Note: this is a reflection on Ali Farjami’s Up operator — I would like to think about it in a very simple context, to begin.
That logic is mainly about consequence relations, and that these correspond to closure operators, was a theme introduced, I believe, by Tarski. A closure operator C on sets of sentences has these characteristics:
(a) X ⊆ C(X)
(b) if Y ⊆ X then C(Y) ⊆ C(X)
(c) C(C(X) ⊆ C(X)
From these it follows that C(C(X) = C(X).
A set X is called C-closed, or a C-theory, exactly if X = C(X).
The most familiar closure operator in logic is the one that corresponds to to the most familiar consequence relation:
Cn(X) = {A: X├A}
and a Cn-theory is just called a theory.
However there is another consequence relation, introduced in discussion of deontic logic, in effect, by Farjami. Here the consequences of any and all single members of the set are gathered together, but there is not ‘putting together of premises’. To indicate the reference to Farjami’s Up operator, I’ll call it Cu:
Cu(X) = the union of the sets {A: B├A} for members B of X
or equivalently:
Cu(X) = ∪{Cn(B): B a member of X}
That Cu has properties (a), (b), and (c) is clear, so it is a closure operator. We can call the relation of B to X if B is in Cu(X) a consequence relation for that reason, though it is very different from the usual one.
The difference from Cn is clearly that, for example, (A & B) is a member of Cn({A, B}) but not of Cu({A, B}).
A Cu-theory is often not consistent, in the usual sense: Cu({A, ~A}) does not contain any explicitly self-contradictions (in general, e.g. if A is atomic), but it is clearly not classically consistent.
But this is just why Cu can represent the proper consequences of a set of commands, imperatives, inputs, or instructions, when that set may harbor conflicts — and hence useful to deontic logics which countenance irresolvable moral conflicts.
There is an definition of consistency, however, that can apply non-trivially to a Cu-theory. Call it A-consistency: a Cu-theory is A-consistent iff it does not contain all sentences (equivalently, it does not contain any sentence that is a classical self-contradiction).
So let us see how that can work, let us focus on the following minimal deontic logic which I will call VHC:
A1. Axiom and rule schemata for classical sentential logic
A2. ├~O(~A &A)
R1. if ├A and ├A ⊃ B, then ├ B
R2. if ├ A ⊃ B, then ├ O(A) ⊃ O(B)
There is a corresponding consequence relation, ‘ ├ in VHC‘ .
It is quite clear what those axioms and rules tell us:
a theory in VHC is a theory X in classical sentential logic with this characteristic, that the set {A: OA is in X} is an A-consistent Cu-theory.
A simple way to model this would be to think of a possible world model: in each world there is an agent who recognizes a certain set of sentences as expressing the primary obligations in force — thus, OA is true in this situation exactly if one of those primary obligation sentences implies A.
But this is too simple, it ignores infinity. A Cu-theory may not be ‘axiomatizable’, it may not be possible to sum it up in that way. Suppose a1, a2, a3, … are the countably many atomic sentences in the language, and let Y = {a1, (a1 & a2), (a1, & a2 & a3), ….}. Then there is no sentence B such that Cu(Y) = Cu(B). In fact there is no finite set Z of sentences such that Cu(Y) = Cu(Z).
We may call a set like Cu({a,b}) finitely generated, but if a, b are logically independent, such as two atomic sentences, then there is no sentence B such that Cu({a, b}) is the same as either Cu(B) or Cn(B). We are used to finite descriptions allowing for a ‘summing up’ in a single sentence, but that is not the case for Cu-theories.
Here we have the motivation to think in terms of the algebra of propositions instead of logic of sentences. In a possible world semantics the propositions are the sets of worlds, hence form a Boolean algebra which is complete. Even in infinitely descending chain of ever stronger propositions has an infimum which is a proposition (every maximal filter is a principal filter).
So the way to set up a possible world model structure is to associate with each world α a set I(α) of propositions. Then when the truth conditions of sentences are spelled out, so that each sentence A expresses a proposition |A|, the condition for “it ought to be” is:
OA is true in α if and only if there is a proposition Q in I(α) such that Q ⊆ |A|
or equivalently
|OA| = {α: there is a proposition Q in I(α) such that Q ⊆ |A|}
which shows clearly that the connective O corresponds to an operator on propositions.
Soundness and completeness for VHC can be discussed with this clue:
the set of sentences true in a world is a maximal theory in VHC, and that is a set of sentences X which is a theory in classical sentential logic, negation complete, and such that {A: OA is in X} is an A-consistent Cu-theory.
Now, how shall we think about those cases in which the Cu-theory in question is not ‘axiomatizable’? It is a situation in which there are more primary obligations than the agent could have spelled out for himself, even in principle, in the language s/he has.
It seems to me that this is the sort of world we live in. In Roman times even the Christians did not realize that slavery is wrong — that was a moral insight that we, Western people, did not yet have. Perhaps this is typical. Not only perhaps, it seems to me, but likely, and I would frame this as something that philosophers writing on ethics, who are not logicians, do not seem to have:
Infinity: for every moral norm in which we gain insight, there is yet another one.
Bas van Fraassen's philosophy blog 