ckaun%aids-unix@sri-unix.UUCP (03/08/84)
From: Carl Kaun <ckaun@aids-unix> Before I say anything, you all should know that I consider myself at best naive concerning formal logic. Having thus outhumbled myself relative to anyone who might answer me and having laid a solid basis for my subsequent fumbling around, I give you my comments about Laws of Form. I do so with the hope that it stirs fruitful discussion. First, as concerns notation. LoF uses a symbol called at one point a "distinction" consisting of a horizontal bar above the scope of the distinction, ending in a vertical bar. Since I can't reproduce that very well here, I will use parentheses to designate scope where the scope is otherwise ambiguous. Also, LoF uses a blank space which can be confusing. I will use an underline "_" in its place. And LoF places symbols in an abutting position to indicate disjunction. I will use a comma to separate disjunctive terms. In Lof, the string of symbols " (a)|, b ", or equivalently, " a|, b" is equivalent logically to the statement " a implies b". The comparison with the equivalent statement " (not a) or b" is also obvious. The "|" symbol seems to be used as a postfix unary [negation] operator. "a" and "b" in the formulae are either "_" or "_|" or any allowable combination of these in terms of the constructions available through the finite application of the symbols "|" and "_". LoF goes on to talk about this form and what it implies at some length. Although it derives some interesting looking formulae (such as the one for distribution), I could find nothing that cannot be equivalently derived from Boolean Algebra. Eventually, LoF comes around to the discussion of paradoxical forms, of which the statement "this sentence is false" is the paradigm. As I follow the discussion at this point, what one really wants is some new distinction (call it "i") which satisfies the formula " (i|)|, i". At least I think it should be a distinction, perhaps it should also be considered simply to be a symbol. The above form purports to represent the sentence "this sentence is false". The formulation in logic is similar to the way one arrives at complex numbers, so LoF also refers to this distinction as being "imaginary". At this point I am very excited, I think LoF is going to explore the formula, create an algebra that one can use to determine paradoxical forms, etc. But no development of an algebra occurs. I played around with this some years ago trying to get a consistent algebra, but I didn't really get anywhere (could well be because I don't know what I'm doing). Lof goes on to describe the distinction "i" in terms of alternating sequences of distinctions, supposedly linking the imaginary distinction to the complex number generator exp(ix), however I find this discussion most unconvincing and unenlightening. Now LoF returns to the subject of distinction again, describing distinctions as circles in a plane (topologically deformable), where distinction occurs when one crosses the boundary of a circle. In this description, the set of distinctions one can make is firmly specified by the number of circles, and the ways that circles can include other circles, etc. LoF gives a most suggestively interesting example of how the topology of the surface might affect the distinctions, and even states that different distinctions result on spheres than on planes, and on toroids than on either, etc. Unfortunately he does not expound in this direction either, and does not link it to his "imaginary" form above, and I think I might have given up on LoF at this time. LoF doesn't even discuss intersecting circles/distinctions. The example that LoF gives is of a sphere where one distinction is the equator, and where there are two additional distinctions (circles, noninclusive one of the other) in the southern hemisphere. Then the structure of the distinctions one can make depends on whether one is in the northern hemisphere, or in the southern hemisphere external to the two distinctions there, or inside one of the circles/distinctions in the southern hemisphere. As I say, I really thought (indeed think today) that perhaps there is some meat to be found in the approach, but I don't have the time to pursue it. I realize that I have mangled LoF pretty considerably in presenting my summary/assessment/impressions of it. This is entirely in accordance with my expertise established above. Still, this is about how much I got out of LoF. I found some suggestive ideas, but nothing new that I (as a definite non-logician) could work with. I would dearly love it if someone would show me how much more there is. I suspect I am not alone in this. Carl Kaun ( ckaun@AIDS-unix )