Positive antecedent theory (P → Q); P ├ q if p is q; p; So, ask
Negative aftereffect theory (P → Q); ? ├? P if p is q; Non-q; So, non-P
Hypothetical syllogism (p → q); (q → r) ├ (p → r) If p, then q; R if q; So if p is r.
Selective syllogism (p ∨ q); ? P ├ q is either p or q; Not p; So, ask
Creative dilemma formula (P → Q) ∧ (R → S); (p ∨ r) ├ (q ∨ s) If P, then Q; If r is s; But it is either p or r; So, it's either Q or S.
Destructive dilemma formula (p → q) ∧ (r → s); (? q ∨? s) ├(? p ∨? R) q if p; If r is s; But it's either q or s; So, it's either p or R.
Simplified formula (p ∧ q) ├ p p and q are true; So, P is true.
Conjunctive formulas p, q ├ (p ∧ q) p and q are true respectively; So, when they are combined, they are true.
Increase the argument p ├ (p ∨ q) p to be true; So the disjunction formula (p or q) is true.
Synthetic formula (p → q) ∧ (p → r) ├ p → (q ∧ r) If p is q; If p, r; So, if p is true, then q and r are true.
De Morgan Law (1)? (p ∧ q) ├(? p ∨? Q) The negation of (p and q) is equivalent to (non-p or non-q).
De Morgan law (2)? (p ∨ q) ├(? p ∧? Q) The negation of (p or q) is equivalent to (non-p and non-q).
Commutative law (1) (p ∨ q) ├ (q ∨ p) (p or q) is equivalent to (q or p).
Commutative laws (2) (p ∧ q) ├ (q ∧ p) (p and q) are equivalent to (q and p).
The associative law (1) p ∨ (q ∨ r) ├ (p ∨ q) ∨ r p or (q or r) is equivalent to (p or q) or r.
The law of association (2) p ∧ (q ∧ r) ├ (p ∧ q) ∧ r p and (q and r) are equivalent to (p and q) and r.
The distribution law (1) p ∧ (q ∨ r) ├ (p ∧ q) ∨ (p ∧ r) p and (q or r) are equivalent to (p and q) or (p and r).
The distribution law (2) p ∨ (q ∧ r) ├ (p ∨ q) ∧ (p ∨ r) p or (q and r) is equivalent to (p or q) and (p or r).
The law of double negation, p ├ p p is equivalent to non-p negation.
The law of transposition (p → q) ├ (? q →? P) if p, then q is equivalent to if it is not q, then it is not p.
Law of substantial implication (p → q) ├ (p ∨ q) If P, then Q is equivalent to P or Q.
Law of substantial equivalence (1) (p? Q) ├ (p → q) ∨ (q → p) (p is equivalent to q) means either (if p is true, then q is true) or (if q is true, then p is true).
Law of Substantial Equivalence (2) (p? q) ├ (p ∧ q) ∨(? q ∧? P) (p is equivalent to Q) means either (both P and Q are true) or (both P and Q are false).
Output law (p ∧ q) → r ├ p → (q → r) From (if P and Q are true, then R is true), we can prove (if Q is true, then P is true and then R is true).