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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.
On behalf of this object I have been spinning in the form of this thread a developing example base of propositional expressions, in the data structures of graphs and strings, along with many examples of step-wise transformations on
On behalf of this object I have been spinning in the form of this thread a developing example base of propositional expressions, in the data structures of graphs and strings, along with many examples of step-wise transformations on these expressions that preserve something of significant logical import, something that might be referred to as their ''logical equivalence class'' (LEC), and that we could as well call the ''constraint information'' or the ''denotative object'' of the expression in view.
these expressions that preserve something of significant logical import, something that might be referred to as their ''logical equivalence class'' (LEC), and that we could as well call the ''constraint information'' or the ''denotative object'' of the expression in view.
To focus still more, let's return to that ''Splendid Theorem'' noted by Leibniz, and let's look more carefully at the two distinct ways of transforming its initial expression that were used to arrive at an equivalent expression, each of which, in its own way, made its tautologous character, or its theorematic nature, as evident as it could be.
To focus still more, let's return to that ''Splendid Theorem'' noted by Leibniz, and let's look more carefully at the two distinct ways of transforming its initial expression that were used to arrive at an equivalent expression, each of which, in its own way, made its tautologous character, or its theorematic nature, as evident as it could be.
Just to remind you, here is the ''Splendid Theorem'' again:
Just to remind you, here is the ''Splendid Theorem'' again:
{| align="center" cellpadding="10"
| [[Image:Logical_Graph_Figure_33.jpg|500px]]
| Praeclarum Theorema (PT)` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
|}
| `((a(b))(d(c))((ad(bc)))) ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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. This appeared in Note 35.
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. This appeared in Note 35.
* FOLG 35. http://stderr.org/pipermail/inquiry/2005-November/003214.html
:* [http://stderr.org/pipermail/inquiry/2005-November/003214.html FOLG 35]
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. This appeared in Note 50.
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. This appeared in Note 50.
* FOLG 50. http://stderr.org/pipermail/inquiry/2005-November/003238.html
:* [http://stderr.org/pipermail/inquiry/2005-November/003238.html FOLG 50]
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 ''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.
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.
For a salient example, consider the existential graph for "p ⇔ q", to wit:
For a salient example, consider the existential graph for "p ⇔ q", to wit:
{| align="center" style="text-align:center; width:90%"
|
<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
|}
This can be read as "not p without q and not q without p", or what's the same, "[p ⇒ q] and [q ⇒ p]", or "p ⇔ q".
This can be read as "not p without q and not q without p", or what's the same, "[p ⇒ q] and [q ⇒ p]", or "p ⇔ q".
Graphing the topological dual form, one obtains this rooted tree:
Graphing the topological dual form, one obtains this rooted tree:
{| align="center" style="text-align:center; width:90%"
|
<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
|}
Now it is not the sort of thing that I ever noticed until it came time to program a theorem prover for logical graphs at Peirce's alpha level, but expressions like these, that mention each variable twice simply in order to express a basic 2-variate operator, are extremely inefficient forms of representation, and their use is enough to bog down a routine logical modeler or an automatic theorem prover in a slough of despond.
Now it is not the sort of thing that I ever noticed until it came time to program a theorem prover for logical graphs at Peirce's alpha level, but expressions like these, that mention each variable twice simply in order to express a basic 2-variate operator, are extremely inefficient forms of representation, and their use is enough to bog down a routine logical modeler or an automatic theorem prover in a slough of despond.
However, the cactus graph expression for equivalence works much better:
However, the cactus graph expression for equivalence works much better:
{| align="center" style="text-align:center; width:90%"
|
<pre>
<pre>
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
</pre>
</pre>
|}
The cactus language syntax also improves the ''reflective capacities'' of the logical calculus, in particular, it facilitates our ability to use the calculus to reflect on the process of proof, that is, the process of establishing equivalences between expressions.
The cactus language syntax also improves the ''reflective capacities'' of the logical calculus, in particular, it facilitates our ability to use the calculus to reflect on the process of proof, that is, the process of establishing equivalences between expressions.