In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. Logic, from Classical Greek λόγος logos (the word), is the study of patterns found in reasoning. ... In mathematical logic, a sentence is a formula with no free variables; therefore, a sentence is either true or false in a given structure. ... In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B = {0, 1} is a boolean domain and where k is a nonnegative integer. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ...

For example, the two sentences, "It's raining" and "I'm indoors", can be combined by means of various connectives to form the following compound sentences:

• It's not raining so I'm indoors.
• It's raining and I'm indoors.
• If it's raining then I'm indoors.

The basic operators are "not" (¬ or ~), "and" (∧ or &), "or" (∨), "conditional" (→ or ⊃), and "biconditional" (iff) (↔). "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). Negation (i. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ... In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

Note the similarity between the symbols for "and" (∧) and set-theoretic intersection (∩); likewise for "or" (∨) and set-theoretic union (∪). This is not a coincidence: the definition of the intersection uses "and" and the definition of union uses "or". In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Truth tables for these connectives: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

In order to reduce the number of necessary parentheses, one introduces 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.

Not all of these operators are necessary for a full-blooded 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). In logic, statements p and q are logically equivalent if they have the same logical content. ...

For the sake of convenience (and brevity), only the five most-commonly used operators (in math) are listed above. One can also consider other connectives, such as NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or"). NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ...

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 2-input gates. All other operators are implemented by breaking them down into a logically equivalent combination of 2 or more of the above logic gates. A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. ... Digital circuits are electric circuits based on a number of discrete voltage levels. ... Dram can mean several things: For the imperial unit of volume see dram (unit), commonly used to describe a measure of Scotch whisky For the imperial unit of weight or mass see avoirdupois and apothecaries system (of mass) For the Armenian monetary unit see dram (currency) DRAM is a type... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ... Negation (i. ... CMOS Transmission Gate A transmission gate is an electronic element. ...

If you throw away all the operators that are not necessary, what operators are you left with? Which conditionals are the crucial must-have ones? Surprisingly, there is more than one answer to that question.

• All connectives can be expressed with NAND alone.

• All connectives can be expressed with NOR alone, as demonstrated by the Apollo guidance computer.

• Since NAND can be built from NOT and AND, and we know that all connectives can be built from NAND alone, clearly all connectives can be built from combinations of NOT and AND.

• ... and several other answers.

The "logical equivalence" of "NAND alone", "NOR alone", and "NOT and AND" is similar to Turing equivalence. The Apollo Guidance Computer (AGC) was the first recognizably modern embedded system, used in real-time by astronaut pilots to collect and provide flight information, and to automatically control all of the navigational functions of the Apollo spacecraft. ... In computability theory, an abstract machine or programming language is called Turing-complete, Turing-equivalent, or (computationally) universal if it has a computational power equivalent to a universal Turing machine (a simplified model of a programmable computer). ...

Is some new technology (such as reversible computing, clockless logic, quantum dots computing, or Tinker Toys) "logically complete", in that it can be used to build computers that can do all the sorts of computation that CMOS-based computers can do? If it can implement the NAND operator, only then is it logically complete. The term reversible computing refers to any computational process that is (at least to some close approximation) reversible, i. ... An asynchronous circuit is a circuit in which the parts are largely autonomous. ... Fluorescence induced by exposure to ultraviolet light in vials containing various sized Cadmium selenide (CdSe) quantum dots. ... Static CMOS Inverter Complementary-symmetry/metal-oxide semiconductor (CMOS) (see-moss, IPA:), is a major class of integrated circuits. ...

References

• Article on logical constants at the Stanford Encyclopedia of Philosophy.

The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ...

See also