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==Logic as sign transformation==
==Logic as sign transformation==
We have been looking at various ways of transforming propositional expressions, expressed in the parallel formats of character strings and graphical structures, all the while preserving certain aspects of their "meaning" — and here I risk using that vaguest of all possible words, but only as a promissory note, hopefully to be cached out in a more meaningful species of currency as the discussion develops.
We have been looking at various ways of transforming propositional expressions, expressed in the parallel formats of character strings and graphical structures, all the while preserving certain aspects of their "meaning" — and here I risk using that vaguest of all possible words, but only as a promissory note, hopefully to be cached out in a more meaningful species of currency as the discussion develops.
I cannot pretend to be acquainted with or to comprehend every form of intension that others might find of interest in a given form of expression, nor can I speak for every form of meaning that another might find in a given form of syntax. The best that I can hope to do is to specify what my object is in using these expressions, and to say what aspects of their syntax are meant to serve this object, lending these properties the interest I have in preserving them as I put the expressions through the paces of their transformations.
I cannot pretend to be acquainted with or to comprehend every form of intension that others might find of interest in a given form of expression, nor can I speak for every form of meaning that another might find in a given form of syntax. The best that I can hope to do is to specify what my object is in using these expressions, and to say what aspects of their syntax are meant to serve this object, lending these properties the interest I have in preserving them as I put the expressions through the paces of their transformations.
Just to remind you, here is the ''Splendid Theorem'' again:
Just to remind you, here is the ''Splendid Theorem'' again:
<pre>
o-----------------------------------------------------------o
| Praeclarum Theorema (PT)` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` = ` ` ` ` ` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `((a(b))(d(c))((ad(bc)))) ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
</pre>
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.
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.
* PERS 5. http://forum.wolframscience.com/showthread.php?postid=957#post957
:*[http://forum.wolframscience.com/showthread.php?postid=957#post957 PERS Note 5].
The other way of transforming the expression that appears on the left hand side of the equation can be described as ''model-theoretic'' in character. That was given in Note 9.
The other way of transforming the expression that appears on the left hand side of the equation can be described as ''model-theoretic'' in character. That was given in Note 9.
* PERS 9. http://forum.wolframscience.com/showthread.php?postid=962#post962
:*[http://forum.wolframscience.com/showthread.php?postid=962#post962 PERS Note 9].
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 ''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n''</sub>, 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.
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.
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.
My broader interest lies in the theory of inquiry as a special application or a special case of the theory of signs. Another name for the theory of inquiry is ''logic'' and another name for the theory of signs is ''semiotics''. So I might as well have said that I am interested in logic as a special application or a special case of semiotics. But what sort of a special application? What sort of a special case? Well, I think of logic as ''formal semiotics'' — though, of course, I am not the first to have said such a thing — and by ''formal'' we say, in our etymological way, that logic is concerned with the ''form'', indeed, with the ''animate beauty'' and the very ''life force'' of signs and sign actions. Yes, perhaps that is far too Latin a way of understanding logic, but it's all I've got.
My broader interest lies in the theory of inquiry as a special application or a special case of the theory of signs. Another name for the theory of inquiry is ''logic'' and another name for the theory of signs is ''semiotics''. So I might as well have said that I am interested in logic as a special application or a special case of semiotics. But what sort of a special application? What sort of a special case? Well, I think of logic as ''formal semiotics'' — though, of course, I am not the first to have said such a thing — and by ''formal'' we say, in our etymological way, that logic is concerned with the ''form'', indeed, with the ''animate beauty'' and the very ''life force'' of signs and sign actions. Yes, perhaps that is far too Latin a way of understanding logic, but it's all I've got.
Now, if you think about these things just a little more, I know that you will find them just a little suspicious, for what besides logic would I use to do this theory of signs that I would apply to this theory of inquiry that I'm also calling ''logic''? But that is precisely one of the things signified by the word ''formal'', for what I'd be required to use would have to be some brand of logic, that is, some sort of innate or inured skill at inquiry, but a style of logic that is casual, catch-as-catch-can, formative, incipient, inchoate, unformalized, a work in progress, partially built into our natural language and partially more primitive than our most artless language. In so far as I use it more than mention it, mention it more than describe it, and describe it more than fully formalize it, then to that extent it must be consigned to the realm of unformalized and unreflective logic, where some say "there be oracles", but I don't know.
Now, if you think about these things just a little more, I know that you will find them just a little suspicious, for what besides logic would I use to do this theory of signs that I would apply to this theory of inquiry that I'm also calling ''logic''? But that is precisely one of the things signified by the word ''formal'', for what I'd be required to use would have to be some brand of logic, that is, some sort of innate or inured skill at inquiry, but a style of logic that is casual, catch-as-catch-can, formative, incipient, inchoate, unformalized, a work in progress, partially built into our natural language and partially more primitive than our most artless language. In so far as I use it more than mention it, mention it more than describe it, and describe it more than fully formalize it, then to that extent it must be consigned to the realm of unformalized and unreflective logic, where some say "there be oracles", but I don't know.
