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Continuous Predicate
{{technical, date=January 2020 ''Continuous predicate'' is a term coined by Charles Sanders Peirce (1839–1914) to describe a special type of relational predicate that results as the limit of a recursive process of hypostatic abstraction. Here is one of Peirce's definitive discussions of the concept: When we have analyzed a proposition so as to throw into the subject everything that can be removed from the predicate, all that it remains for the predicate to represent is the form of connection between the different subjects as expressed in the propositional ''form''. What I mean by "everything that can be removed from the predicate" is best explained by giving an example of something not so removable. But first take something removable. "Cain kills Abel." Here the predicate appears as "— kills —." But we can remove killing from the predicate and make the latter "— stands in the relation — to —." Suppose we attempt to remove more from the predicate and put the last ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for thirty years, Peirce made major contributions to logic, a subject that, for him, encompassed much of what is now called epistemology and the philosophy of science. He saw logic as the formal branch of semiotics, of which he is a founder, which foreshadowed the debate among logical positivists and proponents of philosophy of language that dominated 20th-century Western philosophy. Additionally, he defined the concept of abductive reasoning, as well as rigorously formulated mathematical induction and deductive reasoning. As early as 1886, he saw that logic gate, logical operations could be carried out by electrical switching circuits. The same idea was used decades later to produce digital computers. See Also In 1934, the philosopher Paul W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate (mathematics)
In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula P(a), the symbol P is a predicate which applies to the individual constant a. Similarly, in the formula R(a,b), R is a predicate which applies to the individual constants a and b. In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R(a,b) would be true on an interpretation if the entities denoted by a and b stand in the relation denoted by R. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation used to interpret them. While first-order logic only includes predicates which apply to individual constants, other logics may allow predicates which apply to other predicates. Predicates in different systems * In propositional logic, atomic formulas are sometimes regarded as zero-place predicates In a sense, these are nullary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit of a function is usually written as : \lim_ f(x) = L, (although a few authors may use "Lt" instead of "lim") and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or \rightarrow), as in :f(x) \to L \text x \to c, which reads "f of x tends to L as x tends to c". History Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work ''Opus Geometricum'' (1647): "The ''terminus'' of a pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypostatic Abstraction
Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal operation that transforms a predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ... into a Relation (philosophy), relation; for example "Honey ''is'' sweet" is transformed into "Honey ''has'' sweetness". The relation is created between the original subject and a new term that represents the Property (philosophy), property expressed by the original predicate. Description Technical definition Hypostasis changes a propositional formula of the form ''X is Y'' to another one of the form ''X has the property of being Y'' or ''X has Y-ness''. The logical functioning of the second object ''Y-ness'' consists solely in the truth-values of those propositions that have the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bradley's Regress
Bradley's regress is a philosophical problem concerning the nature of relations. It is named after F. H. Bradley who discussed the problem in his 1893 book ''Appearance and Reality''. It bears a close kinship to the issue of the unity of the proposition. Description Bradley raises the problem while discussing the bundle theory of objects, according to which an object is merely a "bundle" of properties. This theory raises the question of how the various properties that together comprise an object are related when they in fact comprise an object. More generally, the question that arises is what has to be the case for any two things to be related. Bradley's Regress appears to show that the notion of two things being related generates an infinite regress. Suppose, for example, that ''a'' respects ''b''. This state of affairs seems to involve three things: ''a'', ''b'', and the relation of respecting. For the state of affairs of ''a'' respecting ''b'' to obtain, it doesn't, ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unity Of The Proposition
In philosophy, the unity of the proposition is the problem of explaining how a sentence in the indicative mood expresses more than just what a list of proper names expresses. History The problem was discussed under this name by Bertrand Russell, but can be traced back to Plato. In Plato's ''Sophist'', the simplest kind of sentence consists of just a proper name and a universal term (i.e. a predicate). The name refers to or picks out some individual object, and the predicate then says something about that individual. The difficulty is to explain how the predicate does this. If, as Plato thinks, the predicate is the name of some universal concept or form, how do we explain how the sentence comes to be true or false? If, for example, "Socrates is wise" consists of just a name for Socrates, and a name for the universal concept of Wisdom, how could the sentence be true ''or'' false? In either case, the "Socrates" signifies Socrates, and the predicate signifies Wisdom. But the sen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip P
Philip, also Phillip, is a male given name, derived from the Greek (''Philippos'', lit. "horse-loving" or "fond of horses"), from a compound of (''philos'', "dear", "loved", "loving") and (''hippos'', "horse"). Prominent Philips who popularized the name include kings of Macedonia and one of the apostles of early Christianity. ''Philip'' has many alternative spellings. One derivation often used as a surname is Phillips. It was also found during ancient Greek times with two Ps as Philippides and Philippos. It has many diminutive (or even hypocoristic) forms including Phil, Philly, Lip, Pip, Pep or Peps. There are also feminine forms such as Philippine and Philippa. Antiquity Kings of Macedon * Philip I of Macedon * Philip II of Macedon, father of Alexander the Great * Philip III of Macedon, half-brother of Alexander the Great * Philip IV of Macedon * Philip V of Macedon New Testament * Philip the Apostle * Philip the Evangelist Others * Philippus of Croton (c. 6th centur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstraction
Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstraction" is the outcome of this process—a concept that acts as a common noun for all subordinate concepts and connects any related concepts as a ''group'', ''field'', or ''category''. Suzanne K. Langer (1953), ''Feeling and Form: a theory of art developed from Philosophy in a New Key'' p. 90: " Sculptural form is a powerful abstraction from actual objects and the three-dimensional space which we construe ... through touch and sight." Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only those aspects which are relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, 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 Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, 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 pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
