Is there canonical terminology for logical connectives between more than two propositions?

Question

My problem is with the definition of exclusive disjunction, at least according to Wikipedia. The Wikipedia page for Exclusive Or, at the time of writing, states that "More generally, XOR is true whenever an odd number of inputs is true." This seems useful for computer science, but in logical reasoning I would want a connective that means "only one of these propositions is true" no matter how many propositions there are. For instance, I would like a connective that says "You can either order milk, coffee, or orange juice with your breakfast."

But all of the connectives that Wikipedia lists are binary but yields strange results like that exclusive disjunction is true when an odd number of propositions are true. Is there canonical terminology for what I'm talking about or is there no standard terminology? I want to still call this exclusive disjunction, but it contradicts with the usage on Wikipedia as referenced above.

Also...is there terminology for other connectives over an indeterminate number of propositions?

Answer 1

You need to be careful. There is no such thing as a logical connective between more than two propositions. There are no ternary (or n-ary where n > 2) connectives in non-exotic logics, only unary (negation) and binary (conjunction, disjunction, implication, etc.) ones. As always, doing truth tables helps illustrate the concept:

First, this is how you want XOR to work with two inputs (in fact, this is how it actually works):

Table 1: | A | B | XOR | |-------------| | F | F | F | | F | T | T | | T | F | T | | T | T | F |

Next, this is how you want XOR to work with three inputs (but there's an error here):

Table 2: | A | B | C | XOR | |-----------------| | F | F | F | F | | F | F | T | T | | F | T | F | T | | F | T | T | F | | T | F | F | T | | T | F | T | F | | T | T | F | F | | T | T | T | F | (*)

Let's look at the last line marked with an asterisk. XOR(A,B) will yield F per Table 1. Similarly, XOR(B,C) as well as XOR(C,A) will also yield F per Table 1. So, no matter how we decide to commute, we will have to deal with an F and the remaining T. And XOR(T,F) will always yield a T per Table 1. This is why we observe the odd-even alternation of truth values when all operands are true.

It turns out that you don't want to use a XOR in examples like the one in your OP. What you actually want to do is something like (A v B v C) & ~(A & B & C) or (via DeMorgans) (A v B v C) & (~A v ~B v ~C). There is no logical connective that does this since this would require an n-ary connector, e.g. something like atLeastOneButNotAll(A,B,C,...,n) which would only work as a function in FOL.

Here is an interesting paper discussing the ternary exclusive or.