Predicate Logic with Definitions

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专家系统谓词逻辑推理

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Propositional Logic
Biconditional p q

states that p implies q and q implies p (p → q) ∧ (q → p) has the following meanings:
p if and only if q q if and only if p if p then q, and if q then p

In particular, it deals with the manipulation of logical variables, which represent propositions
P: A square has four equal sides Q: George Washington was the second president
The conditional does not mean exactly the same
as the IF-THEN in a procedural language or a rulebased expert system

IF-THEN means to execute the actions following the THEN if the conditions of the IF are true In logic, the conditional is defined by its truth table （真值表）:p → q is false only when p is true and q is false otherwise it is true.
Its meaning can be translated into natural

形式方法--讲义1

• , set theory and propositional and predicate logic and functions集合论，命集合论，集合论题逻辑和谓词逻辑，函数. 题逻辑和谓词逻辑，函数 • The unit also gives learners experience of proof by induction (a principle of functional programming) and relationship matrices数学归纳法证明数学归纳法证明和关系矩阵. 和关系矩阵 The latter, as well as reinforcing set and logic theory, will provide an alternative application of the rules of matrix algebra.
• Logic（数理逻辑）: 数理逻辑） • definition of proposition（命题的定义）, （命题的定义） propositional logic laws (and, or, not, eor, equivalence, implication)（命题逻（辑）, laws of propositional logic, tautologies（重言式）, truth tables for （重言式） compound statements（真值表）, （真值表） simplification of logic expressions（逻（辑表达式及其化简）辑表达式及其化简）, predicate logic (universal and existential qualifier) 谓词逻辑：全称量词和存在量词）（谓词逻辑：全称量词和存在量词）

WhatisLogic（什么是逻辑）

LogicThis article is about reasoning and its study. For other uses, see Logic (disambiguation).Logic (from the Ancient Greek: λογική, logike)[1] is the use and study of valid reasoning.[2][3] The study of logic features most prominently in the subjects of philosophy, mathematics, and computer science.Logic was studied in several ancient civilizations, includingIndia,[4] China,[5] Persia and Greece. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Buddhists and Jains.Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.The study of logic“Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you alreadyincline to think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy: Do not block the way of inquiry.”—Charles Sanders Peirce, "First Rule of Logic" The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics.∙Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato[6] are goodexamples of informal logic.∙Formal logic is the study of inference with purely formal content. An inference possessesa purely formal content if it can be expressed as a particular application of a whollyabstract rule, that is, a rule that is not about any particular thing or property. The worksof Aristotle contain the earliest known formal study of logic. Modern formal logic followsand expands on Aristotle.[7] In many definitions of logic, logical inference and inferencewith purely formal content are the same. This does not render the notion of informallogic vacuous, because no formal logic captures all of the nuances of natural language.∙Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[8][9] Symbolic logic is often divided into two branches: propositionallogic and predicate logic.∙Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.Logical formMain article: Logical formLogic is generally considered formal when it analyzes and represents the form of any valid argument type. The form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. If one considers the notion of form too philosophically loaded, one could say that formalizing simply means translating English sentences into the language of logic.This is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a considerable variety of form and complexity that makes their use in inference impractical. It requires, first, ignoring those grammatical features irrelevant to logic (such as gender and declension, if the argument is in Latin), replacing conjunctions irrelevant to logic (such as "but") with logical conjunctions like "and" and replacing ambiguous, or alternative logical expressions ("any", "every", etc.) with expressions of a standard type (such as "all", or the universalquantifier ∀).Second, certain parts of the sentence must be replaced with schematic letters. Thus, for example, the expression "all As are Bs" shows the logical form common to the sentences "all men are mortals", "all cats are carnivores", "all Greeks are philosophers", and so on.That the concept of form is fundamental to logic was already recognized in ancient times. Aristotle uses variable letters to represent valid inferences in Prior Analytics, leading JanŁukasiewicz to say that the introduction of variables was "one of Aristotle's greatest inventions".[10] According to the followers of Aristotle (such as Ammonius), only the logical principles stated inschematic terms belong to logic, not those given in concrete terms. The concrete terms "man", "mortal", etc., are analogous to the substitution values of the schematic placeholders A, B, C, which were called the "matter" (Greek hyle) of the inference.The fundamental difference between modern formal logic and traditional, or Aristotelian logic, lies in their differing analysis of the logical form of the sentences they treat.∙In the traditional view, the form of the sentence consists of (1) a subject (e.g., "man") plus a sign of quantity ("all" or "some" or "no"); (2) the copula, which is of the form "is"or "is not"; (3) a predicate (e.g., "mortal"). Thus: all men are mortal. The logicalconstants such as "all", "no" and so on, plus sentential connectives such as "and" and"or" were called "syncategorematic" terms (from the Greek kategorei – to predicate,and syn – together with). This is a fixed scheme, where each judgment has an identifiedquantity and copula, determining the logical form of the sentence.∙According to the modern view, the fundamental form of a simple sentence is given by a recursive schema, involving logical connectives, such as a quantifier with its boundvariable, which are joined by juxtaposition to other sentences, which in turn may havelogical structure.∙The modern view is more complex, since a single judgement of Aristotle's system involves two or more logical connectives. For example, the sentence "All men aremortal" involves, in term logic, two non-logical terms "is a man" (here M) and "is mortal"(here D): the sentence is given by the judgement A(M,D). In predicate logic, the sentenceinvolves the same two non-logical concepts, here analyzed as and , and thesentence is given by , involving the logical connectives foruniversal quantification and implication.∙But equally, the modern view is more powerful. Medieval logicians recognized the problem of multiple generality, where Aristotelian logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and"some" may be relevant in an inference, but the fixed scheme that Aristotle used allowsonly one to govern the inference. Just as linguists recognize recursive structure innatural languages, it appears that logic needs recursive structure.Deductive and inductive reasoning, and abductive inferenceDeductive reasoning concerns what follows necessarily from given premises (if a, then b). However, inductive reasoning—the process of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Similarly, it is important to distinguish deductive validity and inductive validity (called "cogency"). An inference is deductively valid if and only if there isno possible situation in which all the premises are true but the conclusion false. An inductive argument can be neither valid nor invalid; its premises give only some degree of probability, but not certainty, to its conclusion.The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providingthis definition may be approached in various ways, some less formal than others; some of these definitions may use mathematical models of probability. For the most part this discussion of logic deals only with deductive logic.Abduction[11] is a form of logical inference that goes from observation to a hypothesis that accounts for the reliable data (observation) and seeks to explain relevant evidence. The American philosopher Charles Sanders Peirce (1839–1914) first introduced the term as "guessing".[12] Peirce said that to abduce a hypothetical explanation from an observed surprising circumstance is to surmise that may be true because then would be a matter of course.[13] Thus, to abduce from involves determining that is sufficient (or nearly sufficient), but not necessary, for .Consistency, validity, soundness, and completenessAmong the important properties that logical systems can have:∙Consistency, which means that no theorem of the system contradicts another.[14]∙Validity, which means that the system's rules of proof never allow a false inference from true premises. A logical system has the property of soundness when the logical systemhas the property of validity and uses only premises that prove true (or, in the case ofaxioms, are true by definition).[14]∙Completeness, of a logical system, which means that if a formula is true, it can be proven (if it is true, it is a theorem of the system).∙Soundness, the term soundness has multiple separate meanings, which creates a bit of confusion throughout the literature. Most commonly, soundness refers to logicalsystems, which means that if some formula can be proven in a system, then it is true inthe relevant model/structure (if A is a theorem, it is true). This is the converse ofcompleteness. A distinct, peripheral use of soundness refers to arguments, which means that the premises of a valid argument are true in the actual world.Some logical systems do not have all four properties. As an example, Kurt Gödel's incompleteness theorems show that sufficiently complexformal systems of arithmetic cannot be consistent and complete;[9] however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.[15]Rival conceptions of logicMain article: Definitions of logicLogic arose (see below) from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations".[16]By contrast, Immanuel Kant argued that logic should be conceived as the science of judgement, an idea taken up in Gottlob Frege's logical and philosophical work. But Frege's work is ambiguous in the sense that it is both concerned with the "laws of thought" as well as with the "laws of truth", i.e. it both treats logic in the context of a theory of the mind, and treats logic as the study of abstract formal structures.HistoryMain article: History of logicAristotle, 384–322 BCE.In Europe, logic was first developed by Aristotle.[17] Aristotelian logic became widely accepted in science and mathematics and remained in wide use in the West until the early 19th century.[18] Aristotle's system of logic was responsible for the introduction of hypothetical syllogism,[19] temporal modal logic,[20][21] and inductive logic,[22] as well as influential terms such as terms, predicables, syllogisms and propositions. In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith. During the High Middle Ages, logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of scholasticism. In 1323, William of Ockham's influential Summa Logicae was released. By the 18th century, the structured approach to arguments had degenerated and fallen out of favour, as depicted in Holberg's satirical play Erasmus Montanus.The Chinese logical philosopher Gongsun Long (c. 325–250 BCE) proposed the paradox "One and one cannot become two, since neither becomes two."[23] In China, the tradition of scholarly investigation into logic, however, was repressed by the Qin dynasty following the legalist philosophy of Han Feizi.In India, innovations in the scholastic school, called Nyaya, continued from ancient times into the early 18th century with the Navya-Nyaya school. By the 16th century, it developed theories resembling modern logic, such as Gottlob Frege's "distinction between sense and reference of proper names" and his "definition of number", as well as the theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory.[24] Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and George Boole.[25] In the20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff have explored Indian logic more extensively.The syllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics stimulated the development of symbolic logic (now called mathematical logic). In 1854, George Boole published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is now known as Boolean logic. In 1879, Gottlob Frege published Begriffsschrift, which inaugurated modern logic with the invention of quantifier notation. From 1910 to 1913, Alfred North Whitehead and Bertrand Russell published Principia Mathematica[8] onthe foundations of mathematics, attempting to derive mathematical truths from axioms and inference rules in symbolic logic. In 1931,Gödel raised serious problems with the foundationalist program and logic ceased to focus on such issues.The development of logic since Frege, Russell, and Wittgenstein had a profound influence on the practice of philosophy and the perceived nature of philosophical problems (see Analytic philosophy), and Philosophy of mathematics. Logic, especially sentential logic, is implemented in computer logic circuits and is fundamental to computer science. Logic is commonly taught by university philosophy departments, often as a compulsory discipline.Types of logicSyllogistic logicMain article: Aristotelian logicThe Organon was Aristotle's body of work on logic, with the Prior Analytics constituting the first explicit work in formal logic, introducing the syllogistic.[26] The parts of syllogistic logic, also known by the name term logic, are the analysis of the judgements into propositions consisting of two terms that are related by one of a fixed number of relations, and the expression of inferences by means of syllogisms that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises.Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out system. However, it was not alone: the Stoics proposed a system of propositional logic that was studied by medieval logicians. Also, the problem of multiple generality was recognized in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary solutions.Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest in extending term logics), regarded as made obsolete by the advent of propositional logic and the predicate calculus. Others use Aristotle in argumentation theory to help develop andcritically question argumentation schemes that are used in artificial intelligence and legal arguments.Propositional logic (sentential logic)Main article: Propositional calculusA propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and in which a system of formal proof rules establishes certain formulae as "theorems".Predicate logicMain article: Predicate logicPredicate logic is the generic term for symbolic formal systems such as first-order logic, second-order logic, many-sorted logic, and infinitary logic.Predicate logic provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. Aristotelian syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take. Predicatelogic allows sentences to be analysed into subject and argument in several additional ways—allowing predicate logic to solve the problem of multiple generality that had perplexed medieval logicians.The development of predicate logic is usually attributed to Gottlob Frege, who is also credited as one of the founders of analytical philosophy, but the formulation of predicate logic most often used today is the first-order logic presented in Principles of Mathematical Logic by David Hilbert and Wilhelm Ackermann in 1928. The analytical generality of predicate logic allowed theformalization of mathematics, drove the investigation of set theory, and allowed the development of Alfred Tarski's approach to model theory. It provides the foundation of modern mathematical logic.Frege's original system of predicate logic was second-order, rather than first-order. Second-order logic is most prominently defended (against the criticism of Willard Van Orman Quine and others) by George Boolos and Stewart Shapiro.Modal logicMain article: Modal logicIn languages, modality deals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles. For example, "We go to the games" can be modified to give "We should go to the games", and "We can go to the games" and perhaps "We will go to the games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.Aristotle's logic is in large parts concerned with the theory of non-modalized logic. Although, there are passages in his work, such as the famous sea-battle argument in De Interpretatione § 9, that are now seen as anticipations of modal logic and its connection with potentiality and time, the earliest formal system of modal logic was developed by Avicenna, whom ultimately developed a theory of "temporally modalized" syllogistic.[27]While the study of necessity and possibility remained important to philosophers, little logical innovation happened until the landmark investigations of Clarence Irving Lewis in 1918, who formulated a family of rival axiomatizations of the alethic modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to include deontic logic and epistemic logic. The seminal work of Arthur Prior applied the same formal language totreat temporal logic and paved the way for the marriage of the two subjects. Saul Kripke discovered (contemporaneously with rivals) his theory of frame semantics, which revolutionized the formal technology available to modal logicians and gave a new graph-theoretic way of looking at modality that has driven many applications in computational linguistics and computer science, such as dynamic logic.Informal reasoningMain article: Informal logicThe motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good from bad arguments, and so become more effective in argument and oratory, and perhaps also to become a better person. Half of the works of Aristotle's Organontreat inference as it occurs in an informal setting, side by sidewith the development of the syllogistic, and in the Aristotelian school, these informal works on logic were seen as complementary to Aristotle's treatment of rhetoric.This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typically dialectical logic forms the heart of a course in critical thinking, a compulsory course at many universities.Argumentation theory is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations. Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied in artificial intelligence and law.Mathematical logicMain article: Mathematical logicMathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and mathematical reasoning, and the second, in the other direction, the application of mathematical techniques to the representation and analysis of formal logic.[28]The earliest use of mathematics and geometry in relation to logic and philosophy goes back to the ancient Greeks such as Euclid, Plato, and Aristotle.[29] Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.[30]One of the boldest attempts to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic.[8] The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems.Both the statement of Hilbert's program and its refutation by Gödel depended upon their work establishing the second area of mathematical logic, the application of mathematics to logic in the form of proof theory.[31] Despite the negative nature of the incompleteness theorems,Gödel's completeness theorem, a result in model theory and another application of mathematics to logic, can be understood as showing how close logicism came to being true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it. Thus we see how complementary the two areas of mathematical logic have been.[citation needed]If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pillars of the subject. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms.Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church–Turing thesis.[32] Today recursion theory is mostly concerned with the more refined problem of complexity classes—when is a problem efficiently solvable?—and the classification of degrees of unsolvability.[33]Philosophical logicMain article: Philosophical logicPhilosophical logic deals with formal descriptions of ordinary, non-specialist ("natural") language. Most philosophers assume that the bulk of everyday reasoning can be captured in logic if a method or methods to translate ordinary language into that logic can be found. Philosophical logic is essentially a continuation of the traditional discipline called "logic" before the invention of mathematical logic. Philosophical logic has a much greater concern with the connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard logics (e.g. free logics, tense logics) as well as various extensions of classical logic (e.g. modal logics) and non-standard semantics for such logics (e.g. Kripke's supervaluationism in the semantics of logic).Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can help a person better structure his own arguments and critique the arguments of others. Many popular arguments arefilled with errors because so many people are untrained in logic and unaware of how to formulate an argument correctly.Computational logicMain article: Logic in computer scienceLogic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems. The notion of the general purpose computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s.In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic with mathematical notation, it would be possible to create a machine that reasons, or artificial intelligence. This was more difficult than expected because of the complexity of human reasoning. In logic programming, a program consists of a set of axioms and rules. Logic programming systems such as Prolog compute the consequences of the axioms and rules in order to answer a query.Today, logic is extensively applied in the fields of Artificial Intelligence, and Computer Science, and these fields provide a rich source of problems in formal and informal logic. Argumentation theory is one good example of how logic is being applied to artificial intelligence. The ACM Computing Classification System in particular regards:∙Section F.3 on Logics and meanings of programs and F.4 on Mathematical logic and formal languages as part of the theory of computer science: this work covers formalsemantics of programming languages, as well as work of formal methods such as Hoarelogic;∙Boolean logic as fundamental to computer hardware: particularly, the system's sectionB.2 on Arithmetic and logic structures, relating to operatives AND, NOT, and OR;。

胡壮麟语言学名词解释总结

胡壮麟语言学名词解释总结1.design feature: are features that define our human languages,such asarbitrariness,duality,creativity,displacement,cultural transmission,etc.2.function: the use of language tocommunicate,to think ,nguage functions inclucle imformativefunction,interpersonal function,performative function, emotive function,phaticcommunion,recreational function and metalingual function.3.etic: a term in contrast with emic which originates from American linguist Pike’s distinction ofphonetics and phonemics.Being etic mans making far too many, as well as behaviouslyinconsequential,differentiations,just as was ofter the case with phonetic vx.phonemic analysis in linguistics proper.4.emic: a term in contrast with etic which originates from American linguist Pike’s distinction ofphonetics and phonemics.An emic set of speech acts and events must be one that is validated as meaningful via final resource to the native members of a speech communith rather than via qppeal to the investigator’s ingenuith or intuition alone.5.synchronic: a kind of description which takes a fixed instant(usually,but not necessarily,thepresent),as its point of observation.Most grammars are of this kind.6.diachronic:study of a language is carried through the course of its history.7.prescriptive: a kind of linguistic study in which things are prescribed how ought to be,yingdown rules for language use.8.descriptive: a kind of linguistic study in which things are just described.9.arbitrariness: one design feature of human language,which refers to the face that the forms oflinguistic signs bear no natural relationship to their meaning.10.duality: one design feature of human language,which refers to the property of having two levels ofare composed of elements of the secondary.level and each of the two levels has its own principles of organization.11.displacement: one design feature of human language,which means human language enable theirusers to symbolize objects,events and concepts which are not present c in time and space,at themoment of communication.12.phatic communion: one function of human language,which refers to the social interaction oflanguage.13.metalanguage: certain kinds of linguistic signs or terms for the analysis and description of particularstudies.14.macrolinguistics: the interacting study between language and language-related disciplines such aspsychology,sociology,ethnograph,science of law and artificial intelligence etc.Branches ofmacrolinguistics include psycholinguistics,sociolinguistics, anthropological linguistics,etpetence: language user’s underlying knowledge about the system of rules.16.performance: the actual use of language in concrete situation.ngue: the linguistic competence of the speaker.18.parole: the actual phenomena or data of linguistics(utterances).19.Articulatory phonetics: the study of production of speechsounds.20.Coarticulation: a kind of phonetic process in which simultaneous or overlapping articulations areinvolved..Coarticulation can be further divided into anticipatory coarticulation and perseverative coarticulation.21.Voicing: pronouncing a sound (usually a vowel or a voiced consonant) by vibrating the vocal cords.22.Broad and narrow transcription: the use of a simple set of symbols in transcription is called broadtranscription;while,the use of more specific symbols to show more phonetic detail is referred to as narrow transcription.23.Consonant: are sound segments produced by constricting or obstructing the vocal tract at some placeto divert,impede,or completely shut off the flow of air in the oral cavity.24.Phoneme: the abstract element of sound, identified as being distinctive in a particular language.25.Allophone:any of the different forms of a phoneme(eg.<th>is an allophone of /t/in English.When/t/occurs in words like step,it is unaspirated<t>.Both<th>and <t>are allophones of the phoneme/t/. 26.Vowl:are sound segments produced without such obstruction,so no turbulence of a total stopping ofthe air can be perceived.27.Manner of articulation: in the production of consonants,manner of articulation refers to the actualrelationship between the articulators and thus the way in which the air passes through certain parts of the vocal tract.28.Place of articulation: in the production of consonants,place of articulation refers to where in thevocal tract there is approximation,narrowing,or the obstruction of air.29.Distinctive features: a term of phonology,i.e.a property which distinguishes one phoneme fromanother.plementary distribution: the relation between tow speech sounds that never occur in the sameenvironment.Allophones of the same phoneme are usually in complementary distribution.31.IPA: the abbreviation of International Phonetic Alphabet,which is devised by the InternationalPhonetic Association in 1888 then it has undergong a number of revisions.IPA is a comprised system employing symbols of all sources,such as Roman small letters,italics uprighted,obsolete letters,Greek letters,diacritics,etc.32.Suprasegmental:suprasegmental features are those aspects of speech that involve more than singlesound segments.The principal supra-segmental features are syllable,stress,tone,,and intonation.33.morpheme:the smallest unit of language in terms of relationship between expression and content,aunit that cannot be divided into further small units without destroying or drastically altering themeaning,whether it is lexical or grammatical.pound oly morphemic words w hich consist wholly of free morphemes,such asclassroom,blackboard,snowwhite,etc.35.inflection: the manifestation of grammatical relationship through the addition of inflectionalaffixes,such as number,person,finiteness,aspect and case,which do not change the grammatical class of the stems to which they are attached.36.affix: the collective term for the type of formative that can be used only when added to anothermorpheme(the root or stem).37.derivation: different from compounds,derivation shows the relation between roots and affixes.38.root: the base from of a word that cannot further be analyzed without total lass of identity.39.allomorph:any of the different form of a morpheme.For example,in English the plural mortheme isbut it is pronounced differently in different environments as/s/in cats,as/z/ in dogs and as/iz/ inclasses.So/s/,/z/,and /iz/ are all allomorphs of the plural morpheme.40.Stem: any morpheme or combination of morphemes to which an inflectional affix can be added.41.bound morpheme: an element of meaning which is structurally dependent on the world it is addedto,e.g. the plural morpheme in “dog’s”.42.free morpheme: an element of meaning which takes the form of an independent word.43.lexeme:A separate unit of meaning,usually in the form of a word(e.g.”dog in the manger”)44.lexicon: a list of all the words in a language assigned to various lexical categories and provided withsemantic interpretation.45.grammatical word: word expressing grammatical meanings,such conjunction,prepositions,articlesand pronouns.46.lexical word: word having lexical meanings,that is ,those which refer to substance,action andquality,such as nouns,verbs,adjectives,and verbs.47.open-class: a word whose membership is in principle infinite or unlimited,such asnouns,verbs,adjectives,and many adverbs.48.blending: a relatively complex form of compounding,in which two words are blended by joining theinitial part of the first word and the final part of the second word,or by joining the initial parts of the two words.49.loanword: a process in which both form and meaning are borrowed with only a slight adaptation,insome cases,to eh phonological system of the new language that they enter.50.loanblend: a process in which part of the form is native and part is borrowed, but the meaning is fullyborrowed.51.leanshift: a process in which the meaning is borrowed,but the form is native.52.acronym: is made up form the first letters of the name of an organization,which has a heavilymodified headword.53.loss: the disappearance of the very sound as a morpheme in the phonological system.54.back-formation: an abnormal type of word-formation where a shorter word is derived by deleting animagined affix from a long form already in the language.55.assimilation: the change of a sound as a result of the influence of an adjacent sound,which is morespecifically called.”contact”or”contiguous”assimilation.56.dissimilation: the influence exercised.By one sound segment upon the articulation of another, so thatthe sounds become less alike,or different.57.folk etymology: a change in form of a word or phrase,resulting from an incorrect popular nation ofthe origin or meaning of the term or from the influence of more familiar terms mistakenly taken to be analogous58.category:parts of speech and function,such as the classification of words in terms of parts ofspeech,the identification of terms of parts of speech,the identification of functions of words in term of subject,predicate,etc.59.prepositional logic: also known as prepositional calculus or sentential calculus,is the study of thetruth conditions for propositions:how the truth of a composite propositions and the connectionbetween them.60.Proposition:what is talk about in an utterance,that part of the speech act which has to do withreference.61.predicate logic: also predicate calculus,which studies the internal structure of simple.62.assimilation theory: language(sound,word,syntax,etc)change or process by which features of oneelement change to match those of another that precedes or follows.63.cohort theory: theory of the perception of spoken words proposed in the mid-1980s.It saaumes a“recognition lexicon”in which each word is represented by a full and independent”recognistionelement”.When the system receives the beginning of a relevant acoustic signal,all elements matching it are fully acticated,and,as more of the signal is received,the system tries to match it independently with each of them,Wherever it fails the element is deactivated;this process continues until only one remains active.64.context effect: this effect help people recognize a word more readily when the receding wordsprovide an appropriate context for it.65.frequency effect: describes the additional ease with which a word is accessed due to its morefrequent usage in language.66.inference in context: any conclusion drawn from a set of proposition,from something someone hassaid,and so on.It includes things that,while not following logically,are implied,in an ordinarysense,e.g.in a specific context.67.immediate assumption: the reader is supposed to carry out the progresses required to understandeach word and its relationship to previous words in the sentence as soon as that word in encountered.nguage perception:language awareness of things through the physical senses,esp,sight.nguage comprehension: one of the three strand of psycholinguistic research,which studies theunderstanding of language.nguage production: a goal-directed activety,in the sense that people speak and write in orde tomake friends,influence people,convey information and so on.71.lexical ambiguity:ambiguity explained by reference to lexical meanings:e.g.that of I saw a bat,wherea bat might refer to an animal or,among others,stable tennis bat.72.macroproposition:general propositions used to form an overall macrostructure of the story.73.modular:which a assumes that the mind is structuied into separate modules or components,eachgoverned by its own principles and operating independently of others.74.parsing:the task of assigning words to parts of speech with their appropriate accidents,traditionallye.g.to pupils learning lat in grammar.75.propositions:whatever is seen as expressed by a sentence which makes a statement.It is a property ofpropositions that they have truth values.76.psycholinguistics: is concerned primarily with investigating the psychological reality of linguisticstructure.Psycholinguistics can be divided into cognitive psycholing uistics(being concerned above all with making inferences about the content of human mind,and experimental psycholinguistics(being concerned somehow whth empirical matters,such as speed of response to a particular word).77.psycholinguistic reality: the reality of grammar,etc.as a purported account of structures representedin the mind of a speaker.Often opposed,in discussion of the merits of alternative grammars,to criteria of simplicity,elegance,and internal consistency.78.schemata in text: packets of stored knowledge in language processing.79.story structure: the way in which various parts of story are arranged or organized.80.writing process: a series of actions or events that are part of a writing or continuing developmeng.municative competence: a speaker’s knowledge of the total set ofrules,conventions,erning the skilled use of language in a society.Distinguished by D.Hymes in the late 1960s from Chomsley’s concept of competence,in the restricted sense of knowledge of a grammar.82.gender difference: a difference in a speech between men and women is”genden difference”83.linguistic determinism: one of the two points in Sapir-Whorf hypothesis,nguage determinesthought.84.linguistic relativity: one of the two points in Spir-Whorf hypotheis,i.e.there’s no limit to thestructural diversity of languages.85.linguistic sexism:many differences between me and women in language use are brought about bynothing less than women’s place in society.86.sociolinguistics of language: one of the two things in sociolinguistics,in which we want to look atstructural things by paying attention to language use in a social context.87.sociolinguistics of society:one of the two things in sociolinguistics,in which we try to understandsociological things of society by examining linguistic phenomena of a speaking community.88.variationist linguistics: a branch of linguistics,which studies the relationship between speakers’socialstarts and phonological variations.89.performative: an utterance by which a speaker does something does something,as apposed to aconstative,by which makes a statement which may be true or false.90.constative: an utterance by which a speaker expresses a proposition which may be true or false.91.locutionary act: the act of saying something;it’s an act of conveying literal meaning by means ofsyntax,lexicon,and ly.,the utterance of a sentence with determinate sense andreference.92.illocutionary act: the act performed in saying something;its force is identical with the speaker’sintention.93.perlocutionary act: the act performed by or resulting from saying something,it’s the consequenceof,or the change brought about by the utterance.94.conversational implicature: the extra meaning not contained in the literal utterances,underatandableto the listener only when he shares the speaker’s knowledge or knows why and how he violates intentionally one of the four maxims of the cooperative principle.95.entailment:relation between propositions one of which necessarily follows from the other:e.g.”Maryis running”entails,among other things,”Mary is not standing still”.96.ostensive communication: a complete characterization of communication is that it is ostensive-infer-ential.municative principle of relevance:every act of ostensive communication communicates thepresumption of its own optimal relevance.。

人工智能4第5章

一般地，表达式 P(x1，x2，……，xn) 在谓词逻辑中称为n元谓词。其中P是谓词符号，也称谓词，代表一个确定的特征或关系。x1， x2，…，xn称为谓词的参量或者项，一般表示个体。
§5.1.1. 谓词、函数、量词
设a1，a2，…，an表示个体对象，A表示它们的属性、状态或关系，则表达式 A(a1，a2，…，an)。在谓词逻辑中就表示一个(原子)命题。例如， (1)素数(2)，就表示命题“2是个素数”。
§5.1.2. 谓词公式量词分为：全称量词记为∀x和存在量词记为∃y。紧接于量词之后被量词作用(即说明)的谓词公式称为该量词的辖域。例： (1) ∀xP(x) P(x)为∀x的辖域， (2) ∀x(H(x) → G(x,y)) (H(x) → G(y,x)) 为∀x的辖域， (3) ∃xA(x) ∧B(x) A(x)为∃x的辖域，但B(x)并非∃x的辖域。量词后的变元如∀x，∃y中的x， y称为量词的指导变元(或作用变元)，而在一个量词的辖域中与该量词的指导变元相同的变元称为约束变元，其他变元(如果有的话)称为自由变元，例如(2)中的x为约束变元，而y为自由变元，(3)中A(x)中x的为约束变元，但B(x)中x的为自由变元。例如(3)，一个变元在一个公式中既可约束出现，又可自由出现，但为了避免混淆，通常通过改名规则，使得一个公式中一个变元仅以一种形式出现。
§5.1.1. 谓词、函数、量词
常用的逻辑联结词有下列五个： 1）联结词“非”(Negation)，记作“⇁ ”； 2）联结词“与”或者“合取” (Conjunction) ，记作 “∧”； 3）联结词“或”或者“析取” ( (Implication) ，记作 4）联结词“蕴含”或者“蕴涵” Di sjuncti on) ，记作 “∨”； “→”；它表示被它连接的两个命题的“蕴含”关系。如P→Q表示“P蕴含Q”，即“如果P，则Q”，其中P称为前提，Q称为后件。 5）联结词“等价” (Equivalence) ，记作“n”。联结词的优先级别是： ⇁ ， ∧ ， ∨ ， → ， n 。逻辑联结词又称真值联结词。联接词又称联接词、连词、连接词。

谓词逻辑

值时,则这个式子表示“若x大于y 且y 大于z，则x大于z” 。这是一个永真式。如果H(x,y)解释为: “x是y的儿子”, 当x,y,z都指人时，则这个式子表示“若x为y的儿子且y 是 z的儿子，则x是z的儿子” 。这是一个永假式。如果H(x,y)解释为: “x距y10米”, 当x,y,z为平面上的点，则这个式子表示“若x距y10米且y距 z10米，则x距z10米” 。这个命题的真值将由 x,y,z的具体位置而定，它可能是1，也可能是0。
• 命题逻辑的局限性：在命题逻辑中，命题是命题演算的基本单位，不再对原子命题进行分解，因而无法研究命题的内部结构、成分及命题之间的内在联系，甚至无法处理一些简单而又常见的推理过程。
第二章谓词逻辑（Predicate Logic）
• 例如，下列推理：所有的人都是要死的。苏格拉底是人。苏格拉底是要死的。众所周知 , 这是真命题。但在命题逻辑中,如果用P,Q,R表示以上三个命题，则上述推理过程为：（P∧Q）R。借助命题演算的推理理论不能证明其为重言式。
第二章谓词逻辑（Predicate Logic）
2.1 谓词的概念与表示(Predicate and Its Expression)
2.存在量词(The Existential Quantifiers) 对日常语言中的“有一个”、“有的”、 “存在着”、“至少有一个”、 “存在一些”等词，用符号“”表示，ｘ表示存在个体域里的个体，ｘＦ(ｘ)表示存在个体域里的个体具有性质F.符号“”称为存在量词. 例4：在谓词逻辑中将下列命题符号化. （1）一些数是有理数。（2）有些人活百岁以上。
第二章谓词逻辑（Predicate Logic）
2.1 谓词的概念与表示(Predicate and Its Expression)

第二章谓词逻辑法

3 谓词演算 predicate calculus
3.1 语法和符号 syntax and notation 3.2 连词 conjunctions 3.3 量词 quantifiers
谓词
谓词
在谓词逻辑中，命题是用形如P(x1,x2,…,xn)的谓词来表述的。一个谓词可分为谓词名与个体两个部分
通常用大写英文字母表示一个命题，例如：
P：西安是座古老的城市
命题逻辑的局限性？
客观事物的结构及逻辑特征？不同事物间的共同特征？
2 命题 (Proposition)
命题逻辑的局限性？
命题这种表示方法无法把它所描述的客观事物的结构及逻辑特征反映出来，也不能把不同事物间的共同特征表述出来
– ③任何整数或者为正或者为负: » (x)(I(x) →(P(x) ∨ N(x)))
4 谓词公式
谓词公式
例2：用谓词逻辑描述右图中的房子的概念
个体：A , B 谓词：
Support( x,y )：表示 x 被 y支撑着 Wedge ( x )：表示 x 是楔形块 Brick( y )：表示 y 是长方块其中 x , y是个体变元，它们的个体域｛A,B｝房子的概念可以表示成一组合式谓词公式的合取式：
例如，用字母P表示“小张是老张的儿子”这一命题，则无法表述出老张与小张是父子关系
又如，“张三是学生”，“李四是学生”这两个命题，用命题逻辑表示时，无法把两者的共同特征“都是学生”形式的表示出来
可否用 Student（“张三”）， Student（“李四”）表示上述命题？——谓词逻辑
“每个雇员都有一个经理。”
( y)( x)(Employee(x) ∧ Manager(y, x)):

DMch2-3

第二章谓词逻辑（Predicate Logic）谓词逻辑（ Logic）
结
束
谢谢!
第二章谓词逻辑（Predicate Logic） Logic）
2.7 谓词演算的推理理论
2.7.2证明举例 (Examples of proof) 证明举例 )
例1 证明苏格拉底三段论:凡是人都是要死的。苏格证明苏格拉底三段论凡是人都是要死的。凡是人都是要死的拉底是人。苏格拉底是要死的。拉底是人。苏格拉底是要死的。是人。是要死的。苏格拉底苏格拉底。设：M(x):x是人。D(x):x 是要死的。a:苏格拉底。是人前提：则前提：x(M(x)→D(x))，M(a）. → ，）结论：结论： D(a) . 证明：证明：① x (M(x)→D(x)) P → US ① ② M(a)→D(a) → P ③ M(a) T② ③ I11 ④ D(a) ② (直接证法直接证法) 直接证法
第二章谓词逻辑（Predicate Logic）谓词逻辑（ Logic）
2.7 谓词演算的等价式与蕴含式
小结：小结：本节介绍了谓词演算的推理规则,并举例说明本节介绍了谓词演算的推理规则并举例说明了它的应用. 了它的应用深刻理解四个推理规则,会应用它们推理证明会应用它们推理证明. 四个推理规则会应用它们推理证明作业: 作业 P79: (1):b,d ,(2)；(3):a,b；；；
第二章谓词逻辑（Predicate Logic） Logic）
2.7 谓词演算的推理理论
2.7.1推理规则推理规则(Rules of inference) ) 推理规则在谓词演算中，在谓词演算中，推理的形式结构仍为 H1∧H2∧H3∧....∧HnC ∧ 是永真式，若 H1∧H2∧H3∧....∧Hn→C是永真式，则称由前提 ∧ 是永真式 H1,H2,H3,.…,Hn逻辑的推出结论，但在谓词逻逻辑的推出结论C，辑中, 均为谓词公式。辑中 H1,H2,H3,.…,Hn , C均为谓词公式。均为谓词公式命题演算中的推理规则，命题演算中的推理规则，可在谓词推理理论中应用。

George Yule《语言研究》章节题库-第9~12章【圣才出品】

第9章语义学I. Fill in the blanks.1. According to G Leech ____ meaning is the communicative value an expression has by virtue of what it refers to, over and above its purely conceptual content. (北二外2006研)【答案】Connotative【解析】利奇认为内涵意义是指通过语言所指传达的意义，是位于纯粹的概念意义之上的。

2. According to G Leech ____ meaning refers to logic, cognitive, or denotative content. (北二外2005研)【答案】Conceptual【解析】利奇认为概念意义是指逻辑的、认知的、外延的内容。

3. According to G. Leech _____ meaning refers to what is communicated of the feelings and attitudes of the speaker／writer. (北二外2007研)【答案】Affective【解析】利奇认为感情意义是指所传达的关于说话人/作者感情、态度方面的意义。

4. The theory of meaning which relates the meaning of a word to the thing it refers to, or stands for, is known as the _____ theory. (中山大学2008研)【答案】Referential【解析】把词语意义跟它所指称或所代表的事物联系起来的理论，叫做指称理论。

5. _____ is the technical name for the sameness relation. (北二外2007研)【答案】Synonymy【解析】同义关系是相同关系的专业术语，完全的同义关系是很少的。

SYMBOLICLOGICPredicateLogic

SYMBOLIC LOGIC: Predicate LogicIDENTITYTHREE WAYS OF USING THE VERB “To be”In English we use the verb “to be” in three main ways.I) The “is” of existence.Sometimes we assert the existence of a certain kind of thing.Cats exist, at least one cat exists, some cats exist, there is a cat (∃x)(Cx) Unicorns do not exist ~(∃x)(Ux) II) The Is of predication.Sometimes we attribute properties to a thing or assign an individual or group of individuals to a class. For this kind of use of “to be” we don’t need a special sign since it is built into the predicate letter.Alex is smart. Sa S –is smart Moab is a whale. Wm W –is a whale III) The “is” of identityBut we also use the verb “to be” to represent the relationship of identity. For example, we can say “Mark Twain is Samuel Clemens” meaning that the person named by the name “Mark Twain” is identical to the person named by “Samuel Clemens.” This is the easiest type of identity statement to represent since it contains only proper names and the “is” of identity. There is some debate about exactly what information is contained in such a sentence. But in the logical system we are studying here we have an identity predicate (represented by “=”), that takes two objects, the two objects about which identity is being predicated (Mark Twain and Samuel Clemens).In our system we would translate each proper name using a constant and use the identity predicate (=) to claim that the individuals named by each constant is identical. m = cIdentity statements containing only proper names are the most particular kind of sentences, since they refer to individuals. However, sometimes we want to abstract. For example, we might want to say that every individual is identical to itself. To do this we would use the identity predicate (=) combined with a variable and a quantifier.PROPERTIES OF IDENTITYThe identity relation has several properties. It is•reflexive•symmetrical•transitiveTo state these rules we need to write more abstract formulas, using variables and quantifiers.I) REFLEXIVITYNotice that every individual is identical to itself. If there were a finite number of individuals in the universe we could assert these truths one by one?m = ms = setc.However, since this relation holds between everything in our universe of discourse, we can use the variable x and a quantifier to state the general rule:(x)(x = x) Everything is identical to itselfII) SYMMETRY (Or Commutativity)Note that if Mark Twain is identical to Samuel Clemens, Samuel Clemens is also identical to Mark Twain (i.e. m = s ⊃ s = m). We can state this abstractly as follows:(x)(y)(x = y ⊃ y = x)III) TRANSITIVITYNote that if Anna is the same person as Carol and if Carol is the same person as Karen, then Anna is the same person as Karen. We can state this as follows: (x)(y)(z)(x=y & y=z ⊃ x = z)USEFULNESS OF THE IDENTITY PREDICATEThe identity predicate allows us to do many things, including, but not limited to, the following:1. to represent that there are at least 1 (or 2 or 3, etc.) of a certain kind ofthing.2. to represent that there are at most 1 (or 2 or 3, etc.) of a certain kind ofthing.3. to represent that there are exactly 1 (or 2 or 3, etc.) of a certain kind ofthing.4. to represent sentences containing references to “another.”5. to represent superlatives (fastest, most beautiful, etc.)REPRESENTING QUANTITIES OF THINGS (using Identity)A MINIMUM QUANTITY (existential)There are at least 2 children in the family.(∃x)(∃y)[(Fx&Cx) & (Fy&Cy) & x /= y)]A MAXIMUM QUANTITY (universal)There are at most 3 children in the family.There are no more than 3 children in the family.(x)(y)(z)(v)[(Fx&Cx & Fy&Cy & Fz&Cz & Fv&Cv) ⊃ (x = y V x =z V x = v V y = z V y = v V z = v)AN EXACT QUANTITY (at least + no more than)There are exactly 2 children in the Duggan family.Universe of Discourse: the Duggan family(∃x)(∃y)(Cx & Cy & x /= y) & (x)(y)(z)[(Cx & Cy & Cz) ⊃ x = y V x = z V y = z) There are at least 2 and there are no more than 2REFERENCES TO ANOTHER OR SELFUniverse of Discourse: peopleJack photographs everyone but himself. (x)(x /= j ⊃ Pjx)SUPERLATIVESUniverse of Discourse: peopleJack is the fastest swimmer. Sj & (x)((Sx & x /= j) ⊃ Fjx)。

Predicate logic谓词逻辑课件

Predicate logic
• Bruce is a moose. • This sentence says something about an individual. • This individual is Bruce, and he has the property of being a moose. • Such sentences are called predications (谓语句)—— one predicates something (e.g. a property) of an individual.
•The smallest English sentence is formed by combining a verb with a subject.
•In Predicate Logic, the smallest proposition is formed by combining a predicate with an individual.
M (x) In this expression, x is not an individual constant but an individual variable. x does not stand for a particular individual but for any individual.(x stands for an arbitrary individual.)
“主目”这个术语，在这里用来指那些在语法术语中所讲的主语和宾语。
Predicate logic
• The individual terms that follow the predicate in an expression in predicate is called the arguments of that predicate. • If a predicate takes one argument, it is called a one-place predicate; if it takes two arguments, it is called a two-place predicate, and so on. • P(t) one-place predicate • P(t1, t2) two-place predicate • P(t1, t2…t3) three-place predicate • P(t1, t2, t3…tn) n-place predicate

第十一章谓词逻辑

“是个劳动模范”、“是个大学生”、“…比…高 2cm”、 “…在…与…之间”都是谓词。
个体的一定变化范围叫个体域所有的个体聚集在一起所构成的个体域叫全总个体域以某个个体域为变域的变元叫个体变元. 个体变元可用小写拉丁字母表示刻划一个个体性质的词称之为一元谓词,刻划个个体之间关系的词称之为n元谓词. 由n元谓词及n个个体变元所组成的命题变元可以表示成： F(x1,x2,…,xn).
第十一章谓词逻辑（Predicate Logic）
11.1谓词的概念与表示(Predicate and its expression) 11.2 命题函数与量词 (Propositional functions &
Quantifiers)
11.3谓词公式与翻译(Predicate formulae) 11.4变元的约束(Bound of variable) 11.5谓词演算的等价式与蕴含式(Equivalences &
小结: 本节将原子命题进行分解,分为个体和
谓词两部分.进而介绍了个体和谓词、一元谓词和n元谓词的概念。
重点：掌握一元谓词和n元谓词的概念。
11.2 命题函数与量词(Propositional functions & Quantifiers)
11.2.1 量词(Quantifiers) 11.2.2 命题函数 (Propositional functions)
命题逻辑的局限性：在命题逻辑中，命题是命题演算的基本单位，不再对原子命题进行分解，因而无法研究命题的内部结构、成分及命题之间的内在联系，甚至无法处理一些简单而又常见的推理过程。例如，下列推理：
所有的人都是要死的。苏格拉底是人。苏格拉底是要死的。众所周知,这是真命题。但在命题逻辑中,如果用P,Q,R表示以上三个命题，则上述推理过程为：（P∧Q）R。借助命题演算的推理理论不能证明其为重言式。

predicatedefinition 例子

predicatedefinition 例子摘要：I.引言- 介绍主题- 给出predicate 的定义II.例子- 给出具体的例子- 详细解释例子III.分析- 分析predicate 的性质- 讨论predicate 与其他概念的关系IV.结论- 总结主要观点- 对predicate 的进一步思考正文：I.引言在本篇文章中，我们将探讨predicate 的定义以及其具体的例子。

首先，我们需要了解predicate 是什么。

Predicate 是逻辑学、语言学和计算机科学中的一个重要概念。

在传统的逻辑学中，predicate 通常被翻译为“谓词”或“命题”。

它用来描述或判断一个主语的性质或状态。

简单来说，predicate 定义了一个事物或概念的特征。

II.例子为了更好地理解predicate，我们可以通过一个具体的例子来说明。

假设我们有一个句子：“猫是哺乳动物。

”在这个句子中，“猫”是主语，而“是哺乳动物”就是predicate。

它描述了主语“猫”的一个特性，即它是哺乳动物。

III.分析Predicate 具有以下几个重要的性质：1.谓词性：predicate 主要用来描述主语的性质或状态，如上述例子中的“是哺乳动物”。

2.量化性：predicate 可以对主语进行量化，如“所有的猫都是哺乳动物”。

3.原子性：predicate 通常被认为是语言表达的基本单位，无法再分解为更小的单位。

此外，predicate 与其他概念，如名词和动词，有着密切的关系。

名词通常表示事物的概念，而动词则表示事物之间的关系。

在某些情况下，名词和动词可以充当predicate 的角色。

IV.结论总之，predicate 是一个十分重要的概念，它在逻辑学、语言学和计算机科学中都有着广泛的应用。

通过理解predicate 的定义和性质，我们可以更好地理解这些领域中的相关知识。

谓词逻辑(PredicateLogic)

Q：苏格拉底是人； R：苏格拉底是要死的。前提：P,Q，结论：R。则(P∧Q)→R表示上述推理，这个命题公式不是重言式。
前言
在谓词逻辑中，如果设： H(x)： x是人。
M(x)： x是要死的。 a：苏格拉底。
前提：(x)(H(x) →M(x))，H(a) 结论：M(a) (x)(H(x)→M(x))∧H(a)M(a)
前言
主语
谓语
客（个）体
谓词
客体可以独立存在，它可以是具体的，也可以是抽象的。
而用来描述客体的性质或关系的即是谓词。
为了刻画命题内部的逻辑结构，就需要研究谓词逻辑（Predicate Logic）。
前言
比如： P：张三是大学生 Q：李四是大学生以上这些命题都具备有一个共同的特征就是： x是大学生。 P(x)就可以代表这一类的命题。
第二章谓词逻辑 Predicate Logic
前言
苏格拉底三段论(Socrates syllogism)：所有人都是要死的。苏格拉底是人。所以苏格拉底是要死的。（Socrates, 古希腊哲学家,公元前470~前 399） (孔子,中国伟大哲学家,公元前551~前479)
前言
在命题逻辑中，如果设： P：凡人都是要死的；
P(x) ： x是大学生，a：张三， b：李四， P(a)：张三是大学生 P(b)：李四是大学生
2-1 谓词的概念与表示
2-1.1 谓词的概念定义1：谓词（predicate）在命题中，用以刻画客体词的性质或客体词之间关系
的词即是谓词，谓词相当于命题中的谓语部分。例如： (1) 他是三好学生
比如L(x,y)表示“x小于y”，那么L(2,3)表示了一个真命题“2小于3”。而 L(5,1)表示了一个假命题“5小于1”

逻辑的起源英文作文介绍

逻辑的起源英文作文介绍英文回答：The origins of logic can be traced back to the ancient Greeks, particularly to the philosopher Aristotle (384-322 BCE). Aristotle developed a system of formal logic that became the foundation for Western thought. This system, known as Aristotelian logic, is based on the idea of syllogism, which is a deductive argument that consists of two premises and a conclusion. The validity of a syllogism depends on the relationship between the premises and the conclusion. If the premises are true, then the conclusion must also be true.In addition to Aristotle, other Greek philosophers such as Plato and Socrates also contributed to the development of logic. Plato's theory of Forms, which posits that there is a realm of perfect ideas that exist independently of the physical world, influenced the development of logic by providing a foundation for abstract reasoning. Socrates'method of questioning, known as the Socratic method, also played a role in the development of logic by encouraging critical thinking and the search for truth.The development of logic continued in the Middle Ages, particularly in the work of Islamic philosophers such as Avicenna (980-1037) and Averroes (1126-1198). These philosophers translated Aristotle's works into Arabic and developed their own commentaries on them. This work helpedto preserve and spread Aristotle's ideas throughout the Islamic world and eventually into Europe.In the Renaissance, the study of logic was revived in Europe, particularly in the work of scholars such as Peter Abelard (1079-1142) and William of Ockham (c. 1285-1349). These scholars developed new logical techniques and applied them to the study of theology and other subjects.The development of logic continued in the modern period, with the work of philosophers such as René Descartes(1596-1650), Gottfried Leibniz (1646-1716), and GeorgeBoole (1815-1864). These philosophers developed new systemsof logic that were more powerful and expressive than Aristotelian logic.In the 20th century, logic underwent a major transformation with the development of mathematical logic. This new field of logic, which is based on the use of mathematical symbols and techniques, has revolutionized the study of logic and has led to the development of newlogical systems that are more powerful and expressive than ever before.中文回答：逻辑的起源。

离散数学 Predicates and Quantifiers(期望与量词)

How about ∀x P(x) ?
Solution:
∀x P(x) is the same as
P(1) /\ P(2) /\ P(3) /\ P(4)
--------false
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9
(7) Example 13 (page 36) ∀x (x2≥x) the domain ---- all integers
--------denoted as ∃x P(x) “There is an x such that P(x)” “There is at least one x such that P(x)” “For some x, P(x)”
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P(x) is false for every x
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4. Other quantifiers The most often quantifier is uniqueness quantifier ---denoted by ∃!xP(x), or ∃1x P(x)
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3
(5) In general,
A statement of the form P(x1,x2,…,xn) is the
value of the propositional function P at the

predicate机制

predicate机制English Answer:Predicate Mechanism.In computer science, a predicate is a logical expression that evaluates to either true or false. Predicates are commonly used in programming to control the flow of execution or to make decisions based on certain conditions.Predicate logic is a formal system for representing and reasoning about predicates. It provides a set of rules for combining predicates into more complex expressions and for determining the truth value of these expressions.Predicate mechanisms are used in a variety of programming languages and applications. For example, predicates can be used to:Check the validity of input data.Determine if a certain condition is met.Control the flow of execution in a program.Make decisions based on user input.There are two main types of predicates:First-order predicates are predicates that can be applied to any object in the domain of discourse.Higher-order predicates are predicates that can be applied to other predicates.Predicate mechanisms can be implemented in a variety of ways. One common implementation is to use a truth table, which is a table that lists all possible combinations of input values and the corresponding output values.Another common implementation is to use a decision tree,which is a hierarchical structure that represents thelogical relationships between different predicates.Predicate mechanisms are a powerful tool for representing and reasoning about logical expressions. They are used in a variety of programming languages and applications, and they can be implemented in a variety of ways.Chinese Answer:谓词机制。

Propositionalvs.PredicateLogic

1First-Order Logic(First-Order Predicate Calculus)2Propositional vs. Predicate Logic •In propositional logic, each possible atomic fact requires aseparate unique propositional symbol.•If there are n people and m locations, representing the factthat some person moved from one location to another requires nm 2 separate symbols.•Predicate logic includes a richer ontology:-objects (terms)-properties (unary predicates on terms)-relations (n -ary predicates on terms)-functions (mappings from terms to other terms)•Allows more ﬂexible and compact representation ofknowledgeMove(x, y, z) for person x moved from location y to z.Syntax for First-Order LogicSentence →AtomicSentence| Sentence Connective Sentence | Quantiﬁer Variable Sentence |¬Sentence | (Sentence)AtomicSentence →Predicate(Term, Term, ...) | Term=Term Term →Function (Term,Term,...)| Constant |Variable Connective → ∨ | ∧ | ⇒ | ⇔Quanitﬁer → ∃ | ∀Constant → A | John | Car1Variable → x | y | z |...Predicate → Brother | Owns | ...Function → father-of | plus | ...First-Order Logic:Terms and Predicates•Objects are represented by terms :-Constants : Block1, John-Function symbols: father-of, successor, plusAn n -ary function maps a tuple of n terms to another term: father-of(John), succesor(0), plus(plus(1,1),2)•Terms are simply names for objects. Logical functions arenot procedural as in programming languages. They do not need to be deﬁned,and do not really return a value.Allows for the representation of an inﬁnite number of terms.•Propositions are represented by a predicate applied to atuple of terms. A predicate represents a property of or relation between terms that can be true or false:Brother(John, Fred), Left-of(Square1, Square2)GreaterThan(plus(1,1), plus(0,1))•In a given interpretation, an n -ary predicate can deﬁned asa function from tuples of n terms to {T rue, False} or equivalently, a set tuples that satisfy the predicate:{<John, Fred>, <John, T om>, <Bill, Roger>, ...}5Sentences in First-Order Logic •An atomic sentence is simply a predicate applied to a set ofterms.Owns(John,Car1)Sold(John,Car1,Fred)Semantics is True or False depending on the interpretation,i.e. is the predicate true of these arguments.•The standard propositional connectives (∨ ¬ ∧ ⇒ ⇔)can be used to construct complex sentences:Owns(John,Car1)∨ Owns(Fred, Car1)Sold(John,Car1,Fred)⇒¬Owns(John, Car1)Semantics same as in propositional logic.6Quantiﬁers•Allows statements about entire collections of objects ratherthan having to enumerate the objects by name.•Universal quantiﬁer:∀xAsserts that a sentence is true for all values of variable x ∀x Loves(x, FOPC)∀x Whale(x)⇒ Mammal(x)∀x Grackles(x)⇒ Black(x)∀x (∀y Dog(y) ⇒ Loves(x,y))⇒(∀z Cat(z)⇒ Hates(x,z))•Existential quantiﬁer:∃Asserts that a sentence is true for at least one value of a variable x∃x Loves(x, FOPC)∃x(Cat(x)∧ Color(x,Black)∧ Owns(Mary,x))∃x(∀y Dog(y)⇒ Loves(x,y))∧ (∀z Cat(z)⇒ Hates(x,z))Use of Quantiﬁers•Universal quantiﬁcation naturally uses implication:∀x Whale(x)∧Mammal(x)Says that everything in the universe is both a whale and a mammal.•Existential quantiﬁcation naturally uses conjunction:∃x Owns(Mary,x)⇒ Cat(x)Says either there is something in the universe that Mary does not own or there exists a cat in the universe.∀x Owns(Mary,x)⇒ Cat(x)Says all Mary owns is cats (i.e. everthing Mary owns is a cat). Also true if Mary owns nothing.∀x Cat(x)⇒ Owns(Mary,x)Says that Mary owns all the cats in the universe.Also true if there are no cats in the universe.Nesting Quantiﬁers•The order of quantiﬁers of the same type doesn’t matter∀x ∀y(Parent(x,y)∧ Male(y)⇒ Son(y,x))∃x ∃y(Loves(x,y)∧Loves(y,x))•The order of mixed quantiﬁers does matter:∀x ∃y(Loves(x,y))Says everybody loves somebody, i.e. everyone has someone whom they love.∃y ∀x(Loves(x,y))Says there is someone who is loved by everyone in the universe.∀y ∃x(Loves(x,y))Says everyone has someone who loves them.∃x ∀y(Loves(x,y))Says there is someone who loves everyone in the universe.9Variable Scope•The scope of a variable is the sentence to which thequantiﬁer syntactically applies.•As in a block structured programming language, a variablein a logical expression refers to the closest quantiﬁer within whose scope it appears.∃x (Cat(x)∧∀x(Black (x)))The x in Black(x) is universally quantiﬁed Says cats exist and everything is black•In a well-formed formula (wff ) all variables should beproperly introduced:∃xP(y)not well-formed•A ground expression contains no variables.10Relation Between Quantiﬁers•Universal and existential quantiﬁcation are logically relatedto each other:∀x ¬Love(x,Saddam)⇔ ¬∃x Loves(x,Saddam)∀x Love(x,Princess-Di)⇔ ¬∃x ¬Loves(x,Princess-Di)•General Identities - ∀x ¬P ⇔¬∃x P - ¬∀x P ⇔∃x ¬P - ∀x P ⇔¬∃x ¬P - ∃x P⇔¬∀x ¬P-∀x P(x)∧Q(x)⇔∀xP(x)∧∀xQ(x)-∃x P(x)∨Q(x)⇔∃xP(x) ∨∃xQ(x)Equality •Can include equality as a primitive predicate in the logic,orrequire it to be introduced and axiomitized as the identity relation .•Useful in representing certain types of knowledge:∃x ∃y(Owns(Mary, x)∧ Cat(x) ∧Owns(Mary,y)∧ Cat(y)∧ ¬(x=y))Mary owns two cats. Inequality needed to insure x and y are distinct.∀x ∃y married(x, y)∧∀z(married(x,z)⇒ y=z)Everyone is married to exactly one person. Secondconjunct is needed to guarantee there is only one unique spouse.Higher-Order Logic•FOPC is called ﬁrst-order because it allows quantiﬁers torange over objects (terms) but not properties, relations, or functions applied to those objects.•Second-order logic allows quantiﬁers to range overpredicates and functions as well:∀x ∀y [ (x=y)⇔ (∀p p(x)⇔ p(y)) ]Says that two objects are equal if and only if they have exactly the same properties.∀f ∀g [ (f=g)⇔ (∀x f(x)= g(x)) ]Says that two functions are equal if and only if they have the same value for all possible arguments.•Third-order would allow quantifying over predicates ofpredicates, etc.For example, a second-order predicate would be Symetric(p) stating that a binary predicate p represents a symmetric relation.13Notational Variants •In Prolog, variables in sentences are assumed to beuniversally quantiﬁed and implications are represented in a particular syntax.son(X, Y) :- parent(Y ,X), male(X).•In Lisp, a slightly different syntax is common.(forall ?x (forall ?y (implies (and (parent ?y ?x) (male ?x)) (son ?x ?y)))•Generally argument order follows the convention that P(x,y)in English would read “x is (the) P of y”14Logical KB•KB contains general axioms describing the relationsbetween predicates and deﬁnitions of predicates using ⇔.∀x,y Bachelor(x)⇔ Male(x)∧ Adult(x)∧¬∃yMarried(x,y).∀x Adult(x)⇔Person(x)∧ Age(x) >=18.•May also contain speciﬁc ground facts.Male(Bob), Age(Bob)=21, Married(Bob, Mary)•Can provide queries or goals as questions to the KB:Adult(Bob) ?Bachelor(Bob) ?•If query is existentially quantiﬁed, would like to returnsubstitutions or binding lists specifying values for the existential variables that satisfy the query.∃x Adult(x) ?∃x Married(Bob,x) ?{x/Bob} {x/Mary}∃x,y Married(x,y) ?{x/Bob, y/Mary}Sample Representations•There is a barber in town who shaves all men in town whodo not shave themselves.∃x (Barber(x)∧ InT own(x)∧∀y (Man(y)∧ InTown(y)∧ ¬Shave(y,y)⇒ Shave(x,y)))•There is a barber in town who shaves only and all men intown who do not shave themselves.∃x (Barber(x)∧ InT own(x)∧∀y (Man(y)∧ InTown(y)∧ ¬Shave(y,y) ⇔Shave(x,y)))•Classic example of Bertrand Russell used to illustrate aparadox in set theory:Does the set of all sets contain itself?。

谓词逻辑自然语言翻译例句

谓词逻辑自然语言翻译例句
谓词逻辑（predicate logic）是一门表示量化关系的语言，其主要用于表达不可直接表达的语言表达及抽象事物关系。

举个例子，以下是一句谓词逻辑表达式及其自然语言翻译：
∀x∃y(Fx⇒Gy)。

对于每个x，存在一个y，如果x是F，那么y就是G。

谓词逻辑是一个基于逻辑的语言，可用来表达抽象的逻辑表达式，并可翻译成自然语言。

根据谓词逻辑，有如下例句：
∀x∃y(Px=>Qy)。

对于任何x，存在一个y，如果x是P，那么y就是Q。

∃x∀z(Mx⇒Nz)。

存在一个x，对任何z，如果x是M，那么z就是N。

∀x∃y(Lx&Fy)。

对于所有x，存在一个y，同时x是L和y是F。

谓词逻辑语言可以帮助人们更清晰地表达一些不能直接表达的逻辑表达式并把它们翻译成自然语言。

下面是另一句谓词逻辑表达式及其自然语言翻译：
∃x∀y(Rx⇒Sy)。

存在一个x，对任何y，如果x是R，那么y就是S。

亚里士多德解释逻辑学

亚里士多德解释逻辑学
亚里士多德是古希腊哲学家和逻辑学家，被认为是逻辑学的奠基者之一。

他在《论演绎推理》（Prior Analytics）和《论辩证法》（Topics）等作品中提出了对逻辑学的解释和系统化。

亚里士多德的逻辑学主要涉及以下几个方面：
1. Categorical Syllogisms（范畴三段论）：亚里士多德系统地研究了三段论的形式推理，其中包括如“A是B”和“B是C”这样的命题，并从中推导出结论。

他将命题分为四种类型，即A型、E型、I 型和O��，并探讨了它们之间的关系。

2. Terms and Predication（术语和谓词）：亚里士多德强调了术语和谓词的重要性。

他认为术语是构建命题的基本单位，而谓词则用来描述和判断术语之间的关系。

3. Logic of Definition（定义的逻辑）：亚里士多德认为定义是确立概念的关键，因此他详细研究了定义的逻辑。

他提出了实质定义和属性定义的区别，并探讨了定义与分类、泛化和特殊化等概念之间的关系。

4. Fallacies（谬误）：亚里士多德将一些常见的论证错误归类为谬
误，并详细研究了它们的形式和特征。

他认为通过理解和识别谬误，可以提高逻辑推理的准确性和有效性。

亚里士多德的逻辑学对后来的哲学和逻辑学发展产生了深远影响。

尽管其观点在现代逻辑学中已经得到修正和扩展，但他的基本思想和方法仍然被广泛研究和借鉴。

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c s .L O /9906010 7 J u n 1999Predicate Logic with DefinitionsVictor MakarovEMD Inc186 Bay 31st StreetBrooklyn, NY 11214USAvedasystem@AbstractPredicate Logic with Definitions (PLD or D-logic) is a modification of first-order logic intended mostly for practical formalization of mathematics. The main syntactic constructs of D-logic are terms, formulas and definitions. A definition is a definition of variables, a definition of constants, or a composite definition (D-logic has also abbreviation definitions called abbreviations). Definitions can be used inside terms and formulas. This possibility alleviates introducing new quantifier-like names. Composite definitions allow constructing new definitions from existing ones.1. Introduction The importance of practical formalization of mathematics has been widely recognized now – both for mathematics itself and especially for various applications such as, for instance, computer hardware and software design [Friedman 97, Qed Manifesto 94].Following [Harrison 96], by practical formalization of mathematics “we mean expressing mathematics, both statements and proofs, in a (usually small and simple) formal language with strict rules of grammar and unambiguous semantics”. Such a language can benaturally called as a practical formal mathematical language [Glushkov 72].One of the most well-known such languages is the Mizar language [Trybulec 93](though the language is not very simple – a BNF description of the Mizar syntax is 10pages long, there are about 140 nonterminals in the BNF grammar, and no complete description of the Mizar “semantics” has been yet published). The Mizar language is based on first-order logic (the language has also some not first-order features such as free second order variables used in axiom schemas). The Ontic language [McAllister 88] is another such language based on first-order logic. But most other such languages are based on higher order logic, e.g. [Gordon 93] (HOL).From the other side, the development of specification languages such as Z [Spivey 92],VDM SL [Jones 90], and modern object – oriented programming languages such as C++[Stroustrup 97], Eifell [Meyer 92], Smalltalk [Kay 96] is very interesting for the design of practical logical languages . In this connection, see, e.g. [Tseytin 98], where the term “object-oriented logic” has been coined. In particular, Tseytin writes:“The fundamental concepts of object-oriented programming like object , class ,i nheritance … belong to description of thinking rather than to programming techniques”.Because of their importance, let us discuss in some details the main concepts of object-oriented programming (OOP). The main concepts of OOP are the concepts of object and class. An object is a k-tuple of values (k ≥ 1). A value can be, for example, an integer or real number, a character or a string of characters or another object. Any object is an instance of a class. A class is a description (in an object-oriented programming language) of the structure of its objects, some conditions (called class invariant in the Eifell language) that all objects of the class must satisfy and some operations on its objects. A class is to an object (of its class) as a blueprint of a real object (a bicycle, for example) is to the real object.Note the similarity between the concept of object in OOP and the concept of mathematical object. A mathematical object, such as a group, a graph and so on, is also usually defined as a k-tuple of sets, functions or other mathematical objects. For example, a group is a pair( or a 2-tuple) <G, f> where G is the carrier set and f is a function from G×G to G, satisfying certain conditions – group axioms. Note that a concrete group is a model of the group theory and, more generally, a concrete mathematical object is a model of the corresponding theory (excluding primitive mathematical objects such as, for example, integer numbers). So we may say that a class is to an object (of the class) as a mathematical theory is to a model of the theory (and the class invariant is the analogue of the axioms of a theory).As mathematical theories are the main way of structuring of the mathematical knowledge, so classes are the main way of structuring of the “programming knowledge”.A very important feature of OOP is the possibility to build class hierarchies by defining a class as a heir of other class (simple inheritance) or classes (multiple inheritance). In mathematics, a similar method is used when we define, for example, the theory of linear ordered groups referring to the theory of linear order and the group theory.Unfortunately, classical logical languages (such as first-order logic) have no special syntactic constructs for definitions of mathematical theories (actually, they have no syntactic constructs for definitions at all). It is the main obstacle in using a classical logical language as a practical formal mathematical language.In this paper, Predicate Logic with Definitions (PLD or D-logic) intended mostly for practical formalization of mathematics is introduced. As Tseytin’s “object-oriented logic”, D-logic has been also developed under influence of object-oriented programming. But, unlike Tseytin’s “object-oriented logic” (which is not based on the standard (i.e. first-order) logic), the D – logic is a modification of first-order logic by adding to the main syntactic constructs of first-order logic (terms and formulas) new syntactic constructs called definitions. A definition is a definition of variables, a definition of constants, or a composite definition. Definitions of constants in D-logic are in a sense analogous to classes in object-oriented programming languages and they are mostly used for defining mathematical theories. Definitions of variables are mostly used in quantified formulas and terms. Such an approach (using definitions inside formulas and terms) alleviates introducing new quantifier-like names. Composite definitions allow constructing new definitions from existing ones. It corresponds (in the case of definitions of constants) to defining a new class in object-oriented programming as a heir of early defined classes.D-logic is intended to be used as the input language of a new version of the Veda proof-checking system (main ideas of the Veda project have been described in [Makarov 92]).2. From first-order logic to D – logicThe symbols used in the language of first-order logic (we follow basically to [Davis 93] and [Shoenfield 73]) are usually divided into the following classes:1.Logical symbols: ~ (negation), → (implication), & (conjunction), ∨ (disjunction),≡ (equivalence), ∀ (universal quantifier), ∃ (existential quantifier).2.Constant symbols.3.Function symbols.4.Relation symbols.For each function and relation symbol, a natural number k, called arity, must be assigned. The notions of terms and formulas can then be defined [Davis 93]. Fixing the sets of constant, function and relation symbols we receive a concrete first-order language – for instance, the language of the ZFC set theory.Let us consider the formulas of the form ∃xP and ∀xP, where x is a variable, P is a formula. Let us slightly change the syntax of such formulas and write ∃(x|P) and ∀(x|P), respectively, where ‘|’ is a punctuation symbol. Let us call (in this section) the expression x|P as a definition of variables (actually, a definition of variables in D-logic can contain many variables – see below).An immediate benefit of this approach is the following: it is easier to introduce new “quantifier-like” names. To show it, let us generalize the notion of arity in the following way. Let us call a generalized arity (or g-arity or simply arity) any finite sequence of the letters F, T, D (from Formula, Term, Definition). We assign a g-arity to each symbol as follows:1.1Unary logical symbols (~): FF.1.2Binary logical symbols (→, &, ∨, ≡ ): FFF.1.3Quantifiers (∀,∃): DF.2. Constant and variable symbols: T.3. Function symbols of arity k: T k+1 (i.e. the sequence of k+1 letters T).4. Relation symbols of arity k: T k F. For instance, the g-arity TTF will be assigned to the symbols “=” (equality) and “∈” (is a member of).Now it is obvious that for introducing, for example, Hilbert’s ε-symbol or the symbol λused in λ-notation, it is sufficient to assign to the symbols ε and λ the g-arities DT and DTT respectively (and, of course, provide defining axioms for the symbols). So the expressions εxP(x) and λxf(x) could be written in the following way: ε(x|P(x)) and λ(x|true, f(x)) respectively. But actually, due to practical considerations, a slightlydifferent syntax is used: H[x|P(x)] and F[x|true, f(x)] respectively (typed λ-terms of the form λx:t f(x) can be written as F[x|x∈t, f(x)] or simply as F[x:t, f(x)] ).Besides of definitions of variables, D-logic has also definitions of constants which have (in the simplest case) the form def[c1, ..., c k | P] (i.e. they have the same abstract syntax as the definitions of variables) where k > 0, c1, ..., c k are some constant symbols, P is the defining axiom of the constant symbols.Note that it is possible to introduce symbols of the g-arities, for example, DD, DDD or DFD. Such symbols can be used for denoting operations on definitions and constructing composite definitions.For example, if d is a definition of variables of the form (x|P), R is a formula then the negation of the definition d (symbolically, ~d) is the definition (x|~P). And the conjunction of the definition d and the formula R (symbolically, d&R) is the definition (x|P&R). Another useful operations on definitions are concatenation and hereditary concatenation of two definitions (the names “!” and “%” will be used for these operations, respectively).If d1 is (x1, … ,x k | P), d2 is (y1, … ,y m | Q) then:d1 ! d2 is (x1, … ,x k, y1, … ,y m | P & Q),d1 % d2 is (x1, … ,x k, y2, … ,y m P & Q’),where Q’ is the result of substitution, in the formula Q, the variable x1 instead of the variable y1. Though definitions of variables are used in the examples above, the same operations are also can be used for definitions of constants (because both definitions of constants and definitions of variables have the same abstract syntax).It is widely accepted now that every mathematical theory T can be considered as an extension of the ZFC set theory by adding to ZFC (or an extension of ZFC) the constants and axioms of the theory T [Dieudonne 82, p.215 ]. So definitions of constants can be used for defining theories.For example a definition of group can have the following form:Group := def[G, * ; <group axioms>]where the constants G and * represent respectively the carrier set and the law of composition (a function from G x G to G ), the symbol “:=” means that the name Group is an abbreviation of the expression “def[G, * ; <group axioms>]” – see below a description of abbreviations. Note that the formula E[Group] expresses the following statement: the group theory is consistent.Using the & operation on definitions, a definition of commutative group can be written in the following form:CommutativeGroup := Group & A[x, y | x ∈ G & y ∈ G → x * y = y * x].Using the definition of group Group and a definition of linear order (let us name it as LinearOrder), a definition of linear ordered group can be written in the following way:LinearOrderedGroup := Group % LinearOrder & Rwhere % is the operation of hereditary concatenation (see above) and R is an additional axiom.For denoting symbols we shall use both internal and external names. An internal name can be identified with the symbol itself. All internal names must be unique. Because it is inconvenient to use internal names, external (usually more mnemonic but not necessary unique) names are used. It is always must be possible for every external name to determine its corresponding internal name (as a rule, using the context). In a computerimplementation of D–logic, internal names can be chosen automatically, when the system handles definitions. Usually, an external name is an identifier, an operator or a bracketed name. An identifier is a sequence of letters or digits beginning from a letter. Usually, an operator is a sequence of such characters as +, −, *, /, %, &, ~, #, =, ^. Mostly, operators are used for constructing infix expressions. A bracketed name is an identifier after which a left bracket follows. Internal names can be considered as related to abstract syntax [Sethi 96], whereas external names are related to concrete syntax.3.D-languagesTo specify a (logical) language L we must specify the set of its symbols and the set of formulas of L (the language L can also have other syntactic constructs such as e.g. terms). The main syntactic constructs of a D-language are formulas, terms and definitions which are called basic constructs (it is convenient to include into a D-language some auxiliary syntactic constructs called declarations – see below). For each basic construct, we assign a mode – one of the letters T, F, D in the following way:Terms are assigned the mode T, formulas are assigned the mode F, and definitions are assigned the mode D.A D-language L has as symbols the following:a)Variables – an infinite set V;b)Primary symbols – a nonempty set S, such that S ∩ V = ∅. For each primary symbols, a g-arity g(s) must be assigned – see section 2 (from now on we shall write simply “arity” instead of “g-arity”). The set S must include the set of logical symbols LS (the arities of the logical symbols are shown on the right): ~ (negation) FF, & (conjunction) FFF, ∨ (disjunction) FFF, → (implication) FFF, ≡ (equivalence) FFF, if (if-then-else) FTTT, true (a truth value) F, false (a truth value) F, ∀ (the universal quantifier)DF, ∃ (the existential quantifier) DF, H (Hilbert’s epsilon symbol) DT, = (equality) TTF. The symbols A, E may be used instead of the symbols ∀, ∃ due to practical considerations. The other (i.e. not logical) symbols are called nonlogical symbols. We assume that every D-language has the symbol ∈ with the arity TTF and the symbol → of the arity TTT among its nonlogical symbols (A→B will denote the set of all functions from A to B).c)Punctuation symbols – the set P of such symbols as parenthesis (‘(‘,’)’), brackets (‘[‘,‘]’), comma (‘,’), semicolon (‘;’), vertical bar (‘|’) and some other symbols (see below).Let us call the quadruple U = (V, S, P, g) as a vocabulary (of the D-language). An expression in the vocabulary U is a finite sequence of symbols from the set V∪ S ∪ P. Given the vocabulary U, we define the following sets (below last(g(s)) denotes the last element of the sequence g(s) ):C = {s:S | g(s) = T} – the set of constant symbols (or simply constants);R = {s:S | last(g(s)) = ‘F’} – the set of relation symbols;F = {s:S | last(g(s)) = ‘T’} – the set of function symbols;D = {s:S | last(g(s)) = ‘D’} – the set of definition symbols.3.1 DeclarationsDeclarations are auxiliary syntactic constructs (actually, they should be considered as metalanguage constructs). As definitions, declarations are used for introducing new names. But, unlike definitions, declarations can not be used inside terms, formulas or definitions. A declaration can be an abbreviation, a declaration of primary names, a declaration of variables, a declaration of an axiom or a declaration of a theorem.An abbreviation has the form N := B, where N is an identifier (i.e. a sequence of letters and digits beginning from a letter), B is a term or a formula or a definition. The purpose of abbreviations is obvious: to use a possibly short name N instead of possibly large terms, formulas or definitions. The name N can not occur (directly or indirectly) in B.If B is a formula then the abbreviation is called a formula abbreviation.A declaration of primary names can have one of the following forms:def[N : A] ordef[N : A; u1 = u2] ordef[N : A; p1≡ p2]where N is a primary name (i.e. the name denoting a primary symbol) – an identifier or operator or a bracketed name, A is its arity, u1 is a term of the form N or of the form N(v1, …, v k) where v1, …, v k (k > 0) are some variables (including propositional and definitional ones), u2 is a term, p1 is a formula of the form N(v1, …, v k), p2 is a formula. All free variables of u2 (or p2) must be among the variables v1, …, v k. The formula u1 = u2 (or p1≡ p2) is called the defining axiom of the name N. A declaration of primary names can be understood as a command to extend the current theory T by adding to the primary names of the theory T the new primary name N and the corresponding defining axiom. The form of the defining axioms guarantees that the extension of the theory T will be a conservative [Shoenfield 73] one.A declaration of variables is used for introducing syntactic variables [Shoenfield 73] and has the form V1, …, V k : A where V1, …, V k are syntactic variable names (some identifiers), A is an one-letter arity (i.e. A is T, F, or D).Syntactic variables are used for writing axiom schemas. Syntactic variables of the arity T are called term variables, of the arity F – propositional variables and of the arity D – definitional variables.A declaration of an axiom has the form Axiom A, where A is a formula abbreviation of the form N := Q. The declaration Axiom N := Q states that the formula Q is an axiom (and N is its name).A declaration of a theorem has the form Theorem T, where T is a formula abbreviation of the form N := Q. The declaration Theorem N := Q states that the formula Q is possibly a theorem (and N is its name). A computer implementation of D-logic may try to find a proof of the formula Q.3.2 DefinitionsA definition can be a simple definition or a composite definition. A simple definition is a definition of constants or a definition of variables.A definition of constants can be typed or untyped. An untyped definition of constants has the following form:(1)def[c1, …, c k; P]where c1, …, c k (k > 0) are constant names (identifiers or operators), P is a formula called the defining formula of the definition (the defining formula can not be considered as an axiom, because, e.g. P may be the formula “false”). If P is the formula “true”, then the definition (1) can be written as def[c1, …, c k].A typed definition of constants has the following form:(2)def[c1:t1, …, c k:t k; P]where c1, …, c k (k > 0) and P have the same meaning as above and t1, … , t k are types. Each type is a term or a definition (if P is the formula “true”, then it can be omitted).. A typed definition of the form (2) corresponds to the following untyped definition of constants:(3)def[c1, …, c k; c1∈t1& …& c k∈t k&P]A definition of variables can be typed or untyped or short. An untyped definition of variables has the following form:(4)dev[x1, …, x k; P]where x1, …, x k (k > 0) are variable names (some identifiers), P is a formula called the defining formula of the definition (again, the defining formula can not be considered as an axiom). If P is the formula “true”, then the definition (4) can be written as dev[x1, …, x k].A typed definition of variables has the following form:(5)def[x1:t1, …, x k:t k; P]where x1, …, x k (k > 0) and P have the same meaning as above and t1, … , t k are types (if P is the formula “true”, then it can be omitted). A typed definition of the form (5) corresponds to the following untyped definition of constants:(6)def[c1, …, c k; c1∈t1& …& c k∈t k & P]A short definition of variables has the form x:t where x is a variable name (an identifier) and t is a type. The definition x:t denotes the definition dev[x:t].A composite definition has the form g(z1, …, z k), where g is a definition symbol of the arity m1…m k D (k > 0), and z1, …, z k are basic constructs of the modes m1, … ,m k respectively. If g is an infix name (or unary prefix name) then the notation z1 g z2 (or respectively g z1) must be used. We shall assume that for each definition symbol is given a computable function that converts the composite definition into a simple one (i.e. definition symbols are eliminable).It will be also assumed that there is a predefined definition symbol Rep of the arity DTDD. If d, d1 are definitions, z = [z1, … ,z k] is a model of d (for a definition of model in D-logic, see below section 6), then Rep(d, z, d1) denotes the result of substitution in the definition d1 the terms z1, … ,z k instead of the corresponding d-names in d1. Rep(d, z, d1) can be written as d(z, d1).3.3 Terms(1)Every variable introduced in a definition of variables is a term.(2)Every primary name of the arity T is a term.(3)Every constant introduced in a definition of constants is a term.(4)If f is a function symbol of the arity m1…m k T (k > 0), and z1, … ,z k are basicconstructs of the modes m1, … ,m k respectively then f(z1, … ,z k) is a term. If f is an infix name (or an unary prefix name or a bracketed name “h[“ - such names are called nonstandard) then the notation z1 f z2 (or respectively f z1 or h[z1, … ,z k]) must be used.(5)If d is a definition, z and x are terms, then d(z, x) is a term. Usually, z = [z1, … ,z k ]is a model of d; d(z, x) is the result of substitution the terms z1, … ,z k instead of the corresponding d-names in x ; in other words, d(z, x) is “the value of x in the model z”).(6)If z, x are terms, then z.x is a term. Usually, z is a model of a definition d and thedefinition d can be found from the context. Then z.x denotes the term d(z, x).(7)If z1, …, z k are terms (k ≥ 0) then { z1, … ,z k } is a term (“the set consisting of z1, …,z k”).(8)If z1, …, z k are terms (k ≥ 0) then [ z1, … ,z k ] is a term (“the tuple (z1, … ,z k)”).(9)if d is a definition then [d] is a term. Suppose d has the form (x1, …, x k|P).Then if k = 1 then [d] denotes x1, otherwise [d] denotes the tuple [x1, …, x k].(10)There are no other terms.3.4 Formulas(1)Every primary name of the arity F (i.e. true and false) is a formula.(2)If r is a relation symbol of the arity m1…m k F (k > 0), and z1, … ,z k are basicconstructs of the modes m1, … ,m k respectively then r(z1, … ,z k) is a formula. If r is a nonstandard name then the corresponding notation must be used.(3)If d is a definition, z is a term, and p is a formula, then d(z, p) is a formula.Usually, z = [z1, … ,z k ] is a model of d; d(z, p) is the result of substitution the terms z1, … ,z k instead of the corresponding d-names in p ; in other words, d(z, p) is “the value of p in the model z”.(4)If d is a definition, z is a term, then d(z) is a formula (“ z is a model of d”). If d is(x|P(x)) then d(z) denotes the formula P(z).(5)There are no other formulas.4. D – theoriesA D-theory T is a triple <L, A, R> where L is a D-language, A is a set of axioms (every axiom is a formula of L), R is a set of inference rules (or simply rules). Eachinference rule is a computable function from the set Form(L)n to the set Form(L) where n is a natural number given for each inference rule.The set A (of axioms) is the union of two sets, A L and A N – the sets of logical and nonlogical axioms, respectively. The logical axioms of a D-theory are the following formulas (below d is a definitional variable, z is a term variable, p is a propositional variable; note that if an axiom contains syntactic variables then it actually represents an axiom schema and denotes an infinite set of axiom):1)Tautologies: every tautology is an axiom;2)Each formula of the form d(z) → d(H[d]) is an axiom; (ε-axiom)3)Each formula of the form E[d] ≡ d(H[d]) is an axiom; (the defining axiom for theexistential quantifier)4)Each formula of the form A[d] ≡ d(H[~d]) is an axiom; (the defining axiom for theuniversal quantifier)5)Each formula of the form E[d, p] ≡ E[d & p] is an axiom; (the defining axiom forthe bounded existential quantifier)6)Each formula of the form A[d, p] ≡ A[d → p]) is an axiom; (the defining axiom forthe bounded universal quantifier)The set R (of inference rules) consists of the following rule:1)Modus ponens (MP): from the formulas P and P → Q infer the formula Q.The set of theorems of the D-theory T can be defined in the usual way:1)The axioms of T are theorems of T;2)If all of the hypotheses of a rule of T are theorems of T then the conclusion of the ruleis a theorem of T;3)Any theorem of T may be obtained only using these definitions.A D-theory T is called consistent if the formula “false” is not a theorem of T.A D-calculus is a D-theory without nonlogical axioms.Theorem. Any D-calculus is a consistent D-theory.Proof: in a similar way as for first-order logic [Mendelson 63].5.Some theorems of D-logicMost of theorems in first-order logic have natural analogues in D-logic. In this section, such analogues of some theorems from [Hilbert 68, chapter 4, §3] will be listed (mostly without proofs, with the same numeration). The original theorems (maybe in a slightly different notation) will be written on the right.a) A[d] → d(z); ( ∀xP(x) → P(z) )b) d(z) → E[d]; ( P(z) →∃xP(x) )Proof: immediately from the ε-axiom d(z) → d(H[d]) and the defining axiomE[d] ≡ d(H[d]);1. A[d] → E[d]; ( ∀xP(x) →∃xP(x) )2. ~A[d] ≡ E[~d]; ( ~∀xP(x) ≡∃x ~P(x) )2.a ~A[d, P] ≡ E[d, ~P]3`. ~E[d] ≡ A[~d]; ( ~∃x P(x) ≡∀x ~P(x) )3.a ~E[d, P] ≡ A[d, ~P]4. A[P → d] ≡ P → A[d] (if P is d-free); (∀x (P → Q(x)) ≡ P →∀xQ(x))1A[P ∨ d] ≡ P ∨ A[d]; (if P is d-free); (∀x (P ∨ Q(x)) ≡ P ∨∀xQ(x))2A[P & d] ≡ P & A[d]; (if P is d-free); (∀x (P & Q(x)) ≡ P & ∀xQ(x))3A[d, P & Q] ≡ A[d, P] & A[d, Q]; (∀x (P(x) & Q(x)) ≡∀xP(x) & ∀xQ(x))4E[P ∨ d] ≡ P ∨ E[d]; (if P is d-free); (∃x (P(x) ∨ Q(x)) ≡ P ∨∃xQ(x))5E[d, P ∨ Q] ≡ E[d, P] ∨ E[d, Q]; (∃x (P(x) ∨ Q(x)) ≡∃xP(x) ∨∃xQ(x))6E[P & d] ≡ P & E[d]; (if P is d-free); (∃x (P & Q(x)) ≡ P & ∃xQ(x) )7A[d, P → Q] → (A[d, P] → A[d, Q]); (∀x (P(x) → Q(x)) → (∀x P(x) →∀x Q(x)) ) 8A[d, P → Q] → (E[d, P] → E[d, Q]); (∀x (P(x) → Q(x)) → (∃x P(x) →∃x Q(x)) ) 9A[d1, A[d2, P]] ≡ A[d2, A[d1, P]]; (∀x∀y P(x, y) ≡∀y∀x P(y, x))10(no natural analogue); (∀x∀y P(x, y) →∀x P(x, x))11A[d1, A[d2, P & Q]] ≡ A[d1, P] & A[d2, P]; (∀x∀y (P(x) & Q(y)) ≡∀x P(x) &∀y Q(y))16a. A[d, P → Q] ≡ E[d, P] → Q; (Q is d-free); (∀x(P(x) → Q) ≡∃xP(x) → Q)16b. E[d, P → Q] ≡ A[d, P] → Q; (Q is d-free); (∃x(P(x) → Q) ≡ (∀xP(x) → Q) )17. A[d, P ≡ Q] → (A[d, P] → A[d, Q]); (∀x (P(x) ≡ Q(x) → (∀xP(x) ≡∀xQ(x)))18. E[d1, A[d2, P]] → A[d2, E[d1, P]]; (∃x∀y P(x, y) →∀y∃x P(x, y) )The following formulas are also theorems:21. (d → P)(z) ≡ d(z) → d(P, z)22. (d & P)(z) ≡ d(z) & d(P, z)23. (d ∨ P)(z) ≡ d(z) ∨ d(P, z)24. (~d)(z) ≡ ~ (d(z))25. A[d, P] & E[d] → P25a. A[d, P] & d(z) → P26.A[d, false] ≡ ~E[d] (from 16.a, taking P = true, Q = false)27.A[d, P ∨ Q] ≡ A[d&~P, Q]28.A[d, P→Q] ≡ A[d& P, Q]Some derived inference rules of D-logic are as follows:1)from P → d(z) infer P → A[d], if P does not contain free occurrences of z (α-rule in[Hilbert 68] );Proof: substitute z = H[~d] in the formula “P → d(z)” and then use the defining axiom “A[d] ≡ d(H[~d])”;2)from d(z) → P infer E[d] → P, if P does not contain free occurrences of z (β-rule in[Hilbert 68] );Proof: substitute z = H[d] in the formula “P → d(z)” and then use the defining axiom “E[d] ≡ d(H[d])”;3)from d(z) infer A[d] ( γ` - rule in [Hilbert 68]);Proof: substitute z = H[~d] in the formula “d(z)” and the use the defining axiom “A[d] ≡ d(H[~d])”;4)from P infer A[d, P];Proof:。