# Rules of inference

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Welcome to the forum.

I am having a bit of challenge with the following question.
It would help if you write exactly what the problem is asking you to do. And I mean as precisely as possible: saying "Prove that the hypotheses imply the conclusion" is not precise enough because the word "imply" can have several different meanings in mathematical logic.

What seems confusing to me is the relationship between hypothesis 1 and 2.
Would you be puzzled if I said that I find confusing the relationship between the following statements: "All my pets are dogs or cats" and "If a pet is a cat, it is a feline"? I suspect you would because it is not even clear what I mean by a relationship: these are two separate statements, nobody claims that they have to be consistent, or follow from each other, or anything like that. What relationship did you find or want to find between hypotheses 1 and 2, and why is this relationship confusing?

#### Alan

##### Member
Here's a proof (fill out the details).
1. Ax (Px v Qx)
2. Ax (~Qx v Sx)
3. Ax (Rx ->~Sx)
4. Ex ~Px
5. Pa v Qa , universal, 1.
6. ~Qa v Sa universal, 2.
7. Ra ->~Sa , universal, 3.
8. ~Pb, existential, 4.
9.Qb ,5,8 DS.
10. ~~Qb, 9, DN.
11. Sb 7,6, DS.
12. ~~Sb ->~Rb 7, MT.
13. ~Rb , 11,12, MP.
14. Ex ~Rx, QED.