# Disjunctive syllogism

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

In classical logic, **disjunctive syllogism**^{[1]}^{[2]} (historically known as * modus tollendo ponens* (

**MTP**),

^{[3]}Latin for "mode that affirms by denying")

^{[4]}is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

^{[5]}

^{[6]}

An example in English:

- 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.

## Contents

## Propositional logic[edit]

In propositional logic, **disjunctive syllogism** (also known as **disjunction elimination** and **or elimination**, or abbreviated **∨E**),^{[7]}^{[8]}^{[9]}^{[10]} 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 *P* is true 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. "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:

where the rule is that whenever instances of "", and "" appear on lines of a proof, "" 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, one of the three traditional laws of thought.

## Formal notation[edit]

The *disjunctive syllogism* rule may be written in sequent notation:

where is a metalogical symbol meaning that is a syntactic consequence of , and in some logical system;

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

where , and are propositions expressed in some formal system.

## Natural language examples[edit]

Here is an example:

- 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 red or it is blue.
- It is not blue.
- Therefore, it is red.

## Inclusive and exclusive disjunction[edit]

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:

- 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 (only) P or (only) Q.
- P.
- Therefore, not Q.

However, if the fact is true it does not commit the fallacy.

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

## Related argument forms[edit]

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 include:

Disjunctive syllogism holds in classical propositional logic and intuitionistic logic, but not in some paraconsistent logics.^{[11]}

## See also[edit]

## References[edit]

**^**Copi, Irving M.; Cohen, Carl (2005).*Introduction to Logic*. Prentice Hall. p. 362.**^**Hurley, Patrick (1991).*A Concise Introduction to Logic 4th edition*. Wadsworth Publishing. pp. 320–1.**^**Lemmon, Edward John. 2001.*Beginning Logic*. Taylor and Francis/CRC Press, p. 61.**^**Stone, Jon R. (1996).*Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language*. London: Routledge. p. 60. ISBN 0-415-91775-1.**^**Hurley**^**Copi and Cohen**^**Sanford, David Hawley. 2003.*If P, Then Q: Conditionals and the Foundations of Reasoning*. London, UK: Routledge: 39**^**Hurley**^**Copi and Cohen**^**Moore and Parker**^**Chris Mortensen, Inconsistent Mathematics,*Stanford encyclopedia of philosophy*, First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008