Possessing negative truth value
In
logic
,
false
[1]
or
untrue
is the state of possessing negative
truth value
and is a
nullary
logical connective
. In a
truth-functional
system of propositional logic, it is one of two postulated truth values, along with its
negation
,
truth
.
[2]
Usual notations of the false are
0
(especially in
Boolean logic
and
computer science
), O (in
prefix notation
, O
pq
), and the
up tack
symbol
.
[3]
[4]
Another approach is used for several
formal theories
(e.g.,
intuitionistic propositional calculus
), where a propositional constant (i.e. a nullary connective),
, is introduced, the truth value of which being always false in the sense above.
[5]
[6]
[7]
It can be treated as an absurd proposition, and is often called absurdity.
In classical logic and Boolean logic
[
edit
]
In
Boolean logic
, each variable denotes a
truth value
which can be either true (1), or false (0).
In a
classical
propositional calculus
, each
proposition
will be assigned a truth value of either true or false. Some systems of classical logic include dedicated symbols for false (0 or
), while others instead rely upon formulas such as
p
∧ ¬
p
and
¬(
p
→
p
)
.
In both Boolean logic and Classical logic systems, true and false are opposite with respect to
negation
; the negation of false gives true, and the negation of true gives false.
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true
|
false
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false
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true
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The negation of false is equivalent to the truth not only in classical logic and Boolean logic, but also in most other logical systems, as explained below.
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(
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False, negation and contradiction
[
edit
]
In most logical systems,
negation
,
material conditional
and false are related as:
- ¬
p
⇔ (
p
→ ⊥)
In fact, this is the definition of negation in some systems,
[8]
such as
intuitionistic logic
, and can be proven in propositional calculi where negation is a fundamental connective. Because
p
→
p
is usually a theorem or axiom, a consequence is that the negation of false (
¬ ⊥
) is true.
A
contradiction
is the situation that arises when a
statement
that is assumed to be true is shown to
entail
false (i.e.,
φ ? ⊥
). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from
? ¬φ
. A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the
Latin
term
falsum
being used in English to denote either, but false is one specific
proposition
.
Logical systems may or may not contain the
principle of explosion
(
ex falso quodlibet
in
Latin
),
⊥ ? φ
for all
φ
. By that principle, contradictions and false are equivalent, since each entails the other.
Consistency
[
edit
]
A
formal theory
using the "
" connective is defined to be consistent, if and only if the false is not among its
theorems
. In the absence of
propositional constants
, some substitutes (such as the ones
described above
) may be used instead to define consistency.
See also
[
edit
]
Wikiquote has quotations related to
Falsehood
.
References
[
edit
]
- ^
Its noun form is
falsity
.
- ^
Jennifer Fisher,
On the Philosophy of Logic
, Thomson Wadsworth, 2007,
ISBN
0-495-00888-5
,
p. 17.
- ^
Willard Van Orman Quine
,
Methods of Logic
, 4th ed, Harvard University Press, 1982,
ISBN
0-674-57176-2
,
p. 34.
- ^
"Truth-value | logic"
.
Encyclopedia Britannica
. Retrieved
2020-08-15
.
- ^
George Edward Hughes
and D.E. Londey,
The Elements of Formal Logic
, Methuen, 1965,
p. 151.
- ^
Leon Horsten and Richard Pettigrew,
Continuum Companion to Philosophical Logic
, Continuum International Publishing Group, 2011,
ISBN
1-4411-5423-X
,
p. 199.
- ^
Graham Priest
,
An Introduction to Non-Classical Logic: From If to Is
, 2nd ed, Cambridge University Press, 2008,
ISBN
0-521-85433-4
,
p. 105.
- ^
Dov M. Gabbay and Franz Guenthner (eds),
Handbook of Philosophical Logic, Volume 6
, 2nd ed, Springer, 2002,
ISBN
1-4020-0583-0
,
p. 12.
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Formal:
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Negation
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