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==Analysis of contingent propositions==

==Analysis of contingent propositions==

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, let's now take up a much simpler example of a contingent proposition:

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For the sake of simplicity in discussing this example, I will revert to the existential interpretation (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings.

For the sake of simplicity in discussing this example, let's stick with the existential interpretation (<math>\operatorname{Ex}</math>) of logical graphs and their corresponding parse strings.  Under <math>\operatorname{Ex}</math> the formal expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> translates into the vernacular expression <math>{}^{\backprime\backprime} p ~\operatorname{implies}~ q ~\operatorname{and}~ p ~\operatorname{implies}~ r {}^{\prime\prime},</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 <math>\operatorname{Ex}</math> 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, we may refer to the propositional expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math> under the name <math>f\!</math> by making use of the following definition:

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>

| <math>f ~=~ \texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</math>

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.

Since the expression <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))}</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&nbsp;1 indicates the points of the universe of discourse ''X'' for which the proposition ''f'' : ''X'' &rarr; '''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 [&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;] for 0 and [&nbsp;`&nbsp;` &nbsp;`&nbsp;] for 1.

Figure&nbsp;1 indicates the points of the universe of discourse ''X'' for which the proposition ''f'' : ''X'' &rarr; '''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 [&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;] for 0 and [&nbsp;`&nbsp;` &nbsp;`&nbsp;] for 1.