Ternary Logic
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'', and some third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for ''true'' and ''false''. Emil Leon Post is credited with first introducing additional logical truth degrees in his 1921 theory of elementary propositions. The conceptual form and basic ideas of three-valued logic were initially published by Jan Łukasiewicz and Clarence Irving Lewis. These were then re-formulated by Grigore Constantin Moisil in an axiomatic algebraic form, and also extended to ''n''-valued logics in 1945. Pre-discovery Around 1910, Charles Sanders Peirce defined a many-valued logic system. He never published it. In fact, he did not even number the three pages of notes where he defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Balanced Ternary
Balanced ternary is a ternary numeral system (i.e. base 3 with three Numerical digit, digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalanced) ternary system, in which digits have values 0, 1 and 2. The balanced ternary system can represent all integers without using a separate minus sign; the value of the leading non-zero digit of a number has the sign of the number itself. The balanced ternary system is an example of a Non-standard positional numeral systems, non-standard positional numeral system. It was used in some early computers and has also been used to solve balance puzzles. Different sources use different glyphs to represent the three digits in balanced ternary. In this article, T (which resembles a typographical ligature, ligature of the minus sign and 1) represents −1, while 0 and 1 represent themselves. Other conventions include using '−' and '+ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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And (logic)
In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or \times or \cdot in which \wedge is the most modern and widely used. The ''and'' of a set of operands is true if and only if ''all'' of its operands are true, i.e., A \land B is true if and only if A is true and B is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English language, English "Conjunction (grammar), and"; * In programming languages, the Short-circuit evaluation, short-circuit and Control flow, control structure; * In set theory, Intersection (set theory), intersection. * In Lattice (order), lattice theory, logical conjunction (Infimum and supremum, greatest lower bound). Notati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unary Operator
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to ''binary operations'', which use two operands. An example is any function , where is a set; the function is a unary operation on . Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial ), functional notation (e.g. or ), and superscripts (e.g. transpose ). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument. Examples Absolute value Obtaining the absolute value of a number is a unary operation. This function is defined as , n, = \begin n, & \mbox n\geq0 \\ -n, & \mbox n<0 \end where is the absolute value of . Negation |
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Connectives
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective \lor can be used to join the two atomic formulas P and Q, rendering the complex formula P \lor Q . Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations in nonclassical logics. Their classical interpretations are similar to the meanings of natural language expressions such as English "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning as well as approaches which pair a cla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Propositional Logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ternary Signal
In telecommunications, a ternary signal is a signal that can assume, at any given instant, one of three states or significant conditions, such as power level, phase position, pulse duration, or frequency. Examples of ternary signals are (a) a pulse that can have a positive, zero, or negative voltage value at any given instant ( PAM-3), (b) a sine wave that can assume phases of 0°, 120°, or 240° relative to a clock pulse (3- PSK), and (c) a carrier signal that can assume any one of three different frequencies depending on three different modulation signal significant conditions (3- FM). Some examples of PAM-3 line codes that use ternary signals are: * hybrid ternary code * bipolar encoding * MLT-3 encoding used in 100BASE-TX Ethernet * B3ZS * 4B3T used in some ISDN basic rate interface * 8B6T used in 100BASE-T4 Ethernet * return-to-zero * SOQPSK-TG uses ternary continuous phase modulation 3-PSK can be seen as falling between "binary phase-shift keying" ( BPSK), w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ternary Computer
A ternary computer, also called trinary computer, is one that uses ternary logic (i.e., base 3) instead of the more common binary system (i.e., base 2) in its calculations. Ternary computers use trits, instead of binary bits. Types of states Ternary computing deals with three discrete states, but the ternary digits themselves can be defined differently: Ternary quantum computers use qutrits rather than trits. A qutrit is a quantum state that is a complex unit vector in three dimensions, which can be written as , \Psi\rangle = \alpha, 0\rangle + \beta, 1\rangle + \gamma, 2\rangle in the bra-ket notation. The labels given to the basis vectors (, 0\rangle, , 1\rangle, , 2\rangle) can be replaced with other labels, for example those given above. History One early calculating machine, built entirely from wood by Thomas Fowler in 1840, operated in balanced ternary. The first modern, electronic ternary computer, Setun, was built in 1958 in the Soviet Union at the Moscow St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sigma Xi
Sigma Xi, The Scientific Research Honor Society () is an international non-profit honor society for scientists and engineers. Sigma Xi was founded at Cornell University by a faculty member and graduate students in 1886 and is one of the oldest honor societies. Membership in Sigma Xi is by invitation only, where members nominate others on the basis of their research achievements or potential. The society was a founding member of the Association of College Honor Societies in 1925, but withdrew in 1933 and much later was a founder of to form the Honor Society Caucus. History Sigma Xi was founded in November 1886 at the Sibling College of Mechanical Engineering at Cornell University in Ithaca, New York.Shepardson, Francis Wayland, ed. Baird's Manual of American College Fraternities, 12th edition'. Menasha, Wisconsin: The Collegiate Press/George Banta Publishing Company, 1930. pp. 369-371. ''via'' Hathi Trust. Its founders were Henry Shaler Williams, a Cornell faculty member, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Scientist
''American Scientist'' (informally abbreviated ''AmSci'') is an American bimonthly science and technology magazine published since 1913 by Sigma Xi, The Scientific Research Honor Society. In the beginning of 2000s the headquarters was moved to Research Triangle Park, (Durham), North Carolina. Each issue includes feature articles written by scientists and engineers who review research in fields from molecular biology to computer engineering. Each issue also includes the work of cartoonist A cartoonist is a visual artist who specializes in both drawing and writing cartoons (individual images) or comics (sequential images). Cartoonists differ from comics writers or comics illustrators/artists in that they produce both the litera ...s, including those of Sidney Harris, Benita Epstein, and Mark Heath. Also included is the ''Scientists' Nightstand'' that reviews a vast range of science-related books and novels. ''American Scientist Online'' () was launched in May 2003. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Undecidable Problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether an arbitrary program eventually halts when run. Background A decision problem is a question which, for every input in some infinite set of inputs, requires a "yes" or "no" answer. Those inputs can be numbers (for example, the decision problem "is the input a prime number?") or values of some other kind, such as strings of a formal language. The formal representation of a decision problem is a subset of the natural numbers. For decision problems on natural numbers, the set consists of those numbers that the decision problem answers "yes" to. For example, the decision problem "is the input even?" is formalized as the set of even numbers. A decision pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Skew Binary Number System
The skew binary number system is a non-standard positional numeral system in which the ''n''th digit contributes a value of 2^ - 1 times the digit (digits are indexed from 0) instead of 2^ times as they do in binary. Each digit has a value of 0, 1, or 2. A number can have many skew binary representations. For example, a decimal number 15 can be written as 1000, 201 and 122. Each number can be written uniquely in skew binary canonical form where there is only at most one instance of the digit 2, which must be the least significant nonzero digit. In this case 15 is written canonically as 1000. Examples Canonical skew binary representations of the numbers from 0 to 15 are shown in following table: Arithmetical operations The advantage of skew binary is that each increment operation can be done with at most one carry operation. This exploits the fact that 2 (2^ - 1) + 1 = 2^ - 1 . Incrementing a skew binary number is done by setting the only two to a zero and incrementing the ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |