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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)

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<p>The conjugative term involves the conception of ''third'', the relative that of second or ''other'', the absolute term simply considers ''an'' object.  No fourth class of terms exists involving the conception of ''fourth'', because when that of ''third'' is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.  Whether this ''reason'' for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.  (Peirce, CP 3.63).</p>
 
<p>The conjugative term involves the conception of ''third'', the relative that of second or ''other'', the absolute term simply considers ''an'' object.  No fourth class of terms exists involving the conception of ''fourth'', because when that of ''third'' is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.  Whether this ''reason'' for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.  (Peirce, CP 3.63).</p>
 
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==Commentary Note 1==
      
I am going to experiment with an interlacing commentary on Peirce's 1870 "Logic of Relatives" paper, revisiting some critical transitions from several different angles and calling attention to a variety of puzzles, problems, and potentials that are not so often remarked or tapped.
 
I am going to experiment with an interlacing commentary on Peirce's 1870 "Logic of Relatives" paper, revisiting some critical transitions from several different angles and calling attention to a variety of puzzles, problems, and potentials that are not so often remarked or tapped.
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==Commentary Note 2==
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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.
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<pre>
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Peirce's remarks at CP 3.65 are so replete with remarkable ideas,
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some of them so taken for granted in mathematical discourse that
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they usually escape explicit mention, and others so suggestive
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of things to come in a future remote from his time of writing,
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and yet so smoothly introduced in passing that it's all too
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easy to overlook their consequential significance, that I
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can do no better here than to highlight these ideas in
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other words, whose main advantage is to be a little
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more jarring to the mind's sensibilities.
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| Numbers Corresponding to Letters
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|
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| I propose to use the term "universe" to denote that class of individuals
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| 'about' which alone the whole discourse is understood to run.  The universe,
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| therefore, in this sense, as in Mr. De Morgan's, is different on different
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| occasions.  In this sense, moreover, discourse may run upon something which
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| is not a subjective part of the universe;  for instance, upon the qualities
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| or collections of the individuals it contains.
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|
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| I propose to assign to all logical terms, numbers;  to an absolute term,
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| the number of individuals it denotes;  to a relative term, the average
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| number of things so related to one individual.  Thus in a universe of
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| perfect men ('men'), the number of "tooth of" would be 32.  The number
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| of a relative with two correlates would be the average number of things
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| so related to a pair of individuals;  and so on for relatives of higher
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| numbers of correlates.  I propose to denote the number of a logical term
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| by enclosing the term in square brackets, thus ['t'].
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|
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| C.S. Peirce, 'Collected Papers', CP 3.65
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1.  This mapping of letters to numbers, or logical terms to mathematical quantities,
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    is the very core of what "quantification theory" is all about, and definitely
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    more to the point than the mere "innovation" of using distinctive symbols
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    for the so-called "quantifiers".  We will speak of this more later on.
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2.  The mapping of logical terms to numerical measures,
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    to express it in current language, would probably be
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    recognizable as some kind of "morphism" or "functor"
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    from a logical domain to a quantitative co-domain.
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3.  Notice that Peirce follows the mathematician's usual practice,
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    then and now, of making the status of being an "individual" or
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    a "universal" relative to a discourse in progress.  I have come
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    to appreciate more and more of late how radically different this
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    "patchwork" or "piecewise" approach to things is from the way of
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    some philosophers who seem to be content with nothing less than
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    many worlds domination, which means that they are never content
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    and rarely get started toward the solution of any real problem.
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    Just my observation, I hope you understand.
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4.  It is worth noting that Peirce takes the "plural denotation"
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    of terms for granted, or what's the number of a term for,
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    if it could not vary apart from being one or nil?
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5.  I also observe that Peirce takes the individual objects of a particular
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'''Numbers Corresponding to Letters'''
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    universe of discourse in a "generative" way, not a "totalizing" way,
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# 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.
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    and thus they afford us with the basis for talking freely about
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# 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.
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    collections, constructions, properties, qualities, subsets,
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# 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.
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    and "higher types", as the phrase is mint.
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# 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?
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</pre>
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# 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==
Jon Awbrey
12,080

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