Argument whose conclusion must be true if its premises are
In
logic
, specifically in
deductive reasoning
, an
argument
is
valid
if and only if
it takes a form that makes it impossible for the
premises
to be
true
and the conclusion nevertheless to be
false
.
[1]
It is not required for a valid argument to have premises that are actually true,
[2]
but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called
well-formed formulas
(also called
wffs
or simply
formulas
).
The
validity
of an argument can be tested, proved or disproved, and depends on its
logical form
.
[3]
Arguments
[
edit
]
In logic, an
argument
is a set of statements expressing the
premises
(whatever consists of empirical evidences and axiomatic truths) and an
evidence-based conclusion.
An argument is
valid
if and only if it would be contradictory for the conclusion to be false if all of the premises are true.
[3]
Validity does not require the truth of the premises, instead it merely
necessitates
that conclusion follows from the formers without violating the correctness of the
logical form
. If also the premises of a valid argument are proven true, this is said to be
sound
.
[3]
The
corresponding conditional
of a valid argument is a
logical truth
and the negation of its corresponding conditional is a
contradiction
. The conclusion is a
logical consequence
of its premises.
An argument that is not valid is said to be "invalid".
An example of a valid (and
sound
) argument is given by the following well-known
syllogism
:
- All men are mortal. (
True
)
- Socrates is a man. (
True
)
- Therefore, Socrates is mortal. (
True
)
What makes this a valid argument is not that it has true premises and a true conclusion, but the logical necessity of the conclusion, given the two premises. The argument would be just as valid were the premises and conclusion false. The following argument is of the same
logical form
but with false premises and a false conclusion, and it is equally valid:
- All cups are green. (
False
)
- Socrates is a cup. (
False
)
- Therefore, Socrates is green. (
False
)
No matter how the universe might be constructed, it could never be the case that these arguments should turn out to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with the following invalid one:
- All men are immortal. (
False
)
- Socrates is a man. (
True
)
- Therefore, Socrates is mortal. (
True
)
In this case, the conclusion contradicts the deductive logic of the preceding premises, rather than deriving from it. Therefore, the argument is logically 'invalid', even though the conclusion could be considered 'true' in general terms. The premise 'All men are immortal' would likewise be deemed false outside of the framework of classical logic. However, within that system 'true' and 'false' essentially function more like mathematical states such as binary 1s and 0s than the philosophical concepts normally associated with those terms.
A standard view is that whether an argument is valid is a matter of the argument's logical form. Many techniques are employed by logicians to represent an argument's logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:
- All P are Q.
- S is a P.
- Therefore, S is a Q.
Similarly, the third argument becomes:
- All P's are not Q.
- S is a P.
- Therefore, S is a Q.
An argument is termed formally valid if it has structural self-consistency, i.e. if when the operands between premises are all true, the derived conclusion is always also true. In the third example, the initial premises cannot logically result in the conclusion and is therefore categorized as an invalid argument.
Valid formula
[
edit
]
A formula of a
formal language
is a valid formula if and only if it is true under every possible
interpretation
of the language. In propositional logic, they are
tautologies
.
Statements
[
edit
]
A statement can be called valid, i.e. logical truth, if it is true in all interpretations.
Soundness
[
edit
]
Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:
- All animals live on Mars. (
False
)
- All humans are animals. (
True
)
- Therefore, all humans live on Mars. (
False
)
The problem with the argument is that it is not
sound
. In order for a deductive argument to be sound, the argument must be valid
and
all the premises must be true.
[3]
Satisfiability
[
edit
]
Model theory
analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, formula is valid if all such interpretations make it true. An inference is valid if all interpretations that validate the premises validate the conclusion. This is known as
semantic validity
.
[4]
Preservation
[
edit
]
In
truth-preserving
validity, the interpretation under which all variables are assigned a
truth value
of 'true' produces a truth value of 'true'.
In a
false-preserving
validity, the interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false'.
[5]
See also
[
edit
]
References
[
edit
]
- ^
Validity and Soundness ? Internet Encyclopedia of Philosophy
- ^
Jc Beall and Greg Restall,
"Logical Consequence"
, The Stanford Encyclopedia of Philosophy (Fall 2014 Edition).
- ^
a
b
c
d
Gensler, Harry J. (January 6, 2017).
Introduction to logic
(Third ed.). New York: Routledge.
ISBN
978-1-138-91058-4
.
OCLC
957680480
.
- ^
L. T. F. Gamut
,
Logic, Language, and Meaning: Introduction to Logic
, University of Chicago Press, 1991, p. 115.
- ^
Robert Cogan,
Critical Thinking: Step by Step
, University Press of America, 1998,
p. 48
.
Further reading
[
edit
]
Wiktionary has definitions related to
Validity
.
|
---|
|
Major fields
|
|
---|
Foundations
| |
---|
Lists
| |
---|
|