Even so, a count of the number of
individual
signs needed for the two may well not turn out unfavourably for the formula.
Gottlob-Frege-Posthumous-Writings
But that even Boole on his npproach does not establish an organic connection between the two parts cun be seen from the fact that he does not use the equations of the first part us constituents of equations of the second part, and, if you hold strictly to Iheir meanings, cannot so use them.
For in the first part A = B is a judge- ment.
whereas if it were made a constituent of an equation of the second purl, as say in
( A = B )C = D
? This is not meant to imply that concept formation takes up a great deal uf space in their presentations. Rather, that their logics are essentially doctrines of inference, in which the formation of concepts is presupposed AN something that has already been completed.
** The relation of identity has greater content and so is less extensive nru. l general than that of subordination.
? ? ? 16 Boole's logical Calculus and the Concept-script
A = B would mean the class of time instants at which the content of the judgement 'A = B' was to be affirmed. On the other hand, if, to avoid this, you do not resort to the expedient of the idea of a class of time instants, then you get in return the double sense of the 0, the 1 etc. in the same formula, which would then, to say the least, become unperspicuous.
As opposed to this, I start out from judgements and their contents, and not from concepts. The precisely defined hypothetical relation between contents of possible judgement has a similar significance for the foundation of my concept-script to that which identity of extensions has for Boolean logic. I only allow the formation of concepts to proceed from judgements. If, that is, you imagine the 2 in the content of possible judgement
24 = 16
to be replaceable by something else, by (-2) or by 3 say, which may be
indicated by putting an x in place of the 2: x4 = 16,
the content of possible judgement is thus split into a constant and a variable part. The former, regarded in its own right but holding a place open for the latter, gives the concept '4th root of 16'. ?
We may now express
by the sentences '2 is a fourth root of 16' or 'the individual 2 falls under the concept "4th root of 16"' or 'belongs to the class of 4th roots of 16'. But we
? The reader may be surprised I don't put this in the form
2 = {116.
The reason is that {116 may not in the same context mean now this, now
that, 4th root of 16. Otherwise you could form the inference 2 = y/i6
- 2 = ~16 2=-2.
By using ~ on one side of an identity sign, you have laid it down that \li6 is to mean a particular 4th root of 16, just as the letter a too must be given the same meaning throughout a given context. I f it is in fact stipulated that ~is to mean the positive real root then -2 = ~is false and 2 =
~ isnottoberead:
'2 is a 4th root of 16', but
'2 is the positive real root of 16'.
We can see from this that the root sign cannot properly be used to help express that an individual falls under the concept of a root.
? Boole's logical Calculus and the Concept-script 17
may also just as well say '4 is a logarithm of 16 to the base 2'. Here the 4 is being treated as replaceable and so we get the concept 'logarithm of 16 to the base 2':
2X = 16.
lhc x indicates here the place to be occupied by the sign for the individual lulling under the concept. We may now also regard the 16 in x4 = 16 as replaceable in its turn, which we may represent, say, by x4 = y. In this way we arrive at the concept of a relation, namely the relation of a number to its ? lth power. And so instead of putting a judgement together out of an utdividual as subject* and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of possible judgement. ** Of course, if the expression of the content of possible judgement is to be analysable in this way, it must already be 1tsclf articulated. We may infer from this that at least the properties and 1l'lations which are not further analysable must have their own simple dl'signations. But it doesn't follow from this that the ideas of these properties 1111d relations are formed apart from objects: on the contrary they arise . \lmultaneously with the first judgement in which they are ascribed to things. llcnce in the concept-script their designations never occur on their own, but always in combinations which express contents of possible judgement. I could compare this with the behaviour of the atom: we suppose an atom nl'vcr to be found on its own, but only combined with others, moving out of uuc combination only in order to enter immediately into another. *** A sign ll>r a property never appears without a thing to which it might belong being ut least indicated, a designation of a relation never without indication of the things which might stand in it.
In contrast with Boole, I now reduce his primary propositions to the ,\'t'condary ones. I construe the subordination of the concept 'square root of
* The cases where the subject is not an individual are completely different from these and are here left out of consideration.
? ? A great deal of tedious discussion about negative concepts such as 'not-triangle' will, as I hope, be rendered redundant by the conception of the relation of judgement and concept outlined here. In such a case one simply doesn't have anything complete, but only the predicate of a judgement which as yet lacks a subject. The difficulties arise when people treat such a fru~ment as something whole.
In this connection, I find it extraordinary that some linguists have recently viewed a 'Satzwort' (sentence-word), a word expressing a whole judgement, as the primitive form of speech and ascribe no independent existence to the roots, as mere abstractions. I note this from the KMtlngschen ge/ehrten Anzeigen 6 April 1881: A. H. Sayee, Introduction to the? Science ofLanguage 1880 by A. Fick.
? ? ? As I have since seen, Wundt makes a similar use of this image in his l. oglk.
? 18 BooZe's logical Calculus and the Concept-script
4' to the concept '4th root of 16' as meaning: if something is a square root of
4 it is a 4th root of 16:
trx4 = 16 Lx2 = 4.
I believe that in this way I have set up a simple and appropriate organic relation between Boole's two parts. Moreover, on this view we do justice to the distinction between concept and individual, which is completely obliterated in Boole. Taken strictly, his letters never mean individuals but always extensions of concepts. That is, we must distinguish between concept and thing, even when only one thing falls under a concept. The concept 'planet whose distance from the sun lies between that of Venus and that of Mars' is still something different from the individual object the Earth, even though it alone falls under the concept. Otherwise you couldn't form concepts with different contents whose extensions were all limited to this one thing, the Earth. In the case of a concept it is always possible to ask whether something, and if so what, falls under it, questions which are senseless in the case of an individual. We must likewise distinguish the case of one concept being subordinate to another from that of a thing falling under a concept although the same form of words is used for both. The examples given above
tx4 = 16 24
x = 4 and f--2 = 16
show the distinction in the concept-script. The generality in the judgement
'TX4 = 16 Lx2 = 4
'All square roots of 4 are 4th roots of 16' is expressed by means of the letter x, in that the judgement is put forward as holding no matter what one understands by x. I stipulated that roman letters used in the expression of judgements should always have this sense.
Let us now look at the case where the content of such a general affirmative judgement occurs as part of a compound judgement, say as the antecedent of an hypothetical judgement; e. g. :
If every square root of 4 is a 4th root of m, then m must be 16. The expression
does not correspond to the sentence, and is even false, which is why the judgement stroke has been left off the left-hand end of the uppermost horizontal stroke; for we may substitute numbers for x and m which falsify this content. Thus if we take m to be 17, then the consequent m= 16 would
? ? ? Boole's logical Calculus and the Concept-script 19 hecome 17 = 16 and so would be false. Of course that doesn't yet
necessarily make the whole
false; for if the antecedent
17=16 1(x4 = 17
x2 =4
x4 = 17 l x2 = 4
were also false, the whole would be true despite the falsity of the conse- lJUent. *
But we may take a value for x, 3 say, which satisfies the condition x' = 17
for
lx2 = 4 134 = 17
2 3=4
is true, since not only 34 = 17 but also 32 = 4 is false. Thus ifwe give mthe value 17 and x the value 3, the antecedent
is satisfied but the consequent
x'=m
1x2 = 4 m= 16
m= 16
x=m ~4
2 x=4
is false. Thus
is not true for all values of x and m, which is what would be asserted by prefacing the formula with the judgement-stroke. But the sentence
'If every square root of 4 is a 4th root of m, then m must be 16'
Nays something different. You could also express its content as follows:
'If, whatever you understand by x it holds that x4 = m must be true ifx2 = 4, then m= 16'.
We can see: the generality to be expressed by means of the x must not KOvern the whole
*Cf. Footnote above p. lOfT. and Begriffschrift ? 5, where the latter atdmittedly contains the mistake pointed out by Schroder in his review. This however had no effect on what followed.
? ? 20 Boole's logical Calculus and the Concept-script but must be restricted to
x4 =m Lx2 = 4
I designate this by supplying the content-stroke with a concavity in which I put a gothic letter which also replaces the x:
-6-r-a' = m La2 = 4.
I thus restrict the scope of the generality designated by the gothic letter to the content, into whose content stroke the concavity has been introduced (? 11 of the BegrifJsschrift). * So our judgement is given the following expression:
m= 16 a4 =m a2 = 4.
By means of this notation, I am now also able to express particular and existential judgements. I render the sentence 'Some 4th roots of 16 are square roots of 4' thus:
For
~a2 =4
La4 = 16.
n a2=4 -vra4 = 16
means the content of possible judgement
'If anything is a 4th root of 16, it is not a square root of 4', or 'No 4th roots of 16 are square roots of 4'.
We now designate this content as a false generalization by prefacing it with a negation-stroke, and present the result as an assertion by means of the judgement-stroke. Analogously
~a2 = 4means: 'There is at least one square root of 4'.
This is the negation of the generalization of the negation of the equation a2 = 4.
We can now also easily show the link between particular and existential judgements. For in
tr0,-ra2 = 4 La4 = 16
* In discussing my monograph, E. Schroder made the proposal that the designation of generality be introduced into Boolean logic by the use of gothic letters. However that is inadequate, since the scope over which the generality is supposed to extend is still left open. The drawback that a second negation sign is needed is connected with this.
? BooZe's logical Calculus and the Concept-script 21 we may insert two negation-strokes in immediate succession, which thus
nmcel each other out
IT&mra2 = 4 La4 = 16
nnd think of this as concatenated as indicated here: I~"T1Ta2 = 4
tust as you may analyse
La4 = 16
tr&ra2=4 into l~a2 =4.
Thus the only distinction between
IN lhat
IT(j;,(T a2 = 4
La4 = 16 and
'[a2 = 4 a4 = 16
t11kcs the place of a2 = 4. But according to ? 7 of the Begriffsschrift,* this mcnns that a is a square root of 4 and a 4th root of 16. So just as you can lrnnslate
~a2 =4 as
'There is at least one square root of 4', you may express
IT6-rrrra2 =4 or t-r&Tra2 =4 La4 = 16 La4 = 16
by Ihe sentence
'There is at least one number which is both a square root of 4 and a 4th ruul of 16'. But this is equivalent to the expression:
'At least one 4th root of 16 is a square root of 4'.
I have now shown that the difference in extent of the domains governed by 8oolean logic and by my concept-script, extraneous as it might at first llaht appear, is in fact as closely as possible bound up with their original uunatruction. A few examples may now serve to illustrate how the uonatruction of the concept-script enables it when combined with the signs orarithmetic to achieve the more far-reaching goals it set itself.
(I) There are at least two different square roots of 4. The sentence 'a- b follows from a2 = 4 and b2 = 4' is denied in its generality.
? Sec also above, footnote on p. IOff.
,
ba=b a2 = 4 b2 = 4
? ? ? 22 BooZe's logical Calculus and the Concept-script (2) there is at most one number whose double is 4
1f;a:! [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.
( A = B )C = D
? This is not meant to imply that concept formation takes up a great deal uf space in their presentations. Rather, that their logics are essentially doctrines of inference, in which the formation of concepts is presupposed AN something that has already been completed.
** The relation of identity has greater content and so is less extensive nru. l general than that of subordination.
? ? ? 16 Boole's logical Calculus and the Concept-script
A = B would mean the class of time instants at which the content of the judgement 'A = B' was to be affirmed. On the other hand, if, to avoid this, you do not resort to the expedient of the idea of a class of time instants, then you get in return the double sense of the 0, the 1 etc. in the same formula, which would then, to say the least, become unperspicuous.
As opposed to this, I start out from judgements and their contents, and not from concepts. The precisely defined hypothetical relation between contents of possible judgement has a similar significance for the foundation of my concept-script to that which identity of extensions has for Boolean logic. I only allow the formation of concepts to proceed from judgements. If, that is, you imagine the 2 in the content of possible judgement
24 = 16
to be replaceable by something else, by (-2) or by 3 say, which may be
indicated by putting an x in place of the 2: x4 = 16,
the content of possible judgement is thus split into a constant and a variable part. The former, regarded in its own right but holding a place open for the latter, gives the concept '4th root of 16'. ?
We may now express
by the sentences '2 is a fourth root of 16' or 'the individual 2 falls under the concept "4th root of 16"' or 'belongs to the class of 4th roots of 16'. But we
? The reader may be surprised I don't put this in the form
2 = {116.
The reason is that {116 may not in the same context mean now this, now
that, 4th root of 16. Otherwise you could form the inference 2 = y/i6
- 2 = ~16 2=-2.
By using ~ on one side of an identity sign, you have laid it down that \li6 is to mean a particular 4th root of 16, just as the letter a too must be given the same meaning throughout a given context. I f it is in fact stipulated that ~is to mean the positive real root then -2 = ~is false and 2 =
~ isnottoberead:
'2 is a 4th root of 16', but
'2 is the positive real root of 16'.
We can see from this that the root sign cannot properly be used to help express that an individual falls under the concept of a root.
? Boole's logical Calculus and the Concept-script 17
may also just as well say '4 is a logarithm of 16 to the base 2'. Here the 4 is being treated as replaceable and so we get the concept 'logarithm of 16 to the base 2':
2X = 16.
lhc x indicates here the place to be occupied by the sign for the individual lulling under the concept. We may now also regard the 16 in x4 = 16 as replaceable in its turn, which we may represent, say, by x4 = y. In this way we arrive at the concept of a relation, namely the relation of a number to its ? lth power. And so instead of putting a judgement together out of an utdividual as subject* and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of possible judgement. ** Of course, if the expression of the content of possible judgement is to be analysable in this way, it must already be 1tsclf articulated. We may infer from this that at least the properties and 1l'lations which are not further analysable must have their own simple dl'signations. But it doesn't follow from this that the ideas of these properties 1111d relations are formed apart from objects: on the contrary they arise . \lmultaneously with the first judgement in which they are ascribed to things. llcnce in the concept-script their designations never occur on their own, but always in combinations which express contents of possible judgement. I could compare this with the behaviour of the atom: we suppose an atom nl'vcr to be found on its own, but only combined with others, moving out of uuc combination only in order to enter immediately into another. *** A sign ll>r a property never appears without a thing to which it might belong being ut least indicated, a designation of a relation never without indication of the things which might stand in it.
In contrast with Boole, I now reduce his primary propositions to the ,\'t'condary ones. I construe the subordination of the concept 'square root of
* The cases where the subject is not an individual are completely different from these and are here left out of consideration.
? ? A great deal of tedious discussion about negative concepts such as 'not-triangle' will, as I hope, be rendered redundant by the conception of the relation of judgement and concept outlined here. In such a case one simply doesn't have anything complete, but only the predicate of a judgement which as yet lacks a subject. The difficulties arise when people treat such a fru~ment as something whole.
In this connection, I find it extraordinary that some linguists have recently viewed a 'Satzwort' (sentence-word), a word expressing a whole judgement, as the primitive form of speech and ascribe no independent existence to the roots, as mere abstractions. I note this from the KMtlngschen ge/ehrten Anzeigen 6 April 1881: A. H. Sayee, Introduction to the? Science ofLanguage 1880 by A. Fick.
? ? ? As I have since seen, Wundt makes a similar use of this image in his l. oglk.
? 18 BooZe's logical Calculus and the Concept-script
4' to the concept '4th root of 16' as meaning: if something is a square root of
4 it is a 4th root of 16:
trx4 = 16 Lx2 = 4.
I believe that in this way I have set up a simple and appropriate organic relation between Boole's two parts. Moreover, on this view we do justice to the distinction between concept and individual, which is completely obliterated in Boole. Taken strictly, his letters never mean individuals but always extensions of concepts. That is, we must distinguish between concept and thing, even when only one thing falls under a concept. The concept 'planet whose distance from the sun lies between that of Venus and that of Mars' is still something different from the individual object the Earth, even though it alone falls under the concept. Otherwise you couldn't form concepts with different contents whose extensions were all limited to this one thing, the Earth. In the case of a concept it is always possible to ask whether something, and if so what, falls under it, questions which are senseless in the case of an individual. We must likewise distinguish the case of one concept being subordinate to another from that of a thing falling under a concept although the same form of words is used for both. The examples given above
tx4 = 16 24
x = 4 and f--2 = 16
show the distinction in the concept-script. The generality in the judgement
'TX4 = 16 Lx2 = 4
'All square roots of 4 are 4th roots of 16' is expressed by means of the letter x, in that the judgement is put forward as holding no matter what one understands by x. I stipulated that roman letters used in the expression of judgements should always have this sense.
Let us now look at the case where the content of such a general affirmative judgement occurs as part of a compound judgement, say as the antecedent of an hypothetical judgement; e. g. :
If every square root of 4 is a 4th root of m, then m must be 16. The expression
does not correspond to the sentence, and is even false, which is why the judgement stroke has been left off the left-hand end of the uppermost horizontal stroke; for we may substitute numbers for x and m which falsify this content. Thus if we take m to be 17, then the consequent m= 16 would
? ? ? Boole's logical Calculus and the Concept-script 19 hecome 17 = 16 and so would be false. Of course that doesn't yet
necessarily make the whole
false; for if the antecedent
17=16 1(x4 = 17
x2 =4
x4 = 17 l x2 = 4
were also false, the whole would be true despite the falsity of the conse- lJUent. *
But we may take a value for x, 3 say, which satisfies the condition x' = 17
for
lx2 = 4 134 = 17
2 3=4
is true, since not only 34 = 17 but also 32 = 4 is false. Thus ifwe give mthe value 17 and x the value 3, the antecedent
is satisfied but the consequent
x'=m
1x2 = 4 m= 16
m= 16
x=m ~4
2 x=4
is false. Thus
is not true for all values of x and m, which is what would be asserted by prefacing the formula with the judgement-stroke. But the sentence
'If every square root of 4 is a 4th root of m, then m must be 16'
Nays something different. You could also express its content as follows:
'If, whatever you understand by x it holds that x4 = m must be true ifx2 = 4, then m= 16'.
We can see: the generality to be expressed by means of the x must not KOvern the whole
*Cf. Footnote above p. lOfT. and Begriffschrift ? 5, where the latter atdmittedly contains the mistake pointed out by Schroder in his review. This however had no effect on what followed.
? ? 20 Boole's logical Calculus and the Concept-script but must be restricted to
x4 =m Lx2 = 4
I designate this by supplying the content-stroke with a concavity in which I put a gothic letter which also replaces the x:
-6-r-a' = m La2 = 4.
I thus restrict the scope of the generality designated by the gothic letter to the content, into whose content stroke the concavity has been introduced (? 11 of the BegrifJsschrift). * So our judgement is given the following expression:
m= 16 a4 =m a2 = 4.
By means of this notation, I am now also able to express particular and existential judgements. I render the sentence 'Some 4th roots of 16 are square roots of 4' thus:
For
~a2 =4
La4 = 16.
n a2=4 -vra4 = 16
means the content of possible judgement
'If anything is a 4th root of 16, it is not a square root of 4', or 'No 4th roots of 16 are square roots of 4'.
We now designate this content as a false generalization by prefacing it with a negation-stroke, and present the result as an assertion by means of the judgement-stroke. Analogously
~a2 = 4means: 'There is at least one square root of 4'.
This is the negation of the generalization of the negation of the equation a2 = 4.
We can now also easily show the link between particular and existential judgements. For in
tr0,-ra2 = 4 La4 = 16
* In discussing my monograph, E. Schroder made the proposal that the designation of generality be introduced into Boolean logic by the use of gothic letters. However that is inadequate, since the scope over which the generality is supposed to extend is still left open. The drawback that a second negation sign is needed is connected with this.
? BooZe's logical Calculus and the Concept-script 21 we may insert two negation-strokes in immediate succession, which thus
nmcel each other out
IT&mra2 = 4 La4 = 16
nnd think of this as concatenated as indicated here: I~"T1Ta2 = 4
tust as you may analyse
La4 = 16
tr&ra2=4 into l~a2 =4.
Thus the only distinction between
IN lhat
IT(j;,(T a2 = 4
La4 = 16 and
'[a2 = 4 a4 = 16
t11kcs the place of a2 = 4. But according to ? 7 of the Begriffsschrift,* this mcnns that a is a square root of 4 and a 4th root of 16. So just as you can lrnnslate
~a2 =4 as
'There is at least one square root of 4', you may express
IT6-rrrra2 =4 or t-r&Tra2 =4 La4 = 16 La4 = 16
by Ihe sentence
'There is at least one number which is both a square root of 4 and a 4th ruul of 16'. But this is equivalent to the expression:
'At least one 4th root of 16 is a square root of 4'.
I have now shown that the difference in extent of the domains governed by 8oolean logic and by my concept-script, extraneous as it might at first llaht appear, is in fact as closely as possible bound up with their original uunatruction. A few examples may now serve to illustrate how the uonatruction of the concept-script enables it when combined with the signs orarithmetic to achieve the more far-reaching goals it set itself.
(I) There are at least two different square roots of 4. The sentence 'a- b follows from a2 = 4 and b2 = 4' is denied in its generality.
? Sec also above, footnote on p. IOff.
,
ba=b a2 = 4 b2 = 4
? ? ? 22 BooZe's logical Calculus and the Concept-script (2) there is at most one number whose double is 4
1f;a:! [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.
