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Universal Elimination

Understand the concept of Universal Elimination in predicate logic with examples and the official definition. Learn how to apply this inference rule correctly.

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Universal Elimination

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

  2. Overview • What is Universal Elimination? • 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… • 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 Imprudent. WbQb

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

  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 3 E • 5. |Ga & ~Ga 2,5 &I • 6. ~xGx 3-5 ~I • 7. xFx 1,6 DS

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