Write the argument below in symbols to determine whether it is valid or invalid. State a reason for your conclusion. Specify the p and q you used. Submit your full detailed solution to the dropbox.
If the gazebo is made of wood, then the vine is growing on the gazebo.
The vine is not growing on the gazebo.
Therefore The gazebo is not made of wood.
To determine the validity of the argument, we first need to translate the statements into symbolic notation and assess the logical structure.
Let's assign:
p: The gazebo is made of wood.
q: The vine is growing on the gazebo.
The argument can be represented as:
If p, then q.
Not q.
Therefore, not p.
Using these symbolic expressions, the argument can be written as:
p -> q
~q
-------
~p
Now, let's apply a truth table to check the validity of the argument.
p | q | ~q | p -> q | ~p
-------------------------
T | T | F | T | F
T | F | T | F | F
F | T | F | T | T
F | F | T | T | T
Examining the truth table, we can see that when both premises are true, the conclusion is always true. Therefore, the argument is valid.
The reason for its validity lies in the logical structure of the statements, which follow the form of a valid rule of inference called modus tollens: If p implies q, and q is false (~q), then the negation of p (~p) must be true.
Hence, based on the truth table and the logical structure of the argument, the argument is considered valid.