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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
Revision as of 19:42, 3 November 2008
, 19:42, 3 November 2008→Analysis of contingent propositions: HTML → TeX
For all of the reasons mentioned above, and for the sake of a more compact illustration of the in and outs of a typical propositional equation reasoning system (PERS), let's now take up a much simpler example of a contingent proposition:
For all of the reasons mentioned above, and for the sake of a more compact illustration of the in and outs of a typical propositional equation reasoning system (PERS), let's now take up a much simpler example of a contingent proposition:
<pre>
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` q o ` o r ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` p o ` o p ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `(p (q)) (p (r))` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
</pre>
For the sake of simplicity in discussing this example, I will revert to the existential interpretation (''Ex'') of logical graphs and their corresponding parse strings.
For the sake of simplicity in discussing this example, I will revert to the existential interpretation (''Ex'') of logical graphs and their corresponding parse strings.
Under ''Ex'' the expression "(p (q))(p (r))" interprets as the vernacular expression "''p'' implies ''q'' and ''p'' implies ''r''", in symbols, {''p'' ⇒ ''q''} ∧ ''p'' ⇒ ''r'', 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 "(p (q))(p (r))" under the name "''f''" by making use of the following definition: "f = (p (q))(p (r))".