In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, a vacuous truth is a
conditional
Conditional (if then) may refer to:
*Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a co ...
or
universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a t ...
statement (a universal statement that can be converted to a conditional statement) that is true because the
antecedent cannot be
satisfied.
For example, the statement "she does not own a cell phone" will imply that the statement "all of her cell phones are turned off" will be assigned a
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
. Also, the statement "all of her cell phones are turned ''on''" would also be vacuously true, as would the
conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
of the two: "all of her cell phones are turned on ''and'' turned off", which would otherwise be incoherent and false. For that reason, it is sometimes said that a statement is vacuously true because it is meaningless.
More formally, a relatively
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
usage refers to a conditional statement (or a universal conditional statement) with a false
antecedent.
One example of such a statement is "if Tokyo is in France, then the Eiffel Tower is in Bolivia".
Such statements are considered vacuous truths, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the
consequent
A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
. In essence, a conditional statement, that is based on the
material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...
, is true when the antecedent ("Tokyo is in France" in the example) is false regardless of whether the conclusion or
consequent
A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in that way.
Examples common to everyday speech include conditional phrases used as
idioms of improbability like "when hell freezes over..." and "when pigs can fly...", indicating that not before the given (impossible) condition is met will the speaker accept some respective (typically false or absurd) proposition.
In
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
. This notion has relevance in
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, as well as in any other field that uses
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
.
Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about
qualified objects which
do not actually exist. For example, a child might truthfully tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In this case, the parent can believe that the child has actually eaten some vegetables, even though that is not true. In addition, a vacuous truth is often used colloquially with absurd statements, either to confidently assert something (e.g. "the dog was red, or I'm a monkey's uncle" to strongly claim that the dog was red), or to express doubt, sarcasm, disbelief, incredulity or indignation (e.g. "yes, and I'm the King of England" to disagree a previously made statement).
Scope of the concept
A statement
is "vacuously true" if it
resembles a
material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...
statement
, where the
antecedent is known to be false.
Vacuously true statements that can be reduced (
with suitable transformations) to this basic form (material conditional) include the following
universally quantified
In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pr ...
statements:
*
, where it is the case that
.
*
, where the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is
empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
.
**This logical form
can be converted to the material conditional form in order to easily identify the
antecedent. For the above example
"all cell phones in the room are turned off", it can be formally written as
where
is the set of all cell phones in the room and
is "
is turned off". This can be written to a material conditional statement
where
is the set of all things in the room (including cell phones if they exist in the room), the antecedent
is "
is a cell phone", and the
consequent
A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
is "
is turned off".
*
, where the symbol
is restricted to a
type that has no representatives.
Vacuous truths most commonly appear in
classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this class ...
with
two truth values. However, vacuous truths can also appear in, for example,
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, in the same situations as given above. Indeed, if
is false, then
will yield a vacuous truth in any logic that uses the
material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...
; if
is a
necessary falsehood, then it will also yield a vacuous truth under the
strict conditional.
Other non-classical logics, such as
relevance logic Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. It is generally, but ...
, may attempt to avoid vacuous truths by using alternative conditionals (such as the case of the
counterfactual conditional
Counterfactual conditionals (also ''subjunctive'' or ''X-marked'') are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactua ...
).
In computer programming
Many programming environments have a mechanism for querying if every item in a collection of items satisfies some predicate. It is common for such a query to always evaluate as true for an empty collection. For example:
* In
JavaScript
JavaScript (), often abbreviated as JS, is a programming language that is one of the core technologies of the World Wide Web, alongside HTML and CSS. As of 2022, 98% of websites use JavaScript on the client side for webpage behavior, of ...
, the
array
An array is a systematic arrangement of similar objects, usually in rows and columns.
Things called an array include:
{{TOC right
Music
* In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
method
every
executes a provided callback function once for each element present in the array, only stopping (if and when) it finds an element where the callback function returns false. Notably, calling the
every
method on an empty array will return true for any condition.
* In
Python, the
all
function returns
True
if all of the elements of the given iterable are
True
. The function also returns
True
when given an iterable of zero length.
* In
Rust
Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxide-hydroxide (FeO( ...
, the
Iterator::all
function accepts an iterator and a predicate and returns
true
only when the predicate returns
true
for all items produced by the iterator, or if the iterator produces no items.
Examples
These examples, one from
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and one from
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...
, illustrate the concept of vacuous truths:
* "For any integer x, if x > 5 then x > 3."
– This statement is
true
True most commonly refers to truth, the state of being in congruence with fact or reality.
True may also refer to:
Places
* True, West Virginia, an unincorporated community in the United States
* True, Wisconsin, a town in the United States
* ...
non-vacuously (since some
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s are indeed greater than 5), but some of its implications are only vacuously true: for example, when x is the integer 2, the statement implies the vacuous truth that "if 2 > 5 then 2 > 3".
* "All my children are goats" is a vacuous truth, when spoken by someone without children. Similarly, "None of my children are goats" would also be a vacuous truth, when spoken by someone without children (possibly the same person).
See also
*
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
– specifically the law that a universal statement is true just in case no counterexample exists:
*
Empty sum
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.
The natural way to extend non-empty sums is to let the empty sum be the additive identity.
Let a_1, a_2, a_3, ... be a sequence of numbers, and let
...
and
empty product
*
Empty function
*
Paradoxes of material implication
The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formu ...
, especially the
principle of explosion
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a ...
*
Presupposition
In the branch of linguistics known as pragmatics, a presupposition (or PSP) is an implicit assumption about the world or background belief relating to an utterance whose truth is taken for granted in discourse. Examples of presuppositions include ...
,
double question
*
State of affairs (philosophy) In philosophy, a state of affairs (german: Sachverhalt), also known as a situation, is a way the actual world must be in order to make some given ''proposition'' about the actual world true; in other words, a state of affairs is a ''truth-maker'', w ...
*
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always ...
– another type of true statement that also fails to convey any substantive information
*
Triviality (mathematics) and
degeneracy (mathematics)
References
Bibliography
* Blackburn, Simon (1994). "vacuous," ''
The Oxford Dictionary of Philosophy
''The Oxford Dictionary of Philosophy'' (1994; second edition 2008; third edition 2016) is a dictionary of philosophy by the philosopher Simon Blackburn, published by Oxford University Press.
References
* Blackburn, Simon ( 0052008), 2nd re ...
''. Oxford: Oxford University Press, p. 388.
*
David H. Sanford
David H. Sanford (born 1937-2022) was a professor of philosophy at Duke University. He specializes in perception and metaphysics.
Sanford studied at Cass Technical High School, Oberlin College and at Wayne State University. He received his Ph ...
(1999). "implication." ''
The Cambridge Dictionary of Philosophy'', 2nd. ed., p. 420.
* {{cite conference , last1=Beer , first1=Ilan , last2=Ben-David , first2=Shoham , last3=Eisner , first3=Cindy , last4=Rodeh , first4=Yoav , contribution=Efficient Detection of Vacuity in ACTL Formulas , year=1997, title=Computer Aided Verification: 9th International Conference, CAV'97 Haifa, Israel, June 22–25, 1997, Proceedings , series=
Lecture Notes in Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973.
Overview
The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorial ...
, volume=1254 , pages=279–290 , url=http://citeseer.ist.psu.edu/beer97efficient.html , doi=10.1007/3-540-63166-6_28, isbn=978-3-540-63166-8, doi-access=free
External links
Conditional Assertions: Vacuous truth
Logic
Mathematical logic
Truth
Informal fallacies