inferred about B, C and D from the presence of A,
(2) Whether any relations whatever hold between the presence or absence of B, C and D independently of the presence or absence of the remaining properties,
(3) What follows for A, C and D from the presence of B, (4) What follows for A, C and D considered in themselves.
In the solution I use the corresponding Greek capitals so that e. g. A means the circumstance that the property A is to be found in the object under consideration.
I first translate the individual data.
(a) The denial ofA and Fhas a consequence the
affirmation of E (1).
The denial of A and r has as a consequence the affirmation of one of the two B or Ll (2);
but it is impossible to have B and Ll together with the denial of A and F(3).
({J) If A and Ll are both to be affirmed and E denied, B and rare either both to be affirmed or both denied; that is, if B is affirmed, Fis also to be affirmed (4);
(1)
(2)
(3)
(4)
? Boole's logical Calculus and the Concept-script 41 but ifB is denied, ris also to be denied (5).
(y) I begin by breaking this down.
(y1)
( y ) 2
(y3)
IfA and Bare affirmed, For Ll is to be affirmed (6) but not both (7).
! ~(6) ! ~(7) The same result holds if A and E are affirmed:
(8) and (9).
~~(8) ! ~(9)
If r is affirmed and Ll denied, A is to be affirmed (10). Since r is already a condition one of the tWO iS alSO tO be affirmed, for at leaSt F iS. I
' y1 and y4 are evidently faulty: no single interpretation will make them consistent l'lthcr with the problem or for that matter internally. They do not even as they stand 111akc clear sense. The most likely hypothesis to explain Frege's mistakes here is that while working at the problem, he sometimes approached it as it stands in the tl'xt, and sometimes following Schroder's version which contained a misprint of 'C' Im ? E' in the last clause of (y). E. g. the prose of y3 tallies with the Schroder misprinted version, but y4 is a hybrid ofthe two readings. The resulting formulae are
w""'? ;n the . en. e that (10) need? ? upplementing by (10)' ~~ond (12) ''"hodog by (12)'~-(11) ;, written he<e, but no"ub. eq~clJ~yF<ege ;n
tlu: form 12! which we have followed the German editors in emending as above.
there is a further slip in that Frege here writes his (12) with A for the Ll, where we huve similarly corrected it. Luckily, despite this morass of confusion it has no effect on the solution of the problem: e. g. Frege makes no further use of his defective (12) 1111d ns far as we can see the formulae omitted by Frege would contribute nothing to the resolution of this problem. He does indeed get the problem right, although the correctness of his solution is put in jeopardy by the fact that he does not of course Mhow that ( 10)' and ( 12)' yield nothing further of relevance to the questions asked. 1t does not seem worth exploring this minor and irritating matter further. (trans. )
(5)
(10)
? ? ? ? ? ? 42 Boole's logical Calculus and the Concept-script
(y4) If Fis denied and Ll affirmed, A is to be affirmed
(11) and one ofthe two BorE is also to be
affirmed. Here that can only be B (12). (11)
(12)
These are the data. The first question is already answered in part by (6) and (7). The remaining data yield no answers to this question, either because, as in (2) or (3) they do not contain the affirmation of A but its denial as a condition, or, as in (10), (11), (12), do not contain A as a condition at all, or because, as in (4), (5), (8), (9) they contain E as well as A, B, r, Ll. The question arises whether E could perhaps be eliminated from some of these last. This can be done ifEisa consequence in one judgement, as it is in (1), and a condition in another as it is in (9). We then write (9) unaltered as far as the condition E and replace this by the two conditions on which E depends in (1). This yields (13). This judgement
is satisfied no matter what the meanings of A, r
and Ll, first because--. r appears as a condition
o f - - . r itself, and secondly because two of the
conditions, A and-. -A, are contradictory. For
you obtain a truth by putting an arbitrary content
of possible judgement as the consequence of two
contradictory conditions. ? Thus (13) gives no information about the con- tents A, rand Lf. We may proceed with (1) and (8) as with (1) and (9). But we only have to glance at the formula to convince ourselves that we do not get any information by doing this either, since the resulting formula once again contains two contradictory conditions. Whereas in (8) and (9) the affirmation of E occurs as a condition in both judgements, it cannot be eliminated, but where its affirmation is a condition in one and its denial in the other, as in (8) and (4) it may. That is, you can transform a judgement with something denied as a condition by presenting the affirmation of the condition denied as a consequence and making the denial of the original consequence a condition. ? ? So (4) and (5) may be transformed into (14) and (15). Each of these two
judgements can now be com- bined with either of the judge- ments (8) and (9) in order to eliminate E.
We need only glance at the formulae to
(14)
? Begriffssc~rift Formula (36), p. 45.
? ? Begriffsschrift Formulae (33) and (34), pp. 44 f.
(13)
? yield no result of value. (7) and (16) give us the formula alongside; which gives us (17) when simplified as above. This tells us that when the property A is present, the properties C and D exclude one another. (6) then shows that one of the two prop- erties C and D is present when, besides A, B is also present.
r
Ll
A
r
Ll
A
Boole's logical Calculus and the Concept-script 43
convince ourselves that, as above, the results to be obtained from (8) and (14), (8) and (15), and (9) and (14) do not tell us anything which doesn't hold independently of
the contents. But from (9) and
Ll
A
r
B A Ll
(15) we obtain the formula
opposite. Here the antecedents
that occur twice over, A and A,
can be assimilated. The two r
can also be assimilated by first,
as before making B the con-
sequence and r the condition
and now dropping one of the
rs (16). This is the third answer
to the first question and the
judgements (6), (7) and (16)
contain everything that is to
be obtained from the data in answer to the first question. At most, it might still be possible to give the results in a different form by eliminating a letter-B, say. (6) and (16)
I move on to the second question. To decide whether any relations hold between B, rand L1 independently of A and E, we must eliminate the latter from the data and see whether the results contain anything other than logical platitudes. Instead of the data containing E we may straight off use the formula (16) we have al-
ready discovered. We have ac- cordingly to eliminate A from (2), (3), (6), (7), (10), (11) and ( 16). We begin by transforming (2) and (3) into (18) and (19).
We may now combine
(18) (19)
(6) with (10) or (7) with (10) or (16) with (10) or (6) with (11) or (7) with (11) or (16) with (ll) or
r
(16)
(17)
? 44
Boole's logical Calculus and the Concept-script
(6) with (17) or (7) with (17) or (16) with (17) or (6) with (18) or (7) with (18) or (16) with (18).
A glance at the formulae shows that these pairs yield results that hold independently of the contents. Hence the second question is to be answered negatively.
The answer to the third question is contained in (6), (7) and (19). We may infer from (7) and (19), that if, as well as B, the property D is present, one of the properties A and C must be present, but not both. (6) shows that if D is absent, either A is also absent or A and C are both present.
To answer the fourth question we need to eliminate B from (2), (3), (6), (7) and (16). We adopt (19) in place of (3). Of the possible combinations
(2) with (6), (2) with (7), (2) with (19),
(16) with (6), (16) with (7), (16) with (19)
only the last but one is of any
So the answer to the fourth
cannot all be present at once, and that as (10) and (11) show, the presence of one of the properties C and D without the other implies the presence of A.
This solution requires practically no theoretical preparation at all. I have accompanied the account with every rule required for solving the problem, and this may have created the impression of a greater length than the true one. So I will now collate the data and computation in brief and in a surveyable form:
Data
use, and has already been used to give us (17). question is that the properties A, C and D
~~(! )
n~cr~ ~r~ ~r~ ~. r~
~~(3) ~(6) ~(7)
~(4) ~(8) ~(9)
A
! r ! ~(12) ~ (11)
? ? ? ? ? Boole's logical Calculus and the Concept-script 45 Computation
4 [5] 3
~~
X
~~(19) (7): ? - ? -? -?
! ? (17)
Here the x between two formulae indicates the transition spelt out above. The sign ? - ? - ? - ? that stands between (5) and (16') and between (16') and (17) indicates a rule that abbreviates the other route followed above. It runs as follows:
If two judgements (e. g. (5) and (9)) have a common consequence (-,-I'), and one has a condition contradicting a condition of the other (E and -. -E), we may form a new judgement (16'), by attaching to the common consequence (-,-I'), the conditions of the two original judgements ((5) and (9)) minus the contradictory ones (E and -,-E), but in which conditions
common to both judgements are only written once (A and Ll). (16') isn't essentially different from (16).
The answer to the first question is contained in (6) and (17);
The answer to the third question is contained in (6), (7) and (19); The answer to the fourth question is contained in (10), (11) and (17);
The answer to the second question is in the negative.
Whereas the dominant procedure in Boole is the unification of different
judgements into a single expression, I analyse the data into simple judgements, which are then in part already answers to the questions. I then select from the simple judgements those lending themselves to the climinations needed, and so arrive at the rest of the answers. These will then only contain what we wanted to find out.
l believe that I have in this way shown that if in fact science were to require the solution of such problems, the concept-script can cope with them without any difficulty. But we see too that, in all this, its real power, which resides in the designation of generality, the concept of a function, in the possibility of putting more complicated expressions in the positions here tlccupied by simple letters, in no way comes into its own.
! ~
X
IL4)
~~
(9): ? - ? -? -?
! t6')
? 46 Boole's logical Calculus and the Concept-script
We may finally add a remark about the externals of my concept-script.
Schroder reproaches me for deviating from the normal way of writing from left to right in writing from above downwards. In fact I am in complete accord with usual practice; for in an arithmetical derivation too we put the individual equations in succession one beneath the other. But every equation is a content of possible judgement, or a judgement, as is every inequality, congruence etc. Now what I set beneath one another are also contents of possible judgement, or judgements. It is only when the simple contents of possible judgement are indicated by single letters that we have this appearance of something odd. As soon as they are spelt out, as they almost always are in actual use, each one is extended in a line from left to right, and they are severally written one beneath another. We thus make use of the advantage that a formal language, laid out in two dimensions on the written page, has over spoken language, which unfolds in the one dimension of time. Boole does not need to take up a line for each single content of possible judgement, because he has no thought of presenting them at greater length than by a single letter. This has the consequence that it would be extremely difficult to grasp what was going on, if one wished subsequently to introduce whole formulae in place of these single letters.
I believe in this essay I have shown:
(1) My concept-script has a more far-reaching aim than Boolean logic, in that it strives to make it possible to present a content when combined with arithmetical and geometrical signs.
(2) Disregarding content, within the domain of pure logic it also, thanks to the notation for generality, commands a somewhat wider domain than Boole's formula-language.
(3) It avoids the division in Boolean logic into two parts (primary and secondary propositions) by construing judgements as prior to concept formation.
(4) It is in a position to represent the formations of the concepts actually needed in science, in contrast to the relatively sterile multiplicative and additive combinations we find in Boole.
(5) It needs fewer primitive signs for logical relations and hence fewer primitive laws.
(6) It can be used to solve the sort of problems Boole tackles, and even do so with fewer preliminary rules for computation. This is the point to which I attach least importance, since such problems will seldom, if ever, occur in science.
? ? Boole's logical Formula-language and my Concept-script1 [1882]
Since this journaJ2 has already devoted some attention to the Boolean presentation of logical laws by means of equations, I hope that a comparison of it with another way of designating logical relations, proposed by me,* will not be unwelcome. First, however, I would like to stress that the aim of my concept-script is different from that of Boolean logic. I wanted to supplement the formula-language of mathematics with signs for logical relations so as to create a concept-script which would make it possible to Jispense with words in the course of a proof, and thus ensure the highest degree of rigour whilst at the same time making the proofs as brief as possible. For this purpose the signs I introduced had to be such as were suitable for combining of themselves with those ordinarily employed in mathematics. The Boolean signs (in part stemming from Leibniz) are completely unsuited to this, which is scarcely to be wondered at when you consider their purpose; they are merely meant to present the logical form with no regard whatever for the content. I think it is necessary to preface my remarks in this way to guard against the false impression that one could validly compare the two scripts in every respect.
It is only the second part of Boolean logic-the part dealing with secondary propositions-that I wish to investigate here, leaving open the possibility of my taking the comparison further on another occasion. By secondary propositions Boole understands hypothetical, disjunctive, and in general such judgements as state a relation between contents of possible judgement, as opposed to the primary propositions, in which concepts are set in relation to one another. I make a distinction between judgement and content of possible judgement, reserving the first word for cases where such n content is put forward as true.
If, therefore, it is a question of setting two contents of possible judgement
* Begriffsschrift, Halle a. S. 1879.
1 In all probability this essay contains the content of a manuscript returned to hegc by R. Avenarius with the letter of 20/4/1882. Avenarius there cites the title us 'Boole's logical formula-language' and rejects the manuscript for the Vlmeljahrsschrijlfiir wissenschaftliche Philosophie (ed. ).
2 Vierteljahrsschrij/ fiir wissenschqftliche Philosophle.
? 48 Boole's logical Formula-language and my Concept-script
A and B in relation, we have to hold before our minds the following
cases:
the equals sign
the addition sign
the multiplication sign
and subtraction sign
A=B,
A +B,
A ? B,
A-B.
A andB,
A andnotB, notA andB,
not A and not B.
These cases can for their part be either affirmed or denied. Now Boole has
We may here disregard logical division as less important.
The equals sign for Boole affirms the denial of the two middle cases, so
that the cases left open are 'A and B' and 'not A and not B'. In 'A = 1' and 'B = 0' the assertoric force of the identity sign stands out unalloyed. The first equation puts A forward as true, the second B as false. For Boole 'A + B' means the denial of the first and the last cases, leaving open the two middle ones. You can translate it as 'A orB', where 'or' is to be understood in the exclusive sense. Leibniz and some of Boole's followers, such as S. Jevons and E. Schroder, have kept the meaning of the inclusive 'or' for the + sign. In that case only the last of the four cases is denied by 'A + B'. 'A ? B' means the first case, 'A and B'. The denial of a content of possible judgementisexpressedbyBoolebymeansof'1- A',andbyothersinother ways. To this is added the above-mentioned 'A = 0' for the case where the denial is expressed as a judgement. Some people have a further sign for inequality, which also includes a denial. What strikes one in all this is the superfluity of signs. This, in its turn, entails a superfluity of primitive rules for computation. The reason for this lies doubtless in the desire to force on logic signs borrowed from an alien discipline, instead of taking one's departure from logic itself and its own requirements.
I have followed another path, by giving to each primitive sign as simple a meaning as possible. If, given two designations, one says all that is meant by the other, but not conversely, I call the meaning of the second simpler than that of the first, because it has less content. If we now apply this yardstick we see that the simplest relation of two contents of possible judgement results from denying one of the four cases
? Boole's logical Formula-language and my Concept-script 49
A andB,
A andnotB, notA andB,
not A and not B,
for the denial of two of these cases says more than that of one on its own, and the denial of three even more: it is tantamount to the affirmation of the fourth case. None of the Boolean signs meets the requirement that the meaning is the simplest possible. It is only met by the + sign in the sense adopted E. Schroder,? of the inclusive 'or'. The advantage of the latter makes itself immediately felt in the greater adaptability of the formulae when compared with Boole's. I don't understand how W. Schlotel** can find anything slovenly in this. This objection would only be justified if the meaning, once adopted, were not adhered to in what followed. Whether a special sign is adopted for the inclusive 'or' or not is merely a question of convenience. Now the exclusive 'A orB' contains two things:
1. That one of the two obtains 2. That they don't both obtain.
Since these two don't always go together, since rather, by the laws of probability, their combination is rarer than either on its own, it is more convenient for the individual cases which occur more frequently to have signs of their own that it is for their combination, which is relatively uncommon. And even when A and B do stand in the exclusive 'or' relation, almost invariably only one of the two will come into consideration in a given inference: either that A and B are not both false, or that they are not both true. For the rarer exclusive 'A or B', Boole has the simple designation
'A + B', for the commoner inclusive 'A or B', the complex expression 'A +- B(l -A)'. In the case of Schroder the converse is true: he renders the former by 'AB1 + A1B', the latter by 'A + B'. The suffix 1 here means a
denial, so that A
The Boolean 'A = B' contains three things:
1. that A doesn't obtain without B, 2. that B doesn't obtain without A; 3. the judgement that this is so.
I Iere too the combination is given the honour of a simple designation, while the constituent elements have to be content with complex expressions.
As the affirmation of the first case, Boole's logical product 'A? B' means the denial of the last three of our cases, and so is very rich in content. llowever this designation is more convenient than the others, since we ohtain a simple content by mere denial of such a product.
If one wishes to avoid the defect in the Boolean signs we have just
? Der Operationskreis des Logikkalkuls. Leipzig 1877. ? ? Vierteljahrsschriftfiir wissenschqftl. Phi/os. I, p. 456.
1
means the denial of A.
? 50 BooZe's logical Formula-language and my Concept-script
emphasized, one must introduce a special designation for the denial of one of the four cases set out above. It is enough for this purpose to select a single one of tliose cases for by using a negation sign one can obtain from each of the four cases any other. If, for instance, we put r for the denial of A those four cases then run:
not rand B,
not rand not B,
randB, rand notB;
that is the first case has assumed the form of the third, the second that of the fourth, and conversely the third that of the first, the fourth that of the second. If now it is possible to manage with one single sign which denies one of the four cases, we ought to do so, for the fewer primitive signs one introduces, the fewer primitive laws one needs, and the easier it will be to master the formulae.
Now, I have chosen the third case 'not A and B' as the one whose denial receives a special sign:
To form the denials of the remaining cases, I make use of the negation- stroke, a small vertical stroke attached beneath one of the horizontal strokes. So
means the denial of 'not not A and B', i. e. of 'A and B',
the denial of the case 'A and not B',
the denial of the case 'not A and not B'.
If the cases are to be affirmed, instead of denied, this is done by means of
a negation-stroke attached to the left hand end of the uppermost horizontal stroke. Accordingly
means 'not A and B',
? r
BooZe's logical Formula-language and my Concept-script 51
'A and B',
? A and not B', and finally
'not A and not B'. It is easy to see that you can translate
by 'A orB' in the inclusive sense, and
by 'neither A nor B'.
Nothing is yet asserted, no judgement is made, by all these designations;
only a new content of possible judgement is formed from given ones. Now in order to put a content forward as true, I make use of a small vertical stroke, the judgement stroke, as in
whereby the truth of the equation is asserted, whereas in - 3 2 = 9 no judgement has been made. Hence since the judgement stroke is lacking, we can even write down - 3 2 = 4 without saying anything untrue. If we include the negation-stroke, we can add the judgement stroke too without falling into error
means: 32 is not equal to 4 By
l2=4 t1+3=5
the fact that the case that 12 is not equal to 4 and that 1 + 3 = 5 does not obtain is asserted; and this rightly, since 1 + 3 simply is not equal to 5. Like- wise it is correct to put the judgement stroke in
22 = 4 t2+3=5
because the case that 22 is not equal to 4 and 2 + 3 = 5, does not obtain; for, of course 22 = 4. Similarly
? 52 Boole's logical Formula-language and my Concept-script
is right for two reasons: because (-2)2 = 4, and because -2 + 3 does not
equal 5. Whatever number you may put in the place of the 1 in
12=4 T1+3=s
the content is always correct. To express this general assertion, I use a roman letter:
Youcanalsorenderthis:ifx+3= 5,thenx2= 4. Andsowehaveherea hypothetical judgement. And the outstanding importance of this judgement has persuaded me to give the sign
precisely the meaning of the denial of the case 'not A and B'. Of course this alone doesn't yet give us a genuine hypothetical judgement: that arises only when A and B have in common an indefinite component which makes the situation described general.
I believe I have now adequately shown, that, as is proper, I divide the different tasks with which Boole burdens the one sign among several signs, without thereby increasing the total number of signs. The signs which I have also introduced elsewhere may be ignored in this context, since I am restricting myself to what corresponds to Boole's secondary propositions. As against his addition, subtraction, multiplication and equals signs, and his 0 and 1, we have:
1.
A andB
11 AandnotB Ill notAandB
IV not A and not B,
my T~ denies the third, Boole's identity sign the middle two; for Boole
himself the addition sign denies the first and the last, for Leibniz and Stanley Jevons only the last; and finally the multiplication sign affirms the first possibility, and so denies the other three. The first thing one notices is that Buole uses a greater number of signs. Indeed I too have an identity sign, but I use it between contents of possible judgement almost exclusively to
? ? 36
Boole's logical Calculus and the Concept-script
stipulate the sense of a new designation. Furthermore I now no longer regard it as a primitive sign but would define it by means of others. In that case there would be one sign of mine to three of Boole's. I follow the basic principle of introducing as few primitives as possible, not from any aversion to new signs-in that case, I would, like Boole, have endowed old ones with new meanings-but because it makes it difficult to survey the state of a science if the same thing is dressed up in different garbs. That seems to me the only reasonable ground for resisting new designations. This does not prevent the subsequent introduction of a simple sign for a very complicated
combination of signs that occurs frequently. But then you don't lay down as primitive the sentences which hold for such signs, you derive them from their meanings. The more primitive signs you introduce, the more axioms you need. But it is a basic principle of science to reduce the number of axioms to the fewest possible. Indeed the essence of explanation lies precisely in the fact that a wide, possibly unsurveyable, manifold is governed by one or a few sentences. The value of an explanation can be directly measured by this condensation and simplification: it is zero if the number of assumptions is as great as the number of facts to be explained. Now, to arrive at the fewest possible primitive signs, I must choose those with the simplest possible meanings, just as in chemistry the only hope of decreasing the number of elements is further analysis. But the simpler a content is, the less it says. For instance, my conditional stroke, which only denies the third of the four cases, says less than the Boolean identity sign which denies the second as well. The multiplication sign says even more, because it denies the fourth possibility as well, eliminating all choice. Only the addition sign, like my conditional stroke, excludes only one case, if you adopt Stanley Jevons' improvement,* and it only is an improvement because it diminishes the content of the sign. Of course, in some cases, the result is more cumbersome formulae. The exclusive 'a orb', which Boole can simply express by a + b, has to be written by Schroder in the form ab, + a, b. But this only concerns particular cases. In general it is always the sign with the simplest content which is the most widely applicable and leads to the clearest way of putting things. A content which is a component part of another, as that of my conditional stroke is of Boole's identity sign (if we simply ignore the idea of the class of time instants), as that of the inclusive 'or' is ofthe exclusive, will probably occur in several other contexts beside this one; indeed it will probably occur more often in other contexts. Even if two contents of possible judgement do in fact stand in the exclusive 'or' relation, in many inferences what matters is that one of the two contents holds; for others it's only essential that they don't both hold; finally there will be a few inferences, but probably the smallest number, in which both facts are needed. And this still doesn't take any account of the fact that for the most part contents of possible judgement only stand in one of the two relations
*See note above on p. 10 [**I.
? Boole's logical Calculus and the Concept-script 37
anyway. Boole's identity sign does the work of two of my conditional strokes: TA a n d T B Here, as above, it is true that in many cases use will
B A.
only be made ofT~ or only o f T ! , in a few use will be made of both, and
the fact is that the combination of precisely these two assumptions may not occur appreciably more frequently than any others. It might be held that we have to choose primitive signs with a simple content if only because we cannot express a content by means of signs with more content. But in fact it 1sn't impossible; it is only that a frequent construction will then be expressed hy a more complex formula than one that is relatively uncommon. E. g. Boole for his part has to use a more cumbersome expression for Schroder's 11 t b, the inclusive 'a orb'. But the exclusive 'or' perhaps only occurs once l11r every ten occurrences of the inclusive. So in chemistry everyone will regard it as more appropriate to represent the elements hydrogen and oxygen by single letters H and 0, and to form OH from them, than to designate the hydroxyl complex OH by a single letter, while using a L"ombination of signs to designate hydrogen as de-oxidized hydroxyl.
Now to obtain a sign joining two contents of possible judgement whose 111caning was as simple as possible, I had four choices open, all from this point of view equally justified: I could have adopted as the meaning of such n sign the denial of any one of the four cases mentioned above. But it Mulliced to choose one, since the four cases can be converted into one unother by replacing A and B by their denials. To use a chemical metaphor, Ihey are only allotropes of the same element. I chose the denial of the third ense, because of the ease with which it can be used in inference, and because IIN content has a close affinity with the important relation of ground and ~onsequent.
The fundamental principle of reducing the number of primitive laws as far IN possible wouldn't be fully satisfied without a demonstration that the few lcl\ are also sufficient. It is this consideration which determined the form of lhc second and third sections of my book. Here too it would be wrong to 1uppose that a direct comparison with Boole's work is possible. In his case lhcre is nothing remarkable in the attempt to manage everything with the fcwcst possible primitive laws. His only object is to find a brief and practical Wily to solve his problems. I sought as far as possible to translate into furrnulae everything that could also be expressed verbally as a rule of lnrcrence, so as not to make use of the same thing in different forms. ltccuuse modes of inference must be expressed verbally, I only used a single unc by giving as formulae what could otherwise have also been introduced ? ? modes of inference. This admittedly gave rise to a longwindedness which mliht appear pedantic. Not that it would not have been a simple matter for me tu give the transitions a briefer form, as I have done in the examples alvcn here and already indicated in the preface to my book. But it wasn't my Intention to provide a sample of how to carry out such derivations in a brief
? 38 Boole's logical Calculus and the Concept-script
and practical way: it was to show that I can manage throughout with my basic laws. Of course the fact that I managed with them in several cases could not render this more than probable. But it wasn't a matter of indifference which example I chose for my demonstration. So as not perhaps to overlook precisely those transformations which are of value in scientific use, I chose the step by step derivation of a sentence which, it seems to me, is indispensable to arithmetic, although it is one that commands little attention, being regarded as self-evident. The sentence in question is the following:
If a series is formed by first applying a many-one operation to an object (which need not belong to arithmetic), and then applying it successively to its own results, and if in this series two objects follow one and the same object, then the first follows the second in the series, or vice versa, or the two objects are identical.
I proved this sentence from the definitions of the concepts of following in a series, and of many-oneness by means of my primitive laws. In the process I derived the sentence that if in a series one member follows a second, and the second a third, then the first follows the third. Apart from a few formulae introduced to cater for Aristotelian modes of inference, I only assumed such as appeared necessary for the proof in question.
These were the principles which guided me in setting up my axioms and in the choice and derivation of other sentences. It was a matter of complete indifference to me whether a formula seemed interesting or to say nothing. That my sentences have enough content, in so far as you can talk of the content of sentences of pure logic at all, follows from the fact that they were adequate for the task. Sentences that were indispensable links in a chain of inference had to be assumed even if they contained superfluous conditions. We have a similar situation in Boolean computation. If there you multiply an equation through by a letter, you introduce into it something which is superfluous for its validity, and so reduce its content, just as when you add an unnecessary condition to a judgement. But there are times when such a diminution in content, far from being a loss, is a necessary point of transition in the development.
In accordance with my guiding principles, I also had to assume formulae which merely express the different ways in which you may alter the order of a number of conditions. Instead of giving a general rule that conditions may be ordered at random, I only introduced a much weaker axiom that two conditions may be interchanged, and then derived from this the permissi- bility of other transpositions. We have something similar in Boolean computation where it is a matter of changing the order of factors or summands. Schroder lays down the commutative and associative laws of multiplication and addition as axioms in his 'Operationskreise des Logik- kalkuls', but d,oesn't derive from it for the case of more than three factors or summands that the order and grouping is arbitrary. But such proofs would be necessary, if you wished to prove in Boole's formal logic, as far as
? Boole's logical Calculus and the Concept-script 39
this is possible, the sentences derived by me, with an equally complete chain of inference. This wouldn't be afforded by 'mental multiplying out'. You also need the sentence that you may interchange two sides of an equation, and that equals may always be substituted for equals. Schroder does not include these among his thirteen axioms, although there is no justification for leaving them out, even if you regard them as self-evident truths of logic. And so he really uses fifteen axioms. In my Begrif. fsschrift I laid down nine axioms, to which we must add the rules set out in words, than in essentials are determined by the modes of designation adopted. They are as follows:
(I) What follows the content-stroke must be a content of possible judgement (p. 2).
(2) Theruleofinference.
(3) Different gothic letters are to be chosen when one occurs within the
scope of another* (p. 21 ).
(4) A rule for replacing roman letters by gothic (p. 21).
(5) A rule for exporting a condition outside the scope of a gothic letter
(p. 21).
We may ignore here what I have to say about the use of Greek small
letters, since it lies outside the domain within which we may compare the mncept-script and Boole's formula-language. So with 14 primitive sentences I command a somewhat wider domain than does Schroder with 15. But I have since seen that the two basic laws for identity are completely dispensable, and that we may reduce the three basic laws for negation to two. After this simplification I need only 11 basic sentences. I see in this the success of my endeavour to have simple primitive constituents and proofs free from gaps. And so I replace the logical forms which in prose proliferate indefinitely by a few. This seems to me essential if our trains of thought are lo he relied on; for only what is finite and determinate can be taken in at once, and the fewer the number of primitive sentences, the more perfect a
mastery can we have of them.
Since, then, Boolean computations cannot be compared with the
llerivations I gave in the Begrif. fsschrift, it may not be out of place to introduce here an example where there can be a comparison. It would not be Nurprising and I could happily concede the point, if Boolean logic were hetter suited than my concept-script to solve the kind of problems it was Npecifically designed for, or for which it was specifically invented. But 11111yhe not even this is the case. Since the question involved is for me one of Nli~ht importance, I will confine myself to using the concept-script to solve a prohlem that has been treated by Boole,** then by Schroder,*** and then Wundt,**** while very briefly indicating how it differs from Boole's method.
? Strictly, this rule is implicit in the first.
? ? Op. cif. pp. 146 f.
? ? ? Der Operationskreis des Logikkalkuls, pp. 25 f. ? ? ? ? Logik l, p. 356.
? 40 Boole's logical Calculus and the Concept-script
In Schroder's formulation, the problem is as follows. Suppose we observe a class of phenomena (natural kinds or artefacts, e. g. substances) and arrive at the following general results:
(a) If the characteristics or properties A and C are simultaneously absent from any of the phenomena, the property E is found together with either the property B or the property D but not both.
(/J) Wherever A and D are found together in the absence of E, B and C are either both present or both absent.
(y) Wherever A is found together with BorE or both, either C or D is to be found but not both. And conversely wherever one of C, D is found without the other, then A is to be found together with BorE or both.
We now have to find:
(1) What in general can be inferred about B, C and D from the presence of A,
(2) Whether any relations whatever hold between the presence or absence of B, C and D independently of the presence or absence of the remaining properties,
(3) What follows for A, C and D from the presence of B, (4) What follows for A, C and D considered in themselves.
In the solution I use the corresponding Greek capitals so that e. g. A means the circumstance that the property A is to be found in the object under consideration.
I first translate the individual data.
(a) The denial ofA and Fhas a consequence the
affirmation of E (1).
The denial of A and r has as a consequence the affirmation of one of the two B or Ll (2);
but it is impossible to have B and Ll together with the denial of A and F(3).
({J) If A and Ll are both to be affirmed and E denied, B and rare either both to be affirmed or both denied; that is, if B is affirmed, Fis also to be affirmed (4);
(1)
(2)
(3)
(4)
? Boole's logical Calculus and the Concept-script 41 but ifB is denied, ris also to be denied (5).
(y) I begin by breaking this down.
(y1)
( y ) 2
(y3)
IfA and Bare affirmed, For Ll is to be affirmed (6) but not both (7).
! ~(6) ! ~(7) The same result holds if A and E are affirmed:
(8) and (9).
~~(8) ! ~(9)
If r is affirmed and Ll denied, A is to be affirmed (10). Since r is already a condition one of the tWO iS alSO tO be affirmed, for at leaSt F iS. I
' y1 and y4 are evidently faulty: no single interpretation will make them consistent l'lthcr with the problem or for that matter internally. They do not even as they stand 111akc clear sense. The most likely hypothesis to explain Frege's mistakes here is that while working at the problem, he sometimes approached it as it stands in the tl'xt, and sometimes following Schroder's version which contained a misprint of 'C' Im ? E' in the last clause of (y). E. g. the prose of y3 tallies with the Schroder misprinted version, but y4 is a hybrid ofthe two readings. The resulting formulae are
w""'? ;n the . en. e that (10) need? ? upplementing by (10)' ~~ond (12) ''"hodog by (12)'~-(11) ;, written he<e, but no"ub. eq~clJ~yF<ege ;n
tlu: form 12! which we have followed the German editors in emending as above.
there is a further slip in that Frege here writes his (12) with A for the Ll, where we huve similarly corrected it. Luckily, despite this morass of confusion it has no effect on the solution of the problem: e. g. Frege makes no further use of his defective (12) 1111d ns far as we can see the formulae omitted by Frege would contribute nothing to the resolution of this problem. He does indeed get the problem right, although the correctness of his solution is put in jeopardy by the fact that he does not of course Mhow that ( 10)' and ( 12)' yield nothing further of relevance to the questions asked. 1t does not seem worth exploring this minor and irritating matter further. (trans. )
(5)
(10)
? ? ? ? ? ? 42 Boole's logical Calculus and the Concept-script
(y4) If Fis denied and Ll affirmed, A is to be affirmed
(11) and one ofthe two BorE is also to be
affirmed. Here that can only be B (12). (11)
(12)
These are the data. The first question is already answered in part by (6) and (7). The remaining data yield no answers to this question, either because, as in (2) or (3) they do not contain the affirmation of A but its denial as a condition, or, as in (10), (11), (12), do not contain A as a condition at all, or because, as in (4), (5), (8), (9) they contain E as well as A, B, r, Ll. The question arises whether E could perhaps be eliminated from some of these last. This can be done ifEisa consequence in one judgement, as it is in (1), and a condition in another as it is in (9). We then write (9) unaltered as far as the condition E and replace this by the two conditions on which E depends in (1). This yields (13). This judgement
is satisfied no matter what the meanings of A, r
and Ll, first because--. r appears as a condition
o f - - . r itself, and secondly because two of the
conditions, A and-. -A, are contradictory. For
you obtain a truth by putting an arbitrary content
of possible judgement as the consequence of two
contradictory conditions. ? Thus (13) gives no information about the con- tents A, rand Lf. We may proceed with (1) and (8) as with (1) and (9). But we only have to glance at the formula to convince ourselves that we do not get any information by doing this either, since the resulting formula once again contains two contradictory conditions. Whereas in (8) and (9) the affirmation of E occurs as a condition in both judgements, it cannot be eliminated, but where its affirmation is a condition in one and its denial in the other, as in (8) and (4) it may. That is, you can transform a judgement with something denied as a condition by presenting the affirmation of the condition denied as a consequence and making the denial of the original consequence a condition. ? ? So (4) and (5) may be transformed into (14) and (15). Each of these two
judgements can now be com- bined with either of the judge- ments (8) and (9) in order to eliminate E.
We need only glance at the formulae to
(14)
? Begriffssc~rift Formula (36), p. 45.
? ? Begriffsschrift Formulae (33) and (34), pp. 44 f.
(13)
? yield no result of value. (7) and (16) give us the formula alongside; which gives us (17) when simplified as above. This tells us that when the property A is present, the properties C and D exclude one another. (6) then shows that one of the two prop- erties C and D is present when, besides A, B is also present.
r
Ll
A
r
Ll
A
Boole's logical Calculus and the Concept-script 43
convince ourselves that, as above, the results to be obtained from (8) and (14), (8) and (15), and (9) and (14) do not tell us anything which doesn't hold independently of
the contents. But from (9) and
Ll
A
r
B A Ll
(15) we obtain the formula
opposite. Here the antecedents
that occur twice over, A and A,
can be assimilated. The two r
can also be assimilated by first,
as before making B the con-
sequence and r the condition
and now dropping one of the
rs (16). This is the third answer
to the first question and the
judgements (6), (7) and (16)
contain everything that is to
be obtained from the data in answer to the first question. At most, it might still be possible to give the results in a different form by eliminating a letter-B, say. (6) and (16)
I move on to the second question. To decide whether any relations hold between B, rand L1 independently of A and E, we must eliminate the latter from the data and see whether the results contain anything other than logical platitudes. Instead of the data containing E we may straight off use the formula (16) we have al-
ready discovered. We have ac- cordingly to eliminate A from (2), (3), (6), (7), (10), (11) and ( 16). We begin by transforming (2) and (3) into (18) and (19).
We may now combine
(18) (19)
(6) with (10) or (7) with (10) or (16) with (10) or (6) with (11) or (7) with (11) or (16) with (ll) or
r
(16)
(17)
? 44
Boole's logical Calculus and the Concept-script
(6) with (17) or (7) with (17) or (16) with (17) or (6) with (18) or (7) with (18) or (16) with (18).
A glance at the formulae shows that these pairs yield results that hold independently of the contents. Hence the second question is to be answered negatively.
The answer to the third question is contained in (6), (7) and (19). We may infer from (7) and (19), that if, as well as B, the property D is present, one of the properties A and C must be present, but not both. (6) shows that if D is absent, either A is also absent or A and C are both present.
To answer the fourth question we need to eliminate B from (2), (3), (6), (7) and (16). We adopt (19) in place of (3). Of the possible combinations
(2) with (6), (2) with (7), (2) with (19),
(16) with (6), (16) with (7), (16) with (19)
only the last but one is of any
So the answer to the fourth
cannot all be present at once, and that as (10) and (11) show, the presence of one of the properties C and D without the other implies the presence of A.
This solution requires practically no theoretical preparation at all. I have accompanied the account with every rule required for solving the problem, and this may have created the impression of a greater length than the true one. So I will now collate the data and computation in brief and in a surveyable form:
Data
use, and has already been used to give us (17). question is that the properties A, C and D
~~(! )
n~cr~ ~r~ ~r~ ~. r~
~~(3) ~(6) ~(7)
~(4) ~(8) ~(9)
A
! r ! ~(12) ~ (11)
? ? ? ? ? Boole's logical Calculus and the Concept-script 45 Computation
4 [5] 3
~~
X
~~(19) (7): ? - ? -? -?
! ? (17)
Here the x between two formulae indicates the transition spelt out above. The sign ? - ? - ? - ? that stands between (5) and (16') and between (16') and (17) indicates a rule that abbreviates the other route followed above. It runs as follows:
If two judgements (e. g. (5) and (9)) have a common consequence (-,-I'), and one has a condition contradicting a condition of the other (E and -. -E), we may form a new judgement (16'), by attaching to the common consequence (-,-I'), the conditions of the two original judgements ((5) and (9)) minus the contradictory ones (E and -,-E), but in which conditions
common to both judgements are only written once (A and Ll). (16') isn't essentially different from (16).
The answer to the first question is contained in (6) and (17);
The answer to the third question is contained in (6), (7) and (19); The answer to the fourth question is contained in (10), (11) and (17);
The answer to the second question is in the negative.
Whereas the dominant procedure in Boole is the unification of different
judgements into a single expression, I analyse the data into simple judgements, which are then in part already answers to the questions. I then select from the simple judgements those lending themselves to the climinations needed, and so arrive at the rest of the answers. These will then only contain what we wanted to find out.
l believe that I have in this way shown that if in fact science were to require the solution of such problems, the concept-script can cope with them without any difficulty. But we see too that, in all this, its real power, which resides in the designation of generality, the concept of a function, in the possibility of putting more complicated expressions in the positions here tlccupied by simple letters, in no way comes into its own.
! ~
X
IL4)
~~
(9): ? - ? -? -?
! t6')
? 46 Boole's logical Calculus and the Concept-script
We may finally add a remark about the externals of my concept-script.
Schroder reproaches me for deviating from the normal way of writing from left to right in writing from above downwards. In fact I am in complete accord with usual practice; for in an arithmetical derivation too we put the individual equations in succession one beneath the other. But every equation is a content of possible judgement, or a judgement, as is every inequality, congruence etc. Now what I set beneath one another are also contents of possible judgement, or judgements. It is only when the simple contents of possible judgement are indicated by single letters that we have this appearance of something odd. As soon as they are spelt out, as they almost always are in actual use, each one is extended in a line from left to right, and they are severally written one beneath another. We thus make use of the advantage that a formal language, laid out in two dimensions on the written page, has over spoken language, which unfolds in the one dimension of time. Boole does not need to take up a line for each single content of possible judgement, because he has no thought of presenting them at greater length than by a single letter. This has the consequence that it would be extremely difficult to grasp what was going on, if one wished subsequently to introduce whole formulae in place of these single letters.
I believe in this essay I have shown:
(1) My concept-script has a more far-reaching aim than Boolean logic, in that it strives to make it possible to present a content when combined with arithmetical and geometrical signs.
(2) Disregarding content, within the domain of pure logic it also, thanks to the notation for generality, commands a somewhat wider domain than Boole's formula-language.
(3) It avoids the division in Boolean logic into two parts (primary and secondary propositions) by construing judgements as prior to concept formation.
(4) It is in a position to represent the formations of the concepts actually needed in science, in contrast to the relatively sterile multiplicative and additive combinations we find in Boole.
(5) It needs fewer primitive signs for logical relations and hence fewer primitive laws.
(6) It can be used to solve the sort of problems Boole tackles, and even do so with fewer preliminary rules for computation. This is the point to which I attach least importance, since such problems will seldom, if ever, occur in science.
? ? Boole's logical Formula-language and my Concept-script1 [1882]
Since this journaJ2 has already devoted some attention to the Boolean presentation of logical laws by means of equations, I hope that a comparison of it with another way of designating logical relations, proposed by me,* will not be unwelcome. First, however, I would like to stress that the aim of my concept-script is different from that of Boolean logic. I wanted to supplement the formula-language of mathematics with signs for logical relations so as to create a concept-script which would make it possible to Jispense with words in the course of a proof, and thus ensure the highest degree of rigour whilst at the same time making the proofs as brief as possible. For this purpose the signs I introduced had to be such as were suitable for combining of themselves with those ordinarily employed in mathematics. The Boolean signs (in part stemming from Leibniz) are completely unsuited to this, which is scarcely to be wondered at when you consider their purpose; they are merely meant to present the logical form with no regard whatever for the content. I think it is necessary to preface my remarks in this way to guard against the false impression that one could validly compare the two scripts in every respect.
It is only the second part of Boolean logic-the part dealing with secondary propositions-that I wish to investigate here, leaving open the possibility of my taking the comparison further on another occasion. By secondary propositions Boole understands hypothetical, disjunctive, and in general such judgements as state a relation between contents of possible judgement, as opposed to the primary propositions, in which concepts are set in relation to one another. I make a distinction between judgement and content of possible judgement, reserving the first word for cases where such n content is put forward as true.
If, therefore, it is a question of setting two contents of possible judgement
* Begriffsschrift, Halle a. S. 1879.
1 In all probability this essay contains the content of a manuscript returned to hegc by R. Avenarius with the letter of 20/4/1882. Avenarius there cites the title us 'Boole's logical formula-language' and rejects the manuscript for the Vlmeljahrsschrijlfiir wissenschaftliche Philosophie (ed. ).
2 Vierteljahrsschrij/ fiir wissenschqftliche Philosophle.
? 48 Boole's logical Formula-language and my Concept-script
A and B in relation, we have to hold before our minds the following
cases:
the equals sign
the addition sign
the multiplication sign
and subtraction sign
A=B,
A +B,
A ? B,
A-B.
A andB,
A andnotB, notA andB,
not A and not B.
These cases can for their part be either affirmed or denied. Now Boole has
We may here disregard logical division as less important.
The equals sign for Boole affirms the denial of the two middle cases, so
that the cases left open are 'A and B' and 'not A and not B'. In 'A = 1' and 'B = 0' the assertoric force of the identity sign stands out unalloyed. The first equation puts A forward as true, the second B as false. For Boole 'A + B' means the denial of the first and the last cases, leaving open the two middle ones. You can translate it as 'A orB', where 'or' is to be understood in the exclusive sense. Leibniz and some of Boole's followers, such as S. Jevons and E. Schroder, have kept the meaning of the inclusive 'or' for the + sign. In that case only the last of the four cases is denied by 'A + B'. 'A ? B' means the first case, 'A and B'. The denial of a content of possible judgementisexpressedbyBoolebymeansof'1- A',andbyothersinother ways. To this is added the above-mentioned 'A = 0' for the case where the denial is expressed as a judgement. Some people have a further sign for inequality, which also includes a denial. What strikes one in all this is the superfluity of signs. This, in its turn, entails a superfluity of primitive rules for computation. The reason for this lies doubtless in the desire to force on logic signs borrowed from an alien discipline, instead of taking one's departure from logic itself and its own requirements.
I have followed another path, by giving to each primitive sign as simple a meaning as possible. If, given two designations, one says all that is meant by the other, but not conversely, I call the meaning of the second simpler than that of the first, because it has less content. If we now apply this yardstick we see that the simplest relation of two contents of possible judgement results from denying one of the four cases
? Boole's logical Formula-language and my Concept-script 49
A andB,
A andnotB, notA andB,
not A and not B,
for the denial of two of these cases says more than that of one on its own, and the denial of three even more: it is tantamount to the affirmation of the fourth case. None of the Boolean signs meets the requirement that the meaning is the simplest possible. It is only met by the + sign in the sense adopted E. Schroder,? of the inclusive 'or'. The advantage of the latter makes itself immediately felt in the greater adaptability of the formulae when compared with Boole's. I don't understand how W. Schlotel** can find anything slovenly in this. This objection would only be justified if the meaning, once adopted, were not adhered to in what followed. Whether a special sign is adopted for the inclusive 'or' or not is merely a question of convenience. Now the exclusive 'A orB' contains two things:
1. That one of the two obtains 2. That they don't both obtain.
Since these two don't always go together, since rather, by the laws of probability, their combination is rarer than either on its own, it is more convenient for the individual cases which occur more frequently to have signs of their own that it is for their combination, which is relatively uncommon. And even when A and B do stand in the exclusive 'or' relation, almost invariably only one of the two will come into consideration in a given inference: either that A and B are not both false, or that they are not both true. For the rarer exclusive 'A or B', Boole has the simple designation
'A + B', for the commoner inclusive 'A or B', the complex expression 'A +- B(l -A)'. In the case of Schroder the converse is true: he renders the former by 'AB1 + A1B', the latter by 'A + B'. The suffix 1 here means a
denial, so that A
The Boolean 'A = B' contains three things:
1. that A doesn't obtain without B, 2. that B doesn't obtain without A; 3. the judgement that this is so.
I Iere too the combination is given the honour of a simple designation, while the constituent elements have to be content with complex expressions.
As the affirmation of the first case, Boole's logical product 'A? B' means the denial of the last three of our cases, and so is very rich in content. llowever this designation is more convenient than the others, since we ohtain a simple content by mere denial of such a product.
If one wishes to avoid the defect in the Boolean signs we have just
? Der Operationskreis des Logikkalkuls. Leipzig 1877. ? ? Vierteljahrsschriftfiir wissenschqftl. Phi/os. I, p. 456.
1
means the denial of A.
? 50 BooZe's logical Formula-language and my Concept-script
emphasized, one must introduce a special designation for the denial of one of the four cases set out above. It is enough for this purpose to select a single one of tliose cases for by using a negation sign one can obtain from each of the four cases any other. If, for instance, we put r for the denial of A those four cases then run:
not rand B,
not rand not B,
randB, rand notB;
that is the first case has assumed the form of the third, the second that of the fourth, and conversely the third that of the first, the fourth that of the second. If now it is possible to manage with one single sign which denies one of the four cases, we ought to do so, for the fewer primitive signs one introduces, the fewer primitive laws one needs, and the easier it will be to master the formulae.
Now, I have chosen the third case 'not A and B' as the one whose denial receives a special sign:
To form the denials of the remaining cases, I make use of the negation- stroke, a small vertical stroke attached beneath one of the horizontal strokes. So
means the denial of 'not not A and B', i. e. of 'A and B',
the denial of the case 'A and not B',
the denial of the case 'not A and not B'.
If the cases are to be affirmed, instead of denied, this is done by means of
a negation-stroke attached to the left hand end of the uppermost horizontal stroke. Accordingly
means 'not A and B',
? r
BooZe's logical Formula-language and my Concept-script 51
'A and B',
? A and not B', and finally
'not A and not B'. It is easy to see that you can translate
by 'A orB' in the inclusive sense, and
by 'neither A nor B'.
Nothing is yet asserted, no judgement is made, by all these designations;
only a new content of possible judgement is formed from given ones. Now in order to put a content forward as true, I make use of a small vertical stroke, the judgement stroke, as in
whereby the truth of the equation is asserted, whereas in - 3 2 = 9 no judgement has been made. Hence since the judgement stroke is lacking, we can even write down - 3 2 = 4 without saying anything untrue. If we include the negation-stroke, we can add the judgement stroke too without falling into error
means: 32 is not equal to 4 By
l2=4 t1+3=5
the fact that the case that 12 is not equal to 4 and that 1 + 3 = 5 does not obtain is asserted; and this rightly, since 1 + 3 simply is not equal to 5. Like- wise it is correct to put the judgement stroke in
22 = 4 t2+3=5
because the case that 22 is not equal to 4 and 2 + 3 = 5, does not obtain; for, of course 22 = 4. Similarly
? 52 Boole's logical Formula-language and my Concept-script
is right for two reasons: because (-2)2 = 4, and because -2 + 3 does not
equal 5. Whatever number you may put in the place of the 1 in
12=4 T1+3=s
the content is always correct. To express this general assertion, I use a roman letter:
Youcanalsorenderthis:ifx+3= 5,thenx2= 4. Andsowehaveherea hypothetical judgement. And the outstanding importance of this judgement has persuaded me to give the sign
precisely the meaning of the denial of the case 'not A and B'. Of course this alone doesn't yet give us a genuine hypothetical judgement: that arises only when A and B have in common an indefinite component which makes the situation described general.
I believe I have now adequately shown, that, as is proper, I divide the different tasks with which Boole burdens the one sign among several signs, without thereby increasing the total number of signs. The signs which I have also introduced elsewhere may be ignored in this context, since I am restricting myself to what corresponds to Boole's secondary propositions. As against his addition, subtraction, multiplication and equals signs, and his 0 and 1, we have:
1.
