The book I'm reading says that the natural English sentence

 (38) - Today is either Monday or Tuesday

is not an example of an exclusive "Or" in natural English.

As I understand it, his argument goes like this:

1- In order for an exclusive or to exist, it must exist the first row of its truth table, that is, it must be possible for A and B to be true.

2- Since the statement (38) makes it impossible for A and B to be true, then the first row does not exist, then the exclusive or does not exist either.

Is this a good argument? What happens to a statement in which some row is not possible to evaluate because the scenario is not possible?

  • 3
    $\begingroup$ This is not maths, alas. $\endgroup$ – Angina Seng Feb 17 '18 at 19:45
  • 1
    $\begingroup$ Not applied math of course. Do you think it has nothing to do in this exchange? $\endgroup$ – César D. Vázquez Feb 17 '18 at 19:48
  • $\begingroup$ It seems to me like a silly argument. Would the book be willing to accept "You can have soup or salad with that " as an example? "Can I have both?" "No, sir. Just one or the other." That's okay, because it's logically possible to get both, even if the restaurant categorically refuses to do it? What if they will serve both, but only for an extra charge? This is an issue of pragmatics, not semantics. $\endgroup$ – MJD Feb 17 '18 at 20:04
  • $\begingroup$ This is a false statement. Today is Saturday.. This is also a horrible argument - the truth or falsity of the statement doesn't depend on whether "or" is exclusive or inclusive because the options are mutually exclusive. But that is a feature of the options and not of what "or" means if this is not the case. $\endgroup$ – Mark Bennet Feb 17 '18 at 20:12
  • $\begingroup$ "The book I'm reading ..." Which book is that? Either it is indeed advancing a really terrible argument, or (perhaps a bit more likely) you've got the wrong end of the stick? $\endgroup$ – Peter Smith Feb 17 '18 at 20:25

It's a terrible argument.

First of all, the rows in a truth-table reflect the possible truth-values one could assign to the statements involved, where the whole point is that one does not look at the meaning of the atomic claims involved: a truth-functional analysis is just that: looking at how the truths of different statements relate to each other in terms of the truth-functional operators involved. So, given that there are no operators involved in either 'Today is Monday' and 'Today is Tuesday', a truth-functional analysis will simply treat these statements like two independent statements $P$ and $Q$', and thus under this analysis, it is possible for both statements to be true at the same time just fine. In fact, even a statement like $1=1$ will be treated as a single statement $P$, and therefore will be considered a statement that can be false under a truth-functional analysis. So you always have all possible rows for the truth-table, including the one where both statements are true, even if in our specific world the two statements cannot both be true. So this goes against claim 2.

Claim 1 makes even less sense: in order for there to be some logical connective there must exist certain rows in the truth-table?! No, logical connectives exist whether or not there is some kind of analysis in terms of truth-tables or not.

  • $\begingroup$ There is no possible world in which it could be simultaneously both Monday and Tuesday, by definition: Tuesday begins when Monday ends. ( I agree with you that it is a terrible argument.) $\endgroup$ – MJD Feb 17 '18 at 20:00
  • $\begingroup$ Now I'm not sure. In some standard timekeeping systems, Monday 24:00 and Tuesday 00:00 are logically identical. If you accept that, the book's example fails for a different reason: it is an inclusive, not exclusive or! $\endgroup$ – MJD Feb 17 '18 at 20:10
  • $\begingroup$ @MJD True, but as I said in the post, truth-tables don't look at the meaning of atomic statements: the truth-functional analysis they perform is blind to this. Even a claim like $1=1$ will be seen as a claim that could be false. $\endgroup$ – Bram28 Feb 17 '18 at 20:12
  • $\begingroup$ @MJD Oh, that's interesting about Monday 24:00 and Tuesday 0:00! :) $\endgroup$ – Bram28 Feb 17 '18 at 20:13

LePore and Cumming aren't ideally clear, perhaps. But the point they are making is actually a straightforward one, and fairly clearly correct too.

Suppose a claim A or B is made which is naturally understood, in the context, as telling us overall that either $A$ or $B$ but not both.

Does this give us evidence that "or" has an exclusive meaning in English (alongside the inclusive meaning it undoubtedly has)?

Well plainly not, if there is another explanation of our reading in this case, that is compatible with "or" having an uniquely inclusive meaning.

And that will be the situation, at least in a range of examples.

In particular, suppose that $A$ and $B$ are already logically incompatible. Then in this case, of course the disjuncts of A or B can't be true together -- and if we know this, then we will understand A or B as conveying the overall message that $A$ or $B$ but not both. In other words, in this sort of case, we have no reason to suppose that our exclusive reading is due to a special exclusive meaning of "or", when it can simply be due to our understanding that $A$ rules out $B$.

So, take the example "Today is Monday or today is Tuesday". Ok, we hear this as conveying an exclusive message, let's agree. But the disjuncts are incompatible, and we know them to be so. So we can explain why we read the overall message in that claim to be that only one of the disjuncts holds without supposing that that is part of the meaning of "or" itself.

More generally -- and this is LePore and Cumming's correct point -- disjunctions A or B with evidently incompatible disjuncts aren't good examples to use to try to show that ordinary-language "or" itself has a special exclusive meaning, since in these cases we have another explanation of why we construe the overall message as exclusive. (But their mode of presentation of this correct point, I certainly agree, is pretty poor.)


I believe the truth table of XOR is this:

T T |    F
T F |    T
F T |    T
F F |    F

Because in exclusive-OR, only one operand must be true, so the provided reasoning doesn't follow.

I would argue that the original sentence is XOR, because saying

Today is either Monday or Tuesday

Would essentially be the same as:

Today is only Monday (A) or today is only Tuesday (B)

Where, using the truth table just stated, would match, as today cannot be both Monday and Tuesday (T XOR T = F).

If today is neither Monday nor Tuesday, the given statement is false (F XOR F = F).

If today is Monday, but not Tuesday, the given statement is true (T XOR F = T).

And if today is not Monday, but Tuesday, the given statement is true (F XOR T = T).


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