Listing of the rules of inference and equivalence

Rules of inference  (8)

Modus Ponens

MP       p > q  / p  // q

Modus Tollens

MT       p > q / ~q // ~p

Hypothetical syllogism

HS       p > q / q > r //  p > r

.

Disjunctive syllogism

DS       p v q / ~p // q

Simplification

SM       p . q  //   p

Conjunction

CN      p  / q //  p. q

AD      p // p v q

Constructive Dilemma

CD     (p > q) . (r > s) /  p v r  //  q v s

Rules of Equivalence/ Replacement  (10)

Double Negation

DN      p ::  ~~p

Commutation

CM     (p . q)  :: (q . p)        (p v q) :: (q v p)

Association

AS      ((p . q) . r) ::  (p . ( q . r))     ((p v q) v r) :: (p v (q v r))

DeMorgan’s Rule

DM     ~(p v q) :: (~p . ~q)          ~(p .q) :: (~p v ~q)

Distribution

DIST  (p v (q . r))  :: ((p v q) . (p v r))      (p . (q v r)) :: ((p . q) v (p . r))

Transposition

TRAN  (p > q) :: (~q > ~p)

Implication

IMP     (p v q) :: (~p > q)

Equivalence

EQ      (p ≡ q) :: ((p > q) . (q > p))         (p ≡ q) :: ((p . q) v (~p . ~q))

Exportation

EXP    (p > (q > r)) :: ((p . q) > r)

Tautology

TAUT  (p v p) :: p      (p . p) :: p

4 Responses to Listing of the rules of inference and equivalence

1. Modus ponens is the elimination rule for the logical connective in formal logic (forming conditional statements).

This is function application on more broad types in type theory.

Even in Hilbert-style logic, where there are essentially no rules of inference (apart from the numerous axioms), a rule of modus ponens is generally present. (In particular, in Hilbert’s version of the propositional calculus, modus ponens is the only non-axiom rule of inference.)

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