Changes

<p>It is to be observed that:</p>

<p>It is to be observed that:</p>

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| align="center" | <math>[\mathit{1}] ~=~ \mathfrak{1}.</math>

| align="center" | <math>[\mathit{1}] ~=~ 1.</math>

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<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.</p>

<p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.</p>

<p>(Peirce, CP 3.76).</p>

<p>(Peirce, CP 3.76 and CE 2, 376).</p>

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There appears to be a problem with the printing of the text at this point.  Let us first recall the conventions that I am using in this transcription:  <math>\mathfrak{1}</math> for the "antique figure one" that Peirce defines as <math>\mathit{1}_\infty = \text{something},</math> and <math>\mathit{1}\!</math> for the italic 1 that signifies the ordinary 2-adic identity relation.

There are problems with the printing of the text at this point.  Let us first recall the conventions that I am using in this transcription:  <math>\mathit{1}\!</math> for the italic 1 that signifies the 2-adic identity relation and <math>\mathfrak{1}</math> for the "antique figure one" that Peirce defines as <math>\mathit{1}_\infty = \text{something}.</math>

CP&nbsp;3 gives <math>[\mathit{1}] = \mathfrak{1},</math> which I cannot make any sense of.  CE&nbsp;2 gives <math>[\mathit{1}] = 1,\!</math> which makes sense on the reading of "1" as denoting the natural number 1, and not as the absolute term "1" that denotes the universe of discourse.  On this reading, <math>[\mathit{1}]\!</math> is the average number of things related by the identity relation <math>\mathit{1}\!</math> to one individual, and so it makes sense that <math>[\mathit{1}] = 1 \in \mathbb{N},</math> where <math>\mathbb{N}</math> is the set of non-negative integers <math>\{ 0, 1, 2, \ldots \}.</math>

CP&nbsp;3 gives <math>[\mathit{1}] = \mathfrak{1},</math> which I cannot make any sense of.  CE&nbsp;2 gives the 1's in different styles of italics, but reading the equation as <math>[\mathit{1}] = 1,\!</math> makes the best sense if the "1" on the right hand side is read as the numeral "1" that denotes the natural number 1, and not as the absolute term "1" that denotes the universe of discourse.  Read this way, <math>[\mathit{1}]\!</math> is the average number of things related by the identity relation <math>\mathit{1}\!</math> to one individual, and so it makes sense that <math>[\mathit{1}] = 1 \in \mathbb{N},</math> where <math>\mathbb{N}</math> is the set of non-negative integers <math>\{ 0, 1, 2, \ldots \}.</math>

With respect to the 2-identity !1! in the syntactic domain ''S'' and the number 1 in the non-negative integers '''N'''&nbsp;&sub;&nbsp;'''R''', we have:

With respect to the 2-identity !1! in the syntactic domain ''S'' and the number 1 in the non-negative integers '''N'''&nbsp;&sub;&nbsp;'''R''', we have: