A few definitions:
DEFINITION 1: A system S is a collection of propositions, along with a system of inference. We'll be using standard logic. If p is a proposition in S, write p ∈ S. If p is a proposition not in S, write p ∉ S.
DEFINITION 2: A system is closed if (p ∈ S and p ⇒ q ∈ S) ⇒ q ∈ S.
Example: Suppose that "Jane is in France" ∈ S, and "If Jane is in France, then Jane is in Europe" ∈ S. If S is closed, then "Jane is in Europe" ∈ S.
DEFINITION 3: The closure of a system S, Cl S, is the collection of all p ∈ S together with all propositions entailed by propositions in S.
Proposition 1: By definition of a system, if S is a system then Cl S is itself a system.
Proposition 2: Cl S is clearly closed, by definition of closed.
Example: Suppose "Jane is taller than George" ∈ S, and "George is 5 feet tall" ∈ S. Then "Jane is taller than 5 feet" ∈ Cl S.
DEFINITION 4: A system S is inconsistent if Cl S contains a contradiction. Otherwise, S is called consistent.
Example: Suppose "John is taller than Kate" ∈ S, "John is 5 feet tall" ∈ S, and "Kate is 6 feet tall" ∈ S. Then S is inconsistent.
HERE'S THE CHALLENGE:
Does there exist a system S satisfying the following properties:
i) S is consistent;
ii) "At least one god exists" ∉ Cl S
iii) "Objective moral values exist" ∈ Cl S
---------- Post added at 11:45 AM ---------- Previous post was at 11:38 AM ----------
So a nice thing about this terminology is that it gives you a nice way of talking about systems that implement other systems, or that extend other systems.