This example came up in class:

I love all logic, but I don’t love deductive reasoning. Therefore, the moon is made of green cheese.

I understand the premise is contradictory and the conclusion is false, but the prof said the argument is valid, which I don't understand.

The definition of validity was taught as: if premises are true, the conclusion must be true or it is impossible for the premises to be true and the conclusion to be false.

Isn't the premise false?

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    There is a difference between the logical argument being valid and the logical argument being "correct" in the usual sense of the word. You have to abstract away what the words actually say when playing with formal logic. – T. Sar Sep 12 at 10:58
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    One way to think about it is: Is there any possible universe, where you love all logic, yet don't love deductive logic, in which the moon might possibly not be made of green cheese? There are no such universes, therefore there are no universes where the moon is not green cheese, therefore the implication is valid.... "All my cars are red" and "all my cars are black" can both be true, if, and only if, I have no cars. – Ben Sep 12 at 13:36
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    Wikipedia says on Validity(logic): "It is not required that a valid argument have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion." - A world that makes it possible for the first two to be true, does not automatically include a green-cheese moon though, does it? ---- Also: Everybody here renders the First sentence als two statements :"I love all A" and "I do not love all A" - but is it not also readable as "I love all A with the Exception of a" ? – bukwyrm Sep 13 at 12:52
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    This is clearly fallacious reasoning. Everyone knows the moon is made of Wensleydale. – Jules Sep 13 at 15:06
up vote 37 down vote accepted

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.

Classical implication P → Q is equivalent to ¬P ∨ Q

 [P → Q]  P ¬P
       Q  1  1
      ¬Q  0  1

and equivalently

[¬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 and Q.

P ∧ ¬P
------
Q

Since P and 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 ----- with 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.

The definition of validity was taught as: if premises are true, the conclusion must be true or it is impossible for the premises to be true and the conclusion to be false.

Isn't the premise false?

The premises are contradictory.

So it is impossible for the premises to all be true.

Thus it is impossible for the premises to all be true and the conclusion false.

Therefore the argument is valid.

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    Is that merely because it is a logical implication p ⇒ q where it's only false if p is true and q is false? Is "therefore" what indicates that it is an implication? And would 'p' stand for both premises here? – Battle Sep 12 at 9:26
  • This "answer" doesn't explain anything. I would -1 it if I had the rep. – Pod Sep 12 at 10:00
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    This answer is spot on - have an upvote – Rory Alsop Sep 13 at 11:18
  • Maybe I'm an idiot, but: it is also impossible for the premises to all be true and the conclusion true. Therefore the argument is invalid. – Tanath yesterday
  • No, @Tanath , the "or" in the definition means at least one criteria need be met; one is. The argument would be invalid only if both were not met. – Graham Kemp yesterday

It's not perfectly clear, but my best guess is that the instructor intends:

I love all logic, but I don’t love deductive reasoning.

to contain two incompossible premises.

meaning if "I love all logic" is true, it requires "I love deductive reasoning" to be true.

and conversely if "I don't love deductive reasoning", then "I love all logic is false".

Ergo, this argument can never have both of its premises true.

If the premises can never be true at the same time, then an argument is valid (at least on the definition of validity where an argument is valid if it can never have true premises and a false conclusion).


Where the argument suffers a bit is on clarity. There's three issues here.

First, if the proof is at the level of sentential logic, then it's not clear that the first and second premises are such that they cannot both be true. To understand their incompatibility we need to use "all" and other concepts you may not have learned. We can formalize the argument as :

  1. ∀x (Lx -> Vx) [for any x if x is a logic, then love x]
  2. La -> ~Va [a = "deductive reasoning, it's a logic but "you don't love it]

These could not both be true. Thus with reference to the definition of validity (impossible to have all true premises and a false conclusion), this argument is valid.

Second, the argument uses "I". This gets a bit dicey due to two issues:

  1. Pronouns are always a worry. (this article by David Kaplan looks rather thick on demonstratives but there are some issues with using pronouns)
  2. "X loves" - this is dicey because it complicates things. A common whipping boy for the 20th century logicians was to look at how one could (a) love Sartre's fiction and (b) hate Sartre's philosophy. And to ask if this is simultaneously possible. Here, similarly, "I" could be confused about the meaning of "logic" and/or "deductive reasoning" and thus believe to hold both claims.

Third, while it seems pretty obvious, it's not perfectly clear that the two premises are entangled. This seems to be what's tripping up a commentor on my answer -- they are pointing out something true: 1. P. 2. Q. Therefore, R is invalid, because R can be FALSE when both P and Q are TRUE. This example is, however, not that. Instead, the two premises are related such that we can't just view them as completely separate -- thus, the validity. But that should be made explicit.

Sometimes this sort of thing depends on knowledge beyond just the rules. For instance,

  1. The moon is a giant tomato.
  2. The moon is a big piece of yellow cheddar cheese.
  3. Therefore, I ate salad for breakfast.

When each premise is considered independently as P,Q, and R. This argument is invalid. But if we accept that "P" and "Q" here are related such that if P is true, then Q is not true and if Q is true, then P is not true, then the argument would be valid because the two variables are not independent.

The question on the third point is how far do we expect someone looking at an argument to be able to connect these things before we're just being absurd and our argument is actually invalid.

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    I'm not sure if my wording was confusing or if you don't understand validity. The standard go to definition of validity is: "if all of the premises are true, then the conclusion must be true." Ergo, if it is never the case that all premises can be true, then an argument is valid -- because it can never produce a situation where the premises are true but the conclusion is false. – virmaior Sep 12 at 0:11
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    The composition of the moon has dependency on someone's opinion of logic therefore this argument is invalid. many statements depend on facts about the world (or people's opinions about them), but an argument can be valid even if it deals in things we know to be false -- because validity looks at the forms -- not the truth or falsity of the statements with respect to the world. – virmaior Sep 12 at 0:13
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    @Cell What virmaior is saying is standard textbook logic. It's also known as the principle of explosion: en.wikipedia.org/wiki/Principle_of_explosion – Eliran H Sep 12 at 0:25
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    @Cell I'm doing my best to make sense of the problem the OP was given. The OP was told by their professor that the argument is valid. The way to make sense of this is not P & Q. Therefore, R. Instead, it's ∀x (Bx) & ~Ba. Therefore, R. – virmaior Sep 12 at 0:36
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    @Cell you're confusing "the argument is valid" with "the conclusion is true". They aren't the same thing. A valid argument only implies a true conclusion if the premises are true, which they aren't in this case. Saying that "the Moon is made of green cheese" is a valid conclusion here is not saying anything at all about the composition of the Moon. – Mike Scott Sep 12 at 13:58

The way I think of this is similar to how Ben describes it in the comments to the question i.e. it's similar to a vacuous truth. Formally, I guess it's not exact the same but it might be easier to understand coming from that direction.

In a nutshell you can make any assertion about the properties of members of an empty set and it's true. For example, if I say "all flying elephants have gossamer wings", the statement is true. How so? Well if it's not true, then there need be one flying elephant whose wings are not gossamer. It's also true that no flying elephants have gossamer wings for the same reason.

The most common way you might encounter such a construction is when someone says somoething like "if he's a doctor then I'm Santa Claus" which is ultimately a fancy way of saying the premise (i.e. he is a doctor) must be false.

It's a little arbitrary but if logic wasn't defined this way, I recall that it creates issues in more complex situations, the details of which I can't remember.

The answer to your question, "How is the argument...valid?," is simply - by definition!

If you examine the argument, it meets the second definition of valid. It is impossible for both premises to be true, and the conclusion is false. Therefore the argument is valid - by definition!

To me, the premises, being true, are excluding deductive reasoning from the set of things called "logic". First, that exclusion is incoherent and therefore allows anything to be deduced from it. Second, those premises are actively changing what rules of logic are to be considered valid when evaluating the sentence (a rather amusing prospect). In both cases the reasoning is correct.

To clarify, I see two ways to interpret the question. The first one which is already pointed out by most answers is that the premises are false therefore anything can be deduced. The second interpretation is that the premises are true. If the premises are true, then deductive reasoning is not part of logic. If deductive reasoning is not part of logic, then (and this is probably where many will disagree), it is plausible that with this particular definition of logic, the sentence is logically correct. Obviously this hinges on the semantics of what logic and deductive reasoning are.

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    I am finding it hard to understand this answer. In the first sentence I don't see how the premises are "excluding deductive reasoning from the set of things called 'logic'". Regardless, welcome to this SE! – Frank Hubeny Sep 13 at 13:38
  • I hope my edit was able to clarify your meaning. – Mark Andrews Sep 14 at 1:18

protected by Philip Klöcking Sep 13 at 19:24

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