# Do statements about borderline cases hold for both the vague term and its negation?

I read subvaluationists think that P can be both true and false (unlike supervaluationists, who think that P is neither true nor false), but it's completely unclear (because I can't read symbolic logic and haven't found an introduction) whether they - or indeed anyone else - claim that borderline cases are both P and not P.

For subvaluationists are logical quantifiers of borderline cases true for both P and not P? e.g. if some people in my family are borderline bald then are some people in my family borderline not bald?

I am asking because I think it likely (won't bother saying why) that my consciousness is vague and necessarily not everything. I am trying to work out whether (for any treatments of vagueness) with borderline cases of consciousness a quantifier is true of both consciousness and its negation.

Because if so, borderline cases of consciousness are necessarily not absent from everything. And I think I find that interesting (and equivalent to saying that borderline states of consciousness necessarily exist), perhaps pending finding out that it also isn't.

• See Subvaluationism: "Whereas the supervaluationist analyzes borderline cases in terms of truth-value gaps the dialetheist analyzes them in terms of truth-value gluts. A glut is a proposition that is both true and false. The rule for assigning gluts is the mirror image of the rule for assigning gaps: A statement is true exactly if it comes out true on at least one precisification. The statement is false just if it comes out false on at least one precisification. 1/2 Aug 28, 2023 at 9:33
• So if the statement comes out true under one precisification and false under another precisification, the statement is both true and false." 2/2 Aug 28, 2023 at 9:33
– user67521
Aug 28, 2023 at 9:38
• Yes, borderline cases of vague predicates are one of the standard candidates for dialetheias, a.k.a. truth value gluts, they are both true and false, see SEP, Other Motivations for Dialetheism. In subvaluationism, this is because a statement is true/false when it is so on at least one precisification, and borderline cases, by definition, admit opposite precisifications. Aug 28, 2023 at 9:50
• The simple answer is YES, because negation in dialetheical logic works as usual: reversing the truth value. Aug 28, 2023 at 9:51

The comments here are quite good, but I want to pick up on something that I was interested in.

Consider the quantifier exchange rules:

¬∃xPx ↔ ∀x¬Px (there's not some P iff everything is not P) ∃x¬Px ↔ ¬∀xPx (there's some P iff not everything is P)

if some people in my family are borderline bald then are some people in my family borderline not bald?

Consider now two incompatible properties, Px and Gx, where to be Gx is just to not be Px.

i.e., we could say:

Px↔¬Gx and Gx↔¬Px

Maybe we could interpret Gx as a vague predicate and interpret Px as a precise predicate' there are now two ways to show their incompatibility, using quantifiers:

¬∃xGx ↔ ∀xPx (there's not some borderline case iff everything is precise)

∃xGx ↔ ¬∀xPx (there's some borderline case iff not everything is precise)

and

¬∃xPx ↔ ∀xGx (there's not some precise case iff everything is borderline)

∃xPx ↔ ¬∀xGx (there's some precise case iff not everything is borderline)

I think it likely (won't bother saying why) that my consciousness is vague and necessarily not everything. I am trying to work out whether (for any treatments of vagueness) with borderline cases of consciousness a quantifier is true of both consciousness and its negation. Because if so, borderline cases of consciousness are necessarily not absent from everything.

my consciousness is vague and necessarily not everything.

seems equivalent to saying there's some borderline case (∃xGx). Given the incompatible predicates, this is true iff: not everything is precise (¬∀xPx)

borderline cases of consciousness are necessarily not absent from everything.

This is a bit confusing, but here I think you're saying: (necessarily) everything is borderline (∀xGx). This is true iff ¬∃xPx.

In other words without having to worry about what to believe regarding super/subvaluationism or vagueness or the like, your inference "∃xGx, therefore ¬∃xPx" will not work because: ¬∃xPx ↔ ∀xGx

My takeaway: you will not have to worry about the unrestricted quantifiers handling matters vague and precise at once, unless you think everything is precise or everything is vague. In which case unrestricted quantification over vagueness will tell you what is precise: nothing. Mutatis mutandis for the precise.

• err. i was saying that if borderline cases hold for both the vague term and its negation, then supposing not everything is borderline conscious, then something is borderline conscious. it may not be obvious why! can't read symbolic logic, sorry
– user67675
Sep 12, 2023 at 18:23
• Using the quantifier exchange rules: not everything is borderline conscious iff there's something that is not borderline conscious. Sep 12, 2023 at 18:28
• OK. does subvaluationism get what i wanted?
– user67675
Sep 12, 2023 at 18:29
• I gave it some thought and I... am not sure. Here's a negative view: it won't give you what you wanted in that inference. It's the analogous to saying that not everything is an apple, so there's an apple. It won't matter if we can't tell if some particular thing is an apple or not, it's just a matter of how the quantifiers work. Sep 12, 2023 at 18:43
• apples aren't borderline states, but yeah thanks. dunno why i can't just intuitively pick up symbolic logic
– user67675
Sep 12, 2023 at 18:45

if some people in my family are borderline bald then are some people in my family borderline not bald?

Yes, but be careful not to switch "borderline not bald" with "not borderline bald".