In a possible world where there are no bachelors is the proposition "bachelors are unmarried" vacuously true? Or is it tautologically true and you would then need to quantify bachelors as "all bachelors in this world are unmarried" to make it vacuously true? Does "all bachelors are unmarried" quantify over the real world or all possible worlds?
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"Bachelors are unmarried" must be formalized with ∀x (Bachelor(x) → Married(x)). Thus, if there are no bachelors at all: YES, it is vacuously true.– Mauro ALLEGRANZACommented Feb 7, 2017 at 14:45
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At least on the traditional account analytic propositions do not quantify over anything, they simply express conventions about using words. Since the same conventions are presumably used to speak of all possible worlds "bachelors are unmarried" remains true even when there are no bachelors. You may want to look at two-dimensional semantics, "the context dimension is apt for tracking analytic knowledge obtained from the mastery of our language. On the other hand, the possible-worlds dimension keeps track of what is necessary".– ConifoldCommented Feb 7, 2017 at 19:01
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This depends on your use of constants. There is a big debate about this very question and I am going to present to you the most popular alternatives:
- Constant domain modal propositional logic
This system assumes that every constant you can speak about exists in every possible world. Analytical statements might be tautologically true.
- Variable domain modal propositional logic (VMPL)
This system takes some assuption from free logic and uses them for MPL. In free logic any allquantified statement can either be true about existing constants or it is vacuously ture about non existing objects.
For this distinction free logic and VMPL fix one predicate to be Existance predicate. The valuation function of this predicate maps to the n-tupel of existing constans in the domain of a world. This means that analytical statements can be vacuously or tautologically true.
Some people argue for a negative constraint on VMPl. They want statements for nonexisting constants to be false. For them your analystical statement is vacuously false.
