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I am reading How to prove it by Velleman, and something is bothering me. On page 74, he states that "all unicorns are green" is vacuously true and so is "all unicorns are purple." He goes on to state that this two statements do not contradict each other and does not give any reasons why. This does not make sense to me. If both statements are considered to be true, the conclusion is that they contradict each other.

I understand the mathematical reasoning behind the conclusion, but that still does not explain why we should accept that both of the statements that are supposedly true do not contradict each other. Does this signal to a problem in the theory of quantificational logic mainly in how the conditional connector is defined?

SebastianLinde
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    To get a contradiction, you need a unicorn first. No unicorn, no contradiction. – Thomas Jan 26 '22 at 20:59
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    I think the feeling of contradiction comes from our brains that really can conceive of unicorns (despite none existing). Here's a pair of true, purely mathematical implications with the same relationship: "Every number that's both even and odd is prime" and "Every number that's both even and odd is composite". Do you feel that those two statements contradict each other? – Greg Martin Jan 26 '22 at 21:11
  • @Thomas If we just consider our definition for the conditional connector, we must say that those two statements are true. But it is obvious that they can't be true at the same time. What am I missing? – SebastianLinde Jan 26 '22 at 21:28
  • @GregMartin I see your point, but my doubt remains. No matter the statements, I just don't see how we can say that two statements that contradict each other can be true at the same time. – SebastianLinde Jan 26 '22 at 21:30
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    That's fair. All I can suggest is that whenever our intuition and the mathematical truth are in conflict, we have to modify our intuition. Your intuition tells you that those statements contradict each other, but that doesn't mean the statements actually contradict each other; this is a learning moment, and time to modify our intuition. – Greg Martin Jan 26 '22 at 21:42
  • For "all unicorns are green" and "all unicorns are purple" to be contradictory (both green and purple) there must be something that is green and purple. As "all unicorns" is not anything at all, it is not a thing that is both green and purple. .... or consider this: "A is green and purple" is a contradictory statement. Replace "A" with "nothing" and we have "nothing is green and purple". Is that a contradictory statement? How can nothing be both green and purple if being green and purple is contradictory. Well, nothing is not a thing and does not exist. – fleablood Jan 26 '22 at 21:42
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    "all unicorns are green" $\iff$ "there are no unicorns that are not green". Well, are there any unicorns that are not green? – fleablood Jan 26 '22 at 21:53
  • Well, unicorns are essentially just imaginary horses that happen to have a horn (as far as appearance is concerned), correct? A horse can be brown and black, so in the magical made-up land where unicorns actually exist - it stands to reason that hypohetical unicorns could also have varying colours. For instance "all zebras are white" is arguably true, as it doesn't say they're not also black. – Andrew Corrigan Jan 27 '22 at 08:47
  • You can claim "all airplanes I have bought are jumbo jets" and also "all airplanes I have bought are Pipers" if you have not bought any planes. Those are of course then very misleading statements , but logically undoubtly true. The contradiction only occurs if you actually have bought at least one plane. Then , of course , it is impossible that both statements are true. – Peter Jan 27 '22 at 12:45

4 Answers4

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Both statements are true. So of course they do not contradict each other.

Perhaps say it another way: From the two statements "all unicorns are green" and "all unicorns are purple", we may conclude "there are no unicorns".

GEdgar
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    Well I guess you conclude that either there are no unicorns or that green is purple, and since green is not purple (at least to my understanding of color), we then conclude that there are no unicorns. – Steven Creech Jan 26 '22 at 21:35
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To get a contradiction, you must have a pair of sentences which are the negations of one another. You seem to believe that "all unicorns are green" and "all unicorns are not-green" are negations of one another, but this is not true. For comparison, consider "all numbers are even" and "all numbers are odd." Both of these sentences are obviously false, which means they can't be negations of one another, because the negation of a false statement should be a true statement.

Instead, the negation of "all numbers are even" is "at least one number is odd," which is a true statement, as we would expect from negating a false statement. Similarly, the negation of "all unicorns are green" is "at least one unicorn is not green," but that can't be true unless at least one non-green unicorn actually exists.

Kevin
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For "all unicorns are green" and "all unicorns are purple" to be contradictory (both green and purple being an impossibility) there must be something that is green and purple. As "all unicorns" is not anything at all, it is not a thing that is both green and purple.

....

or consider this: "A is green and purple" is a contradictory statement.

Replace "A" with "nothing" and we have "nothing is green and purple". Is that a contradictory statement? How can nothing be both green and purple if being green and purple is contradictory.

Well, nothing is not a thing and does not exist. Saying "nothing is green and purple" is not saying "there is something that is nothing, and it is green and purple".

After all, what is "all unicorns". Well, it's nothing.

Lest that seems like joking word play what does "nothing is X" or "all of a nonexisting entity are X" mean? Well, "nothing is X" means "all things that are, are not X". And "all of nonexisting entity are X" means "there do not exist only of the non-existing things that are not X".... and that is certainly true! There do not exist any unicorns that are not green. And there do not exist any unicorns that are not purple.

fleablood
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"all unicorns are green"

"all unicorns are purple"

Two sentences being in contradiction entails that they are inconsistent with each other.

  1. Since the given sentences are jointly true (it's immaterial that the truthhood is vacuous), they are jointly satisfiable, which means that they are certainly not in contradiction.

  2. Assume, for the sake of contradiction, that the given sentences contradict each other. Then $$∀u\,G(u)∧∀v\,P(v),$$ i.e., $$∀u,v\:\big(G(u)∧P(v)\big),$$ is false for some $(u,v).$ So, the discourse domain is nonempty; but unicorns do not actually exist. Hence, the given sentences in fact don't contradict each other.

ryang
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