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==Selection 4==
==Selection 4==
<pre>
<blockquote>
<p>'''The Signs for Addition'''</p>
</pre>
<p>The sign of addition is taken by Boole so that</p>
<pre>
<p>''x'' + ''y''</p>
<p>denotes everything denoted by ''x'', and, ''besides'', everything denoted by ''y''.</p>
<p>Thus</p>
<p>''m'' + ''w''</p>
Other than that, I just want to red flag the beginnings
<p>denotes all men, and, besides, all women.</p>
of another one of those "failures to communicate" that
so dogged the disciplines in the 20th Century, namely,
<p>This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken ''besides'' the objects denoted by the other.</p>
the fact that Peirce seemed to have an inkling about
the problems that would be caused by using the plus
<p>For example,</p>
sign for inclusive disjunction, but, as it happens,
his advice was overridden by the usages in various
<p>''f'' + ''u''</p>
different communities, rendering the exchange of
information among engineering, mathematical, and
<p>means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are ''besides themselves''.</p>
philosophical specialties a minefield in place
of mindfield to this very day.
<p>For this reason alone, in a paper which is published in the Proceedings of the Academy for March 17, 1867, I preferred to take as the regular addition of logic a non-invertible process, such that</p>
<p>''m'' +, ''b''</p>
<p>stands for all men and black things, without any implication that the black things are to be taken besides the men; and the study of the logic of relatives has supplied me with other weighty reasons for the same determination.</p>
<p>Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in a tract called ''Pure Logic, or the Logic of Quality'' [1864], had anticipated me in substituting the same operation for Boole's addition, although he rejects Boole's operation entirely and writes the new one with a "+" sign while withholding from it the name of addition.</p>
<p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of 'taking together' involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.</p>
<p>Addition being taken in this sense, 'nothing' is to be denoted by 'zero', for then</p>
<p>''x'' +, 0 = ''x'',</p>
whatever is denoted by ''x''; and this is the definition of 'zero'. This interpretation 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</p>
<p>[0] = 0.</p>
(Peirce, CP 3.67).
</blockquote>
A wealth of issues arise here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing.
The two papers that precede this one in CP 3 are Peirce's papers of March and September 1867 in the 'Proceedings of the American Academy of Arts and Sciences', titled "On an Improvement in Boole's Calculus of Logic" and "Upon the Logic of Mathematics", respectively. Among other things, these two papers provide us with further clues about the motivating considerations that brought Peirce to introduce the "number of a term" function, signified here by square brackets. I have already quoted from the "Logic of Mathematics" paper in a related connection. Here are the links to those excerpts:
* [http://suo.ieee.org/ontology/msg04350.html]
* [http://suo.ieee.org/ontology/msg04351.html]
In setting up a correspondence between "letters" and "numbers", my sense is that Peirce is "nocking an arrow", or constructing some kind of structure-preserving map from a logical domain to a numerical domain, and this interpretation is here reinforced by the careful attention that he gives to the conditions under which precisely which aspects of structure are preserved, plus his telling recognition of the criterial fact that zeroes are preserved by the mapping. But here's the catch, the arrow is from the qualitative domain to the quantitative domain, which is just the opposite of what I tend to expect, since I think of quantitative measures as preserving more information than qualitative measures. To curtail the story, it is possible to sort this all out, but that is a story for another day.
Other than that, I just want to red flag the beginnings of another one of those "failures to communicate" that so dogged the disciplines in the 20th Century, namely, the fact that Peirce seemed to have an inkling about the problems that would be caused by using the plus sign for inclusive disjunction, but, as it happens, his advice was overridden by the usages in various different communities, rendering the exchange of information among engineering, mathematical, and philosophical specialties a minefield in place of mindfield to this very day.
==Selection 5==
==Selection 5==
