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Propositional Formula
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as ''p'' and ''q'', using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example: : (''p'' AND NOT ''q'') IMPLIES (''p'' OR ''q''). In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that ''denotes'' a proposition, a formal object under discussion, just like an expression such as "" is not a value, but denotes a value. In some contexts, maintaining the distincti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
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 (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. Propositions that contain no logical connectives are called atomic propositions. 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 logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate (grammar)
The term predicate is used in one of two ways in linguistics and its subfields. The first defines a predicate as everything in a standard declarative sentence except the subject, and the other views it as just the main content verb or associated predicative expression of a clause. Thus, by the first definition the predicate of the sentence ''Frank likes cake'' is ''likes cake''. By the second definition, the predicate of the same sentence is just the content verb ''likes'', whereby ''Frank'' and ''cake'' are the arguments of this predicate. Differences between these two definitions can lead to confusion. Syntax Traditional grammar The notion of a predicate in traditional grammar traces back to Aristotelian logic. A predicate is seen as a property that a subject has or is characterized by. A predicate is therefore an expression that can be ''true of'' something. Thus, the expression "is moving" is true of anything that is moving. This classical understanding of predicates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Circuits
A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and unlimited fan-out, or it may refer to a non-ideal physical device (see Ideal and real op-amps for comparison). Logic gates are primarily implemented using diodes or transistors acting as electronic switches, but can also be constructed using vacuum tubes, electromagnetic relays (relay logic), fluidic logic, pneumatic logic, optics, molecules, or even mechanical elements. Now, most logic gates are made from MOSFETs (metal–oxide–semiconductor field-effect transistors). With amplification, logic gates can be cascaded in the same way that Boolean functions can be composed, allowing the construction of a physical model of all of Boolean logic, and therefore, all of the algorithms and mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Test
The Turing test, originally called the imitation game by Alan Turing in 1950, is a test of a machine's ability to artificial intelligence, exhibit intelligent behaviour equivalent to, or indistinguishable from, that of a human. Turing proposed that a human evaluator would judge natural language conversations between a human and a machine designed to generate human-like responses. The evaluator would be aware that one of the two partners in conversation was a machine, and all participants would be separated from one another. The conversation would be limited to a text-only channel, such as a computer keyboard and screen, so the result would not depend on the machine's ability to render words as speech. If the evaluator could not reliably tell the machine from the human, the machine would be said to have passed the test. The test results would not depend on the machine's ability to give correct question answering, answers to questions, only on how closely its answers resembled t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artificial Intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech recognition, computer vision, translation between (natural) languages, as well as other mappings of inputs. The ''Oxford English Dictionary'' of Oxford University Press defines artificial intelligence as: the theory and development of computer systems able to perform tasks that normally require human intelligence, such as visual perception, speech recognition, decision-making, and translation between languages. AI applications include advanced web search engines (e.g., Google), recommendation systems (used by YouTube, Amazon and Netflix), understanding human speech (such as Siri and Alexa), self-driving cars (e.g., Tesla), automated decision-making and competing at the highest level in strategic game systems (such as chess and Go). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Table
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. A truth table has one column for each input variable (for example, P and Q), and one final column showing all of the possible results of the logical operation that the table represents (for example, P XOR Q). Each row of the truth table contains one possible configuration of the input variables (for instance, P=true Q=false), and the result of the operation for those values. See the examples below for further clarification. Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is ''valid'' and all its premises are true. Some theorists define deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Psychology is interested in deductive reasoning as a psychological process, i.e. how people ''actually'' draw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x, y, \text z of S, x * (y + z) = (x * y) + (x * z); *the operation \,*\, is over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
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 (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. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or Quantifier (logic), quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (logical conjunction, conjunction), "or" (lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ..., linguistics and computer science. History In English, the study of meaning in language has been known by many names that involve the Ancient Greek word (''sema'', "sign, mark, token"). In 1690, a Greek rendering of the term ''semiotics'', the interpretation of signs and symbols, finds an early allusion in John Locke's ''An Essay Concerning Human Understanding'': The third Branch may be called [''simeiotikí'', "semiotics"], or the Doctrine of Signs, the most usual whereof being words, it is aptly enough ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntax
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituency), agreement, the nature of crosslinguistic variation, and the relationship between form and meaning (semantics). There are numerous approaches to syntax that differ in their central assumptions and goals. Etymology The word ''syntax'' comes from Ancient Greek roots: "coordination", which consists of ''syn'', "together", and ''táxis'', "ordering". Topics The field of syntax contains a number of various topics that a syntactic theory is often designed to handle. The relation between the topics is treated differently in different theories, and some of them may not be considered to be distinct but instead to be derived from one another (i.e. word order can be seen as the result of movement rules derived from grammatical relations). Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Quantifier
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. Other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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