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===Commentary Note 8.6===
===Commentary Note 8.6===
The foregoing has hopefully filled in enough background that we can begin to make sense of the more mysterious parts of CP 3.73.
The foregoing has hopefully filled in enough background that we
can begin to make sense of the more mysterious parts of CP 3.73.
<blockquote>
<p>Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p>
<p>Now the absolute term "man" is really exactly equivalent to the relative term "man that is ---", and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.</p>
<p>Then:</p>
: <p>"man that is black"</p>
<p>will be written</p>
: <p>m,b.</p>
<p>(Peirce, CP 3.73).</p>
</blockquote>
In any system where elements are organized according to types,
In any system where elements are organized according to types, there tend to be any number of ways in which elements of one type are naturally associated with elements of another type. If the association is anything like a logical equivalence, but with the first type being "lower" and the second type being "higher" in some sense, then one frequently speaks of a "semantic ascent" from the lower to the higher type.
there tend to be any number of ways in which elements of one
type are naturally associated with elements of another type.
If the association is anything like a logical equivalence,
but with the first type being "lower" and the second type
being "higher" in some sense, then one frequently speaks
of a "semantic ascent" from the lower to the higher type.
For instance, it is very common in mathematics to associate an element m
For instance, it is very common in mathematics to associate an element ''m'' of a set ''M'' with the constant function ''f''<sub>''m''</sub> : ''X'' → ''M'' such that ''f''<sub>''m''</sub>(''x'') = ''m'' for all ''x'' in ''X'', where ''X'' is an arbitrary set. Indeed, the correspondence is so close that one often uses the same name "''m''" for the element ''m'' in ''M'' and the function ''m'' = ''f''<sub>''m''</sub> : ''X'' → ''M'', relying on the context or an explicit type indication to tell them apart.
of a set M with the constant function f_m : X -> M such that f_m (x) = m
for all x in X, where X is an arbitrary set. Indeed, the correspondence
is so close that one often uses the same name "m" for the element m in M
and the function m = f_m : X -> M, relying on the context or an explicit
type indication to tell them apart.
<pre>
For another instance, we have the "tacit extension" of a k-place relation
For another instance, we have the "tacit extension" of a k-place relation
L c X_1 x ... x X_k to a (k+1)-place relation L' c X_1 x ... x X_k+1 that
L c X_1 x ... x X_k to a (k+1)-place relation L' c X_1 x ... x X_k+1 that
