What I mean is, for example, we can understand A xor B as "true if A is different from B, false otherwise".
Is it possible to understand the "collective xor" of multiple logic variables
A1 xor A2 xor ... xor An
with a single short sentence? The task is trivial for and or or.
There are two ways of interpreting it, one in the spirit of the binary operation asking "are these mutually exclusive and exhaustive options", in which case the sentence is
... is true when one, and only one, of the n sentences is true.
as Christopher says.
The other is like the result of putting this into a computer. As XOR is both commutative and associative, we can put any statement of the form
A1 xor A2 xor ... xor An
into a cannonical form, putting all the trues first and all the falses last
T xor T xor F xor T xor F
could be written
(((T xor T) xor T) xor F) xor F
so it is only the number of trues and falses that matters. XOR can be identified with addition mod 2, with false being 0 and true being 1 . So it is the number of ones (trues) makes a difference to the final result, i.e.
X xor F -> X (nothing changes)
X xor T -> not X (value flips)
so, the truth is determined by whether there is an odd or even number of "trues". Then we have
T xor T is false
so even numbers of trues give false, and the sentence you would be looking for is
... is true when the number of true sentences is odd?
See Wiki : Exclusive or :
Exclusive disjunction or exclusive or is a logical operation that outputs true whenever both inputs differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR, EOR, EXOR. The opposite of XOR is logical biconditional, which outputs true whenever both inputs are the same.
It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; exclusive or excludes that case. This is sometimes thought of as "one or the other but not both".
More generally, XOR is true whenever an odd number of inputs is true. A chain of XOR's — a XOR b XOR c XOR d (and so on) — is true whenever an odd number of the inputs are true and is false whenever an even number of inputs are true.
If you "restore" the parentheses in it : (((a XOR b) XOR c) XOR d) and evaluate it staritng from the inner one, of course you will get the same result: after the first T, each time you meet a new T it "flips" from T to F and from F to T. So it will be T if an odd number of T is present.
... is true when one, and only one, of the n sentences is true.
This is not right, because the following should turn out to be true (substituting truth values for sentence names: (T xor F) xor (T xor T). The right term there is False, so the whole is True. My error!
