If we interpret two possible worlds, one with rain, and one without rain, with "I" as a rigid designator, then the law of noncontradiction is violated, because "I" gets wet and does not get wet, at the same time. The qualification of "different worlds" is just a facade to hide the contradiction. If we look at what Banno called "counterpart theory", then we have no continuity of the object "I", from the present time into the future, only possible similar objects in the future, therefore the law of identity is not applicable. — Metaphysician Undercover
The claim that "I get wet and do not get wet" violates the law of noncontradiction misunderstands how modal logic works. These are not simultaneous truths in a single world, but distinct evaluations across possible worlds, which is actually the reason modal logic exists. The law of noncontradiction applies within worlds, not between them.
Additionally, the entirety of the "different worlds" enterprise must be jettisoned and the resultant collapse of modal logic as well if we follow out your logic. The term "different" as applied here by you includes any dissimilarity whatsoever, even the simple fact they are in different locations. That is, it is impossible under your reasoning to have any metaphysically related universes because everything within each one would be relevantly different.
To make my point clearer: Suppose you had Universe #1, and within it you get wet and in Universe #2, you also get wet. In fact, every single thing within #1 and #2 are the "same," they would still not bear any metaphysical relationship to each other because they are all necessarily different since they occupy different time and space. That is, #1 and #2 do not collapse into being the same thing because they are not identical under your view. They are just curiously similar.
When we chart out all possible worlds, under your reasoning, an infinite number could be the same in every apparent regard because you deny the concept of rigid designation in theory.
This is to say that if you deny a rigid designation for "I," you must do it for all things. That means that not only does the fact that you're not the same you in #1 and #2, the rain isn't the same in #1 and #2. They must be different. You can't have a different you in #1 and #2 and share the same rain. When we say it will rain in #1, while that sounds like any old generic rain will do, if we were being more precise, we'd describe the exact identity of the rain that would strike you in #1 versus #2.
This I suggest is the logical consequence of demanding cross universe consistency.
This is why
@Banno brings in counterpart theory, which holds there is a
similar counterpart in another universe that satisfies the conditions needed to perform the modal logic. It dispenses with the impossible mental gymnastics needed to comprehend what it could possibly mean to have identical entities in different locations at the same time, when location is a component of identity. This concept of possible worlds is injected to expand our epistimological understanding of the hypothetical, but, if taken literally, it cannot be sustained because there really aren't multiverses outside of science fiction.
All of this is just to say:even if we allow for a rigid designator of "I" across multiverses, that does not create a contradiction as long as we assume contradiction applies only intra-universe and not inter-universe.
But back to the classic versus modal logic discussion:
If in classic logic I say:
All glurgs are glogs
I am a glurg
Therefore I am a glog
That is true, despite the fact there is no referent for any of this gibberish. That is why we can use symbols to represent these entities because their existence is irrelevant for the analysis.
Under modal logic we say:
It is necessary all glurgs are glogs
It is possible I am a glurg
It is therefore possible I am a glog
This is true as well, purely from a formal level, despite there are no glogs here, there, are anywhere.
The issue then becomes providing a definition of "possible," as you allow for pure meaningless formality under classic logic but not under modal logic. Since "possible" is the only new thing inserted, that must be the reason you treat these two systems different. What you then do is require metaphysical grounding in order for the possible to occur, but that I challenge. You no more need semantical validity for modal logic to work than classic. It's good to have semantically meaningful statements, but not required.
That is, the same tension occurs in classic logic. Typically those syllogisms do more than just mindlessly maintain truth value through vacuous symbols. Typically they have semantical truth, but not always. I would say the same of modal logic as well. But to demand that modal logic always be semantical results in its collapse, despite its pragmatic value. I think the discussion of impossible worlds makes that clear. You can use modal logic to consider events that did not and cannot occur.
My position is that we must fictionalize the idea of multiverses for the purposes of gaining epistimological clarity, without regard to whether they really exist. Maybe explain what is gained or lost by this approach.