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==Preliminaries==

==Preliminaries==

Peirce's text employs a number of different typefaces to denote different types of logical entities.  The following Tables indicate the LaTeX typefaces that we will use for Peirce's stock examples.

Peirce's text employs a number of different typefaces to denote different types of logical entities.  The following Tables indicate the LaTeX typefaces that we will use for Peirce's stock examples.

==Selection 1==

==Selection 1==

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

==Selection 2==

==Selection 2==

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

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<p>I propose to use the term "universe" to denote that class of individuals ''about'' which alone the whole discourse is understood to run.  The universe, therefore, in this sense, as in Mr.&nbsp;De&nbsp;Morgan's, is different on different occasions.  In this sense, moreover, discourse may run upon something which is not a subjective part of the universe;  for instance, upon the qualities or collections of the individuals it contains.</p>

<p>I propose to use the term "universe" to denote that class of individuals ''about'' which alone the whole discourse is understood to run.  The universe, therefore, in this sense, as in Mr.&nbsp;De&nbsp;Morgan's, is different on different occasions.  In this sense, moreover, discourse may run upon something which is not a subjective part of the universe;  for instance, upon the qualities or collections of the individuals it contains.</p>

Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse that they usually escape explicit mention, and others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly introduced in passing that it's all too easy to overlook their consequential significance, that I can do no better here than to highlight these ideas in other words, whose main advantage is to be a little more jarring to the mind's sensibilities.

Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse that they usually escape explicit mention, and others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly introduced in passing that it's all too easy to overlook their consequential significance, that I can do no better here than to highlight these ideas in other words, whose main advantage is to be a little more jarring to the mind's sensibilities.

:* This mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what "quantification theory" is all about, and definitely more to the point than the mere "innovation" of using distinctive symbols for the so-called "quantifiers".  We will speak of this more later on.

# This mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what "quantification theory" is all about, and definitely more to the point than the mere "innovation" of using distinctive symbols for the so-called "quantifiers".  We will speak of this more later on.

 

# The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as some kind of "morphism" or "functor" from a logical domain to a quantitative co-domain.

:* The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as some kind of "morphism" or "functor" from a logical domain to a quantitative co-domain.

# Notice that Peirce follows the mathematician's usual practice, then and now, of making the status of being an "individual" or a "universal" relative to a discourse in progress.  I have come to appreciate more and more of late how radically different this "patchwork" or "piecewise" approach to things is from the way of some philosophers who seem to be content with nothing less than many worlds domination, which means that they are never content and rarely get started toward the solution of any real problem.  Just my observation, I hope you understand.

 

# It is worth noting that Peirce takes the "plural denotation" of terms for granted, or what's the number of a term for, if it could not vary apart from being one or nil?

:* Notice that Peirce follows the mathematician's usual practice, then and now, of making the status of being an "individual" or a "universal" relative to a discourse in progress.  I have come to appreciate more and more of late how radically different this "patchwork" or "piecewise" approach to things is from the way of some philosophers who seem to be content with nothing less than many worlds domination, which means that they are never content and rarely get started toward the solution of any real problem.  Just my observation, I hope you understand.

# I also observe that Peirce takes the individual objects of a particular universe of discourse in a "generative" way, not a "totalizing" way, and thus they afford us with the basis for talking freely about collections, constructions, properties, qualities, subsets, and "higher types", as the phrase is mint.

 

:* It is worth noting that Peirce takes the "plural denotation" of terms for granted, or what's the number of a term for, if it could not vary apart from being one or nil?

 

:* I also observe that Peirce takes the individual objects of a particular universe of discourse in a "generative" way, not a "totalizing" way, and thus they afford us with the basis for talking freely about collections, constructions, properties, qualities, subsets, and "higher types", as the phrase is mint.

==Selection 3==

==Selection 3==

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

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<p>I shall follow Boole in taking the sign of equality to signify identity.  Thus, if <math>\mathrm{v}\!</math> denotes the Vice-President of the United States, and <math>\mathrm{p}\!</math> the President of the Senate of the United States,</p>

<p>I shall follow Boole in taking the sign of equality to signify identity.  Thus, if <math>\mathrm{v}\!</math> denotes the Vice-President of the United States, and <math>\mathrm{p}\!</math> the President of the Senate of the United States,</p>

|-

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==Selection 4==

==Selection 4==

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

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<p>The sign of addition is taken by Boole so that</p>

<p>The sign of addition is taken by Boole so that</p>

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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:

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/msg04350.html]

* [http://suo.ieee.org/ontology/msg04351.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.

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.