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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. |
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| 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. |
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− | <pre>
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| A few examples will suffice to anchor these ideas. | | A few examples will suffice to anchor these ideas. |
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| Absolute terms: | | Absolute terms: |
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− | m = "man" = C +, I +, J +, O | + | : m = "man" = C +, I +, J +, O |
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− | n = "noble" = C +, D +, O | + | : n = "noble" = C +, D +, O |
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− | w = "woman" = B +, D +, E | + | : w = "woman" = B +, D +, E |
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| Diagonal extensions: | | Diagonal extensions: |
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− | m, = "man that is ---" = C:C +, I:I +, J:J +, O:O | + | : m, = "man that is ---" = C:C +, I:I +, J:J +, O:O |
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− | n, = "noble that is ---" = C:C +, D:D +, O:O | + | : n, = "noble that is ---" = C:C +, D:D +, O:O |
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− | w, = "woman that is ---" = B:B +, D:D +, E:E | + | : w, = "woman that is ---" = B:B +, D:D +, E:E |
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| Sample products: | | Sample products: |
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− | m,n = "man that is noble" | + | : m,n = "man that is noble" |
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− | = (C:C +, I:I +, J:J +, O:O)(C +, D +, O)
| + | :: = (C:C +, I:I +, J:J +, O:O)(C +, D +, O) |
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− | = C +, O
| + | :: = C +, O |
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− | n,m = "noble that is man" | + | : n,m = "noble that is man" |
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− | = (C:C +, D:D +, O:O)(C +, I +, J +, O)
| + | :: = (C:C +, D:D +, O:O)(C +, I +, J +, O) |
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− | = C +, O
| + | :: = C +, O |
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− | n,w = "noble that is woman" | + | : n,w = "noble that is woman" |
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− | = (C:C +, D:D +, O:O)(B +, D +, E)
| + | :: = (C:C +, D:D +, O:O)(B +, D +, E) |
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− | = D
| + | :: = D |
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− | w,n = "woman that is noble" | + | : w,n = "woman that is noble" |
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− | = (B:B +, D:D +, E:E)(C +, D +, O)
| + | :: = (B:B +, D:D +, E:E)(C +, D +, O) |
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− | = D
| + | :: = D |
− | </pre>
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| ==Selection 9== | | ==Selection 9== |