inference rule
 schema
 example

Modus Ponens
  1. If p then q.
  2. p.
  3. Therefore, q.
  1. If today is Sunday, then we are in Texas.
  2. Today is Sunday.
  3. Therefore, we are in Texas.

Modus Tollens
(Contraposition)
  1. If p then q.
  2. Not-q.
  3. Therefore, not-p.
  1. If today is Sunday, then we are in Texas.
  2. We're not it Texas.
  3. Therefore, today isn't Sunday.

Hypothetical Syllogism
(Transitivity)
  1. If p then q.
  2. If q then r.
  3. Therefore, if p then r.
  1. If today is Sunday, then we are in Texas.
  2. If we are in Texas, then we are below the Mason-Dixon line.
  3. Therefore, if today is Sunday, we are below the Mason-Dixon line.

Disjunctive Syllogism
(Cancellation)
  1. Either p or q.
  2. Not-p.
  3. Therefore, q.
  1. Either today is Sunday, or we are in Texas.
  2. Today isn't Sunday.
  3. Therefore, We are in Texas.

Dilemma
  1. Either p or q.
  2. If p then r.
  3. If q then s.
  4. Therefore, r or s.
  1. Either today is Sunday, or we are in Texas.
  2. If today is Sunday, then we are in Arizona.
  3. If we are in Texas, then today is Monday.
  4. Therefore, either we are in Arizona or today is Monday.

Reductio ad absurdum
(indirect proof)
(prove p by assuming not-p and deriving a contradiction)
  1. Nothing can do what is physically impossible.
  2. It is physically impossible for bumblebees to fly.
  3. Therefore, bumblebees cannot fly.

However,

  1. Bumblebees do in fact fly.
  2. Therefore, either premise one or two must be false (in this case, premise 2).