1. Inference Rules: Tautologies
(p  (p  q))  q Modus Ponens
(q  (p  q))  p Modus Tollens
p  (p  q) Addition
(p  q)  p Simplification
((p)  (q))  p  q Conjunction
((p  q)  q)  p Disjunctive Syllogism
((p  q)  (q  r))  (p  r) Hypothetical Syllogism
(aka, the transitive rule)
((p  q )  ( p  r))  (q  r) Resolution
((p  q )  (p  r)  (q  r))  r Cases

2. Fallacies (q  (p  q))  p. Fallacy of Affirming the Conclusion
(p  (p  q))  q.
Fallacy of Denying the Hypothesis

3. Inference Rules: Quantified Statements
Universal Instantiation
(for an individual object c from UoD)
x P(x)  P(c)
Universal Generalization
(for any arbitrary element c from UoD)
P(c)___
 x P(x)
x P(x)  P(c)
Existential Instantiation
(for some specific object c from UoD)
Existential Generalization
(for some object c from UoD)
P(c)__
 x P(x)

4. Proof Methods Trivial Proof of p  q given q (see truth table).
Vacuous Proof of p  q given p (see truth table).
Direct Proof of p  q given p, show q.
Indirect Proof of p  q (proof by contraposition, so
show q  p starting with q).
Proof of p  q by Contradiction: recall that p  q Ξ
p  q. Assume the negation, p  q, is true and
show a contradiction.
Proof of p1  p2  p3  … pn  q by Cases: show that
pi  q is true for all i, where 1 ≤ i ≤ n.
Proof of x P(x) by example: just find an instance.
Proof that x P(x) is false by counterexample: just
find an instance c where P(c) is false.