Truth function

In logic, a truth function^[1] 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.^[2]

Classical propositional logic is a truth-functional logic,^[3] 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), thus every compound statement is a truth function.^[4] On the other hand, modal logic is non-truth-functional.

: 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.

associativity

: The operands of the connective may be swapped without affecting the truth-value of the expression.

commutativity

: A connective denoted by · distributes over another connective denoted by +, if a · (b + c) = (a · b) + (a · c) for all operands a, b, c.

distributivity

: 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).

idempotence

: 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.

absorption

f_nand(T,T) = F; f_nand(T,F) = f_nand(F,T) = f_nand(F,F) = T

Instead of using truth tables, 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 I be an interpretation function, let Φ, Ψ be any two sentences and let the truth function f_nand be defined as:

Then, for convenience, f_not, f_or f_and and so on are defined by means of f_nand:

or, alternatively f_not, f_or f_and and so on are defined directly:

Then

etc.

Thus if S is a sentence that is a string of symbols consisting of logical symbols v₁...v_n representing logical connectives, and non-logical symbols c₁...c_n, then if and only if $I (v 1)... I (v n)$ have been provided interpreting v₁ to v_n by means of f_nand (or any other set of functional complete truth-functions) then the truth-value of $I(s)$ is determined entirely by the truth-values of c₁...c_n, i.e. of $I (c 1)... I (c n)$ . In other words, as expected and required, S is true or false only under an interpretation of all its non-logical symbols.

Computer science[edit]

Logical operators are implemented as logic gates in digital circuits. 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.

This article incorporates material from TruthFunction on , which is licensed under the Creative Commons Attribution/Share-Alike License.

PlanetMath

(1959), A Précis of Mathematical Logic, translated from the French and German versions by Otto Bird, Dordrecht, South Holland: D. Reidel.

Józef Maria Bocheński

(1944), Introduction to Mathematical Logic, Princeton, NJ: Princeton University Press. See the Introduction for a history of the truth function concept.