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{{DISPLAYTITLE:Precursors}}

{{DISPLAYTITLE:Precursors}}

'''Note.'''  I was going to go with ''The Fruit Of Our Purloins'', but I reckon this is more succinct.  The way that I normally start an inquiry like this is just to collect a sample of source materials that seem like they belong together.  Still traveling, so this will be sporadic at first.  —[[User:Jon Awbrey|JA]]

'''Note.'''  I was going to go with ''The Fruit Of Our Purloins'', but I reckon this is more succinct.  The way that I normally start an inquiry like this is just to collect a sample of source materials that seem like they belong together.  Still traveling, so this will be sporadic at first.  —[[User:Jon Awbrey|JA]]

==Preamble==

==Preamble==

<p>On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into ''Realms'' for the different Predicaments.</p>

<p>On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into ''Realms'' for the different Predicaments.</p>

<p>Peirce, CP&nbsp;4.549, &ldquo;Prolegomena to an Apology for Pragmaticism&rdquo;, ''The Monist'' 16, 492&ndash;546 (1906), CP&nbsp;4.530&ndash;572.</p>

<p>C.S. Peirce, CP&nbsp;4.549, &ldquo;Prolegomena to an Apology for Pragmaticism&rdquo;, ''The Monist'' 16, 492&ndash;546 (1906), CP&nbsp;4.530&ndash;572.</p>

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===Selection 1===

===Selection 1===

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<p>For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates.  Now, however, there is nothing to prevent us from ''considering the predicates and sentences themselves as individuals which may serve as arguments of predicates''.</p>

<p>For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates.  Now, however, there is nothing to prevent us from _considering the predicates and sentences themselves as individuals which may serve as arguments of predicates_.</p>

<p>Consider, for example, a logical expression of the form <math>(x)(A \rightarrow F(x)).</math> This may be interpreted as a predicate <math>P(A, F)</math> whose first argument place is occupied by a sentence <math>A,</math> and whose second argument place is occupied by a monadic predicate <math>F.</math></p>

<p>Consider, for example, a logical expression of the form $(x)(A \rightarrow F(x))$.  This may be interpreted as a predicate $P(A, F)$ whose first argument place is occupied by a sentence $A$, and whose second argument place is occupied by a monadic predicate $F$.</p>

<p>A false sentence <math>A</math> is related to every <math>F</math> by the relation <math>P(A, F);</math>  a true sentence <math>A</math> only to those <math>F</math> for which <math>(x)F(x)</math> holds.</p>

<p>A false sentence $A$ is related to every $F$ by the relation $P(A, F)$;  a true sentence $A$ only to those $F$ for which $(x)F(x)$ holds.</p>

<p>Further examples are given by the properties of ''reflexivity'', ''symmetry'', and ''transitivity'' of dyadic predicates.  To these correspond three predicates:  <math>\operatorname{Ref}(R),</math> <math>\operatorname{Sym}(R),</math> and <math>\operatorname{Tr}(R),</math> whose argument <math>R</math> is a dyadic predicate. These three properties are expressed in symbols as follows:</p>

<p>Further examples are given by the properties of _reflexivity_, _symmetry_, and _transitivity_ of dyadic predicates.  To these correspond three predicates:  $\mathop{Ref}(R)$, $\mathop{Sym}(R)$, and $\mathop{Tr}(R)$, whose argument $R$ is a dyadic predicate. These three properties are expressed in symbols as follows:</p>

<math>\begin{array}{l}

\operatorname{Ref}(R) \colon (x)R(x, x),

\\[6pt]

\operatorname{Sym}(R) \colon (x)(y)(R(x, y) \rightarrow R(y, x)),

\operatorname{Tr}(R) \colon (x)(y)(z)(R(x, y) \And R(y, z) \rightarrow R(x, z)).

\end{array}</math>

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<p>All three properties are possessed by the predicate <math>\equiv(x, y)</math> (<math>x</math> is identical with <math>y</math>).  The predicate <math><(x, y),</math> on the other hand, possesses only the property of transitivity.  Thus the formulas <math>\operatorname{Ref}(\equiv),</math> <math>\operatorname{Sym}(\equiv),</math> <math>\operatorname{Tr}(\equiv),</math> and <math>\operatorname{Tr}(<)</math> are true sentences, whereas <math>\operatorname{Ref}(<)</math> and <math>\operatorname{Sym}(<)</math> are false.</p>

<td style="border:none">

<p>$\mathop{Ref}(R): (x)R(x, x)$,<br>

$\mathop{Sym}(R): (x)(y)(R(x, y) \rightarrow R(y, x))$,<br>

$\mathop{Tr}(R): (x)(y)(z)(R(x, y)$&nbsp;&amp;&nbsp;$R(y, z) \rightarrow R(x, z))$.</p></td></table>

<p>Such ''predicates of predicates'' will be called ''predicates of second level''.  (p.&nbsp;135).</p>

 

<p>Such _predicates of predicates_ will be called _predicates of second level_.  (p. 135).</p>

 

===Selection 2===

===Selection 2===

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<p>We have, first, predicates of individuals, and these are classified into predicates of different categories, or types, according to the number of their argument places.  Such predicates are called ''predicates of first level''.</p>

<p>We have, first, predicates of individuals, and these are classified into predicates of different categories, or types, according to the number of their argument places.  Such predicates are called <i>predicates of first level</i>.</p>

<p>By a ''predicate of second level'', we understand one whose argument places are occupied by names of individuals or by predicates of first level, where a predicate of first level must occur at least once as an argument.  The categories, or types, of predicates second level are differentiated according to the number and kind of their argument places.  (p.&nbsp;152).</p>

<p>Hilbert and Ackermann, ''Principles of Mathematical Logic'', Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950.  First published, ''Grundzüge der Theoretischen Logik'', 1928.  Second edition, 1938.  English translation with revisions, corrections, and added notes by Robert E. Luce, 1950.</p>

 

<p>Hilbert and Ackermann, <i>Principles of Mathematical Logic</i>, Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950.  First published, <i>Grundzüge der Theoretischen Logik</i>, 1928.  Second edition, 1938.  English translation with revisions, corrections, and added notes by Robert E. Luce, 1950.</p>

 

==References==

==References==

* Aristotle, &ldquo;The Categories&rdquo;, Harold P. Cooke (trans.), pp. 1&ndash;109 in ''Aristotle, Volume&nbsp;1'', Loeb Classical Library, William Heinemann, London, UK, 1938.

* Aristotle, "The Categories", Harold P. Cooke (trans.), pp. 1&ndash;109 in _Aristotle, Volume&nbsp;1_, Loeb Classical Library, William Heinemann, London, UK, 1938.

* Aristotle, "Categories", E.M. Edghill (trans.), eBooks@Adelaide, University of Adelaide, South Australia, 2007.  [Online](http://ebooks.adelaide.edu.au/a/aristotle/categories/).

* Aristotle, &ldquo;Categories&rdquo;, E.M. Edghill (trans.), eBooks@Adelaide, University of Adelaide, South Australia, 2007.  [http://ebooks.adelaide.edu.au/a/aristotle/categories/ Online].

* van Heijenoort, Jean (1967/1977), _From Frege to Gödel : A Source Book in Mathematical Logic, 1879&ndash;1931_, Harvard University Press, Cambridge, MA, 1967.  2nd printing, 1972.  3rd printing, 1977.

* van Heijenoort, Jean (1967/1977), ''From Frege to Gödel : A Source Book in Mathematical Logic, 1879&ndash;1931'', Harvard University Press, Cambridge, MA, 1967.  2nd printing, 1972.  3rd printing, 1977.

* Hilbert, D., and Ackermann, W. (1938/1950), _Principles of Mathematical Logic_, Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950.  First published, _Grundzüge der Theoretischen Logik_, 1928.  Second edition, 1938.  English translation with revisions, corrections, and added notes by Robert E. Luce, 1950.

* Hilbert, D., and Ackermann, W. (1938/1950), ''Principles of Mathematical Logic'', Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950.  First published, ''Grundzüge der Theoretischen Logik'', 1928.  Second edition, 1938.  English translation with revisions, corrections, and added notes by Robert E. Luce, 1950.

* Kant

* Kant

* C.S. Peirce, &ldquo;[On a New List of Categories](http://www.cspeirce.com/menu/library/bycsp/newlist/nl-frame.htm)&rdquo;

* C.S. Peirce, &ldquo;On a New List of Categories&rdquo;, [http://www.cspeirce.com/menu/library/bycsp/newlist/nl-frame.htm Online].

* Carnap, _[The Logical Syntax of Language](http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover)_, _cf._ &ldquo;[Functor](http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover#v=onepage&q=Functor&f=false)&rdquo;

* Carnap, ''[http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover The Logical Syntax of Language]'', ''vide'' [http://books.google.com/books?id=Yf9R6WFFLhYC&printsec=frontcover#v=onepage&q=Functor&f=false &ldquo;Functor&rdquo;]

==Related Topics==

==Related Topics==

==Document History==

==Document History==

* [http://ncatlab.org/nlab/revision/precursors+%3E+history/15 Precursors @ nLab]

* [http://ncatlab.org/nlab/revision/precursors/15 Precursors @ nLab]