# Are all answers to a contradictory question correct? Or are all wrong? Or is it something in between?

Suppose we have a contradictory question. For example

What is the sum of the angles in a triangle with sides 1 cm, 1 cm and 10 cm?

The question doesn't make sense because there is no such triangle (at least if we assume Euclidean plane geometry).

100°

correct? Is it wrong? Is there any correct answer?

I feel like I can argue either way:

It's wrong, for the same reason "1/0 = x" is false, for any value of xR. 1/0 is undefined and nothing else, and so is any property of a non-existent triangle. Put differently, the only correct answer to the question is "The sum of the angles is undefined." and all other answers are wrong.

It's correct, since you can derive a contradiction from the premise of the question, and you can derive any statement from a contradiction. (At least that's what I learned when studying natural deduction.)

So, is there a way to clearly argue one way or the other here? Is any of the reasoning above flawed?

• Welcome to Philosophy.SE. You seem to be confusing right/wrong and true/false. A statement in logic can be true or false; the answer to a question right or wrong. You could rephrase the answer to the form "The sum of the angles in a triangle with sides 1, 1 and 10 is 100°" which is a statement and thus has a truth value. I would suggest you remove this ambiguity from your question. Wrong/correct aren't really logical categories.
– user2953
Aug 17 '16 at 12:30
• In any case, this is related to Wittgenstein's logical atomism. He would argue that the sentence doesn't refer to an object, hence the sentence is meaningless. On the other hand, you could formalise this as (for all triangles: if its sides are 1, 1 and 10, then the sum of its angles is 100) or (for all triangles with sides 1, 1 and 10: the sum of its angles is 100); both are vacuously true.
– user2953
Aug 17 '16 at 12:32
• Right. I felt I had a tension between true/correct vs false/wrong. I still think that "the sum of the angles in a triangle with sides 1, 1 and 10" is undefined (regardless of the fact that I wrote "1/0 = x" is false). Aug 17 '16 at 12:35
• Yes, that is a valid theory. It is related to Verificationism. According to my textbook, "such a view of meaning enabled one to rule out of court as meaningless all statements which could neither be verified nor falsified by experience." (Anthony Kenny. An Illustrated Brief History of Western Philosophy. Blackwell Publishing, 2006).
– user2953
Aug 17 '16 at 12:41
• See Is the use of inconsistent definitions a logical fallacy? philosophy.stackexchange.com/questions/31058/… The answer depends on whether the question is existential (presupposes existence of the inconsistent object) or universal (if such an inconsistent object were to exist then it would be thus and so). Your formulation of the question is ambiguous on this score, and which way it goes depends on whether you formalize it with or without the existential claim. Aug 17 '16 at 20:38

As you correctly observe, the question asks whether something that doesn't exist has certain properties. This is related to the more general problem of talking about non-existent things. One prominent solution to this problem is given by Russell's theory of descriptions.

Allow me to use a different example:

Is the current king of France bald?

As there's no such person, would answers to this question be true or false?

The king of France is bald.

And a 'no' answer amounts to the claim:

The king of France is not bald.

Russell's solution is to construe such apparently referring expressions as existence claims. That is, the above turn into:

There is something which is a king of France, and it is the only such thing, and it is bald.

There is something which is a king of France, and it is the only such thing, and it is not bald.

Construed thus, both of these are false, since both of them claim that something exists ('a king of France') while in fact there is no such thing.

Accordingly, any answer to your example question would be rephrased, e.g. as:

There is something which is a triangle with sides 1 cm, 1 cm and 10 cm ... and the sum of its angles is X.

For any X, that statement is false.

• Thank you. Very informative. A follow up: Suppose I massage the question the same way. I think it would become something like "There is something which is a triangle with sides 1,1,10. What is the sum of it's angles?" By referring to a theorem stating that a+b>=c for the sides of all triangles, you can derive a contradiction. And by contradiction elimination (⊥-elim in Fitch style) you can then derive any conclusion. Wouldn't it be valid to derive the conclusion stating that the answer is 100°? Aug 17 '16 at 13:17
• I think in the English language "the current king of France" and "every current king of France" or not the same thing and would need to be treated differently. "Every" doesn't claim that there is one. Aug 17 '16 at 14:56
• @gnasher729, I think that is pretty obvious. Besides it has already been pointed out by Keelan in his second comment to the question. Aug 17 '16 at 15:15
• @aioobe I'm not sure you can derive a contradiction that way. Questions don't have truth values, so you cannot use them as premises in arguments, and so you cannot derive anything from them. Aug 17 '16 at 17:07
• Excellent answer. I would notice that Russell's method is contradictory with "logical atomism" however (There is something which is a king of France < there is something that is France and is a country, and it has a king; and so on, infinetely). Aug 18 '16 at 13:46

Eliran's answer is great, but it isn't the only answer. Russell indeed took the view that descriptions that refer to nothing render any surrounding statement false.

Another major school of thought, though, and the one more often taught in contemporary philosophy of language and linguistics, is rooted in Frege's dissenting view: that such descriptors render statements meaningless.

Thus, "the king of France is bald" is neither true nor false, but simply fails to have a meaning. No truth value, just a big question mark.

The intuition used to promote this view is that there's something different about "there is a king of France" and "the King of France is bald". The former prompts a "no", while the latter tends to evoke a "...what?" Because it features a failed presupposition: that there even is a king of France.

There is philosophy, mathematical logic, and English language... And they don't agree.

In the triangle question, to avoid endless discussions, I wouldn't see "100 degrees". I'd say "In any triangle with sides 1cm, 10cm and 100cm, the sum of angles is 100 degrees" or better yet "there is no triangle with sides 1cm, 10cm and 100cm where the sum of angles isn't 100 degrees". The last one is beyond discussion in philosophy, mathematical logic, and plain English.

Is the current king of France bald? In the English language, this is both a claim and a question. The claim is "there is exactly one person who is currently king of France". And the question: "Is the unique person who is currently king of France bald?". The correct reply is to state that the claim is wrong and the question therefore doesn't make sense.

You could also say "If there was a current king of France then he would be bald" and you could just as well say "If there was a current king of France then you would have immense hair" or "nobody with hair on his head is the current king of France".

• I understand how to rephrase the question / statement into one that is vacuously true. That's not what I'm asking about. Aug 17 '16 at 15:09

The official answer is that any deduction from a false premise is true. It is false that you have the triangle, so anything about it is true.

At the same time, the official answer is that any truth is implied by any other truth.

One of the reasons for moving away from classical logic in mathematics, in general, is that both of those positions are preposterous.

And the problem does not hinge on the problem of modal existence.

Just as it is not true that there is a sum of the angles of your nonexistent triangle, it is just not the fact that "If puppies are cute, then the U.S. is a democratic republic." The two have nothing to do with one another, and that 'if' and that 'then' have no business being there.

Positions like Intuitionism, Constructivism, and even Fictionalism formalized through proof theory take restoring the natural connection between statements seriously, whether they are motivated by distrust of classical foundations, an imperative toward usefulness, or a feeling of basic integrity.

The problem is that it is so convenient to render formal logic into an algebra, that thinking through the real rules is, by comparison, a muddle. Intuitionism, for instance, proposes that the real numbers, in the form of an endless sequence of digits actually exist. But constructivists respond, "Not in the real world, they don't. If you are actually interested in such things, you might well have already gone around the bend that leads down the path to untrustworthy conventions." Even if you back off to the modified formalism of proof theory, can proofs be arbitrarily long? By arbitrarily, can we mean implicitly infinitely long with big infinite gaps? Unfortunately, it matters.

So, yes, there are many arguments against this kind of logic. But there are so many it is hard to choose one. And very few people are worried enough about it to fight for a given solution, even if they find winding out special solutions can be fun, (and sometimes helps improve engineering tasks like computer science.)

• This is an interesting answer, and I agree with most of with. Just to nitpick, At the same time, the official answer is that any truth is implied by any other truth is not the whole "truth". The official position is that a true proposition is implied by any other proposition, true or false. "if puppies are repugnant, the United States is a democratic republic" is as valid as "if puppies are cute, the United States is a democratic republic" Aug 18 '16 at 13:56
• @LuísHenrique The first half of that is already covered by the other rule, So putting it that way seemed redundant to me. 'At the same time' means at the same time.
– user9166
Aug 18 '16 at 13:59
• in common parlance, we would say that the United States is a democratic republic, whether puppies are cute or not, and this seems to make more sense. The semi-formal language makes it more difficult to understand, which is perhaps another argument against formalisations. Aug 18 '16 at 14:03
• @LuísHenrique The question is in the context of math, where formalizations have already paid off big-time. The examples in casual form are meant to point out that we don't really accept the rules of classical logic in everyday conversation, as we do in math. The 'if puppies are cute' sentence could be in a Lewis Carrol novel, or a logic or math text, but not in a normal conversation.
– user9166
Aug 18 '16 at 15:13

• TRUE
• FALSE
• Meaningless (nonsense, gibberish, unintelligible, malformed, ERROR in programming)

What does purple taste like?

...it is meaningless.

• `NULL` is not unknown; it is known to be `NULL`. Whether you use a `NULL` value for something unknown, is an interpretation issue. You could also interpret `NULL` as 5, if that is somehow useful in your program. Also, unanswered is not the same as unknown (I may know something but decide not to answer about it).