Hmmmm ... I thought the original post was very interesting and even understood most of it (I even spotted the mistake, sort of, but assumed that the reason I couldn't see how to get 1111 and 0000 from paths was that I was being brainless).
But don't these results all come from the huge degree of generality of group theory, which in turn is a result of the high degree of abstraction? If you start introducing any degree of intensional content (and certainly if you start having "T" and "F" referring to true and false rather than to formal properties), then I wonder whether you can still make the assumptions necessary to get one theory to be a subgroup of another.
I didn't understand why "scientific truth" was represented as the truth table for a conjunction operator, and would be grateful to anyone who could explain why.
I also seem to remember that any truth table can be generated by repeated use of either the "nand" or the "nor" operator, so the underlying symoblic logic might be even simpler?
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David,
I did that post about three in the morning and just got back from working all day tonight, so this may not be coherent or correct either.
Operator question. First I had to fudge on the operator notation because straight ascii doesn't have the 'implied', 'contained' operators (the sideways 'U') and doesn't have the direct sum, direct product, or equivalence symbols (plus in a circle, x in a circle, dot in a circle). So, ignore 'v', 'u' and 'n'. The notation is off. Supply the correct symbol based on the operator name.
Nand/nor question. I wrote out the complete group of operators (F0-F15). There are numerous ways to combine operators to get the same result as using just one particular operator. This follows, just as there is a way to transform one permutation into another, so there will be certain combinations of operators that duplicate the result of using one.
There is a group theorem to the effect that there is a kernel which, when combined with a co-set will generate the entire group. (But I don't remember which of the operators corresponds to the kernel and which to the generating co-sets).
Group generality question. Yes, in a way that was the point. Because groups are so effective in ordering any collection with a binary rule, they are in a sense meaningless. That is one representation has to resemble another. Everything starts to look alike after awhile--because every representation is the same from a group-theoretic point of view.
Truth as intension question. Truth as in 'T' is obviously just a symbol that could be replaced with x or 1. However, assume that T means truth. The group structure is the same either way. So, in what sense does it matter what the assignment is, if the underlying structure is uneffected? That means that some part of the structure of the truth table is shared by Lacan's symbol system, even if they have nothing else in common. The point was only minor and it was intended to show that Lacan wasn't making a great escape from the 'system'.
On the other hand, this raises an historical point which was that the systems employed by the structuralists ignored content or intrinsic meanings entirely. This really pissed off a lot of people and probably explains why they generated such a strong reaction.
Chuck Grimes