In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, a truth function is a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 from a set of truth values to truth-values. Classically the domain and range

Range (mathematics)

In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...

 of a truth function are {truth,falsehood}, but they may have any number of truth-values, including an infinity of these.

A sentence is truth-functional if the truth-value of the sentence is a function of the truth-value of its subsentences. A class of sentences is truth-functional if each of its members is. For example, the sentence "Apples are fruits and carrots are vegetables" is truth-functional since it is true just in case each of its subsentences "apples are fruits" and "carrots are vegetables" is true, and it is false otherwise. Not all sentences of a natural language, such as English, are truth-functional.

Sentences of the form "x believes that..." are typical examples of sentences that are not truth-functional. Let us say that Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese. Then the sentence

"Mary believes that Al Gore was President of the USA on April 20, 2000"

is true while

"Mary believes that the moon is made of green cheese"

is false. In both cases, each component sentence (i.e. "Al Gore was president of the USA on April 20, 2000" and "the moon is made of green cheese") is false, but each compound sentence formed by prefixing the phrase "Mary believes that" differs in truth-value. That is, the truth-value of a sentence of the form "Mary believes that..." is not determined solely by the truth-value of its component sentence, and hence the (unary) connective

Logical connective

In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

 (or simply operator since it is unary) is non-truth-functional.

In classical logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

 a truth function is a compound proposition whose truth or falsity is unequivocally determined by the truth or falsity of its components for all cases, the class of its formulas (including sentences) is truth-functional since every sentential connective (e.g. &, →, etc.) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system

Formal system

In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

 whose formulas may be interpreted as either true or false.

See also

• Bertrand Russell
  Bertrand Russell
  Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
  
   and Alfred North Whitehead
  Alfred North Whitehead
  Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
  
  , Principia Mathematica
  Principia Mathematica
  The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
  
  , 2nd edition.
• Wittgenstein, Tractatus Logico-Philosophicus
  Tractatus Logico-Philosophicus
  The Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. It was an ambitious project: to identify the relationship between language and reality and to define the limits of science...
  
  , Proposition 5.101.
• Boolean function
• Boolean-valued function
  Boolean-valued function
  A boolean-valued function, in some usages is a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain....
  
  
• Binary function
  Binary function
  In mathematics, a binary function, or function of two variables, is a function which takes two inputs.Precisely stated, a function f is binary if there exists sets X, Y, Z such that\,f \colon X \times Y \rightarrow Z...
  
  
• Truth-functional propositional logic

Further reading

• Church, Alonzo
  Alonzo Church
  Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...
  
  (1944), Introduction to Mathematical Logic. See the Introduction for a history of the truth function concept.