In logic, a logical connective (also called a logical operator, sentential connective, or sentential 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 value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective. The most common logical connectives are binary connectives (also called dyadic connectives) which join two sentences which 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. Semantics of a logical connective is often, but not always, presented as a truth function. A logical connective is similar to but not equivalent to a conditional operator.[1] Contents 1 In language 1.1 Natural language 1.2 Formal languages 2 Common logical connectives 2.1 List of common logical connectives 2.2 History of notations 2.3 Redundancy 3 Properties 4 Order of precedence 5 Computer science 6 See also 7 Notes 8 References 9 Further reading 10 External links In language[edit] Natural language[edit] 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: A: Jack went up the hill. B: Jill went up the hill. C: Jack went up the hill and Jill went up the hill. D: Jack went up the hill so Jill went up the hill. 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 logical connectives, and some of them are synonymous. Examples are: Word Connective Symbol Logical Gate and conjunction "∧" AND and then conjunction "∧" AND and then within conjunction "∧" AND or disjunction "∨" OR either, but not both exclusive disjunction "⊻" XOR implies implication "→" "←" if...then implication "→" "←" if and only if biconditional "↔" XNOR only if implication "→" "←" just in case biconditional "↔" XNOR but conjunction "∧" AND however conjunction "∧" AND not both alternative denial "" NAND neither...nor joint denial "↓" NOR not negation "¬" NOT it is false that negation "¬" NOT it is not the case that negation "¬" NOT although conjunction "∧" AND even though conjunction "∧" AND therefore implication "→" "←" so implication "→" "←" that is to say biconditional "↔" XNOR furthermore conjunction "∧" AND Formal languages[edit] In formal languages, truth functions are represented by unambiguous symbols. These symbols are called "logical connectives", "logical operators", "propositional operators", or, in classical logic, "truth-functional connectives". See well-formed formula for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives. Logical connectives can be used to link more than two statements, so one can speak about "n-ary logical connective". Common logical connectives[edit] Name / Symbol
P = 0 1 Truth/Tautology ⊤ 1 1 0 1 False/Contradiction ⊥ 0 0 Negation ¬ 1 0 Binary connectives P = 0 0 1 1 Q = 0 1 0 1 Conjunction ∧ 0 0 0 1 Alternative denial ↑ 1 1 1 0 Disjunction ∨ 0 1 1 1 Joint denial ↓ 1 0 0 0 Material conditional → 1 1 0 1 Exclusive or ↮ displaystyle nleftrightarrow 0 1 1 0 Biconditional ↔ 1 0 0 1 Converse implication ← 1 0 1 1 0 0 1 1 0 1 0 1 More information List of common logical connectives[edit] Commonly used logical connectives include
Alternative names for biconditional are "iff", "xnor" and "bi-implication". 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: It is not raining ( ¬ displaystyle neg P) It is raining and I am indoors ( P ∧ Q displaystyle Pwedge Q ) It is raining or I am indoors ( P ∨ Q displaystyle Plor Q ) If it is raining, then I am indoors ( P → Q displaystyle Prightarrow Q ) If I am indoors, then it is raining ( Q → P displaystyle Qrightarrow P ) I am indoors if and only if it is raining ( P ↔ Q displaystyle Pleftrightarrow Q ) It is also common to consider the always true formula and the always false formula to be connective: True formula (⊤, 1, V [prefix], or T) False formula (⊥, 0, O [prefix], or F) History of notations[edit] Negation: the symbol ¬ appeared in
P ¯ displaystyle overline P ; another alternative notation is to use a prime symbol as in P'.
Conjunction: the symbol ∧ appeared in
⋀ displaystyle bigwedge to be found in Peano. False: the symbol 0 comes also from Boole's interpretation of logic as a ring; other notations include ⋁ displaystyle bigvee to be found in Peano. Some authors used letters for connectives at some time of the history: u. for conjunction (German's "und" for "and") and o. for disjunction (German's "oder" for "or") in earlier works by Hilbert (1904); Np for negation, Kpq for conjunction, Dpq for alternative denial, Apq for disjunction, Xpq for joint denial, Cpq for implication, Epq for biconditional in Łukasiewicz (1929);[13] cf. Polish notation. Redundancy[edit] Such a logical connective as converse implication "←" is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic) certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the classical equivalence between ¬P ∨ Q and P → Q. Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugar for a compound having one negation and one disjunction. There are sixteen Boolean functions associating the input truth values P and Q with four-digit binary outputs.[14] These correspond to possible choices of binary logical connectives for classical logic. Different implementations of classical logic can choose different functionally complete subsets of connectives. One approach is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above. The following are the minimal functionally complete sets of operators in classical logic whose arities do not exceed 2: One element ↑ , ↓ . Two elements ∨ , ¬ displaystyle vee ,neg , ∧ , ¬ displaystyle wedge ,neg , → , ¬ displaystyle to ,neg , ← , ¬ displaystyle gets ,neg , → , ⊥ displaystyle to ,bot , ← , ⊥ displaystyle gets ,bot , → , ↮ displaystyle to ,nleftrightarrow , ← , ↮ displaystyle gets ,nleftrightarrow , → , ↛ displaystyle to ,nrightarrow , → , ↚ displaystyle to ,nleftarrow , ← , ↛ displaystyle gets ,nrightarrow , ← , ↚ displaystyle gets ,nleftarrow , ↛ , ¬ displaystyle nrightarrow ,neg , ↚ , ¬ displaystyle nleftarrow ,neg , ↛ , ⊤ displaystyle nrightarrow ,top , ↚ , ⊤ displaystyle nleftarrow ,top , ↛ , ↔ displaystyle nrightarrow ,leftrightarrow , ↚ , ↔ displaystyle nleftarrow ,leftrightarrow . Three elements ∨ , ↔ , ⊥ displaystyle lor ,leftrightarrow ,bot , ∨ , ↔ , ↮ displaystyle lor ,leftrightarrow ,nleftrightarrow , ∨ , ↮ , ⊤ displaystyle lor ,nleftrightarrow ,top , ∧ , ↔ , ⊥ displaystyle land ,leftrightarrow ,bot , ∧ , ↔ , ↮ displaystyle land ,leftrightarrow ,nleftrightarrow , ∧ , ↮ , ⊤ displaystyle land ,nleftrightarrow ,top . See more details about functional completeness in classical logic at Functional completeness in truth function. Another approach is to use with equal rights connectives of a certain convenient and functionally complete, but not minimal set. This approach requires more propositional axioms, and each equivalence between logical forms must be either an axiom or provable as a theorem. The situation, however, is more complicated in intuitionistic logic. Of its five connectives, ∧, ∨, →, ¬, ⊥ , only negation "¬" can be reduced to other connectives (see details). Neither conjunction, disjunction, nor material conditional has an equivalent form constructed of the other four logical connectives. Properties[edit] Some logical connectives possess properties which may be expressed in the theorems containing the connective. Some of those properties that a logical connective may have are: Associativity: 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 preserving logical equivalence to the original 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 compound is logically equivalent to the operand. Absorption: A pair of connectives ∧, ∨ satisfies the absorption law if a ∧ ( a ∨ b ) = a displaystyle aland (alor b)=a for all operands a, b. Monotonicity: If f(a1, ..., an) ≤ f(b1, ..., bn) for all a1, ..., an, b1, ..., bn ∈ 0,1 such that a1 ≤ b1, a2 ≤ b2, ..., an ≤ bn. E.g., ∨, ∧, ⊤, ⊥. Affinity: Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, ↔, ↮ displaystyle nleftrightarrow , ⊤, ⊥. Duality: To read the truth-value assignments for the operation from top to bottom on its truth table is the same as taking the complement of reading the table of the same or another connective from bottom to top. Without resorting to truth tables it may be formulated as g̃(¬a1, ..., ¬an) = ¬g(a1, ..., an). E.g., ¬. Truth-preserving: The compound all those argument are tautologies is a tautology itself. E.g., ∨, ∧, ⊤, →, ↔, ⊂. (see validity) Falsehood-preserving: The compound all those argument are contradictions is a contradiction itself. E.g., ∨, ∧, ↮ displaystyle nleftrightarrow , ⊥, ⊄, ⊅. (see validity) Involutivity (for unary connectives): f(f(a)) = a. E.g. negation in classical logic. For classical and intuitionistic logic, the "=" symbol means that corresponding implications "…→…" and "…←…" for logical compounds can be both proved as theorems, and the "≤" symbol means that "…→…" for logical compounds is a consequence of corresponding "…→…" connectives for propositional variables. Some many-valued logics may have incompatible definitions of equivalence and order (entailment). Both conjunction and disjunction are associative, commutative and idempotent in classical logic, most varieties of many-valued logic and intuitionistic logic. The same is true about distributivity of conjunction over disjunction and disjunction over conjunction, as well as for the absorption law. In classical logic and some varieties of many-valued logic, conjunction and disjunction are dual, and negation is self-dual, the latter is also self-dual in intuitionistic logic. This section needs expansion. You can help by adding to it. (March 2012) Order of precedence[edit] 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 displaystyle Pvee Qwedge neg R rightarrow S is short for ( P ∨ ( Q ∧ ( ¬ R ) ) ) → S displaystyle (Pvee (Qwedge (neg R)))rightarrow S . Here is a table that shows a commonly used precedence of logical operators.[15] Operator Precedence ¬ 1 ∧ 2 ∨ 3 → 4 ↔ 5 displaystyle begin array cc text Operator & text Precedence \hline neg &1\land &2\vee &3\to &4\leftrightarrow &5end array However, not all compilers use the same order; for instance, an ordering in which disjunction is lower precedence than implication or bi-implication has also been used.[16] Sometimes precedence between conjunction and disjunction is unspecified requiring to provide it explicitly in given formula with parentheses. The order of precedence determines which connective is the "main connective" when interpreting a non-atomic formula. Computer science[edit] This section needs expansion. You can help by adding to it. (March 2012) A truth-functional approach to logical operators is 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; see more details in
Boolean domain
Boolean function
Boolean logic
Boolean-valued function
List of
Logical constant
Modal operator
Propositional calculus
Notes[edit] ^ Cogwheel. "What is the difference between logical and conditional
/operator/". Stack Overflow. Retrieved 9 April 2015.
^ a b
References[edit] Bocheński, Józef Maria (1959), A Précis of Mathematical Logic,
translated from the French and German editions by Otto Bird, D.
Reidel, Dordrecht, South Holland.
Enderton, Herbert (2001), A Mathematical Introduction to
Further reading[edit] Lloyd Humberstone (2011). The Connectives. MIT Press. ISBN 978-0-262-01654-4. External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "Propositional connective",
Encyclopedia of Mathematics,
v t e Logical connectives Tautology/True ⊤ displaystyle top Alternative denial (NAND gate) ↑ displaystyle uparrow Converse implication ← displaystyle leftarrow Implication (IMPLY gate) → displaystyle rightarrow Disjunction (OR gate) ∨ displaystyle lor Negation (NOT gate) ¬ displaystyle neg Exclusive or (XOR gate) ↮ displaystyle nleftrightarrow Biconditional (XNOR gate) ↔ displaystyle leftrightarrow Statement Joint denial (NOR gate) ↓ displaystyle downarrow Nonimplication ↛ displaystyle nrightarrow Converse nonimplication ↚ displaystyle nleftarrow Conjunction (AND gate) ∧ displaystyle land Contradiction/False ⊥ displaystyle bot v t e Mathematical logic General Formal language Formation rule Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Rule of inference Relation Theorem Logical consequence Type theory Symbol Syntax Theory Systems Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus Traditional logic Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram Propositional calculus Boolean logic Boolean functions
Propositional calculus
Propositional formula
Logical connectives
Predicate logic First-order Quantifiers Predicate Second-order Monadic predicate calculus Naive set theory Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair Set theory Foundations of mathematics
Zermelo–Fraenkel set theory
Model theory Model
Interpretation
Non-standard model
Finite model theory
Proof theory Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax Computability theory Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive |