## praeclarum-theorema

Noun

- (logic) The following theorem of propositional calculus: (A → B) ∧ (C → D) → (A ∧ C → B ∧ D).
^{ }^{ }^{ }^{ }*What is now called the praeclarum theorema is actually "one half" of Leibniz's original theorem, which was like so: if A = B and C = D, then AC = BD, whose appearance is splendidly algebraic. (It can also be stated as*(A ↔ B) ∧ (C ↔ D) → (A ∧ C ↔ B ∧ D).*)*- The praeclarum theorema can be seen to correspond with the logical rule of sequent calculus; given two sequents and one may infer (through the sequent calculus) that , where the comma on the left side of the turnstile can be interpreted as a kind of conjunction. So perhaps the rule, together with the rule: , and the disjunctive analogue , can help to interpret the sequent calculus as being rather "algebraic" (esp. if the syntactic consequence (represented by the turnstile) is compared to a preorder).

Origin

So named by G.W. Leibniz in his unpublished papers of 1690 (later published as *Leibniz: Logical Papers* in 1966), meaning "splendid theorem" in Latin.