Inference rules

Below are all the most common inference rules for propositional logic listed. I have also given an example of them.

Modus Ponens (MP)

Proposition	Explanation	Example
P→Q	Premise	If Socrates is a man, then Socrates is mortal.
P	Premise	Socrates is a man.
Q	Inference	Socrates is mortal.

Modus Tollens (MT)

Proposition	Explanation	Example
P→Q	Premise	If Socrates is a man, then Socrates is mortal.
¬Q	Premise	Socrates is not mortal.
¬P	Inference	Socrates is not a man.

Hypothetical Syllogism (HS)

Proposition	Explanation	Example
P→Q	Premise	If Socrates is a man, then Socrates is mortal.
Q→R	Premise	If Socrates is mortal, then Socrates is dead.
P→R	Inference	If Socrates is a man, then Socrates is dead.

Disjunctive Syllogism (DR)

Proposition	Explanation	Example
P∨Q	Premise	Socrates is alive or Socrates is dead.
¬P	Premise	Socrates is not alive.
Q	Inference	Socrates is dead.

Constructive Dilemma (CD)

Proposition	Explanation	Example
(P→Q)∧(R→T)	Premise	If Socrates is a man, then Socrates is mortal, and if Plato is a man, then Plato is a mortal.
P∨R	Premise	Socrates is a man or Plato is a man.
Q∨T	Inference	Socrates is mortal or Plato is mortal.

Destructive Dilemma (DD)

Proposition	Explanation	Example
(P→Q)∧(R→T)	Premise	If Socrates is a man, then Socrates is mortal, and if Plato is a man, then Plato is a mortal.
¬Q∨¬T	Premise	Socrates is not mortal or Plato is not mortal.
¬P∨¬R	Inference	Socrates is not a man or Plato is not a man.

Conjunction Introduction (Conj.) or Adjunction

Proposition	Explanation	Example
P	Premise	Socrates is a man.
Q	Premise	Plato is a man.
P∧Q	Inference	Socrates is a man and Plato is a man.

Simplification (Simp.) or Conjunction Elimination

Proposition	Explanation	Example
P∧Q	Premise	Socrates is a man and Plato is a man.
P	Inference	Socrates is a man.

Addition (Add.) or Disjunction Introduction.

Proposition	Explanation	Example
P	Premise	Socrates is a man.
P∨Q	Inference	Socrates is a man or Plato is a man.

Reductio Ad Absurdum (RAA.) or Negation Introduction

Proposition	Explanation	Example
P→Q	Premise	If a round square exists, then that square has edges.
P→¬Q	Premise	If a round square exists, then that square has no edges.
¬P	Inference	A round square does not exist.

Replacement rules

Below are some of the most common replacement rules listed. I have not given examples of these due to the complexity the examples would need to have.

De Morgan’s Theorem (DM)

¬(P∧Q)⇔(¬P∨¬Q)

¬(P∨Q)⇔(¬P∧¬Q)

Commutation (Com.)

P∧Q⇔Q∧P

P∨Q⇔Q∨P

Association (Assoc.)

[(P∨Q)∨R]⇔[P∨(Q∨R)]

[P∧(Q∧R)]⇔[(P∧Q)∧R]

Distribution (Dist.)

[P∧(Q∨R)]⇔[(P∧Q)∨(P∧R)]

[P∨(Q∧R)]⇔[(P∨Q)∧(P∨R)]

Double Negation (DN)

P⇔¬¬P

Material Implication (M. Imp.)

(P→Q)⇔(¬P∨Q)

(P→Q)⇔¬(P∧¬Q)

Transposition (Trans.) or Contra-Position (CP.)

(P→Q)⇔(¬Q→¬P)

Material Equivalence (M. Equiv.)

(P↔Q)⇔[(P→Q)∧(Q→P)]

(P↔Q)⇔[(P∧Q)¬(¬P∧¬Q)]

Law of Exportation

[(P∧Q)→R]⇔[(P→(Q→R)]

References

http://www.mathpath.org/proof/proof.inference.htm

http://en.wikipedia.org/wiki/List_of_rules_of_inference

I. M. Copi, Symbolic Logic, 5th Edition. (New York: Macmillian, 1979)

N. Swartz, R. Bradley, Possible Worlds, 1979, pp. 198-200.