Exclusivity is the title of a number-one R&B single by Damian Dame. The hit song spent two weeks at number-one on the US R&B chart. Rhythm and blues (or R & B) is a musical marketing term introduced in the United States in the late 1940s by Billboard magazine. ... Damian Dame was an R&B duo consisting of Debra Jean Deah Dame Hurd (September 20, 1958-June 27, 1994) and Bruce Edward Damian Broadus (September 13, 1966-June 27, 1996). ...
Exclusive or (usual symbol XOR occasionally EOR), which is sometimes called exclusivedisjunction, is a logical operator that results in true if one of the operands, but not both of them, is true.
The exclusivedisjunction of propositions A and B is usually called A xor B, where "xor" stands for "exclusive or" and is pronounced "eks-or" or "zor".
Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network systems.
"Exclusivedisjunction is the sense of the word or as in the proverb you can have your cake or eat it (but not both)."
which is not supported by the definition of exclusivedisjunction.
In other words, if the meaning in the proverb were really the exclusivedisjunction, it would follow that I had permission to take one of the two actions, but you wouldn't tell me which.