107 Sentences With "logical system" | Random Sentence Generator

Ironically, by designing a logical system focused on health, we actually get a system that costs less.

Our system of admittance is a decades old program based largely on nepotism and family ties, rather than any logical system of governance that benefits Americans.

When that is done, your product and investor materials need to be sufficiently robust to allow their apparently logical System 2 to confirm their bias toward you.

In 2015, GM was the first automaker to integrate Apple CarPlay and Android Auto across its entire product lineup, which was a massive step toward making a more logical system that customers found intuitive.

There is over $2 trillion in so-called "unremitted earnings" in foreign countries, some of which, under a more logical system, would be brought back to the U.S. The second effect is more drastic.

Sure, they crashed sometimes, but losing a little unsaved work or having to restart occasionally seemed like a small price to pay for not having to tediously spell out every little piece of a program in the language of a formal logical system.

Conversely, a logic system is complete if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms. An example of a logical system is Peano arithmetic.

A logical system or language (not be confused with the kind of "formal language" discussed above which is described by a formal grammar), is a deductive system (see section above; most commonly first order predicate logic) together with additional (non-logical) axioms and a semantics. According to model-theoretic interpretation, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure. A structure that satisfies all the axioms of the formal system is known as a model of the logical system. A logical system is sound if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.

While Russell's formalization did not contain such paradoxes, Kurt Gödel showed that it must contain independent statements. Any logical system that is rich enough to contain elementary arithmetic contains at least one proposition whose interpretation is this proposition is unprovable (from within the logical system concerned), and hence no such system can be both complete and consistent.

The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved. Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem.

In order to avoid a trivial logical system and still allow certain contradictions to be true, dialetheists will employ a paraconsistent logic of some kind.

Logical harmony, a name coined by Michael Dummett, is a supposed constraint on the rules of inference that can be used in a given logical system.

There is also such a thing as a logical system. The most obvious example is the calculus developed simultaneously by Leibniz and Isaac Newton. Another example is George Boole's Boolean operators. Other examples have related specifically to philosophy, biology, or cognitive science.

The records management phase of the records life-cycle consists of creation, classification, maintenance and disposition. Creation occurs during the receipt of information in the form of records. Records or their information is classified in some logical system. As records are used they require maintenance.

He says "a logical system is possible, but an existential system is impossible."Jean T.Wilde and William Kimmel, eds.,The Search for Being (1962) New York:Twayne, p. 51-52 In 1963 Kenneth Hamilton described Paul Tillich as an individual who was as anti-Hegel as Kierkegaard was.

The modus ponens rule may be written in sequent notation as :P \to Q,\; P\;\; \vdash\;\; Q where P, Q and P → Q are statements (or propositions) in a formal language and ⊢ is a metalogical symbol meaning that Q is a syntactic consequence of P and P → Q in some logical system.

The fragment of TL without weak negation and the implication operator is classical logic. TL is thus a logical blend or rather a crossbreed. Peña's plan to investigate the grounds of his logical system as a nonclassical combinatory logic has thus far remained programmatic, but the combinatory account fits his metaphysical approach.

These show Blackstone's attempts to reduce English law to a logical system, with the division of subjects later being the basis for his Commentaries.Prest (2008) pp. 115-7Simpson (1981) p. 652 The lecture series brought him £116, £226 and £111 a year respectively from 1753 to 1755 — a total of £ in terms.

In contrast to the Planning Domain Definition Language, Golog supports planning and scripting as well. Planning means that a goal state in the world model is defined, and the solver brings a logical system into this state. Behavior scripting implements reactive procedures, which are running as a computer program. For example, suppose the idea is to authoring a story.

W. V. Quine's Mathematical Logic also made much of the Sheffer stroke. A Sheffer connective, subsequently, is any connective in a logical system that functions analogously: one in terms of which all other possible connectives in the language can be expressed. For example, they have been developed for quantificational and modal logics as well. Sheffer was a dedicated teacher of mathematical logic.

Formal ethics is a formal logical system for describing and evaluating the "form" as opposed to the "content" of ethical principles. Formal ethics was introduced by Harry J. Gensler, in part in his 1990 logic textbook Symbolic Logic: Classical and Advanced Systems,Gensler, Harry J. Symbolic logic: Classical and advanced systems. Prentice Hall, 1990. but was more fully developed and justified in his 1996 book Formal Ethics.

John Ruskin's Study of Gneiss Rock, Glenfinlas, 1853. Pen and ink and wash with Chinese ink on paper, Ashmolean Museum, Oxford, England. Ruskin's views on art, wrote Kenneth Clark, "cannot be made to form a logical system, and perhaps owe to this fact a part of their value." Ruskin's accounts of art are descriptions of a superior type that conjure images vividly in the mind's eye.

Transparent intensional logic (frequently abbreviated as TIL) is a logical system created by Pavel Tichý. Due to its rich procedural semantics TIL is in particular apt for the logical analysis of natural language. From the formal point of view, TIL is a hyperintensional, partial, typed lambda calculus. TIL applications cover a wide range of topics from formal semantics, philosophy of language, epistemic logic, philosophical, and formal logic.

The disjunction introduction rule may be written in sequent notation: : P \vdash (P \lor Q) where \vdash is a metalogical symbol meaning that P \lor Q is a syntactic consequence of P in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: :P \to (P \lor Q) where P and Q are propositions expressed in some formal system.

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions.

Although some basic novelties syncretizing mathematical and philosophical logic were shown by Bolzano in the early 1800s, it was Ernst Mally, a pupil of Alexius Meinong, who was to propose the first formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's propositional calculus. Another logical system founded after World War II was fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965.

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry Gensler, Introduction to Logic, Routledge, 2001, p. 336. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

It is a work (he says against Ong) of a rooted scholar with a "method" but turning Ramism back on itself.Geoffrey Hill, Collected Critical Writings (2008), editor Kenneth Haynes, p. 298 and p. 332. Samuel Taylor Coleridge combined Aristotelian logic with the Holy Trinity to create his "cinque spotted spider making its way upstream by fits & starts," his logical system based on Ramist logic (thesis, antithesis, synthesis, mesothesis, exothesis).

Dialetheic logics, which are also many-valued, are paraconsistent, but the converse does not hold. Intuitionistic logic allows A ∨ ¬A not to be equivalent to true, while paraconsistent logic allows A ∧ ¬A not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the "dual" of intuitionistic logic. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems.

In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire space or Cantor space) is called universally Baire if it has a certain strong regularity property. Universally Baire sets play an important role in Ω-logic, a very strong logical system invented by W. Hugh Woodin and the centerpiece of his argument against the continuum hypothesis of Georg Cantor.

Most of Lewis’ lasting interests originated during his Harvard years. The most important was thermodynamics, a subject in which Richards was very active at that time. Although most of the important thermodynamic relations were known by 1895, they were seen as isolated equations, and had not yet been rationalized as a logical system, from which, given one relation, the rest could be derived. Moreover, these relations were inexact, applying only to ideal chemical systems.

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Propositional and predicate calculi are examples of formal systems.

In 1922 de Hevesy co-discovered (with Dirk Coster) the element hafnium (72Hf) (Latin Hafnia for "Copenhagen", the home town of Niels Bohr). Mendeleev's 1869 periodic table arranged the chemical elements into a logical system, but a chemical element with 72 protons was missing. Hevesy determined to look for that element on the basis of Bohr's atomic model. The mineralogical museum of Norway and Greenland in Copenhagen furnished the material for the research.

The notion of institution was created by Joseph Goguen and Rod Burstall in the late 1970s, in order to deal with the "population explosion among the logical systems used in computer science". The notion tries to capture the essence of the concept of "logical system".J. A. Goguen and R. M. Burstall, Institutions: Abstract Model Theory for Specification and Programming, Journal of the Association for Computing Machinery 39, pp. 95–146, 1992.

Because a function may be expressed as a composition, a truth-functional logical calculus does not need to have dedicated symbols for all of the above-mentioned functions to be functionally complete. This is expressed in a propositional calculus as logical equivalence of certain compound statements. For example, classical logic has equivalent to . The conditional operator "→" is therefore not necessary for a classical-based logical system if "¬" (not) and "∨" (or) are already in use.

The same logical system was used to communicate between programs running on one transputer, implemented as virtual network links in memory. So programs asking for any input or output automatically paused while the operation completed, a task that normally required an operating system to handle as the arbiter of hardware. Operating systems on the transputer did not need to handle scheduling; the chip could be considered to have an OS inside it.

The disjunctive syllogism rule may be written in sequent notation: : P \lor Q, \lnot P \vdash Q where \vdash is a metalogical symbol meaning that Q is a syntactic consequence of P \lor Q, and \lnot P in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: : ((P \lor Q) \land eg P) \to Q where P, and Q are propositions expressed in some formal system.

Most of mathematics can be implemented in standard set theory or one of its large alternatives. Set theories, on the other hand, are introduced in terms of a logical system; in most cases it is first-order logic. The syntax and semantics of first-order logic, on the other hand, is built on set-theoretical grounds. Thus, there is a foundational circularity, which forces us to choose as weak a theory as possible for bootstrapping.

In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, xy, together with induction for formulas with bounded quantifiers. EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.

Matching bias is a non-logical heuristic. The matching bias is described as a tendency to use lexical content matching of the statement about which one is reasoning, to be seen as relevant information and do the opposite as well, ignore relevant information that doesn't match. It mostly affects problems with abstract content. It doesn't involve prior knowledge and beliefs but it is still seen as a System 1 heuristic that competes with the logical System 2.

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations.

The mirror neuron system would be a logical system to derive primary benefit from an increase in plasticity under the conditions proposed. If this adaptation were indeed a result of the mastery of fire and the additional nutrition and food security brought about by cooking, changes in communal relations would be a logical co-requisite development, as fire promised warmth and security, and the invention of cooking would have provided an easier way of feeding larger groups.

In logic, William of Ockham wrote down in words the formulae that would later be called De Morgan's Laws,In his Summa Logicae, part II, sections 32 and 33.Translated on page 80 of Philosophical Writings, tr. P. Boehner, rev. S. Brown, (Indianapolis, IN, 1990) and he pondered ternary logic, that is, a logical system with three truth values; a concept that would be taken up again in the mathematical logic of the 19th and 20th centuries.

Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers. Giuseppe Peano (1889) published a set of axioms for arithmetic that came to bear his name (Peano axioms), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time.

Most criticism of Gödel's proof is aimed at its axioms: as with any proof in any logical system, if the axioms the proof depends on are doubted, then the conclusions can be doubted. It is particularly applicable to Gödel's proof – because it rests on five axioms, some of which are questionable. A proof does not necessitate that the conclusion be correct, but rather that by accepting the axioms, the conclusion follows logically. Many philosophers have called the axioms into question.

A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistency-tolerant") systems of logic. Inconsistency- tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ("beside the consistent") was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada Cantuarias.Priest (2002), p.

The conjunction elimination sub- rules may be written in sequent notation: : (P \land Q) \vdash P and : (P \land Q) \vdash Q where \vdash is a metalogical symbol meaning that P is a syntactic consequence of P \land Q and Q is also a syntactic consequence of P \land Q in logical system; and expressed as truth-functional tautologies or theorems of propositional logic: :(P \land Q) \to P and :(P \land Q) \to Q where P and Q are propositions expressed in some formal system.

The significance of the document lies in the longevity and wide application of its logical system of diacritics. Its impact is apparent in the Náměšťská mluvnice ("Grammar of Náměšť"), the first grammar of the Czech language, published in 1533, but the adoption of the new rules was relatively slow and far from uniform. Throughout the 16th century, some printers and typesetters ignored the prescriptions of Orthographia bohemica and continued to maintain some digraphs (e.g. ss for instead of š), although their use became considerably more uniform.

The disjunction elimination rule may be written in sequent notation: : (P \to Q), (R \to Q), (P \lor R) \vdash Q where \vdash is a metalogical symbol meaning that Q is a syntactic consequence of P \to Q, and R \to Q and P \lor R in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: :(((P \to Q) \land (R \to Q)) \land (P \lor R)) \to Q where P, Q, and R are propositions expressed in some formal system.

The fact that every non-negative integer has a unique representation in bijective base-k (k ≥ 1) is a "folk theorem" that has been rediscovered many times. Early instances are for the case k = 10, and and for all k ≥ 1. Smullyan uses this system to provide a Gödel numbering of the strings of symbols in a logical system; Böhm uses these representations to perform computations in the programming language P′′. mentions the special case of k = 10, and discusses the cases k ≥ 2.

The constructive dilemma rule may be written in sequent notation: : (P \to Q), (R \to S), (P \lor R) \vdash (Q \lor S) where \vdash is a metalogical symbol meaning that Q \lor S is a syntactic consequence of P \to Q, R \to S, and P \lor R in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: :(((P \to Q) \land (R \to S)) \land (P \lor R)) \to (Q \lor S) where P, Q, R and S are propositions expressed in some formal system.

The absorption rule may be expressed as a sequent: : P \to Q \vdash P \to (P \land Q) where \vdash is a metalogical symbol meaning that P \to (P \land Q) is a syntactic consequence of (P \rightarrow Q) in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: :(P \to Q) \leftrightarrow (P \to (P \land Q)) where P, and Q are propositions expressed in some formal system.

In logic, a rule of replacementMoore and Parker is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other.

Such a logical connective as converse implication "←" is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic), certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the classical equivalence between and . Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugar for a compound having one negation and one disjunction.

The use of institutions makes it possible to develop concepts of specification languages (like structuring of specifications, parameterization, implementation, refinement, development), proof calculi and even tools in a way completely independent of the underlying logical system. There are also morphisms that allow to relate and translate logical systems. Important applications of this are re-use of logical structure (also called borrowing), heterogeneous specification and combination of logics. The spread of institutional model theory has generalized various notions and results of model theory, and institutions themselves have impacted the progress of universal logic.

Sri Aurobindo believes this is due to the fact that the intuitive knowledge sees things in the whole and only sides of an indivisible whole, while reason on the contrary proceeds by analysis, dissemination and assembles its facts to form a whole. leading into assembly of facts so formed would contain opposites, anomalies, logical incompatibilities and tendencies to affirm some and negate others which conflict with its chosen conclusions so that it may form a flawlessly logical system, which has resulted in different school of thoughts in Hinduism.

The Cambodian hydro- logical system is dominated by the Mekong River and Tonle Sap Great Lake. From July to the end of October, when the level of the Mekong is high, water flows via the Tonle Sap River, increasing the size of the lake from 2,600 km2 to about at its maximum extent. The storage capacity of Tonle Sap Great Lake is about . When the level of the Mekong decreases, the Tonle Sap River reverses its flow and water flows back from the Tonle Sap Lake into the Mekong River.

Detail from Raphael's The School of Athens featuring a Greek mathematician – perhaps representing Euclid or Archimedes – using a compass to draw a geometric construction. Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.

The metamathematical value of the principle of explosion is that for any logical system where this principle holds, any derived theory which proves ⊥ (or an equivalent form, \phi \land \lnot \phi) is worthless because all its statements would become theorems, making it impossible to distinguish truth from falsehood. That is to say, the principle of explosion is an argument for the law of non-contradiction in classical logic, because without it all truth statements become meaningless. Reduction in proof strength of logics without ex falso are discussed in minimal logic.

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Formal systems, like other syntactic entities may be defined without any interpretation given to it (as being, for instance, a system of arithmetic).

It is a formal logical system with its content targeted at the real things, information and thoughts that we experienced. As Francis Bacon pointed out in the 17th century, experimental verification of the propositions must be carried out rigorously and cannot take logic itself as the way to draw conclusions in nature. Formal science is a method that is helpful to science but cannot replace science. Although formal sciences are conceptual systems, lacking empirical content, this does not mean that they have no relation to the real world.

In 1847 Boole published the pamphlet Mathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system and wanted An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities to be seen as the mature statement of his views. Contrary to widespread belief, Boole never intended to criticise or disagree with the main principles of Aristotle's logic. Rather he intended to systematise it, to provide it with a foundation, and to extend its range of applicability.

The Common Law has been continuously in print since 1881, and remains an important contribution to jurisprudence. The book also remains controversial, for Holmes begins by rejecting various kinds of formalism in law. In his earlier writings he had expressly denied the utilitarian view that law was a set of commands of the sovereign, rules of conduct that became legal duties. He rejected as well the views of the German idealist philosophers, whose views were then widely held, and the philosophy taught at Harvard, that the opinions of judges could be harmonized in a purely logical system.

This level of abstraction coincides with the view that the correctness of the input/output behaviour of a program takes precedence over all its other properties. In the property- oriented approach to specification (taken e.g. by CASL), specifications of programs consist mainly of logical axioms, usually in a logical system in which equality has a prominent role, describing the properties that the functions are required to satisfy — often just by their interrelationship. This is in contrast to so-called model-oriented specification in frameworks like VDM and Z, which consist of a simple realization of the required behaviour.

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described (although non-rigorously by modern standards) in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof.

From earliest infancy children were attracted by colors; Hughey had simply taken advantage of this fact and developed a natural and logical system from it. A thorough test of the color method had been made before presenting it to the public, and the eagerness with which children took up the lessons, and their enthusiasm over the work, had amply demonstrated its practicability. By its means the drudgery attendant upon the first period of the child's musical studies was entirely eliminated. Work became play; rapid progress was made in ear training, in accuracy in determining intervals, and in technic.

When viewed as a programming language, Coq implements a dependently typed functional programming language;A short introduction to Coq when viewed as a logical system, it implements a higher-order type theory. The development of Coq has been supported since 1984 by INRIA, now in collaboration with École Polytechnique, University of Paris-Sud, Paris Diderot University, and CNRS. In the 1990s, ENS Lyon was also part of the project. The development of Coq was initiated by Gérard Huet and Thierry Coquand, and more than 40 people, mainly researchers, have contributed features to the core system since its inception.

Zinoviev was engaged in logic not just as a scientific discipline, but reconsidered its foundations as part of the creation of a new field of intellectual activity. According to Konstantin Krylov, he experienced a temporary stage of creating a "general theory of everything", which, however, he quickly passed. It is noted that in logical studies Zinoviev was clearly vain, which led to imprudent steps and embarrassment: for example, he insistently published proof of Fermat's unprovability in the framework of the logical system he built. At Moscow State University, Zinoviev formed a group of followers from domestic and foreign students and graduate students.

Earlier (colonial) observers on Azande witchcraft frequently cast the practice as belonging to a primitive people. Anthropologist E. E. Evans-Pritchard (who acknowledged the importance of the work done by Claude Lévi-Strauss) argued that the pervasive belief in witchcraft was a belief system not essentially different from other world religions; Azande witchcraft is a coherent and logical system of ideas. Evans-Pritchard's Witchcraft, Oracles, and Magic among the Azande (1937) is a standard reference work on Azande witchcraft. It has been subjected to a number of reviews, and is seen as a "turning point in the evaluation of 'primitive thought'".

GameDaily's Big Download considered Simutrans to be one of the best freeware games, highlighting the logical system of routing passengers and freight to their destinations, decent AI opponents and the support for custom aesthetics or rules-sets. However, the sometimes unreliable vehicle pathfinding was criticized, particularly with respect to alternate routes and switches for train lines. The sound effects were deemed to be unengaging, and new players may be baffled by the range of transportation possibilities. Another review from Amiga Future came to very similar conclusions (apart from the lack of sound support on Amiga OS).

As is common in slums and ghettos worldwide, a new road network planned to encircle Lusaka will likely displace residents in many kombonis and bifurcate communities. Roads in the wealthier planned neighborhoods tend to be logical, laid out following a grid or other logical system, whereas in the kombonis the roads tend to be irregular, often lacking names or signs. Commercial maps often place advertisements over kombonis, making it impossible to use the maps to navigate those areas. Public transportation in Lusaka tends to take passengers from the kombonis to downtown, but not to other kombonis.

The biconditional elimination rule may be written in sequent notation: :(P \leftrightarrow Q) \vdash (P \to Q) and :(P \leftrightarrow Q) \vdash (Q \to P) where \vdash is a metalogical symbol meaning that P \to Q, in the first case, and Q \to P in the other are syntactic consequences of P \leftrightarrow Q in some logical system; or as the statement of a truth- functional tautology or theorem of propositional logic: :(P \leftrightarrow Q) \to (P \to Q) :(P \leftrightarrow Q) \to (Q \to P) where P, and Q are propositions expressed in some formal system.

The Pythagoreans regarded numerology and geometry as fundamental to understanding the nature of the universe and therefore central to their philosophical and religious ideas. They are credited with numerous mathematical advances, such as the discovery of irrational numbers. Historians credit them with a major role in the development of Greek mathematics (particularly number theory and geometry) into a coherent logical system based on clear definitions and proven theorems that was considered to be a subject worthy of study in its own right, without regard to the practical applications that had been the primary concern of the Egyptians and Babylonians.

In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics (Katz 1998, p. 686). Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift, published in 1879, a work generally considered as marking a turning point in the history of logic.

The destructive dilemma rule may be written in sequent notation: : (P \to Q), (R \to S), ( eg Q \lor eg S) \vdash ( eg P \lor eg R) where \vdash is a metalogical symbol meaning that eg P \lor eg R is a syntactic consequence of P \to Q, R \to S, and eg Q \lor eg S in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: :(((P \to Q) \land (R \to S)) \land ( eg Q \lor eg S)) \to ( eg P \lor eg R) where P, Q, R and S are propositions expressed in some formal system.

The ergonomists and psychologists believed in analysis and experiment as the basis for design, whereas the stylists were mostly concerned with form, and had evolved design rules about proportion, colour and texture which they thought of as a logical system for creating the cool, minimalist look for which Ulm became famous.Krampen, M and Hőrmann, 'The Ulm School of Design - Beginnings of a Project of Unyielding Modernity', Ernst&Sohn;, Berlin, 2003. . Archer tried to convey each side's belief systems across the divide, but each group thought he had aligned himself with the other. Maldonado had left Ulm even before Archer arrived, and he found himself isolated.

In a formal logical system, that is, a set of propositions that are consistent with one another, it is possible that some of the statements can be deduced from other statements. For example, in the syllogism, "All men are mortal; Socrates is a man; Socrates is mortal" the last claim can be deduced from the first two. A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of geometric propositions can be deduced from a set of definitions, postulates, and common notions: all three types constitute first principles.

A belief bias is the tendency to judge the strength of arguments based on the plausibility of their conclusion rather than how strongly they support that conclusion. Some evidence suggests that this bias results from competition between logical (System 2) and belief-based (System 1) processes during evaluation of arguments. Studies on belief-bias effect were first designed by Jonathan Evans to create a conflict between logical reasoning and prior knowledge about the truth of conclusions. Participants are asked to evaluate syllogisms that are: valid arguments with believable conclusions, valid arguments with unbelievable conclusions, invalid arguments with believable conclusions, and invalid arguments with unbelievable conclusions.

In programming languages, a type system is a logical system comprising a set of rules that assigns a property called a type to the various constructs of a computer program, such as variables, expressions, functions or modules. These types formalize and enforce the otherwise implicit categories the programmer uses for algebraic data types, data structures, or other components (e.g. "string", "array of float", "function returning boolean"). The main purpose of a type system is to reduce possibilities for bugs in computer programs by defining interfaces between different parts of a computer program, and then checking that the parts have been connected in a consistent way.

Boolean algebra is the starting point of mathematical logic and has important applications in computer science. Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. The German mathematician Gottlob Frege (1848–1925) presented an independent development of logic with quantifiers in his Begriffsschrift (formula language) published in 1879, a work generally considered as marking a turning point in the history of logic. He exposed deficiencies in Aristotle's Logic, and pointed out the three expected properties of a mathematical theory # Consistency: impossibility of proving contradictory statements.

As noted by Weyl, formal logical systems also run the risk of inconsistency; in Peano arithmetic, this arguably has already been settled with several proofs of consistency, but there is debate over whether or not they are sufficiently finitary to be meaningful. Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency. What Hilbert wanted to do was prove a logical system S was consistent, based on principles P that only made up a small part of S. But Gödel proved that the principles P could not even prove P to be consistent, let alone S.

A conceptual model is a representation of some phenomenon, data or theory by logical and mathematical objects such as functions, relations, tables, stochastic processes, formulas, axiom systems, rules of inference etc. A conceptual model has an ontology, that is the set of expressions in the model which are intended to denote some aspect of the modeled object. Here we are deliberately vague as to how expressions are constructed in a model and particularly what the logical structure of formulas in a model actually is. In fact, we have made no assumption that models are encoded in any formal logical system at all, although we briefly address this issue below.

Logic is concerned with the patterns in reason that can help tell us if a proposition is true or not. Logicians use formal languages to express the truths which they are concerned with, and as such there is only truth under some interpretation or truth within some logical system. A logical truth (also called an analytic truth or a necessary truth) is a statement which is true in all possible worldsLudwig Wittgenstein, Tractatus Logico-Philosophicus. or under all possible interpretations, as contrasted to a fact (also called a synthetic claim or a contingency) which is only true in this world as it has historically unfolded.

And the whole Universe or Substance is conceived as one dynamic system of which the various Attributes are the several world-lines along which it expresses itself in all the infinite variety of events.See also The Short Treatise on God, Man and his Well-being, London: A. & C. Black, 2006 – scanned, University of Toronto, Internet Archive. Given the persistent misinterpretation of Spinozism it is worth emphasizing the dynamic character of reality as Spinoza conceived it. The cosmic system is certainly a logical or rational system, according to Spinoza, for Thought is a constitutive part of it; but it is not merely a logical system — it is dynamic as well as logical.

Stephen Crain is the director of the ARC Centre of Excellence in Cognition and its Disorders (CCD),ARC Centre of Excellence in Cognition and its Disorders and a distinguished professor at Macquarie University in the Department of Linguistics. He is a well-known researcher specializing in language acquisition, focusing specifically on syntax and semantics. Crain views language acquisition as based on language-specific faculties, and he conducts his research in the tradition of Chomskyan generative grammar. Recently, Crain has proposed that language is based on a universal logical system, and he has begun to explore the neural correlates of language acquisition from a cross- linguistic perspective using magnetoencephalography (MEG).

A paradox usually involves contradictory-yet- interrelated elements that exist simultaneously and persist over time. In logic, many paradoxes exist which are known to be invalid arguments, but which are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions which were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system.

Some proof calculi will only deal with a theory whose formulae are written in prenex normal form. The concept is essential for developing the arithmetical hierarchy and the analytical hierarchy. Gödel's proof of his completeness theorem for first-order logic presupposes that all formulae have been recast in prenex normal form. Tarski's axioms for geometry is a logical system whose sentences can all be written in universal- existential form, a special case of the prenex normal form that has every universal quantifier preceding any existential quantifier, so that all sentences can be rewritten in the form \forall u \forall v \ldots \exists a \exists b \phi, where \phi is a sentence that does not contain any quantifier.

Why is it that according to Nyaya logic, it is possible to establish the existence of God while in the Buddhist logical system it is not possible to establish the existence of God? An adequate answer to this question lies in the concept of knowledge of the different systems leading to different kinds of understanding of human beings themselves. Thus Nyaya system has as horizon a theory of knowledge which renders possible the discourse about God; it could even be asserted that according to Nyaya, the Absolute becomes the horizon of all knowledge and therefore also of all human activities. Such an understanding of Nyaya helped him to develop his own philosophy.

Books such as those by H.S.M. Coxeter routinely used the Erlangen program approach to help 'place' geometries. In pedagogic terms, the program became transformation geometry, a mixed blessing in the sense that it builds on stronger intuitions than the style of Euclid, but is less easily converted into a logical system. In his book Structuralism (1970) Jean Piaget says, "In the eyes of contemporary structuralist mathematicians, like Bourbaki, the Erlangen Program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of structure." For a geometry and its group, an element of the group is sometimes called a motion of the geometry.

In propositional logic, tautological consequence is a strict form of logical consequenceBarwise and Etchemendy 1999, p. 110 in which the tautologousness of a proposition is preserved from one line of a proof to the next. Not all logical consequences are tautological consequences. A proposition Q is said to be a tautological consequence of one or more other propositions (P_1, P_2, ..., P_n) in a proof with respect to some logical system if one is validly able to introduce the proposition onto a line of the proof within the rules of the system and in all cases when each of those one or more other propositions (P_1, P_2, ..., P_n) are true, the proposition Q also is true.

Rear view of a stack of 8 switches The ERS 2500 Series of Switches can be stacked with Flexible Advanced Stacking Technology (FAST) to allows eight switches to operate as single logical system with a 32 Gbit/s virtual backplane. The stack operates on a bi-directional and shortest path forwarding star topology that allows traffic to flow either 'upstream' or downstream' simultaneously from every switch allowing packets to take the optimal forwarding path (shortest path). The bi-directional paths allow the traffic to automatically redirect around any switch in the stack that is not operating properly. This stacking technology allows stackable switches to operate with the same performance and resiliency as chassis solution.

Beginning in 1528 he immersed himself in comparisons and tests on what had been written about mineralogy and mining and his own observations of the local materials and the methods of their treatment. He constructed a logical system of the local conditions, rocks and sediments, the minerals and ores, explained the various terms of general and specific local territorial features. He combined this discourse on all natural aspects with a treatise on the actual mining, the methods and processes, local extraction variants, the differences and oddiities he had learnt from the miners. For the first time, he tackled questions on the formation of ores and minerals, attempted to bring the underlying mechanisms to light and introduce his conclusions in a systematic framework.

Classical algebraic logic, which comprises all work in algebraic logic until about 1960, studied the properties of specific classes of algebras used to "algebraize" specific logical systems of particular interest to specific logical investigations. Generally, the algebra associated with a logical system was found to be a type of lattice, possibly enriched with one or more unary operations other than lattice complementation. Abstract algebraic logic is a modern subarea of algebraic logic that emerged in Poland during the 1950s and 60s with the work of Helena Rasiowa, Roman Sikorski, Jerzy Łoś, and Roman Suszko (to name but a few). It reached maturity in the 1980s with the seminal publications of the Polish logician Janusz Czelakowski, the Dutch logician Wim Blok and the American logician Don Pigozzi.

The commentary stresses the importance of a number of aspects in Quranic commentary which were thought a novel approach at the time of its publication such as the inter-relationship of the text of the entire Quran and of each Surah to the preceding, the themes of the Quran are connected and all chapters, verses and words are perfectly and purposefully arranged according to a coherent and logical system. It also presents a distinctive eschatological reading of the Qur'an, applying many of its prophecies to the present times, as per Ahmadiyya beliefs, such as with reference to Surah 18 (al-Kahf) and especially the latter chapters of the Quran.Abdul Basit Shahid. Swaneh Fazl-i-Umar vol III, Fazl-i-Umar Foundation, 2006, p.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions (well-formed formulas), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions. When the formal system is intended to be a logical system, the expressions are meant to be interpreted as statements, and the rules, known to be inference rules, are typically intended to be truth-preserving. In this setting, the rules, which may include axioms, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements. The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see axiom schema).

His works included two volumes on Deductive and Inductive Logic respectively, which have passed through many editions, and are, in the main, a reproduction for Oxford use of the logical system of John Stuart Mill; an elaborate edition of Francis Bacon's Novum Organon, with introduction and notes' an edition of Locke's Conduct of the Understanding; monographs on John Locke, Bacon and Shaftesbury and Hutcheson; Progressive Morality, an Essay in Ethics; and The Principles of Morality, an important and original work, which incorporates as much of the thought of J. M. Wilson as Wilson ever managed to put on paper. The work is Fowler's own, but it was largely inspired by Wilson, and in some few parts it was written by him. In 1886, he was awarded a Bachelor and Doctor of Divinity.

Despite the fact that we lack background knowledge to indicate that there are dramatically fewer men than short people, we still find ourselves inclined to reject the conclusion. Hintikka's example is: "... a generalization like 'no material bodies are infinitely divisible' seems to be completely unaffected by questions concerning immaterial entities, independently of what one thinks of the relative frequencies of material and immaterial entities in one's universe of discourse." His solution is to introduce an order into the set of predicates. When the logical system is equipped with this order, it is possible to restrict the scope of a generalization such as "All ravens are black" so that it applies to ravens only and not to non-black things, since the order privileges ravens over non- black things.

It is not, in general, possible for a logical system to have a formal negation operator such that there is a proof of "not" P exactly when there isn't a proof of P ; see Gödel's incompleteness theorems. The BHK interpretation instead takes "not" P to mean that P leads to absurdity, designated \bot, so that a proof of ¬P is a function converting a proof of P into a proof of absurdity. A standard example of absurdity is found in dealing with arithmetic. Assume that 0 = 1, and proceed by mathematical induction: 0 = 0 by the axiom of equality. Now (induction hypothesis), if 0 were equal to a certain natural number n, then 1 would be equal to n+1, (Peano axiom: Sm = Sn if and only if m = n), but since 0=1, therefore 0 would also be equal to n + 1\.

Uşūl al-Fiqh, the methodology of jurisprudence, which is usually – and inaccurately, if not incorrectly – translated “principles of jurisprudence,” is an Islamic science which is developed by Shiite scholars in two recent centuries into an unparalleled intellectual, logical system of thought and a comprehensive branch of knowledge which not only serves as the logic of jurisprudence but as an independent science dealing with some hermeneutical problems. Lack of precise English equivalents to expressions and terms of this complicated science indicates the least difficulties of preparing the first English version of Shiite uşūl al-fiqh. "An Introduction to Methodology of Islamic Jurisprudence (Uşūl al-Fiqh)-A Shiite Approach" is the first English version of Shiite uşūl al-fiqh. This book is written by Alireza Hodaee, Professor of Jurisprudence and the Essentials of Islamic Law, University of Tehran.

Propositional logic is a logical system that is intimately connected to Boolean algebra. Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean algebras in a way that the tautologies (theorems) of propositional logic correspond to equational theorems of Boolean algebra. Syntactically, every Boolean term corresponds to a propositional formula of propositional logic. In this translation between Boolean algebra and propositional logic, Boolean variables x,y... become propositional variables (or atoms) P,Q,..., Boolean terms such as x∨y become propositional formulas P∨Q, 0 becomes false or ⊥, and 1 becomes true or T. It is convenient when referring to generic propositions to use Greek letters Φ, Ψ,... as metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions.

A few years after, he got a government position as assistant president at the Imperial Medical Institute in Beijing. However, even though he had climbed up the social ladder, as his father had originally wanted, he left a year later to return to being a doctor. In his government position, Li was able to read many rare medical books; he also saw the disorder, mistakes, and conflicting information that were serious problems in most medical publications of the time and soon began the book Compendium of Materia Medica to compile correct information with a logical system of organization. A small part was based on another book which had been written several hundred years earlier, Jingshi Zhenglei Beiji Bencao (Ching-hsih Cheng-lei Pei-chi Pen-tsao; "Classified Materia Medica for Emergencies") – which, unlike many other books, had formulas and recipes for most of the entries.

In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed. If the axiomatization of set theory is formulated in such a logical system with the axiom schema of separation as axioms, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem. If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement schema. The formulation used in the axiom schema of replacement article only allows to construct the image F[a] when a is contained in the domain of the class function F; then the derivation of separation requires the axiom of empty set.

In 1941, Shigekuni Honda is sent to Bangkok as legal counsel for Itsui Products in a case involving a spoilt shipment of antipyretic drugs. He takes advantage of the trip to see as much as he can of Thailand. After touring many great buildings, he visits the Temple of Dawn and is deeply impressed by its sumptuous architecture, which to the sober lawyer represents "golden listlessness", the luxurious feel of anti-rationalism and of "the constant evasion of any organized logical system". Mentioning to his translator, Hishikawa, that he went to school with two Siamese princes (Pattanadid, a younger brother of Rama VI, and his cousin Kridsada, a grandson of Rama IV-- both in Lausanne with their uncle Rama VIII), a short meeting is arranged with Pattanadid's seven-year-old daughter, Princess Chantrapa (Ying Chan), who claims to be the reincarnation of a Japanese boy, much to the embarrassment of her relatives, who keep her isolated in the Rosette Palace.

The focus of abstract algebraic logic shifted from the study of specific classes of algebras associated with specific logical systems (the focus of classical algebraic logic), to the study of: #Classes of algebras associated with classes of logical systems whose members all satisfy certain abstract logical properties; #The process by which a class of algebras becomes the "algebraic counterpart" of a given logical system; #The relation between metalogical properties satisfied by a class of logical systems, and the corresponding algebraic properties satisfied by their algebraic counterparts. The passage from classical algebraic logic to abstract algebraic logic may be compared to the passage from "modern" or abstract algebra (i.e., the study of groups, rings, modules, fields, etc.) to universal algebra (the study of classes of algebras of arbitrary similarity types (algebraic signatures) satisfying specific abstract properties). The two main motivations for the development of abstract algebraic logic are closely connected to (1) and (3) above.

Grzegorczyk's undecidability of Alfred Tarski's concatenation theory is based on the philosophical motivation claiming that investigation of formal systems should be done with a help of operations on visually comprehensible objects, and the most natural element of this approach is the notion of text. On his research, Tarski's simple theory is undecidable although seems to be weaker than the weak arithmetic, whereas, instead of computability, he applies more epistemological notion of the effective recognizability of properties of a text and relationships between different texts. In 2011, Grzegorczyk introduced yet one more logical system, which today is known as the Grzegorczyk non-Fregean logic or the logic of descriptions (LD), to cover the basic features of descriptive equivalence of sentences, wherein he assumed that a human language is applied primarily to form descriptions of reality represented formally by logical connectives. According to this system, the logical language is equipped in at least four logical connectives negation (¬), conjunction (∧), disjunction (∨), and equivalence (≡).

Kierkegaard criticized Hegel's idealist philosophy in several of his works, particularly his claim to a comprehensive system that could explain the whole of reality. Where Hegel argues that an ultimate understanding of the logical structure of the world is an understanding of the logical structure of God's mind, Kierkegaard asserts that for God reality can be a system but it cannot be so for any human individual because both reality and humans are incomplete and all philosophical systems imply completeness. For Hegel, a logical system is possible but an existential system is not: "What is rational is actual; and what is actual is rational".G.W.F. Hegel, Elements of the Philosophy of Right (1821) Hegel's absolute idealism blurs the distinction between existence and thought: our mortal nature places limits on our understanding of reality; > So-called systems have often been characterized and challenged in the > assertion that they abrogate the distinction between good and evil, and > destroy freedom.

Original plan of the extension of Barcelona (1859) the plan for new barcelona a plan for street Cerdà focused on key needs: chiefly, the need for sunlight, natural lighting and ventilation in homes (he was heavily influenced by the sanitarian movement), the need for greenery in people's surroundings, the need for effective waste disposal including good sewerage, and the need for seamless movement of people, goods, energy, and information. His designs belie a network-oriented approach far ahead of his time. His street layout and grid plan were optimized to accommodate pedestrians, carriages, horse-drawn trams, urban railway lines (as yet unheard-of), gas supply and large-capacity sewers to prevent frequent floods, without neglecting public and private gardens and other key amenities. The latest technical innovations were incorporated in his designs if they could further the cause of better integration, but he also came up with remarkable new concepts of his own, including a logical system of land readjustment that was essential to the success of his project, and produced a thorough statistical analysis of working-class conditions at the time, which he undertook in order to demonstrate the ills of congestion.