Frames in the semantics of modal logic
In normal modal logic a frame is a couple <W, R>, with R a map of W into the powerset P(W) of W and with the operator □ on P(W) defined by
□X = {w: R(w) ⊆ X}
In neighborhood semantics a frame is a couple <W, N>, with N a map of W into the powerset P(P(W)) of P(W), and with the operator □ on P(W) defined by
□X = {w: X is in N(w)}.
That is a generalization: in a normal modal logic frame we can define N(w) = {X: w is in □X}, which is of course the family of supersets of R(w).
A general frame is a triple <W, N, P>, where <W, N> is as above, P is a Boolean algebra of subsets of W (‘the propositions’), and the operator □ is defined as above but only on P, not on P(W). (When the corresponding syntax is interpreted in a general frame, the semantic values assigned to sentences are all in P.)
That too is a generalization: When P = P(W) we have the special case of an ordinary neighborhood frame.
These three types of frames are clearly special instances of a more general type, which I will call the truly general frames:
A truly general frame is a triple <W, 𝜙, P> where 𝜙 is some mathematical entity or other, P is a Boolean algebra of subsets of W, and the operator □ is an monotone operator on P which is a function of W and 𝜙.
(I take the monotonicity of □ to be the most basic, and inalienable, characteristic of the subject of modal logic, however broadly construed.)
This truly general type can be put to work: I will give a simple example of its use for showing the independence of certain basic principles of modal logic.
Preliminary: a little possible world story
In one kind of world all the inhabitants are enormously cheerful, elated, they think everything is just great. I call these the Elated worlds. In another kind of world the inhabitants are odious for their negativity and pessimism, they think everything sucks, nothing is great. These are the Odious worlds. To describe what is the case in these worlds we introduce sentential unary connective □, read as “It is considered great that”. Note that for any sentence A, □A is true in all and only the Elated worlds. So for instance □(A v ~A) is not true in all worlds. But in other respects it’s all pretty normal. For instance, □ is monotone: if A implies B then □A implies □B.
Representation of the example
For convenience I will represent these worlds by natural numbers, with even numbers representing the Elated worlds and odd numbers the Odious worlds.
Frame ARITH = <W, 𝜙, P> with W the set of natural numbers (starting with 1), 𝜙 its natural ordering, P just P(W), and the operator □ on P defined by □X = the set of even numbers that are members of X.
Before interpreting the modal logic syntax in ARITH let’s just note a few features of this operator:
- T1. □ is a monotone operator on P: if X ⊆ Y then □X ⊆ □Y
- T2. □X ⊆ X
- T3. (□X ∩ □Y) = □(X ∩ Y)
- T4. □X= □□X
- T5. □⊥ = ⊥
- T6. □W ≠ W
(The proofs are straightforward, but see Appendix for some details.)
Interpretation, logic
I take the modal sentential syntax to be familiar and will use ‘□’ for the modal connective, the symbols ‘&’ and ‘~’ for conjunction and negation, ‘⊥’ for the falsum and ‘T’ for ‘~⊥’ (with the context preventing confusion).
A model is a frame together with an interpretation of this syntax in this frame, which is an assignment of propositions (subsets of W) to the sentences, by the recipe:
|A & B| = |A| ∩ |B|
|~A| = W minus |A|
|□A| = □|A|
|⊥| = Λ, the empty set
Validity in a model. If M is a model, with frame <W, 𝜙, P> and interpretation | |, we write ‘╞M A’ for |A| = W, and ‘A1, A2, …╞M B’ for (|A1| ∩ |A2| ∩ … ) ⊆ |B|.
Let us consider the following principles familiar in modal logic, dividing them into two groups (roughly following Chellas’ (1980) nomenclature).
Group One.
RM. If A├B then □A ⊢ □B (Monotonicity)
M. □(A & B) ⊢ (□A & □B)
C. (□A & □B) ⊢ □(A & B)
K. □(A ⊃ B) ⊢ (□A ⊃ □B)
N◊. ⊢ ~□ ⊥
D. □A ⊢ ~□~A
T. □A ⊢ A
S4. □A ⊢ □ □A
Group Two.
RN. If ├A then ⊢ □A
N. ⊢ □T
B. A ⊢ □ ~□~A
S5. ~□~A ⊢ □~□~A
Satisfaction and Violation in a model
I will call any such principle satisfied in model M if replacing ‘⊢’ in the principle by ‘╞M’ yields a true statement about M, and violated in M if it is not satisfied in M.
Independence.
A principle Q of modal logic is independent of principles Q1, Q2, Q3, … if there is a model M with a truly general frame such that Q1, Q2, Q3, … are sound in M and Q is violated in M.
Theorem. The modal logic principles in Group Two are independent of the principles in Group One.
Let M be a model with frame ARITH and interpretation | |. The proof is that all principles in Group One are satisfied in M, and all the principles in Group Two are violated in M.
The satisfaction of Group One follows in the main from facts T1. through T5. about ARITH. The violation of N, which implies the violation of RN, is due to fact T6., that □W is the set of even numbers, while W is the set of all natural numbers. B is similarly violated because |□~□~T| is some set of even numbers. For the violation of S5 we have to look to a proposition other than W, e.g the set <4] = {1, 2, 3, 4}. If |A| = <4] then |~□~A| contains all the odd numbers (as well as 2 and 4), while |□~□~A| ={2, 4}. (For more details see Appendix.)
APPENDIX
Let E be the set of even numbers. Then to prove T1. – T5., replace in each case ‘□’ by ‘E ∩’. The crucial fact that □W¹ W is simply that E is a proper subset of W. Note that if we look at ARITH as a augmented neighborhood frame, N(x) is not closed under supersets but under supersets within E, with E as fixed point, and that suffices to make □ monotone.
To prove the theorem, the remarks there suffice to show that the principles in Group Two are violated. To spell out the argument for S5, I used the intuitive notation <n] for the initial segment {1, …, n} of W, and we can equally set [ n> for the final segment {n+1, …} of W.
Then ~□~<4] = ~□[5> = ~{6, 8, …} = {1,2,3,4, 5, 7, 9, …}= the union of <4] with the odd numbers. But then □~□~<4] = {2, 4}, which includes no odd numbers.
That the principles in Group One are all satisfied in model M follows mainly from facts T1.-T5 about M. For N◊, see from T5 that |~□ ⊥| = W. For K, suppose that n is in |□(A ⊃ B)|. Then n is even and belongs either to |~A| or to |B|. If it belongs to |~A| then it is not in |A|, and hence not in |□A|. Therefore it either does not belong to |□A| or belongs |□B|.
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