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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 ''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
we get by letting ''L''′ = ''L'' × ''X''<sub>''k''+1</sub>, that is, by maintaining the constraints
we get by letting ''L''′ = ''L'' × ''X''<sub>''k''+1 </sub>, that is, by maintaining the constraints
of ''L'' on the first ''k'' variables and letting the last variable wander freely.
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 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.
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.
A few examples will suffice to anchor these ideas.
A few examples will suffice to anchor these ideas.
Absolute terms:
Absolute terms:
m = "man" = C +, I +, J +, O
: m = "man" = C +, I +, J +, O
n = "noble" = C +, D +, O
: n = "noble" = C +, D +, O
w = "woman" = B +, D +, E
: w = "woman" = B +, D +, E
Diagonal extensions:
Diagonal extensions:
m, = "man that is ---" = C:C +, I:I +, J:J +, O:O
: m, = "man that is ---" = C:C +, I:I +, J:J +, O:O
n, = "noble that is ---" = C:C +, D:D +, O:O
: n, = "noble that is ---" = C:C +, D:D +, O:O
w, = "woman that is ---" = B:B +, D:D +, E:E
: w, = "woman that is ---" = B:B +, D:D +, E:E
Sample products:
Sample products:
m,n = "man that is noble"
: m,n = "man that is noble"
:: = (C:C +, I:I +, J:J +, O:O)(C +, D +, O)
:: = C +, O
n,m = "noble that is man"
: n,m = "noble that is man"
:: = (C:C +, D:D +, O:O)(C +, I +, J +, O)
:: = C +, O
n,w = "noble that is woman"
: n,w = "noble that is woman"
:: = (C:C +, D:D +, O:O)(B +, D +, E)
:: = D
w,n = "woman that is noble"
: w,n = "woman that is noble"
:: = (B:B +, D:D +, E:E)(C +, D +, O)
:: = D
==Selection 9==
==Selection 9==
