Disjunctive syllogism

A disjunctive syllogism, also known as disjunction-elimination and or-elimination (∨E), and historically known as modus tollendo ponens,[1], is a classically valid, simple argument form:

A is B or C
A is not C
Therefore, A is B

In logical operator notation:

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

where $\vdash$ represents the logical assertion.

Roughly speaking, we are told that at least one of two statements is true; then we are told that it is not the former that is true; so we infer that it has to be the latter that 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 disjunction, which means simply an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts.

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

Here is an example:

Either I will choose soup or I will choose salad.
I will not choose soup.

Here is another example:

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

## Inclusive versus exclusive disjunction

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.

## 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:

Disjunctive syllogism holds in classical propositional logic and intuitionistic logic, but not in some paraconsistent logics.[2]

## References

1. ^ Sanford, David Hawley. 2003. If P, Then Q: Conditionals and the Foundations of Reasoning. London, UK: Routledge: 39
2. ^ Chris Mortensen, Inconsistent Mathematics, Stanford encyclopedia of philosophy, First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008

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### Look at other dictionaries:

• Disjunctive syllogism — Disjunctive Dis*junc tive, a. [L. disjunctivus: cf. F. disjonctif.] 1. Tending to disjoin; separating; disjoining. [1913 Webster] 2. (Mus.) Pertaining to disjunct tetrachords. Disjunctive notes. Moore (Encyc. of Music). [1913 Webster]… …   The Collaborative International Dictionary of English

• Disjunctive — Dis*junc tive, a. [L. disjunctivus: cf. F. disjonctif.] 1. Tending to disjoin; separating; disjoining. [1913 Webster] 2. (Mus.) Pertaining to disjunct tetrachords. Disjunctive notes. Moore (Encyc. of Music). [1913 Webster] {Disjunctive… …   The Collaborative International Dictionary of English

• Disjunctive conjunction — Disjunctive Dis*junc tive, a. [L. disjunctivus: cf. F. disjonctif.] 1. Tending to disjoin; separating; disjoining. [1913 Webster] 2. (Mus.) Pertaining to disjunct tetrachords. Disjunctive notes. Moore (Encyc. of Music). [1913 Webster]… …   The Collaborative International Dictionary of English

• Disjunctive proposition — Disjunctive Dis*junc tive, a. [L. disjunctivus: cf. F. disjonctif.] 1. Tending to disjoin; separating; disjoining. [1913 Webster] 2. (Mus.) Pertaining to disjunct tetrachords. Disjunctive notes. Moore (Encyc. of Music). [1913 Webster]… …   The Collaborative International Dictionary of English

• Syllogism — A syllogism (Greek: συλλογισμός – syllogismos – conclusion, inference ) is a kind of logical argument in which one proposition (the conclusion) is inferred from two or more others (the premises) of a certain form. In antiquity, there were… …   Wikipedia

• syllogism — /ˈsɪlədʒɪzəm / (say siluhjizuhm) noun 1. Logic an argument with two premises and a conclusion. Both the premises of a categorical syllogism are categorical propositions, containing just three distinct terms between them, e.g. all men are mortal… …   Australian English dictionary

• disjunctive — dis•junc•tive [[t]dɪsˈdʒʌŋk tɪv[/t]] adj. 1) serving or tending to disjoin 2) gram. a) syntactically setting two or more expressions in opposition to each other, as but in poor but happy, or expressing an alternative, as or in this or that[/ex]… …   From formal English to slang

• disjunctive — disjunctively, adv. /dis jungk tiv/, adj. 1. serving or tending to disjoin; separating; dividing; distinguishing. 2. Gram. a. syntactically setting two or more expressions in opposition to each other, as but in poor but happy, or expressing an… …   Universalium

• Hypothetical syllogism — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) …   Wikipedia

• Paraconsistent logic — A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or… …   Wikipedia