Hypothetical Syllogism  Definition & Examples
Hypothetical syllogisms are arguments in formal logic that deduce conclusions from conditional (if–then) statements.
Syllogisms are typically found in contexts such as formal logic, but syllogistic reasoning is also applied in fields like science and mathematics.
What is a hypothetical syllogism?
Hypothetical syllogisms are arguments that explore the logical implications of at least one conditional statement, typically expressed as an if–then proposition. They are commonly known as conditional syllogisms.
Like other syllogisms, hypothetical syllogisms express deductive reasoning and consist of two premises and a conclusion, with the conclusion following as an inevitable logical
consequence of the premises.
Hypothetical syllogisms are one of three main categories of syllogisms, along with disjunctive syllogisms and categorical syllogisms.
Modus ponens and modus tollens are two types of hypothetical syllogisms that are widely used:
 Modus ponens (or affirming the antecedent) confirms the result to infer the initial condition is true.
 Formulation: If P, then Q. P. Therefore, Q.
 Example: “If it snows, the school will close. It is snowing. Therefore, the school is closed.”
 Modus tollens (or denying the consequent) negates the result to infer the initial condition is false.
 Formulation: If P, then Q. Not Q. Therefore, not P.
 Example: “If the light is on, the switch is up. The switch is not up. Therefore, the light is not on.”
Hypothetical syllogism examples
Examples of hypothetical syllogisms are typically found in studies of philosophy, in domains such as formal logic and ethics.
However, the thought process used in hypothetical syllogisms is also crucial to other fields, such as science.
Similarly, this reasoning plays a fundamental role in mathematics.
Hypothetical syllogism truth table
Truth tables can be used to test the validity of hypothetical syllogisms. To make a truth table, first assign a truth value (true or false, expressed as T or F) to each element of the argument: variables, premises, and conclusion.
The truth table below shows that whenever both premises of a hypothetical syllogism are true, the conclusion is also true (rows one and four). This proves that the structure is valid.
P  Q  Premise 1: If P, then Q. (P ⟹ Q) 
Premise 2: If Q, then R. (Q ⟹ R) 
Conclusion: If P, then R. (P ⟹ R) 
T
T F F 
T
F T F 
T
F T T 
T
T F T 
T
F F T 
Hypothetical syllogisms vs logical fallacies
Hypothetical syllogisms are valid when formed correctly, but structural flaws result in formal logical fallacies (or non sequitur fallacies).
Two logical fallacies often result from invalid hypothetical syllogisms:
 Affirming the consequent wrongly concludes a condition is true because a consequence is true.
 Formulation: If P, then Q. Q. Therefore, P.

 Example: “If an animal is a rabbit, then it has fur. This animal has fur. Therefore, it is a rabbit.”
 Denying the antecedent wrongly concludes a result is false because the initial condition is false.
 Formulation: If P, then Q. Not P. Therefore, not Q.

 Example: “If a shape is a rectangle, then it has a perimeter. This shape is not a rectangle. Therefore, this shape does not have a perimeter.”
Hypothetical syllogism formula
In symbolic logic, two common formulas are used to represent hypothetical syllogisms. Each highlights different aspects of the logical progression:
 Sequential implication form: P ⟹ Q, Q ⟹ R ∴ P ⟹ R
 Lists conditions in sequence, ending with a conclusion marked by the symbol ∴ (“therefore”)
 Conjunctive implication form: (P → Q) ∧ (Q → R) ⟹ (P → R)
 Shows two conditions leading to a third by using a conjunction symbol ∧ (“and”) and an implication symbol ⟹ (“logically implies”)
Symbol  Meaning 

∧  and 
→  materially implies: expresses a conditional truth 
⇒  logically implies: expresses an absolute truth 
∴  therefore 
Frequently asked questions about hypothetical syllogisms
 Is a hypothetical syllogism a fallacy?

A hypothetical syllogism is a valid argument form, not a fallacy. However, syllogisms can result in formal logical fallacies (or non sequitur fallacies) if they have structural errors that render them invalid.
The fallacies of affirming the consequent and denying the antecedent are especially likely to occur in failed attempts at forming hypothetical syllogisms.
 Are hypothetical syllogisms inductive or deductive?

Hypothetical syllogisms express deductive reasoning, beginning with relatively general premises and inferring specific conclusions. All three major categories of syllogisms (hypothetical syllogisms, disjunctive syllogisms, and categorical syllogisms) are deductive.
In contrast, inductive reasoning begins with specific observations and infers relatively broad conclusions.