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| | ''F<sub>3</sub> || 1 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1 | | | ''F<sub>3</sub> || 1 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1 |
− | |} | + | |}<br> |
− | <br> | |
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| I am going to put off explaining Table 2, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix. | | I am going to put off explaining Table 2, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix. |
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| |''m''<sub>15</sub>||anything happens|| || || || || | | |''m''<sub>15</sub>||anything happens|| || || || || |
− | |} | + | |}<br> |
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| ====Higher Order Propositions and Logical Operators (''n'' = 2)==== | | ====Higher Order Propositions and Logical Operators (''n'' = 2)==== |
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− | | <math>X = \{ (\lnot x, \lnot y), (\lnot x, y), (x, \lnot y), (x, y) \}</math> | + | | <math>X = \{ (\lnot x, \lnot y), (\lnot x, y), (x, \lnot y), (x, y) \}.</math> |
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− | | <math>X \cong \{ (0, 0), (0, 1), (1, 0), (1, 1) \}</math> | + | | <math>X \cong \{ (0, 0), (0, 1), (1, 0), (1, 1) \}.</math> |
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| + | | <math>X \cong \mathbb{B}^2.</math> |
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| | <math>X^\uparrow = (X \to \mathbb{B}) = \{ f : X \to \mathbb{B} \} \cong (\mathbb{B}^2 \to \mathbb{B}).</math> | | | <math>X^\uparrow = (X \to \mathbb{B}) = \{ f : X \to \mathbb{B} \} \cong (\mathbb{B}^2 \to \mathbb{B}).</math> |
− | |} | + | |}<br> |
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| As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again. | | As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again. |
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− | The next higher order universe of discourse that is built on ''X''° is ''X''°2 = [''X''°] = <nowiki>[[</nowiki>''x'', ''y''<nowiki>]]</nowiki>, which may be developed in the following way. The propositions of ''X''° become the points of ''X''°2, and the mappings of the type ''m'' : (''X'' → '''B''') → '''B''' become the propositions of ''X''°2. In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form ''w'' : ('''B'''<sup>2</sup> → '''B''')<sup>''k''</sup> → '''B'''. | + | The next higher order universe of discourse that is built on <math>X^\circ</math> is <math>X^{\circ 2} = [X^\circ] = [[x, y]],</math> which may be developed in the following way. The propositions of <math>X^\circ</math> become the points of <math>X^{\circ 2},</math> and the mappings of the type <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> become the propositions of <math>X^{\circ 2}.</math> In addition, it is convenient to equip the discussion with a selected set of higher order operators on propositions, all of which have the form <math>w : (\mathbb{B}^2 \to \mathbb{B})^k \to \mathbb{B}.</math> |
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| To save a few words in the remainder of this discussion, I will use the terms "measure" and "qualifier" to refer to all types of higher order propositions and operators. To describe the present setting in picturesque terms, the propositions of [''x'', ''y''] may be regarded as a gallery of sixteen venn diagrams, while the measures ''m'' : (''X'' → '''B''') → '''B''' are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not. In this way, each judge m_j partitions the gallery of pictures into two aesthetic portions, the pictures (''m''<sub>''j''</sub>)<sup>–1</sup>(1) that ''m''<sub>''j''</sub> likes and the pictures (''m''<sub>''j''</sub>)<sup>–1</sup>(0) that ''m''<sub>''j''</sub> dislikes. | | To save a few words in the remainder of this discussion, I will use the terms "measure" and "qualifier" to refer to all types of higher order propositions and operators. To describe the present setting in picturesque terms, the propositions of [''x'', ''y''] may be regarded as a gallery of sixteen venn diagrams, while the measures ''m'' : (''X'' → '''B''') → '''B''' are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not. In this way, each judge m_j partitions the gallery of pictures into two aesthetic portions, the pictures (''m''<sub>''j''</sub>)<sup>–1</sup>(1) that ''m''<sub>''j''</sub> likes and the pictures (''m''<sub>''j''</sub>)<sup>–1</sup>(0) that ''m''<sub>''j''</sub> dislikes. |