Statements P->Q, ~R->(S->T), R v(P v T), and ~R are true. What is the truth value for the statement, (Q v S)?
A nice puzzle.
The truth table for ~R is
R ~R
T F
F T
Since ~R=true, therefore R=false.
Given R is false, and
R v(P v T)=R ∨ P ∨ T is true, we conclude that P∨T is true, i.e.
P=true ∨ T=true .... (1)
Then we need the table for the conditional (->)
A B A->B
T T T
T F F
F T T
F F T
This means that the conditional is false if and only if the first proposition (A) is true and the second (B) is false.
Consequently, given that
~R->(S->T) is true
and that ~R=true, we conclude that
S->T = true .....(2)
We are also given that
P->Q = true ....(3)
From (1), we have two cases, either P is true or T is true.
case 1. P=true
From (3), P->Q = true, so if P is true, Q must be true.
case 2: T=true
(2) does not guarantee that S is true. So S can be true or false.
Hence, we do not know if (Q∨S) is true or false.
Can you check the question to see if there was a typo, such as
R v(P v T) was meant to be R∨(P∧T) ?
You came up with everything I came up with. I, too, believe something must be incorrect. I met with my professor today, and showed him my 25+ pages of attempts at solving it. He didn't want to go through the whole thing, but said it had to do with focusing on showing (QvS) could not be False (Q and S both being False). I didn't press him any further because I was still getting an A out of the course, but I'll probably still work on the problem to satisfy my own curiosity. Will try to post if I solve it.
Sorry, I came up with the same conclusion:
when P=false, T=true, we cannot determine the truth value of neither Q nor S.
To determine the truth value for the statement (Q v S), we need to evaluate the truth values for Q and S separately and then apply the logical OR (v) operation.
Given statements:
1) P -> Q
2) ~R -> (S -> T)
3) R v (P v T)
4) ~R
Let's evaluate the truth values step by step:
From statement 4) ~R, we know that R is false.
Now, let's use statement 3) R v (P v T) with R being false:
R v (P v T) translates to (False) v (P v T), which simplifies to P v T.
Since R is false, P v T is equivalent to P v True, which is always True.
Next, let's apply statement 1) P -> Q with the value of P being True:
P -> Q translates to True -> Q, which simplifies to Q.
So, Q is True.
Now, let's go back to statement 2) ~R -> (S -> T) with the value of R being false:
~R -> (S -> T) translates to ~False -> (S -> T), which simplifies to True -> (S -> T).
Since ~R (not R) is True, the resulting expression is always True regardless of the values of S and T.
With Q = True, we can conclude that (Q v S) simplifies to (True v S), which is always True.
Therefore, the truth value for the statement (Q v S) is True, regardless of the specific truth values of S.