How do we test the validity of the following

Question

(~p^ q) , (p--> r) , (~r-->s) , (s-->t) :- t

Answer 1

From 3 and 4 using law of syllogism , we can get~r-->t is true , but how do we test the validity of t only??

Answer 2

I don't think we can say anything about t
we know ~p but we have no information about ~p -> ??

Answer 3

Since ~p^q is true can we take ~p is true and because of that ~p-->~r is true(inverse of 2)
And then using the law of syllogism repetedly, we get ~p--> t is true.
And finally using law of detachment can we get t is true(~p-->~t is true and ~p is true)?

Answer 4

~p--> t is true*

Answer 5

I don't like it.
p -> r
does not mean that ~p -> ~r
The converse is not always true.
All we know is that ~r -> ~p

From #1, we know that ~p is true, but we know nothing about what happens for ~p