# Is it okay that vacuously true statements can contradict each other?

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?

• To get a contradiction, you need a unicorn first. No unicorn, no contradiction. Jan 26 at 20:59
• 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? Jan 26 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? Jan 26 at 21:28
• 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. Jan 26 at 21:42
• "all unicorns are green" $\iff$ "there are no unicorns that are not green". Well, are there any unicorns that are not green? Jan 26 at 21:53

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".

• 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. Jan 26 at 21:35

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.

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.

I understand the mathematical reasoning behind the conclusion

I'll provide two perspectives anyway:

1. Since the conjunction, "all unicorns are green and purple", of "all unicorns are green" and "all unicorns are purple" is again vacuously true, the two given sentences don't contradict each other.

2. Assume that "all unicorns are green" and "all unicorns are purple" contradict each other. Then, letting the discourse domain be the set of unicorns, $$∀u\,G(u)∧∀v\,P(v),$$ i.e., $$∀u,v\:\big(G(u)∧P(v)\big),$$ is false for some $$(u,v).$$ So, the discourse domain is nonempty; since the discourse domain is in fact empty, we have a contradiction. Thus, the assumption is false: the two given sentences don't contradict each other.