# existential import by Corcoran and Masoud: odd assertions

In the paper by John Corcoran & Hassan Masoud (2014): Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of Logic, already in the introduction it says, as self-evident, that

"The universalized conditional $∀x(x = 0 → x = (x + x))$ implies the corresponding existentialized conjunction $∃x(x = 0 \text { & } x = (x + x))$. And $∃x(x = 0)$ is tautological (in the broad sense, i.e. logically true)."

There are two assertions here, and I have difficulty with each one. In the first assertion I do not see why a model in which there is no $0$ (e.g., $\mathsf {ZFC^*}$ obtained by negating all the axioms of $\mathsf {ZFC}$), so that the conclusion of the implication is false and the premise would be (vacuously) true, would not be a counter-example. In the second assertion, I don't see why the statement is tautological (since it is not satisfied by all models, such as $\mathsf {ZFC^*}$).

As soon as you use a constant symbol, any interpretation will need to map that symbol to some object of its domain. So even if you say something like $\neg \exists x x=0$, you still need some object that $0$ denotes ... And since that object is of course identical to itself, this statement is a logical contradiction ... Meaning that $\exists x x=0$ is a tautology.
Something similar hold for the first one: whatever the $0$ denotes, it is of course identica to itself, and so by the universal it must then also be true that $0 = 0+0$, and hence the existential is true.
• @nomadreid Existential import simply means that the domain of any interpretation is not empty, i.e. that there is always something that exists. Or, in your words, that the logic is not a free logic (where you can have empty domains). And it is that assumption that allows us to require that any constant denotes some object in the domain, and that is exactly what standard logics do. Hence, as soon as you use $unicorn$ in a sentence, it will denote something, meaning that $\exists x x =unicorn$ is a tautology, and hence $\neg \exists x x=unicorn$ is always a contradiction in standard logic. Jan 9, 2017 at 12:02