Mathematical Logic unusual question

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.]

Tags:

Logic

Related

Why is compactness required in Arzela Ascoli? is this true for an induction proof Why do we need the local triviality condition when working with vector bundles? Prove $\lim_{x\to 0} f(x)$ exists $\implies\lim_{x\to 0} f(x^2)$ exists Improper integral of $\int_0^\infty \frac{dx}{(x^2+{\pi}^2)\cosh(x)}$ Function defined everywhere but continuous nowhere How do I solve this integral via integration by parts when it shows a recursion-like behavior? Prove/disprove: $ I_1 \cong I_2 \iff R/I_1 \cong R/I_2 $ On the Equivalent Norms for Subspaces Prove that $\varphi$ is continuous on the real numbers and $\varphi(x) = x$ $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ is Galois over $\mathbb{Q} \iff p=n^2+1$ for a prime $p$. Is $\mathbb{Q} \times \mathbb{Q} \cup \mathbb{I} \times \mathbb{I}$ disconnected?

Recent Posts