(Trans) Transitivity

p □→ q, q □→ r, therefore p □→ r

(Contr) Contraposability

p □→ q, therefore ~q □→ ~p

(Stren) Strengthening

p □→ q, therefore (p & r) □→ q

System T

(1) □ p → p

(2) □ (p → q) → (□ p → □ q)

System B (Brouwerian)

(3) p → □ ◊ p

System S4 (Lewis)

(4) □ p → □ □ p

System S5 (Lewis)

(5) ◊ p → □ ◊ p

“[…] the laws of logic which ultimately govern the world of the mind are, by their nature, essentially unvariable; they are common not only to all periods and places but to all subjects of whatever kind, without any distinction even between those that we call the real and the chimerical; they are to be seen even in dream.”

– Auguste Comte, Cours de Philosophie positive, 52e leçon.