Changes

We have enough material on morphisms now to go back and cast a more studied eye on what Peirce is doing with that "number of" function, the one that we apply to a logical term <math>t\!</math> by writing it in square brackets, as <math>[t].\!</math>  It is convenient to have a prefix notation for this function, and since Peirce reserves <math>\mathit{n}\!</math> for <math>\operatorname{not},\!</math> let's use <math>v(t)\!</math> as a variant for <math>[t].\!</math>

We have enough material on morphisms now to go back and cast a more studied eye on what Peirce is doing with that "number of" function, the one that we apply to a logical term <math>t\!</math> by writing it in square brackets, as <math>[t].\!</math>  It is convenient to have a prefix notation for this function, and since Peirce reserves <math>\mathit{n}\!</math> for <math>\operatorname{not},\!</math> let's use <math>v(t)\!</math> as a variant for <math>[t].\!</math>

My plan will be nothing less plodding than to work through all of the principal statements that Peirce has made about the "number of" function up to our present stopping place in the paper, namely, those that I collected once at this location:

My plan will be nothing less plodding than to work through all of the principal statements that Peirce has made about the "number of" function up to our present stopping place in the paper, namely, those collected in [[Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives#Commentary_Note_11.2|Section 11.2]].

 

:* [[Directory:Jon_Awbrey/Papers/Peirce%27s_1870_Logic_Of_Relatives#Commentary_Note_11.2|Commentary Note 11.2]]

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We may formalize the role of the "number of" function by assigning it a name and a type as <math>v : S \to \mathbb{R},</math> where <math>S\!</math> is a suitable set of signs, a so-called ''syntactic domain'', that is ample enough to hold all of the terms whose numbers we might wish to evaluate in a given discussion, and where <math>\mathbb{R}</math> is the real number domain.

We may formalize the role of the "number of" function by assigning it a name and a type as <math>v : S \to \mathbb{R},</math> where <math>S\!</math> is a suitable set of signs, a so-called ''syntactic domain'', that is ample enough to hold all of the terms whose numbers we might wish to evaluate in a given discussion, and where <math>\mathbb{R}</math> is the real number domain.

Transcribing Peirce's example, we let <math>\mathrm{m} = \text{man}\!</math> and <math>\mathit{t} = \text{tooth of}\,\underline{~~~~}.</math> Then <math>v(\mathit{t}) = [\mathit{t}] = \tfrac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.\!</math> That is to say, in a universe of perfect human dentition, the number of the relative term <math>\text{tooth of}\,\underline{~~~~}.</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math>

Transcribing Peirce's example:

 

| <math>\mathrm{m} = \text{man}\!</math>

| and

| <math>\mathit{t} = \text{tooth of}\,\underline{~~~~}.</math>

| Then

| <math>v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}.\!</math>

 

That is, in a universe of perfect human dentition, the number of the relative term <math>\text{tooth of}\,\underline{~~~~}</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math>

The 2-adic relative term ''t'' determines a 2-adic relation ''T''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', where ''U'' and ''V'' are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively.

The 2-adic relative term ''t'' determines a 2-adic relation ''T''&nbsp;&sube;&nbsp;''U''&nbsp;&times;&nbsp;''V'', where ''U'' and ''V'' are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively.