Conjunctions and logical connectives
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In my posting on and/or, I gave an informal (but precise) account of what I take to be the semantics of expressions of the form X1 or X2 … or Xn, where or is understood exclusively: the disjunction is true if and only if exactly one Xi is true and the rest are false. (Compare the inclusive understanding, where the disjunction is true if and only if at least one Xi is true.) My inbox is now filling up with mail from people explaining to me that I'm wrong about the semantics of exclusive or. Well, actually, they're telling me that I'm wrong about the semantics of the binary logical connective of exclusive disjunction + (or however you want to represent this logical connective). I'm perfectly clear about the semantics of +, but that's not what I was talking about.
The initial question is: what does the English conjunction or mean? The answer I gave, which is a claim about how speakers of English understand or, is that sometimes or is understood inclusively (at least one Xi is true) — this is the understanding that many linguists would take to be the primary one — but that sometimes it's understood exclusively (exactly one Xi is true), as in the disjunctive offer
With your breakfast, you can have orange juice or grapefruit juice or tomato juice.
I stand by this as a description of how English speakers use the conjunction or.
My correspondents object to this account of exclusive disjunction — one referred to the "strange semantics" I gave to exclusive or in my posting — and explain to me how the logical connective + works. Now, + is a BINARY connective, so that p+q+r, with no grouping indicated, must be understood as either (p+q)+r or p+(q+r), but these are truth-functionally equivalent (that is, + is an associative operator), so there's no problem if we dispense with grouping and treat + as an n-ary operator. What is the semantics of +? Michael Watts summarized it in his mail to me:
Logical xor is an associative operator, and when applied to many arguments it is true when an odd number of the arguments are true and false otherwise.
What this means is that + is simply not a satisfactory translation of English exclusive or. In particular, for a sentence with three disjuncts, like the breakfast-juice offer, translating or as + says that the sentence is true when exactly one of the disjuncts is true and when all three of them are — and that second claim is just wrong; the offer is not of any one juice or all three, but just of any one juice. That's why I gave a different account of the meaning of exclusive or.
It's true that my account doesn't correspond to any binary logical connective, but that just means we need to define a new n-ary operator, call it !, with the value true if and only if exactly one operand is true.
My previous posting treated exclusive or in English as translated by !, so that if English or is always exclusive, X or Y or both translates as p!q!(p&q), which is, unfortunately, equivalent to p!q, not the intended p|q (where | represents inclusive disjunction).
Two complications. First, as several correspondents have pointed out, translating exclusive or as + gives the right results for X or Y or both (though it doesn't for X or Y or Z in general), because p+q+(p&q) is in fact equivalent to p|q (thanks to the fact that with three disjuncts, + is true when all three of the disjuncts are true — that is, when both p and q are true). I'm not quite sure what to make of this; it seems like a bizarre accident to me.
Second, and more significantly, I think, Ryan Denzer-King notes that if X or Y or both is parsed as (X or Y) or both, rather than as a three-way disjunction, then ! does fine as a translation of or: (p!q)!(p&q) is equivalent to p|q. This parsing is not implausible — though at the moment I have no evidence favoring one or the other of the parsings — and if we accept it, then my argument against always-exclusive or from the semantics of or both is undercut. Of course, there are other good arguments against always-exclusive or, as Geoff Pullum noted a few days ago, in particular an argument from denials as responses to disjunctive questions. (Victoria Martin has now added a famous example to the ones in Geoff's posting: the HUAC question, "Are you now, or have you ever been, a member of the Communist Party?" If or were always exclusive, then someone who was a member of the CP at the time of questioning and had been a member before that time could have truthfully answered "no". I don't think so.)