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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:  `1` for the "antique 1" that Peirce defines as !1!_∞ = "something", and !1! for the "bold 1" that signifies the ordinary 2-identity relation.

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.

CP 3 gives [!1!] = `1`, which I cannot make any sense of.  CE 2 gives [!1!] = 1 , 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, [!1!] is the average number of things related by the identity relation !1! to one individual, and so it makes sense that [!1!] = 1&nbsp;:&nbsp;'''N''', where '''N''' is the set or the type of the natural numbers {0,&nbsp;1,&nbsp;2,&nbsp;&hellip;}.

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>

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: