Substitutional definition of $\forall$

Question

In [1] it is written (p. 116):

The only slight subtlety in the business arises at the level of quantification. Here is a simple, tempting, and wrong approach to defining truth for the case of quantification, called the substitutional approach: $$M \models \forall x F(x) \iff \text{ for every closed term } t, M \models F(t) $$

The author wrote that it is wrong approach. But why?

---

References

[1] George S. Boolos, John P. Burgess, Richard C. Jeffrey. Computability and Logic. 5th Edition. 2007. Cambridge University Press. ISBN-13: 978-0521701464.

Answer 1

See page 116-117 [4th and 5th editions] :

In other words, under this definition a universal quantification is true if and only if every substitution instance is true, and an existential quantification is true if and only if some substitution instance is true. This definition in general produces results not in agreement with intuition, unless it happens that every individual in the domain of the interpretation is denoted by some term of the language. If the domain of the interpretation is enumerable, we could always expand the language to add more constants and extend the interpretation so that each individual in the domain is the denotation of one of them. But we cannot do this when the domain is nonenumerable. (At least we cannot do so while ontinuing to insist that a language is supposed to involve only a finite or enumerable set of symbols. Of course, to allow a ‘language’ with a nonenumerable set of symbols would involve a considerable stretching of the concept.)

The concepts are explained with the example of the formula $\exists x \ (x \cdot x = 2)$. With the substitutional approach the formula

would come out false [in $\mathbb R$]. But intuitively, though ‘there is something (in the domain) that multiplied by itself yields two’ is false on the rational interpretation, it is true on the real interpretation. We could try to fix this by adding yet more terms to the language, but by Cantor’s theorem there are too many real numbers to add a term for each of them while keeping the language enumerable.

Answer 2

To bring out succinctly the salient point in Mauro's correct answer:

There may be (and often are) members of the model that are not expressible as closed terms.

In fact, if the the model has no constants or functions (just relations), then there are no closed terms at all, so the statement $$\text{"for every closed term } t, M \models F(t)\!"$$ is vacuously true for any formula $F;$ it follows that the alternate definition wouldn't be satisfactory!