Question on modal logic

Suppose the sentence p is possible, and -p is impossible.

Is the disjunction p v -p necessary?

• -p is impossible means necessarily --p, i.e. necessarily p.
– Era
Commented Feb 29, 2016 at 19:05

Formal tautologies are always logically necessary. Standard FOL has p v -p as a formal tautology, hence it is necessary regardless of whether p is necessary or possible. This breaks down if you consider alternative logics that don't have the Law of Excluded Middle as a tautology.

• Thanks Era for the reply. Subsidiary question: if 'p' is neither necessary, nor impossible. Is p = p v p necessary?
– user13738
Commented Feb 29, 2016 at 20:24
• @student Can you clarify exactly what you mean by `p = p v p`?
– Era
Commented Feb 29, 2016 at 20:42
• OK, let's say that, on one hand, p is not necessary, and, on the other hand, p is not impossible. Is p necessary too?
– user13738
Commented Feb 29, 2016 at 21:30
• @student not impossible means possible. `p` can be possible without being necessary.
– Era
Commented Feb 29, 2016 at 21:56
• So, if p is either not necessary or possible, is p necessary?
– user13738
Commented Feb 29, 2016 at 22:32