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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, a truth function is a function that accepts
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''). Truth values are used in ...
s 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 The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
, wherein a compound statement is constructed using individual statements connected by
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...
s; 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. Classical propositional logic is a truth-functional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
), thus every compound statement is a truth function. On the other hand,
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
is non-truth-functional.


Overview

A
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...
is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "''and''" is truth-functional since a sentence like "''Apples are fruits and carrots are vegetables''" is true '' if, and only if,'' each of its sub-sentences "''apples are fruits''" and "''carrots are vegetables''" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional. Connectives of the form "x ''believes that'' ..." are typical examples of connectives that are not truth-functional. If e.g. 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 (or simply ''operator'' since it is unary) is non-truth-functional. The class of
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 c ...
connectives (e.g. &, ) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
s. Truth-functional propositional calculus is a
formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
whose formulae may be interpreted as either true or false.


Table of binary truth functions

In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs ''P'' and ''Q''. Any of these functions corresponds to a truth table of a certain
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...
in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood are denoted as 1 and 0, respectively, in the following truth tables for sake of brevity.


Functional completeness

Because a function may be expressed as a composition, a truth-functional logical calculus does not need to have dedicated symbols for all of the above-mentioned functions to be functionally complete. This is expressed in a
propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
as
logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending ...
of certain compound statements. For example, classical logic has equivalent to . The conditional operator "→" is therefore not necessary for a classical-based logical system if "¬" (not) and "∨" (or) are already in use. A minimal set of operators that can express every statement expressible in the
propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
is called a ''minimal functionally complete set''. A minimally complete set of operators is achieved by NAND alone and NOR alone . The following are the minimal functionally complete sets of operators whose arities do not exceed 2:Wernick, William (1942) "Complete Sets of Logical Functions," ''Transactions of the American Mathematical Society 51'': 117–32. In his list on the last page of the article, Wernick does not distinguish between ← and →, or between \nleftarrow and \nrightarrow. ;One element: , . ;Two elements: \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \. ;Three elements: \, \, \, \, \, \.


Algebraic properties

Some truth functions possess properties which may be expressed in the theorems containing the corresponding connective. Some of those properties that a binary truth function (or a corresponding logical connective) may have are: *''
associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
'': Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed. *'' commutativity'': The operands of the connective may be swapped without affecting the truth-value of the expression. *'' distributivity'': A connective denoted by · distributes over another connective denoted by +, if ''a'' · (''b'' + ''c'') = (''a'' · ''b'') + (''a'' · ''c'') for all operands ''a'', ''b'', ''c''. *'' idempotence'': Whenever the operands of the operation are the same, the connective gives the operand as the result. In other words, the operation is both truth-preserving and falsehood-preserving (see below). *'' absorption'': A pair of connectives \land, \lor satisfies the absorption law if a\land(a\lor b)=a\lor(a\land b)=a for all operands ''a'', ''b''. A set of truth functions is functionally complete if and only if for each of the following five properties it contains at least one member lacking it: *''
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
'': If ''f''(''a''1, ..., ''a''''n'') ≤ ''f''(''b''1, ..., ''b''''n'') for all ''a''1, ..., ''a''''n'', ''b''1, ..., ''b''''n'' ∈ such that ''a''1 ≤ ''b''1, ''a''2 ≤ ''b''2, ..., ''a''''n'' ≤ ''b''''n''. E.g., \vee, \wedge, \top, \bot. *''
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...
'': For each variable, changing its value either always or never changes the truth-value of the operation, for all fixed values of all other variables. E.g., \neg, \leftrightarrow, \not\leftrightarrow, \top, \bot. *''self dual'': To read the truth-value assignments for the operation from top to bottom on its
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
is the same as taking the complement of reading it from bottom to top; in other words, ''f''(¬''a''1, ..., ¬''a''''n'') = ¬''f''(''a''1, ..., ''a''''n''). E.g., \neg. *''truth-preserving'': The interpretation under which all variables are 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''). Truth values are used in ...
of ''true'' produces a truth value of ''true'' as a result of these operations. E.g., \vee, \wedge, \top, \rightarrow, \leftrightarrow, \subset. (see validity) *''falsehood-preserving'': The interpretation under which all variables are 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''). Truth values are used in ...
of ''false'' produces a truth value of ''false'' as a result of these operations. E.g., \vee, \wedge, \nleftrightarrow, \bot, \not\subset, \not\supset. (see validity)


Arity

A concrete function may be also referred to as an ''operator''. In two-valued logic there are 2 nullary operators (constants), 4 unary operators, 16 binary operators, 256 ternary operators, and 2^ ''n''-ary operators. In three-valued logic there are 3 nullary operators (constants), 27 unary operators, 19683 binary operators, 7625597484987 ternary operators, and 3^ ''n''-ary operators. In ''k''-valued logic, there are ''k'' nullary operators, k^k unary operators, k^ binary operators, k^ ternary operators, and k^ ''n''-ary operators. An ''n''-ary operator in ''k''-valued logic is a function from \mathbb_k^n \to \mathbb_k. Therefore, the number of such operators is , \mathbb_k, ^ = k^, which is how the above numbers were derived. However, some of the operators of a particular arity are actually degenerate forms that perform a lower-arity operation on some of the inputs and ignore the rest of the inputs. Out of the 256 ternary Boolean operators cited above, \binom\cdot 16 - \binom\cdot 4 + \binom\cdot 2 of them are such degenerate forms of binary or lower-arity operators, using the inclusion–exclusion principle. The ternary operator f(x,y,z)=\lnot x is one such operator which is actually a unary operator applied to one input, and ignoring the other two inputs. "Not" is a unary operator, it takes a single term (¬''P''). The rest are binary operators, taking two terms to make a compound statement (). The set of logical operators may be partitioned into disjoint subsets as follows: ::: \Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m \,. In this partition, \Omega_j is the set of operator symbols of ''
arity In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...
'' . In the more familiar propositional calculi, \Omega is typically partitioned as follows: :::nullary operators: \Omega_0 = \ :::unary operators: \Omega_1 = \ :::binary operators: \Omega_2 \supset \


Principle of compositionality

Instead of using
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
s, logical connective symbols can be interpreted by means of an interpretation function and a functionally complete set of truth-functions (Gamut 1991), as detailed by the principle of compositionality of meaning. Let be an interpretation function, let be any two sentences and let the truth function ''f''nand be defined as: * ''f''nand(T,T) = F; ''f''nand(T,F) = ''f''nand(F,T) = ''f''nand(F,F) = T Then, for convenience, ''f''not, ''f''or ''f''and and so on are defined by means of ''f''nand: * ''f''not(''x'') = ''f''nand(''x'',''x'') * ''f''or(''x'',''y'') = ''f''nand(''f''not(''x''), ''f''not(''y'')) * ''f''and(''x'',''y'') = ''f''not(''f''nand(''x'',''y'')) or, alternatively ''f''not, ''f''or ''f''and and so on are defined directly: * ''f''not(T) = F; ''f''not(F) = T; * ''f''or(T,T) = ''f''or(T,F) = ''f''or(F,T) = T; ''f''or(F,F) = F * ''f''and(T,T) = T; ''f''and(T,F) = ''f''and(F,T) = ''f''and(F,F) = F Then etc. Thus if ''S'' is a sentence that is a string of symbols consisting of logical symbols ''v''1...''v''''n'' representing logical connectives, and non-logical symbols ''c''1...''c''''n'', then if and only if have been provided interpreting ''v''1 to ''v''''n'' by means of ''f''nand (or any other set of functional complete truth-functions) then the truth-value of is determined entirely by the truth-values of ''c''1...''c''''n'', i.e. of . In other words, as expected and required, ''S'' is true or false only under an interpretation of all its non-logical symbols.


Definition

Using the functions defined above, we can give a formal definition of a proposition's truth function. Let ''PROP'' be the set of all propositional variables, : PROP = \ We define a truth assignment to be any function \phi:PROP\to \. A truth assignment is therefore an association of each propositional variable with a particular truth value. This is effectively the same as a particular row of a proposition's truth table. For a truth assignment, \phi, we define its extended truth assignment, \overline\phi, as follows. This extends \phi to a new function \overline \phi which has domain equal to the set of all propositional formulas. The range of \overline\phi is still \. # If A \in PROP then \overline\phi(A) = \phi(A). # If ''A'' and ''B'' are any propositional formulas, then ## \overline\phi(\neg A) = f_ (\overline\phi(A)). ## \overline\phi(A\land B) = f_ (\overline\phi(A),\overline\phi(B)). ## \overline\phi(A\lor B) = f_ (\overline\phi(A),\overline\phi(B)). ## \overline\phi(A\to B) = \overline\phi(\neg A\lor B). ## \overline\phi(A\leftrightarrow B) = \overline\phi((A\to B)\land (B\to A)). Finally, now that we have defined the extended truth assignment, we can use this to define the truth-function of a proposition. For a proposition, ''A'', its truth function, f_A, has domain equal to the set of all truth assignments, and range equal to \. It is defined, for each truth assignment \phi, by f_A(\phi) = \overline\phi(A). The value given by \overline\phi(A) is the same as the one displayed in the final column of the truth table of ''A'', on the row identified with \phi.


Computer science

Logical operators are implemented as
logic gate A logic gate is a device that performs a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, one that has, for ...
s in
digital circuit In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematica ...
s. Practically all digital circuits (the major exception is DRAM) are built up from NAND, NOR, NOT, and transmission gates. NAND and NOR gates with 3 or more inputs rather than the usual 2 inputs are fairly common, although they are logically equivalent to a cascade of 2-input gates. All other operators are implemented by breaking them down into a logically equivalent combination of 2 or more of the above logic gates. The "logical equivalence" of "NAND alone", "NOR alone", and "NOT and AND" is similar to Turing equivalence. The fact that all truth functions can be expressed with NOR alone is demonstrated by the Apollo guidance computer.


See also

*
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
and
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...
,
''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
'', 2nd edition *
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...
,
''
Tractatus Logico-Philosophicus The ''Tractatus Logico-Philosophicus'' (widely abbreviated and Citation, cited as TLP) is the only book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein that was published during his lifetime. The project had a broad goal ...
'', Proposition 5.101 *
Bitwise operation In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operatio ...
* Binary function *
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
*
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
* Boolean-valued function * List of Boolean algebra topics * Logical constant * Modal operator *
Propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
* Truth-functional propositional logic


Notes


References

*


Further reading

*
Józef Maria Bocheński Józef Maria Bocheński or Innocentius Bochenski (30 August 1902 – 8 February 1995) was a Polish Dominican, logician and philosopher. Biography Bocheński was born on 30 August 1902 in Czuszów, then part of the Russian Empire, to a fami ...
(1959), ''A Précis of Mathematical Logic'', translated from the French and German versions by Otto Bird, Dordrecht, South Holland: D. Reidel. *
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
(1944), ''Introduction to Mathematical Logic'', Princeton, NJ: Princeton University Press. See the Introduction for a history of the truth function concept. {{Logical truth Mathematical logic Logical truth