Universal Elimination  Kareem Khalifa Department of Philosophy Middlebury College

Overview What is Universal Elimination? Examples 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 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

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

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.

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.

Samples: Nolt 8.3.1.1 ├ xFx → Fa 1. | xFx H for →I 2. | Fa 1 E 3. xFx→Fa 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) 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 3-9 ~I 7. | | xGx 6I 8. | |P&~P 2,7 EFQ 9. | P&~P 3,4-8 E 10.~xFx 3-9 ~I (Alternative Proof)

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