The
formulae
(5) and (11) are left to be derived by the reader, which is a simple problem.
Gottlob-Frege-Posthumous-Writings
[2b =4
(3) 4 is a positive whole number (including 0). That is, 4 belongs to the series beginning with 0, in which the immediate successor of any member is obtained by adding 1. *
(4) 12 is a multiple of 4; that is, 12 follows 0 in the arithmetical progression with difference 4. ** Two numbers with opposed signs are here not counted as multiples of one another, and 0 only counted as a multiple of itself.
(5) 12 is a common multiple of 4 and 6; that is, 12 is a multiple of 4, and*** 12 is a multiple of 6.
1~(0, +4 ~12,)
~(07 +6 = 1211)
(6) 4 is a common (aliquot) factor of 12 and 20.
y
1~ (0, +4 ~ 12,) p(07 +4 = 2011)
(7) The multiple of a multiple of a number is a multiple of that number.
T~(07+a= c11)
[5(0,+a~ b1)
~(Oy+b) = Cp)
(8) A and B are congruent modulo M. Here A, B and M need not be
whole numbers; nothing further than their addibility is presupposed. This
*Cf. Begriffss. :hrift ? 29, where in the gloss on formula (99) for;f(x10z6 ) one should re~d~/(x1,z6)?
. . Begrilfsschrift ? 26.
. . . See footnote above, p. lOfT.
? BooZe's logical Calculus and the Concept-script 23 way of speaking would be inconvenient for ordinary use, but it is only meant
to have application when it is necessary to go back to the concept. ~(A,+ M~B1)
t{ J ( B 7 + M = A p )
(9) 13 is prime. Here 1 is counted as a prime number, 0 not. More
explicitly: 13 is a positive non-zero whole number(~(07 +1 = 13p)),and"'
whatever positive number b may be which is greater than 1(p(17 + 1 = bp)) und different from 13 (--. - b = 13), 13 cannot be a multiple of b ( --,~(07 + b = 1 3 p ) ) -
y
{i(07+b= 13p)
y
( IO) A and B are positive non-zero whole numbers that are eo-prime. IIere 1is treated as prime to every number.
p(07 +b = Ap) p(07 +b=Bp) ; (1y +1=bp) p(07 -+- 1= Ap)
7i (1y
b = 13
+ 1 = bp) 5(07 + 1 = 13p)
. . . . . . . __p(0
( 11) A is a positive non-zero rational number; that is, there is"'"' at least
+ 1 =Bp)
one positive non-zero whole number which is a multiple ofA.
~;(o, +A~ n,) X(0,-+- 1 = np)
{J
"' See footnote above, p. lOfT.
"'"' See above p. 14 and Begr(ffsschrift ? 12.
7
? ? ? ? ? 24 Boole's logical Calculus and the Concept-script
(12) A is the least common multiple of B and r. More explicitly: every common multiple of B and r is greater than or equal to A, and* A is a common multiple of B and r.
~(Or+B =Ap)
~(Or+F = A11) A . :;;:: a
~(Or+B = a11)
~(Oy +F = a11)
In neither this nor the preceding examples is the concept of a product
presupposed.
(13) The real function <P(x) is continuous at x = A; that is, given any positive non-zero number n, there is** a positive non-zero g such that any number b lying between +g and -g satisfies the inequality - n. :;;::<P(A+b)- <P(A)::;;n
n o b -n. :;;:: <P(A + b ) _ <P(A) :? : n -g. S:b. S:g
g>O n>0
I have assumed here that the signs <, >. :;; mark the expressions they stand between as real numbers.
(14) The real function <P(x) of a real variable xis continuous throughout the interval from A to B.
- n. :;;::<P(c+b)-<P(c). :;;::n -g. S:b. S:g A. S:c+b. S:B
g>0
. . . . _______n >0 '------A . :;;:: c . :;;:: B
If in this case the formula seems longwinded by comparison with the verbal expression, you must always bear in mind that it gives the definition of a concept which the latter only names. Even so, a count of the number of individual signs needed for the two may well not turn out unfavourably for the formula.
* See footn'ote p. I Off.
**See above p. 14 and Begriffsschrift ? 12.
? Boole's logical Calculus and the Concept-script 25 ( 15) <P(x,y) is a real function of x and y continuous at x = A, y = B.
n n r b - n;:;;:;: (/)(A+ b,B +e) -<P(A,B):::;;; n m -g;:;;:;:b:::;;;g
-g::S:e;:;;:;:g g>0
n>O
(16) A is the limit of the </>-series beginning with B (Cf. BegrifJsschrift ~? IJ, 10, 26, 29).
A+n:;;::a:;;::A-n k <P (by, ap)
{3
k <P (By, bp) {3
1-----n >0
F. g. 1 is the limit approached by members of the series beginning with 0, 111 which the successor (y) of each member (x) is derived by the rule
1/. 1I 2/3X=y
11
l+n:2:a:2:1-n
~(~+~by=ap) ~(~+~OY= b p )
. . . _____n >0
This is the series: 0, l/3, 1/3 + 2/9, l/3 + 2/9 + 4/27, . . .
( 17) A is the limit approached by the value of the real function <P(x) as &he nrgument approaches B from above.
n b n n;;;;:::(/)(B+a)-A ;;;;::: - n b>a>O
b>O n>O
( IH) A is the limit approached by the value of the real function <P(x) as x &tndN through real values towards plus infinity.
n o b - n ;:;;:;: (/) (b) - A ;:;;:;: n
b>g g>O n>O
( IIJ) A is the least upper bound of the numbers falling under the concept K; that is, every number with the property X is less than or equal to A
? ? ? ? 26 . 8oole's logical Calculus and the Concept-script
( ~~ ~)b)1and* for every positive non-zero n there is** a number with the property X greater than A - n
" b A-n<b
Tfi;~)o
qA ;:;::;b
X (b)
Here the b in('L~(b) n < b)has nothing to do with the bin ( 1~(;)b)
so that you could replace the second b by a different gothic letter. Use is made here of the generalized concept of a function explained in ? ? 9 and 10 of the 'BegriffSschrift. According to that, you can render X(Lf) :Lf has the property X, or falls under the concept X.
X might e. g. be the property of being a multiple of Fless than B. Then
(1~:~+F~b,))
takes the place of X(b) and we have: A is the least upper-bound of multiples
ofX that are less than B.
* See note o'n p. lOfT. **See above p. 14.
A-n<b
~ (Oy + r = b p )
b<B
'----n > 0 --~. . . --A;:;::;b
~(O +r =bp)
b<B
(20) Given an arbitrary positive non-zero number (b), we may find a positive non-zero number (n) such that if it is greater than the absolute value of the real number c, and if r lies in the interval [A,B), the absolute value of the real function (})(r, c) is less than b.
b n c ' _ b<(/)(! ,c)<b A:;;;;;:! :;;;;;: B
-n<c<n
n>O b>O
? Boole's logical Calculus and the Concept-script 27
(21) Given an arbitrary positive non-zero number (b), we may find for every value of r within the interval [A,B] a positive non-zero number such that if the absolute value of c is smaller than it, the absolute value of the real
function cJ>(r, c) is less than b.
? b n c - b<tP(! ,c)<b -n<c<n
n>O b>O A:;;;;! :;;;; B
If I stress that the Boolean formula-language cannot match this, it is only in order to point out the more far-reaching goals of my concept-script. The formulae just given would be of slight value if particular signs had to be invented for each one of them. But this is so far from being the case that on the contrary nothing is invented in setting up a single one of them. A few new signs suffice to present a wide variety of mathematical relations which it has hitherto only been possible to express in words. This of itself justifies their introduction, since the formulae are much briefer and more perspicuous than the equivalent definitions of the concepts in words. Too great a horror of new signs, leading to the old ones being made to carry more meanings than they can bear, is far more damaging than an over- fertile delight in invention, since anything superfluous soon disappears of its own accord, leaving what is of value behind. But the usefulness of such formulae only fully emerges when they are used in working out inferences, nnd we can only fully appreciate their value in this regard with practice. A longer connected passage would really be demanded to give at least something approaching an idea of this. Nevertheless the following example I hnve chosen may tempt people to experiment with the concept-script. It is of little significance which topic I choose, since the inference is always of the Nnme sort and is always governed by the same few laws, whether one is working in the elementary or the advanced regions of the science. But in the ! utter case more would probably be needed by way of preparation.
If I often refer to my Begriffsschrift in what follows, I will nevertheless try ll! l far as possible to make myself understood without recourse to it. This of ~:ourse obliges me to accompany formulae that really ought to speak for themselves with continual prose glosses.
I wish to prove the theorem that the sum of two multiples of a number is in its turn a multiple of that number. Here, as above, I count a number as a multiple of itself; nought or a number with opposed sign do not count as multiples. The numbers whose multiples are to be considered are subject tu no conditions other than that the following addition theorems:
f-- (n+b)+a=n+(b+a) (1) (2)
? ? ? ? 28 Boole's logical Calculus and the Concept-script
hold for them. Not only are we not presupposing any multiplication theorem, we are not even to assume the concept of multiplication. Of the theorems of pure logic we principally require that introduced as (84) on p. 65 of the BegrijJsschrift, which we may first reproduce as it stands. We may express it in words as follows: if the property F is hereditary in the f-series, then if x has the property F and precedes y in the f-series, then y has the property F.
r~:x,y,)
~F (x) ! 5(F(a) ~ f(! 5, a)
How )-series and 'hereditary' are to be understood will become clear from their application. * I now regard it as superfluous to introduce the com- bination of signs
! 5( F (a) ) (~ f(! 5,a)
and will once more replace it by the original expression
b a F(a) (~~(~bt))
used to define it in ? 24 of the Begriflsschrift. Our formula now assumes the form:
n--,. ,- F(y) y
pf(xy,yp)
F (x)
F (a)
L f(b, a)
F (b) (3)
In addition we need the formula (4) which is introduced as (96) on p. 71 of the Begriffsschrift. It means: ify follows x in thef-series, then every result of applying the operation/toy follows x in thefseries:
(4)
*See also? ? 24 and 26 of the Begrilfsschrift.
? ? ? ? Boole's logical Calculus and the Concept-script 29
In the preface of my Begriffsschrift I already said that the restriction to a single rule of inference which I there laid down was to be dropped in later developments. This is achieved by converting what was expressed as a judgement in a formula into a rule of inference. I do this with formulae (52) and (53) of the Begriffsschrift, whose content I render by the rule: in any judgement you may replace one symbol by another, if you add as a wndition the equation between the two. We now make use of (3), taking
the formula~(07+a= (n +x)11)for the function F(x), and the formula
r + a = y for f(x,y). What we referred to above as 'the property F' is now the property of a number yielding a multiple of a when added to n; thef- scries is now an arithmetical progression with difference a. I substitute 0 for x. (3) then becomes (5):
IT------. . . ,-#(01 +a= (n +y)11) #(01 +a=Yp)
ba
} (07 + a = (n + 0)11) #(01 +a=(n+a)11)
b+a=a
#(01 +a = (n +b)11)
We must first rid this of the bottom-most condition
~b a i(07+a=(n+a)p)) b+a=a
} (01 +a = (n +b)11)
(
which states that the property of yielding a multiple of a when added to n is hereditary in our arithmetical progression; i. e. if one member of this series hns this property its successor has it too. As above, we substitute x + a =y for f(x,y), 0 for x, (n +b) for y and (n + m) for z in (4), giving us (6):
j(01 +a= (n+m)11) ~( n + b ) + a = n + m
#(0 +a=(n+b)) 7 11
(6)
We apply to this the rule established above by substituting (b + a) for m in 1he second line, and at the same time adding the condition b + a = m. This "ivcs us (7).
(5)
? ? ? 30 Boole's logical Calculus and the Concept-script
i(07 a= (n +m)p)
! (n + b) + a = n + (b + a)
b+a=m
~(07 +a= (n+b)p) (7)
From which together with (1) there follows (8)
tri(07 +a= (n +m)p) [b+a=m
i(07 +a= (n +b)p) letters band m* and obtain (9):
(8) Here we may now introduce gothic letters, b and a in place of the roman
~i (07 +a= (n +a)p) Lb+a=a
i(01 +a= (n +b)p)
the whole judgement, whereas the scope of the generality designated by m
(9) In this, the scope of the generality designated by b (or b) remains as before
(or a) does not include the condition~(01 +a= (n +b)p), which is
possible since it does not contain m. (9) asserts the inheritance already mentioned.
Hence from (5) and (9) we may infer (10): ~;(0, +a~(n +y)1)
[f(O, +a~y,)
i(01 +a= (n +O)p) (10)
We once more apply our rule to this by substituting n for n + 0 and adding the condition n = n + 0.
i(01+a= (n+Y)p) y
7J(01 +a =Yp) ~(07 +a= np)
n=n+O (11) ? Begriffsschrift ? 11, pp. 21 and 22.
? ? ? Boole's logical Calculus and the Concept-script 31 But we may immediately drop this condition again because of (2). This gives
us (12), the theorem to be proved:
(12)
Continuing in a similar way you may also easily derive the theorem that the multiple of the multiple of a number is a multiple of that number. For this, you only require the addition theorem t--n + 0 = n and formula (78) of the Begriffsschrift. Since nothing fundamentally new would emerge in the process, I will not carry out the derivation, but instead will repeat the preceding computation as it appears when no words are interpolated, and complete familiarity with the concept-script is assumed. The numbers on the right name the formulae, those on the left refer back to earlier ones. The different sorts of line drawn between the individual judgements are to indicate the mode of inference.
The formulae (5) and (11) are left to be derived by the reader, which is a simple problem. The formula (3) here represents a form of mathematical induction. It follows from ? ? 24 and 26 of my Begriffsschrift that this mode of inference is not, as one might suppose, one peculiar to mathematics, but rests on general laws of logic.
(6)
(7)
y
p(01 +a= (n +m)p)
4
~(n + b) + a = n + m
y
p(01 +a= (n +b)11)
l~(01 +a=(n+m)p)
t(n + b) +a= n + (b +a)
b+a=m
~(01 +a=(n+b)11) (1)[::]---------
lr~(01+a= (n+m)p) [b+a=m
~(01 +a= (n+b)p) (8)
? ? ? ? ? 32
Boole's logical Calculus and the Concept-script
~;(01 +a=(n+a)p) Lb+a=a
; (01 +a=(n+b)p) (5):---------
y
~(01 +a= (n +Y)p)
(9)
(10)
(12)
You may be inclined to regard such a derivation as longwinded in comparison with other proofs unless you consider the demands which this proof satisfies and which are to be made of those other proofs if there is to be any point in the comparison. These demands are as follows:
(1) One may not stop at theorems less simple than those used above. If e. g. someone wished to use multiplication theorems here, he would first have to prove them from our two addition theorems.
(2) One may not appeal to intuition as a means of proof;* for it is a law of scientific economy to use no inore devices than necessary.
(3) One must take care there are no gaps in the chain of inference. This would e. g. be violated even by the fact that only an example of a theorem was strictly speaking proved and its generalization left to the reader.
Precision and rigour are the prime aims of the concept-script; brevity will only be sought after if it can be achieved without jeopardizing those aims. I now return once more to the examples mentioned earlier, so as to point
out the sort of concept formation that is to be seen in those accounts. The fourth example gives us the concept of a multiple of 4, if we imagine the 12
in ~ (0 1 + 4 = 12p) as replaceable by something else; the concept of the
* Whereas if is permissible to use intuition as a helpful expedient in pinning down an idea.
~p(01 +a=Jp) y
p(01 +a= (n +O)p) (2):: ===============
? Boole's logical Calculus and the Concept-script
33
relation of a number of its multiple if we imagine the 4 as also replaceable; and the concept of a factor of 12 if we imagine the 4 alone as replaceable. The 8th example gives us the concept of the congruence of two numbers with respect to a modulus, the 13th that of the continuity of a function at a point etc. All these concepts have been developed in science and have proved their fruitfulness. For this reason what we may discover in them has a far higher claim on our attention than anything that our everyday trains of thought might offer. For fruitfulness is the acid test of concepts, and scientific workshops the true field of study for logic. Now it is worth noting in all this, that in practically none of these examples is there first cited the genus or class to which the things falling under the concept belong and then the characteristic mark of the concept, as when you define 'homo' as 'animal rationale'. Leibniz has already noted that here we may also conversely construe 'rationale' as genus and 'animal' as species. In fact, by this definition 'homo' is to be whatever is 'animal' as well as being 'rationale'. * If the circle A represents the extension of the concept 'animal' and B that of 'rationale', then the region common to the two circles cor-
responds to the extension of the concept
'homo'. And it is all one whether I think of
that as having been formed from the circle A
hy its intersection with B or vice versa. This
construction corresponds to logical multipli-
cation. Boole would express this, say, in the
form C =AB, where C means the extension of
the concept 'homo'. You may also form con- A ccpts by logical addition. We have an example
of this if we define the concept 'capital offence' as murder or the attempted murder of the Kaiser or of the ruler of one's own Land or of a German prince in his own Land. The area A signifies the extension of the concept 'murder', the area B that of the concept 'attempted murder of the Kaiser or of the ruler of one's own Land or of a German prince in his own Land'. Then the whole area of the two circles, whether they have a region in common or not, will represent the extension of the con- cept 'capital offence'. If we look at what we hnve in the diagrams, we notice that in both cases the boundary of the ~:uncept, whether it is one formed by logical multiplication or addition is
* Wundt in his Logik I, p. 224 does not concede this, but his own geo- metrical representation on p. 252 refutes him. One must always hold fast to the fact that a difference is only logically significant if it has an effect on possible inferences.
A B
? ? 34 Boole's logical Calculus and the Concept-script
made up of parts of the boundaries of the concepts already given. This holds for any concept formation that can be represented by the Boolean notation. This feature of the diagrams is naturally an expression of something inherent in the situation itself, but which is hard to express without recourse to imagery. In this sort of concept formation, one must, then, assume as given a system of concepts, or speaking metaphorically, a network of lines. These really already contain the new concepts: all one has to do is to use the lines that are already there to demarcate complete surface areas in a new way. It is the fact that attention is principally given to this sort of formation of new concepts from old ones, while other more fruitful ones are neglected which surely is responsible for the impression one easily gets in logic that for all our to-ing and fro-ing we never really leave the same spot. Obviously, the more finely the original network of lines is drawn, the greater the possible set of new concepts. We might now fancy we could obtain all possible concepts if we took as our system of given concepts that of the individual objects (or, more precisely, a system of concepts under each of which only one object falls). This is in fact the course adopted by R. Grassmann. He forms classes or concepts by logical addition. He would e. g. define 'continent' as 'Europe or Asia [or Africa] or America or Australia'. But it is surely a highly arbitrary procedure to form concepts merely by assembling individuals, and one devoid of significance for actual thinking unless the objects are held together by having characteristics in common. It is precisely these which constitute the essence of the concept. Indeed one can form concepts under which no object falls, where it might perhaps require lengthy investigation to discover that this was so. Moreover a concept, such as that of number, can apply to infinitely many individuals. Such a concept would never be attained by logical addition. Nor finally may we presuppose that the individuals are given in toto, since some, such as e. g. the numbers,* are only yielded by thought.
If we compare what we have here with the definitions contained in our examples, of the continuity of a function and of a limit, and again that of following a series which I gave in ? 26 of my Begriffsschrift, we see that there's no question there of using the boundary lines of concepts we already have to form the boundaries of the new ones. Rather, totally new boundary lines are drawn by such definitions-and these are the scientifically fruitful ones. Here too, we use old concepts to construct new ones, but in so doing we combine the old ones together in a variety of ways by means of the signs for generality, negation and the conditional.
I believe almost all errors made in inference to have their roots in the imperfection of the concepts. Boole presupposes logically perfect concepts as ready to hand, and hence the most difficult part of the task as having been
*That is to say, the number 3 is not to be regarded as a concept, since the question that falls under it is nonsense. Whereas tripleness-the property of being composed of three things-is a concept.
? Boole's logical Calculus and the Concept-script 35
already discharged; he can then draw his inferences from the given assumptions by a mechanical process of computation. Stanley Jevons has in fact invented a machine to do this. But if we have perfect concepts whose content we do not need to refer back to, we can easily guard ourselves from error, even without computation. This is why Boolean logic disappoints the hopes which, in the light of all that has been achieved by using symbolism in mathematics, we might entertain of it; and not because those achievements are linked to the concept of magnitude. That is a view which has surely only arisen as a result of an over-hasty generalization from past experience. Boolean formula-language only represents a part of our thinking; our thinking as a whole can never be coped with by a machine or replaced by purely mechanical activity. It is true that the syllogism can be cast in the rorm of a computation, albeit one which cannot be performed without thinking. Still the fact that it follows a few fixed and perspicuous forms gives it a high degree of certainty. But we can only derive any real benefit from doing this, if the content is not just indicated but is constructed out of its constituents by means of the same logical signs as are used in the computation. In that case, the computation must quickly bring to light any llaw in the concept formations. But neither does this form any part of Boole's original plan, nor can his formula-language be subsequently adapted ror this purpose. For even if its form made it better suited to reproduce a content than it is, the lack of a representation of generality corresponding to mine would make a true concept formation-one that didn't use already existing boundary lines-impossible. It was certainly also this defect which hindered Leibniz from proceeding further.
Now that I have spelled out the ways in which my concept-script goes hcyond Boolean logic, and the consequences this brings in its wake, I will continue with my comparison, confining my attention to the domain common to the two formal languages. In this I can ignore Boole's first part.
In my case contents of possible judgements A and B are connected by the conditional stroke as in L~? in Boole's by equations, addition and
multiplication. 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.
? ?
(3) 4 is a positive whole number (including 0). That is, 4 belongs to the series beginning with 0, in which the immediate successor of any member is obtained by adding 1. *
(4) 12 is a multiple of 4; that is, 12 follows 0 in the arithmetical progression with difference 4. ** Two numbers with opposed signs are here not counted as multiples of one another, and 0 only counted as a multiple of itself.
(5) 12 is a common multiple of 4 and 6; that is, 12 is a multiple of 4, and*** 12 is a multiple of 6.
1~(0, +4 ~12,)
~(07 +6 = 1211)
(6) 4 is a common (aliquot) factor of 12 and 20.
y
1~ (0, +4 ~ 12,) p(07 +4 = 2011)
(7) The multiple of a multiple of a number is a multiple of that number.
T~(07+a= c11)
[5(0,+a~ b1)
~(Oy+b) = Cp)
(8) A and B are congruent modulo M. Here A, B and M need not be
whole numbers; nothing further than their addibility is presupposed. This
*Cf. Begriffss. :hrift ? 29, where in the gloss on formula (99) for;f(x10z6 ) one should re~d~/(x1,z6)?
. . Begrilfsschrift ? 26.
. . . See footnote above, p. lOfT.
? BooZe's logical Calculus and the Concept-script 23 way of speaking would be inconvenient for ordinary use, but it is only meant
to have application when it is necessary to go back to the concept. ~(A,+ M~B1)
t{ J ( B 7 + M = A p )
(9) 13 is prime. Here 1 is counted as a prime number, 0 not. More
explicitly: 13 is a positive non-zero whole number(~(07 +1 = 13p)),and"'
whatever positive number b may be which is greater than 1(p(17 + 1 = bp)) und different from 13 (--. - b = 13), 13 cannot be a multiple of b ( --,~(07 + b = 1 3 p ) ) -
y
{i(07+b= 13p)
y
( IO) A and B are positive non-zero whole numbers that are eo-prime. IIere 1is treated as prime to every number.
p(07 +b = Ap) p(07 +b=Bp) ; (1y +1=bp) p(07 -+- 1= Ap)
7i (1y
b = 13
+ 1 = bp) 5(07 + 1 = 13p)
. . . . . . . __p(0
( 11) A is a positive non-zero rational number; that is, there is"'"' at least
+ 1 =Bp)
one positive non-zero whole number which is a multiple ofA.
~;(o, +A~ n,) X(0,-+- 1 = np)
{J
"' See footnote above, p. lOfT.
"'"' See above p. 14 and Begr(ffsschrift ? 12.
7
? ? ? ? ? 24 Boole's logical Calculus and the Concept-script
(12) A is the least common multiple of B and r. More explicitly: every common multiple of B and r is greater than or equal to A, and* A is a common multiple of B and r.
~(Or+B =Ap)
~(Or+F = A11) A . :;;:: a
~(Or+B = a11)
~(Oy +F = a11)
In neither this nor the preceding examples is the concept of a product
presupposed.
(13) The real function <P(x) is continuous at x = A; that is, given any positive non-zero number n, there is** a positive non-zero g such that any number b lying between +g and -g satisfies the inequality - n. :;;::<P(A+b)- <P(A)::;;n
n o b -n. :;;:: <P(A + b ) _ <P(A) :? : n -g. S:b. S:g
g>O n>0
I have assumed here that the signs <, >. :;; mark the expressions they stand between as real numbers.
(14) The real function <P(x) of a real variable xis continuous throughout the interval from A to B.
- n. :;;::<P(c+b)-<P(c). :;;::n -g. S:b. S:g A. S:c+b. S:B
g>0
. . . . _______n >0 '------A . :;;:: c . :;;:: B
If in this case the formula seems longwinded by comparison with the verbal expression, you must always bear in mind that it gives the definition of a concept which the latter only names. Even so, a count of the number of individual signs needed for the two may well not turn out unfavourably for the formula.
* See footn'ote p. I Off.
**See above p. 14 and Begriffsschrift ? 12.
? Boole's logical Calculus and the Concept-script 25 ( 15) <P(x,y) is a real function of x and y continuous at x = A, y = B.
n n r b - n;:;;:;: (/)(A+ b,B +e) -<P(A,B):::;;; n m -g;:;;:;:b:::;;;g
-g::S:e;:;;:;:g g>0
n>O
(16) A is the limit of the </>-series beginning with B (Cf. BegrifJsschrift ~? IJ, 10, 26, 29).
A+n:;;::a:;;::A-n k <P (by, ap)
{3
k <P (By, bp) {3
1-----n >0
F. g. 1 is the limit approached by members of the series beginning with 0, 111 which the successor (y) of each member (x) is derived by the rule
1/. 1I 2/3X=y
11
l+n:2:a:2:1-n
~(~+~by=ap) ~(~+~OY= b p )
. . . _____n >0
This is the series: 0, l/3, 1/3 + 2/9, l/3 + 2/9 + 4/27, . . .
( 17) A is the limit approached by the value of the real function <P(x) as &he nrgument approaches B from above.
n b n n;;;;:::(/)(B+a)-A ;;;;::: - n b>a>O
b>O n>O
( IH) A is the limit approached by the value of the real function <P(x) as x &tndN through real values towards plus infinity.
n o b - n ;:;;:;: (/) (b) - A ;:;;:;: n
b>g g>O n>O
( IIJ) A is the least upper bound of the numbers falling under the concept K; that is, every number with the property X is less than or equal to A
? ? ? ? 26 . 8oole's logical Calculus and the Concept-script
( ~~ ~)b)1and* for every positive non-zero n there is** a number with the property X greater than A - n
" b A-n<b
Tfi;~)o
qA ;:;::;b
X (b)
Here the b in('L~(b) n < b)has nothing to do with the bin ( 1~(;)b)
so that you could replace the second b by a different gothic letter. Use is made here of the generalized concept of a function explained in ? ? 9 and 10 of the 'BegriffSschrift. According to that, you can render X(Lf) :Lf has the property X, or falls under the concept X.
X might e. g. be the property of being a multiple of Fless than B. Then
(1~:~+F~b,))
takes the place of X(b) and we have: A is the least upper-bound of multiples
ofX that are less than B.
* See note o'n p. lOfT. **See above p. 14.
A-n<b
~ (Oy + r = b p )
b<B
'----n > 0 --~. . . --A;:;::;b
~(O +r =bp)
b<B
(20) Given an arbitrary positive non-zero number (b), we may find a positive non-zero number (n) such that if it is greater than the absolute value of the real number c, and if r lies in the interval [A,B), the absolute value of the real function (})(r, c) is less than b.
b n c ' _ b<(/)(! ,c)<b A:;;;;;:! :;;;;;: B
-n<c<n
n>O b>O
? Boole's logical Calculus and the Concept-script 27
(21) Given an arbitrary positive non-zero number (b), we may find for every value of r within the interval [A,B] a positive non-zero number such that if the absolute value of c is smaller than it, the absolute value of the real
function cJ>(r, c) is less than b.
? b n c - b<tP(! ,c)<b -n<c<n
n>O b>O A:;;;;! :;;;; B
If I stress that the Boolean formula-language cannot match this, it is only in order to point out the more far-reaching goals of my concept-script. The formulae just given would be of slight value if particular signs had to be invented for each one of them. But this is so far from being the case that on the contrary nothing is invented in setting up a single one of them. A few new signs suffice to present a wide variety of mathematical relations which it has hitherto only been possible to express in words. This of itself justifies their introduction, since the formulae are much briefer and more perspicuous than the equivalent definitions of the concepts in words. Too great a horror of new signs, leading to the old ones being made to carry more meanings than they can bear, is far more damaging than an over- fertile delight in invention, since anything superfluous soon disappears of its own accord, leaving what is of value behind. But the usefulness of such formulae only fully emerges when they are used in working out inferences, nnd we can only fully appreciate their value in this regard with practice. A longer connected passage would really be demanded to give at least something approaching an idea of this. Nevertheless the following example I hnve chosen may tempt people to experiment with the concept-script. It is of little significance which topic I choose, since the inference is always of the Nnme sort and is always governed by the same few laws, whether one is working in the elementary or the advanced regions of the science. But in the ! utter case more would probably be needed by way of preparation.
If I often refer to my Begriffsschrift in what follows, I will nevertheless try ll! l far as possible to make myself understood without recourse to it. This of ~:ourse obliges me to accompany formulae that really ought to speak for themselves with continual prose glosses.
I wish to prove the theorem that the sum of two multiples of a number is in its turn a multiple of that number. Here, as above, I count a number as a multiple of itself; nought or a number with opposed sign do not count as multiples. The numbers whose multiples are to be considered are subject tu no conditions other than that the following addition theorems:
f-- (n+b)+a=n+(b+a) (1) (2)
? ? ? ? 28 Boole's logical Calculus and the Concept-script
hold for them. Not only are we not presupposing any multiplication theorem, we are not even to assume the concept of multiplication. Of the theorems of pure logic we principally require that introduced as (84) on p. 65 of the BegrijJsschrift, which we may first reproduce as it stands. We may express it in words as follows: if the property F is hereditary in the f-series, then if x has the property F and precedes y in the f-series, then y has the property F.
r~:x,y,)
~F (x) ! 5(F(a) ~ f(! 5, a)
How )-series and 'hereditary' are to be understood will become clear from their application. * I now regard it as superfluous to introduce the com- bination of signs
! 5( F (a) ) (~ f(! 5,a)
and will once more replace it by the original expression
b a F(a) (~~(~bt))
used to define it in ? 24 of the Begriflsschrift. Our formula now assumes the form:
n--,. ,- F(y) y
pf(xy,yp)
F (x)
F (a)
L f(b, a)
F (b) (3)
In addition we need the formula (4) which is introduced as (96) on p. 71 of the Begriffsschrift. It means: ify follows x in thef-series, then every result of applying the operation/toy follows x in thefseries:
(4)
*See also? ? 24 and 26 of the Begrilfsschrift.
? ? ? ? Boole's logical Calculus and the Concept-script 29
In the preface of my Begriffsschrift I already said that the restriction to a single rule of inference which I there laid down was to be dropped in later developments. This is achieved by converting what was expressed as a judgement in a formula into a rule of inference. I do this with formulae (52) and (53) of the Begriffsschrift, whose content I render by the rule: in any judgement you may replace one symbol by another, if you add as a wndition the equation between the two. We now make use of (3), taking
the formula~(07+a= (n +x)11)for the function F(x), and the formula
r + a = y for f(x,y). What we referred to above as 'the property F' is now the property of a number yielding a multiple of a when added to n; thef- scries is now an arithmetical progression with difference a. I substitute 0 for x. (3) then becomes (5):
IT------. . . ,-#(01 +a= (n +y)11) #(01 +a=Yp)
ba
} (07 + a = (n + 0)11) #(01 +a=(n+a)11)
b+a=a
#(01 +a = (n +b)11)
We must first rid this of the bottom-most condition
~b a i(07+a=(n+a)p)) b+a=a
} (01 +a = (n +b)11)
(
which states that the property of yielding a multiple of a when added to n is hereditary in our arithmetical progression; i. e. if one member of this series hns this property its successor has it too. As above, we substitute x + a =y for f(x,y), 0 for x, (n +b) for y and (n + m) for z in (4), giving us (6):
j(01 +a= (n+m)11) ~( n + b ) + a = n + m
#(0 +a=(n+b)) 7 11
(6)
We apply to this the rule established above by substituting (b + a) for m in 1he second line, and at the same time adding the condition b + a = m. This "ivcs us (7).
(5)
? ? ? 30 Boole's logical Calculus and the Concept-script
i(07 a= (n +m)p)
! (n + b) + a = n + (b + a)
b+a=m
~(07 +a= (n+b)p) (7)
From which together with (1) there follows (8)
tri(07 +a= (n +m)p) [b+a=m
i(07 +a= (n +b)p) letters band m* and obtain (9):
(8) Here we may now introduce gothic letters, b and a in place of the roman
~i (07 +a= (n +a)p) Lb+a=a
i(01 +a= (n +b)p)
the whole judgement, whereas the scope of the generality designated by m
(9) In this, the scope of the generality designated by b (or b) remains as before
(or a) does not include the condition~(01 +a= (n +b)p), which is
possible since it does not contain m. (9) asserts the inheritance already mentioned.
Hence from (5) and (9) we may infer (10): ~;(0, +a~(n +y)1)
[f(O, +a~y,)
i(01 +a= (n +O)p) (10)
We once more apply our rule to this by substituting n for n + 0 and adding the condition n = n + 0.
i(01+a= (n+Y)p) y
7J(01 +a =Yp) ~(07 +a= np)
n=n+O (11) ? Begriffsschrift ? 11, pp. 21 and 22.
? ? ? Boole's logical Calculus and the Concept-script 31 But we may immediately drop this condition again because of (2). This gives
us (12), the theorem to be proved:
(12)
Continuing in a similar way you may also easily derive the theorem that the multiple of the multiple of a number is a multiple of that number. For this, you only require the addition theorem t--n + 0 = n and formula (78) of the Begriffsschrift. Since nothing fundamentally new would emerge in the process, I will not carry out the derivation, but instead will repeat the preceding computation as it appears when no words are interpolated, and complete familiarity with the concept-script is assumed. The numbers on the right name the formulae, those on the left refer back to earlier ones. The different sorts of line drawn between the individual judgements are to indicate the mode of inference.
The formulae (5) and (11) are left to be derived by the reader, which is a simple problem. The formula (3) here represents a form of mathematical induction. It follows from ? ? 24 and 26 of my Begriffsschrift that this mode of inference is not, as one might suppose, one peculiar to mathematics, but rests on general laws of logic.
(6)
(7)
y
p(01 +a= (n +m)p)
4
~(n + b) + a = n + m
y
p(01 +a= (n +b)11)
l~(01 +a=(n+m)p)
t(n + b) +a= n + (b +a)
b+a=m
~(01 +a=(n+b)11) (1)[::]---------
lr~(01+a= (n+m)p) [b+a=m
~(01 +a= (n+b)p) (8)
? ? ? ? ? 32
Boole's logical Calculus and the Concept-script
~;(01 +a=(n+a)p) Lb+a=a
; (01 +a=(n+b)p) (5):---------
y
~(01 +a= (n +Y)p)
(9)
(10)
(12)
You may be inclined to regard such a derivation as longwinded in comparison with other proofs unless you consider the demands which this proof satisfies and which are to be made of those other proofs if there is to be any point in the comparison. These demands are as follows:
(1) One may not stop at theorems less simple than those used above. If e. g. someone wished to use multiplication theorems here, he would first have to prove them from our two addition theorems.
(2) One may not appeal to intuition as a means of proof;* for it is a law of scientific economy to use no inore devices than necessary.
(3) One must take care there are no gaps in the chain of inference. This would e. g. be violated even by the fact that only an example of a theorem was strictly speaking proved and its generalization left to the reader.
Precision and rigour are the prime aims of the concept-script; brevity will only be sought after if it can be achieved without jeopardizing those aims. I now return once more to the examples mentioned earlier, so as to point
out the sort of concept formation that is to be seen in those accounts. The fourth example gives us the concept of a multiple of 4, if we imagine the 12
in ~ (0 1 + 4 = 12p) as replaceable by something else; the concept of the
* Whereas if is permissible to use intuition as a helpful expedient in pinning down an idea.
~p(01 +a=Jp) y
p(01 +a= (n +O)p) (2):: ===============
? Boole's logical Calculus and the Concept-script
33
relation of a number of its multiple if we imagine the 4 as also replaceable; and the concept of a factor of 12 if we imagine the 4 alone as replaceable. The 8th example gives us the concept of the congruence of two numbers with respect to a modulus, the 13th that of the continuity of a function at a point etc. All these concepts have been developed in science and have proved their fruitfulness. For this reason what we may discover in them has a far higher claim on our attention than anything that our everyday trains of thought might offer. For fruitfulness is the acid test of concepts, and scientific workshops the true field of study for logic. Now it is worth noting in all this, that in practically none of these examples is there first cited the genus or class to which the things falling under the concept belong and then the characteristic mark of the concept, as when you define 'homo' as 'animal rationale'. Leibniz has already noted that here we may also conversely construe 'rationale' as genus and 'animal' as species. In fact, by this definition 'homo' is to be whatever is 'animal' as well as being 'rationale'. * If the circle A represents the extension of the concept 'animal' and B that of 'rationale', then the region common to the two circles cor-
responds to the extension of the concept
'homo'. And it is all one whether I think of
that as having been formed from the circle A
hy its intersection with B or vice versa. This
construction corresponds to logical multipli-
cation. Boole would express this, say, in the
form C =AB, where C means the extension of
the concept 'homo'. You may also form con- A ccpts by logical addition. We have an example
of this if we define the concept 'capital offence' as murder or the attempted murder of the Kaiser or of the ruler of one's own Land or of a German prince in his own Land. The area A signifies the extension of the concept 'murder', the area B that of the concept 'attempted murder of the Kaiser or of the ruler of one's own Land or of a German prince in his own Land'. Then the whole area of the two circles, whether they have a region in common or not, will represent the extension of the con- cept 'capital offence'. If we look at what we hnve in the diagrams, we notice that in both cases the boundary of the ~:uncept, whether it is one formed by logical multiplication or addition is
* Wundt in his Logik I, p. 224 does not concede this, but his own geo- metrical representation on p. 252 refutes him. One must always hold fast to the fact that a difference is only logically significant if it has an effect on possible inferences.
A B
? ? 34 Boole's logical Calculus and the Concept-script
made up of parts of the boundaries of the concepts already given. This holds for any concept formation that can be represented by the Boolean notation. This feature of the diagrams is naturally an expression of something inherent in the situation itself, but which is hard to express without recourse to imagery. In this sort of concept formation, one must, then, assume as given a system of concepts, or speaking metaphorically, a network of lines. These really already contain the new concepts: all one has to do is to use the lines that are already there to demarcate complete surface areas in a new way. It is the fact that attention is principally given to this sort of formation of new concepts from old ones, while other more fruitful ones are neglected which surely is responsible for the impression one easily gets in logic that for all our to-ing and fro-ing we never really leave the same spot. Obviously, the more finely the original network of lines is drawn, the greater the possible set of new concepts. We might now fancy we could obtain all possible concepts if we took as our system of given concepts that of the individual objects (or, more precisely, a system of concepts under each of which only one object falls). This is in fact the course adopted by R. Grassmann. He forms classes or concepts by logical addition. He would e. g. define 'continent' as 'Europe or Asia [or Africa] or America or Australia'. But it is surely a highly arbitrary procedure to form concepts merely by assembling individuals, and one devoid of significance for actual thinking unless the objects are held together by having characteristics in common. It is precisely these which constitute the essence of the concept. Indeed one can form concepts under which no object falls, where it might perhaps require lengthy investigation to discover that this was so. Moreover a concept, such as that of number, can apply to infinitely many individuals. Such a concept would never be attained by logical addition. Nor finally may we presuppose that the individuals are given in toto, since some, such as e. g. the numbers,* are only yielded by thought.
If we compare what we have here with the definitions contained in our examples, of the continuity of a function and of a limit, and again that of following a series which I gave in ? 26 of my Begriffsschrift, we see that there's no question there of using the boundary lines of concepts we already have to form the boundaries of the new ones. Rather, totally new boundary lines are drawn by such definitions-and these are the scientifically fruitful ones. Here too, we use old concepts to construct new ones, but in so doing we combine the old ones together in a variety of ways by means of the signs for generality, negation and the conditional.
I believe almost all errors made in inference to have their roots in the imperfection of the concepts. Boole presupposes logically perfect concepts as ready to hand, and hence the most difficult part of the task as having been
*That is to say, the number 3 is not to be regarded as a concept, since the question that falls under it is nonsense. Whereas tripleness-the property of being composed of three things-is a concept.
? Boole's logical Calculus and the Concept-script 35
already discharged; he can then draw his inferences from the given assumptions by a mechanical process of computation. Stanley Jevons has in fact invented a machine to do this. But if we have perfect concepts whose content we do not need to refer back to, we can easily guard ourselves from error, even without computation. This is why Boolean logic disappoints the hopes which, in the light of all that has been achieved by using symbolism in mathematics, we might entertain of it; and not because those achievements are linked to the concept of magnitude. That is a view which has surely only arisen as a result of an over-hasty generalization from past experience. Boolean formula-language only represents a part of our thinking; our thinking as a whole can never be coped with by a machine or replaced by purely mechanical activity. It is true that the syllogism can be cast in the rorm of a computation, albeit one which cannot be performed without thinking. Still the fact that it follows a few fixed and perspicuous forms gives it a high degree of certainty. But we can only derive any real benefit from doing this, if the content is not just indicated but is constructed out of its constituents by means of the same logical signs as are used in the computation. In that case, the computation must quickly bring to light any llaw in the concept formations. But neither does this form any part of Boole's original plan, nor can his formula-language be subsequently adapted ror this purpose. For even if its form made it better suited to reproduce a content than it is, the lack of a representation of generality corresponding to mine would make a true concept formation-one that didn't use already existing boundary lines-impossible. It was certainly also this defect which hindered Leibniz from proceeding further.
Now that I have spelled out the ways in which my concept-script goes hcyond Boolean logic, and the consequences this brings in its wake, I will continue with my comparison, confining my attention to the domain common to the two formal languages. In this I can ignore Boole's first part.
In my case contents of possible judgements A and B are connected by the conditional stroke as in L~? in Boole's by equations, addition and
multiplication. 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.
? ?
