In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. [1] It is sometimes said that a statement is vacuously true because it does not really say anything. [2] For example, the statement "all cell phones in the room are turned off" will be true when no cell phones are present in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on and turned off", which would otherwise be incoherent and false.
More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. [1] [3] [2] [4] One example of such a statement is "if Tokyo is in Spain, 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. In essence, a conditional statement, that is based on the material conditional, is true when the antecedent ("Tokyo is in Spain" in the example) is false regardless of whether the conclusion or consequent ("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, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction. [5] This notion has relevance in pure mathematics, as well as in any other field that uses classical logic.
Outside of mathematics, statements in the form of a vacuous truth, while logically valid, can nevertheless 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.
A statement is "vacuously true" if it resembles a material conditional statement , where the antecedent is known to be false. [1] [3] [2]
Vacuously true statements that can be reduced (with suitable transformations) to this basic form (material conditional) include the following universally quantified statements:
Vacuous truths most commonly appear in classical logic with two truth values. However, vacuous truths can also appear in, for example, intuitionistic logic, 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; 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, may attempt to avoid vacuous truths by using alternative conditionals (such as the case of the counterfactual conditional).
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:
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. [6] 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. [7] 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. [8] These examples, one from mathematics and one from natural language, illustrate the concept of vacuous truths:
First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In logic, a logical connective is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective can be used to join the two atomic formulas and , rendering the complex formula .
In propositional logic, modus ponens, also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore, Q must also be true."
In propositional logic, modus tollens (MT), also known as modus tollendo tollens and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. Either way, the truth of the converse is generally independent from that of the original statement.
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", standing for " is not true", written , or . It is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P. Similarly, P is sufficient for Q, because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement " if and only if ", where is known as the antecedent, and the consequent.
In propositional logic, material implication is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- or and that either form can replace the other in logical proofs. In other words, if is true, then must also be true, while if is not true, then cannot be true either; additionally, when is not true, may be either true or false.
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than to David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems in a first-order language rather than conditional tautologies.
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic, it provides a canonical normal form useful in automated theorem proving.
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.
The material conditional is an operation commonly used in logic. When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction.
Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Avicenna, Ockham, and Duns Scotus developed many of their observations, it was C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Kripke.
The drinker paradox is a theorem of classical predicate logic that can be stated as "There is someone in the pub such that, if he or she is drinking, then everyone in the pub is drinking." It was popularised by the mathematical logician Raymond Smullyan, who called it the "drinking principle" in his 1978 book What Is the Name of this Book?
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below.