I was thinking about what happened to my theory of vagueness if one dropped the constraint that all contradictions are perfectly false. The logic that one gets is actually rather interesting, but I don’t see any reason to believe that it’s the right theory of vagueness.
Note that there are two ways one could go about dropping non-contradiction. First, one could drop it because one reinstates all the intuitions behind M, and in particular the idea that truer is a linear order. That would be a bad idea for several reasons. I want to write this argument up in more detail, but I’m starting to think th deep problem with linearity is that once it is imposed one cannot handle higher-order vagueness. The problem arises from the combination of the following three points:
- If we want linearity in the theory, presumably we want it to be determinately true that truer is linear
- What it is for truer to be linear is for a quantified disjunction to be true: for all A, B either A is truer than B or B is truer than A or A is as true as B
- If truer is linear, then if a disjunction is true, then one of its disjuncts must be true
The first two points combine to tell us that for any A, it is determinate that A is as true as 0=0 or A is less true than 0=0. (I assume here what should be utterly uncontroversial, that a universally quantified disjunction is only determinately true if every instance of it is determinately true.) The third point tells that this means one or other of these disjuncts must be determinately true, if the disjunction is to be determinately true. But this is to rule out a certain kind of higher-order vagueness, the kind where it is indeterminate whether A is determinately true. Or, more precisely, it is to rule out an analysis of this possibility in terms of truer. I think this is a Very Bad Thing. Note that this is a reason to abandon linearity that is independent of worries about contradictions. (I might yet revise the truer paper to include this argument, though to be honest the thing is getting pretty long as it is.)
But that’s not the only way to abandon non-contradiction. We could simply drop the idea that contradictions are perfectly false without insisting on any positive theory. My overall position will be that such a position is coherent but under-motivated. I think the only reason one might want non-contradiction is that one thinks truer is linear, and linearity is a Very Bad Idea. But if one has independent motivation for scrapping non-contradiction, then the resulting position looks perfectly coherent.
In my theory I assume three things about negation:
- Self-inverting: ~~A is as true as A
- Order-inverting: A is truer than B iff ~B is truer than A
- Non-Contradiction: Nothing is less true than both A and ~A
It’s certainly possible to keep the last two and drop the first. That way lies intuitionism. In general it’s possible to drop the second and keep the first and third, but the only examples of that I can come up with violate distributivity. I’m not sure whether the other constraints I put in to get distributivity mean that the first and third entail the second. There’s an interesting little research project there, but not one I want to finish today. What I want to sketch today is how to keep the first and second without keeping the third.
Assume that the truth values are subsets of [0,1), with A being truer than B iff V(A) is a proper superset of V(B). For ease of notation, I’ll write c(S) for the complement of S. Assume we have a self-inverting function f from [0,1) onto itself. Identity is such a function, but it’s not the only such function. Plus 0.5 mod 1 (if that makes sense) is also such a function. I’ll write f(S) for {f(x): x is an element of S}. I won’t assume anything about f other than that it is self-inverting. Intuitively, the models for the logic are all models satisfying the following constraints where f is a self-inverting function. Now define the connectives the following way:
- V(A and B) = V(A) intersection V(B)
- V(A or B) = V(A) union V(B)
- V(If A then B) = c(V(A)) union V(B)
- V(Not A) = c(f(V(A)))
As far as I can tell, the logic you get is just classical logic with reductio replaced with a rule form of contraposition: If A entails B then ~B entails ~A. (The sequent form of contraposition need not hold. If A then B does not entail If not B then not A.) We keep all the other inference rules, including double negation elimination. (We don’t have ex falso, but I’m assuming that isn’t a primitive rule in any case.) It’s easiest to think about the logic I find if you assume f is Plus 0.5 mod 1, or intuitively a rotation by 0.5. This makes it easy to see that applying the rule for not twice gets you back to where you started, that negation is order-inverting, and that apart from that there’s really very little we can tell about negation.
This is a moderately interesting logic, though for me its main interest is psychological. I really don’t understand how this logic behaves despite knowing both its semantics and the natural deduction rules for it. I would have thought antecedently that once I knew those things about a logic then at some level I understood it. But not here.
Anyway, if one simply wants to drop non-contradiction, here’s a logic that does it. As you might imagine, excluded middle is not a theorem of this logic. Personally, I think the only motivation to drop non-contradiction is some left over linearity intuition, and that’s a bad intuition, so I think this theory is undermotivated. But if it really is motivated, here’s a way to formally capture the intuitions.