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Let $x, y, z$ denote (bound) variables and $a, b, c$ free variables or individual parameters. Gentzen’s rules are the following:

$$ \begin{array}{ll} (\forall E) & \forall x\varphi \vdash \varphi[x/a] \\ (\exists I) & \varphi [x/a] \vdash \exists x\varphi \\ (\forall I) & \text{If }\Gamma \vdash \varphi [x/a] \text{, then }\Gamma \vdash \forall x\varphi \\ (\exists E) & \text{If }\Gamma \vdash \exists x\varphi\text{ and } \Delta, \varphi[x/a] \vdash \psi\text{, then }\Gamma, \Delta \vdash \psi \end{array} $$

Here $\varphi [x/a]$ denotes the operation of substitution, that is, of replacing all free occurrences of $x$ in $\varphi$ with a parameter $a$. In case of ($\forall I$) and ($\exists E$) a parameter $a$ is required to be “fresh” in the sense of having no other occurrences in $\Gamma , \Delta, \varphi, \psi$. Such a fresh a is sometimes called an ‘eigenvariable’ or a ‘proper variable’.

It is from here.

What does $(\forall I)$ mean to say? It seems to me counter-intuitive.

Is the requirement on $a$ that it must not occur in $\phi$, neither bound nor free variables?

$(\exists I)$ and $(\forall I)$ are very similar, but why does $(\forall I)$ achieve stronger than $(\exists I)$?

vitamin d
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Mary
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  • The basic of idea of Universal Introduction is: If something is true for any object, then it is is true for all objects. We just have to make sure that the any is truly any arbitrary one: an object for which we know that it is part of the domain, but about which we assumed nothing else when we introduced it. – Bram28 Mar 26 '21 at 18:24

1 Answers1

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The idea is the following. Suppose, from a set of formulas $\Gamma$, I've managed to conclude that object $a$ has property $\varphi$ - but $a$ itself was never referred to in $\Gamma$. Then $a$ can't be at all special in this respect: every object has to have property $\varphi$, assuming $\Gamma$. Basically, if I prove something about an individual without ever using any specific properties of that individual, the thing I've proved actually holds in general.

This is the rule underlying natural-language arguments of the form:

"Let $c$ be an arbitrary [foo]. Then (argument involving set of assumptions $\Gamma$), So $c$ has the [bleen] property. Since $c$ was arbitrary, everything has the [bleen] property."

Of course when we introduce $c$, that symbol has to be "fresh." We can't say e.g. "let $2$ be an arbitrary number," since the symbol $2$ already means something.

Noah Schweber
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  • Thank you! (∃I) and (∀I) are very similar, but why does (∃I) achieve weaker than (∀I)? In (∃I), why is $a$ not arbitrary? – Mary Mar 27 '21 at 03:27
  • @Mary $(\exists I)$ and $(\forall I)$ are two completely different types of rule. The rule $(\exists I)$ gives us a sequent "for free," while the rule $(\forall I)$ gives us a sequent assuming we've already gotten another sequent. In Gentzen's system we're not really proving theorems, we're deriving sequents. (Both introduction rules give us sequents for free, and both elimination rules give us sequents contingent on already having other sequents.) – Noah Schweber Mar 27 '21 at 04:02
  • Even if we just pay attention to the conclusions of the sequents involved (which we should not do) there's still a balance here. While the conclusion of the sequent we get from $(\exists I)$ is weaker than the conclusion of the sequent we get from $(\forall I)$ (namely, "$\exists x\varphi$" and "$\forall x\varphi$" respectively), the $(\exists I)$ rule is easier to apply than the $(\forall I)$ rule. So it's a case of "less resources buys you less stuff." – Noah Schweber Mar 27 '21 at 04:02
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    Why this it so subtle? A problem that any treatment of quantifier introduction and elimination has to face is how to prohibit the following line of attack: "Suppose $\exists x \varphi$. Let $a$ be such that $\varphi(a)$ holds. Since $a$ was not mentioned in the hypothesis ($\exists x\varphi$), we've established $\varphi(a)$ for $a$ arbitrary and so we can conclude $\forall x\varphi$." This will be permitted if you're not careful setting up the quantifier rules for your system. It takes some care to "untangle" existential instantiation and universal generalization from each other correctly. – Noah Schweber Mar 27 '21 at 04:05