praeclarum theorema

Translingual

Etymology

So named by G.W. Leibniz in his unpublished papers of 1690 (later published as Leibniz: Logical Papers in 1966), meaning "splendid theorem" in Latin.

Noun

praeclarum theorema

1. (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 $\wedge R$ of sequent calculus; given two sequents $A \vdash B$ and $C \vdash D$ one may infer (through the sequent calculus) that $A, C \vdash B \wedge D$, where the comma on the left side of the turnstile can be interpreted as a kind of conjunction. So perhaps the $\wedge R$ rule, together with the $I$ rule: $A \vdash A$, and the disjunctive analogue $\vee L$, 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).

See also

• constructive dilemma

References