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Directory:Jon Awbrey/Papers/Information = Comprehension × Extension (view source)
Revision as of 17:45, 30 March 2009
, 17:45, 30 March 2009→Discussion
Figure 1 depicts the situation that is being contemplated here.
Figure 1 depicts the situation that is being contemplated here.
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Figure 1. Conjunctive Term z, Taken as Predicate
Figure 1. Conjunctive Term z, Taken as Predicate
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What Peirce is saying about <math>z\!</math> not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction <math>z,\!</math> in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, <math>z = \operatorname{glb}( \{ t_j : j = 1 ~\text{to}~ 6 \}),</math> and what we might regard as the ''natural conjunction'' or the ''natural glb'' of these terms, namely, <math>y := \text{an orange}.\!</math> That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose. The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between <math>z\!</math> and <math>y.\!</math>
What Peirce is saying about <math>z\!</math> not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction <math>z,\!</math> in lattice terms, the ''greatest lower bound'' (''glb'') of the conjoined terms, <math>z = \operatorname{glb}( \{ t_j : j = 1 ~\text{to}~ 6 \}),</math> and what we might regard as the ''natural conjunction'' or the ''natural glb'' of these terms, namely, <math>y := \text{an orange}.\!</math> That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose. The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between <math>z\!</math> and <math>y.\!</math>