Why is $\alpha \rightarrow \forall x(\alpha)$ not generally correct in first-order logic?

Question

Why is $\alpha \rightarrow \forall x(\alpha)$ not generally correct in first-order logic?

i.e., when there are free occurrences of $x$ in $\alpha$, and, on the same point, why is the formula scheme correct when there are no free occurrences of $x$ in $\alpha$?

Answer 1

About the rule :

if $\Gamma \vdash \varphi$, then $\Gamma \vdash \forall x \varphi$ provided that $x$ does not occur free in any assumption in $\Gamma$,

consider the following example :

let $\varphi$ be $x = 0$, where $0$ is an individual constant, and $\Gamma = \{ \varphi \}$. Then, from $\varphi$ we can construct the following derivation :

$ x = 0$ (assumption)

$\forall x (x = 0)$ (illegal : $x$ is free in the assumption).

The last statement is false in any structure with at least two elements. [See also Stephen Cole Kleene, Mathematical Logic (1967), page 119].

If we have $\Gamma = \emptyset$ we can derive the following rule :

if $\vdash \varphi$, then $\vdash \forall x \varphi$,

but in this case $\varphi$ is nor more an assumption : it must be a theorem.

In order to show that the formula $\varphi \rightarrow \forall x \varphi$ is not in general correct, consider the following example :

$\varphi$ is $P(x)$

and take as the domain of the interpretation the set $\mathbb Z$ of integers, and let the interpretation of the predicate $P$ be “$x$ is greater than $0$”.

The formula :

$P(x) \rightarrow \forall x P(x)$

is not valid, because in $\mathbb Z$ not all the numbers are greater than $0$.

We want that our calculus is sound, i.e. that it allows us to prove only valid formulas, where valid means true in every interpretation. The above formula is not true in the aforesaid interpretation, so it is not valid.

Consequently, we do not want it in our calculus, and we must set up the rules accordingly, so that it should not be derivable.

Answer 2

A good question. The difference between implication and rule of inference is clear in formal logic. However confusions in everyday reasoning are not rare because both constructions are usually pronounced as "if... then". That is why it is not easy to explain to pupils the difference between them but a few example may be useful. When we prove a theorem of geometry, it is a common manner to "take" a concrete triangle, say ABC. After investigating some its properties we formulate the theorem: "For all triangles.." However it would not be a correct analogy to "take" a concrete man, say Mack the Knife, and on the base of his life to conclude that "All men are killers". I think that such notices help understanding the logical principles and their teaching.