Disjunctive syllogism
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A
disjunctive syllogism, historically known as
modus tollendo ponens,
(1) is a
classically valid, simple
argument form:
A is B or C or D
A is not C or D
Therefore, A is B
In
logical operator notation:
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.
Therefore, I will choose salad.
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
-
[Sanford, David Hawley. 2003. If P, Then Q: Conditionals and the Foundations of Reasoning. London, UK: Routledge: 39]
-
[Chris Mortensen, Inconsistent Mathematics, Stanford encyclopedia of philosophy, First published Tue Jul 2, 1996; substantive revision Thu Jul 31, 2008]
Modus tollendo ponensModus tollendo ponensJákvæð neitunarregla選言三段論法Silogismo disjuntivo选言三段论
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