Not
The NOT operation, also known as logical negation or inversion, is a fundamental unary logical connective in Boolean algebra and digital electronics that reverses the truth value of its single input operand, producing an output of true (1) when the input is false (0) and false (0) when the input is true (1).[1] This operation is represented symbolically by a straight line over the input variable (e.g., ¬A or \overline{A}) in mathematical notation[2] or by a triangular gate with a small circle at the output in circuit diagrams.[3] The truth table for the NOT gate is simple: for input 0, output is 1; for input 1, output is 0.[3] The concept of negation traces its origins to classical logic, with early formulations in Aristotle's work on syllogisms around 350 BCE,[4] but it was formalized in modern terms through George Boole's development of Boolean algebra in his 1847 book The Mathematical Analysis of Logic and 1854 treatise An Investigation of the Laws of Thought.[5] In the 20th century, Claude Shannon's 1937 master's thesis at MIT demonstrated how Boolean operations, including NOT, could be applied to electrical switching circuits, laying the groundwork for digital computing.[6] The physical implementation of the NOT gate emerged in the mid-20th century with the advent of transistor-based electronics; early versions used vacuum tubes or diodes,[7] but modern NOT gates are typically constructed from a complementary pair of CMOS transistors for efficient inversion in integrated circuits.[8] As one of the three basic logic gates (alongside AND and OR), the NOT gate is essential for constructing all other logic functions and forms the basis of digital systems, from simple inverters in power electronics to complex processors in computers and microcontrollers.[9] In programming languages like C# and Python, the NOT operator (often denoted as ! or not) performs the same negation on boolean values, enabling conditional logic and data manipulation in software.[10] Its simplicity and universality make it indispensable in fields ranging from computer science to electrical engineering, with ongoing relevance in quantum computing where the NOT gate equates to the Pauli-X operator for qubit flipping.[11]Language and Grammar
Negation in English
In English grammar, "not" functions as the principal adverb of negation, primarily modifying verbs to express denial or reversal of a proposition, but also capable of negating adjectives, adverbs, or entire clauses. For instance, the affirmative statement "The team plays well" is negated as "The team does not play well," where "not" inverts the meaning to indicate the absence of the described action or quality. This adverbial role allows "not" to scope over various syntactic elements, such as in "That is not impossible" (negating an adjective) or "Not quickly enough" (negating an adverb phrase).[12] The word "not" evolved from Old English nāht or nōht, a contraction meaning "nothing" or "not a thing," derived from ne ("not") + āht ("aught" or "anything"). By the mid-13th century, during the transition to Middle English around the 12th century, it had shortened to its modern unstressed form "not," serving as a standalone negative particle independent of "nought" or "naught." This development marked a shift from the Old English prefixal negation ne- (enclitic on verbs) to a more analytic, post-verbal adverb in Middle English texts.[13] Syntactically, "not" typically follows auxiliary or modal verbs in declarative sentences, as in "She will not attend," but requires do-support in simple present or past tenses lacking other auxiliaries to avoid adjacency issues with the main verb. Do-support emerged gradually in Early Modern English (ca. 1500–1700), becoming obligatory by the 18th century for negated declaratives and questions without auxiliaries, exemplified by "You do not see the problem?" rather than the archaic "You see not the problem?" This construction inserts the dummy auxiliary "do" to host tense and enable "not" placement immediately after it, reflecting a broader loss of verb movement in English clause structure. In infinitival clauses, "not" could historically adjoin in multiple positions (e.g., "not to go" or "to not go"), but modern usage favors pre-infinitive placement.[12] In formal speech and writing, "not" appears in its full form, often after auxiliaries like "is not ready" or "cannot not agree," emphasizing clarity and precision. Informal contexts, however, frequently employ contractions where "not" fuses with preceding auxiliaries, such as "isn't," "don't," "willn't" (or more commonly "won**'t**"), and "can**'t**," which shorten utterances for conversational efficiency—e.g., "She isn't coming" instead of "She is not coming." These contractions, recorded from the 1630s in dramatic texts, are generally avoided in formal registers to maintain a professional tone, as they can obscure parsing in complex sentences.[14] The Norman Conquest of 1066 profoundly influenced English negation patterns by accelerating the language's analytic evolution during Middle English (ca. 1100–1500), where multiple negation for emphasis—such as "I ne see not nothing" (I see nothing)—became widespread, drawing partly from Old French bipartite negation (ne...pas). This period saw a proliferation of double (or multiple) negation as a standard intensifier, contrasting with Old English's simpler ne-prefixation. However, by Early Modern English (ca. 1500–1800), prescriptive grammarians, influenced by Latin's single-negation model, promoted standard single negation, leading to the decline of multiple forms in educated speech and the establishment of "not" as the sole sentential negator in Modern English—e.g., shifting from Middle English "He ne lovede not hire" to "He did not love her."[15]Emphatic and Sarcastic Constructions
In English, the "...not!" tag construction serves as an emphatic ironic device, appending "not!" to a statement to contradict it humorously, as in "That was awesome...not!" This form gained widespread popularity in the 1990s through sketches on Saturday Night Live and the 1992 film Wayne's World, where characters Wayne Campbell and Garth Algar frequently employed it to reverse positive assertions for comedic effect.[16] The construction exemplifies non-literal negation, deviating from standard syntactic negation by prioritizing rhetorical punch over literal meaning.[17] Sarcastic uses of "not" often build on this pattern, such as "He's so smart...not," to mock or undermine a preceding claim through implied opposition. These expressions trace roots to 19th-century traditions of British understatement, where negation subtly conveys irony or disdain without overt confrontation, a style prominent in Victorian literature and social discourse.[18] Understatement via negation allows speakers to signal insincerity while maintaining politeness, a hallmark of English ironic expression.[19] Regional variations appear in African American Vernacular English (AAVE), where emphatic negation with "not" reinforces denial in constructions like "He ain't coming, not!" This pattern intensifies negation for rhetorical emphasis, distinct from standard English double negatives, and reflects AAVE's syntactic flexibility in conveying strong assertions. Linguistically, the psychological role of "not" in these constructions involves prosody—specifically stress and intonation—to cue sarcasm or humor, enabling listeners to detect insincerity through vocal delivery rather than words alone. Studies show that exaggerated intonation on "not" activates contextual inference for ironic intent, enhancing communicative efficiency in social interactions.[20] The "...not!" tag's prominence waned after the 2000s, supplanted by internet-era alternatives like "just kidding" or abbreviations such as "jk," as digital communication favored concise meme-based irony over 1990s pop culture references. This shift reflects broader evolution in online humor toward brevity and shareability.[21]Logic and Mathematics
Logical Negation
Logical negation is a fundamental unary connective in propositional logic that reverses the truth value of a proposition, such that if a proposition P is true, its negation \neg P is false, and vice versa.[22] This operation inverts the semantic value of the operand without altering its internal structure, serving as the basic mechanism for expressing denial or opposition in formal reasoning systems.[23] The philosophical foundations of logical negation trace back to Aristotle's formulation in the 4th century BCE, particularly in his Metaphysics Book Gamma, where he articulates the law of non-contradiction: it is impossible for the same attribute to belong and not to belong simultaneously to the same thing in the same respect.[24] This principle establishes negation as a safeguard against contradictions, ensuring that a statement and its direct opposite cannot both hold true, thereby underpinning the coherence of rational discourse and metaphysical inquiry.[24] In propositional logic, negation plays a central role in compound formulas, enabling the distribution over other connectives as described by De Morgan's laws, which state that the negation of a conjunction is equivalent to the disjunction of the negations, and vice versa—for instance, \neg (P \land Q) \equiv \neg P \lor \neg Q, and \neg (P \lor Q) \equiv \neg P \land \neg Q.[25] These equivalences, formalized by Augustus De Morgan in 1847, facilitate the transformation and simplification of logical expressions, highlighting negation's distributive properties in Boolean structures.[25] Unlike a contradiction, which refers specifically to a compound statement that is always false—such as P \land \neg P, formed by a proposition and its own negation—logical negation itself is a general operator applied to any proposition to yield its complement, without inherently producing falsity.[26] This distinction underscores that negation preserves the potential for truth or falsity depending on the operand, whereas a contradiction is semantically vacuous by design.[27] Common symbols for negation include \neg, \sim, and !, varying by logical tradition.Symbols and Formal Representation
In formal logic and mathematics, the representation of negation has evolved through various symbols, each adopted in specific contexts to denote the unary operation that reverses the truth value of a proposition. Gottlob Frege introduced negation as a primitive operator in his 1879 work Begriffsschrift, using a short vertical stroke to modify the content stroke of a judgment, thereby expressing the denial of a condition.[28] This marked a significant historical shift toward treating negation as a fundamental connective in symbolic logic, distinct from earlier rhetorical or algebraic notations. The symbol ¬ (not sign) was introduced in the early 20th century, first used in print as a logical symbol by Alessandro Padoa in 1912.[29] It became the standard for negation in classical propositional logic, where ¬P asserts the falsehood of P. Other common symbols include the tilde ~, which Peano himself adopted in 1897 for negation in later works, and variants like the exclamation mark ! or the stroke /, used in certain historical or specialized notations to indicate denial.[30] The Sheffer stroke |, introduced by Henry Sheffer in 1913, serves as a binary connective equivalent to the negation of conjunction (NAND), from which unary negation can be derived as (P | P); it represents a variant emphasizing functional completeness in Boolean algebra.[31] The usage of these symbols varies across logical systems. In classical propositional logic, ¬ is the predominant notation for negation, as seen in standard formulations where it inverts truth values. The tilde ~ appears in some modal logics to denote negation, particularly in contexts distinguishing it from modal operators like necessity (□). In intuitionistic logic, negation is often represented by ¬ but semantically defined as implication to falsehood (¬P ≡ P → ⊥), avoiding the law of excluded middle; the exclamation mark ! occasionally appears in formal systems influenced by intuitionistic principles, such as certain proof-theoretic contexts.[32] The semantics of negation are captured succinctly in its truth table, which demonstrates its unary operation on propositional variables:| P | ¬P |
|---|---|
| True | False |
| False | True |
Computing and Electronics
NOT Operator in Programming
The NOT operator in programming serves as the practical implementation of logical negation, inverting boolean values or bit patterns in code to enable conditional logic and data manipulation.Logical NOT
The logical NOT operator inverts the truth value of its operand, converting true to false and false to true, and is typically used in conditional expressions for boolean operations. In C-like languages such as C, C++, and Java, this is represented by the unary ! operator, which evaluates to 1 (true) if the operand is false (0) and 0 (false) otherwise. For example, in C,!true evaluates to false, and !0 evaluates to true since 0 is treated as false.[33] This operator performs short-circuit evaluation in compound expressions but as a unary operator, it always evaluates its single operand.
In Python, the logical NOT is implemented as the keyword not, which also inverts the boolean value of its argument based on Python's truthiness rules—where False, None, zero, and empty collections are falsy, and others are truthy.[34] For instance, not True returns False, and not 0 returns True.[34] The not operator has lower precedence than non-boolean operators and supports short-circuit evaluation in chained logical expressions like not a or b.[34]
In SQL, the NOT operator negates the result of a boolean condition in WHERE clauses or other predicates, allowing queries to filter rows where the condition is false. For example, SELECT * FROM users WHERE NOT active = true retrieves inactive users, reversing the typical equality check. This operator handles NULL values by propagating UNKNOWN results in three-valued logic, where NOT NULL is NULL.
Bitwise NOT
The bitwise NOT operator inverts every bit in the binary representation of an integer operand, flipping 0s to 1s and 1s to 0s, and is commonly used for bit manipulation tasks like masking or sign extension. In languages like Java, C, and Python, it is denoted by the unary ~ operator and operates on the two's complement representation of integers. For example, in an 8-bit signed integer system, ~5 (binary 00000101) becomes 11111010, which is -6 in two's complement. In Python, due to arbitrary-precision integers, ~5 evaluates to -6, following the formula ~x = -x - 1 for positive x.[35] Language-specific behaviors vary; for instance, in Java, the ~ operator promotes the operand to int if it's a byte or short, performing the inversion on 32 bits, while in C, it applies to the promoted integer type. These operations do not short-circuit, as they are arithmetic rather than logical.[35]Common Errors
A frequent mistake is confusing logical and bitwise NOT, leading to unexpected results since they operate on different levels—logical on truth values and bitwise on binary digits. For example, in C or Java, !0 yields 1 (true), but ~0 yields -1 (all bits set to 1 in two's complement), which may cause issues in conditional statements if an integer is misused as a boolean.[36] Programmers often overlook integer promotion or sign extension in bitwise NOT, such as when applying ~ to a small positive number and getting a large negative result, altering program logic unintentionally.Performance Implications
Bitwise NOT operations are generally faster than equivalent logical checks for integer manipulations because they involve simple bit inversion without branching or type conversions, executing in constant time on the CPU's ALU. In contrast, logical NOT may incur minor overhead from boolean coercion in languages like Python, though modern compilers optimize both to near-equivalent assembly instructions for simple cases. For performance-critical code, such as in embedded systems, bitwise NOT is preferred for direct bit flipping over logical inversion followed by casting.NOT Gate in Digital Logic
The NOT gate, also known as an inverter, is a basic building block in digital electronics that performs logical negation on a binary input signal, producing an output that is the inverse of the input. It features a single input terminal and a single output terminal, making it the simplest unary logic gate. In practice, the NOT gate inverts a logic low (0) to a logic high (1) and vice versa, enabling signal reversal in circuit designs.[37] Electronically, the NOT gate is commonly implemented using transistor-based circuits, with the complementary metal-oxide-semiconductor (CMOS) configuration being a standard approach due to its low power consumption and high noise immunity. In a CMOS NOT gate, a p-type MOSFET (PMOS) transistor connects the output to the positive supply voltage, while an n-type MOSFET (NMOS) transistor connects the output to ground; both share the same input signal, which turns one on and the other off depending on the input level, thereby inverting the signal.[38] The logical behavior of the NOT gate is described by the Boolean equation Q = \overline{A}, where A is the input and Q is the output. This operation is summarized in the following truth table:| Input A | Output Q |
|---|---|
| 0 | 1 |
| 1 | 0 |