Basic Inference Rules

1. Basic Inference Rules Kareem Khalifa Department of Philosophy Middlebury College

2. Overview • Why this matters • Proofs are like games • Procedure for constructing formal proofs • Sample Exercises

3. Why this matters • We now have two ways of ascertaining validity: • Constructing counterexamples, a fairly natural way of ascertaining validity, but only as reliable as your insight and imagination. • Truth tables, a very reliable way of ascertaining validity, but is both unwieldy and not very natural. • Formal proofs of validity (“natural deductions,”“formal inferences”) are more rule-governed than constructing counterexamples, but less unwieldy than truth-tables. • A nice balance!

4. Football Initial field position Rules of play (offside, holding, no forward lateral) End zone Proof Premises Basic rules of inference (→E, &E, &I, ↔E, ↔I, vE, vI, ~E) Conclusion Proofs are like games

5. The Basic Inference Rules

6. Proofs are really like games • In football, you move from your initial field position, in accordance with the rules of play, to the end zone. • In formal proofs, you move from your premises, in accordance with the rules of inference, to the conclusion.

7. Playing the game • You will be given an argument. It is your task to show that the conclusion follows validly from the premises. To do this: • List the premises as the first lines of the proof. Mark them with an “A” for Assumption. • Apply the basic rules of inference to the premises and then to the subconclusions that result from those applications. • Every line in the proof should have a proposition and a rationale for why you are entitled to assert that proposition. • Follow step 2 until you get the desired conclusion. Then you win!

8. A simple example Recall: E says that from (Φ →Ψ) and Φ, you should infer Ψ. In this case, Φ = P and Ψ = Q. So, Line 1 thus gives you (Φ →Ψ) and Line 3 gives you Φ. • PQ, QR, P├ R Recall: E says that from (Φ →Ψ) and Φ, you should infer Ψ. In this case, Φ = Q and Ψ = R. So, Line 2 thus gives you (Φ →Ψ) and Line 4 gives you Φ. 4. Q 1, 3 E 5. R 2, 4 E  This is the desired conclusion. So you win!

9. One rule students are eternally tempted to break • So this is okay: • P  ~~Q A • P A • ~~Q 1, 2 E • Q 3 ~E • But this isn’t: • P  ~~Q A • P A • P  Q 1, ~E • Q 2,3E • The basic rules of inference apply only to whole lines of a proof, not parts of a proposition. X

10. Proofs are really, really like games • Many people can learn to play chess correctly, but it takes some talent and practice to play chess strategically. • Many people can use the basic rules of inference properly, but it takes some talent and practice to prove things strategically.

11. Some helpful strategies • Recognize patterns. • Think about the big picture, then worry about the details. • Reverse engineer. • Using “clean-up” procedures, i.e., try to establish common patternsbetween different premises and intermediate conclusions in the proof. • Tease things out of the premises, i.e., use the rules of inference to draw interesting conclusions. • Cut the fat, i.e., use the rules of inference to eliminate statements that occur in the premises but not in the conclusion. • Know the rules that cause roadblocks for you.

12. Recognizing patterns • {P  {Q  [R v (S & ~T)]}}  (P v S), {P  {Q  [R v (S & ~T)]}} ├ P v S Recall: E says that from (Φ →Ψ) and Φ, you should infer Ψ. In this case, Φ = {P  {Q  [R v (S & ~T)]}} and Ψ = P v S. So, Line 1 thus gives you (Φ →Ψ) and Line 2 gives you Φ.

13. Reverse engineer • Look at the conclusion, and ask yourself how you might get there from the premises you have. • In other words, imagine your 2nd to last step, 3rd to last step, etc. until you get back to the premises.

14. A slightly more challenging example • ~Q→P, (R & S) →~Q, R, ~T↔S, ~T ├ P • A counterexample here is going to be very hard to follow. • A truth table will require 32 rows. • So, what can a formal proof do with this puppy?

15. Reverse engineering: For your scratch paper • ~Q → P, (R & S) →~Q, R, ~T↔S ~T ├ P 6. ~T→S 4 ↔E 7. S5,~T →S?? →E 8. R&S 3,S?? &I 9. ~QR&S??, 2 →E 10. P 1, ~Q?? → E

16. Example: The final result • ~Q → P, (R & S) →~Q, R, ~T↔S ~T ├ P 6. ~T→S 4 ↔E 7. S 5,6 →E 8. R&S 3,7 &I 9. ~Q 8,2 →E 10. P 1,9 → E

17. Sample Exercise, Nolt 4.2.2

18. Nolt, 4.2.18