Hypothetical syllogism

Rules of inference
Propositional calculus
Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) Modus ponendo tollens (¬(A∧B), A ⊢ ¬B) Conjunction introduction (A, B ⊢ A∧B) Simplification (A∧B ⊢ A) Disjunction introduction (A ⊢ A∨B) Disjunction elimination (A∨B, A→C, B→C ⊢ C) Disjunctive syllogism (A∨B, ¬A ⊢ B) Hypothetical syllogism (A→B, B→C ⊢ A→C) Constructive dilemma (A→P, B→Q, A∨B ⊢ P∨Q) Destructive dilemma (A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B) Biconditional introduction (A→B, B→A ⊢ A↔B) Biconditional elimination (A↔B ⊢ A→B)
Predicate calculus
Universal generalization Universal instantiation Existential generalization Existential instantiation
v · d · e

In logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.

Propositional logic

Hypothetical syllogism is one of the proof rules in classical logic that may or may not be available in a non-classical logic. The hypothetical syllogism (abbr. H.S.) is a valid argument of the following form:

If P → Q.

If Q → R.

____________________

Then P → R.

Symbolically, this is expressed:

$P \rightarrow Q, Q \rightarrow R \vdash P \rightarrow R$

Example of use:

If I do not wake up, then I cannot go to work.

If I cannot go to work, then I will not get paid.

Therefore, if I do not wake up, then I will not get paid.

See also

• Modus ponens
• Modus tollens
• Modus tollendo ponens
• Affirming the consequent
• Denying the antecedent
• Disjunctive syllogism
• Inference rule
• Transitive relation

References