If Tim’s coat is in the hall and Tim is in the building then Tim is in his office.
If Tim is not in the building then his coat is not in the hall.
Tim’s coat is in the hall.
Therefore Tim is in his office.
I've just learnt something that I'm not so comfortable with.......Does any one know how to translate them and use the syntactic method to prove the validity???
A thousand thanks!
Tc = Tim’s coat is in the hall .
Tb = Tim is in the building.
To = Tim is in his office.
1. (Tc & Tb) -> To.
2. ~Tb -> ~Tc.
By truth tables..
(((Tc & Tb) -> To) & (~Tb -> ~Tc)) -> To, ..is a tautology.
1. (Tc & Tb) -> To. Premise
2. ~Tb -> ~Tc. Premise
2a. Tc -> Tb.
By: 2, (~q -> ~p) <-> (p -> q).
3. Tc. Premise
By: 3, 2a, (p & (p -> q)) -> q.
By: 3, 3a, 1, ((p & (p -> q) -> q.