I've got a set of propositions (can I call them "disjuncts"?) such that:

  • they are mutually exclusive: at most one can be true; and
  • they are collectively exhaustive: exactly one must be true to make the formula true.

Is there a notation to express that succinctly? Looking at math.stackexchange answers, it seems to take a lot of symbols and parentheses.

Simple example (and making up some notation):

$$ p \lor q \equiv ( (p \land q) \;\mathbf{|}\; (p \land \neg q) \;\mathbf{|}\; (q \land \neg p) ) $$

I've used a frame $( ... | ... | ... )$ to separate the disjuncts.

(In this example, the mutual exclusion is apparent by inspecting the disjuncts. In real examples there's arbitrary quantification and predicates inside each disjunct.)

If there's exactly 2 disjuncts, I can use exclusive or $\dot\lor$. But that doesn't extend to 3 or more: exclusive or is associative, but a chain of them $( p \dot\lor q \dot\lor r)$ would be true if any odd number of disjuncts is true. If all 3 are true I want the overall formula to be false. For the 3 case I could put $\dot\lor$ in place of $|$, and append $(...) \land \neg (p \land q \land r)$. That's not going to scale up for larger numbers of disjuncts.


You could use the Iverson bracket: if $P$ is a proposition, then $[P]=1$ if $P$ is true, and $[P]=0$ if $P$ is false. So your example would be:


and it should be clear how to extend this. (The only issue is that this is not too common of a notation, so you'll want to define it first.)


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