[A]

(1) p [premise]

(2) p v q [1, disjunction introduction]

[B]

(1) p v q [premise]

(2) ~p [premise]

(3) q [1,2 disjunctive syllogism]

[C]

(1) p

__v__q [premise]

(2) p [premise]

(3) ~q [1,2, exclusive disjunctive syllogism]

Each of these three represents a different sort of disjunction. It's pretty clear that [C] is different from [A] and [B]; truth functionally, exclusive disjunction is 0110 (fttf), whereas the other two are inclusive, 1110 (tttf). That is to say, in exclusive disjunction if both are true, the statement is false, whereas with inclusive disjunction if both are true, the statement is true. (As a complete sidenote, can I say I'm actually somewhat impressed with the Wikipedia article on exclusive disjunction, if only for the fact that it follows Barrett and Stenner in recognizing that English does not naturally have an exclusive or. You can, of course, represent it by circumlocutions, but the English 'or' is never exclusive, because English 'or' is never strong enough on its own to require that the truth of both disjuncts makes the disjunction false. This is something that many philosophers never learn. The SEP article on disjunction recognizes this, too, but as it's by Jennings that's unsurprising, since he wrote a very good book,

*The Genealogy of Disjunction*, in which this is discussed at length.*)

But it may seem less obvious that [A] and [B] are different kinds of disjunction. After all, they are truth-functionally the same, and isn't that supposed to be all that you need?

Obviously, I don't think so. My reasoning for treating these as different is this. Even though they have the same truth-functional properties, A2 and B1 cannot be used in argument in the same way. A2 is a weak disjunction for reasoning with, because it arises in the argument solely from disjunction introduction, and the only reason for concluding it is that p is taken is true. And the only thing one can conclude from it is that p is true, unless you also argue that p&q is true. The q has no reason for being there except that you have performed a disjunction introduction on p. If you isolate A2 from its context, and add to it the premise ~p, you cannot conclude q. Your only grounds for accepting p v q was p, and therefore the 'v q' has no independent authority as a disjunct, so to speak; p v q is just a more complicated p.

This is very different in the disjunction syllogism. B1 is a premise; it does not depend on anything. Because of this, it is a very robust disjunction for reasoning, since its existence in the argument, and its truth, does not depend on either the truth of p or the truth of q. Each disjunct stands in its own right.

Thus, I would suggest, contrary to truth-functional orthodoxy, that truth-functional disjunction is rationally ambiguous, i.e., it is ambiguous for the purposes of reasonable argument. We cannot take a disjunctive sentence in propositional logic and know how to understand it for the purposes of actually arguing with it unless we know why that sentence was introduced. Of course, if this is right, it means that every sentence in propositional logic exhibits this ambiguity, because every other operation is related to disjunction by a set of equivalences that are blind to everything except truth values.

What say you?

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* Jennings, of course, is right in his critique, as were Barrett and Stenner. And yet you still find philosophers saying that "Cream or sugar" on a menu, understood naturally as saying that you can have cream or you can have sugar, but not both, is an instance of the exclusive or. This, of course, is nonsense. The choice really is, as a list:

You may have cream,

you may have sugar,

you may not have both cream and sugar.

And the option, "You may have cream or you may have sugar" is not false if both 'disjuncts' are true. Quite the contrary. For the choice to make any sense at all, the 'disjuncts'

*must*both be true --, i.e., it must be true that you may have cream, and it must be true that you may have sugar, otherwise you've been given no choice at all. The 'or' here actually indicates a conjunction:

You may have cream (to the exclusion of sugar) & you may have sugar (to the exclusion of cream)

and both conjuncts are true.