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In a text I am reading, the section on Propositional Logic says that a proposition is a statement that is either true or false, but not both true and false. Also, from this lecture online, the instructor says that we must be able to associate a truth value to a proposition.

The text I mentioned contains as an example of an assertion that is not a proposition the following:

(1) "this statement is false."

In the margin, the text says that the form of this statement makes it impossible to designate a truth value to it and the instructor in the lecture says simply that, "if [the statement] is true, then it is false, and if it is false, then it is true."

However, why exactly is it impossible to for (1) to have a truth value? What does it mean to say that if (1) is true, it is false, and conversely?

Response to Asaf Karagila
As has been pointed out, I have already asked this question very recently yesterday but it has not received proper attention. This question is one that I feel can be put to rest if only someone would provide an explanation that is direct and suitable for my level, which is that of a novice.

Community
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    Honestly, this is a bad example that's not really worth considering. It won't hinder your understanding of the subject to ignore this. – Kaynex Mar 16 '16 at 03:31
  • @Kaynex It bothers me tremendously to skip it though. –  Mar 16 '16 at 03:32
  • I would say that we CAN assign a truth value to it, but neither would be correct. Others might say that because we cannot have a correct truth value, we can't assign one. Either way, as long as you understand what a proposition is, you're fine. – Kaynex Mar 16 '16 at 03:34
  • @Kaynex It's excruciating to ignore this but I think you're right. This is not an example that's helpful for a beginner. –  Mar 16 '16 at 03:43
  • @Kaynex, philosophically, I'd say its pretty interesting. It shows that the English language is altogether too expressive for the purposes of mathematics, and thereby provides one justification of the use of formal languages in mathematics. – goblin GONE Mar 16 '16 at 04:30
  • What even is an example of an assertion that is both true and false? –  Mar 16 '16 at 04:53
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    You should at least mention that you've already asked about this. Yesterday. http://math.stackexchange.com/questions/1697691/this-statement-is-false – Asaf Karagila Mar 16 '16 at 05:45
  • @AsafKaragila Fixed. –  Mar 16 '16 at 05:57
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    @AsafKaragila, should we close one of them? – goblin GONE Mar 16 '16 at 06:00
  • @goblin Yes, the other one. –  Mar 16 '16 at 06:01
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    Ha! Not the proper attention? You received three answers that had to guess your knowledge and mathematical aptitude. You want better suited answers? Write better questions. – Asaf Karagila Mar 16 '16 at 06:15
  • @AsafKaragila You're right, which is why I tried to ask a better one in this post. I'm genuinely just wanting to put this question behind me, but via an explanation that is appropriate for someone at my level. –  Mar 16 '16 at 06:19

2 Answers2

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Let me toss my 2 cents for what your instructor said.

Suppose (1) "this statement is false." holds. Then the assertion inside "" is false. Thus this statement is false does not hold, or (if we abide by the binary logic) this statement is true.

Now suppose (1) "this statement is false." does not hold. Then that statement must be true (as long as we abide by the binary logic.) So the assertion this statement is false is true.

The bottom line is, the statement inside "" does not conform to the binary logic.

eltonjohn
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  • Right, if we suppose that (1) is true, then what it claims is the case. And what it claims is that it is false. The same sort of thing occurs when we suppose that (1) is false. Now what's happening here? What does it mean for something to be false if and only if its is true? –  Mar 16 '16 at 04:24
  • It means that it does not conform to the binary logic. – eltonjohn Mar 16 '16 at 04:29
  • I agree that it doesn't "conform to the binary logic," by which I interpret as meaning that it can't be said to be definitely true or definitely false. But a fear that I have is being asked about this and having to say that without really understanding why. I mean, does the statement have an "unstable" truth value? –  Mar 16 '16 at 04:41
  • I don't know if the statement conforms to an $n$-ary logic for some suitable $n \geq 3$ (I am sceptical if there exists such an $n$, though.) I would reply to your instructor that we cannot create a truth table since we cannot decide the truth value for that statement. – eltonjohn Mar 16 '16 at 10:27
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We shouldn't even start to work with it like a propositional variable and use rules of inference with it, that would just be confusing as it is not a proposition to begin with for the following reason:

A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.

Now the key point here is that we must be able to assign a truth value to a statement in order for it to be considered a proposition. In other words, we have to be able to check with reality to see whether the statement accurately represent it or not (because that's how we know if a claim is true). Now "this statement is false." is simply saying-if we replace "false" with its definition-that "this statement does not accurately represent reality". But which claim is that? There are none.

It is not a proposition for the same reason that "$x + 1 = 2$" is not. (we can't assign any truth value to it)

The fact that it is trying to ascertain it's own truth value doesn't make it any special because all statements do that implicitly. (for example "Toronto is the capital of Canada." really is saying "The statement: "Toronto is the capital of Canada." is true", if we were to take out the claim, it would just be "this statement is true" which doesn't make any sense to say as there is no claim.)

Hopefully this clears out the confusion.

GSmith
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Ilyas Ait Ameur
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