How is $\exists$ formulated in a FOL proof system?

Question

$\exists x \phi(x)$ in mathematics means there is something $c$, s.t. $\phi(c)$ holds. What is the formulation of that same idea as a inference rule, an axiom, or something else in a proof system in FOL?

Particularly what is the elimination rule for $\exists$? (It is not correct that if $\Phi \vdash \exists x \phi$, there exists a term $t$ so $\Phi \vdash \phi[t/x]$.)

What is the similar one for $\forall$?

Mauro ALLEGRANZA · Accepted Answer · 2021-03-26T15:05:08.490

2

In the Natural Deduction proof system we have a pair of inference rules for the existential (as well as the universal) quantifier:

$(\exists \text I) \ \varphi [x/a] \vdash \exists x \varphi$

$(\exists \text E) \ \text { if } \Gamma \vdash \exists x \varphi \text { and } \Delta, \varphi [x/a] \vdash \psi, \text { then } \Gamma, \Delta \vdash \psi$.

In an Hilbert-style proof system we have at least one axiom for it, unless it is defined in terms of the universal one.

See e.g. J. Shoenfield, Mathematical Logic for the axiom

$A[x/a] \to \exists x A$

and the rule:

$\dfrac {A \to B}{\exists x A \to B}$, provided that $x$ is not free in $B$.

In Enderton's system (see page 78) the existential quantifier is an abbreviation: $∃x α$ abbreviates $(¬∀x(¬α))$.

edited Mar 26 '21 at 15:05

answered Mar 26 '21 at 13:05

Mauro ALLEGRANZA

99,247

can you list the rules or axioms? Are they for both introduction and elimination? – Mary Mar 26 '21 at 13:07
How about elimination? – Mary Mar 26 '21 at 13:11
Thanks. in https://iep.utm.edu/nat-ded/#H7, What does the introduction rule $(\forall I)$ for $\forall$ say? If $\Gamma \vdash \phi[x/a]$, then $\Gamma \vdash \forall x \phi$. Does it say that if there is a term $a$ so that $\Gamma \vdash \phi[x/a]$, then $\Gamma \vdash \forall x \phi$? That is not true though – Mary Mar 26 '21 at 13:51
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@Mary - you have forgotten the proviso: provided that $a$ is not free in $\Gamma$. – Mauro ALLEGRANZA Mar 26 '21 at 14:00
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See also this post as well as some posts regarding UG and $\forall \text I$ – Mauro ALLEGRANZA Mar 26 '21 at 14:08
Do you mean $a$ must be free in $\Gamma$, and $\phi$? – Mary Mar 26 '21 at 14:11
Why must $a$ be free (or not free) in Γ, and ϕ? – Mary Mar 26 '21 at 14:51

score 1 · Answer 2 · answered Mar 26 '21 at 13:05

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There are different ways to introduce $\exists$ in a FOL proof system.

I think the most common approach is that $\exists$ is defined via $\forall$:

$$\exists x \phi(x) \stackrel{\text{def}}\iff \lnot\forall x \lnot \phi(x).$$

And $\forall$ is introduced axiomatically before that.

answered Mar 26 '21 at 13:05

supinf

13,593

Are there rules/axioms for both introduction and elimination? – Mary Mar 26 '21 at 13:08

How is $\exists$ formulated in a FOL proof system?

2 Answers2