(1) ∼ p ∨ q → r
(2) s ∨ ∼ q
(3) ∼ t
(4) p → t
(5) ∼ p ∧ r →∼ s
Conclusions: q∧ r
We will start by making conclusion false try to make all the premeses true .
If we can make the premises true and conclusion false that means this is invalid argument , if not then it is valid.
If we make conclusion false q^r= false ,
this means that both q and r are false .
Now we try to make all the arguments true:
~p^r -> ~s
since r is false so r and anything is false
false implies anything is true
so we made (5) ∼ p ∧ r →∼ s == True
sv~q
q is false from conclusion so ~q is true ,
and we know that true and anything is true
so we made (2) s ∨ ∼ q == True
∼ p ∨ q → r
q is false and r is false from conclusion , so to make ~p∨ q → r as true , we need false implies false so ~p should also be false .
from above we can see that p=true.
so we made (1) ∼ p ∨ q → r == True
p → t
here p=true , to make this argument as true we make t = true , so true implies true is true.
so we made (4) p → t == True
but then ∼ t is false
(3) ∼ t == false
Hence we are unable to make all premises as true .
So this is a valid argument.
