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The statement

$1$ can be irrational.

means that $1$ is either rational, or irrational. So that statement should be true. The negation of that statement is

$1$ cannot be irrational.

But that should also be true, because $1$ is a natural number, hence $1$ cannot be irrational. I'm very confused.

Peter
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BIRA
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    Spoken language is usually confusing. But the thing is that the second sentence is not the logical negation of the first. – peek-a-boo Aug 26 '20 at 15:37
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    What does "can" mean ? – TheSilverDoe Aug 26 '20 at 15:38
  • @peek-a-boo So what should be the logical negation of the first statement? – BIRA Aug 26 '20 at 15:38
  • @TheSilverDoe "can" means that it is either possible for $1$ to be rational, or it is possible for $1$ to be irrational. – BIRA Aug 26 '20 at 15:39
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    "$1$ can be rational" I would interpret as "$1$ is rational or $1$ is irrational". The logical negation of this is "$1$ is not rational and $1$ is not irrational"; i.e "$1$ is irrational and $1$ is rational" (which is clearly false, and this also makes sense because it is the negation of a true statement). – peek-a-boo Aug 26 '20 at 15:40
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    It is "1 is neither rational nor irrational" since the first sentence means "1 is rational or irrational" – Peter Aug 26 '20 at 15:41
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    Since the first sentence is (logically interpreted) a tautology , the negation must be a contradiction. – Peter Aug 26 '20 at 15:44
  • @BIRA but it is not possible for $1$ to be irrational. We know that $1$ is rational. – Radial Arm Saw Aug 26 '20 at 15:48
  • If something "can be", the opposite of that "can be" as well. So the qualifier "can be" doesn't satisfy what a usual logical word like "is" does : if something "is", the opposite, is not. Basically, the set of things that "can be" would have events along with their complements, something that is not true for the set of things that "are", or "are not". That's why phrases like "is / are / are not/is not" are logically translatable into language , while "can" isn't. – Sarvesh Ravichandran Iyer Aug 26 '20 at 15:49
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    @BIRA "can" is usually used in a random experiment : "The next coin-throw can be heads". But whether $1$ is irrational is determined, so "can" does not actually make sense here. $1$ is not "sometimes rational and sometimes not rational" – Peter Aug 26 '20 at 15:53
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    Even more tricky is : "The Goldbach conjecture can be false". Since we do not know whether it is true this is a true statement although the conjecture is either true or false, so the truthness is not depending on randomness. Should someone prove it in the future, the statement becomes false. So "it can be" semantically means "we cannot rule it out". – Peter Aug 26 '20 at 16:00
  • @Peter Doesn't it all boil down to the fact that "$A$ cannot be $B$" is actually not the negation of "$A$ can be $B$"? – BIRA Aug 26 '20 at 16:37

1 Answers1

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If we accept the interpretation of "$x$ can have property $P$" as "$x$ either has property $P$ or doesn't have property $P$," and the interpretation of "$x$ cannot have property $P$" as "it is not the case that $x$ can have property $P$," then $(1)$ is true and $(2)$ is false - for silly reasons.

However, this is sort of missing the real issue: the premise of the question - that "$1$ can(not) be irrational" can be faithfully translated into classical logic in a straightforward way - is incorrect.

This is a situation where the simplest logical framework - straightforward classical logic - does a bad job of implementing natural reasoning. The issue is the idea of possibility implicit in the word "can(not):" there's a sense of quantifying over "possible worlds" here, which is hard to do in standard classical logic. And we can tell that there's something wrong with ignoring this feature: if we interpret "$x$ can have property $P$" as "either $x$ has property $P$ or $x$ doesn't have property $P$," then that's always trivially true - so this is good evidence that that does a terrible job of accurately interpreting the word "can." (And it gets even worse when we think about "cannot.")

A better framework for dealing with this sort of assertion is provided by modal logic, or more generally the possible world semantices (not a typo: I mean the plural of "semantics") and their relatives. According to this approach, we roughly have the following situation: the sentence

"$1$ can be irrational"

would mean

"There is some possible world in which $1$ is irrational,"

whereas

"$1$ cannot be irrational"

would mean

"In every possible world, $1$ is rational"

or if you prefer,

"There is no possible world in which $1$ is irrational."

(I said "roughly" above because there's some nuance to the notion of possibility here: we really have a notion of relative possibility, or accessibility, at play which I'm ignoring here.)

Of course as you probably suspect the devil is in the details, but this is ultimately an approach which makes sense and does a much better job of capturing how expressions like the above are treated in natural language.

Noah Schweber
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  • Did you meant to say "1 
    cannot be irrational"

    would mean

    "In every possible world, 1

    is RATIONAL" ?

     – imranfat Aug 26 '20 at 15:52
  • @imranfat Whoopsydoodle. Fixed! – Noah Schweber Aug 26 '20 at 15:53
  • So, $1$ can be irrational, but is not. Is that correct? – BIRA Aug 26 '20 at 15:59
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    @BIRA Correct according to what interpretation of "can be"? According to the interpretation in the OP, yes; but I would argue that that's not actually a reasonable interpretation, and that in fact according to any reasonable interpretation the answer would be no. (That is, in a reasonable possible-worlds approach, $1$ would be rational in every possible world.) – Noah Schweber Aug 26 '20 at 17:15