What is "disjunctive syllogism"

Disjunctive \Dis*junc"tive\, a. [L. disjunctivus: cf. F. disjonctif.]

1. Tending to disjoin; separating; disjoining.

2. (Mus.) Pertaining to disjunct tetrachords. ``Disjunctive notes.''
   --Moore (Encyc. of Music).

   Disjunctive conjunction (Gram.), one connecting grammatically two words or clauses, expressing at the same time an opposition or separation inherent in the notions or thoughts; as, either, or, neither, nor, but, although, except, lest, etc.

   Disjunctive proposition, a proposition in which the parts are connected by disjunctive conjunctions, specifying that one of two or more propositions may hold, but that no two propositions may hold at the same time; as it is either day or night.

   Disjunctive syllogism (Logic), one in which the major proposition is disjunctive; as, the earth moves in a circle or an ellipse; but in does not move in a circle, therefore it moves in an ellipse.

n. (label en logic) (l en A logical argument of the form that if there are only two possibility possibilities, and one of them is rule out ruled out, then the other must take place.) In symbols: $P\; vee\; Q,\; neg\; P\; vdash\; Q$

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 not a safety violation. Therefore, it is not subject to fines.

In propositional logic, disjunctive syllogism (also known as disjunction elimination, kneecapper's argument, 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:

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

where the rule is that whenever instances of "$P \or Q$", and "¬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. It is also related to the Law of noncontradiction and the Law of excluded middle, two of the three traditional laws of thought.