Logical connective

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Hasse diagram of logical connectives. Logical connectives Hasse diagram.svg
Hasse diagram of logical connectives.

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.

Contents

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]

In language

Natural language

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 functional. For example, consider the following sentences:

  1. Jack went up the hill.
  2. Jill went up the hill.
  3. Jack went up the hill and Jill went up the hill.
  4. 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:

WordConnectiveSymbolLogical gate
and conjunction "∧" AND
and then conjunction "∧" AND
and then within conjunction "∧" AND
or disjunction "∨" OR
either...or exclusive disjunction "⊻" XOR
either, but not both exclusive disjunction "⊻" XOR
implies material implication "→" IMPLY
is implied by converse implication "←"
if...then material implication "→" IMPLY
...if converse implication "←"
if and only if biconditional "↔" XNOR
just in case biconditional "↔" XNOR
but conjunction "∧" AND
however conjunction "∧" AND
not both alternative denial "↑" NAND
neither...nor joint denial "↓" NOR
not, not that negation "¬" NOT
it is false that negation "¬" NOT
it is not the case that negation "¬" NOT
although conjunction "∧" AND
even though conjunction "∧" AND
therefore material implication "→" IMPLY
so material implication "→" IMPLY
that is to say biconditional "↔" XNOR
furthermore conjunction "∧" AND
but not material nonimplication "↛" NIMPLY
not...but converse nonimplication "↚"
no...without material implication "→" IMPLY
without...there is no converse implication "←"

Formal languages

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

Symbol, nameTruth
table
Venn
diagram
Unary connectives
P =01
Truth/tautology 11 Venn11.svg
Proposition P01 Venn01.svg
Falsity/contradiction 00 Venn00.svg
¬ Negation 10 Venn10.svg
Binary connectives
P =01
Q =0101
Proposition P0011 Venn0101.svg
Proposition Q0101 Venn0011.svg
Conjunction 0001 Venn0001.svg
Alternative denial 1110 Venn1110.svg
Disjunction 0111 Venn0111.svg
Joint denial 1000 Venn1000.svg
Material conditional 1101 Venn1011.svg
Exclusive or 0110 Venn0110.svg
Biconditional 1001 Venn1001.svg
Converse implication 1011 Venn1101.svg
More information

List of common logical connectives

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 also common to consider the always true formula and the always false formula to be connective:

History of notations

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

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
, , , , , , , , , , , , , , , , , .
Three elements
, , , , , .

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

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 for all operands a, b.
Monotonicity
If f(a1, ..., an) ≤ f(b1, ..., bn) for all a1, ..., an, b1, ..., bn ∈ {0,1} such that a1b1, a2b2, ..., anbn. E.g., , , ⊤, ⊥.
Affinity
Each variable always makes a difference in the truth-value of the operation or it never makes a difference. E.g., ¬, , , ⊤, ⊥.
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 g̃a1, ..., ¬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., , , , ⊥, ⊄, ⊅ (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.

Order of precedence

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, is short for .

Here is a table that shows a commonly used precedence of logical operators. [15]

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

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 Truth function in computer science. Logical operators over bit vectors (corresponding to finite Boolean algebras) are bitwise operations.

But not every usage of a logical connective in computer programming has a Boolean semantic. For example, lazy evaluation is sometimes implemented for P ∧ Q and P ∨ Q, so these connectives are not commutative if either or both of the expressions P, Q have side effects. Also, a conditional, which in some sense corresponds to the material conditional connective, is essentially non-Boolean because for if (P) then Q; the consequent Q is not executed if the antecedent  P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and constructivist views on the material conditional, rather than to classical logic's ones.

See also

Notes

  1. Cogwheel. "What is the difference between logical and conditional /operator/". Stack Overflow. Retrieved 9 April 2015.
  2. 1 2 Heyting (1929) Die formalen Regeln der intuitionistischen Logik.
  3. Denis Roegel,;l,;l.';, (2002), Petit panorama des notations logiques du 20e siècle (see chart on page 2).
  4. 1 2 3 4 Russell (1908) Mathematical logic as based on the theory of types (American Journal of Mathematics 30, p222–262, also in From Frege to Gödel edited by van Heijenoort).
  5. Peano (1889) Arithmetices principia, nova methodo exposita .
  6. 1 2 Schönfinkel (1924) Über die Bausteine der mathematischen Logik, translated as On the building blocks of mathematical logic in From Frege to Gödel edited by van Heijenoort.
  7. Peirce (1867) On an improvement in Boole's calculus of logic.
  8. Hilbert (1917/1918) Prinzipien der Mathematik (Bernays' course notes).
  9. Vax (1982) Lexique logique, Presses Universitaires de France.
  10. Tarski (1940) Introduction to logic and to the methodology of deductive sciences.
  11. Gentzen (1934) Untersuchungen über das logische Schließen.
  12. Chazal (1996) : Éléments de logique formelle.
  13. See Roegel
  14. Bocheński (1959), A Précis of Mathematical Logic, passim.
  15. O'Donnell, John; Hall, Cordelia; Page, Rex (2007), Discrete Mathematics Using a Computer, Springer, p. 120, ISBN   9781846285981 .
  16. Jackson, Daniel (2012), Software Abstractions: Logic, Language, and Analysis, MIT Press, p. 263, ISBN   9780262017152 .

Related Research Articles

Logical disjunction logical connective OR

In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is typically written as ∨ or +.

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists x such that x is Socrates and x is a man" and there exists is a quantifier while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

Logical conjunction logical connective AND

In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is typically written as or .

Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and argument flow. Compound propositions are formed by connecting propositions by logical connectives. The propositions without logical connectives are called atomic propositions.

De Morgans laws pair of transformation rules that are both valid rules of inference

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

In logic, negation, also called the logical complement, is an operation that takes a proposition to another proposition "not ", written , which is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

Logical NOR

In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form is true precisely when neither p nor q is true—i.e. when both of p and q are false. In grammar, nor is a coordinating conjunction.

Logical biconditional term

In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective used to conjoin two statements and to form the statement " if and only if ", where is known as the antecedent, and the consequent. This is often abbreviated as " iff ". The operator is denoted using a doubleheaded arrow, a prefixed E "Epq", an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to both and , and the XNOR boolean operator, which means "both or neither".

In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and that either form can replace the other in logical proofs.

Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the style of natural deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies.

A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.

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In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression. A well-known complete set of connectives is { AND, NOT }, consisting of binary conjunction and negation. Each of the singleton sets { NAND } and { NOR } is functionally complete.

Conditioned disjunction

In logic, conditioned disjunction is a ternary logical connective introduced by Church. Given operands p, q, and r, which represent truth-valued propositions, the meaning of the conditioned disjunction [p, q, r] is given by:

Monoidal t-norm based logic, the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices by the axiom of prelinearity.

Material nonimplication

Material nonimplication or abjunction is the negation of material implication. That is to say that for any two propositions and , the material nonimplication from to is true if and only if the negation of the material implication from to is true. This is more naturally stated as that the material nonimplication from to is true only if is true and is false.

T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning.

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.

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Further reading