Disjunctive syllogism

In classical logic disjunctive syllogism (historically known as modus tollendo ponens) is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

Either the breach is a safety violation, or it is not subject to fines.

The breach is a not safety violation.

Therefore, it is not subject to fines.

In propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid rule of inference. If we are told that at least one of two statements is true; and also told that it is not the former that is true; we can infer that it has to be the latter that is true. If either P or Q is true and P is false, then Q is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism, a three-step argument, and second, it contains a logical disjunction, which simply means an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts. The rule makes it possible to eliminate a disjunction from a logical proof. It is the rule that:

${\frac {P\lor Q,\neg P}{\therefore Q}}$

where the rule is that whenever instances of "$P\lor Q$", and "$\neg P$" appear on lines of a proof, "$Q$" can be placed on a subsequent line.

Disjunctive syllogism is closely related and similar to hypothetical syllogism, in that it is also type of syllogism, and also the name of a rule of inference.

Formal notation

The disjunctive syllogism rule may be written in sequent notation:

$P\lor Q,\lnot P\vdash Q$

where $\vdash$ is a metalogical symbol meaning that $Q$ is a syntactic consequence of $P\lor Q$, and $\lnot P$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

$((P\lor Q)\land \neg P)\to Q$

where $P$, and $Q$ are propositions expressed in some formal system.

Natural language examples

Here is an example:

Either I will choose soup or I will choose salad.

I will not choose soup.

Therefore, I will choose salad.

Here is another example:

It is either red or blue.

It is not blue.

Therefore, it is red.

Inclusive and exclusive disjunction

Please observe that the disjunctive syllogism works whether 'or' is considered 'exclusive' or 'inclusive' disjunction. See below for the definitions of these terms.

There are two kinds of logical disjunction:

• inclusive means "and/or" - at least one of them is true, or maybe both.
• exclusive ("xor") means exactly one must be true, but they cannot both be.

The widely used English language concept of or is often ambiguous between these two meanings, but the difference is pivotal in evaluating disjunctive arguments.

This argument:

Either P or Q.

Not P.

Therefore, Q.

is valid and indifferent between both meanings. However, only in the exclusive meaning is the following form valid:

Either P or Q (exclusive).

P.

Therefore, not Q.

With the inclusive meaning you could draw no conclusion from the first two premises of that argument. See affirming a disjunct.

Proof

Proposition	Derivation
$P\lor Q$	Given
$\neg P$	Given
$\neg P\rightarrow Q$	Material implication
$Q\,\!$	Modus ponens

Proposition	Derivation
$P\lor Q$	Given
$\neg Q$	Given
$\neg P\rightarrow Q$	Material implication
$\neg Q\rightarrow P$	Transposition
$P\,\!$	Modus ponens

Related argument forms

Unlike modus ponendo 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 (slightly devious) combination of reductio ad absurdum and disjunction elimination.

Other forms of syllogism:

• hypothetical syllogism
• categorical syllogism

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