Solved mathematical logic problems to practice reasoning

 

(0) Mathematical logic problems : Solution 0.1

0.1 A few days ago I was told about a man named Roque, a barber in Batuecas. They had only said two sentences when I exclaimed: Impossible!

"Roque lives in Batuecas"

"Roque shaves the inhabitants of Batuecas who do not shave themselves and only them."

 

INTUITIVE ANSWER:

 

Suppose Roque shaves himself,

in that case, since he is an inhabitant of Batuecas, he should not shave himself. But the truth is that he does shave himself.

 

Now, suppose Roque does not shave himself,

Well, in that case Roque shaves himself for that very reason.

 

MATHEMATICAL ANSWER:

 

1. Hrb ⊦ Arr ∧ ¬Arr
2. ∀x(Hxb ∧ ¬Axx <-->Arx)
3. Hrb ∧ ¬Arr <--> Arx) E∀2 (x:a)
┌ 4. Arr
| 5. Arr -->Hrb ∧¬Arr E<-->3
| 6. Hrb ∧ ¬Arr E-->4,5
┗ 7. ¬Arr E^6
8. Arr --> ¬Arr I-->4-7
┌ 9. ¬Arr
| 10. Hrb ∧¬Arr -->Arr E<-->3
| 11. Hrb ∧ ¬Arr I^1,9
┗ 12. Arr E--> 10,11
13. ¬Arr --> Arr I--> 9-12
14. ¬Arr ∨ ¬Arr Def -->/v 8
15. Arr ∨ Arr Def -->/v 131
16. ¬Arr Idem 14
17. Arr Idem 15
18. Arr ∧ ¬Arr I^ 16, 17

 

This demonstrates that these premises lead to a contradiction, that is, that Roque shaves and does not shave himself. Since these two premises lead us to a contradiction, we can affirm that both premises are inconsistent. Impossible!