7.4 Existential Entailment II

Most English verbs have existential entailments in the following sense:

In logic, the first translation entails the second:

$$\begin{array}[t]{l} \mathrm{\:\sc see\:}(\mathrm{\:j\:},\:m)... ...ists x \mathrm{\:\sc see\:}(\mathrm{\:j\:},\:x) \\ \end{array}$$

Another case:

In logic, the two sentences have the same translation:
$$\begin{array}[t]{l} \exists x \mathrm{\:\sc eat\:}(\mathrm{\... ...rm{\:j\:},\:x) \wedge \mathrm{\:\sc apple\:}(x) \\ \end{array}$$

Even more simply:

In logic, again, the first translation entails the second:

$$\begin{array}[t]{l} \exists x \mathrm{\:\sc eat\:}(\mathrm{\... ...ists x \mathrm{\:\sc eat\:}(\mathrm{\:j\:},\:x) \\ \end{array}$$

In general,

$$\begin{array}[t]{l} p \wedge q \Rightarrow p \end{array}$$

Read $\Rightarrow$ as ``entails'' in logic too. It is different from $\rightarrow$. Technically, $\alpha \Rightarrow \beta$ means $\alpha \rightarrow \beta$ is a tautology.

The following existential entailment also holds:

In logic, the first translation again entails the second:

$$\begin{array}[t]{l} \exists x \mathrm{\:\sc eat\:}(\mathrm{\... ... eat\:}(y,\:x) \wedge \mathrm{\:\sc apple\:}(x) \\ \end{array}$$

So eat has existential entailments for both its subject position and object position.

You can think of existential entailment as semantic obligatoriness. Eating can't go on without something filling both the eater and the eaten roles. But remember: existential entailment is quite different from syntactic obligatoriness:

(i) John devoured the apple.
(ii) John devoured something.
(iii) * John devoured.

The verb devour has an existential entailment on the direct object position. And, independently of that, that second argument position is obligatory.

Not every verb gives an existential entailment for every argument position:

Because of this, it's not clear how to translate John is looking for a unicorn into predicate logic. This translation

$$\exists x \:\mathrm{\:unicorn\:}(x) \wedge \mathrm{\:look\_for\:}(\mathrm{\:j\:},\:x)$$

is wrong because it immediately entails that what John is looking for exists. Under standard assumptions about what the translations mean, this sentence can't be translated into predicate-logic.