The claim that only analytical truths are necessary is itself not an analytical proposition. This self-contradiction is just part of the reason verificationism has been almost universally rejected. There are necessary synthetic propositions. For example:
*Something cannot come from nothing.
*It is morally wrong to torture children for fun.
*"Hey Jude" is more beautiful than "Pants on the Ground."
first synthetic proposition is metaphysical. The second and third are
ethical and aesthetic, respectively. Now, suppose one accepts a version
of PSR wherein all synthetic truths have an explanation. The
explanation will either be contingent or necessary.
1. "Something cannot come from nothing" is a necessary synthetic truth. (Premise)
2. Every synthetic truth has an explanation. (Premise, PSR)
3. Hence, "something cannot come from nothing" has an explanation. (From 1 and 2)
4. Every explanation is either contingent or necessary. (Definition)
5. Nothing contingent can explain a necessary truth. (Premise)
6. Therefore, "something cannot come from nothing" is explained by something necessary. (From 3 - 5)
could justify (5) by pointing out that there are possible worlds at
which nothing contingent exists, but "something cannot come from
nothing" still has an explanation. Since only existing things can serve
as actual explanations, it follows that only something necessary can
explain a necessary truth.