Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. It only takes a minute to sign up.
Sign up to join this community
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?
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.
Modus ponens 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 false.
Another way is to replace it with a single primitive variable P.
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.
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.
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?
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 :
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:
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,
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.
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.
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.
