Non-normal Worlds. Daniel Bonevac. February 5, 2012

Transcription

1 Non-normal Worlds Daniel Bonevac February 5, 2012 Lewis and Langford (1932) devised five basic systems of modal logic, S1 - S5. S4 and S5, as we have seen, are normal systems, equivalent to K ρτ and K ρστ, while T = K ρ. What about S1, S2, and S3? They are non-normal systems. They allow worlds where anything is possible. 1 Dead Ends and Non-normal Worlds Say that a world w is a dead end iff no worlds are accessible to w: x wrx. All necessity statements are true vacuously in such worlds; all possibility statements are false in them. If w is a dead end, for any formula A, v w ( A) = 0 v w ( A) = 1 Conditions η and ρ rule dead ends out, of course, but they are possible in K, and show that K has no valid formulas of the form A. The opposite of a dead end is a non-normal world (Kripke 1965), a world in which all necessity statements are false and all possibility statements are true. If w is a non-normal world, for any formula A, v w ( A) = 1 v w ( A) = 0 In normal logics such as K and its extensions, there are no such worlds. There are no worlds in which (p p) is true, for example; that would require a world in which p p was true. At non-normal worlds, however, everything is possible. The laws of logic are in effect suspended there. (There is thus a sense in which non-normal worlds give the effect of adding impossible worlds to our semantics, without actually doing that.) 1

2 Non-normal modal logics allow the possibility of non-normal worlds. So, we need to expand our concept of an interpretation. A non-normal interpretation or model I =< W, N, R, ν > is just as before, but with N W. N is a set of normal worlds; W N, a set of non-normal worlds. There are now options for defining validity. Is it truth-preservation in all worlds in all structures? Or only in the normal worlds? System N results from choosing the latter. (If we were to choose the former, we would have no validities of forms A or A. Dead-end worlds would rule out all validities of the form A, and non-normal worlds would rule out all validities of the form A. This is thus a very weak logic, significantly weaker than N.) Say that ν w (X) = 1 for all B X, then ν w (B) = 1. X = N A for all I =< W, N, R, ν > and all w N, ν w (X) = 1 ν w (A) = 1. By adding nonnormal worlds, we add models. We thereby reduce the set of validities. Whenever we expand the class of models, we make it harder for something to hold in all models, so we shrink the set of validities. Conversely, whenever we reduce the class of models, we expand the set of validities. The system N that results from adding nonnormal worlds to K structures is thus weaker than K. Anything that holds in all non-normal models holds in all normal models, so everything valid in N is valid in K. But the reverse is not true. In N, for example, necessitation fails. = (p p) but not = (p p). At any nonnormal world, (p p) holds, because all possibility statements hold. It holds at no normal worlds, so (p p) is valid; but because of the non-normal worlds, (p p) is not valid. (p p) fails at a non-normal world, so its necessitation fails at a normal world. The simplest K-valid formula that fails in N is (p p). All boxed formulas are false at non-normal worlds, so, at a normal world having access to a non-normal world, a double-boxed formula is bound to fail. (Worlds like that normal worlds with access to non-normal worlds thus play a big role in non-normal modal logics. They re called, unimmaginatively, standard worlds. I think of them as Susan worlds, after the movie Desperately Seeking Susan. Susan is the non-normal one; all the other characters begin to act in unusual ways because of their access to her.) What about distribution, the characteristic K axiom A, (A B) = B? It still holds. A and A B must hold at every accessible world, so B must hold there as well. We can generalize the point. Claim: Any validity without nested modal operators that is valid in K is also valid in N. Why? When in a normal world, a modal operator sends us to an accessible world. It might be normal or non-normal. But, if there are no nested modalities, we are looking only at non-modal features of such a world. And on non-modal features, normal and non-normal worlds are alike; the non-normality makes no difference. Are there any N-validities with nested modal operators? Yes, but they are boring: p p, for instance. There are no validities of the form A. Moreover, validities in non-normal systems never take the form A A, in which the consequent adds more modal operators than are there in the antecedent, for there is a risk of encountering a non-normal world. 2

3 2 Lewis Systems We can add constraints to N to obtain additional and more familiar systems of modal logic. S2 and S3 are non-normal systems extending N. S2 = N ρ S2 is the extension of N determined by the class of all interpretations whose accessibility relations are reflexive. Like T, it distinguishes infinitely many distinct modalities. And, like T, it has as axioms A A and (A B) ( A B)). But it does not contain an unrestricted rule of necessitation. = A = A holds only where A is a propositional tautology or an axiom of S2. Why does the latter work? Because A A and (A B) ( A B)) hold at non-normal as well as normal worlds, since A and (A B) are false at all non-normal worlds. S3 = N ρτ S3, which Lewis himself had formulated as early as 1918, is the extension of N determined by the class of all interpretations whose accessibility relations are reflexive and transitive. It differs from S2 only in adding a box to each of S2 s axioms, thus: ( A A) and ( (A B) ( A B)). Necessitation holds only for propositional tautologies or axioms of S3. Note that we do not get A A; non-normal worlds can make the consequent false even in the presence of transitivity. We do, however, get ( A A) B. Any normal world is such that either it has access to a non-normal world or it doesn t. If it does, B holds there; if not, A A does. This is an odd result, however, since A and B don t necessarily have any relation to each other. This makes S3 unreasonable in the sense of Hallden. S3 distinguishes 42 irreducible modalities. S3.5 = N ρστ S3.5, formulated by Aqvist in 1964, adds to S3 the axiom schema A A, which is familiar from S5. It is determined by the class of interpretations with equivalence relations as accessibility relations. S3.5 has 26 irreducible modalities: A, A, A, A, A, A, A, A, A, A, A, A, A, and their negations. Logicians have developed a variety of other systems extending S3, most notably S3.1, which adds ( A A); S3.2, which adds ( A A), and S3.3, which adds ( A A). This discussion might make one wonder about Lewis and Langford s S1. It is so weak that it doesn t even treat its own axioms as necessary. It therefore requires a different approach. Cresswell gave it a semantics in It has as axioms A A and (A B) (B C) (A C). 3

4 3 Non-normal Tableaux To devise tableaux for non-normal systems, we need the notion of an inhabited world. Non-normal worlds count all possibility statements true and all necessity statements false. So, if a world w makes any formula of the form A true, w is normal. If i occurs on b, i is -inhabited iff some node on b has the form B, i. i is -inhabited iff w i is established as normal in the tableau. Tableaux rules for N are just the same as for K, except that the rule applies only to normal worlds. Any world that is -inhabited is normal. So, we apply the possibility rule to A, i only if i = 0 or i is -inhabited. And we can infer a world s normality from its making something of the form A true. A, i Ni Ni A, i A, i A, i Ni It is easy to explain the rationale for the restriction. A might be true for the wrong reason. Non-normal worlds make all possibility statements true. So, in a non-normal world, nothing can be inferred from the truth of A; it is true automatically, whether there are any accessible worlds in which A or not. Recall that X = A iff every normal world making all of X true makes A true. We begin a tableau, therefore, by assuming that there is a normal world in which X, A holds. So, world 0 is always normal. We thus begin a tableau: X, 0 A, 0 N0 We can read countermodels off open branches as before, except that world 0 and -inhabited worlds are normal, and all others are non-normal. A K-interpretation is in effect an N-interpretation in which all worlds are normal: W = N. So, K is an extension of N. 4 Normal v. Non-normal Logics How do non-normal logics differ from normal ones? Primarily, on the rule of necessitation, as we have seen. Consider a tautology such as A A. It holds in all worlds of all models, so = N A A. Since it holds in all normal worlds of all models, = N (A A). But any non-normal world makes (A A) false, since it makes all necessity statements false. So, if there are any non-normal worlds, (A A) fails, even in normal worlds, so it is not valid in N. This shows not only that necessitation fails but also that we cannot add the characteristic S4 axiom A A to a non-normal logic without collapsing it into a normal 4

5 one. Since (A A) is valid but (A A) is not, clearly (A A) (A A) fails. That is why S3.1, S3.2, etc. all add weakenings of the characteristic S4 axiom rather than the axiom itself. Notice that the generalizations we formulated about validity in K do not hold in N. X = A does not imply X = A. Suppose that in some normal world w, X holds. Then X holds at every world accessible to w. X = A allows us to conclude, not that A holds in every such world, but only that A holds in every such normal world. So, we cannot conclude that A holds in w. Similarly, A = B does not imply A = B. Suppose A is true in some normal world w. Then A holds in some w accessible to w. If w is normal, then B must also hold in w, and so B must hold in w. But we have no guarantee of that. 5