Laura Dvorak 5:47 p.m., Dec. 6
A while back, there was a discussion of what an ethic is. I can't seem to find the discussion here anymore, but I spent some time thinking about it.
For now, my definition of an ethic is a formal logic that contains "If A, then B" statements, where A is a set of conditions that are met and B is the right, proper action to take with respect to the met conditions.
On page 76 of Elliott Mendelson's third edition of Introduction to Mathematical Logic, there is Exercise 2.64, describing the formal system K2 which consists of the usual symbols for a first-order predicate calculus, five logical axioms, 8 proper axioms, and two rules on inference to give us a theory of densely ordered sets with neither first or last element. I believe that this formal theory K2 is sufficient to model various forms and augmentations of utilitarianism, and because it is not so complex as a Peano-style formal number system, it does not run into the problems of Gödel's Incompleteness Theorem for undecidable number-theoretic formulas. In other words, K2 provides an unambiguous foundation for further defining a specific ethic through the introduction of appropriate proper axioms, and if the negation of those added axioms cannot be proven in K2, then the ethic developed from K2 must be consistent. It would then be needed to arrive at a completeness proof similar to Gödel's Completeness Proof for First-Order Predicate Calculus as offered by Mendelson.
The reason for jumping through the completeness/consistency hoop is to get rid of all of the semantic ambiguity that a lot of people love to introduce into ethical arguments. The necessity for developing a formal definition of an ethic came up as part of seeking a grand unification theory encompassing Judaism, Christianity, and Islam.
I'm still working on the ethics paper. I'll probably dump a PDF copy of it somewhere on the Internet, as it has a lot of symbols that would be a royal pain to reproduce correctly in plain HTML.