Regarding the existence of Santa Claus and the excluded middle, Bertrand Russell dealt with puzzles like these in his paper 'On Denoting'.
The classic example is 'The King of France is bald' and 'The King of France is not bald'. Russell distinguishes two scopes in which it can be read, narrow and wide*.
For example, Russell identifies a sentence 'I thought your yacht was larger than it is' as to have:
A narrow scope reading:
- I thought the size of your yacht is bigger than the size of your yacht.
A wide scope reading:
- There is a size x, such that the size of your yacht is x, and I
thought that the size of your yacht is bigger than x.
So, the proposition:
- Either Santa Claus is hungry, or Santa Claus is not hungry.
Expresses truth only if two disjuncts contradict each other. The left disjunct is obviously false, but the right one has two possible readings:
1. A wide scope reading:
- ∃x (Sx ∧ ∀y (Sy → y = x) ∧ ¬Hx)
- There is a unique x who is Santa Claus, and x is not hungry.
2. A narrow scope reading:
- ¬∃x (Sx ∧ ∀y (Sy → y = x) ∧ Hx)
- It is not the case that there is a unique x who is Santa Claus, and x is hungry.
If the scope is wide (1.), it is simply false. If the scope is narrow (2.), it is true. Only the narrow reading (2.) is the negation of the left disjunct, so the law of excluded middle is preserved.
(For further reading, I encourage engaging with the paper itself.)
*- A narrow scope is also called a secondary occurrence, and the wide scope is called a primary occurrence.