Disjunctive syllogism

Formal notation

For a logical system that validates it, the disjunctive syllogism may be written in sequent notation as $${\displaystyle P\vee Q,\mathrm{\neg }P⊢Q}$$ where ${\displaystyle ⊢}$ is a metalogical symbol meaning that ${\displaystyle Q}$ is a syntactic consequence of ${\displaystyle P\vee Q}$, and ${\displaystyle \mathrm{\neg }P}$.

It may be expressed as a truth-functional tautology or theorem in the object language of propositional logic as $${\displaystyle ((P\vee Q)\wedge \mathrm{\neg }P)\to Q}$$ where ${\displaystyle P}$, and ${\displaystyle Q}$ are propositions expressed in some formal system.

Natural language examples

Here is an example:

1. It is red or it is blue.
2. It is not blue.
3. Therefore, it is red.

Here is another example:

1. The breach is a safety violation, or it is not subject to fines.
2. The breach is not a safety violation.
3. Therefore, it is not subject to fines.

Strong form

Modus tollendo ponens can be made stronger by using exclusive disjunction instead of inclusive disjunction as a premise: $${\displaystyle \frac{P\underset{\_}{\vee }Q,\mathrm{\neg }P}{\therefore Q}}$$

Related argument forms

Unlike modus ponens and modus ponendo tollens, with which it should not be confused, disjunctive syllogism is often not made an explicit rule or axiom of logical systems, as the above arguments can be proven with a combination of reductio ad absurdum and disjunction elimination.

Other forms of syllogism include:

• hypothetical syllogism
• categorical syllogism

Disjunctive syllogism holds in classical propositional logic and intuitionistic logic, but not in some paraconsistent logics.