Basic Inference Rules

Kareem Khalifa

Department of Philosophy

Middlebury College

Overview

• Why this matters

• Proofs are like games

• Procedure for constructing formal proofs

• Sample Exercises

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!

Proofs are like games

• 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

The Basic Inference RulesName Abbreviation Rule

Conditional Elimination (Modus ponens)

→E From (Φ →Ψ) and Φ, infer Ψ.

Conjunction Elimination (Simplification)

&E From (Φ&Ψ), infer either Φ or Ψ.

Conjunction Introduction (Adjunction)

&I From Φ and Ψ, infer (Φ&Ψ).

Biconditional Elimination ↔E From (Φ↔Ψ), infer (Φ→Ψ) or (Ψ→Φ).

Biconditional Introduction ↔I From (Φ→Ψ) and (Ψ→Φ), infer (Φ↔Ψ).

Disjunction Elimination (Dilemma)

vE From (Φ v Ψ), (Φ→θ), and (Ψ→θ), infer θ.

Disjunction Introduction (Addition)

vI From Φ, infer (Φ v Ψ) or (Ψ v Φ).

Negation Elimination (Double Negation)

~E From ~ ~Φ, infer Φ.

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.

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:

1. List the premises as the first lines of the proof. Mark them with an “A” for Assumption.

2. 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.

3. Follow step 2 until you get the desired conclusion. Then you win!

A simple example• PQ, QR, P├ R

1. PQ A

2. QR A

3. P A

4. Q 1, 3 E

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.

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.

5. R 2, 4 E

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.

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.

This is the desired conclusion. So you win!

One rule students are eternally tempted to break

• So this is okay:

1.P ~~Q A

2.P A

3.~~Q 1, 2 E

4.Q 3 ~E

• But this isn’t:

1. P ~~Q A

2. P A

3. P Q 1, ~E

4. Q 2,3E

• The basic rules of inference apply only to whole lines of a proof, not parts of a proposition.

X

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.

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 patterns between 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.

Recognizing patterns

• {P {Q [R v (S & ~T)]}} (P v S), {P {Q [R v (S & ~T)]}} ├ P v S

1. {P {Q [R v (S & ~T)]}} (P v S) A

2. {P {Q [R v (S & ~T)]}} A

3. P v S 1, 2 ERecall: 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.

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.

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.

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?

Reverse engineering: For your scratch paper

• ~Q → P, (R & S) →~Q, R, ~T↔S ~T ├ P

1. ~Q → P A

2. (R & S) →~Q A

3. R A

4. ~T↔S A

5. ~T A

6. ~T→S 4 ↔E

7. S 5,~T →S?? →E 8. R&S 3,S?? &I

9. ~Q R&S??, 2 →E 10. P 1, ~Q?? → E

Example: The final result• ~Q → P, (R & S) →~Q, R, ~T↔S ~T ├ P

1. ~Q → P A

2. (R & S) →~Q A

3. R A

4. ~T↔S A

5. ~T A

6. ~T→S 4 ↔E

7. S 5,6 →E

8. R&S 3,7 &I9. ~Q 8,2 →E

10. P 1,9 → E

Sample Exercise, Nolt 4.2.2

1. P → (Q →R) A

2. P A

3. Q AR To be proven

4. Q → R 1, 2 →E

5. R 3, 4 →E

Nolt, 4.2.181. (P & (Q v R)) S A

2. P A

3. ~~R AS To be proven

4. R 3 ~E

5. Q v R 4 v I

6. P & (Q v R) 2, 5 &I

7. (P & (Q v R)) S 1 E

8. S 6, 7 E