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Glossary Of Logic
This is a glossary of logic. Logic is the study of the principles of valid reasoning and argumentation. A B C D E F ... [...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 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] |
Analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 BC), though ''analysis'' as a formal concept is a relatively recent development. The word comes from the Ancient Greek (''analysis'', "a breaking-up" or "an untying" from ''ana-'' "up, throughout" and ''lysis'' "a loosening"). From it also comes the word's plural, ''analyses''. As a formal concept, the method has variously been ascribed to René Descartes ('' Discourse on the Method''), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name). The converse of analysis is synthesis: putting the pieces back together again in a new or different whole. Science and technology Chemistry The field of chemistry uses analysis in three ways: to i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotle's Sea Battle Argument
Future contingent propositions (or simply, future contingents) are statements about states of affairs in the future that are '' contingent:'' neither necessarily true nor necessarily false. The problem of future contingents seems to have been first discussed by Aristotle in chapter 9 of his ''On Interpretation'' (''De Interpretatione''), using the famous sea-battle example. Roughly a generation later, Diodorus Cronus from the Megarian school of philosophy stated a version of the problem in his notorious '' master argument''. The problem was later discussed by Leibniz. The problem can be expressed as follows. Suppose that a sea-battle will not be fought tomorrow. Then it was also true yesterday (and the week before, and last year) that it will not be fought, since any true statement about what will be the case in the future was also true in the past. But all past truths are now necessary truths; therefore it is now necessarily true in the past, prior and up to the original state ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotelian Logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge. Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it. Aristotle's system Aristotle's logical work is collected in the six texts that are collectively known as the '' Organon''. Two of these texts in particular, namely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, quantifies over relations. For example, the second-order sentence \forall P\,\forall x (Px \lor \neg Px) says that for every formula ''P'', and every individual ''x'', either ''Px'' is true or not(''Px'') is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, but no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persuasion. Arguments are intended to determine or show the degree of truth or acceptability of another statement called a conclusion. The process of crafting or delivering arguments, argumentation, can be studied from three main perspectives: the logical, the dialectical and the rhetorical perspective. In logic, an argument is usually expressed not in natural language but in a symbolic formal language, and it can be defined as any group of propositions of which one is claimed to follow from the others through deductively valid inferences that preserve truth from the premises to the conclusion. This logical perspective on argument is relevant for scientific fields such as mathematics and computer science. Logic is the study of the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Priori And A Posteriori
('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on experience. knowledge is independent from any experience. Examples include mathematics,Some associationist philosophers have contended that mathematics comes from experience and is not a form of any ''a priori'' knowledge () tautologies and deduction from pure reason. Galen Strawson has stated that an argument is one in which "you can see that it is true just lying on your couch. You don't have to get up off your couch and go outside and examine the way things are in the physical world. You don't have to do any science." () knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge. The terms originate from the analytic methods found in '' Organon'', a collection of works by Aristotle. Prior analytics () is about deductive logic, which comes from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antisymmetric Relation
In mathematics, a binary relation R on a set X is antisymmetric if there is no pair of ''distinct'' elements of X each of which is related by R to the other. More formally, R is antisymmetric precisely if for all a, b \in X, \text \,aRb\, \text \,a \neq b\, \text \,bRa\, \text, or equivalently, \text \,aRb\, \text \,bRa\, \text \,a = b. The definition of antisymmetry says nothing about whether aRa actually holds or not for any a. An antisymmetric relation R on a set X may be reflexive (that is, aRa for all a \in X), irreflexive (that is, aRa for no a \in X), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive. Examples The divisibility relation on the natural numbers is an important example of an antisymmetric relation. In this context, antisymmetry means that the only way each of two numbers can be divisible by the other is if the two are, in fact, the same number; equivalently, if n and m are distinct and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antinomy
In philosophy, an antinomy (Ancient Greek: 'against' + 'law') is a real or apparent contradiction between two conclusions, both of which seem justified. It is a term used in logic and epistemology, particularly in the philosophy of Immanuel Kant. Antinomy is a common form of argument in the dialogues of Plato. Kant credited Zeno of Elea (see Zeno's paradoxes) as the inventor of the antinomic mode of argumentation, which he described as a "skeptical method" of "watching, or rather provoking, a conflict of assertions, not for the purpose of deciding in favor of one or the other side, but of investigating whether the object of the controversy is not perhaps a deceptive appearance which each vainly tries to grasp, and in regard to which, even if there were no opposition to overcome, neither can arrive at any result". The antinomic procedure was further developed by Fichte, Schelling and Hegel. Hegel said that Kant was in error when he limited the antinomies to cosmological ideas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antecedent (logic)
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the ''protasis''. Examples: * If P, then Q. This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q. In the implication "\phi implies \psi", \phi is called the antecedent and \psi is called the consequent.Sets, Functions and Logic - An Introduction to Abstract Mathematics, Keith Devlin, Chapman & Hall/CRC Mathematics, 3rd ed., 2004 Antecedent and consequent are connected via logical connective to form a proposition. * If X is a man, then X is mortal. "X is a man" is the antecedent for this proposition while "X is mortal" is the consequent of the proposition. * If men have walked on the Moon, then I am the king of France. Here, "men have walked on the Moon" is the antecedent and "I am the king of France" is the consequent. Let y=x+1. * If x=1 then y=2,. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Closure
In mathematics, the transitive closure of a homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets is the unique minimal element, minimal transitive superset of . For example, if is a set of airports and means "there is a direct flight from airport to airport " (for and in ), then the transitive closure of on is the relation such that means "it is possible to fly from to in one or more flights". More formally, the transitive closure of a binary relation on a set is the smallest (w.r.t. ⊆) transitive relation on such that ⊆ ; see . We have = if, and only if, itself is transitive. Conversely, transitive reduction adduces a minimal relation from a given relation such that they have the same closure, that is, ; however, many differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancestral Relation
In mathematical logic, the ancestral relation (often shortened to ancestral) of a binary relation ''R'' is its transitive closure, however defined in a different way, see below. Ancestral relations make their first appearance in Frege's ''Begriffsschrift''. Frege later employed them in his ''Grundgesetze'' as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicist foundation of arithmetic. Definition The numbered propositions below are taken from his ''Begriffsschrift'' and recast in contemporary notation. A property ''P'' is called ''R''-hereditary if, whenever ''x'' is ''P'' and ''xRy'' holds, then ''y'' is also ''P'': :(Px \land xRy) \rightarrow Py An individual ''b'' is said to be an ''R''-ancestor of ''a'', written ''aR*b'', if ''b'' has every ''R''-hereditary property that all objects ''x'' such that ''aRx'' have: :\mathbf\ \vdash aR^*b \leftrightarrow \forall F forall x (aRx \to Fx) \land \forall x \forall y ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
