I'm trying to wrap my head around this new subject. I have to determine the validity of this argument (using a truth table):
"If Steve went to the movies then Maria's sister would not have stayed home. Either Steve went to the movies or Maria or both. If Maria went, then Maria’s sister would have stayed home. Both Maria's sister and Steve’s sister stayed home. So, Steve did not go to the movies."
So far, I've come up with this, as a first step. I doubt it's correct. Please correct and guide me, as I'm still new to all of this. Thank you
P = Steve went to the movies
Q = Maria's sister stayed home
R = Maria went to the movies
S = Steve's sister stayed home
$(P \rightarrow \lnot Q), ((P \lor R) \lor (P \land R)), (R \rightarrow Q), (Q \land S) ∴ \lnot P$
Best Answer
A reasoning with premises P1, P2, P3, etc. and conclusion C is valid iff its corresponding conditional is valid ( = is aa tautology).
By " corresponding conditional" I mean : " P1 & P2 & P3... --> C"
( For this kind of problem, this definition is perfectly OK).
So build the corresponding conditional of your reasoning by putting an "&" between the premises, adding an arrrow and finally your conclusion.
Build a truth table for this conditional.
Note : here you need a 2 to the 4th power = 16 lines truth table.
In case this conditional has truth value " true" on all lines of the truth table, the reasoning is valid.
Note : apparently, premises 2 and 3 are useless, since the conclusion can be proved using only premises 1 and 4
(1) P --> ~ Q
(2) Q & S
(3) Since Q and S is true, Q is true.
(4) Since P --> ~Q is true , Q --> ~P is true ( by contraposition)
(5) Since Q --> ~P and Q are true, ~ P is true ( as desired).