// laws and rules · Rule of inference
Disjunctive syllogism_
The disjunctive syllogism (also called modus tollendo ponens) starts from a disjunction p ∨ q and the negation of one of its parts, ¬p, to conclude the other, q. It is the reasoning of “ruling out options”: if there are only two possibilities and one is eliminated, the remaining one must hold. The formula ((p ∨ q) ∧ ¬p) ⇒ q is a tautology.
Example
((p ∨ q) ∧ ¬p) ⇒ q
What the variables mean
- ▸ p: “The package came by mail”
- ▸ q: “The package came by courier”
In plain words
The package came by mail or by courier. It did not come by mail. Therefore, it came by courier.
Truth table
| p | q | p ∨ q | ¬p | (p ∨ q) ∧ ¬p | ((p ∨ q) ∧ ¬p) ⇒ q★ |
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | F | T | F | F | T |
| F | T | T | T | T | T |
| F | F | F | T | F | T |
Classification: Tautology · 4 rows
Statement of the rule
Premise 1: p ∨ q. Premise 2: ¬p. Conclusion: q. By the symmetry of disjunction it also works with ¬q to conclude p.
In natural deduction it is known as the simple form of disjunction elimination: a disjunction plus the negation of one disjunct yields the other disjunct.
Why it is valid: reading the table
The conclusion q is F only in the rows where q = F. If additionally p = T (row p = T, q = F), the premise ¬p is F. If p = F (row p = F, q = F), the disjunction p ∨ q is F. Either way the conjunction of premises is F and the whole implication is T.
The only row where both premises are T is p = F, q = T: the disjunction is T thanks to q and ¬p is T. There the conclusion q is T. The argument never goes from true premises to a false conclusion.
How it is used in proofs
It appears whenever a proof by cases narrows the options: “x is positive or x is zero; x is not zero; therefore x is positive.”
Combined with material implication, (p ⇒ q) ⇔ (¬p ∨ q), it lets you rewrite modus ponens as a disjunctive syllogism: from ¬p ∨ q and p (that is, ¬¬p) you get q.
Examples
Everyday: “The keys are in the backpack or in the coat. They are not in the backpack. So they are in the coat.”
Programming: a two-case `switch` with no `default` uses this reasoning: if the value is A or B and we have checked it is not A, the branch that runs is B.
Mathematics: “An integer is either even or odd. 7 is not even. Therefore 7 is odd.”
Relation to other laws
It is a close cousin of modus tollens: both eliminate an option to reach a conclusion. The difference is that here the major premise is a disjunction rather than an implication.
It works whether the disjunction is inclusive or exclusive, because the argument only uses the fact that at least one part is true.
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 “tollendo ponens” mean? ▼
“By denying, it affirms”: one disjunct is denied in order to affirm the other. It is the counterpart of modus ponendo ponens.
Does it work with exclusive or? ▼
Yes. The rule only needs at least one of the two parts to be true, which both ∨ and ⊕ guarantee.
What if I deny q instead of p? ▼
You conclude p. Disjunction is commutative, so ((p ∨ q) ∧ ¬q) ⇒ p is a tautology as well.