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For some reason I always think of this as the way that our DNA would prove it.

For some reason I always think of this as the way that our DNA would prove it.

We are in the process of examining various proofs of the propositional equation "(p (q))(p (r)) = (p (q r))", and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations.

We are in the process of examining various proofs of the propositional equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},</math> and viewing these proofs in the light of their character as semiotic processes, in essence, as sign-theoretic transformations.

Here is a reminder of the equation in question:

Here is a reminder of the equation in question:

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What we have harvested is the succulent equivalent of a disjunctive normal form (DNF) for the proposition with which we started.  Remembering that a blank node is the graphical equivalent of a logical value ''true'', we can read this brand of DNF in the following manner:

What we have harvested is the succulent equivalent of a disjunctive normal form (DNF) for the proposition with which we started.  Remembering that a blank node is the graphical equivalent of a logical value <math>\operatorname{true},</math> we can read this brand of DNF in the following manner:

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{| align="center" cellpadding="10" style="text-align:center; width:90%"

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Sorry, the half-time show was cancelled by the censors.  But I'm guessing that the reader can probably finish off the second half of the proof with a few scribbles on paper faster than I can asciify it on my own, so at least there's that entertaiment to occupy the interval.

The reader can probably finish the second half of the proof with a few scribbles on paper faster than I can asciify it on my own, so at least there's that entertainment to occupy the interval.

We are still in the middle of contemplating a particular example of a propositional equation, namely, "(p (q))(p (r)) = (p (q r))", and we are still considering the second of three formal methods that I intend to illustrate in the process of thrice-over establishing it.

We are still in the middle of contemplating a particular example of a propositional equation, namely, <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))},</math> and we are still considering the second of three formal methods that are illustrated in the process of thrice-over establishing it.

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| align="center" cellpadding="10" style="text-align:center; width:90%"

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I know that it must seem tedious, but I probably ought to go ahead and carry out the second half of this analogically model-theoretic strategy, just so that we will have the security of this concrete and shared experience on which to fall back at every later point in whatmay quickly become a rather abstruse discussion.  Here then is the rest of the necessary chain of equations:

I know that it must seem tedious, but I probably ought to go ahead and carry out the second half of this analogically model-theoretic strategy, just so that we will have the security of this concrete and shared experience on which to fall back at every later point in what may quickly become a rather abstruse discussion.  Here then is the rest of the necessary chain of equations:

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| align="center" cellpadding="10" style="text-align:center; width:90%"

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This is not only a logically equivalent DNF, but exactly the same DNF expression that we obtained before, so we have established the given equation "(p (q))(p (r)) = (p (q r))".

This is not only a logically equivalent DNF but exactly the same DNF expression that we obtained before, so we have established the given equation <math>\texttt{(} p \texttt{(} q \texttt{))(} p \texttt{(} r \texttt{))} = \texttt{(} p \texttt{(} q r \texttt{))}.</math>

Incidentally, one may wish to note that this DNF expression quickly folds into the following form:

Incidentally, one may wish to note that this DNF expression quickly folds into the following form: