## Friday, August 3, 2018

### Existential quantifiers aren't defined by their logic

Start with a first order language L describing the concrete objects of our world and expand L to a language L* by adding a new name “obump”. Given an interpretation I of L in a model M, create a model M* such that:

• The domain of M* is the domain of M with one more object, o, in its domain.

• The non-unary relations of M* are the same as those of M, except that I(=) is replaced by a new relation J = I(=)∪{(o, o)}.

• The unary relations of M* are all and only the relations R* for R a unary relation of M, where R* = R ∪ {o} if R is equal to I(F) either for a physical predicate F of L such that I(trump)∈I(F) or for a mental predicate F of L such that I(obama)∈I(F) and R* = R otherwise.

Define the interpretation I* of L* in M* as follows: I*(a)=I(a) for any name other than obump, I*(obump)=o, I*(F)=(I(F))* for a unary F, and I*(F)=I(F) for any non-unary F.

We can now give a semantics for L*: If Iw is the intended interpretation of L in a world w, then the intended interpretation of L* in w is given by Iw*. We can define validity for L in an analogous way.

The symbols ∃ and ∀ of L* have the same logic as the same symbols of L. But the ∃ of L* is not really an existential quantifier. For if it were, then it would be true that there exists an entity that has all the mental properties of Obama and all the physical properties of Trump, which is false. Thus, logic is not sufficient to make a symbol be an existential quantifier.