Peirce's 1870 “Logic Of Relatives ” • Comment 11.2

Post   : Peirce's 1870 =93Logic Of Relatives=94 =95 Comment 11.2
http://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-%e=
2%80%a2-comment-11-2/
Posted : April 30, 2014 at 12:30 am
Author : Jon Awbrey

NOF Said =85
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D

Let's bring together the various things that Peirce has said
about the ''number of function'' up to this point in the paper.

NOF 1
=3D=3D=3D=3D=3D

<quote>

I propose to assign to all logical terms, numbers;  to an absolute term, th=
e number of individuals =

it denotes;  to a relative term, the average number of things so related to=
 one individual.  Thus in =

a universe of perfect men (''men''), the number of =93tooth of=94 would be =
32.  The number of a relative =

with two correlates would be the average number of things so related to a p=
air of individuals;  and =

so on for relatives of higher numbers of correlates.  I propose to denote t=
he number of a logical =

term by enclosing the term in square brackets, thus [t].

</quote>(Peirce, CP 3.65)

NOF 2
=3D=3D=3D=3D=3D

<quote>

But not only do the significations of  =3D  and  <  here adopted fulfill al=
l absolute requirements, =

but they have the supererogatory virtue of being very nearly the same as th=
e common significations. =

  Equality is, in fact, nothing but the identity of two numbers;  numbers t=
hat are equal are those =

which are predicable of the same collections, just as terms that are identi=
cal are those which are =

predicable of the same classes.  So, to write 5 < 7 is to say that 5 is par=
t of 7, just as to write =

f < m is to say that Frenchmen are part of men.  Indeed, if f < m, then the=
 number of Frenchmen is =

less than the number of men, and if v =3D p, then the number of Vice-Presid=
ents is equal to the number =

of Presidents of the Senate;  so that the numbers may always be substituted=
 for the terms =

themselves, in case no signs of operation occur in the equations or inequal=
ities.

</quote>(Peirce, CP 3.66)

NOF 3
=3D=3D=3D=3D=3D

<quote>

It is plain that both the regular non-invertible addition and the invertibl=
e addition satisfy the =

absolute conditions.  But the notation has other recommendations.  The conc=
eption of ''taking =

together'' involved in these processes is strongly analogous to that of sum=
mation, the sum of 2 and =

5, for example, being the number of a collection which consists of a collec=
tion of two and a =

collection of five.  Any logical equation or inequality in which no operati=
on but addition is =

involved may be converted into a numerical equation or inequality by substi=
tuting the numbers of the =

several terms for the terms themselves =97 provided all the terms summed ar=
e mutually exclusive.

Addition being taken in this sense, ''nothing'' is to be denoted by ''zero'=
', for then

x +, 0 =3D x

whatever is denoted by x;  and this is the definition of ''zero''.  This in=
terpretation is given by =

Boole, and is very neat, on account of the resemblance between the ordinary=
 conception of ''zero'' =

and that of nothing, and because we shall thus have

[0] =3D 0.

</quote>(Peirce, CP 3.67)

NOF 4
=3D=3D=3D=3D=3D

<quote>

The conception of multiplication we have adopted is that of the application=
 of one relation to =

another.  =85

Even ordinary numerical multiplication involves the same idea, for 2 =D7 3 =
is a pair of triplets, and =

3 =D7 2 is a triplet of pairs, where =93triplet of=94 and =93pair of=94 are=
 evidently relatives.

If we have an equation of the form:

xy =3D z

and there are just as many x=92s per y as there are, ''per'' things, things=
 of the universe, then we =

have also the arithmetical equation:

[x][y] =3D [z].

For instance, if our universe is perfect men, and there are as many teeth t=
o a Frenchman (perfect =

understood) as there are to any one of the universe, then:

[t][f] =3D [tf]

holds arithmetically.

So if men are just as apt to be black as things in general:

[m,][b] =3D [m,b]

where the difference between [m] and [m,] must not be overlooked.

It is to be observed that:

[_1_] =3D *1*.

Boole was the first to show this connection between logic and probabilities=
.  He was restricted, =

however, to absolute terms.  I do not remember having seen any extension of=
 probability to =

relatives, except the ordinary theory of ''expectation''.

Our logical multiplication, then, satisfies the essential conditions of mul=
tiplication, has a unity, =

has a conception similar to that of admitted multiplications, and contains =
numerical multiplication =

as a case under it.

</quote>(Peirce, CP 3.76)

Regards,

Jon

-- =


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