// laws and rules · Rule of inference
Hypothetical syllogism_
The hypothetical syllogism (also called transitivity of implication) lets you chain conditionals: if p ⇒ q and q ⇒ r, then p ⇒ r. It is the rule that makes chained reasoning possible, and the formula ((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r) is a three-variable, eight-row tautology.
Example
((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r)
What the variables mean
- ▸ p: “I study”
- ▸ q: “I pass the exam”
- ▸ r: “I pass the course”
In plain words
If I study, I pass the exam. If I pass the exam, I pass the course. Therefore, if I study, I pass the course.
Truth table
| p | q | r | p ⇒ q | q ⇒ r | (p ⇒ q) ∧ (q ⇒ r) | p ⇒ r | ((p ⇒ q) ∧ (q ⇒ r)) ⇒ (p ⇒ r)★ |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | F | F | T |
| T | F | T | F | T | F | T | T |
| T | F | F | F | T | F | F | T |
| F | T | T | T | T | T | T | T |
| F | T | F | T | F | F | T | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | T | T | T |
Classification: Tautology · 8 rows
Statement of the rule
Premise 1: p ⇒ q. Premise 2: q ⇒ r. Conclusion: p ⇒ r. The middle variable q “disappears” in the conclusion, just like the middle term of a classical syllogism.
With three variables the table has 2³ = 8 rows, and the final column is T in every one of them.
Why it is valid: reading the table
The conclusion p ⇒ r is only F when p = T and r = F. Consider those two rows. If additionally q = T (row p = T, q = T, r = F) the premise q ⇒ r is F. If q = F (row p = T, q = F, r = F) the premise p ⇒ q is F. Either way the conjunction of premises is F, so the whole implication is T.
In the other six rows the conclusion p ⇒ r is already T on its own, and an implication with a true consequent is always T. Hence there is no counterexample.
How it is used in proofs
It justifies chained proofs: A ⇒ B, B ⇒ C, C ⇒ D, therefore A ⇒ D. Each link rests on the previous one.
It also explains why “implies” is a transitive relation and lets theorems be organised hierarchically: if a lemma implies another lemma which implies the theorem, the first lemma implies the theorem.
Examples
Everyday: “If the battery dies, the phone turns off. If the phone turns off, I miss calls. Therefore, if the battery dies, I miss calls.”
Programming: in a dependency chain, “if the database fails, the API fails; if the API fails, the app fails.” We conclude that a database outage takes the app down even though they are not directly connected.
Mathematics: “If n is a multiple of 8, it is a multiple of 4; if it is a multiple of 4, it is even. So every multiple of 8 is even.”
Relation to other laws
It can be proved by using modus ponens twice inside a conditional proof: assume p, get q by MP with the first premise, then r by MP with the second.
Do not confuse it with the disjunctive syllogism, which works with a disjunction and a negation, or with exportation, which rearranges nested implications instead of chaining them.
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related operators
Frequently asked questions
Why is it called “hypothetical”? ▼
Because its premises are hypotheses, i.e. conditional statements of the form “if…, then…”, unlike Aristotle's categorical syllogism.
How many rows does its truth table have? ▼
Eight, because three variables (p, q, r) are involved and 2³ = 8.
Can more than two implications be chained? ▼
Yes. Applying the rule repeatedly yields p ⇒ s from p ⇒ q, q ⇒ r and r ⇒ s, and so on.