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Universal Elimination

Kareem Khalifa
Department of Philosophy
Middlebury College
Overview
• What is Universal Elimination?
– A commonsense example
– The official definition
• Examples
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.
The examples examined
•
Ex. Everybody is happy. So John is
happy.xHx ├ Hj
1. xHx
2. Hj
•
A
1 E
Ex. All birds are mortal. Tweety is a bird. So
Tweety is mortal. x(Bx→Mx), Bt ├ Mt
1.
2.
3.
4.
x(Bx→Mx)
Bt
Bt→Mt
Mt
A
A
1 E
2,3 →E
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/.
1.
2.
3.
4.
x(Bx→Mx)
Bt
Bt→Mt
Mt
A
A
1 E
2,3 →E
Some finer points…
•
When you have multiple quantifiers, you
apply E from left to right (outside-in),
e.g.
–
1.
2.
3.
•
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.
Another finer point…
•
•
1.
2.
3.
4.
5.
6.
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
Imprudent.
Wa Qa
2 E
WbQb
~Wb
3,4MT
Wa
1,5 DS
Samples: Nolt 8.3.1.1
├ xFx → Fa
1. | xFx
2. | Fa
3. xFx→Fa
H for →I
1 E
1-2 →I
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
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
8.3.1.8
x(Fx→Gx), ~xGx ├ ~xFx
1. x(Fx→Gx)
2. ~xGx
3. |xFx
4. | |Fa
5. | |Fa→Ga
6. | |Ga
7.
7. || |xGx
| xGx
8.
8. ||xGx
|P&~P
9. ||xGx
P&~P& ~xGx
10.~xFx
10. ~xFx
(Alternative Proof)
A
A
H for ~I
H for E
1 E
4,5→E
6
I
6I
3,4-7
E
2,7 EFQ
2,9
&IE
3,4-8
3-9 ~I
8.3.1.10
•
•
•
•
•
•
•
•
xFx v xGx, ~Ga ├ xFx
1. xFx v xGx
2. ~Ga
3. |xGx
4. |Ga
5. |Ga & ~Ga
6. ~xGx
7. xFx
A
A
H for ~I
3 E
2,5 &I
3-5 ~I
1,6 DS