Let A mean “this integer is even” and B mean “it is odd.” Only two combinations are allowed:
A true, B false: the integer is even.
A false, B true: the integer is odd.
Two combinations are excluded:
Both true: an integer is both even and odd — impossible.
Both false: an integer is neither even nor odd — impossible.
Thus a genuine disjunction allows exactly one true side; this follows from the exclusion of joint truth and joint falsehood, not just from the word “or”.
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Conditional Propositions: Connectives, Disjunctions and Their Main Kinds
Let A mean “this integer is even” and B mean “it is odd.” Only two combinations are allowed:
A true, B false: the integer is even.
A false, B true: the integer is odd.
Two combinations are excluded:
Both true: an integer is both even and odd — impossible.
Both false: an integer is neither even nor odd — impossible.
Thus a genuine disjunction allows exactly one true side; this follows from the exclusion of joint truth and joint falsehood, not just from the word “or”.