It is a logical truth in (standard) first order logic that something exists, i.e., '∃x x=x'.
On his view, two sentences are logically equivalent iff the corresponding biconditional is a logical truth, and what count as a logical truth depends on what one takes to be the logical vocabulary.
Similarly, it is a logical truth in (standard versions of) second order logic that '∃F∀x(Fx ∨ ¬Fx)'.
Thus, if φ is a logical truth, it follows that ~◇~φ is also true (indeed, logically true).
This was essentially how Prior justified the logical truth of '~◇~E!a' and, hence, its theoremhood in his logic Q.
Wilde is clearly not alluding to the logical truth of propositions or the empirical truths of natural science.
All of the axioms must be logical truths, and the rules of inference must preserve logical truth.
That all effects have causes is true by virtue of the (customary) definitions of cause and effect; it is a purely formal or logical truth.
So (3) is a logical truth.
Carnap instead thought of necessity as logical truth.
Quine argued along these lines in "Carnap on Logical Truth".
