We need to be a little bit careful with how we talk about validity.
Using the definition from Wikipedia and ultimately from the Internet Encyclopedia of Philosophy, validity and the related notion of soundness are defined as
A deductive argument is said to be valid if and only if it takes a
form that makes it impossible for the premises to be true and the
conclusion nevertheless to be false. Otherwise, a deductive argument
is said to be invalid.
A deductive argument is sound if and only if it
is both valid, and all of its premises are actually true. Otherwise, a
deductive argument is unsound.
These definitions do not make reference to any particular formal system. What kind of manipulation is valid formally changes from system to system (e.g. in intuitionistic logic vs classical logic).
They also apply to slightly different things, although the definition doesn't present it this way.
An inference rule can be thought of as having abstract or formal premises.
P -> Q P | Q is valid. It does not make sense to talk about whether it is sound or not because it doesn't form a complete argument in the sense of the definition.
A particular application of M.P. like
If it is raining, then the ground is wet. It is raining. Therefore the ground is wet. can be sound or not. In this case, it is not sound, since it's not actually raining where I am right now.
Even though it isn't raining right now, we can still say that
If it is raining, then the ground is wet. It is raining. Therefore the ground is wet. is valid. The truth of the argument flows from its premises. Equivalently, the form of the argument is valid if we replace the premises and conclusion with opaque symbols.
It also makes intuitive sense since the premise is causally related to the conclusion. Classical logic, however, doesn't give us the ability to talk about causality.
If it is raining, then there are clouds would work the same way, even though the causal relationship is completely different.
In the example given
I love all logic, but I don’t love deductive reasoning. Therefore, the
moon is made of green cheese.
It is not immediately obvious how to translate
I love all logic, but I don’t love deductive reasoning into a formal statement in classical logic that we can inspect. We have to make a choice.
One way is just to replace it with
Another way is to replace it with a single primitive variable
We can also replace it with
(P → Q) ∧ P ∧ (¬Q) where
P corresponds to
I love logic and
Q corresponds to
I love deductive reasoning. The relationship between the two
[I love logic] implies [I love deductive reasoning] is "obvious". Supplying it, however, gives us a complete contradiction.
If we choose either the first or the third option, then the premise cannot be satisfied regardless of how we plug in values (true or false) into variables.
This is an example of the principle of explosion or ex falso quodlibet. This is a property of some logical systems (including classical logic) where a false premise, or, equivalently, inconsistent premises, make the inference valid regardless of what the conclusion is.
We can see from the truth table definition of the
→ connective in classical logic that if the left argument is false, then the whole expression is true.
P → Q is equivalent to
¬P ∨ Q
[P → Q] P ¬P
Q 1 1
¬Q 0 1
[¬P ∨ Q] P ¬P
Q 1 1
¬Q 0 1
Just looking at the definition of the connective doesn't prove that the inference is valid on its own. It does suggest why this might be the case.
→ only inspects the truth values of its arguments, not the content.
Let's look at a skeleton inference rule with primitive propositions
P ∧ ¬P
Q are independent of each other, this potential inference rule being valid would show that the system we are working in satisfies the principle of explosion.
To justify the rule we can translate it into a formula involving just the primitive connectives by replacing
→ and determining whether the statement is an unconditional tautology.
(P ∧ ¬P) → Q
[(P ∧ ¬P) → Q] P ¬P
Q 1 1
¬Q 1 1
It is one, therefore classical logic satisfies the principle of explosion because the given rule is admissible / a theorem.
The premise in the statement given
I love all logic, but I don't love deductive reasoning is intended to be manifestly a contradiction (false) and also a joke.
The moon is made of green cheese is also commonly used as an example of an irrelevant conclusion. The example is supposed to highlight the disconnect between everyday reasoning with natural language and classical logic.
Paraphrased using disjunction instead of negation, the sentence would read, roughly
Either it is not the case that I love all logic but hate deductive reasoning,
or the moon is made of green cheese.
This sentence, to me at least, simply seems to be true since it boils down to
Either TRUE or FALSE.