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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 20:46, 3 November 2008
, 20:46, 3 November 2008→Analysis of contingent propositions: HTML → TeX
Under ''Ex'' the expression <math>(p\ (q))(p\ (r))\!</math> interprets as the vernacular expression <math>p\ \operatorname{implies}\ q\ \operatorname{and}\ p\ \operatorname{implies}\ r,</math> in symbols, <math>\{ p \Rightarrow q \} \land \{ p \Rightarrow r \},</math> so this is the reading that we'll want to keep in mind for the present.
Under ''Ex'' the expression <math>(p\ (q))(p\ (r))\!</math> interprets as the vernacular expression <math>p\ \operatorname{implies}\ q\ \operatorname{and}\ p\ \operatorname{implies}\ r,</math> in symbols, <math>\{ p \Rightarrow q \} \land \{ p \Rightarrow r \},</math> so this is the reading that we'll want to keep in mind for the present.
Where brevity is required, and it occasionally is, we may invoke the propositional expression "(p (q))(p (r))" under the name "''f''" by making use of the following definition: "f = (p (q))(p (r))".
Where brevity is required, and it occasionally is, we may invoke the propositional expression <math>(p\ (q))(p\ (r))\!</math> under the name of <math>f\!</math> by making use of the following definition: <math>f = (p\ (q))(p\ (r)).\!</math>
Since the expression "(p (q))(p (r))" involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact a couple of different ways to execute the picture.
Since the expression <math>(p\ (q))(p\ (r))\!</math> involves just three variables, it may be worth the trouble to draw a venn diagram of the situation. There are in fact a couple of different ways to execute the picture.
Figure 1 indicates the points of the universe of discourse ''X'' for which the proposition ''f'' : ''X'' → '''B''' has the value 1 (= true). In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe ''X'', going according to some previously adopted convention, for instance: Let the cells that get the value 0 under ''f'' remain untinted, and let the cells that get the value 1 under ''f'' be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the 0, 1 in '''B''', but in the pattern of regions that they indicate. NB. In this Ascii version, I use [```] for 0 and [^^^] for 1.
Figure 1 indicates the points of the universe of discourse ''X'' for which the proposition ''f'' : ''X'' → '''B''' has the value 1 (= true). In this ''paint by numbers'' style of picture, one simply paints over the cells of a generic template for the universe ''X'', going according to some previously adopted convention, for instance: Let the cells that get the value 0 under ''f'' remain untinted, and let the cells that get the value 1 under ''f'' be painted or shaded. In doing this, it may be good to remind ourselves that the value of the picture as a whole is not in the ''paints'', in other words, the 0, 1 in '''B''', but in the pattern of regions that they indicate. NB. In this Ascii version, I use [```] for 0 and [^^^] for 1.
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o-----------------------------------------------------------o
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^o-------------o^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` ` ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` ` ` ` ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` ` ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^o` ` ` ` ` ` ` ` ` ` ` ` `o^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` P ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ o--o----------o ` o----------o--o ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^/^ ^ \ ` ` ` ` `\`/` ` ` ` ` / ^ ^\^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ / ^ ^ ^\` ` ` ` ` o ` ` ` ` `/^ ^ ^ \ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^/^ ^ ^ ^ \ ` ` ` `/^\` ` ` ` / ^ ^ ^ ^\^ ^ ^ ^ ^ |
| ^ ^ ^ ^ / ^ ^ ^ ^ ^\` ` ` / ^ \ ` ` `/^ ^ ^ ^ ^ \ ^ ^ ^ ^ |
| ^ ^ ^ ^/^ ^ ^ ^ ^ ^ \ ` `/^ ^ ^\` ` / ^ ^ ^ ^ ^ ^\^ ^ ^ ^ |
| ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^o--o-------o--o^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ Q ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ R ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ |
| ^ ^ ^ ^\^ ^ ^ ^ ^ ^ ^ ^ ^\^ ^ ^/^ ^ ^ ^ ^ ^ ^ ^ ^/^ ^ ^ ^ |
| ^ ^ ^ ^ \ ^ ^ ^ ^ ^ ^ ^ ^ \ ^ / ^ ^ ^ ^ ^ ^ ^ ^ / ^ ^ ^ ^ |
| ^ ^ ^ ^ ^\^ ^ ^ ^ ^ ^ ^ ^ ^\^/^ ^ ^ ^ ^ ^ ^ ^ ^/^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ \ ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ / ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^\^ ^ ^ ^ ^ ^ ^ ^/^\^ ^ ^ ^ ^ ^ ^ ^/^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ o-------------o ^ o-------------o ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
o-----------------------------------------------------------o
Figure 1. Venn Diagram for (p (q))(p (r))
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There are a number of standard ways in mathematics and statistics for talking about "the subset ''W'' of the domain ''X'' that gets painted with the value ''z'' by the indicator function ''f'' : ''X'' → '''B'''". The subset
There are a number of standard ways in mathematics and statistics for talking about "the subset ''W'' of the domain ''X'' that gets painted with the value ''z'' by the indicator function ''f'' : ''X'' → '''B'''". The subset