Starting with a given set of set variables, set terms are e.g. the empty set symbol or more complex expressions constructed by means of the intersection, union and difference symbol. Any two of these set terms, combined with the inclusion symbol, make a set formula. More complex set formulas are then constructed by means of the conjunction, disjunction, and negation symbol, as usual.
A set field interpretation assigns a subset of a given carrier set C to each of the set variables. Every set field interpretation induces a function, that turns each set term into a subset of C as well, and another function turning every set formula into an either true or false statement.
This interpretation concept itself induces a boolean order relation on both the set terms and the set formulas. A set term σ is subvalent to a set term θ if and only if every set field interpretation turns σ into a set included by the set of θ. And a set formula φ is subvalent to ψ if and only if, for every set field interpretation, the truth of φ implies the truth of ψ.
The second part of this text demonstrates, how this whole set field logic is embedded into hyper-propositional logic: set terms translate into first degree formulas, set formulas become second degree formulas and the entire boolean order is preserved on both levels.
SetFieldLog.pdf (110 KB),
SetFieldLog.ps (82 KB),
SetFieldLog.dvi (42 KB)
May 2007, 6 pages.
Comprises five figures: 1. Bit values and their algebra; 2. Bit tables and their algebras; 3. Hyper-propositional logic; 4. Set field logic; 4. Embedding set field logic into hyper--propositional logic
The text is a dense summarized overview of the subject, the first three figures are just a repetition of hyper-propositional logic as given in the earlier paper "Hyper-propositional logic: Summarized overview of its syntax and semantics". A better explained version is currently being prepared.