Rules of Inference and Common Fallacies

You must know these by heart. The valid forms and invalid forms in this table can be used to classify certain short arguments. The valid forms can also be combined to construct step-by-step proofs of validity for more complicated arguments.

VALID FORMSINVALID FORMS
Modus Ponens
p→q
p
∴q

One premise is a conditional statement, the other premise affirms the antecedent, and the conclusion affirms the consequent.
Also called Direct Reasoning, or Law of Detachment, among others.
Fallacy of the Converse (Affirming the Consequent)
p→q
q
∴p

One premise is a conditional statement, the other premise affirms the consequent, and the conclusion affirms the antecedent.
Modus Tollens
p→q
¬q
∴¬p

One premise is a conditional statement, the other premise denies the consequent, and the conclusion denies the antecedent..
Also called Indirect Reasoning, or Contrapositive Reasoning, among others.
Fallacy of the Inverse (Denying the Antecedent)
p→q
¬p
∴¬q

One premise is a conditional statement, the other premise denies the antecedent, and the conclusion denies the consequent.
Hypothetical Syllogism
p→q
q→r
∴p→r

One premise is a conditional statement, a second premise is a conditional statement whose antecedent matches the consequent of the other premise, and the conclusion results from this chain of reasoning.
Also called Transitive Reasoning.
Misuse of Hypothetical Syllogism
p→q
p→r
∴q→r


p→q
r→q
∴p→r

An incorrect attempt at Hypothetical Syllogism, in which two conditional premises agree in the antecedent, or agree in the consequent. Also called a False Chain.
Disjunctive Syllogism
p∨q
¬q
∴p


One premise is a disjunction, the other premise denies one of the disjuncts, and the conclusion affirms the other disjunct.
Disjunctive Fallacy (Affirming a Disjunct)
p∨q
q
∴¬p


One premise is a disjunction, the other premise affirms one of the disjuncts, and the conclusion denies the other disjunct.

Other Rules of Inference

The rules of inference below are useful, along with those listed above, in constructing validity proofs for more complicated arguments.
These may be provided on a test.
Note that in this table BOTH columns present VALID forms.
Disjunction Introduction
p
∴p∨q

Conjunction Elimination
p∧q
∴p

Conjunction Introduction
p
q
∴p∧q

Constructive Dilemma
(p→q)∧(r→s)
p∨r
∴q∨s

Universal Instantiation
∀xP(x)
∴P(c)

Existential Instantiation
∃xP(x)
∴P(c)

Existential Generalization
P(c)
∴∃xP(x)