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The statement

$$x>0$$

no doubt implies that

$$x>-10$$

(or that $x$ is greater than any other negative number).

The second statement in turn implies that $x$ can be, say, $-9$. By chaining both implications, we arrive at the assertion that "if $x$ is greater than $0$, then $x$ can be $-9$".

My issue with this is that it looks and sounds senseless. The first statement should not, at least from an intuitive point of view, imply the last statement. To me, $x>0$ implies $x$ cannot be $-9$, the opposite of what has been asserted. What gives? Is this an issue of meaning of the word "can"? Or a misunderstanding on my part of what implication entails in this example?

jvf
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    The statement $x > 0 \implies x > -10$ is true, whereas the converse $x > -10 \implies x > 0$ is not. You appear to be confused at this point. An implication and its converse are distinct statements, and their truth values need not be related at all. – Nick F May 18 '23 at 03:30
  • @NickF But converses don't need to be taken into account when chaining implications, right? I agree that $x>-10$ does not imply $x>0$, but why should that impact the direct implications? For example, $a = b$ implies $a^2 = b^2$, which in turn implies $a^2 - b^2 = 0$. We can then unequivocally assert that $a=b$ implies $a^2 - b^2 = 0$, even though $a^2 = b^2$ does not imply $a = b$. – jvf May 18 '23 at 03:37
  • Right, but you're trying to chain with an implication that is false. You are claiming that $x = -9 \implies x > -10$ (no problem there) and that $x > -10 \implies x > 0$ (which is false); therefore $x = -9 \implies x > 0$. The chain rule requires all elements of the chain to be true. – Nick F May 18 '23 at 03:39
  • @NickF "You are claiming that x=−9⟹x>−10 (no problem there) and that x>−10⟹x>0 (which is false);" Hmm, I don't understand how. The implications I proposed are "$x>0 \implies x>-10$", and "$x>-10 \implies$ $x$ can/could be $-9$". How do those implications follow? – jvf May 18 '23 at 03:42
  • Ah, I see what you mean now. The word "can" in this context really means $x = -9 \implies x > -10$. That is, $-9$ is a possible value of $x$ among those $x$'s which satisfy $x > -10$. – Nick F May 18 '23 at 03:49
  • @NickF Yes, that's true, but should that impact how the implications are chained? – jvf May 18 '23 at 03:53
  • "$x > -10$" does not imply that $x$ can be $-9$. Only that it cannot be, say, $-11$. You may well know other things about $x$ (as you do in your example) that prevent it from being $-9$. – mjqxxxx May 18 '23 at 04:06

2 Answers2

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You're right that the word "can" changes the logical structure of the sentence.

Adding "can" to a propositional phrase is equivalent to adding "it is possible that", and when you say "If $P(x)$, then it is possible that $Q(x)$", what you're really saying is that if you know nothing about $x$ except $P(x)$, then you can't rule out $Q(x)$. In other words: it's not the case that for every possible value of $x$, if $P(x)$ is true then $Q(x)$ is false. Using quantifiers, this translates to:

$$\lnot\forall x(P(x)\to\lnot Q(x))$$

which is equivalent to

$$\exists x(P(x)\land Q(x))$$

and simply expresses that $P$ and $Q$ are consistent with each other. If you think about it, that's what you really mean by "if $P$, then $Q$ is possible."

This is not equivalent to anything of the form $\forall x(P(x)\to\dots)$, so despite the way it's phrased, the sentence isn't really saying that $P$ implies something. You "chained implications" based on the superficial grammar of the statements, but (as Adriano's answer also notes) there's no implication to chain if you properly translate the "can".

(In your example, $P(x)$ is "$x>-10$" and $Q(x)$ is "$x=-9$".)

Karl
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Let's formalize your statements a bit. Define the following sets: $$ A = \{a \in \mathbb R \mid a > 0\} \\ B = \{b \in \mathbb R \mid b > -10\} $$ Then we can say the following:

  • For all $x \in \mathbb R$: if $x \in A$, then $x \in B$.
  • $-9 \in B$

Notice that the first bullet has a quantified implication, while the second bullet does not. The second bullet doesn't really have an implication that could be chained together with the first one.

Adriano
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  • Is it a matter of trying to use implication in a context where it isn't appropriate, then? Admittedly I don't know how to express mathematically the idea that $x$ could be $-9$, though those words spelt out in English have meaning. – jvf May 18 '23 at 03:52