12,089
edits
Changes
Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 19:36, 6 July 2008
, 19:36, 6 July 2008→Special Classes of Propositions: split paragraph
For example, for ''n'' = 3, the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is (''a''<sub>1</sub>)(''a''<sub>2</sub>)(''a''<sub>3</sub>).
For example, for ''n'' = 3, the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is (''a''<sub>1</sub>)(''a''<sub>2</sub>)(''a''<sub>3</sub>).
The coordinate projections or simple propositions ''a''<sub>''i''</sub> : '''B'''<sup>''n''</sup> → '''B''' are both linear and positive. So these two kinds of propositions, the linear or the positive, may be viewed as two different ways of generalizing the class of simple projections. The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {''a''<sub>''i''</sub>}. Therefore, each set of functions can be parameterized by the subsets ''J'' of the basic index set <font face="lucida calligraphy">I</font> = {1, …, ''n''}.
The coordinate projections or simple propositions ''a''<sub>''i''</sub> : '''B'''<sup>''n''</sup> → '''B''' are both linear and positive. So these two kinds of propositions, the linear or the positive, may be viewed as two different ways of generalizing the class of simple projections.
The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {''a''<sub>''i''</sub>}. Therefore, each set of functions can be parameterized by the subsets ''J'' of the basic index set <font face="lucida calligraphy">I</font> = {1, …, ''n''}.
Let us define <font face="lucida calligraphy">A</font><sub>''J''</sub> as the subset of <font face="lucida calligraphy">A</font> that is given by {''a''<sub>''i''</sub> : ''i'' ∈ ''J''}. Then we may comprehend the action of the linear and the positive propositions in the following terms:
Let us define <font face="lucida calligraphy">A</font><sub>''J''</sub> as the subset of <font face="lucida calligraphy">A</font> that is given by {''a''<sub>''i''</sub> : ''i'' ∈ ''J''}. Then we may comprehend the action of the linear and the positive propositions in the following terms: