Second-order logic: change talk of standard semantics to set-theoretic semantics, and contrast to the dependence of Henkin semantics on simply-typed lambda-calculus -- Apr 2009
Boole did not propose a separate syllogistic calculus, rather what he proposed was an interpretation of syllogistic into his algebra. There are a few accounts of the interpretation on the web, of which (Burris 2001) is perhaps the best.
The current article is talking about how this interpretation proposes a resolution to the problem of existential import (see
square of opposition), one incompatible with Aristotle assertions. One of the many problems with the article as it stands is that it isn't clear that that is what it is doing. I suppose no one else is going to fix it but me: I'll get around to it...
Naturally this calculus is propositional logic and not predicate logic. The embedding of syllogistic into propositional logic shows that syllogistic corresponds to a very weak fragment of predicate logic.
What standards do we want for these articles? How about elementary formal reasoning, sufficient for detailed discussion of soundness theorem & deduction theorem, but not as much as needed for completeness theorem?
Tasks for sometime: &c
Free software
Raph Levien: incorporate material on remailers (see talk) --- Mar 2007
In order to show that the axioms of the class of algebras we consider capture exactly the collection of predicates we have in mind, a representation theorem is necessary. A representation theorem is a correspondence between an abstract algebra and its set-theoretical model. The first representation theorem is due to Cayley [Cay78] showing that every abstract group is isomorphic to a concrete group of permutations. A representation theorem for the algebra of all predicates was first proved by Lindenbaum and Tarski [Tar35]. They proved that a Boolean algebra is isomorphic to the collection of all subsets of some set if and only if it is complete and atomic. This general result restricts the class of Boolean algebras for which a concrete representation exists. It was Stone [Sto36] who first saw a connection between algebra and topology. He constructed from a Boolean algebra a set of points using prime ideals which can be made into a topological space in a natural way. Conversely, using a topology on a set of points he was able to construct a Boolean algebra. For certain topological spaces (later called Stone spaces) these constructions give an isomorphism. In a later paper [Sto37], Stone generalized this correspondence from Stone spaces to spectral spaces and from Boolean algebras to distributive lattices. Hofmann and Keimel [HK72] described the Stone representation theorem in a categorical framework showing a duality between the category of Boolean algebras and a sub-category of topological spaces. A representation theorem for Boolean algebras with operators has been considered by J'onsson and Tarski [JT51, JT52]. By means of an extension theorem they proved that operators on a Boolean algebra can be naturally extended to completely additive operators on a complete and atomic Boolean algebra.
Stone's representation theorem leaves open the problem of finding an abstract characterization of topological spaces. For every topological space, its lattice of open sets forms a frame. This fact leads Papert and Papert [PP58] to a representation theorem between spatial frames and sober spaces. Even further, Isbell [Isb72a] gives an adjunction between the category of topological spaces with continuous functions and the opposite category of frames with frame homomorphisms. This adjunction yields a duality between the category of sober spaces and the category of spatial frames.
Computer Science
LISPy things:
LISP macro systems: namespace issues from
RPG's Technical Issues of Separation paper, history of macros (MacLisp, etc.), CL macros (DEFMACRO, FLET), Hygiene (Kohlbecker's algorithm, syntactic closures, macros that work, R4RS syntax-rules, syntax-case) ---- 15 Dec 2004