So a true or false question came in my quiz today. It went like this:-
If the moon is made out of chocolate then I am a purple dinosaur.
This is a p implies q statement but i cant really seem to prove or disprove either so i said that its true. The question that followed this one was:-
If the moon was made out of chocolate then I am not a purple dinosaur.
Again i could not prove or disprove it so i said it's true. Can anyone help out?
A statement $P \Rightarrow Q$ is true if and only if either P is false (an implication with a false antecedent is true) or Q is true.
Since P = "moon is made out of chocolate" is false, the conditional $$P \Rightarrow Q$$ is true for any Q. In particular it is true for Q = "I am a purple dinosaur".
Note: the word "either" in the construction "either P is false or Q is true" serves a bracketing function just like the word "both" that may be used to distinguish between "Both P or Q and R" and "P or Both Q and R." (If we omit "Both", then they look the same: "P or Q and R.) However, the word "either" in this message doesn't indicate "exclusive or" in mathematics. The "exclusive or" operation is sometimes written as "xor", such as if we're talking about the operation is a computer program or an electronic gate in a computer circuit.
I rather like this way of thinking about it:
Suppose the moon is made of chocolate. It follows that the moon is made out of chocolate or that I am a purple dinosaur. On the other hand, it is known that the moon is not made out of chocolate. Since the moon is made out of chocolate or I am a purple dinosaur, and the moon is in fact not made out of chocolate, it must be the case that I am a purple dinosaur.
Thus we have shown that if the moon is made out of chocolate then I am a purple dinosaur.
Gottlob Frege explained it in Logische Untersuchungen, Dritter Teil, see https://digi20.digitale-sammlungen.de/de/fs1/object/display/bsb00047844_00084.html, that way (roughly translated):
Even the thought expressed in the sentence: "If my cock lays an egg today then the Cologne cathedral will crash tommorow." is true. "But the condition and the conclusion lack from any context." one may say. Well, I didn't demand any such context in my explanation and I just ask to understand "If $B$ then $A$" in the way $$\text{not [not $A$ and $B$].}$$
So your example is asked to be understood as
not[I am not a purple dinosaur and
the moon is made out of chocolate.]
Let's look at the truth table for the formula $a \rightarrow b$. $$\begin{array}{|c|c|c|} \hline a & b & a \rightarrow b \\\hline T & T & T \\\hline T & F & F \\\hline F & T & T \\\hline F & F & T \\\hline \end{array}$$
We see that the only case in which $a \rightarrow b$ is false is when $a$ is true and $b$ is false (the intuitive notion of $a$ does not imply $b$). $a \rightarrow b$ is true otherwise.
Let $a$ stand for the statement "the moon is made out of chocolate", and $b$ stand for "I am a purple dinosaur."
$a \rightarrow b$ stands for "If the moon is made out of chocolate, then I am a purple dinosaur."
$a \rightarrow \neg b$ stands for "If the moon is made out of chocolate, then I am not a purple dinosaur."
$a$ is false, so by the above, $a \rightarrow b$ is true, and so is $a \rightarrow \neg b$. Both statements in question are true.
Implication is false if and only if the antecedant is true and the consequent is false; otherwise, it is true. In both cases, your antecedent, which states "the moon is made of chocolate," is false, so the implication is true.
Required to prove:
If the moon is made out of chocolate then I am a purple dinosaur.
Let $C$ be the proposition that the moon is made of chocolate. Let $D$ be the proposition that I am a purple dinosaur.
We know that $C$ is false, i.e. that $\neg C$ is true.
We can prove that $\neg C \implies [C\implies D]$ using a truth table:
We can also prove it by contradiction using natural deduction (in DC Proof 2.0 format):
