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).