In logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, a hypothetical syllogism

Syllogism

A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic

History of logic

The history of logic is the study of the development of the science of valid inference . Formal logic was developed in ancient times in China, India, and Greece...

, 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

Validity

In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

 argument of the following form:

If P → Q.

If Q → R.

____________________

Then P → R.

Symbolically, this is expressed:

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 ponens
  In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
  
  
• Modus tollens
  Modus tollens
  In classical logic, modus tollens has the following argument form:- Formal notation :...
  
  
• Modus tollendo ponens
• Affirming the consequent
  Affirming the consequent
  Affirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....
  
  
• Denying the antecedent
  Denying the antecedent
  Denying the antecedent, sometimes also called inverse error, is a formal fallacy, committed by reasoning in the form:The name denying the antecedent derives from the premise "not P", which denies the "if" clause of the conditional premise....
  
  
• Disjunctive syllogism
  Disjunctive syllogism
  A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion....
  
  
• Inference rule
• Transitive relation
  Transitive relation
  In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....