The sentence:

The difference of two positive numbers is not necessarily positive,

Is expressed as:

$$ (\forall x\in\Bbb Z)((x>0)\land(y>0)\rightarrow(x+y >0))$$

Is this statement correctly expressed? If so, please give a brief(or detailed) explanation.

This statement implies that if x > 0 and y > 0 then, x + y > 0, where's the necessarily part?

If the necessarily part comes from, the fact that if the R.H.S
is False then the statement is false. Then, I'm confused in the contraction with the simple analogy of another statement:

"The product of two negative numbers is always positive", expressed as:
$$ (\forall x\forall y\in\Bbb Z)((x<0) \land (y<0)\rightarrow (x*y) > 0)$$

Which is always true, so, there's no necessarily part in this at all. What I mean is, if the first statement is stated correcly, means that there's a necessarily part in this first statement, in which case, there should also be a necessarily part in the second statement(by comparison), which should not be the case, because the statement is always true.

Of course, if the first statement expressed with logic and quantifiers was wrong, everything is solved.(Note, the example the taken from Rosen's book of discrete mathematics.)

  • $\begingroup$ The first formula says that the sum of $x$ and $y$ is positive $\endgroup$ Jul 13 '17 at 12:48
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    $\begingroup$ To say that "the difference of two positive numbers is not necessarily positive" is the same as "there are two positive numbers whose difference is not positive". $\endgroup$ Jul 13 '17 at 12:49
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    $\begingroup$ Consider the case with $2$ and $3$: $((2>0) ∧ (3>0) \land (2-3 \le 0))$. Now quantify it accordingly. $\endgroup$ Jul 13 '17 at 12:50
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    $\begingroup$ I think that you have misunderstood the locution "not necessarily". It must be read: "it is not true that, if $x$ and $y$ are both positive, then their difference is positive". $\endgroup$ Jul 13 '17 at 13:00
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    $\begingroup$ Here are some fun pop-culture examples of translating assertions from English to first-order logic: jdh.hamkins.org/famous-quotations-in-their-original-language. $\endgroup$
    – JDH
    Jul 13 '17 at 17:05

There are two issues here. One is that the logic statement you offer does not match the English statement. The logic statement you offer (when you add $\forall y$ as Mauro Allegranza suggested in the comments) would be best translated to English as: The sum of any two positive numbers is (necessarily) positive.

For the English statement "The difference of two positive numbers is not necessarily positive", you would want something like: $$\neg \forall x \forall y (x>0 \land y>0 \rightarrow x-y>0)$$ or, equivalently, $$\exists x,y \text{ such that } (x>0 \land y>0 \land x-y <0)$$

The second issue is the question you raise about how "necessarily" figures in these statements. "Necessity" is a phrase used in many ways, often unclearly. For your purposes, in your original English sentences, "necessarily" almost certainly means simply that some positive numbers have negative difference.

In other words, the only reason "necessarily" is there in your English statement is because if it weren't there, many readers might erroneously understand the statement to be claiming that all positive numbers have negative differences. But the statement is not claiming that; it is instead claiming that positive numbers sometimes have negative differences. So "necessarily" is added to make this clear.

The reason why "necessarily" does not occur in your second example is that there is no such danger of ambiguity, so there is no reason to include "necessarily" to prevent misunderstanding.

It is worth adding that some "modal" logical systems -- almost certainly not related to what you are working with -- do incorporate operators for necessity. In such a logic, there is a distinction between truth and necessary truth. (E.g., it is true that my tea is hot, but it is not a necessary truth; my tea can easily become cold. By contrast, it is a necessary truth that 1<2.) In such modal logical languages, necessity is typically denoted by "$\square$". So, we have:

\begin{align*} A \text{ implies }B& & \phantom{\square (}A\rightarrow B\phantom{)}\\ A \text{ necessarily implies }B& &\square (A\rightarrow B) \end{align*}

  • $\begingroup$ I came to this a little bit later, but could you tell me why the negation is not done in y instead or why not in both? $\endgroup$ Jul 15 '17 at 10:17
  • $\begingroup$ @BeshalJaenal Sorry, I don't understand what you mean. Can you rephrase your question? $\endgroup$
    – Ceph
    Jul 16 '17 at 2:56
  • $\begingroup$ below the line: "The difference of two positive numbers is not necessarily positive", you would want something like:, you have given ¬∀x∀y(x>0∧y>0→x−y>0 statement, why is ¬∀x done? Is ¬∀y also OK? How does this alter the meaning? $\endgroup$ Jul 16 '17 at 12:13
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    $\begingroup$ @BeshalJaenal I see now. In my version of the statement, the negation applies to the whole following statement. That is, the statement I wrote is equivalent to: $\neg \left( \forall x \forall y (x>0 \land y>0 \rightarrow x-y>0)\right)$. So in my original version of the statement, the negation doesn't have any particular connection to $x$ rather than $y$. The statement $\forall x \neg \forall y ( x>0\land y>0 \rightarrow x-y>0)$ would be an awkward way of expressing the original -- but yes, it would have the same meaning as my version. $\endgroup$
    – Ceph
    Jul 17 '17 at 20:02

I think the book uses 'necessarily' and 'always' interchangeably. So, that second second could be expressed as 'the product of two negative numbers is always positive', but it can also be expressed as 'the product of two negative numbers is necessarily positive'.

As such, the sentence 'the difference of two positive numbers is not necessarily positive' can be understood as 'the difference of two positive numbers is not always positive'.

To symbolise that first sentence in logic, we can do one of two things:

A first strategy is to symbolise the sentence 'the difference of two positive numbers is always positive', and just negate that. That gives us:

$$\neg \forall x,y \in \mathbb{Z} ((x >0 \land y >0) \rightarrow x-y >0)$$

A second strategy is to rephrase the sentence as 'it is possible for the difference of two positive numbers to not be positive' or what is the same thing, 'there exists two positive numbers such that their difference is not positive'. In logic, that would be:

$$\exists x,y \in \mathbb{Z} (x>0 \land y >0 \land \neg \ x-y >0)$$


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