// laws and rules · Rule of inference
Modus ponens_
Modus ponens (Latin for “the mode that affirms”) is the most basic rule of inference in propositional logic. It states that from an implication p ⇒ q and the truth of p we may conclude q. Its truth table shows that (p ∧ (p ⇒ q)) ⇒ q is a tautology: there is no row where the premises are true and the conclusion is false.
Example
(p ∧ (p ⇒ q)) ⇒ q
What the variables mean
- ▸ p: “It is raining”
- ▸ q: “The street is wet”
In plain words
If it rains, the street gets wet. It is raining. Therefore, the street is wet.
Truth table
| p | q | p ⇒ q | p ∧ (p ⇒ q) | (p ∧ (p ⇒ q)) ⇒ q★ |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
Classification: Tautology · 4 rows
Statement of the rule
Premise 1: p ⇒ q (if p then q). Premise 2: p. Conclusion: q. Written as a single formula the argument becomes (p ∧ (p ⇒ q)) ⇒ q, i.e. “the conjunction of the premises implies the conclusion”.
An argument is valid exactly when that formula is a tautology. That is why the calculator classifies the expression as a tautology: in all four combinations of p and q the final column is T.
Why it is valid: reading the table
The only row that could break the argument is one where the premises are true and the conclusion false. For p ∧ (p ⇒ q) to be T we need p = T and p ⇒ q = T. But with p = T the implication is only true when q = T. So whenever the premises hold, q must hold as well.
Row 1 (p = T, q = T): p ⇒ q is T, the conjunction is T and q is T, so the final implication is T. Row 2 (p = T, q = F): p ⇒ q is F, the conjunction is F, and an implication with a false antecedent is T. Rows 3 and 4 (p = F): the conjunction is F and the final implication is again T.
How it is used in proofs
In a formal proof modus ponens lets you “detach” the consequent from an implication you have already established. If earlier lines contain “p ⇒ q” and “p”, you may write “q” and cite MP as the justification.
Nearly every deductive system (natural deduction, Hilbert-style axiom systems) takes it as a primitive rule; many other rules are derived from it together with equivalences such as the contrapositive or material implication.
Examples
Everyday: “If the light is red, cars stop. The light is red. Therefore, cars stop.”
Programming: `if (isAuthenticated) { showDashboard(); }` applies modus ponens on every run. The program's rule is “if authenticated then show the dashboard”; when the condition evaluates to true, the consequence is executed.
Mathematics: “If n is even, then n² is even. 10 is even. Therefore 10² = 100 is even.”
Relation to other laws
Modus tollens is its mirror image: instead of affirming the antecedent it denies the consequent to conclude the negation of the antecedent. Each rule can be turned into the other with the contrapositive law (p ⇒ q) ⇔ (¬q ⇒ ¬p).
Do not confuse it with the fallacy of affirming the consequent, ((p ⇒ q) ∧ q) ⇒ p, which looks similar but is a contingency: “the street is wet” does not let you conclude that it rained.
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
What does “modus ponens” mean? ▼
It comes from the Latin modus ponendo ponens, “the mode that by affirming affirms”: it affirms the antecedent of an implication in order to affirm its consequent.
Is modus ponens a tautology? ▼
The rule itself is an inference schema, but the formula that represents it, (p ∧ (p ⇒ q)) ⇒ q, is a tautology, and that is exactly what guarantees the rule is valid.
What is the difference between modus ponens and modus tollens? ▼
Modus ponens affirms p to conclude q; modus tollens denies q to conclude ¬p. Both start from the same implication p ⇒ q.
Why does the table have 4 rows? ▼
Because there are two variables, p and q, each of which can be T or F: 2² = 4 possible combinations.