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Consider the following pair of statements:

  1. All multiples of three are odd / Some multiples of three are odd.

  2. No triangle has an interior angle sum of zero degrees / Some triangle has an interior angle sum of zero degrees.

  3. Some dense sets are not infinite / All dense sets are infinite.

The first two pairs of statements would be contradictory because in any given case, The first sentence will never be true and the sentence will always be true.

The second would also be contradictory because you can have a triangle that has a sum of zero degree which would make the second sentence false. If you find at least one triangle that is greater than zero, it would make the second sentence true and the other false.

I'm not sure about the third one.

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Deathslice
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  • In logic, some multiples includes the possibility of all. The first two pair are not contrary. – André Nicolas May 28 '15 at 01:56
  • @AndréNicolas Would you agree then that both pair 1 and pair 2 are contradictory(Those are the ones that I've done so far)? – Deathslice May 28 '15 at 01:58
  • My comment above said that the pair (1) are not contradictory. The pair (2) are contradictory, as are (3). – André Nicolas May 28 '15 at 02:04
  • @AndréNicolas Strange you said contrary at the end of your first comment which meant in my head that you meant that the first two pair are contradictory and not contrary(There is a difference between these two terms). – Deathslice May 28 '15 at 02:09
  • (1) is neither contrary nor contradictory, since both statements could be true, (3) is contradictory – abc May 28 '15 at 02:13
  • @Sebastian How so? 6 is a multiple of 3 and it is not even. Am I missing something? – Deathslice May 28 '15 at 02:16
  • @Deathslice your mistake is that you have an interpretation of "odd" in mind. But there surely is an interpretation of "odd" which would make both statements true. So in there logical form they are not contrary. – abc May 28 '15 at 02:19

1 Answers1

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I think you are mistakenly determining actual truth values for each of the pairs.

In the first question, it isn't important that normally every multiple of three is not odd. I think the real point is that IF it were the case that every multiple of three were odd, then it would also be the case that some multiples of three are odd. Thus the two statements are neither contrary nor contradictory.

In the second question, the two statements are contradictory like you said (and more obviously so because we have immediate examples available to us of models of geometry in which one is true but not the other).

The third pair of statements are also contradictory because they are exact negations of each other.

Jonny
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