construct a truth table to determine whether each argument is valid. Be sure to state whether it is valid or invalid with each question.
1) ~p v q
p
____
triangle of dots p^~q
invalid.
knowing something about ~p tells you nothing about p.
If it's not raining, I'll go to the store.
It's raining.
no telling whether you go to the store or not.
You can say that if you don't go to the store, it's raining. (contrapositive)
To construct a truth table, we need to list all the possible combinations of truth values for the given variables. In this case, we have two variables, p and q, so we need four rows in our truth table.
1) ~p v q:
p | q | ~p | ~p v q
--------------------------
T | T | F | T
T | F | F | F
F | T | T | T
F | F | T | T
2) p:
p | q | p
----------------
T | T | T
T | F | T
F | T | F
F | F | F
3) triangle of dots p^~q:
p | q | ~q | p^~q
------------------------
T | T | F | T
T | F | T | F
F | T | F | F
F | F | T | F
Now that we have constructed the truth table, we can evaluate the validity of each argument.
1) ~p v q:
This argument is valid. In every row of the truth table, when ~p v q is true, both ~p and q are true.
2) p:
This argument is neither valid nor invalid. It simply states the truth value of the variable p in each row of the truth table.
3) triangle of dots p^~q:
This argument is invalid. In the truth table, there are rows where p^~q is false, even if p is true.
Remember, a valid argument means that it follows a correct logical pattern and always produces a true conclusion when all the premises are true. An invalid argument means that it does not follow a correct logical pattern and can produce a false conclusion even if all the premises are true.