1. Universal Elimination
Kareem Khalifa
Department of Philosophy
Middlebury College

2. Overview What is Universal Elimination? Examples A commonsense example
The official definition
Examples

3. What is Universal Elimination?
From a generalization, infer an instance of that generalization.
Ex. Everybody is happy. So John is happy.
Ex. All birds are mortal. Tweety is a bird. So Tweety is mortal.
Perhaps the most basic of our four basic inference rules in predicate logic.

4. The examples examined
Ex. Everybody is happy. So John is happy.xHx ├ Hj
xHx A
Hj 1 E
Ex. All birds are mortal. Tweety is a bird. So Tweety is mortal. x(Bx→Mx), Bt ├ Mt
x(Bx→Mx) A
Bt A
Bt→Mt 1 E
Mt 2,3 →E

5. The official definition
Universal Elimination (E): Let Φ be any universally quantified formula and Φ/ be the result of replacing all occurrences of the variable  in Φ by some name . Then from Φ, infer Φ/.
x(Bx→Mx) A
Bt A
Bt→Mt 1 E
Mt 2,3 →E

6. Some finer points…
When you have multiple quantifiers, you apply E from left to right (outside-in), e.g.
Everyone loves everyone. So Al loves Bob.
xyLxy A
yLay 1 E
Lab 2 E
Note that this is the exact opposite direction as I.

7. Another finer point… Wa v Wb A x(WxQx) A ~Qb A Wa Qa 2 E ~Wb 3,4MT
Be strategic in which name you instantiate when using E.
Example: Either Al or Ben is the winner. All winners must have passed the qualifying round. Ben did not. So Al is the winner.
Wa v Wb A
x(WxQx) A
~Qb A
Wa Qa 2 E
~Wb 3,4MT
Wa 1,5 DS
WbQb
Imprudent.

8. Samples: Nolt 8.3.1.1 ├ xFx → Fa 1. | xFx H for →I 2. | Fa 1 E
3. xFx→Fa →I

9. 8.3.1.4 x(Fx→Gx), Ga→Ha ├ Fa →Ha 1. x(Fx→Gx) A 2. Ga→Ha A
3. Fa→Ga 1 E
4. Fa→Ha 2,3 HS

10. 8.3.1.7 x(Fx→Gx), x~Gx ├ x~Fx 1. x(Fx→Gx) A 2. x~Gx A
3. |~Ga H for E
4. |Fa→Ga 1 E
5. |~Fa 3,4 MT
6. |x~Fx 5 I
7. x~Fx 2,3-6 E

11. 8.3.1.8 x(Fx→Gx), ~xGx ├ ~xFx 1. x(Fx→Gx) A 2. ~xGx A
3. |xFx H for ~I
4. | |Fa H for E
5. | |Fa→Ga 1 E
6. | |Ga 4,5→E
7. | |xGx 6 I 
8. |xGx 3,4-7 E
9. |xGx & ~xGx 2,9 &I
10. ~xFx ~I
7. | | xGx I
8. | |P&~P ,7 EFQ
9. | P&~P ,4-8 E
10.~xFx ~I
(Alternative Proof)

12. 8.3.1.10 xFx v xGx, ~Ga ├ xFx 1. xFx v xGx A 2. ~Ga A
3. |xGx H for ~I
4. |Ga E
5. |Ga & ~Ga 2,5 &I
6. ~xGx ~I
7. xFx 1,6 DS