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While reading an article on logic, there is a sentence "No number is equal to zero" and we have to assign truth values to this sentence. I hope this is true and the article says it as false.

Can someone explain me why it is false? Or the author of the article is wrong?

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    $\begingroup$ Why on earth do you hope that this is true? $\endgroup$ – Gilles 'SO- stop being evil' Dec 20 '13 at 10:55
  • $\begingroup$ @Gilles the model is set of natural numbers $\endgroup$ – Brainy Dec 20 '13 at 10:57
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    $\begingroup$ @babou, it is not that this is non-obvious; it is that the question is not posed very clearly. For instance, the article is not written very clearly or precisely, so we have to make guesses about the intended meaning of that sentence. The fact that different people make different guesses about what precisely that sentence was intended to mean doesn't imply that it's a good question; if it anything, it suggests that maybe it's not such a great question. Just my opinion. $\endgroup$ – D.W.♦ Dec 20 '13 at 23:57
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    $\begingroup$ @D.W. I beg to (dis)agree. What is not clear enough is the paper the OP is trying to understand, not his question. The paper is not self contained and you actually have to look at the hyperlinks, and understand them, though there are not all essential. The OP should probably have given some more details to explain his understanding, but it may have seemed obvious from his point of view: considering only the positive natural numbers. After all, it took a long time historically to consider zero as a number too. The role of the word "zero", and why it should mean anything is not well explained. $\endgroup$ – babou Dec 21 '13 at 16:49
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    $\begingroup$ @TCSL What the article states is that statement (5) is false in all models. I understand that you object because you consider that positive natural numbers do not have a number zero. But the question is whether (together with the usual operation) they constitute a model. And then you have to address the central question: a model of what? It has to be a model of what the old man described. If the description included an element called zero, as hinted by the link referencing rings, then any model must include a zero. And if you call numbers the elements of your model, then zero is a number. $\endgroup$ – babou Dec 21 '13 at 17:25
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It is false if and only if there is a number that is equal to zero.

"There is a number that is equal to zero" is true if and only if zero is a number. Without any context (there might e.g. be some very strange definitions given earlier in the exercise), we cannot say whether zero is considered a number in this case. If there is no other context given in the exercise, it should be clear that zero is a number.

For example, the domain of discourse might be only positive integers*, when zero is not a number, but then it would be very weird to use the word "zero" in the sentence in the first place.

*The set of natural numbers has two contradicting definitions, $\mathbb{N}=\{0,1,2,\dots\}$ (non-negative integers) and $\mathbb{N}=\{1,2,\dots\}$ (positive integers). This might be the source of confusion in your case.

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  • $\begingroup$ Don't you think the answer depends on the model we are using? $\endgroup$ – Brainy Dec 20 '13 at 10:57
  • $\begingroup$ I agree that if zero is not a number, then no number is not equal to zero. On the other hand, the question mentions zero, so I thought it is assumed that zero is a number. Probably the textbook defines the set of natural numbers as $\{0,1,2,\dots\}$, while some define it as $\{1,2,\dots\}$, which may cause confusion in this case. $\endgroup$ – JiK Dec 20 '13 at 11:05
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    $\begingroup$ @TCSLearner A definition of “number” that excludes 0 would be really weird — ”number“ would be the wrong word in that case. Unless the definition is several centuries old, predating not only modern logic but also calculus and even algebra. $\endgroup$ – Gilles 'SO- stop being evil' Dec 20 '13 at 11:22
  • $\begingroup$ I edited the answer to include some of these points. $\endgroup$ – JiK Dec 20 '13 at 11:33
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    $\begingroup$ @TCSLearner I read the original article, but do not see your point. Statement (5) is false considering all models... for the old man's description. A model is something that fits some axiomatic description. There is no such thing as a model without a reference to a description the model must fit. So you must ask yourself what is that description. A statement is true for all models if it can be logically proved from the axioms of the description. Note however that, in some axiomatic theories, some statements may be true of all models without being provable (Gödel incompleteness theorem). $\endgroup$ – babou Dec 22 '13 at 1:22
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What I gathered, form a cursory reading of the paper is that the old man described the algebraic properties of the rational numbers, by actually describing what a ring is, as the corresponding wikipedia article is explicitly referenced in the text (though the word ring is not given). Hence, he must have specified that the addition has a special element called zero wich is an additive identity.

Hence any model that fits the description given by the old man, i.e. that is actually a ring, must have an element called zero with the required properties. This is certainly true of the rationals, and it must be true of what you (as the character of the story) call the integers since you have no disagreement with the old man.

And, as stated in the article, it will actually be true of any model meeting the description of the old man. So, for any model you care to call "numbers", the statement "(5) No number is equal to zero" is clearly false.

However, I am bothered by statements (1), (3), and (4) that refer to the number 2, as the axioms for a ring do not define what 2 might be. It is not supposed to have been defined in the discussion, and it is therefore difficult to assign any meaning to a statement using it. They should have at least agreed that 2 is a notation for the result of "1+1".

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In the part of the article you're referring to, "number" means either "integer" or "rational" (specifically, the conceit is that you're talking to an old man and the old man thinks you're talking about the rationals, while you think you're talking about the integers). There is an integer equal to zero and a rational equal to zero so, in either case, the sentence is true: the number equal to zero is zero itself.

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Your confusion comes from trying to describe a mathematical statement in English -- but not very precisely. English often isn't very precise. Here it isn't clear exactly what the intended translation of the English into mathematics is. Let me outline a few possible translations:

  • $\forall x \in \mathbb{Z} . x \ne 0$. This proposition is false. $\mathbb{Z}$ (the set of all integers) does include $0$.

  • $\forall x \in \mathbb{N} . x \ne 0$. This proposition is false, using the usual definition of $\mathbb{N}$ as the set of all natural numbers (i.e., all non-negative integers, i.e., $\mathbb{N}=\{0,1,2,3,\dots\}$).

  • $\forall x \in \mathbb{Q} . x \ne 0$. This proposition is false. $\mathbb{Q}$ (the set of all rationals) does include $0$.

  • $\forall x \in \{1,2\} . x \ne 0$. This proposition is true, since the set $\{1,2\}$ does not include an element $0$.

  • $\forall x . x \ne 0$. The truth value of this proposition is indeterminate, without knowing the universe over which $x$ is quantified. If we are letting $x$ range over all of $\mathbb{Z}$, this is true. If we are letting $x$ range over $\{1,2\}$, this is false.

From a practical perspective, I think we could say the following: in this particular example, I think nothing very interesting is happening. It's just a case of being imprecise with notation. So, from a practical perspective: take this as a lesson to try to be explicit about what you are quantifying over, to ensure there is no possibility of confusion.

From a theoretical perspective, here is what is going on. That article is talking about propositional logic. Propositional logic is, in some sense, a framework for reasoning about what is true regardless of what universe you are quantifying over. In other words, in some sense propositional logic wants to help us reason about propositions where we don't need to specify the set that $x$ ranges over, because the proposition will be true either way. This is where models come in. A model is basically a set that all quantified variables range over (it also specifies, for each predicate, which elements of that set make the predicate true). Now we can have a propositional formula that is true for all models; false for all models; or true for some models and false for others.

For more, see https://en.wikipedia.org/wiki/Interpretation_%28logic%29 and textbook introductions to propositional logic and first-order logic.

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