logical connective

In propositional logicPlanetmathPlanetmath, the usual way of forming a well-formed formula out of existing ones is by attaching a designated symbol to the existing wffs. This designated symbol is variously known as a logical connective, logical symbol, or simply a connective. Given a connective, the number of wffs needed to form a new wff is a fixed integer, and is called the arity of the connective. For example, if # is a connective of arity 3, and p,q,r are three existing wffs, then


is the wff formed by attaching # to the p,q and r in order. Instead of prefixing to the string of wffs as above, the connective may also be attached at the end (as a suffix), or customarily infixed in case it has arity of 2 (binary). 0-ary connectives are also allowed, in which case it is just a symbol with no wffs attached.

The common classical logical connectives are:

The symbols and are due to Russell.

Remark. “Logical implication and “logical is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to symbols are typically used for logicians. Nevertheless, the symbols for material implication and for material equivalence are commonly used in the literature. In particular, is usually reserved for the concept of limit.

Usually, given a logical connective #, a truth function is associated. The arity of the truth function is defined to be the arity of the connective. When there is no confusion, the symbol for the associated truth function is the same as the symbol of the connective. A truth function of small arity can be easily represented by a table, called the truth tableMathworldPlanetmath of the truth function. The truth functions and truth tables associated with the connectives listed above are

  • ¬:𝟐𝟐 given by ¬(x)=1-x;

    P ¬P
    F T
    T F
  • :𝟐2𝟐 given by (x,y)=max(x,y);

    P Q PQ
    F F F
    F T T
    T F T
    T T T
  • :𝟐2𝟐 given by (x,y)=min(x,y);

    P Q PQ
    F F F
    F T F
    T F F
    T T T
  • :𝟐2𝟐 given by (x,y)=max(1-x,y); and

    P Q PQ
    F F T
    F T T
    T F F
    T T T
  • :𝟐2𝟐 given by (x,y)=(x+y+1)(mod2).

    P Q PQ
    F F T
    F T F
    T F F
    T T T

where 𝟐={0,1}. Note that 0 and 1 have been converted to F and T in the tables above.

Any truth function of any finite arity can be written as a finite combinationMathworldPlanetmathPlanetmath of these connectives (see the entry on functional completeness). However, the collectionMathworldPlanetmath is redundant; the final three symbols, , , and , can be defined in terms of prior ones. By DeMorgan’s law, we can define logical and by


Material implication can be defined by


Finally, material equivalence can be defined by

PQ :=(PQ)(QP)

Hence ¬ and suffice to define all other connectives.

Title logical connective
Canonical name LogicalConnective
Date of creation 2013-03-22 16:26:58
Last modified on 2013-03-22 16:26:58
Owner mps (409)
Last modified by mps (409)
Numerical id 19
Author mps (409)
Entry type Definition
Classification msc 03B05
Synonym logical symbol
Synonym connective
Synonym conjunctive connective
Synonym disjunctive connective
Related topic Ampheck
Related topic ContradictoryStatement
Related topic LogicalAxiom
Related topic SoleSufficientOperator
Related topic PropositionalCalculus
Related topic LogicalImplication
Related topic ZerothOrderLogic