Logical connective
In logic, two sentences (either in a formal language or a natural language) may be joined by means of a logical connective to form a compound sentence. The truthvalue of the compound is uniquely determined by the truthvalues of the simpler sentences. The logical connective therefore represents a function, and since the value of the compound sentence is a truthvalue, it is called a truthfunction and the logical connective is called a "truthfunctional connective." The truthfunctions include conjunction ("and"), disjunction ("or"), and implication ("if … then").
Contents
 1 Truth functions in natural languages
 2 Truth functions in formal languages
 3 Introduction
 4 All binary logical connectives
 5 Functional completeness
 6 Properties
 7 Arity
 8 Order of precedence
 9 Applications in computer science
 10 Basic logic symbols
 11 See also
 12 Notes
 13 References
 14 External links
 15 Credits
Mathematical logic is a mathematical representation of formal rules of human thought, which philosophers have been attempting to develop since Aristotle. Philosophical arguments are often incomprehensible due to obscure or ambiguous expressions. Logical connectives are basic units which constitute the logical structure of an argument. By applying these conceptual tools, arguments can become clearer, communicable, and comprehensible.
Truth functions in natural languages
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 truthfunctions. 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 both 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 fetch a pail of water, not because Jack had gone up the Hill at all. Thus, and is a logical connective but so is not. In the realm of pure logic, (C) is a compound statement but (D) is not. (D) cannot be broken into parts using only the logic of statements, the realm of cause and effect being proper to science rather than logic.
Various English words and word pairs express truthfunctions, and some of them are synonymous. Examples (with the name of the relationship in parentheses) are:
"and" (conjunction), "or" (inclusive or exclusive disjunction), "implies" (implication), "if … then" (implication), "if and only if" (equivalence), "only if" (implication), "just in case" (equivalence), "but" (conjunction), "however" (conjunction) , "not both" (NAND), "neither … nor" (NOR). The word "not" (negation) and "it is false that" (negation) "it is not the case that" (negation) are also English words expressing a logical connective, even though they are applied to a single statement, and do not connect two statements.
Truth functions in formal languages
In formal languages truthfunctions are represented by unambiguous symbols, and 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, can be defined. Not all of them are in common use. These symbols are called "truthfunctional connectives," "logical connectives," "logical operators," or "propositional operators."
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.
Introduction
The basic logical operators are:

Some others are:

For example, the statements it is raining and I am indoors can be reformed using various different connectives to form sentences that relate the two in ways which augment their meaning:
 It is raining and I am indoors.
 If it is raining then I am indoors.
 It is raining if I am indoors.
 It is raining if and only if I am indoors.
 It is not raining.
If one writes "P" for It is raining and "Q" for I am indoors, and uses the usual symbols for logical connectives, then the above examples could be represented in symbols, respectively:
 P & Q
 P Q
 Q P
 P Q
 ¬P
All binary logical connectives
There are sixteen different Boolean functions, associating the inputs P and Q with four digit binary outputs.
The following table shows important equivalences like the De Morgan's laws (lines 1000 and 1110) or the law of Contraposition (line 1101).
Functional completeness
Not all of these 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 So the conditional operator "→" is not necessary if you have "¬" (not) and "∨" (or)
The smallest set of operators which still expresses every statement which is 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 functionally complete sets (of cardinality not exceeding 2) of operators whose arities do not exceed 2:
{ ↓ }, { ↑ }, { , }, { , }, { , ⊂ }, { , ⊄ }, { , }, { , ⊅ }, { ⊄, }, { ⊂, }, { ⊅, }, { ⊂, ⊄ }, { , }, { ⊂, ⊅ }, { , }, { ⊄, }, { ⊅, }
Properties
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:
 Associativity: Within an expression containing two or more of the same associative operators in a row, the order of the operations does not matter as long as the sequence of the operands is not changed. For example:
 (x • y) • z = x • (y • z)
 Commutivity: Each pair of variables connected by the operator may be exchanged for each other without affecting the truthvalue of the expression. For example:
 x • y = y • x
 Distributivity: Given a set S and two binary operations • and + on S, one says that the operation •
 is leftdistributive over + if, given any elements x, y, and z of S,
 x • (y + z) = (x • y) + (x • z);
 is rightdistributive over + if, given any elements x, y, and z of S:
 (y + z) • x = (y • x) + (z • x);
 is distributive over + if it is both left and rightdistributive.
 Idempotency: Multiple applications of the operation does not change the result. For example:
 1 x 1 = 1
 Absorption: An identity linking a pair of binary operations. For example, any two binary operations, say $ and %, are subject to the absorption law if:
 a $ (a % b) = a % (a $ b) = a.
 The operations $ and % are said to form a dual pair.
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:
 monotonic: If f(a_{1}, ..., a_{n}) ≤ f(b_{1}, ..., b_{n}) for all a_{1}, ..., a_{n}, b_{1}, ..., b_{n} ∈ {0,1} such that a_{1} ≤ b_{1}, a_{2} ≤ b_{2}, ..., a_{n} ≤ b_{n}. E.g., , , , .
 affine: Each variable always makes a difference in the truthvalue of the operation or it never makes a difference. E.g., , , , , .
 Self dual: To read the truthvalue assignments for the operation from top to bottom on its truth table is the same as taking the complement of reading it from bottom to top, in other words f(¬a_{1}, ..., ¬a_{n}) = ¬f(a_{1}, ..., a_{n}). For example, .
 Truthpreserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of these operations. E.g., , , , , , ⊂.
 Falsehoodpreserving: The interpretation under which all variables are assigned a truth value of "false" produces a truth value of 'false' as a result of these operations. For example, , , , , ⊄, ⊅.
Arity
In twovalued logic there are 2 nullary operators (constants), 4 unary operators, 16 binary operators, 256 ternary operators, and nary operators. In three valued logic there are 3 nullary operators (constants), 27 unary operators, 19683 binary operators, 7625597484987 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:
 nullary operators:
 unary operators:
 binary operators:
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, 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.
Applications in computer science
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.
In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation and related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.
Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.
Basic logic symbols
Symbol

Name  Explanation  Examples  Unicode Value 
HTML Entity 
LaTeX symbol 

Should be read as  
Category  
⇒
→ ⊃ 
material implication  A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. → may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ may mean the same as ⇒ (the symbol may also mean superset). 
x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2).  8658 8594 8835 
⇒ → ⊃ 
\Rightarrow
\to \supset 
implies; if .. then  
propositional logic, Heyting algebra  
⇔
≡ ↔ 
material equivalence  A ⇔ B means A is true if B is true and A is false if B is false.  x + 5 = y +2 ⇔ x + 3 = y  8660 8801 8596 
⇔ ≡ ↔ 
\Leftrightarrow
\equiv \leftrightarrow 
if and only if; iff  
propositional logic  
¬
˜ 
logical negation  The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. 
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) 
172 732 
¬ ˜ ~ 
\lnot
\tilde{} 
not  
propositional logic  
∧
& 
logical conjunction  The statement A ∧ B is true if A and B are both true; else it is false.  n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.  8743 38 
∧ & 
\land \&^{[1]} 
and  
propositional logic  
∨

logical disjunction  The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.  n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.  8744  ∨  \lor 
or  
propositional logic  
⊕ ⊻ 
exclusive or  The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.  (¬A) ⊕ A is always true, A ⊕ A is always false.  8853 8891 
⊕  \oplus 
xor  
propositional logic, Boolean algebra  
⊤ T 1 
logical truth  The statement ⊤ is unconditionally true.  A ⇒ ⊤ is always true.  8868  T  \top 
top  
propositional logic, Boolean algebra  
⊥ F 0 
logical falsity  The statement ⊥ is unconditionally false.  ⊥ ⇒ A is always true.  8869  ⊥ F 
\bot 
bottom  
propositional logic, Boolean algebra  
∀

universal quantification  ∀ x: P(x) means P(x) is true for all x.  ∀ n ∈ N: n^{2} ≥ n.  8704  ∀  \forall 
for all; for any; for each  
predicate logic  
∃

existential quantification  ∃ x: P(x) means there is at least one x such that P(x) is true.  ∃ n ∈ N: n is even.  8707  ∃  \exists 
there exists  
firstorder logic  
∃!

uniqueness quantification  ∃! x: P(x) means there is exactly one x such that P(x) is true.  ∃! n ∈ N: n + 5 = 2n.  8707 33  ∃ !  \exists ! 
there exists exactly one  
firstorder logic  
:=
≡ :⇔ 
definition  x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence). P :⇔ Q means P is defined to be logically equivalent to Q. 
cosh x := (1/2)(exp x + exp (−x)) A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) 
58 61 8801 58 8660 
:= : ≡ ⇔ 
:=
\equiv \Leftrightarrow 
is defined as  
everywhere  
( )

precedence grouping  Perform the operations inside the parentheses first.  (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  40 41  ( )  ( ) 
everywhere  
⊢

inference  x ⊢ y means y is derived from x.  A → B ⊢ ¬B → ¬A  8866  \vdash  
infers or is derived from  
propositional logic, firstorder logic 
See also
 Formal logic
 Fuzzy logic
 History of logic
 logic
 Metalogic
 Modal logic
 Proof (logic)
 Propositional calculus
Notes
 ↑ Although this character is available in LaTeX, the Mediawiki TeX system doesn't support this character.
References
 Church, Alonzo. Introduction to Mathematical Logic. Princeton mathematical series, 17. Princeton: Princeton University Press, 1956.
 Copi, Irving M. Symbolic Logic. New York: Macmillan, 1954.
 Enderton, Herbert. A Mathematical Introduction to Logic, 2nd ed. Boston, MA: Academic Press, 2001. ISBN 9780122384523.
 Martin, Robert M. Introducing Symbolic Logic. Peterborough, ON: Broadview Press, 2004. ISBN 9781551116358.
 Ochieng'Odhiambo, F. Introductory Symbolic Logic. Guide to philosophy series, no. 7 (rev. ed.). Nairobi: Consolata Institute of Philosophy, 2003. ISBN 9789966820068.
 Quine, W. V. Mathematical Logic. Cambridge: Harvard University Press, 1951.
 Rosenbloom, Paul C. The Elements of Mathematical Logic. The Dover series in mathematics and physics. New York: Dover Publications, 1950.
 Stoli︠a︡r, A. A. Introduction to Elementary Mathematical Logic. Cambridge, Mass: M.I.T. Press, 1970. ISBN 9780262190541.
External links
All links retrieved July 24, 2018.
 "Logical constants," Stanford Encyclopedia of Philosophy.
 Cullinane, Steven H. "The geometry of logic."
 Introduction to logic
General philosophy sources
 Stanford Encyclopedia of Philosophy
 Paideia Project Online
 The Internet Encyclopedia of Philosophy
 Project Gutenberg


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