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.
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 is true while 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 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 tables.
Truth-functional propositional calculus is a formal system whose formulae may be interpreted as either true or false.
Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments.
This is expressed in a propositional calculus as logical equivalence of certain compound statements.
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 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:[5] Some truth functions possess properties which may be expressed in the theorems containing the corresponding connective.
An n-ary operator in k-valued logic is a function from
of them are such degenerate forms of binary or lower-arity operators, using the inclusion–exclusion principle.
"Not" is a unary operator, it takes a single term (¬P).
The rest are binary operators, taking two terms to make a compound statement (P ∧ Q, P ∨ Q, P → Q, P ↔ Q).
is typically partitioned as follows: 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 fnand be defined as: Then, for convenience, fnot, for fand and so on are defined by means of fnand: or, alternatively fnot, for fand 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...vn representing logical connectives, and non-logical symbols c1...cn, then if and only if I(v1)...I(vn) have been provided interpreting v1 to vn by means of fnand (or any other set of functional complete truth-functions) then the truth-value of
In other words, as expected and required, S is true or false only under an interpretation of all its non-logical symbols.
Practically all digital circuits (the major exception is DRAM) are built up from NAND, NOR, NOT, and transmission gates.
The fact that all truth functions can be expressed with NOR alone is demonstrated by the Apollo guidance computer.