Department of Philosophy (PHI)

"Hypothetical Syllogism" (Valid conditional chain argument)

Example: If Fido is a terrier, Fido is a dog; if Fido is a dog, Fido is a mammal; so if Fido is a terrier, Fido is a mammal.

General form in English: if P then Q; if Q then R; therefore if P then R.

Symbolic form: P → Q; Q → R; therefore P → R

Characterization. The component that the premises have in common is the antecedent of one and the consequent of the other. Each of the other components has the same function (antecedent or consequent) in the conclusion that it has in a premise.

Proof of validity of hypothetical syllogism. I will show that the argument cannot have true premises and a false conclusion by showing that if the conclusion is false, there must be a false premise. Assume then that the conclusion is false, that is, P is true and R is false.

• Q must be either true or false.
  
  • Assume first that Q is true. In that case Q is true and R is false, making the second premise false. Hence if Q is true, there is a false premise.
  • Now assume instead that Q is false. In that case P is true and Q is false, making the first premise false. So if Q is false, there is a false premise.
• Therefore, regardless of whether Q is true or false, there is a false premise.

Therefore, if the conclusion is false, there must be a false premise. Hence the argument is valid.

Conditional fallacies (invalid forms):

Fallacy of common antecedent (invalid).

Example: If Bibi is a cat, Bibi has ears;  if Bibi is a cat, Bibi has four legs;  so if Bibi has ears, Bibi has four legs.

General form in English: if P then Q; if P then R; therefore if Q then R.

Symbolic: P → Q; P → R; therefore Q → R

Characterization. The component that the premises have in common is the antecedent of both. One of the other components has a different function in the conclusion than it has in a premise.

Proof of invalidity.  Bibi could be an owl, in which case the premises would be true and the conclusion false.

Fallacy of common consequent (invalid).

Example: If Rex is a cat, Rex likes meat. If Rex is a dog, Rex likes meat. So if Rex is a cat, Rex is a dog.

General form in English: if P then R; if Q then R; therefore if P then Q.

Symbolic: P → R; Q → R; therefore P → Q

Characterization. The component that the premises have in common is the consequent of both. One of the other components has a different function in the conclusion than it has in a premise.

Proof of invalidity.  Rex could be a meat-loving cat, in which case the premises would be true and the conclusion false.

Fallacy of reversed conclusion (invalid).

Example: If Bob has a Pathfinder, he has a 4x4;  if Bob has a 4x4, he can haul plywood;  so if Bob can haul plywood, he has a Pathfinder. 

General form in English: if P then Q; if Q then R; therefore if R then P.

Symbolic: P → Q; Q → R; therefore R → P

Characterization. The component that the premises have in common is the antecedent of one and the consequent of the other. Each of the other components has a different function (antecedent or consequent) in the conclusion than it has in a premise.

Proof of invalidity.  Bob might have a pick-up truck in which he can haul plywood, but not a Pathfinder.  In that case the argument would have true premises and a false conclusion.