Tutorial 21: Existential Elimination

Logical System

2013

Bergmann[2004] The Logic Book Section 10.1.

The Tutorial

Existential Elimination (often called 'Existential Instantiation') permits you to remove an existential quantifier from a formula which has an existential quantifier as its main connective. It is one of those rules which involves the adoption and dropping of an extra assumption (like ∼I,⊃I,∨E, and ≡I).

The circumstance that Existential Instantiation gets invoked looks like this.

 (∃x) ? ? ? << ?

That is, a formula you have has an existential quantifier as its main connective-- and you are trying to obtain some formula which we will call <target>.

For example,

 (∃x)(Fx) ? ? ? << (∃y)(Fy) ?

invites the use of EE. In this case (∃x)(Fx) is the existential formula, (Fx) is its scope, and (∃y)(Fy) is the target.

To use EE you follow a three stage process. First you assume the <scope>[<constant>/<variable>]. The illustrated fragment will then become

Second you continue deriving in this new context until you manage to derive <target> thus

Finally you select <target> at the end of the sub-proof, select the existential formula, and click ∃E to complete the existential instantiation.

In the example, the derivation would look like

There are restrictions on EE. The <constant> must not occur either

• a) in the assumptions of the final <target> line,
• or b) in the existential formula that ∃E is used on,
• or c) in the <target>.

Notice that the constant 'a' does not occur in the assumptions of line 4 (which are (∃x)Fx), it does not occur in the existential formula, line 1, and it does not occur in the target, line 4.

More about these restrictions later. The rule is

The Rule of Elimination of the Existential Quantifier, Existential Elimination ∃E

If a derivation contains a line of the form

n. <assumptions> (∃<variable>)(<scope>) <any justification>

and a line of the form

k. <assumptions+ <scope[<constant>/<variable>]>> <formula> <any justification>

and

the constant <constant> does not occur in ∃<variable>)(<scope>) , in any of the <assumptions> of line k except the additional assumption <scope[<constant>/<variable>]>, or in the formula <formula>

then a line of the form

<next line number> <assumptions> <formula> 'k EI'

may be added to the derivation.

Implementation of EE

You have to have in mind here some existential formula (∃<variable>)<scope> that you are going to instantiate. First use Ass to adopt <scope[<constant>/<variable>]> as a new assumption, You need to pick a 'good' constant here, for it must not occur in all sorts of other places. A good strategy is to pick a brand new constant that does not appear in the derivation at all. Then you derive the <formula> you want as the existential instantiation as the last line of an open sub-proof or sub-derivation. Then select the existential formula (∃<variable>)<scope> and the last line <formula> and press EE. The Rule will check whether <constant> occurs in the existential formula, it will check whether <constant> occurs in the standing assumptions of the intended instantiation <formula>, and it will check whether the <constant> occurs in target <formula>.   The conditions satisfied, the Rule will drop the extra assumption of the sub-derivation and add <formula> as the new line.

Exercise to accompany Predicate Tutorial 11.

Exercise 1 (of 3)

It is probably easier to learn EI using Tactics. After you start each derivation, switch Tactics on, select the existentially quantified formula, and click EI.

Derive the following

a) (∃x)(Fa) ∴ Fa
b) (∃x) (Fx&Gx) ∴ (∃x)(Fx )
c) (∃x) (Fx&Gx) ∴ (∃y)(Gy)
d) (∃x)(Fx)∴ (∃y)(Fy)
e) (∃x)(Fx&Gx) ∴ (∃y)(Fy)&(∃z)(Gz)

Exercise 2 (of 3)

Repeat the derivations of Exercise 1 without using Tactics.

a) (∃x)(Fa) ∴ Fa
b) (∃x) (Fx&Gx) ∴ (∃x)(Fx )
c) (∃x) (Fx&Gx) ∴ (∃y)(Gy)
d) (∃x)(Fx)∴ (∃y)(Fy)
e) (∃x)(Fx&Gx) ∴ (∃y)(Fy)&(∃z)(Gz)

Exercise 3 (of 3)

Each of following derivations will use Existential Instantiation at some stage.

Derive

a) ∴ (∃x)Fx≡(∃y)Fy
b) (∀x)Fx,(∃x)~Fx∴G
c) ∴ (∃x) (Fx∨Gx) ≡ (∃y)Fy∨(∃z)Gz
d) (∃x)(Fx∨Gx),(∀x)(Fx⊃Hx),(∀x)(Gx⊃Hx)∴(∃x)Hx
e) ~(∃x)(Fx&~Gx),~(∃x)(Gx&~Hx)∴(∀x)(Fx⊃Hx)
f) ~(∃x)Gx,(∀x)(Fx⊃Gx)∴(∀x)(Fx⊃Hx)
g) (∃x)Fx∨(∃x)Gx,(∀x)((Fx∨Gx)⊃Hx)∴(∃x)Hx

If you decide to use the web application for the exercises you can launch it from here Deriver[Bergmann] — username 'logic' password 'logic'. Then either copy and paste the above formulas into the Journal or use the Deriver File Menu to Open Web Page with this address https://softoption.us/test/easyDeriver/CombinedExercisesEasyDBergmann.html .

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