>>12211810It's true in classical first order logic at least, yes.
For any predicate ?,
(?x.x=x) ? ?y. (?(y) ? ?z.?(z))
In words,
"If there exists anything at all, then there exists a thing such that if it has the property ?, then everything has the property ?"
PROOF of the logical schema above:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED
Example:
"There is a thing, such that if that thing is a bird, then everything is a bird."
PROOF (same as above, just more specific case)
Not everything is a bird. We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.
Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.
It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.
