1. CS 270 Math Foundations of CS
Natural Deduction
CS 270 Math Foundations of CS
Jeremy Johnson

2. Outline An example What’s a proof Rules of natural deduction
Validity by truth table
Validity by proof
What’s a proof
Proof checker
Rules of natural deduction
Provable equivalence
Soundness and Completeness

3. An Example
If the train arrives late and there are no taxis at the station, then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station.
If it is raining and Jane does not have here umbrella with her, then she will get wet. Jane is not wet. It is raining. Therefore, Jane has her umbrella with her.

4. An Example
If the train arrives late and there are no taxis at the station, then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station.
p = the train arrives late
q = there are taxis at the station
r = John is late for his meeting.
𝑝∧¬𝑞 →𝑟, ¬𝑟,𝑝⊢𝑞 [a sequent]

5. An Example p = it is raining q = Jane has her umbrella
r = Jane gets wet.
𝑝∧¬𝑞 →𝑟, ¬𝑟,𝑝⊢𝑞
If it is raining and Jane does not have here umbrella with her, then she will get wet. Jane is not wet. It is raining. Therefore, Jane has her umbrella with her.

6. Validity by Truth Table
𝑝∧¬𝑞 →𝑟, ¬𝑟,𝑝⊢𝑞 p q r q r
pq
(pq)r F T

7. Proof
By applying rules of inference to a set of formulas, called premises, we derive additional formulas and may infer a conclusion from the premises
A sequent is 1,…,n ⊢ 
Premises 1,…,n
Conclusion 
The sequent is valid if a proof for it can be found

8. Proof
A proof is a sequence of formulas that are either premises or follow from the application of a rule to previous formulas
Each formula must be labeled by it’s justification, i.e. the rule that was applied along with pointers to the formulas that the rule was applied to
It is relatively straightforward to check to see if a proof is valid

9. Validity by Deduction 𝑝∧¬𝑞 →𝑟, ¬𝑟,𝑝⊢𝑞 1 𝑝∧¬𝑞 →𝑟 premise 2 ¬𝑟 3 𝑝 4 ¬𝑞
assumption 5
𝑝∧¬𝑞
∧i 3,4 6 r
→e 1,5 7 ⊥
⊥i 6,2 8 q
¬e 4-7

10. Rules of Natural Deduction
Natural deduction uses a set of rules formally introduced by Gentzen in 1934
The rules follow a “natural” way of reasoning about
Introduction rules
Introduce logical operators from premises
Elimination rules
Eliminate logical operators from premise producing a conclusion without the operator

11. Conjunction Rules Introduction Rule Elimination Rule    i   
 
 i
  
  
 e1 
  
 e2 

12. Implication Rules Introduction Rule Assume  and show 
Elimination Rule (Modus Ponens)  … 
 i
  
   
 e 

13. Disjunction Rules Introduction Rule
Elimination Rule (proof by case analysis) 
 i1
   
 i2
    …   …
  
 e 

14. Negation Rules
Introduce the symbol (⊥ =bottom) to encode a contradiction
Bottom elimination
⊥ can prove anything
Bottom introduction ⊥
⊥ e. 
 
⊥ i ⊥

15. Negation Rules
Introduction and elimination rules (proof by contradiction)
Double negation  … ⊥
 i   … ⊥
 e 
 
  e 

16. Derived Rules Modus Tollens Disjunctive Syllogism DeMorgan’s Law
(P  Q)  P  Q
(P  Q)  P  Q
    MT 
    DS 

17. Alternative Proof 𝑝∧¬𝑞 →𝑟, ¬𝑟,𝑝⊢𝑞 1 𝑝∧¬𝑞 →𝑟 premise 2 ¬𝑟 3 𝑝 4 ¬ 𝑝∧¬𝑞
MT 1,2 5
¬𝑝  ¬¬𝑞
DM 4 6
¬¬𝑞
DS 5,3 7 q
¬¬e 6

18. LogicLab
The LogicLab tool from the Logic and Proofs course from the CMU online learning initiative allows you to experiment with natural deduction proofs

19. LogicLab

20. Provable Equivalence
 and  are provably equivalent,  ⊣⊢ , iff the sequents  ⊢  and  ⊢  are both valid
Alternatively  ⊣⊢  iff the sequent ⊢        is valid
A valid sequent with no premises is a tautology

21. Distributive Law P  (Q  R) ⊢ (P  Q)  (P  R) 1 P  (Q  R) premise2 P
e1 1 3
Q  R
e2 1 4 Q
Assumption 5
P  Q
i 2,4 6
(P  Q)  (P  R)
i1 5 7 R 8
P  R
i 2,7 9
i2 8 10
e 3,4-6,7-9

22. Distributive Law (P  Q)  (P  R) ⊢ P  (Q  R) 1 (P  Q)  (P  R)
premise 2
P  Q
Assumption 3 P
e1 2 4 Q
e2 2 5
Q  R
i1 4 6
P  (Q  R)
i 3,5 7
P  R 8
e1 7 9 R
e2 7 10
i2 9 11
i 8,10 12
e 1,2-6,7-11

23. Semantic Entailment
If for all valuations (assignments of variables to truth values) for which all 1,…,n evaluate to true,  also evaluates to true then the semantic entailment relation 1,…,n ⊨  holds

24. Soundness and Completeness
1,…,n ⊨  holds iff 1,…,n ⊢  is valid
In particular, ⊨ , a tautology, ⊢  is valid. I.E.  is a tautology iff  is provable
Soundness – you can not prove things that are not true in the truth table sense
Completeness – you can prove anything that is true in the truth table sense