The Exclusive Or

Under the Principle of Bivalence, a proposition is either true or false, and is not both true and false. My clarification by adding “and is not both true and false” is already covered if we understand “either true or false” in the exclusive sense. An easy example of what is called the Exclusive Or is that I come to a T-intersection, and I turn either left or right. Here, I do not mean anything that leaves open the possibility that I simultaneously turn right and left. Turning left excludes turning right and vice versa. We may wish to mean something of this exclusive sort either in using ‘or’ more generally, particularly when it is not obvious that the propositions that ‘or’ connects are indeed mutually exclusive (for example, “The morning star rotates clockwise or the evening star rotates counterclockwise relative to Earth’s North”), or to simply flag that they are meant to be mutually exclusive, and so we would not assign a mutually exclusive simultaneity, even if impossible, with true. This is to be prevented in principle.

There’s a connective that is readily available, which I will call EXOR.

P   Q    P EXOR Q

T   T           F                If P is true and Q is true, it will be false that they are mutually exclusive.

T   F           T                If P is true and Q is false, it will be true that they are mutually exclusive.

F   T           T                If P is false and Q is true, it will be true that they are mutually exclusive.

F   F           F                If both are false, it will be false that they are mutually exclusive.

Now, this looks like good news if we limit our mutually exclusive entities to two. For example, at the T-intersection I turn right or left. The above truth table perfectly captures the meaning here. The asserted truth is limited to turing left alone or turning right alone. However, what happens if I am not coming to a T-intersection, but rather a four-way intersection, and coming to it, I either turn left, or I turn right, or I go straight. Once more, I do not mean that I do all three. I mean all these ‘or’s in the exclusive sense.

Unfortunately, EXOR shows itself to be quite limited in providing the exclusive or with its full meaning. EXOR allows that I simultaneously turn left, turn right, and go straight. Check it out:

L  EXOR R  EXOR S

T      F        T       T       T

If ‘L’ and ‘R’ are true, then it is false that L  EXOR R; since ‘L  EXOR R’ is connected to ‘S’ by an ‘EXOR‘ connective, and ‘L  EXOR R’ is false and ‘S’ is true, ‘L  EXOR R  EXOR S’ is true. Thus, our EXOR connective is an impostor: it looks like it means mutual exclusion, yet on connecting odd propositions greater than one together, it returns a value of true if all the propositions are true.

The problem is that mutual exclusion is relative to an entire group of facts. For example, if I am thinking about what my friend in SoCal is doing today, I may come up with a list of a dozen things that are all mutually exclusive in regard to specific places or activities. How they are mutually exclusive is that just one fact, if any are true at all, will be true. It won’t be, or I certainly wouldn’t mean it, that many instances of these events take place simultaneously. I don’t mean that my friend is on PCH and on the 10, 405, 101, and 5 simultaneously. Nor is he bbqing and at work as a store manager simultaneously.

The meaning of the exclusive or is not at the level of connecting two propositions at a time. It is rather at the level of an entire row of the truth table. When multiple propositions are connected with an ‘or’ that’s meant exclusively, this is shorthand for grouping these propositions together and returning true if and only if just one of the propositions is true (and false otherwise). This is the way to evaluate the exclusive or, perhaps dynamically and not statically, since the connective is flexible between two and more propositions (or perhaps holistically rather than atomistically).

Of course, any row of a truth table is describable in terms of the typical connectives (‘and’, ‘negation’, and ‘disjunction’). However, in evaluating the truth value of any number of propositions connected by an exclusive or, we’d always have to at the least consider the number of elements so connected. To begin a restatement of some exclusive or statement or group of exclusive or statements in terms of the typical connectives, we would first restate the proposition by replacing every instance of the exclusive or with the typical two-place connective of the inclusive or (where both can be true); next we’d have to conjoin (using conjunction) this with statements that are the negation of each conjoined combination of the atomic statements of our original (exclusive-or) statement (note: this would not include singletons). This latter step captures the meaning that no combination of them is simultaneously true, be it any pair, triple, n-tuple, or however many are buildable from the original proposition.

For example, take the exclusive-or proposition: (A EXOREXOR C).

To translate this into it’s correct truth value using the typical atomistic connectives, first replace the ‘EXOR‘s of the original with the inclusive or, ‘v’. Thus:

(A v B v C)

Next, conjoin this with every negation of every combination of the atomic propositions that can be combined with conjunction. Like so:

(A v B v C) & ~(A & B) & ~(B & C) & ~(A & C) & ~(A & B & C)

Through this process, one and only one of the exclusive or’s disjuncts will be true, just as we intend its meaning. This is a way that allows for a proper translation of the exclusive or using the normal, atomistic propositional connectives.