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Logical Equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \implies q \equiv \neg p \vee q :#p \implies q \equiv \neg q \implies \neg p :#p \vee q \equiv \neg p \implies q :#p \wedge q \equiv \neg (p \implies \neg q) :#\neg (p \implies q) \equiv p \wedge \neg q :#(p \implies q) \wedge (p \impli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-classical Logic
Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computerized proof methods may not use classical logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrow (symbol)
An arrow is a graphical symbol, such as ← or →, or a pictogram, used to point or indicate direction. In its simplest form, an arrow is a triangle, chevron, or concave kite, usually affixed to a line segment or rectangle, and in more complex forms a representation of an actual arrow (e.g. ➵ U+27B5). The direction indicated by an arrow is the one along the length of the line or rectangle toward the single pointed end. History An older (medieval) convention is the manicule (pointing hand, 👈). Pedro Reinel in c. 1504 first used the fleur-de-lis as indicating north in a compass rose; the convention of marking the eastern direction with a cross is older (medieval). Use of the arrow symbol does not appear to pre-date the 18th century. An early arrow symbol is found in an illustration of Bernard Forest de Bélidor's treatise ''L'architecture hydraulique'', printed in France in 1737. The arrow is here used to illustrate the direction of the flow of water and of the water w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blackboard Bold
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus, they are also referred to as double struck. In typography, such a font with characters that are not solid is called an "inline", "shaded", or "tooled" font. History Origin In some texts, these symbols are simply shown in bold type. Blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from non-bold letters (by using the edge rather than the point of a chalk). It then made its way back into print form as a separate style from ordinary bold, possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis. Use in textbooks In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrows (Unicode Block)
Arrows is a Unicode block A Unicode block is one of several contiguous ranges of numeric character codes ( code points) of the Unicode character set that are defined by the Unicode Consortium for administrative and documentation purposes. Typically, proposals such as the ... containing line, curve, and semicircle symbols terminating in barbs or arrows. Block Emoji The Arrows block contains eight emoji: U+2194–U+2199 and U+21A9–U+21AA. The block has sixteen standardized variants defined to specify emoji-style (U+FE0F VS16) or text presentation (U+FE0E VS15) for the eight emoji, all of which default to a text presentation. History The following Unicode-related documents record the purpose and process of defining specific characters in the Arrows block: See also * Mathematical operators and symbols in Unicode * Unicode input References {{reflist Unicode blocks ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Operators (Unicode Block)
Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation. Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Basic Latin Unicode block, and the plus-or-minus sign (±), multiplication sign (×) and obelus (÷), due to them already appearing in the Latin-1 Supplement block, although a distinct minus sign (−) is included, differing from the Basic Latin hyphen-minus (-). Block Variation sequences The Mathematical Operators block has sixteen variation sequences defined for standardized variants. They use (VS01) to denote variant symbols (depending on the font): History The following Unicode-related documents record the purpose and process of defining specific characters in the Mathematical Operators block: See also * Mathematical operators and symbols in Unicode * Supplemental Mathematical Operators Supplemental Mathematical Operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Equality
Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus. It gives the functional value ''true'' if both functional arguments have the same logical value, and '' false'' if they are different. It is customary practice in various applications, if not always technically precise, to indicate the operation of logical equality on the logical operands ''x'' and ''y'' by any of the following forms: :\begin x &\leftrightarrow y & x &\Leftrightarrow y & \mathrm Exy \\ x &\mathrm y & x &= y \end Some logicians, however, draw a firm distinction between a ''functional form'', like those in the left column, which they interpret as an application of a function to a pair of arguments — and thus a mere indication that the value of the compound expression depends on the values of the component expressions — and an ''equational form'', like those in the right column, which they interpret as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Biconditional
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as the '' antecedent'', and the ''consequent''. This is often abbreviated as " iff ". Other ways of denoting this operator may be seen occasionally, as a double-headed arrow (↔ or ⇔ may be represented in Unicode in various ways), a prefixed E "E''pq''" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡), or ''EQV''. It is logically equivalent to both (P \rightarrow Q) \land (Q \rightarrow P) and (P \land Q) \lor (\neg P \land \neg Q) , and the XNOR (exclusive nor) boolean operator, which means "both or neither". Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis is false but the conclusion is true. In this case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equisatisfiability
In Mathematical logic (a subtopic within the field of formal logic), two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models. Whereas within equisatisfiable formulae, only the primitive proposition the formula imposes is valued. Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable. Examples of translations involving this concept are Skolemization and some translations into conjunctive normal form In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
