Universal Elimination Kareem Khalifa Department of Philosophy Middlebury

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. x. Hx ├ Hj 1. x. Hx 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 /. – – 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 y. Lxy A y. Lay 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, 4 MT Wa 1, 5 DS

Samples: Nolt 8. 3. 1. 1 ├ x. Fx → Fa 1. | x. Fx 2. | Fa 3. x. Fx→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), ~ x. Gx ├ ~ x. Fx 1. x(Fx→Gx) 2. ~ x. Gx 3. | x. Fx 4. | |Fa 5. | |Fa→Ga 6. | |Ga 7. 7. || | x. Gx 8. || x. Gx |P&~P 9. || x. Gx P&~P& ~ x. Gx 10. ~ x. Fx (Alternative Proof) A A H for ~I H for E 1 E 4, 5→E 6 I 3, 4 -7 E 2, 7 EFQ 2, 9 &I E 3, 4 -8 3 -9 ~I

8. 3. 1. 10 • • x. Fx v x. Gx, ~Ga ├ x. Fx 1. x. Fx v x. Gx 2. ~Ga 3. | x. Gx 4. |Ga 5. |Ga & ~Ga 6. ~ x. Gx 7. x. Fx A A H for ~I 3 E 2, 5 &I 3 -5 ~I 1, 6 DS