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The first way of transforming the expression that appears on the left hand side of the equation can be described as ''proof-theoretic'' in character. That was given in Note 5.

| [[Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems#Praeclarum_theorema|The first way of transforming the expression]] that appears on the left hand side of the equation can be described as ''proof-theoretic'' in character.

| The second way of transforming the expression that appears on the left hand side of the equation can be described as ''model-theoretic'' in character.

What we have here amounts to a couple of different styles of communicative conduct, that is, two sequences of signs of the form <math>e_1, e_2, \ldots, e_n,\!</math> each one beginning with a problematic expression and eventually ending with a clear expression of the ''logical equivalence class'' to which every sign or expression in the sequence belongs. Ordinarily, any orbit through a locus of signs can be taken to reflect an underlying sign-process, a case of ''semiosis''.  So what we have here are two very special cases of semiosis, and what we may find it useful to contemplate is how to characterize them as two species of a very general class.

 

 

 

What we have here amounts to a couple of different styles of ''communicational conduct'', or ''conductive communication'', if you prefer, that is to say, two sequences of signs of the form <math>e_1, e_2, \ldots, e_n,\!</math> each one beginning with a problematic expression and eventually ending with a clear expression of the appropriate ''logical equivalence class'' (LEC) to which each and every sign or expression in the sequence belongs.

 

Ordinarily, any orbit through a locus of signs can be taken to reflect an underlying sign-process, a case of ''semiosis''.  So what we have here are two very special cases of semiosis, and what we might just find it useful to contemplate is how to characterize them as two species of a very general class.

We are starting to delve into some fairly picayune details of a particular sign system, non-trivial enough in its own right but still rather simple compared to the types of our ultimate interest, and though I believe that this exercise will be worth the effort in prospect of understanding more complicated sign systems, I feel that I ought to say a few words about the larger reasons for going through this work.

We are starting to delve into some fairly picayune details of a particular sign system, non-trivial enough in its own right but still rather simple compared to the types of our ultimate interest, and though I believe that this exercise will be worth the effort in prospect of understanding more complicated sign systems, I feel that I ought to say a few words about the larger reasons for going through this work.