Translate the following statements to symbolic form.
1. Either both C and
D or F but not G if A implies B.
(A à B) à {[(C . D) v F] . ~G}
2. If neither B nor
W but C, then S if and only if G.
[~(B v W) . C] à
(S ß>
G)
3. Assuming P, Q
unless R.
P à (Q v R)
4. A on condition
that R, but G only if F.
(R à
A) . (G à
F)
5. If both C and D,
then if W, B.
(C . D) à (W à
B)
Determine whether the following argument is valid or invalid
using the shortest truth table you can.
(F à G) . B
~B . F/ ~G
T T F F T F
T F T --------------------------------- T
T F F --------------------------------- T
F T T -- F F
F T F F F
F F T --------------------------------- T
F F F --------------------------------- T
VALID
Or, you could simply do it like this. If the argument were invalid, ~G would have
to be false, which means that G would have to be true. Further, ~B would have to be true, and so
would F. But if that is the case, then
if ~B has to be true, B has to be false.
In the first premise, if B has to be false, the argument cannot be
invalid. This is the case since, if B
is false, the entire first premise would be false. And if the any premise must be false on the assumption of the
falsehood of the conclusion, it is impossible that there can be a case in which
there can be a false conclusion with all true premises. So, the argument must be valid.
The truth
table above shows that there is no case in which, when the conclusion is false,
all the premises are true. If that is
the case, it shows that it is impossible in this argument for the conclusion to
be false and all the premises true.
When that is the case, the argument must be valid.
Determine whether the statement below is a tautology, a
contradiction, or a contingency.
A C| (A
v ~C) à (~A à ~C)
T F T T T
F T F T F
F F T
T T
Tautology