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In Copi's Introduction to Logic,

  • propositions P and Q are called contradictory, if they can't be both true and can't be both false;

  • propositions P and Q are called contrary, if they can't be both true but can be both false;

  • propositions P and Q are called subcontrary, if they can't be both false but can be both true.

Can the three relationships between P and Q be represented in terms of logical operations?

  • propositions P and Q are contradictory, if P and not(Q) are logically equivalent, i.e. P <-> not(Q)?

  • How about that propositions P and Q are contrary?

  • How about that propositions P and Q are subcontrary?

Thanks.

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    You have the contradictory one right: P ↔ ¬Q. Contrary is ¬(P ∧ Q). Subcontrary is P ∨ Q.
    – Bumble
    Sep 30 at 5:41
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    In terms of individual operations, contradictory is XOR (exclusive or, P ⊕ Q), contrary is NAND (Sheffer stroke, P ↑ Q) and subcontrary is OR (disjunction, P ∨ Q).
    – Conifold
    Sep 30 at 5:53

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