# Can the relationships of contradictory, contrary and subcontrary between P and Q be represented in terms of logical operations?

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.

• You have the contradictory one right: P ↔ ¬Q. Contrary is ¬(P ∧ Q). Subcontrary is P ∨ Q. Sep 30 at 5:41
• 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). Sep 30 at 5:53