Disjunctive syllogism

Disjunctive syllogism

Type	Rule of inference
Field	Propositional calculus
Statement	If ${\displaystyle P}$ is true or ${\displaystyle Q}$ is true, and ${\displaystyle P}$ is false, then ${\displaystyle Q}$ is true.
Symbolic statement	${\displaystyle {\frac {P\lor Q,\neg P}{\therefore Q}}}$

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:

1. I will choose soup or I will choose salad.
2. I will not choose soup.
3. Therefore, I will choose salad.

Propositional logic

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 it is known that at least one of two statements is true, and that it is not the former that is true; we can infer that it has to be the latter that is true. Equivalently, if P is true or Q is true and P is false, then Q is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where 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

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

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

Disjunctive syllogism is closely related and similar to hypothetical syllogism, which is another rule of inference involving a syllogism. It is also related to the law of noncontradiction, one of the three traditional laws of thought.

Formal notation

For a logical system that validates it, the disjunctive syllogism may be written in sequent notation as

${\displaystyle P\lor Q,\lnot P\vdash Q}$

where ${\displaystyle \vdash }$ is a metalogical symbol meaning that ${\displaystyle Q}$ is a syntactic consequence of ${\displaystyle P\lor Q}$, and ${\displaystyle \lnot P}$.

It may be expressed as a truth-functional tautology or theorem in the object language of propositional logic as

${\displaystyle ((P\lor Q)\land \neg P)\to Q}$

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:

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

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. ^[11]

See also

• Stoic logic
• Type of syllogism (disjunctive, hypothetical, legal, poly-, prosleptic, quasi-, statistical)

References

1. Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 362.
2. Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition . Wadsworth Publishing. pp. 320–1. ISBN   9780534145156.
3. Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.
4. Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language . London: Routledge. p.  60. ISBN   0-415-91775-1.
5. Hurley
6. Copi and Cohen
7. Sanford, David Hawley. 2003. If P, Then Q: Conditionals and the Foundations of Reasoning. London, UK: Routledge: 39
8. Hurley
9. Copi and Cohen
10. Moore and Parker
11. Chris Mortensen, Inconsistent Mathematics, Stanford encyclopedia of philosophy, First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008