In logic, a logical connective (also called a logical operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) 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 function, called a truth function. For this reason, logical connectives are sometimes called truthfunctional connectives. The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences whose truth values can be thought of as the function's operands. Also commonly, negation is considered to be a unary connective.
Logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic.
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In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a grammatically compound sentence. Some but not all such grammatical conjunctions are truth functions. For example, consider the following sentences:
The words and and so are grammatical conjunctions joining the sentences (A) and (B) to form the compound sentences (C) and (D). The and in (C) is a logical connective, since the truth of (C) is completely determined by (A) and (B): it would make no sense to affirm (A) and (B) but deny (C). However so in (D) is not a logical connective, since it would be quite reasonable to affirm (A) and (B) but deny (D): perhaps, after all, Jill went up the hill to fetch a pail of water, not because Jack had gone up the Hill at all.
Various English words and word pairs express truth functions, and some of them are synonymous. Examples (with the name of the relationship in parentheses) are:
The word "not" (negation) and the phrases "it is false that" (negation) and "it is not the case that" (negation) also express a logical connective – even though they are applied to a single statement, and do not connect two statements.
In formal languages, truth functions are represented by unambiguous symbols; these can be exactly defined by means of truth tables. There are 16 binary truth tables, and so 16 different logical connectives which connect exactly two statements, that can be defined. Not all of them are in common use. These symbols are called "truthfunctional connectives", "logical connectives", "logical operators" or "propositional operators". See wellformed formula for the rules which allow new wellformed formulas to be constructed by joining other wellformed formulas using truthfunctional connectives.
Venn diagrams illustrate the logical connective limitation of all quantifiers to a fixed domain of discourse in a formal language.
Logical connectives can be used to link more than two statements. A more technical definition is that an "nary logical connective" is a function which assigns truth values "true" or "false" to ntuples of truth values.
Commonly used logical connectives include:
For example, the meaning of the statements it is raining and I am indoors is transformed when the two are combined with logical connectives:
For statement P = It is raining and Q = I am indoors.
There are sixteen Boolean functions associating the inputs P and Q with fourdigit binary outputs.























Not all of the abovementioned operators are necessary for a functionally complete logical calculus. Certain compound statements are logically equivalent. For example, ¬P ∨ Q is logically equivalent to P → Q. The conditional operator "→" is therefore not necessary 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:
The logical connectives each possess different set of properties which may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are:
A set of operators is functionally complete if and only if for each of the following five properties it contains at least one member lacking it:
In twovalued logic there are 2 nullary operators (constants), 4 unary operators, 16 binary operators, 256 ternary operators, and nary operators. In threevalued logic there are 3 nullary operators (constants), 27 unary operators, 19683 binary operators, 7625597484987 ternary operators, and nary operators. In kvalued logic, there are k nullary operators, k^{k} unary operators, binary operators, ternary operators, and nary operators. An nary operator in kvalued logic is a function from . Therefore the number of such operators is , which is how the above numbers were derived.
However, some of the operators of a particular arity are actually degenerate forms that perform a lowerarity operation on some of the inputs and ignores the rest of the inputs. Out of the 256 ternary boolean operators cited above, of them are such degenerate forms of binary or lowerarity operators, using the inclusionexclusion principle. The ternary operator 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 (P Q, P Q, P → Q, P ↔ Q).
The set of logical operators may be partitioned into disjoint subsets as follows:
In this partition, is the set of operator symbols of arity .
In the more familiar propositional calculi, is typically partitioned as follows:
As a way of reducing the number of necessary parentheses, one may introduce precedence rules: ¬ has higher precedence than , higher than , and higher than →. So for example, P Q ¬R → S is short for (P (Q (¬R))) → S.
Here is a table that shows a commonly used precedence of logical operators.
Operator  Precedence 

¬  1 
2  
3  
→  4 
5 
The order of precedence determines which connective is the "main connective" when interpreting a nonatomic formula.
Instead of using truth tables, logical connective symbols can be interpreted by means of an interpretation function and a functionally complete set of truthfunctions (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 v_{1}...v_{n} representing logical connectives, and nonlogical symbols c_{1}...c_{n} , then if and only if I(v_{1})...I(v_{n}) have been provided interpreting v_{1} to v_{n} by means of f_{nand} (or any other set of functional complete truthfunctions) then the truthvalue of I(s) is determined entirely by the truthvalues of c_{1}...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 nonlogical symbols.
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 2input 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.
Is some new technology (such as reversible computing, clockless logic, or quantum dots computing) "functionally complete", in that it can be used to build computers that can do all the sorts of computation that CMOSbased computers can do? If it can implement the NAND operator, only then is it functionally complete.
That fact that all logical connectives can be expressed with NOR alone is demonstrated by the Apollo guidance computer.

