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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 17:55, 19 October 2007
, 17:55, 19 October 2007→Commentary Note 8.6: markup
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.
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.
For another instance, we have the "tacit extension" of a ''k''-place relation ''L'' ⊆ ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> to a (''k''+1)-place relation ''L''′ ⊆ ''X''<sub>1</sub> × … × ''X''<sub>''k''+1</sub> that
For another instance, we have the "tacit extension" of a k-place relation
we get by letting ''L''′ = ''L'' × ''X''<sub>''k''+1</sub>, that is, by maintaining the constraints
L c X_1 x ... x X_k to a (k+1)-place relation L' c X_1 x ... x X_k+1 that
of ''L'' on the first ''k'' variables and letting the last variable wander freely.
we get by letting L' = L x X_k+1, that is, by maintaining the constraints
of L on the first k variables and letting the last variable wander freely.
What we have here, if I understand Peirce correctly, is another such
What we have here, if I understand Peirce correctly, is another such type of natural extension, sometimes called the "diagonal extension". This associates a ''k''-adic relative or a ''k''-adic relation, counting the absolute term and the set whose elements it denotes as the cases for ''k'' = 0, with a series of relatives and relations of higher adicities.
type of natural extension, sometimes called the "diagonal extension".
This associates a k-adic relative or a k-adic relation, counting the
absolute term and the set whose elements it denotes as the cases for
k = 0, with a series of relatives and relations of higher adicities.
<pre>
A few examples will suffice to anchor these ideas.
A few examples will suffice to anchor these ideas.