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I was given a statement and then asked for my opinion whether the statement is True or False with some additional proof.

Surprisingly, I found out that the statement was an actual theorem showing that the statement was indeed true.

My question is: Would it be enough, to say that the statement is "True" simply because the statement was an actual theorem?

Git Gud
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Sai82
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    It's not clear what you mean by "the statement was an actual theorem". What makes a statement a theorem is precisely the existence of a proof. How do you know it is a theorem without checking the proof? – mweiss May 29 '14 at 15:40
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    @Sai82 The answer is yes. Note that a true statement and theorem are well defined concepts in the context of mathematical logic. They are different and not always equivalent concepts. However, seeing that you tagged the question abstract algebra, for practical purposes you should think of them as meaning the same. – Git Gud May 29 '14 at 15:41
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    @gitGud It's yes if you assume soundness. – Doug Spoonwood May 29 '14 at 15:52
  • @DougSpoonwood Good point ^_^ – Git Gud May 29 '14 at 15:53

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It really depends on what the examiner intends to test you. The best thing is to write down a proof of the result to show that it is true. If the proof is very complicated, perhaps he just wants you to know that it indeed has been proven already.

please delete me
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In the context of symbolic logic, no, or perhaps better put "not quite".

A theorem consists of a provable statement.

A true statement consists of a statement which holds true.

If you have a theorem, and if your proof system comes as sound in that it enables you to derive only true statement, then by modus ponens it follows that the statement is true. However, you could have a theorem and NOT have a sound proof system. In which case you could prove that statement, but it NOT hold true.

Doug Spoonwood
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