A few definitions:

DEFINITION 1:AsystemSis a collection of propositions, along with a system of inference. We'll be using standard logic. Ifpis a proposition inS, writep∈S. Ifpis a proposition not inS, writep∉S.

DEFINITION 2:A system isclosedif (p∈Sandp⇒q∈S) ⇒q∈S.

Example: Suppose that "Jane is in France" ∈S, and "If Jane is in France, then Jane is in Europe" ∈S. IfSis closed, then "Jane is in Europe" ∈S.

DEFINITION 3:Theclosureof a systemS, ClS, is the collection of allp∈Stogether with all propositions entailed by propositions inS.

Proposition 1: By definition of a system, ifSis a system then ClSis itself a system.

Proposition 2: ClSis 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" ∈ ClS.

DEFINITION 4:A systemSisinconsistentif ClScontains a contradiction. Otherwise,Sis calledconsistent.

Example: Suppose "John is taller than Kate" ∈S, "John is 5 feet tall" ∈S, and "Kate is 6 feet tall" ∈S. ThenSis inconsistent.

HERE'S THE CHALLENGE:

Does there exist a systemSsatisfying the following properties:

i)Sis consistent;

ii) "At least one god exists" ∉ ClS

iii) "Objective moral values exist" ∈ ClS

---------- 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 thatimplementother systems, or thatextendother systems.

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