FOM: Existential commitments in logic

[Neil Tennant]

Vaughan asks

> which is
> the first sentence in the following that is falsified by the assumptions
> of free logic?
> 
> Under (a) the Russellian assumption (all singular terms must denote
> something), (b) the existence of at least one singular term in the
> language, and (c) empty universe, the set of all interpretations of
> the terms of the language is empty.  This in turn implies that when the
> only universe is the empty one, every proposition is vacuously valid.
> Therefore bringing in the empty universe cannot affect any "monotone"
> logic (one that can only lose laws when universes are introduced)
> because no sentence is thereby rendered invalid.

(a) is incorrectly stated. The Russellian assumption is not that the
language contains only denoting terms. Rather, it is that if an atomic
predication P(t_1,...,t_n) is true, then each term t_i denotes something.
That is, the truth conditions of P(t_1,...,t_n) are

	t_1,....,t_n all exist and stand in the relation P.

Thus if t is a non-denoting term and P is an atomic one-place predicate,
P(t) will be false. Note that the Russellian assumption does not rule t
out as ill-formed or otherwise excluded from counting as a term of the
language.

Suppose, then, that we have (a) the Russellian assumption, as now
correctly stated; (b) at least one singular term in the language; and (c)
the empty universe. It does not follow that the set of all interpretations
in the language is empty. Rather, what follows is that there is exactly
one interpretation of the language involving the empty domain. In this
interpretation of the language, every term fails to denote, and the
extension of every predicate is the empty extension.

By taking this (unique) empty model into consideration when defining the
notions of logical truth and logical consequence, we now have the failure
of such inferences as

	(x)F(x), therefore (Ex)F(x)
	(x)F(x), therefore F(t)
	F(t), therefore (Ex)F(x).

These inferences will need extra premisses in order to go through in free
logic. Respectively, they would become

	(x)F(x), (Ex)(x=x), therefore (Ex)F(x)
	(x)F(x), (Ex)(x=t), therefore F(t)
	F(t), (Ex)(x=t), therefore (Ex)F(x).

Thus Vaughan is at odds with the free logician when he claims

> There simply *is* no call for more or different rules, all the old rules
> continue to work as before and no new laws come into existence.

The change can be marked not only by new laws, but also by the failure of
old ones.

> One consequence of my point is that any logic of valid formulas *cannot*
> commit one to the assumption of a nonempty universe.  

Well, no. The free logic I outlined is sound and complete with respect to
the first-order semantics that allows for the empty universe as a
possibility. (Natural Logic, pp. 163-172.) It may be a boring or
frustrating possibility, but it is a possibility nevertheless!

Neil Tennant