# Identity in Quantified Modal Logic

Why is ¬(◇(a=b)∧◇¬(a=b)) a validity in Quantified Modal Logic (QML)? For example, let a:=“the present King of France” and b:=“the richest bald person alive”. Then, it seems ◇(a=b)∧◇¬(a=b) is not a contradiction, since it could have been that a King of France was at a time the richest bald person alive. Clearly, it is false now. So, am I to assume that QML treats the objects in a domain as rigid designators? If so, what is the philosophical motivation?

• Do you have a source for this claim? Thanks! Apr 25 at 20:22
• See also this rather lengthy SEP article for further discussion. Apr 26 at 1:10
• Rigid designator objects of QML's domain since Barcan's formula was motivated to simplify its model theory with constant domain semantic metaphysics across PWs as opposed to the de dicto varying domain semantics which most are accustomed to. OTOH the flexible intensional modal operator can be easily adapted to bisimulate from both constant/varying domain approaches thus they're equiexpressive/isomorphic in a sense with embeddable translations. But Kant heeded long ago beware of any claimed metaphysical necessities though there's no harm to use reason to prove there's no such rationality... Apr 26 at 5:40

That formula is indeed a validity under Kripke's understanding of the necessity of identity, with names as rigid designators.

To understand why your example is not a counterexample, we need to be careful to distinguish between names and definite descriptions. For Kripke, a name identifies an individual across worlds, but definite descriptions typically do not. The expressions "the present King of France" and "the richest bald person alive" are definite descriptions, not names. Definite descriptions and names behave differently when using cross-world identity.

For example, "the inventor of the zip" is a definite description, and let's suppose that Julius is the name of the person who invented the zip in our universe. It makes sense to say, "Julius might not have invented the zip," in the sense that in some other universe somebody other than Julius does. But it does not make sense to say, "the inventor of the zip might not have invented the zip". Whoever invents the zip, in whatever universe, does invent the zip in that universe. There are different ways of expressing this difference, including using an 'actually' operator.

It is possible that the descriptions "King of France" and "richest bald person in the world" apply to the same individual, and also possible that they do not. This can be expressed by saying there is a universe (or possible world) where the same person is both, and there is also a universe where one person is King of France and a different person is the richest bald person.

But if we use names instead, then if Louis is both King of France and richest bald person in our universe, he is so in all universes, since the name Louis rigidly designates this individual across all possible worlds. On the other hand, if in our universe Louis is King and Charlie is the richest bald person, and they are different persons, then they are different in all possible worlds.

In your formula, a and b are names of individuals, so a=b either holds in all possible worlds, or does not hold in all possible worlds, so ◇(a=b) ∧ ◇¬(a=b) always comes out false.

As to the philosophical motivation for Kripke's account of rigid designators, it is based on his theory of the relationship between the concepts of names, identity, reference and necessity. It is expounded in his books, Naming and Necessity, and Reference and Existence. There is also a helpful article on rigid designators in the Stanford Encyclopedia.

Kripke is concerned with what we might call a metaphysical concept of necessity. This is distinguished from what is knowable a priori, or what is provable. The fact that two names identify the same individual does not entail that we know that they do, nor that we can prove that they do. This is why on Kripke's account there can be propositions that are a posteriori necessary (and also contingent a priori).

Kripke holds that it is metaphysically necessary that everything is identical with itself. Thus we have:

1. Kripke postulates the necessity of self-identity, i.e. (∀x)□(x = x)
2. Leibniz's law of the indiscernibility of identicals: (∀x)(∀y)((x = y) → (Fx ↔ Fy))
3. Now instantiate Fz with □(x = z) to get (∀x)(∀y)((x = y) → (□(x = x) ↔ □(x = y)))
4. Hence by classical logic, (∀x)(∀y)((x = y) → □(x = y))

Therefore, if identity holds, it holds necessarily. If we wanted instead to understand □ as an epistemic or proof-theoretic operator, we would not make use of the assumption in 1, since that would amount to assuming that identity relations are always known or provable.

There is some more on this in John Burgess, "On a Derivation of the Necessity of Identity" Synthese (2014) 191, pp. 1567-1585.

• What if we interpret □p as “there is a proof of p”? We wouldn’t want □(a=b)v□~(a=b) in that case. If that were true, then on the mere assumption that some complex number x is a counter-example go the Riemann Hypothesis exists it would follow that for each complex number z=(a+bi) that it is provable that z=x or it is refutable that z=x. This is a problem for Kripke’s interpretation of numbers are rigid designators. Apr 25 at 19:02
• Kripke is concerned with what we might call a metaphysical concept of necessity. This is distinguished from what is knowable a priori, or what is provable. The fact that two names identify the same individual does not entail that we know that they do, nor that we can prove that they do. This is why on Kripke's account there can be propositions that are a posteriori necessary (and also contingent a priori). Apr 25 at 19:19
• @NatalieClarius you can input it to umsu.de/trees Apr 25 at 23:47
• @RW_123 You've already erred subtly in above comment when defining your arbitrary proposition A involving the already bounded variable x at the left occurrence which seems harmless, while you cannot simply substitute same definition to your right occurrence where there's no bounding x. It's already syntactically invalid not to say logically correct. There's no hope here to prove the ontological argument... Apr 26 at 5:24
• @RW_123 of course "x=y" could be said to be a wff, but not closed, and I've hinted the error about the way you substituted it to the 2nd occurrence of A in your above whole wff. You can simply copy to your linked proof tree to verify... Apr 26 at 17:56

I'm not sure about "the" philosophical motivation, but at least one of the motivations is to avoid confusing identical statements with similar statements.

"The present King of France is the richest bald person alive"

"The present King of France was, at some point, the richest bald person alive"

are similar, but not identical from a logical perspective precisely because they have different truth values.

Addendum: the "¬(◇(a=b) ∧ ◇¬(a=b))" phrase is probably a typo of "¬(◇(a=b) ∧ ¬◇(a=b))"; the plain(-ish) English translation of the first one is "a=b can't be both possibly true and possibly false", which is false; the second, "a=b can't be both possible and impossible", which is true.

• That’s fair, but isn’t “possibly” supposed to encompass any kind of situation at any time or place? Apr 25 at 17:06
• Sure, but again, you have to be consistent in how you're applying it. "It's impossible for the King of France to have ever been the richest bald person alive" is what "◇¬(a=b)" would expand out to if that's how you're interpreting it. Also there may have been a typo of it being "¬◇(a=b)" instead. Apr 25 at 17:18
• To conclude: it probably is a typo here. ¬◇ for "it's impossible that" vs "◇¬" for "it's possible that not()". In which case the initial formula you provided is not, in fact, a validity. "Maybe so, maybe not" is a tautology, not a contradiction. "Possible, impossible" is the contradiction probably intended. Apr 25 at 17:26