Chapter1.3-Propositional_logic1(Propositional Equivalences)
Propositional Logic
Syllogisms
• Proposition = Statement that may be either true or false.
• John is in the classroom. • Mary is enrolled in 270A. • If A is true, and A implies B, then B is true. • If some A are B, and some B are C, then
Propositional Logic or how to reason correctly
Chapter 8 (new edition) Chapter 7 (old edition)
Goals
• Feigenbaum: In the knowledge lies the power. Success with expert systems. 70’s.
• If it rains, then the game will be cancelled.
• If the game is cancelled, then we clean house.
• Can we conclude?
– If it rains, then we clean house.
• p = it rains, q = game cancelled r = we clean house.
Single incorrect or false belief will destroy reasoning. No weight of evidence.
Inference
• Does s1,..sk entail s? • Say variables (symbols) v1…vn. • Check all 2^n possible worlds. • In each world, check if s1..sk is true, that s
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;。
第1章 命题逻辑
第一章 命题逻辑（Propositional Logic）
1.2逻辑联结词(Logical Connectives)
联结词“∨”的定义真值表
P
Q
P∨Q
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第一章 命题逻辑（Propositional Logic）
1.2逻辑联结词(Logical Connectives)
1.2 逻辑联结词(Logical Connectives) 1.2.1 否定联结词(Negation) ┐ 1.2.2 合取联结词(Conjunction)∧ 1.2.3 析取联结词(Disjunction)∨ 1.2.4 条件联结词(蕴涵联结词Conditional)→ 1.2.5 双条件联结(等值联结词Biconditional)
1.2逻辑联结词(Logical Connectives)
例3. 将下列命题符号化. (1) 李平既聪明又用功. (2) 李平虽然聪明, 但不用功. (3) 李平不但聪明,而且用功. (4) 李平不是不聪明,而是不用功.
解: 设 P:李平聪明. Q:李平用功. 则 (1) P∧Q (2) P∧┐Q
个值:真（用 T(true)或1 表 示）、假 （用F(false) 或0表 示） 。 ✓ 真命题：判断为正确的命题，即真值为真的命题。 ✓ 假命题：判断为错误的命题，即真值为假的命题。
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第一章 命题逻辑（Propositional Logic）
1.1 命题及其表示
因而又可以称命题是具有唯一真值的陈述句。
说明：“∧” 属于二元(binary)运算符. 合取运算特点：只有参与运算的二命题全为真 时，运算结果才为真，否则为假。
离散数学 Predicates and Quantifiers(期望与量词)共31页
-----predicate logic: predicate, quantifiers
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the Foundations:Logic and Proof Guo Jian
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1.predicate (1) Consider the statement
“x is greater than 3”.
P(x) is also said to be the value of propositional function P at x.
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the Foundations:Logic and Proof Guo-----”x is greater than 3”, propositional function P(4)---------proposition ,true P(2)--------proposition, false
logic that deals with predicate and quantifiers.
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the Foundations:Logic and Proof Guo Jian
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2. Universal Quantifier
(1) domain (or Universe of Discourse )个体域 -----a set containing all the values of a variable
(3)Example2(p31) A(x)---”Computer x is under attack by an
intruder.” CS2, MATH1 are under attack. A(CS1)----false A(CS2)----true A(MATH1)----true (4) Statements can involve more than 1 variable. Example 3 (page 31): Q(x,y)--------”x=y+3” Q(1,2)--------false Q(3,0)--------true
离散数学(1.4真值表与等价公式)
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第一章 命题逻辑（Propositional Logic） 1.4真值 表与等价公式
1.4.1 真值表(Truth Table) 1.4.2 等价公式(Propositional Equivalences) 1.4.1 真值表 前面在定义联结词时,曾经使用过真值表,下面给出 真值表的定义. 定义1.4.1 (对公式的赋值或解释)设P1 , P2 ,…,Pn是出 现在公式A中的全部的命题变元, 给P1 , P2 ,…,Pn各指 定一个真值，称为对A的一个赋值或解释。若指定的 一组值使A的真值为真(假), 称这组值为A的成真(假)赋值.
练习2：构造公式 (P Q ∧Q 真值表。
P
F F T T
Q
F T F T
(P Q )
(P Q)
(P Q) ∧ Q
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第一章 命题逻辑（Propositional Logic） 1.4真值 表与等价公式
练习2：构造公式 (P Q ∧Q 真值表。
P
F F T T
Q
F T F T
F F F F F
Q R P (Q R) (P ∧ Q R
T T F T T T T T T T T T T T T
T F T
T T F
F
T
T
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T
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F
Байду номын сангаас
T T T
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T
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第一章 命题逻辑（Propositional Logic） 1.4真值 表与等价公式
由真值表可知，两个公式为等价式。
F F T T
F T F T
数理逻辑(Mathematical Logic)
数理逻辑（MathematicalLogic）数理逻辑(Mathematical logic)是用数学方法研究诸如推理的有效性、证明的真实性、数学的真理性和计算的可行性等这类现象中的逻辑问题的一门学问。
其研究对象是对证明和计算这两个直观概念进行符号化以后的形式系统。
数理逻辑是数学基础的一个不可缺少的组成部分。
数理逻辑的研究范围是逻辑中可被数学模式化的部分。
以前称为符号逻辑（相对于哲学逻辑），又称元数学，后者的使用现已局限于证明论的某些方面。
历史背景“数理逻辑”的名称由皮亚诺首先给出，又称为符号逻辑。
数理逻辑在本质上依然是亚里士多德的逻辑学，但从记号学的观点来讲，它是用抽象代数来记述的。
某些哲学倾向浓厚的数学家对用符号或代数方法来处理形式逻辑作过一些尝试，比如说莱布尼兹和朗伯（Johann Heinrich Lambert）；但他们的工作鲜为人知，后继无人。
直到19世纪中叶，乔治·布尔和其后的奥古斯都·德·摩根才提出了一种处理逻辑问题的系统性的数学方法（当然不是定量性的）。
亚里士多德以来的传统逻辑得到改革和完成，由此也得到了研究数学基本概念的合适工具。
虽然这并不意味着1900年至1925年间的有关数学基础的争论已有了定论，但这“新”逻辑在很大程度上澄清了有关数学的哲学问题。
在整个20世纪里，逻辑中的大量工作已经集中于逻辑系统的形式化以及在研究逻辑系统的完全性和协调性的问题上。
本身这种逻辑系统的形式化的研究就是采用数学逻辑的方法.传统的逻辑研究（参见逻辑论题列表）较偏重于“论证的形式”，而当代数理逻辑的态度也许可以被总结为对于内容的组合研究。
它同时包括“语法”（例如，从一形式语言把一个文字串传送给一编译器程序，从而转写为机器指令）和“语义”（在模型论中构造特定模型或全部模型的集合）。
数理逻辑的重要著作有戈特洛布·弗雷格（Gottlob Frege）的《概念文字》（Begriffsschrift）、伯特兰·罗素的《数学原理》（Principia Mathematica）等。
哥德尔不完备定理英文原文
哥德尔不完备定理英文原文英文回答：Gödel's incompleteness theorems are two mathematical theorems that demonstrate inherent limitations of axiomatic systems based on first-order logic. The theorems were published by Kurt Gödel in 1931 and are widely acknowledged as foundational results in mathematical logic.Theorem 1 (Incompleteness theorem): Any effectively axiomatizable theory capable of expressing basic arithmetic is either incomplete or inconsistent. That is, there are true statements about the natural numbers that cannot be proven within the theory.Theorem 2 (Undecidability theorem): No consistent, effectively axiomatizable theory capable of expressing basic arithmetic can decide all true statements about the natural numbers. That is, there are statements about the natural numbers that can neither be proven nor disprovenwithin the theory.Implications of Gödel's theorems:Limits of formal systems: Gödel's theorems demonstrate that no formal system can be both complete and consistent if it is capable of expressing basic arithmetic. This has profound implications for the foundations of mathematics and the limits of what can be proven within a given axiomatic system.Creativity and human intelligence: The incompleteness theorems suggest that there are mathematical truths that cannot be discovered through purely mechanical or algorithmic processes. This has led to speculation that human intelligence may involve non-computational elements that allow for creativity and insight.The nature of mathematics: Gödel's theorems have led to a deeper understanding of the nature of mathematics. They have helped to establish the distinction between provability and truth, and have raised questions about therole of intuition and human understanding in mathematical reasoning.中文回答：哥德尔不完备定理。
荣格全集1-17卷目录（中英文对照）
荣格全集1-17卷目录（中英文对照）Volume I: Psychiatric StudiesOn the psychology and pathology of so-called occult phenomena《论所谓神秘现象的心理学和病理学》On hysterical misreading.《癔病性误读》Cryptomnesia.《隐藏的记忆》On manic mood disorder.《狂躁性情绪失调》A case of hysterical stupor in a prisoner in detention.《一名囚犯的癔病性昏迷案例》On simulated insanity.《假性疯狂》A medical opinion on a case of simulated insanity.《假性疯狂的治疗意见》A third and final opinion on two contradictory psychiatric diagnoses.《第三和最后一份关于两种相矛盾的精神病学诊断的意见》On the psychological diagnosis of facts.《对事实的心理学诊断》Volume II: Experimental ResearchesThe associations of normal subjects.《普通被试者的联想》An analysis of the associations of an epileptic.《一例癫痫病患者的联想分析》The reaction-time ratio in the association experiment.《联想实验中的反应时间》Experimental observations on the faculty of memory.《记忆的实验性观察》Psychoanalysis and association experiments.《精神分析与联想实验》The psychological diagnosis of evidence.《心理诊断的证据》Association, dream, and hysterical symptom.《联想，梦，癔病症状》The psychopathological significance of the association experiment.《联想实验在心理病理学中的意义》Disturbances of reproduction in the association experiment.《联想实验中的再现障碍》The association method.《联想法》The family constellation.《家庭排行》On the psychophysical relations of the association experiment.《联想实验的心身关系》Psychophysical investigations with the galvanometer and pneumograph in normal and insane individuals.《用电流计和呼吸描记器对普通人和精神病患者进行的身心调查》Further investigations on the galvanic phenomenon and respiration in normal and insane individuals.《对普通人和精神病患者在电击现象和呼吸作用上的深入调查》Statistical details of enlistment.《兵役期间的统计细节》New aspects of criminal psychology: contribution to the method used for psychological diagnosis of evidence.《犯罪心理学的新特点：用于心理诊断证据之方法的贡献》The psychological methods of investigation used in the psychiatric clinic of the University of Zurich.《用于苏黎世精神病诊所的心理学调查方法》On the doctrine of complexes.《情结学说》On the psychological diagnosis of evidence: the evidence-experiment in the Naf trial.《心理诊断的证据：在Naf审判中的证据实验》Volume III: The Psychogenesis of Mental DiseaseThe psychology of dementia praecox《早发性痴呆心理学》The content of the psychoses《心理学的内容》On psychological understanding《心理学中的理解力》A criticism of Bleuler's theory of schizophrenic negativism.《对布罗伊尔精神分裂症理论的批评》On the importance of the unconscious in psychopathology.《无意识在心理病理学中的重要性》On the problem of psychogenesis in mental disease.《关于精神疾病心理起因的问题》Mental disease and the psyche.《精神疾病与心灵》On the psychogenesis of schizophrenia.《精神分裂症的心理起因》Recent thoughts on schizophrenia.《精神分裂症的新想法》Schizophrenia.《精神分裂症》Letter to the chairman, symposium on chemical concepts of psychosis.《致主席与讨论会关于心灵之化学概念的信》Volume 4: Freud and PsychoanalysisFreud's Theory of Hysteria: A Reply to Aschaffenburg《弗洛伊德的癔病理论：一份给塞芬伯格的回答》The Freudian theory of hysteria.《弗洛伊德的癔病理论》The analysis of dreams.《梦的分析》A contribution to the psychology of rumour.《一个对谣言心理学的贡献》On the significance of number dreams.《数字在梦中的意义》Morton Prince' s "The mechanism and interpretation of dreams": a critical review.《莫顿.普林斯的“梦的机制和解释”：一份批评性评论》On the criticism of psychoanalysis.《对于精神分析学的批评》Concerning psychoanalysis.《关于精神分析学》The theory of psychoanalysis.《精神分析理论》General aspects of psychoanalysis.《精神分析学的基本层面》Psychoanalysis and neurosis.《精神分析与神经症》Some crucial points in psychoanalysis: A correspondence between Dr. Jung and Dr. Loy.《精神分析中的一些关键：荣格和洛伊的通信》Prefaces to "Collected papers on analytical psychology."《分析心理学论文集序言》The significance of the father in the destiny of the individual.《父亲对一个人命运的影响》Introduction to Kranefeldt's "secret ways of the mind."《序克兰费尔德的“精神的秘密方法”》Freud and Jung: Contrasts.《弗洛伊德和荣格的对比》Volume V: Symbols of TransformationSymbols of transformation. Part I. introduction《变形的象征》第一部Two kinds of thinking《两种思考方式》The Miller fantasies: anamnesis.《米勒小姐的幻想：回忆》The hymn of creation《创造的赞美诗》The song of the moth.《蛾之曲》Symbols of transformation. Part II. Introduction.《变形的象征》第二部The concept of libido.《力比多的概念》The transformation of libido《力比多的转化》The origin of the hero.《英雄的起源》Symbols of the mother and of rebirth.《母亲和重生的象征》The battle for deliverance from the mother.《从母亲解脱的斗争》The dual mother.《双重母亲》The sacrifice.《献祭》Symbols of transformation. Epilogue.《变形的象征》结语Symbols of transformation. Appendix: the Miller fantasies.《变形的象征》附录：米勒的幻想Volume VI: Psychological TypesPsychological types. Introduction.《心理类型》The problem of types in the history of classical and medieval thought.《历史上的类型问题》Schiller's ideas on type problem.《席勒关于类型问题的见解》The Apollonian and the Dionysian.《日神型与酒神型》The type problem in human character.《人性格中的类型问题》The type problem in poetry.《诗歌中的类型问题》The type problem in psychopathology.《精神病理学中的类型问题》The type problem in aesthetics.《美学中的类型问题》The type problem in modern philosophy.《现代哲学中的类型问题》General description of the types.《心理类型的基本描述》Psychological types. Definitions.《心理类型》定义Psychological types. Epilogue.《心理类型》结语Appendix附录Volume VII: Two Essays on Analytical Psychology On the psychology of the unconscious.《无意识心理学》The relations between the ego and the unconscious.《自我与无意识的关系》Appendices附录Volume VIII: The Structure and Dynamics of the Psyche On psychic energy.《心理能量》On the nature of the psyche.《心理的性质》General aspects of dream psychology.《梦心理学的基本层面》On the nature of dreams.《梦的性质》The psychological foundation of belief in spirits.《神灵信仰的心理基础》Spirit and life.《精神与生命》Basic postulates of analytical psychology.《分析心理学的基本假设》Analytic psychology and Weltanschauung.《分析心理学和世界观》The real and the surreal.《现实与超现实》The stages of life.《人生诸阶段》The soul and death.《灵魂与死亡》Synchronicity: an acausal connecting principle.《同步性：非因果性联系》The structure of the psyche.《心理的结构》Instinct and the unconscious.《本能与无意识》Psychological factors determining human behavior.《决定人类行为的心理要素》The significance of constitution and heredity in psychology.《体质与遗传在心理学中的意义》A review of the complex theory.《情结理论回顾》The transcendent function.《心理的超越功能》Appendix附录Volume IX.i: The Archetypes of the Collective UnconsciousArchetypes of the collective unconscious.《集体无意识的原型》The concept of the collective unconscious.《集体无意识的概念》Concerning the archetypes, with special reference to the anima concept.《关于原型，特别涉及阿尼玛概念》Psychological aspects of the mother archetype.《母亲原型的心理侧面》Concerning rebirth.《关于再生》The psychology of the child archetype.《儿童原型的心理学》The psychological aspects of the Kore.《Kore的心理学方面》The phenomenology of the spirit in fairytales.《童话中的精神现象学》On the psychology of the trickster-figure.《狡徒形象中的心理学》Conscious, unconscious, and individuation.《意识，无意识，个性化》A study in the process of individuation.《个性化进程的研究》Concerning mandala symbolism.《关于曼荼罗象征》Appendix附录ii: Aion: Researches into the Phenomenology of the SelfThe ego.《自我》The shadow.《阴影》The syzygy: anima and animus.《两极会合：阿尼玛和阿尼姆斯》The self.《自性》Christ, a symbol of the self.《基督，一个自性的象征》The sign of the fishes.《双鱼的记号》The prophecies of Nostradamus.《诺斯卑达穆斯的预言》The historical significance of the fish.《鱼的历史性意义》The ambivalence of the fish symbol.《鱼象征的矛盾现象》The fish in alchemy.《炼金术中的鱼》The alchemical interpretation of the fish.《炼金术中鱼的解释》Background to the psychology of Christian alchemical symbolism.《基督教的炼金术象征主义的背景》Gnostic symbols of the self.《自性的诺斯替教象征》The structure and dynamics of the self.《自性的结构和动力学》Volume X: Civilization in TransitionThe role of the unconscious.《无意识的作用》Mind and earth.《心灵和大地》Archaic man.《原始人》The spiritual problem of modern man.《现代人的精神问题》The love problem of a student.《一个学生的恋爱问题》Woman in Europe.《欧洲的女性》The meaning of psychology for modern man.《心理学的现代意义》The state of psychotherapy today.《今日心理学状况》Preface to "Essays on contemporary events."《<当代事件随笔>序言》Wotan.《沃丁》After the catastrophe.《大难之后》The fight with the shadow.《与阴影之战》Epilogue to "Essays on contemporary events."The undiscovered self (present and future).《未发现的自性：现代与未来》Flying saucers: a modern myth of things seen in the skies. 《飞碟：现代神话》A psychological view of conscience.《心理学的良心观》Good and evil in analytical psychology.《分析心理学中的善与恶》Introduction to Toni Wolff's "Studies in Jungian Psychology."《沃尔夫“荣格心理学研究”之绪论》The Swiss line in the European spectrum.《欧洲范围内的瑞士路线》The rise of a new world.《一个新世界的崛起》La revolution mondiale.《世界革命》The complications of American psychology.《美国心理学的复杂性》The dreamlike world of India.《印度的如梦世界》What India can teach us.《印度能教给我们什么》Volume 11: Psychology and Religion: West and East Psychology and religion.《心理学与宗教》A psychological approach to the dogma of the Trinity.《三位一体观念的心理学考察》Transformation symbolism in the Mass.《弥撒中的转变象征》Foreword to White's "God and the unconscious."《序怀特的“上帝与无意识”》Foreword to Werblowsky's "Lucifer and Prometheus."《序韦尔布娄斯基的“卢西弗和普罗米修斯”》Brother Klaus.《克劳斯兄弟》Psychotherapists or the clergy.《心理治疗者和神职人员》Psychoanalysis and the cure of souls.《精神分析与灵魂治疗》Answer to Job.《答约伯》Psychological commentary on "The Tibetan Book of the Great Liberation."《对西藏大解脱书的心理学评论》Psychological commentary on "The Tibetan Book of the Dead."《对西藏度亡经的心理学评论》Yoga and the West.《瑜伽与西方》Foreword to Suzuki's "Introduction to Zen Buddhism."《序铃木“禅宗导引”》The psychology of Eastern meditation.《东方人的冥思》The holy men of India.《印度圣人》Foreword to the "I-Ching."《序“易经”》Volume 12: Psychology and AlchemyIntroduction to the religious and psychological problems of alchemy.《炼金术的宗教与心理问题介绍》Individual dream symbolism in relation to the alchemy: a study of the unconscious processes at work in dreams.《与炼金术相关的个体梦象征》Religious ideas in alchemy: an historical survey of alchemical ideas.《炼金术中的宗教观念》Epilogue结语Volume 13: Alchemical StudiesCommentary on "The secret of the golden flower."《太乙金华宗旨》注释The visions of Zosimos《左西莫斯幻象》Paracelsus as a spiritual phenomenon.《帕拉塞尔苏斯的精神现象》The spirit Mercurius.《神灵墨丘利》The philosophical tree.《哲学谱系》Volume 14: Mysterium ConiunctionisThe components of the coniunctio.《合体的组成》The paradoxa.《自相矛盾体》The personification of the opposites.《对立面的化身》Rex and regina.《王与王后》Adam and Eve.《亚当和夏娃》The conjunction.《结合》。
英语语言学导论
Course Name:Introduction to LinguisticsSept. 2012, for Grade 2010 Classes 1-6English Undergraduate ProgramDepartment of EnglishSchool of Foreign Language StudiesNanchang University (NCU)Course Instructor: Prof./Dr. JIANGSyllabusCourse Description:This course aims at providing undergraduate juniors of English major with a fundamental and systematic account of the basic knowledge of the studies of linguistics at the modern time with explanations, illustrations, and necessary examples from the course book and also from present English and Chinese language uses, to help develop the students’ interest in this study, to facilitate their understanding of the linguistic terms and theories, and to build a systematic knowledge of the said study.Major Books Used for this Course:1)Hu, Zhuanglin 2006.Linguistics. A Course Book (Third Edition). BeijingUniversity Press, used as students’ course book.2) Robins, R. H. 1967/1997. A Short History of Linguistics (4th edn). London, NewY ork: Longman.4) Yule, George. 2000. The Study of Language.Beijing: Foreign Language Teachingand Research Press.5) Keith Brown et al. (eds.) 2006. Encyclopedia of Language and Linguistics (2nd edn),Oxford: Elsevier.6) Collinge, N. E. (ed.) 2005. An Encyclopaedia of Language. London, New Y ork:Routledge.7) Strazny, Philipp (ed.) 2005. An Encyclopedia of Linguistics.New Y ork, Oxon:Fitzroy Dearborn.8) Wikipedia. /9) Encyclopaedia Britannica. 2007. the electronic edition can be accessible on theinternet.Course Requirements:1)Attentive listening, active participation, quick note-taking and understanding,nice presentation in class and all the related activities2)Complete preview, in-class and after-class assignments3)Pass the final examinationCourse Schedule:1)General Introduction2)Chapter 1—Design features, origin, and functions of language3)Chapter 1 — Main branches, macro-linguistics, and important distinctionsin linguistics4)Chapter 2—Phonetic studies5)Chapter 2—Phonological studies6)Chapter 3—W ord and Formation7)Chapter 3—W ord/lexical changes8)Chapter 4—Syntactic relation, construction, and function9)Chapter 4—Grammatical categories, phrasing and beyond10)Chapter 5—Meaning and sense relation11)Chapter 5—Meaning analyses12)Chapter 8—Pragmatics (1)13)Chapter 8—Pragmatics (2) --Post-Gricean developments14)Chapter 11—Linguistics and foreign language teaching15)SummaryLecture 1General Introduction:Modern Linguistics and Earlier Linguistic StudiesIn this lecture, we shall make a general introduction to the development of modern linguistics and the linguistic studies before that. The students will get a general view/picture of the developments, the main linguistic schools, and their studies through attending this lecture.0.Leading in: What does your mind do most of the time? Or what do you do most ofthe time in your life including the dreaming time when you sleep? What is your major? What is English? Language is what you use most of the time in your life and is also your major. Since you are language majors, very likely you will use it for your future work. Why not learn about it then? What is language? What does it cover? …These are the questions for linguistics.1.The beginning of MODERN linguistics: 1916—(Q1. When did modern linguistics begin?)1) The first modern linguistic approacha. The ―father of modern linguistics‖-- Ferdinand de Saussure索绪尔(1857-1913)(Q2. Who was the father of modern linguistics?)b. The first modern linguistic book ―A Course on General Linguistics‖(1916).《普通语言学教程》Saussure’s two students pieced their notes takenfrom attending their teacher’s lectures and got it published in 1916/Englishversion in 1959.(Q3. What was the first modern linguistic book?)(Q4. Who published it?)2) Why is it ―modern‖?Because of the scientific views and the researchmethods introduced in this book. Eg. Language is a system of signs.Language has a sound system and a meaning system. Language has associative relation and paradigmatic relation.(Q5. In what sense is it called a ―modern‖ linguistic book?)2.What happened before Saussure?Some traditional approaches to language—there are three successive phases before Saussure’s time(Q6. Who were the earliest scholars of language? What did they study?)1) The Greek researchers and their studies--―Traditional Grammar‖ wasinstituted;Based on logic;Aiming solely at providing rules;Distinguishing correct from incorrect forms;Classic study, philosophical, logical, meaning.(Q7. What were studied about language during the Middle Ages?)2) The researches during the Middle Ages--PhilosophyAlexandria, the ―philosophical‖ school, religious, literary, linguistic;Linguistic structure is not the central concern, but meaning is;Seeking primarily to establish, interpret and comment upon texts;Applying the method of criticism;Comparison of texts of different periods and of different writers;Data—written language, exclusively Greek & Roman antiquityPrescriptive grammar;Paved the way for historical linguistics.(Q8. What were the two establishments in the 19th century?)3) Linguistic researches during 1800-1900--―Comparative grammar‖a. The establishment of linguistic family trees—The Indo-European Family Tree印欧语系The Sino-Tibetan Family Tree 汉藏语系, etc*In 1816, ―The Sanskrit Conjugation(动词变位) System‖ by Franz Bopp was a study of the connections between Sanskrit（梵文）, Germanic, Greek, Latin, andother European languages. The primitive elements that Sanskrit maintained are vital for the purposes of reconstruction of Indo-European language family.b. The establishment of the International Phonetic Alphabet (IPA) 国际音标and the publication in 1888, based on French teachers’ practice of teachingLatin and some other related languages.Later representatives—Max Müller, G. Curtius, A. Schleicher;Historical and comparative, etymological (语源学的);Comparativist school was the dominating study of the time.3. What does modern linguistics cover?1) Saussure and his study: the first structuralist approach2) Prague School布拉格学派The leading figure—V. Mathesius马泰休斯(1882-1946);Flourished during 1920s-30s;Mainstream structuralism; synchronic;Approach language structurally and functionally —form-functionalapproach;Followed both Polish B. de Courtenay (库尔特内) and Swiss F. de Saussure;First functionalist;Founded the International Functional Linguistic Association 1976;Distinguished contributions:a) Established the theory of phonology; distinguished between phoneticsand phonology; developed phonology as an independent study in 1939;b) Mathesius’ functional analysis of sentence components—very close tothe present division of given/new or theme/rheme, functional syntacticanalysis.3) American Structuralism/ Also called descriptive linguistics 美国结构主义/美国描写语言学Developed independently from the anthropological studies by Americanscholars such as Boas, Sapir, etc.；The representative figure—Leonard Bloomfield布龙菲尔德(1887-1946),his Language《语言论》(1933)；Flourished during 1930s-1950s；Structural and behavioral；Major focus—syntactic analysis；Contribution—IC analysis (immediate constituent analysis)4) The Chomskian approachThe leading figure—Noam Chomsky乔姆斯基(1928--);Influential during 1960s-1980s;Formal approach, deep structure/surface structure; NP VP;Psychological; we were born with LAD (language acquisition device);Innateness;Major focus—syntax;Contributions— a. Phrase Structure Theory;b. Transformational-Generative Grammar.5) London School 伦敦学派Leading figures:a. Malinowsky马林诺夫斯基(1884-1942) — anthropologist;b. Firth 弗斯(1890-1960) the 1st professor of General Linguistics in GreatBritain;c. M. A. K. Halliday韩礼德(1925-- ) Systemic-Functional Grammar;Influential from the 1980s;Functional approach and anthropological;Major focus—meaning in society and functional grammar;Contribution — functional analysis6) Cognitive Linguistics: a new perspective on how language is used; how weview the world and express it in language; how language tells different cognition of the same world in which human beings live.Leading figures: R. Langacker兰盖克; G. Lakoff 拉可夫;M. A. K. Halliday (partly);N. Chomsky (partly)7) Computational Linguistics: a branch of linguistics about how to teachcomputer to receive, comprehend, produce and translate natural languages. It reflects human ambition.4. Homework:1) Go over the questions discussed.2) Preview 1.1--1.5 of Chapter 1.3) What is language defined by different people? What is linguistics? How does asound come to have meaning?Lecture 2Design Features, Origin and Functions of LanguageIn this section, we shall mainly discuss some important features and functions of language. As widely discussed, there are four features and seven main functions.(Q9. What is linguistics? p14.What is language?p3.)0.Check students’ homework orally in class; ask them to give some presentation;offer them some different definitions (cf. Essentials of Linguistics pp.1 & 14);underline the key words in the definitions; explain them one by one with examples from English and Chinese to facilitate their understanding and memorizing.1) ―Language is a system of arbitrary vocal symbols used for humancommunication. ---It is a system since linguistic elements are arranged systematically rather than randomly. It is arbitrary in the sense that there is no intrinsic connection between a word and the object it refers to. It is symbolic because words are associated with objects, actions, ideas, etc. by convention. It is vocal because sound or speech is the primary medium for a human language. It is human in that no other animals possess such language.‖2) ―Modern linguistics is the scientific study of language. ---It studies the rules and principles whereupon human languages are constructed and operated as systems of communication. ‖(Q10. What is a design feature of language?)(Q11. How many design features are there and what are they?)1. 4 Design features of languageThe features that define human languages are called design features. (p3) (Q12. What is arbitrariness?)1)Arbitrariness 任意性There is no natural relationship between meaningand form, as well as meaning and sound of a language.Eg 1. fish (in English)le poisson (in French)鱼(in Chinese)*The above words and sounds all mean the same and all refer to the same kind of animal living in water, yet they take different forms.Eg 2. eat (in English)manger (in French)吃(in Chinese)(Q13. What is duality?)2)Duality 双重性Word is a combination of sound and meaning. Language has alevel of sounds/ a sound system and a level of meaning/ meaning system.Or: language has two systems—sound system and meaning system.(Q14. What is creativity?)3)Creativity 创造性(productivity 能产性in other linguistic books)a.W e can create new words;b.W e can create endless new sentences with limited number of words.(Q15. What is displacement?)4)Displacement 不受时空限制性a.One can refer to someone/sth. in the past, at the present or in thefuture;b.One can refer to someone/sth in another place or in another world,real or imagined.(Q16. What is convention?)*Convention 约定俗成is a community’s or society’s acceptance, use, and carrying on of a certain sound or form for a meaning.2. Origin of Language(Q17. How did language possibly begin?)1) Language is the very thing that makes us human.2) William C. Stokoe’s interpretation of language origin: language may have begunwith gestural expressions.Instrumental manual actions may have been transformed into symbolic gestures, and vision would have been the key of language evolution.(Q18. What is the possible relationship between language and gesture?)3) The relationship between language and gesture—whether it is one of unity orduality. Should we consider gesture and language as different and independent phenomena?A unity for language and gesture is a more reasonable understanding (Adam Kendon, in McNeill 2000).*Our knowledge and understanding of the nature of language and other related types of communication is limited and calls for ceaseless exploratory endeavor.(Q19. How many functions are there of human language? What are they?)(Q20. What is informative function?)3. 7 Functions of language1) Informative function: 信息功能(also understood as ideational function)Language is used to note down, to carry, and to pass information.(Q21. What is interpersonal function?)2) Interpersonal function: 人际功能Language is used for human communication/for communication amongpeople. It is used to establish and maintain people’s status in society/ orestablish and maintain social rules.(Q22. What is performative function?)3) Performative function: 行事功能W e can use language/words to do things.a. W e can use it to make others do something;b. W e do something ourselves at the time when we are saying something.(Q23. What is emotive function?)4) Emotive function: 情感功能(expressive function in other books)Language can be used to express feelings or emotions.Egs. ―My God.‖―Alas!‖―Ouch!‖―Damn it!‖―Wow.‖(Q24. What is phatic function?)5)Phatic [\feitik] function: 酬应功能Language can be used to indicate or to maintain relationship.This function originated fro m Malinowski’s study of the functions oflanguage.Egs. ―Good morning.‖―God bless you.‖―I’m sorry to hear it.‖―Good day.‖ ―Hello!‖ ―Good-bye.‖(Q25. What is recreational function?)6)Recreational function: 娱乐功能Language can be used for joy, fun, amusement, or recreation.Egs. Jokes,Chinese cross talk,songs and lyrics,poetry in general(Q26. What is metalingual function?)7)Metalingual function: 元语言功能Language can be used to talk about itself.Eg. ―book‖ is a word that we use to refer to something that we read…*What teachers do in class is mainly the use of language of this function--touse language to explain language.4. Homework:1) Go over the questions discussed.2) Preview sections 1.7 -- 1.9 of Chapter 1. Write about differences andsimilarities between phonetics and phonology, morphology and syntax, semantics and pragmatics, and hand in this homework next time.Lecture 3Main Branches, Interdisciplinary Branches and Important Distinctions0.Collect homeworkAn oral check of the questions discussed last time.(Q27. How many main branches of linguistics are there? What are they?)(Q28. What is phonetics?)1. 6 Main branches of linguistics1)Phonetics: 语音学The study of speech sounds. It studies and describes any speech sound whether it distinguishes meaning or not.Eg. three ―p‖sound s are noted in ―speak‖(un-aspirated不送气, as [p=]), ―peak‖(aspirated送气, as indicated by the diacritic h in [p h]), and ―deep‖(the neutral one [p]).(Q29. What is phonology?)2)Phonology: 音位学/音系学The study of the sound system of language--of the minimal/smallest meaningful sounds.--of the minimal sounds that distinguish meaning.Eg. the three ―p‖s in 1) above don’t distinguish meaning. They are of one meaningful phoneme.*Y et, in ―tip‖ and ―sip‖, or ―tip‖ ―dip‖ the change of ―t‖ to ―s‖ or ―t‖ to ―d‖ brings about another word. T herefore, ―t‖ and ―s‖ are two independent phonemes.(Q30. What is morphology?)3)Morphology:形态学The study of the internal structure of words or of the formation of words.Prefix, suffix, root all help to form words.Eg. ab | norm | alfriend | lyglob | al | iz | ationdialogue, monologue, onomatopoeicinternationalism, localization(Q31. What is syntax?)4)Syntax:句法学The study of the structure/formation of sentence.Eg. I speak French.traditional analysis: Chomsky’s analysis:S Pr O SNP VPV NP(Q32. What is semantics?)5)Semantics: 语义学The study of meaning.a.meaning of words and their relations;b.meaning of sentences/ or: sentence meaning.Eg1. flower ( a super-ordinate word)rose lily tulip daffodil (hyperboles)Eg2. buy/purchase; begin/commenceEg3. in/out; give/take(Q33. What is pragmatics?)6)Pragmatics: 语用学The study of meaning in context, or meaning in use.How can people understand the following utterances correctly? Pragmatics tries to explain how and why people get the inference/implied meaning ofutterances other than the superficial/literal meaning expressed by the words.Eg1. A: How do you think of my new dress?B: The one you wore last week is really beautiful.2. A: Shall we go to the cinema?B: I have to complete the homework.3. Butterflies in one’s stomach.4. Apple in one’s eye.5. John is a lion. Queen Victoria was made of iron.(Q34. What is macrolinguistics or interdisciplinary linguistics?)2.Macrolinguistics宏观语言学It is the interdisciplinary(跨学科/跨专业) studies of linguistics, the study of language involving other fields.(Q35. What is psycholinguistics?)1)Psycholinguistics心理语言学: it is the study of the interrelation betweenlanguage and mind(语言与心智), about how language is produced, understood, and acquired/learned.(Q36. What is sociolinguistics?)2)Sociolinguistics社会语言学: it is the study of the characteristics oflanguage varieties, language functions and language speakers within a speech community/society.(Q37. What is anthropological linguistics?)3)Anthropological linguistics人类语言学:it is the study of the unwrittenlanguage, the emergence of language and the divergence of languages over thousands of years through human development.(Q38. What is computational linguistics?)4)Computational linguistics计算（机）语言学: it studies the use of computersto process or produce human language, including machine translation, computer-aided teaching, corpus(语料库), information retrieval(信息提取), and artificial intelligence, etc.3. 4 Important distinctions in linguistics(Q39. What is the distinction between descriptive and prescriptive?)1)Descriptive vs. prescriptive study: 描写性/规定性The former describes how things are; the latter prescribes how thingsought to be.*The 18th century grammar books are mainly prescriptive and the modern onesare mainly descriptive.(Q40. What is the distinction between synchronic and diachronic?)2)Synchronic vs. diachronic: 共时性/历时性The former describes phenomenon of language of a certain/single period;the latter describes language by analyzing its development throughdifferent period of time.Egs: 1) the study of the development of the Chinese ―ba-construction‖;2) the development of the sound“阿”from ―[e]‖ to ―[a]‖;3) meaning changes of words (“小姐”,“老板”, ―girl‖, ―bird‖ etc).(Q41. What is the distinction between langue and parole?)3)Langue & parole: 语言/言语The former refers to the abstract innate system of language; thelatter—the outcome (words and sentences) or what we actually utter/write.(Q42. What is the distinction between competence and performance?)4)Competence & performance: 语言能力/语言使用（或语言行为）The former refers to one’s knowledge or ability of a language; the latterthe use of it.[*The difference between pairs 3) and 4) above: Langue & parole are a pairof notions distinguishing rules and production by people following the ruleswhile competence & performance are a pair of notions focusing on languageuser’s power and the performing of it.]4.Homework: 1) Go over the questions discussed.2) Preview 2.1 & 2.23) What is ―fanqie‖反切? How to use it? What is 注音字母？Howdid it occur? How to use it? When and how did ―pinyin‖拼音begin?And the significance of its occurrence?Lecture 4PhoneticsIn this section, we shall start a new chapter—discussing speech sounds. The students will learn about 1）the main areas of the study；2）the speech organs；3）the manners and places of sound production; and 4）the description of consonants and vowels of English.0.Check students’ homework in class(Q43. What are the three branches of phonetics?)1.Three main areas of phonetics1) Articulatory Phonetics发声语音学--the study of sound production2) Acoustic Phonetics声学语音学--the study of physical properties of sounds3) Auditory Phonetics听觉语音学--the study of how sounds are perceived and understood(Q44. What organs do we use in producing speech sounds?)2. Vocal organs and sound notation1) Speech organs/ vocal (of voice) organs●Lung, trachea (wind pipe), throat, nose, mouth●Tongue, palate (腭roof of the mouth)●pharynx咽, larynx喉●vocal folds (vocal cords)声带, vocal tract 声道●oral cavity, nasal cavity 口腔，鼻腔(Q45. What is a coronal, a dorsal, a radical sound? p25. What is a voiceless sound, a voiced sound? p27.)*In phonetics, the tongue is divided into five parts: the tip, the blade, the front, the back and the root. In phonology, the sound made with the tip and blade is referred to as a coronal sound, with front and back as a dorsal sound, with root as a radical sound.*When the vocal folds are apart, the air can pass through easily and the sound produced is a voiceless sound. When the vocal folds are close together, the airstream causes them to vibrate against each other the resultant sound is voiced.(Q46. What is sound/phonetic notation/transcription? What’s the principle for establishing the IPA?)2) Phonetic transcription/sound notation语音标示/音标●The use of sets of symbols for transcribing speech sounds or torepresent language sounds.●The main principles were that there should be a separate letter foreach distinctive sound, and that the same symbol should be usedfor that sound in any language in which it appears.●International Phonetic Alphabet (IPA), developed first by a group ofFrench language teachers based on their teaching experience around1930s.●Danish grammarian Otto Jesperson 叶斯柏生(1860-1943) formallyproposed it in 1886.●The first publication was in 1888.●Revised and corrected several times afterwards, widely used indictionaries and textbooks.●The very recent version came out in 2008.(Q47. What is a pulmonic sound, a non-pulmonic sound? p29)*Pulmonic sounds are produced by pushing air out of the lung, as in mostcircumstances, while non-pulmonic sounds are produced by either suckingair into the mouth or closing the glottis and manipulating the air betweenthe glottis and a place of articulation.(Q48. What are the manners of articulation? )3. Manner and place of sound production1) Manner of articulation● 1. stop/ plosive塞音/爆破音● 2. nasal 鼻音● 3. fricative 摩擦音● 4. approximant 延续音/畅音● 5. lateral 边音● 6. trill颤音●7. tap and flap触音/闪音eg. better, letter, city, pretty, bottom, button●8. affricate 塞擦音*Find examples for them from English.(Q49. What are the places of articulation? Give examples for each.)2) Place of articulation:● 1. Bilabial 双唇[b] [p] [m][w]● 2. Labio-dental唇齿[f] [v]● 3. Dental 齿[θ][δ]● 4. Alveolar 齿龈[t][d][n][s] [z][l][r]● 5. Postal velar后齿龈[∫] [з][t∫] [dз]egs. chew, true, child, tried, choose, truth, joke, drove, jam, drum● 6. Retroflex卷舌[r]●7. Palatal 颚[j]●8. Velar 软颚[k] [g] [η] eg. English, ink●9. Uvular 小舌[r] in French●10. Pharyngeal咽头Glottal 声门[?] egs. fat[f æ?t], pack[p æ? k], beaten[bi:?n], lantern, button[h] egs. glottal fricative: home, hold, hand, hat4. English speech sounds(Q50. How to describe a consonant sound of English?)1) English consonants (24 symbols according to recent revision)2) The description of consonants usually involves the place and the manner and is made with a sequence of a) the place of articulation; b) the manner.Eg 1. bilabial stop—where bilabial is the place and stop is the manner. [p] [b]Eg 2. bilabial nasal [m]Eg 3. bilabial approximant [w]*Notice: whenever there are two members in the same box in the table above,a third distinction –voice-- is needed and is mentioned in the first place:Eg 4. [p] a voiceless bilabial stop[b] a voiced bilabial stopEg 5. a voiceless labial-dental fricative [f]a voiced labial-dental fricative [v]3) English vowels (20 symbols according to recent revision)front central backhigh i: u:I umid-highз: əo:mid-low e٨ Dlow æ α:(Q51. How to describe a vowel sound of English? p37.)4) The description of English vowels is made in terms of 4 aspects:(1) the height of the tongue (high, mid, low)—tongue height(2) the position of the higher part of the tongue (front, central, back)—tongueposition(3) the length or tenseness of the sound (tense vs. lax, or long vs. short)(4) lip-rounding (rounded vs. un-rounded)Egs. [i:] high front tense un-rounded vowel[I] high front lax un-rounded vowel[α:]low back tense un-rounded[æ]low front un-rounded[ə]mid central un-rounded4. Homework:1) Go over the questions discussed.2) Find how many mistakes you make in your pronunciation of the 44 basicsounds of English. What are they and why do they occur?3) Hand in next time: What is Mandarin Chinese? How many dialects has it? Whatis the difference between language branch(语族), language, and dialect? What is a dialectic island and the significance of its existence?Lecture 5Phonological analysis0. Collect homework and check the other two parts of the homework orally inclass by asking individual students.(Q52. What is co-articulation? What is anticipatory co-articulation and perseverative co-articulation? p38.)1. Co-articulation and different transcriptions1) Co-articulation协同发音Simultaneous/overlapping articulation because of the influence of the neighbor sound(s)Eg. ―map‖ where [æ] is influenced by [m], making it a bit nasalized.―lamb‖ where [æ] becomes more like the following sound [m].*If a sound becomes more like the following sound, it is anticipatoryco-articulation. If a sound shows the influence of the preceding sound, it isperseverative co-articulation.(Q53. What is nasalization? p38.)* When a non-nasal sound carries some feature of a neighboring nasal sound,this phenomenon is called nasalization.(Q54. What is narrow/broad transcription?)2) Narrow/broad transcription: 严式标音/宽式标音The former intends to symbolize all the possible speech sounds while the latter indicates only those capable of distinguishing one word from another.Egs. 1）[p=] [p h] [p] for variations of sounds in ―speak‖ ―peak‖―deep‖2）[p] for all of them(Q55. What is a phone, a phoneme, and an allophone?)2. Phonology—some basic concepts1) Phone, phoneme, and allophone●Phone 音素—a phonetic unit or segment. Any smallest speech soundwe hear and produce.Eg. [pit] [tip] [spit] we can identify three different /p/s;It is what ―n arrow transcription‖ describes;It may or may not distinguish meaning.●Phoneme音位—a phonological unit. It has distinctive value; anabstract unit in the sound system that has no particular sound;represented by a certain phone in a certain phonetic context.Eg. /p/ is realized differently in [pit] [tip] [spi:k]./æ/is realized by an un-nasalized [æ]and a nasalized [æ].。
notes_chapter02逻辑学导论电子文档
C H A P T E R 2Propositional Logic2.1 IntroductionPropositional Logic is concerned with propositions and their interrelationships. The notion of a proposition here cannot be defined precisely. Roughly speaking, a proposition is a possible condition of the world that is either true or false, e.g. the possibility that it is raining, the possibility that it is cloudy, and so forth. The condition need not be true in order for it to be a proposition. In fact, we might want to say that it is false or that it is true if some other proposition is true.In this chapter, we first look at the syntactic rules that define the legal expressions in Propositional Logic. We then look at the semantics of the expressions specified by these rules. Given this semantics, we talk about evaluation, satisfaction, and the properties of sentences. We then define the concept of propositional entailment, which identifies for us, at least in principle, all of the logical conclusions one can draw from any set of propositional sentences.2.2 SyntaxIn Propositional Logic, there are two types of sentences -- simple sentences and compound sentences. Simple sentences express simple facts about the world. Compound sentences express logical relationships between the simpler sentences of which they are composed.Simple sentences in Propositional Logic are often called proposition constants or, sometimes, logical constants. In what follows, we write proposition constants as strings of letters, digits, and underscores ("_"), where the first character is a lower case letter. For example, raining is a proposition constant, as are rAiNiNg, r32aining, and raining_or_snowing. Raining is not a logical constant because it begins with an upper case character. 324567 fails because it begins with a number. raining-or-snowing fails because it contains hyphens.Compound sentences are formed from simpler sentences and express relationships among the constituent sentences. There are five types of compound sentences, viz. negations, conjunctions, disjunctions, implications, and biconditionals.A negation consists of the negation operator ¬ and a simple or compound sentence, called the target. For example, given the sentence p, we can form the negation of p as shown below.(¬p)A conjunction is a sequence of sentences separated by occurrences of the ∧ operator and enclosed in parentheses, as shown below. The constituent sentences are called conjuncts. For example, we can form the conjunction of p and q as follows.(p∧q)A disjunction is a sequence of sentences separated by occurrences of the ∨ operator and enclosed in parentheses. The constituent sentences are called disjuncts. For example, we can form the disjunction of p and q as follows.(p∨q)An implication consists of a pair of sentences separated by the ⇒ operator and enclosed in parentheses. The sentence to the left of the operator is called the antecedent, and the sentence to the right is called the consequent. The implication of p and q is shown below.(p⇒q)An equivalence, or biconditional, is a combination of an implication and a reduction. For example, we can express the biconditional of p and q as shown below.(p⇔q)Note that the constituent sentences within any compound sentence can be either simple sentences or compound sentences or a mixture of the two. For example, the following is a legal compound sentence.((p∨q) ⇒r)One disadvantage of our notation, as written, is that the parentheses tend to build up and need to be matched correctly. It would be nice if we could dispense with parentheses, e.g. simplifying the preceding sentence to the one shown below.p∨q⇒rUnfortunately, we cannot do without parentheses entirely, since then we would be unable to render certain sentences unambiguously. For example, the sentence shown above could have resulted from dropping parentheses from either of the following sentences.((p∨q) ⇒r)(p∨ (q⇒r))The solution to this problem is the use of operator precedence. The following table gives a hierarchy of precedences for our operators. The ¬ operator has higher precedence than ∧; ∧ has higher precedence than ∨; and ∨ has higher precedence than ⇒ and ⇔.¬∧∨⇒⇔In unparenthesized sentences, it is often the case that an expression is flanked by operators, one on either side. In interpreting such sentences, the question is whether the expression associates with the operator on its left or the one on its right. We can use precedence to make this determination. In particular, we agree that an operand in such a situation always associates with the operator of higher precedence. When an operand is surrounded by operators of equal precedence, the operandassociates to the right. The following examples show how these rules work in various cases. The expressions on the right are the fully parenthesized versions of the expressions on the left.¬ p∧q((¬ p) ∧q)p∧ ¬q(p∧ (¬ q))p∧q∨r((p∧q) ∨r)p∨q∧r(p∨ (q∧r)p⇒q⇒r(p⇒ (q⇒r))p⇒q⇔r(p⇒ (q⇔r))Note that just because precedence allows us to delete parentheses in some cases does not mean that we can dispense with parentheses entirely. Consider the example shown earlier. Precedence eliminates the ambiguity by dictating that the unparenthesized sentence is an implication with a disjunction as antecedent. However, this makes for a problem for those cases when we want to express a disjunction with an implication as a disjunct. In such cases, we must retain at least one pair of parentheses.We end the section with two simple definitions that are useful in discussing Propositional Logic. A propositional vocabulary is a set of proposition constants. A propositional language is the set of all propositional sentences that can be formed from a propositional vocabulary.2.3 SemanticsThe treatment of semantics in Logic is similar to its treatment in Algebra. Algebra is unconcerned with the real-world significance of variables. What is interesting are the relationships among the values of the variables expressed in the equations we write. Algebraic methods are designed to respect these relationships, independent of what the variables represent.In a similar way, Logic is unconcerned with the real world significance of proposition constants. What is interesting is the relationship among the truth values of simple sentences and the truth values of compound sentences within which the simple sentences are contained. As with Algebra, logical reasoning methods are independent of the significance of proposition constants; all that matter is the form of sentences.Although the values assigned to logical constants are not crucial in the sense just described, in talking about Logic, it is sometimes useful to make truth assignments explicit and to consider various assignments or all assignments and so forth. Such an assignment is called a truth assignment.Formally, a truth assignment for Propositional Logic is a function assigning a truth value to each of the proposition constants of the language. In what follows, we use the digit 1 as a synonym for true and 0 as a synonym for false; and we refer to the value of a constant or expression under a truth assignment i by superscripting the constant or expression with i as the superscript.The assignment shown below is an example for the case of a logical language with just three proposition constants, viz. p, q, and r.p i = 1q i = 0r i = 1Using our evaluation method, we can see that i satisfies (p∨q) ∧ (¬q∨r).(p∨q) ∧ (¬ q∨r)(1 ∨ 0) ∧ (¬ 0 ∨ 1)1 ∧ (¬ 0 ∨ 1)1 ∧ (1 ∨ 1)1 ∧ 11Now consider truth assignment j defined as follows.p j = 1q j = 1r j = 0In this case, j does not satisfy (p∨q) ∧ (¬q∨r).(p∨q) ∧ (¬q∨r)(1 ∨ 1) ∧ (¬1 ∨ 0)1 ∧ (¬1 ∨ 0)1 ∧ (0 ∨ 0)1 ∧0Using this technique, we can evaluate the truth of arbitrary sentences in our language. The cost is proportional to the size of the sentence.We finish up this section with a few definitions for future use. We say that a truth assignment satisfies a sentence if and only if it is true under that truth assignment. We say that a truth assignment falsifies a sentence if and only if it is false under that truth assignment. A truth assignment satisfies a set of sentences if and only if it satisfies every sentence in the set. A truth assignment falsifies a set of sentences if and only if it falsifies at least one sentence in the set. 2.4 SatisfactionSatisfaction is the opposite of evaluation. We begin with one or more compound sentences and try to figure out which truth assignments satisfy those sentences.One way to do this is using a truth table for the language. A truth table for a propositional language is a table showing all of the possible truth assignments for the propositional constants in the language.The following figure shows a truth table for a propositional language with just three propositional constants (p, q, and r). Each column corresponds to one proposition constant, and each row corresponds to a single truth assignment. The truth assignments i and j defined in the preceding section correspond to the third and second rows of this table, respectively.p q r111110101100011010001000Note that, for a propositional language with n logical constants, there are n columns in the truth tables and 2n rows.Here is an intuitively simple method of finding truth assignments that satisfy given sentences. (1) We start with a truth table for the proposition constants of our language, and we add columns for each sentence in our set. (2) We then evaluate each sentence for each of the rows of the truth table.(3) Finally, any row that satisfies all sentences in the set is a solution to the problem as a whole. Note that there might be more than one.In solving satisfaction problems, we start with a truth table for the proposition constants of our language. We then process our sentences in turn, for each sentence crossing out truth assignments in the truth table that do not satisfy the sentence. The truth assignments remaining at the end of this process are all possible truth assignments of the input sentences.2.5 Logical Properties of Propositional SentencesEvaluation and satisfaction are processes that involve specific sentences and specific truth assignments. In Logic, we are sometimes more interested in the properties of sentences that hold across truth assignments. In particular, the notion of satisfaction imposes a classification of sentences in a language based on whether there are truth assignments that satisfy that sentence.A sentence is valid if and only if it is satisfied by every truth assignment. For example, the sentence p∨ ¬p is valid.A sentence is unsatisfiable if and only if it is not satisfied by any truth assignment. For example, the sentence p⇔ ¬p is unsatisfiable. No matter what truth assignment we take, the sentence is always false.A sentence is contingent if and only if there is some truth assignment that satisfies it and some truth assignment that falsifies it.It is frequently useful to group these classifications into two groups. A sentence is satisfiable if and only if it is valid or contingent. A sentence is falsifiable if and only if it is unsatisfiable or contingent. We have already seen several examples of satisfiable and falsifiable sentences.In one sense, valid sentences and unsatisfiable sentences are useless. Valid sentences do not rule out any possible truth assignments; unsatisfiable sentences rule out all truth assignments; thus they say nothing about the world. On the other hand, from a logical perspective, they are extremely useful in that, as we shall see, they serve as the basis for legal transformations that we can performon other logical sentences.2.6 Propositional EntailmentValidity, satisfiability, and unsatisfiability are properties of individual sentences. In logical reasoning, we are not so much concerned with individual sentences as we are with the relationships between sentences. In particular, we would like to know, given some sentences, whether other sentences are or are not logical conclusions. This relative property is known as logical entailment. When we are speaking about Propositional Logic, we use the phrase propositional entailment.A set of sentences Δlogically entails a sentence φ (written Δ |= φ) if and only if every truth assignment that satisfies Δ also satisfies φ.For example, the sentence p logically entails the sentence (p∨q). Since a disjunction is true whenever one of its disjuncts is true, then (p∨q) must be true whenever p is true. On the other hand, the sentence p does not logically entail (p∧q). A conjunction is true if and only if both of its conjuncts are true, and q may be false. Of course, any set of sentences containing both p and q does logically entail (p∧q).Note that the relationship of logical entailment is a logical one. Even if the premises of a problem do not logically entail the conclusion, this does not mean that the conclusion is necessarily false, even if the premises are true. It just means that it is possible that the conclusion is false.Once again, consider the case of (p∧q). Although p does not logically entail this sentence, it is possible that both p and q are true and, therefore, (p∧q) is true. However, the logical entailment does not hold because it is also possible that q is false and, therefore, (p∧q) is false.Note that logical entailment is not the same as logical equivalence. The sentence p logically entails (p∨q), but (p∨q) does not logically entail p. Logical entailment is not analogous to arithmetic equality; it is closer to arithmetic inequality.One way of determining whether or not a set of premises logically entails a possible conclusion is to check the truth table for the proposition constants in the language. This is called the truth table method. (1) First, we form a truth table for the proposition constants and add a column for the premises and a column for the conclusion. (2) We then evaluate the premises. (3) We evaluate the conclusion. (4) Finally, we compare the results. If every row that satisfies the premises also satisfies the conclusion, then the premises logically entail the conclusion.As an example, let's use this method to show that p logically entails (p∨q). We set up our truth table and add a column for our premise and a column for our conclusion. In this case the premise is just p and so evaluation is straightforward; we just copy the column. The conclusion is true if and only if p is true or q is true. Finally, we notice that every row that satisfies the the premise also satisfies the conclusion.p q p p∨q1111101101010000Now, let's do the same for the premise p and the conclusion (p∧q). We set up our table as before and evaluate our premise. In this case, there is only one row that satisfies our conclusion. Finally, we notice that the assignment in the second row satisfies our premise but does not satisfy our conclusion; so logical entailment does not hold.p q p p∧q1111101001000000Finally, let's look at the problem of determining whether the set of propositions {p, q} logically entails (p∧q). Here we set up our table as before, but this time we have two premises to satisfy. Only one truth assignment satisfies both premises, and this truth assignment also satisfies the conclusion; hence in this case logical entailment does hold.p q p q p∧q11111101000101000000As a final example, let's return to the love life of the fickle Mary. Here is the problem from the course introduction. We know (p⇒q), i.e. if Mary loves Pat, then Mary loves Quincy. We know (m⇒p∨q), i.e. if it is Monday, then Mary loves Pat or Quincy. Let's confirm that, if it is Monday, then Mary loves Quincy. We set up our table and evaluate our premises and our conclusion. Both premises are satisfied by the truth assignments on rows 1, 3, 5, 7, and 8; and we notice that those truth assignments make the conclusion true. Hence, the logical entailment holds.m p q m⇒p∨q p⇒q m⇒q111111110100101111100010011111010101001111000111The disadvantage of the truth table method is computational complexity. As mentioned above, the size of a truth table for a language grows exponentially with the number of proposition constants in the language. When the number of constants is small, the method works well. When the number is large, the method becomes impractical. Even for moderate sized problems, it can be tedious. Even for an application like Sorority World, where there are only 16 proposition constants, there are 65,536 truth assignments.Over the years, researchers have proposed ways to improve the performance of truth table checking. We discuss some of these in Chapter 5. The other approach is to use symbolic manipulation (i.e. logical reasoning and proofs) in place of truth table checking. We discuss these methods in the next chapter.Before we end this section, it is worth noting that logical entailment and satisfiability have an interesting relationship to each other. In particular, if a set Δ of sentences logically entails a sentence φ, then Δ together with the negation of φ must be unsatisfiable. The reverse is also true. If Δ and the negation of φ is unsatisfiable, then Δ must logically entail φ. This is guaranteed by a metatheorem called the unsatisfiability theorem.Unsatisfiability Theorem: Δ |= φ if and only if Δ∪ {¬φ} is unsatisfiable.Proof: Suppose that Δ |= φ. If a truth assignment satisfies Δ, then it must also satisfy φ. But then it cannot satisfy ¬φ. Therefore, Δ∪ {¬φ} is unsatisfiable. Suppose that Δ∪{¬φ} is unsatisfiable. Then every truth assignment that satisfies Δ must fail to satisfy ¬φ, i.e. it must satisfy φ. Therefore, Δ |= φ.An interesting consequence of this result is that we can determine logical entailment by checking for unsatisfiability. This turns out to be useful in various logical proof methods.RecapThe syntax of propositional logic begins with a set proposition constants. Compound sentences are formed by combining simpler sentences with logical operators. In the version of Propositional Logic used here, there are five types of compound sentences - negations, conjunctions, disjunctions, implications, and biconditionals. A truth assignment for Propositional Logic is a mapping that assigns a truth value to each of the propositional constants in the language. The truth or falsity of compound sentences can be determined from a truth assignment using rules based on the six logical operators of the language. A truth assignment i satisfies a sentence if and only if the sentences is true under that truth assignment. A sentence is valid if and only if it is satisfied by every truth assignment. A sentence is unsatisfiable if and only if it is not satisfied by any truth assignment. A sentence is contingent if and only if it is both satisfiable and falsifiable, i.e. it is neither valid nor unsatisfiable. A sentence is satisfiable if and only if it is either valid or contingent. A sentence is falsifiable if and only if it is unsatisfiable or contingent. A set of sentences Δlogically entails a sentence φ (written Δ |= φ) if and only if every truth assignment that satisfies Δ also satisfies φ.。
Chapter1.1-Propositional_logic1(Propositions and Connectives)
Discrete Mathematics
Henan Polytechnic University School of Computer Science and Technology
Statements or propositional variables can be combined by logical connectives（逻辑联结词） to obtain compound statements(复合命题).
p and q: The sun is shining and it is cold.
Logical Connectives（逻辑联结词）
Unary
Negaton（合取） Disjunction （析取） Exclusive OR （异或） Implication （蕴涵） Biconditional （等价，双条件命题）
注意： 一个语句本身是否能分辨真假与我们是否知 道它的真假是两回事。也就是说，对于一个句子， 2050年元旦是晴天。 有时我们可能无法判定它的真假，但它本身却是 有真假的，那么这个语句是命题，否则就不是命 题。 悖论不是命题。
判断下列语句是否是命题，如果是，判断真假。 eg1：我第一次接触《离散数学》，我就立刻 喜欢 上它了。
Foundations of Logic: Overview
Propositional logic:
Basic definitions. Equivalence rules & derivations.
Predicate logic
金字塔原理第一章解读
金字塔原理第一章解读（中英文实用版）Title: The Pyramid Principle - Chapter 1 InterpretationThe Pyramid Principle is a book that delves into the art of logical thinking and effective communication.The first chapter sets the stage for understanding the concept of thinking in a structured manner.在《金字塔原理》这本书中，作者深入探讨了逻辑思维和有效沟通的艺术。
第一章为读者理解结构化思维的概念奠定了基础。
The chapter emphasizes the importance of creating a clear and concise message that is easily understood by others.It highlights the significance of using a top-down approach in presenting information, starting with the main idea and then gradually expanding on supporting details.本章强调了创建清晰简洁的信息对于他人易于理解的重要性。
它突出了在呈现信息时采用自上而下的方法的重要性，从主要观点开始，然后逐渐扩展到支持性细节。
Furthermore, the chapter introduces the concept of "thinking in pyramid structure," which involves organizing ideas in a hierarchical manner.This structure allows for a more focused and organized presentation of thoughts, making it easier for the audience to follow and understand the message being conveyed.此外，本章介绍了“金字塔结构思维”的概念，涉及以层次化的方式组织思想。
重言式与蕴含式
基本永真蕴含式
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等价式与蕴含式的关系: 定理1.5.4: 设P,Q为任意两个命题公式,PQ的充
要条件为PQ且QP. 证：若PQ，则PQ为永真式
因为 PQ （P→Q）（Q→P） 所以 P→Q，Q→P为永真式,从而 PQ，QP. 反之，若PQ，QP，则P→Q，Q→P为永真式, 所以（P→Q）（Q→P）为永真式， 从而 PQ为永真式，即PQ.
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3) （A→B）（A→C） （┐AB）（┐AC） ┐A（BC） A→BC
4) （A→B）（C→B） （┐AB）（┐CB） （┐A┐C）B ┐（AC）B AC→B
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小结：本节介绍了命题公式的分类，重言式、 矛盾式与蕴含式的概念及其性质，等价式与蕴 涵式的关系。
原命题 逆换式 反换式 逆反式 PQ QP P Q Q P
它们之间具有如下关系:
PQ
Q
P
由P21 表1-5.1
QP P Q 可以得出
因此, 要证明P Q有三种方法:
1)真值表法:即列出PQ的真值表,观察其是否永为真。
2)直接证法:假定前件P是真，推出后件Q是真。
3)间接证法:假定后件是假，推出前件是假,即证 Q P
（3）若A不是矛盾式则称A为可满足式(satisfiable)。 注: 由定义可知,重言式一定是可满足式,反之不真.
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判别命题公式的类型有两种方法: 真值表法和等值
演算法.
等值演算法是将所给命题公式通过等值演算化为最
简单的形式, 然后再进行判别.
例1.判别下列命题公式的类型.
(1). Q∨┓((┓P∨Q)∧P)
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?。
THE COMPLEXITY OF TREE AUTOMATA AND LOGICS OF PROGRAMS
THE COMPLEXITY OF TREE AUTOMATA AND LOGICS OFPROGRAMS ∗E.ALLEN EMERSON †AND CHARANJIT S.JUTLA ‡SIAM J.C OMPUT .c1999Society for Industrial and Applied Mathematics Vol.29,No.1,pp.132–158Abstract.The complexity of testing nonemptiness of finite state automata on infinite trees is investigated.It is shown that for tree automata with the pairs (or complemented pairs)acceptance condition having m states and n pairs,nonemptiness can be tested in deterministic time (mn )O (n );however,it is shown that the problem is in general NP-complete (or co-NP-complete,respectively).The new nonemptiness algorithm yields exponentially improved,essentially tight upper bounds for numerous important modal logics of programs,interpreted with the usual semantics over structures generated by binary relations.For example,it follows that satisfiability for the full branching time logic CTL ∗can be tested in deterministic double exponential time.Another consequence is that satisfiability for propositional dynamic logic (PDL)with a repetition construct (PDL-delta)and for the propositional Mu-calculus (Lµ)can be tested in deterministic single exponential time.Key plexity,tree automata,logics of programs,gamesAMS subject classifications.68Q68,68Q60,03B45PII.S00975397933047411.Introduction.There has been a resurgence of interest in automata on infinite objects [1]due to their intimate relation with temporal and modal logics of programs.They provide an important and uniform approach to the development of decision procedures for testing satisfiability of the propositional versions of these logics [43,33].Such logics and their corresponding decision procedures are not only of inherent mathematical interest,but are also potentially useful in the specification,verification,and synthesis of concurrent programs (cf.[27,8,25,21]).In the case of branching time temporal logic,the standard paradigm nowadays for testing satisfiability is the reduction to the nonemptiness problem for finite state au-tomata on infinite trees;i.e.,one builds a tree automaton which accepts essentially all models of the candidate formula and then tests nonemptiness of the tree automaton.Thus in order to improve the complexity there are two issues:(1)the size of the tree automaton and (2)the complexity of testing nonemptiness of the tree automaton.In this paper we obtain new,improved,and essentially tight bounds on testing nonemptiness of tree automata that allow us to close an exponential gap which has existed between the upper and lower bounds of the satisfiability problem of numerous important modal logics of programs.These logics include CTL ∗(the full branch-ing time logic [9]),PDL-delta (propositional dynamic logic with an infinite repetition construct [33]),and the propositional Mu-calculus (Lµ)(a language for characterizing temporal correctness properties in terms of extremal fixpoints of predicate transform-ers [19](cf.[7,2])).∗Receivedby the editors August 15,1993;accepted for publication (in revised form)April 30,1997;published electronically September 14,1999.This work was supported in part by NSF grants CCR-941-5496and CCR-980-4736,ONR URI contract N00014-86-K-0763,and Netherlands NWO grant nf-3/nfb 62-500./journals/sicomp/29-1/30474.html †Department of Computer Sciences,University of Texas at Austin,Austin,TX 78712(emerson @),and Mathematics and Computing Science Department,Eindhoven University of Technology,Eindhoven 5600MB,The Netherlands.‡IBM T.J.Watson Research Center,Yorktown Heights,NY 10598-0218(csjutla@).132COMPLEXITY OF TREE AUTOMATA133 To obtain these improvements,we focus on the complexity of testing nonemptiness of tree automata.Wefirst note,however,that the size of an automaton has two parameters:the number of states in the automaton’s transition diagram and the number of pairs in its acceptance condition.We next make the following important observation:for most logics of programs,the number of pairs is logarithmic in the number of states.We go on to analyze the complexity of testing nonemptiness of pairs tree automata [30]and show that it is NP-complete.However,a multiparameter analysis shows that there is an algorithm that runs in time(mn)O(n)which is polynomial in the number of states m and exponential in the number of pairs n in the acceptance condition of the automaton.The algorithm is based on a type of“pseudomodel checking”for certain restricted Mu-calculus formulae.Moreover,since the problem is NP-complete, it is unlikely to have a better algorithm which is polynomial in both parameters.The previous best known algorithm was in NP[6,41].The above nonemptiness algorithm now permits us to obtain a deterministic dou-ble exponential time decision procedure for CTL∗,by using the reduction from CTL∗to tree automata obtained in[13],in which the size of the automaton is double ex-ponential in the length of the formula and the number of pairs is only exponential in the length of the formula.The bound follows by simple arithmetic,since a double exponential raised to a single exponential power is still a double exponential.This amounts to an exponential improvement over the best previously known algorithm which was in nondeterministic double exponential time[6,41],i.e.,three exponentials when determinized.It is also essentially tight,since CTL∗was shown to be double exponential time hard[41];thus CTL∗is deterministic double exponential time complete.The above result has been obtained using only the classical pairs tree automata of Rabin[30].However,we also consider the complemented pairs tree automata of Streett[33],which were specifically introduced to facilitate formulation of tempo-ral decision procedures.We show that the nonemptiness problem of complemented pairs automata is co-NP-complete by reducing the complement of the problem to nonemptiness of the pairs automata and vice versa.The reduction employs the fact that infinite Borel games are determinate(Martin’s theorem[22]).This reduction also gives a deterministic algorithm which is polynomial in the number of states and exponential in the number of pairs.We can thus reestablish the above upper bound for CTL∗using complemented pairs automata as well.Using the recent single exponential general McNaughton[23]construction of Safra [32](i.e.,construction for determinizing a B¨u chifinite automaton on infinite strings), our new nonemptiness algorithm also gives us a deterministic single exponential time decision procedure for both PDL-delta and the Mu-calculus,since the Safra construc-tion allows us to reduce satisfiability of these logics to testing nonemptiness of a tree automaton with exponentially many states and polynomially many pairs.This rep-resents an exponential improvement over the best known deterministic algorithms for these logics,which took deterministic double exponential time,corresponding to the nondeterministic exponential time upper bounds of[41].The bounds are essentially tight also,since the exponential time lower bound follows from that established for ordinary PDL by Fischer and Ladner[14].It is interesting to note that for most logics(including PDL-delta and the Mu-calculus but excluding CTL∗),our nonemptiness algorithm(s)and Safra’s construc-tion both play a crucial role.Each is independent of the other.Moreover,each is134 E.ALLEN EMERSON AND CHARANJIT S.JUTLAneeded and neither alone suffices.Our algorithm improves the complexity of testing nonemptiness by an exponential factor,while Safra’s construction independently ap-plies to reduce the size of the automaton by an exponential factor.For example,the “traditional”result of Streett[33]gave a deterministic triple exponential algorithm for PDL-delta.Our algorithm alone improves it to deterministic double exponential time.Alternatively,Safra’s construction alone improves it to deterministic double exponential time.As shown in this paper,the constructions can be applied together to get a cumulative double exponential speedup for PDL-delta.In the case of CTL∗, we already had the effect of Safra’s construction,because[13]gave a way to determine with only a single exponential blowup the B¨u chi string automaton corresponding to a linear temporal logic formula by using the special structure of such automata(unique accepting run).Thus Safra’s construction provides no help for the complexity of the CTL∗logic.The remainder of the paper is organized as follows.In section2we give prelimi-nary definitions and terminology.In section3we establish a“small model theorem”for(pairs)tree automata.In section4we give the main technical results on test-ing nonemptiness of pairs tree automata.In section5we give the main results on nonemptiness of complemented pairs tree automata.Applications of the algorithms to testing satisfiability of modal logics of programs,including CTL∗,PDL-delta,and the Mu-calculus,are described in section6.Some concluding remarks are given in section7.2.Preliminaries.2.1.Logics of programs.2.1.1.Full branching time logic.The full branching time logic CTL∗[9]de-rives its expressive power from the freedom of combining modalities which quantify over paths and modalities which quantify states along a particular path.These modal-ities are A,E,F,G,X s,and U w(“for all futures,”“for some future,”“sometime,”“always,”“strong nexttime,”and“weak until,”respectively),and they are allowed to appear in virtually arbitrary combinations.Formally,we inductively define a class of state formulae(true or false of states)and a class of path formulae(true or false of paths):(S1)Any atomic proposition P is a state formula.(S2)If p,q are state formulae,then so are p∧q,¬p.(S3)If p is a path formula,then Ep is a state formula.(P1)Any state formula p is also a path formula.(P2)If p,q are path formulae,then so are p∧q,¬p.(P3)If p,q are path formulae,then so are X s p and pU w q.The semantics of a formula are defined with respect to a structure M=(S,R,L), where S is a nonempty set of states,R is a nonempty binary relation on S,and L is a labeling which assigns to each state a set of atomic propositions true in the state.A fullpath(s1,s2,...)is a maximal sequence of states such that(s i,s i+1)∈R for all i.A fullpath is infinite unless for some s k there is no s k+1such that(s k,s k+1)∈R. We write M,s|=p(M,x|=p)to mean that state formula p(path formula p)is true in structure M at state s(of fullpath x,respectively).When M is understood,we write simply s|=p(x|=p).We define|=inductively using the convention that x=(s1,s2,...)denotes a fullpath and x i denotes the suffix fullpath(s i,s i+1,...), provided i≤|x|,where|x|,the length of x,isωwhen x is infinite and k when x is finite and of the form(s1,...,s k);otherwise x i is undefined.COMPLEXITY OF TREE AUTOMATA 135For a state s ,(S1)s |=P iffP ∈L (s )for atomic proposition P ,(S2)s |=p ∧q iffs |=p and s |=q ,s |=¬p iffnot (s |=p ),(S3)s |=Ep ifffor some fullpath x starting at s ,x |=p .For a fullpath x =(s 1,s 2,...),(P1)x |=p iffs 1|=p for any state formula p ,(P2)x |=p ∧q iffx |=p and x |=q ,x |=¬p iffnot (x |=p ),(P3)x |=X s p iffx 2is defined and x 2|=p ,x |=(p U w q )ifffor all i ∈[1:|x |],if for all j ∈[1:i ]x j |=¬q ,then x i |=p .We say that state formula p is valid ,and write |=p ,if for every structure M and every state s in M ,M,s |=p .We say that state formula p is satisfiable ifffor some structure M and some state s in M ,M,s |=p .In this case we also say that M defines a model of p .We define validity and satisfiability for path formulae similarly.We write f .=g to mean that formula f abbreviates formula g .Other connectives can then be defined as abbreviations in the usual way:p ∨q .=¬(¬p ∧¬q ),p ⇒q .=¬p ∨q ,p ⇔q .=(p ⇒q )∧(q ⇒p ),Ap .=¬E ¬p ,Gp .=p U w false ,and F p .=¬G ¬p .Further operators may also be defined as follows:X w p .=¬X s ¬p is the weak nexttime,pU s q .=(pU w q )∧F q is the strong until,∞F p .=GF X s p means infinitely often p ,∞G p .=F GX s p means almost everywhere p ,inf .=GX s true means the path is infinite,and fin .=F X w false means the path is finite.2.1.2.Propositional dynamic logic plus repeat.As opposed to CTL ∗,in which the models represent behaviors of the programs,in PDL-delta the programs are explicit in the models.The modalities in PDL-delta quantify the states reachable by programs explicitly stated in the modality.Thus,for a program B (which is obtained from atomic programs and tests using regular expressions), B p ([B ]p )states that there is an execution of B leading to p (after all executions of B ,p holds).Also included is the infinite repetition construct delta (△)which makes PDL-delta much more expressive than PDL.△B states that it is possible to execute B repetitively infinitely many times.PDL-delta formulae are interpreted over structures M =(S,R,L ),where S is a set of states,R :P rog →2S ×S is a transition relation,P rog is the set of atomic programs,and L is a labeling of S with propositions in P rop .For more details see [33].2.1.3.Propositional Mu-calculus.A least fixpoint construct can be used to increase the power of simple modal logics.Thus,by adding this construct to PDL,we get Lµ,the propositional Mu-calculus [19],a logic which subsumes PDL-delta.A variant formulation of the Mu-calculus,which we use here,adds the least fixpoint construct to a simple subset of CTL ∗,including just nexttime (AX s ),the boolean connectives,and propositions (cf.[7]).The least fixpoint construct has the syntax µY.f (Y ),where f (Y )is any formula syntactically monotone in the propositional vari-able Y ,i.e.,all occurrences of Y in f (Y )fall under an even number of negations.It is interpreted as the smallest set S of states such that S =f (S ).By the well-known Tarski–Knaster theorem,µY.f (Y )= i f i (false),where i ranges over all ordinals and f i (intuitively)denotes the i -fold composition of f with itself;when the domain136 E.ALLEN EMERSON AND CHARANJIT S.JUTLAisfinite we may take i as ranging over just the natural numbers.Its dual,the greatest fixpoint,is denotedνY.f(Y)(≡¬µY.¬f(¬Y)).Thus,e.g.,µY.[B]Y is equivalent to ¬△B of PDL-delta.Similarly,using temporal logic,µY.P∨AX s Y is equivalent to AF P(i.e.,along all paths P eventually holds).Many correctness properties of con-current programs can be characterized in terms of the Mu-calculus,including all those expressible in CTL∗and PDL-delta.For more details,see,for example,[7,19,35,12].The formulae of the(propositional)Mu-calculus are(1)propositional constants P,Q,...,(2)propositional variables Y,Z,...,(3)¬p,p∨q,and p∧q,where p and q are any formulae,(4)EX s p and AX s p,where p is any formula,(5)µY.f(Y)andνY.f(Y),where f(Y)is any formula syntactically monotone in the propositional variable Y,i.e.,all occurrences of Y in f(Y)fall under even number of negations.In what follows,we will useσas a generic symbol forµorν.In afixed point expressionσY.f(Y),we say that each occurrence of Y is bound toσY.If an occurrence of Y is not bound,then it is free.A sentence(or closed formula)is a formula containing no free propositional variables,i.e.,no variables unbound by aµor aνoperator.Sentences are interpreted over structures M=(S,R,L)as for CTL∗.As usual we will write M,s|=p to mean that in structure M at state s sentence p holds true. To give the technical definition of|=we need some preliminaries.The power set of S,2S,may be viewed as the complete lattice(2S,S,φ,⊆,∪,∩). Intuitively,we identify a proposition with the set of states which make it true.Thus, false,which corresponds to the empty set,is the bottom element,true,which cor-responds to S,is the top element,∪is join,∩is meet,and implication(for all s∈S(P(s)⇒Q(s))),which corresponds to simple set-theoretic containment(P⊆Q), provides the partial ordering on the lattice.Letτ:2S→2S be given;then we say thatτis monotonic provided P⊆Q implies τ(P)⊆τ(Q).A monotonic functionalτalways has both a leastfixpointµX.τ(X) and a greatestfixpointνX.τ(X).For a formula or function p(Y),we write p0(Y)=false,p1(Y)=p(Y), p i+1(Y)=p(p i)(Y)for successor ordinal i+1,and p j(Y)= k<j p k(Y)for limit ordinal j.Theorem2.1(Tarski–Knaster).Letτ:2S→2S be a given monotonic functional. Then(a)µY.τ(Y)= {Y:τ(Y)=Y}= {Y:τ(Y)⊆Y},(b)νY.τ(Y)= {Y:τ(Y)=Y}= {Y:τ(Y)⊇Y},(c)µY.τ(Y)= i≤|S|τi(false),and(d)νY.τ(Y)= i≤|S|τi(true).A formula p with free variables Y0,Y1,...,Y n is thus interpreted as a mapping p M from(2S)n+1to2S,i.e.,it is interpreted as a predicate transformer.We write p(Y0,Y1,...,Y n)to denote that all free variables of p are among Y0,Y1,...,Y n.Let V0,V1,...,V n be subsets of S;then a valuationΥ=V0,V1,...,V n is an assignment of V0,V1,...,V n to the free variables Y0,Y1,...,Y n,respectively.Υ[Y i←V′i]denotes the valuation identical toΥ,except that Y i is assigned V′i.We use p M(Υ)to denote the value of p in structure M on the arguments V0,V1,...,V n.We drop M when it is understood from context.We then let M,s|=p(Υ)iffs∈p M(Υ),and we define |=inductively as follows:(1)s|=P(Υ)iffP∈L(s),COMPLEXITY OF TREE AUTOMATA137(2)s|=Y(Υ)iffs∈Υ(Y),(3)s|=(¬p)(Υ)iffs|=p(Υ),s|=(p∨q)(Υ)iffs|=p(Υ)or s|=q(Υ),s|=(p∧q)(Υ)iffs|=p(Υ)and s|=q(Υ),(4)s|=(EX s p)(Υ)iff∃t(s,t)∈R and t|=p(Υ),s|=(AX s p)(Υ)iff(a)∃u(s,u)∈R and u|=p and(b)for all t(s,t)∈R implies t|=p(Υ),(5)s|=(µY.f(Y))(Υ)iffs∈ {S′⊆S|S′={t:t|=f(Y)(Υ[Y←S′])}},s|=(νY.f(Y))(Υ)iffs∈ {S′⊆S|S′={t:t|=f(Y)(Υ[Y←S′])}}.2.1.4.Conventions.To avoid a proliferation of unnecessary parentheses,we order the connectives from greatest to lowest binding power as follows:¬binds tighter than F,G,X w,X s,∞F,∞G,which bind tighter than∧,which binds tighter than∨,which binds tighter than⇒,which binds tighter than U w,U s,which bind tighter than A,E, which bind tighter thanµ,ν,which bind tighter than⇔.If we write M,s|=p,it is implicit that s is a state of M.That is,s∈S where M=(S,R,L).A convenient abuse of notation is to write s∈M in some places.If p is a formula,then p M denotes{s:M,s|=p},the set of states s in M at which p is true.We can write M,s1,...,s k|=p to abbreviate M,s1|=p and...and M,s k|=p.We write p≡q for|=p⇔q.In the context of a structure M,we can also write p≡M q for p M=q M.If M is understood,then we can drop the M and write just p≡q.It should be clear from context whether equivalence over all structures or over M is meant.We use p1≡p2≡···≡p k as shorthand for p1≡p2and...and p k−1≡p k.Technically,we distinguish between an atomic proposition symbol P and the associated set,viz.,P M,of states which are labeled with it in structure M.It is often convenient notation to use the uppercase sans serif symbol P corresponding to P to denote the set of states that are labeled with P.Remark:We note the following identities:EX w false≡AX w false and asserts that a state has no successors.EX s true≡AX s true and asserts that a state has one or more successors.2.2.Automata on infinite trees.We considerfinite automata on labeled, infinite binary trees.1The set{0,1}∗may be viewed as an infinite binary tree,where the empty stringλis the root node and each node u has two successors:the0-successor u0and the1-successor u1.Afinite(infinite)path through the tree is a finite(respectively,infinite)sequence x=u0,u1,u2,...such that each node u i+1is a successor of node u i.IfΣis an alphabet of symbols,an infinite binaryΣ-tree is a labeling L which maps{0,1}∗−→Σ.Afinite automaton A on infinite binaryΣ-trees consists of a tuple(Σ,Q,δ,q0,Φ), whereΣis thefinite,nonempty input alphabet labeling the nodes of the input tree, Q is thefinite,nonempty set of states of the automaton,δ:Q×Σ→2Q×Q is the nondeterministic transition function,1We consider here only binary trees to simplify the exposition and for consistency with the classical theory of tree automata.CTL∗and the other logics we study have the property that their models can be unwound into an infinite tree.In particular,in[13]it was shown that a CTL∗formula of length k is satisfiable iffit has an infinite tree model withfinite branching bounded by k,i.e.,iffit is satisfiable over a k-ary tree.Our results on tree automata apply to such k-ary trees as explained at the end of the proof of Theorem4.1.138 E.ALLEN EMERSON AND CHARANJIT S.JUTLAq0∈Q is the start state of the automaton,andΦis an acceptance condition described subsequently.A run of A on the inputΣ-tree L is a functionρ:{0,1}∗→Q such that forall v∈{0,1}∗,(ρ(v0),ρ(v1))∈δ(ρ(v),L(v))andρ(λ)=q0.We say that A acceptsinput tree L iffthere exists a runρof A on L such that for all infinite paths x startingat the root of L if r=ρ◦x is the sequence of states A goes through along path x,then the acceptance conditionΦholds along r.For a pairs automaton(cf.[23,30])acceptance is defined in terms of afinite list((RED1,GREEN1),...,(RED k,GREEN k))of pairs of sets of automaton states(whichmay be thought of as pairs of colored lights where Aflashes the red light of thefirstpair upon entering any state of the set RED1,etc.):r satisfies the pairs conditioniffthere exists a pair i∈[1..k]such that RED iflashesfinitely often and GREEN iflashes infinitely often.We assume the pairs acceptance condition is given formallyby a temporal logic formulaΦ= i∈[1..k](GF GREEN i∧¬GF RED i).2Similarly,a complemented pairs(cf.[33])automaton has the negation of the pairs condition as itsacceptance condition;i.e.,for all pairs i∈[1..k],GREEN iflashes infinitely often impliesthat RED iflashes infinitely often,too.The complemented pairs acceptance conditionis given formally by a temporal logic formulaΦ= i∈[1:k]GF GREEN i⇒GF RED i.3.Small model theorems.3.1.Tree automata running on graphs.Note that an infinite binary tree L′may be viewed as a“binary”structure M=(S,R,L),where S={0,1}∗,R=R0∪R1 with R0={(s,s0):s∈S}and R1={(s,s1):s∈S},and L=L′.We could alternatively write M=(S,R0,R1,L).We can also define a notion of a tree automaton running on certain appropriatelylabeled binary,directed graphs that are not binary trees.Such graphs,if accepted,arewitnesses to the nonemptiness of tree automata.We make the following definitions.A binary structure M=(S,R0,R1,L)consists of a state set S and labeling L asbefore,plus a transition relation R0∪R1decomposed into two partial functions:R0:S−→S,where R0(s),when defined,specifies the0-successor of s,and R1:S−→S,where R1(s),when defined,specifies the1-successor of s.We say that M is a fullbinary structure iffR0and R1are total.A run of automaton A on binary structure M=(S,R0,R1,L),if it exists,is amappingρ:S→Q such that for all s∈S,(ρ(R0(s)),ρ(R1(s)))∈δ(ρ(s),L(s)),andρ(s0)=q0.Intuitively,a run is a labeling of M with states of A consistent withthe local structure of A’s transition diagram.It will turn out that if an automatonaccepts some binary tree,there does exist somefinite binary graph on which there isa run that is accepting:all of the paths through the graph define state sequences ofthe automaton meeting its acceptance condition.3.2.The transition diagram of a tree automaton.The transition diagram of A can be viewed as an AND/OR-graph,where the set Q of states of A comprises the set of OR-nodes,while the AND-nodes define the allowable moves of the automaton. Intuitively,OR-nodes indicate that a nondeterministic choice has to be made(depend-ing on the input label),while the AND-nodes force the automaton along all directions. For example,suppose that for automaton A,δ(s,a)={(t1,u1),...,(t m,u m)}and δ(s,b)={(v1,w1),...,(v n,w n)};then the transition diagram contains the portion shown in Figure3.1.2We are assuming that each proposition symbol,such as GREEN i,of formulaΦis associatedCOMPLEXITY OF TREE AUTOMATA 139b b a a Cs ¡¡¡¡ e e e e ¡¡¡ ¡¡¡ ¡¡¡ ¡¡¡ e e e e e e e e e e e e ......00001111s t 1t m u 1u m v 1w 1v n w n=OR-nodes (states of A)a =AND-nodes (transitions on a)Fig.3.1.Formally,given a tree automaton A =(Σ,Q,δ,q 0,Φ)with transition function δ:Q ×Σ−→2Q ×Q :(q,a )−→{(r 1,s 1),...,(r k ,s k )},we may view it as defining a transition diagram T ,which is an AND/OR-graph (D,C,R DC ,R CD 0,R CD 1,L ),whereD =Q is theset of OR-nodes;C = q ∈D a ∈Σ{((q,a ),(r,s )):(r,s )∈δ(q,a )}is the set of AND-nodes.Each AND-node corresponds to a transition.If δ(q,a )={(r 1,s 1),...,(r k ,s k )},then the corresponding AND-nodes are essentially the pairs (r 1,s 1),...,(r k ,s k ).How-ever,since each transition is associated with a unique current state/input symbol pair (q,a ),we formally define the corresponding AND-nodes to be pairs of pairs:((q,a ),(r 1,s 1)),...,((q,a ),(r k ,s k )).R DC ⊆D ×C specifies the AND-node successors of each OR-node.For each q ∈D as above,R DC (q )= a ∈Σ{((q,a ),(r,s )):(r,s )∈δ(q,a )};3R CD 0,R CD 1:C −→D are partial 4functions giving the 0-successor and 1-successor states,respectively:R CD 0(((q,a ),(r,s )))=r and R CD 1(((q,a ),(r,s )))=s ;L is a labeling of nodes.For an AND-node L (((q,a ),(r,s )))={a },where a ∈Σ.For an OR-node,the labeling assigns propositions associated with the acceptance condition Φso that all OR-nodes in the set GREEN i (respectively,RED i )are labeled with the corresponding proposition GREEN i (respectively,RED i ).with exactly the set of states of the corresponding name,in this case GREEN i .3We identify a relation such as R DC ⊆D ×C with the corresponding function R ′DC :D −→2C defined by R ′DC (d )={c ∈C :(d,c )∈R DC }for each d ∈D .4For classically defined tree automata,the functions R CD 0,R CD 1are total so that there is always a 0-successor and 1-successor automaton state.For technical reasons it is convenient to allow R CD 0,R CD 1to be partial.140 E.ALLEN EMERSON AND CHARANJIT S.JUTLAThus,we may write a tree automaton A in the form(T,d0,Φ),where T is the diagram,d0is the start state,andΦis an acceptance condition.3.3.One symbol alphabets.For purposes of testing nonemptiness,without loss of generality,we can restrict our attention to tree automata over a single letter al-phabet and,thereby,subsequently ignore the input alphabet.Let A=(Q,Σ,δ,q0,Φ) be a tree automaton over input alphabetΣ.Let A′=(Q,Σ′,δ′,q0,Φ)be the tree automaton over one letter input alphabetΣ′={c}obtained from A by,intuitively, taking the same transition diagram but now making all transitions on symbol c.For-mally,A′is identical to A except that the input alphabet isΣ′and the transition functionδ′is defined byδ′(q,c)= a∈Σδ(q,a).Observation3.1.The set accepted by A is nonempty iffthe set accepted by A′is nonempty.Henceforth,we shall therefore assume that we are dealing with tree automata over a one symbol alphabet.3.4.Generation and containment.It is helpful to reformulate the notion of run to take advantage of the AND/OR-graph organization of the transition diagram of an automaton.Intuitively,there is a run of diagram T on structure M provided M is“generated”from T by unwinding T so that each state of M is a copy of an OR-node of T.Formally,we say that a binary structure M=(S,R0,R1,L)is generated by a transition diagram T=(D,C,R DC,R CD0,R CD1,L T)(starting at s0∈S and d0∈D) iff∃is a total function h:S−→D such that for all s∈Sif s has any successors in M,then∃c∈R DC(h(s))for all i∈{0,1}R CDi(c)is defined iffR i(s)is defined andR CDi(c)=h(R i(s))when both are defined and L(s)=L T(h(s))(such that h(s0)=d0).We say that a binary structure M=(S,R0,R1,L)is contained in transition diagram T(starting at s0∈M and q0∈T)provided M is generated by T(starting at s0∈M and q0∈T),where the generation function h is the natural injection h:S−→D:s∈S−→s∈D,so that all states of M are OR-nodes of T.Note that if M is a structure generated by(respectively,contained in)T,there is an associated AND/OR-graph H generated by(respectively,contained in)T ob-tained from M by inserting between each state and its successors in M a copy of the AND-node that determines the successors of state via the generation function for M. We write H=ao(M).Conversely,if H is an AND/OR-graph generated by(respec-tively,contained in)T,there is an associated structure M generated by(respectively, contained in)T obtained from H by eliding AND-nodes.Here we write M=o(H).3.5.Linear size model theorems.The following theorem(cf.[6])is the basis of our method of testing nonemptiness of pairs automata.It shows that there is a small binary structure contained in the transition diagram and accepted by the automaton.Theorem3.2(linear size model theorem).Let A be a tree automaton over a one symbol alphabet with pairs acceptance conditionΦ= i∈[1:k](∞F Q i∧∞G P i).Then automaton A accepts some tree T iffA accepts some binary model M of size linear in the size of A,which is a structure contained in the transition diagram of A.。
Propositional-and- First-- Order- Logic课件
• Propositional symbols: P, Q,... (atomic sentences)
• Wrapping parentheses: ( … )
• Sentences are combined by connectives:
and
[conjunction]
or
[disjunction]
Model for a KB
• Let the KB be [PQR, Q P]
• What are the possible models? Consider all possible assignments of T|F to P, Q and R and check truth tables
implies
[implication / conditional]
is equivalent[biconditional]
not
[negation]
• Literal: atomic sentence or negated atomic sentence
P, P
Examples of PL sentences
• An inconsistent sentence or contradiction is a sentence that is False under all interpretations. The world is never like what it describes, as in “It’s raining and it’s not raining.”
A, A B
B
And Introduction A, B
AB
And Elimination A B
Vector Bundles and K-Theory - A. hatcher
ture. Here is the basic definition. An n dimensional vector bundle is a map p : E →B
Vector bundles are special sorts of fiber bundles with additional algebraic struc-
Applications of w1 and c1 72. . . . . . . . . . . . . 63 Axioms and Construction 64. Cohomology of Grassmannians 69.
3.2. The Chern Character
The J–Homomorphism 76.
Version 1.3, July 2001
Allen Hatcher
Copyright c 2001 by Allen Hatcher
Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved.
pairs (x, v) ∈ S n × Rn+1 such that v is perpendicular to the tangent plane to S n at
(4) The normal bundle to S n in Rn+1 , a line bundle p : E →S n with E consisting of
Table of Contents
Chapter 1. Vector Bundles
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( p q) ( p r )
0 0 0 1 1 1 1 1
0 0 0 1 1 1 1 1
De Morgan’s laws: ﹁(p∨q)≡﹁p∧﹁q
2 n and p1 p2 pn We can see that 1 are well defined whenever p1 , p2 ,, pn are propositions. Furthermore, De Morgan’s laws can be extend to
第一章 命题逻辑
2
1.3. Propositional Equivalences
An important method used extensively in the construction of mathematical arguments is to replace a statement by another statement with the same truth value. This is actually a method that produces propositions with the same truth value as a given proposition. The methodology is based on the fact that some wffs have the same truth values in all possible cases even though their expressions are different. For example, the wffs p→q and ﹁p∨q have the same truth values in all possible cases because they have the same truth table. Two wffs are said to be logically equivalent if they have the same truth table. We formally define this notion as below.
( A B) A B
A B ( A B) ( B A) A B A B A B ( A B) (A B)
第一章 命题逻辑 12
3. 重要的等价公式 ⑴ 对合律/双重否定律 PP ⑵ 幂等律/恒等律 P∨PP P∧PP ⑶ 结合律 P∨(Q∨R)(P∨Q)∨R P∧(Q∧R)(P∧Q)∧R (P Q) R P (Q R) ⑷交换律 P∨QQ∨P P∧QQ∧P PQQP
At present there are no other known procedures that a computer can follow to determine in a reasonable amount of time whether a wff in a large number of variables is a tautology. This is the place where logical equivalences are needed. Using the concepts of logically equivalence, tautology, and contradiction, we list the following important equivalences. In these equivalences, T denotes a proposition that is always true and F denotes a proposition that is always false. All these equivalences are easily to be verified by truth tables. In the following, A , B and C denote arbitrary wffs.
第一章 命题逻辑 10
Logical Basic Equivalences
6. Identity laws
7. Domination laws
AT A
AT T
A F A
A F F
A A F
8. Negation laws
A A T
9. Absorption laws
第一章 命题逻辑
3
Definition 1.3.1 Two wffs A and B are called logically equivalent, denoted by A≡B, if the wff A B is a tautology.
第一章 命题逻辑
4
EXAMPLE 1.3.1 (Distributive law of disjunction over conjunction) The wffs p (q r ) and ( p q) ( p r ) are logically equivalent. Proof: We construct the truth table for these wffs in below. Since the truth values of ( p q) ( p r) and p (q r ) agree, these wffs are logically equivalent.
第一章 命题逻辑
5
Truth values of p (q r ) and ( p q) ( p r )
p
0 0 0 0 1 1 1 1
q
0 0 1 1 0 0 1 1
r
0 1 0 1 0 1 0 1
p (q r )
pq
0 0 1 1 1 1 1 1
pr
0 1 0 1 1 1 1 1
第一章 命题逻辑 13
⑸分配律
P∨(Q∧R)(P∨Q)∧(P∨R) P∧(Q∨R)(P∧Q)∨(P∧R) ⑹ 吸收律 P∨(P∧Q)P P∧(P∨Q)P ⑺德-摩根律 (P∨Q)P∧Q (P∧Q)P∨Q ⑻ 同一律 P∨FP P∧TP ⑼ 零律 P∨TT P∧FF ⑽ 互补律 P∨PT律 PQP∨Q QP ⑿双条件转化律 PQ (PQ)∧(QP)(P∨Q)∧(P∨Q) (P∧Q)∨(P∧Q )
(13)输出律 (A∧B)CA(BC)
(14)归谬律 (AB)∧(AB)A
第一章 命题逻辑15
We can use these equivalences to show that other wffs are equivalent without checking truth tables. But first we need to observe two general properties of equivalences.
4. Distributive laws
A ( B C ) ( A B) ( A C ) A ( B C ) ( A B) ( A C )
5. De Morgan’s laws
( A B) A B
( A B) A B
1.3. Propositional Equivalences
WORD AND EXPRESSIONS
idempotent law 幂等律 associative law 结合律 double negation 双重否定 distributive law 分配律 identity law 同一律 domination law 零律 negation law 否定律 absorption law 吸收律 replacement rule 替换法则
( A C ) ( B C ) ( A B) C ( A B) ( A C ) A ( B C ) ( A C ) ( B C ) ( A B) C ( A B) ( A B) A
11. Logical Equivalences involving biconditional
p p p
( p1 p2 pn ) p1 p2 pn
and
( p1 p2 pn ) p1 p2 pn
A truth table can be used to determine whether a wff is a tautology. This can be done by hand for a wff with a small number of variables. But when the number of variables grows, it becomes impractical. For instance there are 220 1,048,576 rows in the truth table for a proposition with 20 variables. In this occasion a computer is needed to help to determine whether a wff in 20 variables is a tautology. But when there are 1000 variables, can even a computer determine in a reasonable amount of time whether a wff is a tautology? Checking every one of the 21000 (a number with more than 300 decimal digits) possible combinations of truth values simply cannot be done by a computer in many years.