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A simple logic proof of the Four Color Theorem

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A simple logic proof of the Four Color TheoremHuang XingbinCollege of Physics Science and Technology,Heilongjiang University,China,Harbin,Xuefu Road74,150086,AbstractThe Four Color Conjecture already was solved under the computer help.In this paper, we unify A.B.Kempe exchange color method and the diagonal­transformation method. Proposed and proved the using diagonal­transformation certainly can change every standard graph into all maximum planar graph with same vertex;if a random maximum planar graph is4­colorable,then the new maximum planar graph which obtained by one time random diagonal­transformation was4­colorable still.Further proved all maximum planar graphs are4­colourable.Therefore has given The Four Color Conjecture simple and direct logical proof.1IntroductionThe story of the Four Color Problem comes from England.In1852,F.Guthrie graduate from University College London was coloring in a map showing the counties of England.He puts forward an interesting problem that the maximum number of colors required coloring any map seemed likely to be four.The coloring has to meet the obvious requirement that no two regions sharing a length of common boundary should be given the same color.In1872,England at that time well­known mathematicianA.Cayley put forward this problem toward the London Mathematical Society formally. From then on Guthrie's question became known as the Four Color Problem but also caused the extensive attention of mathematics field in the world.In the world many top­ranking mathematicians attended to prove the Four Color Problem in succession,but nobody can give a right proof.And it grew to be the second most famous unsolvedproblem in mathematics after Fermat's last theorem.In1976,two American mathematicians,K.Appel and W.Haken,announced that they had solved the problem. But there was a twist.Much of their proof was carried out on a computer,and was far too long for humans to check.To this day it is not known if there can be a short proof of the Four Color Theorem that a human could follow.Most mathematicians think there is not.Here we give a simple logic proof by the method of the diagonal­transformation.2Map’s Dual Graph and All Maximum Planar GraphsWhat makes the four color problem so hard is that it refers to all maps­not just all the maps in all the atlases around the world,but all conceivable maps,maps with millions (and more)of countries of all shapes and sizes.Knowing that you can color some particular map using four colors does not help you at all.You need to produce an argument that will work in all cases[1].Since the sizes and shapes of the map regions do not matter,only the way they join together,the Four Color Problem is a question in topology.We shall see momentarily,it is easily reformulated as a problem about maximum planar graph.Within each region of the given map,you place a single point,known as a vertex of the graph.(You can think of these vertices as the capital cities of the countries,if you wish.)You then join up the vertices to form a graph,in much the same way that you might link two cities by a rail,known as an edge of the graph.The rule is that two vertices are joined together if,and only if,their respective map regions share a common boundary,in which case the line joining them has to lie entirely within the two regions. (In terms of a rail link this would mean that the line couldn’t cross the territory of any third country.)Like this obtains graph named map’s dual graph.Indeed,the Four Color Theorem can be expressed in terms of the dual graph as follows:the dual graph’svertices can be colored with at most four colors such that no two vertices connected by an edge are colored the same color.A map is a planar graph with no“bridges”.Moreover,a planar graph’s dual is also a planar graph.Notice that no map’s dual will have a loop or more than one edge between any two vertices.From now on,we will ignore all graphs that have loops or more than one edge connecting any two vertices.Planar graphs are graphs that can be represented in the plane without any edges intersecting away from the vertices.A simple graph is called maximum planar graph if it is planar but adding any an edge would destroy that planar property.All faces(even the outer one)are then bounded by three edges,explaining the alternative term triangular for these graphs.The Four Color Theorem asserts that any map can have its faces colored with at most four different colors such that no two faces that share an edge are the same color. Thus,to prove the Four Color Theorem,it is equivalent to prove that no maximum planar graph requires more than four colors to color its vertices.For the sake of simplicity,from now on,we will refer to a graph as being4­colorable if its vertices can be colored with4colors.Similarly,we called it is the4­colored graph if its all vertices have colored by4colors(an admissible vertex4­coloring).In this article,uses the maximum planar graph to prove the Four Color Theorem. If all maximum planar graphs can be colored using four colors,so can all planar graphs. The Four Color Theorem is true.All maximum planar graphs meaning be:The vertex quantity changes arbitrarily;if the vertex quantity is a constant so the edge quantity is also the constant but the shape of connecting the vertices with the edges changes arbitrarily in certain scope.In other word,there will be a lot of different configurations of graphs with the vertex of thesame quantity.Therefore the proof all maximum planar graphs are4­colorable is extremely difficult.3Standard Graph and Diagonal­transformationOur proof method and other is completely different.But we can use the predecessor to solve the question experience.It mainly divides three steps.First,Let us choose a special configuration of maximum planar graphs，such as figure1.We called it is the standard graph.It is obvious conclusion that every standard graph that vertex quantity can change arbitrarily is certainly4­colorable.Figure1:The Standard graph is composed by n vertex.Its all vertices are colored into the red,the green,the black and the white such that no two vertices connected by an edge are colored the same color.In the graph n is an even number.If n is an odd number,we can remove vertex n,with all has connectedits edge.Next,consider essences relation of the standard graph and the other graph.We use the conclusion that Professor Liu­Yanpei proved:The step by step using diagonal­transformation[2]change certainly all maximum planar graphs into its standard graph. The diagonal­transformation is shown in Figure2.Because the diagonal­transformation is the reversible transformation,therefore use it to change certainly every standardgraph into its all maximum planar graphs.We will prove the conclusion in the later word.Finally,let us assume a maximum planar graph is4­colorable,using random one time diagonal­transformation change it into a new random maximum planar graph(See figure2).Is the new graph4­colorable？If the answer is affirmative,we could prove that all maximum planar graphs are4­colorable,so are all planar graphs.Figure2:Using diagonal­transformation once to be able to change to a new maximum planar graph from its standard graph(figure1).It is also4­colorable. The so­called diagonal­transformation is:after elimination connection vertex②with④edge after,lays a new edge to connect the vertex③with⑤.4Diagonal­transformation does not Increase the Coloring NumberWe use the conventional method to solve the above these problems successfully. Proving these problems are very easy because had many scientists to do a lot of contributions for proving the Four Color Conjecture.The key part we proved is using Kempe exchange colors the method to complete;this method successfully has proven the Five Color Theorem.Proposition1:All standard graphs are4­colorable.Proof:Standard graph with n vertices is constitutes by two chains.Such as figure1.A chain is from vertex①to vertex②,another chain is from vertex③to vertex n.Firstchain only has two vertices colored with the colors red and green;second chain is composed which by n­2vertices are(alternately)colored with the colors black,white. Two standard graph’s difference is the vertex quantity different,but the vertex quantity change only in second chain.Therefore the vertex quantity change does not affect a standard graph is4­colorable.Hence all standard graphs are4­colorable.Proposition2:The step by step using diagonal­transformation certainly changes every standard graph into its all maximum planar graphs with same vertices.Proof:We proceed by induction on the number of vertices n.Let G(n)is any of the all maximum planar graph with n vertices,G s(n)is their standard graph with n vertices.In fact we will prove using diagonal­transformation can change certainly Gs(n)into allG(n).If the n is smaller,then using diagonal­transformation can change certainly Gs(n) into all G(n).Because it is verify easily.Assume:Step by step using diagonal­transformation can certainly change a standard graph with n­1vertices into all maximum planar graphs with n­1vertices.According to the conclusion of graph’s theory,any planar graph contains a vertex of degree at most5.The degree of a vertex is defined as the number of edges with that vertex as an end[3].Therefore G(n)contains a vertex v of degree at most5.If the degree of v is3,let them respectively is v1、v2and v3.Remove v and all the edges connected to it.The graph left has n­1vertices and the mark is G(n­1).According to assume,step by step using diagonal transformation can certainly change the standard graph Gs(n­1)into G(n­1).See figure3.Place the vertex v now in the v1,v2with v3constitutes of triangle inside in theGs(n­1)and connects them,then getting the Gs(n).Let Gs(n)to duplicate the aboveGs(n­1)diagonal­transformation process,therefore change certainly Gs(n)into theG(n).Figure3:The standard graph with n­1vertices,the mark is G S(n­1).If the degree of v is4,let them respectively is v1,v2,v3and v4,such as figure4A. Then use one time diagonal­transformation changes G(n)into G1(n),such as figure4B. In the G1(n),the degree of v is3.In this case,we run same argument on with above. Hence step by step using the diagonal­transformation can change certainly Gs(n)into G1(n).Next step using the inverse diagonal­transformation change the G1(n)into theG(n).Figure4:The A shows the degree of v is4in G(n).After the diagonal­transformation the B shows the degree of v is3in G1(n).If the degree of v is5,let them respectively is v1,v2,v3,v4and v5.Then use twice diagonal­transformation change G(n)into G2(n).In the G2(n)the degree of v is3.In this case,we run same argument on with above.Hence one time or many times(step by step)using the diagonal­transformation can certainly change Gs(n)into G2(n).Next step using twice the inverse diagonal­transformation change G2(n)into G(n).Hence step by step using the diagonal­transformation certainly changes every standard graph into its all maximum planar graphs with same vertices.Proposition3:Using random diagonal­transformation certainly changes any4­colored maximum planar graphs into another4­colored maximum planar graph.In order to proving the Proposition3we need an arbitrary diagonal­transformation so that it can replace all situations.Hence choose a quadrangle made of arbitrary four vertices in random4­colored maximum planar graph G(n),such as figure5.In an arbitrary4­colored maximum planar graph G(n)figure5A shows an arbitrary diagonal­transformation that certainly can replace all situations.After the diagonal­transformation figure5B shows the quadrangle in another maximum planar graph G d(n).Using a diagonal­transformation once change G(n)into G d(n).If G(n)is4­colorable,so long as we can prove G d(n)also is4­colorable.Hence we had also proven the proposition3.Figure5:The A shows a quadrangle made of arbitrary four vertices in random 4­colored maximum planar graph G(n).The B shows its diagonal­transformationProof:The proposition3will be proved by Kempe method.Let G(n)is a4­colored graph with red,green,black and white(an admissible vertex4­coloring).The vertex①,②with③is the white in turn;red and the black vertex(see figure5A).Our goal removes connection vertex first①with③edge,and then lays a new edge connection vertex②With④.Regardless of vertex④color is green or red,we must guarantee all G d(n)is4­colorable.First we can eliminate an easy case.If the④is green,removes connection the vertex①with③edge,then we achieve the goal immediately.Next if vertex④is red,we must transform the color of vertex④to with the vertex②color is different.Let us take a small detour,and introduce Kempe method.We denote the subgraph of G(n)by G rg.G rg subgraph by all red,the green vertex as well as connects their all edge is composed.A Kempe chain is a chain whose vertices are colored with only two colors.In particular,we speak of an r­g­chain if it is a Kempe chain whose vertices are(alternately)colored with the red and green.A Kempe net is a connected component of a subgraph.In particular,we speak of an r­g­net if it is a Kempe net whose vertices are colored with the red and green.Each pair of vertices of a Kempe net can be joined by a Kempe chain.In fact;Kempe nets can be characterized using Kempe chain.We easy to understand on a Kempe net exchange colors can obtain another admissible vertex4­coloring.Now we return to the proof:If vertex④is red but the vertex②with④must be in different r­g­nets.If so,and then we exchange the colors on the r­g­net of containing vertex④.That assigns green to the④,and the②is red yet.Hence we can connect the ②with the④.Finally if the vertex④is red;the vertex②with④must be in the same r­g­net, therefore there certainly is at least one r­g­chain of containing the vertex②with④. According to figure5there certainly is at the same time at least one r­b­chain of containing the vertex②with④.The r­w–chain is also such.We consider the most complicated case now such as figure6show.Random exists in a lot of the r­x­chains of containing the vertex②with④(x is random one of in the g,b and w).If so,we can exchange the colors on the inside of any r­x­chains.Our goal is to change all r­k­chains into r­p­chains(k and p are random one of in the g,b and w but the p is not equal to the k)so that only retains two kinds of r­x­chains.Figure6:Random exist in a lot of the r­x­chains of containing the vertex②with④(x is random one of in the g,b and w).When the r­x­chains are divided into two parts,some encircle vertex①,else encircle vertex③.In order to let the r­x­chains changed and disappeared in the two parts is consistent colors.First considers left and right outmost r­x­chain color.If the left and right have same colors(e.g.they all are the r­w–chain),we can choose any one of retains two kinds of r­x­chains(e.g.the r­g–chain or r­b–chain).If the left and right have different colors(e.g.the r­w–chain and the r­b–chain),we have no chosen,only can change a surplus r­x­chain(e.g.the r­g–chain).According to above­mentioned principle we handle the left r­x–chain in figure6. Our goal is to change all the r­b­chains into the r­g­chains or the r­w–chains.Let us consider the chain of②­C2­④,they is constituted alternately by the red and green vertices.Attention:Have already removed the edge connected the①with③now.So the②­C2­④­②chain is equal to an r­g­cycle.Therefore we can in②­C2­④­②inside exchange the black and white vertex the color.Now the C3change into the r­w–chain from the r­b­chain,else chain one by one in order is:C1(r­w–chain),C2(r­g–chain),C3(r­w),C4(r­b),C5(r­w),C6(r­g),··①(r­b).Next we can in②­C3­④­②inside exchanges black and the green vertex color. Has:C1(r­w),C2(r­g),C3(r­w),C4(r­g),C5(r­w),C6(r­b),··①(r­g).Repeat above­mentioned process thus，the certainly finally has:C1(r­w),C2(r­g),C3(r­w),C4(r­g),C5(r­w),C6(r­g),··①(r­g or r­w).According to above­mentioned principle we handle the right r­x–chain in figure6. Our goal is same with the left side.We can run same argument on with above,the certainly finally has:C/1(r­g),C/2(r­w),C/3(r­g),C/4(r­w),C/5(r­g),C/6(r­w),··③(r­g or r­w).Now there is no the r­b­chain connected the vertex②with④.So the vertex②with④must be in different r­b­net of G rb.If so,and then we flip the colors on the component containing the vertex④of G rb.That assigns black to the vertex④,and the vertex②is red yet.Hence we can connect the vertex②with④.Hence,G d(n)also is 4­colored graph.Thus using random diagonal­transformation certainly changes any4­colored maximum planar graphs into another random4­colored maximum planar graph.5All Maps are4­colorableConclusion:Every standard graph is four colorable(see figure1).Using step by step the diagonal­transformation must change every standard graph into its all maximum planar graphs with same vertices.Therefore,from a4­colored standard graph transforms one time obtains a new4­colored maximum planar graph.Again transformsone time also obtains another4­colored maximum planar graph.So,continuously transforms it,finally we certainly can obtain all4­colored maximum planar graph.In other word,all maximum planar graphs are4­colorable,so are all maps.Now we can certainly say that the four colors were sufficient.The Four Color Conjecture is true.References[1]Keith st doubts removed about the proof of the Four Color Theorem From:/devlin/devlin_01_05.html[2]Liu Yanpei.Planar graph theory and the Four Color Problem(Ⅰ).Journal of mathematical research and exposition.1983,Vol.3(3):pp.123­136.[3]K.Appel and W.Haken.The solution of the four­color­map problem.Scientific American.1977,Vol.27(4):pp.108­121AcknowledgementsWe thank Mr.Wang Xunye and Miss.Huang Yumeng for translating the paper. Correspondence should be addressed to H.X.(e­mail:huangxingbin@)。

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全新版大学英语(第二版)第一册第八单元短语总结

全新版大学英语(第二版)第一册第八单元短语总结

Useful Expressions in Unit 8 (by Mr. Hu )1. knit one’s brows 皱眉2. scratch one’s head 挠头3. come up with the right answer4. beam at learning the answer5. weave a tale: make up/invent a story6. bring/drive sth home to sb: make sb fully understand/realize sth7. sth hold lessons for sb: sth be instructive/educational to sb8. a fictional story ←→a true story9. as well=, too10. Aesop’s Fables11. run out of: use up12. file folders13. a handful of: a small number of14. before/plus tax: when the tax hasn’t been paid yet15. in amazement/surprise/disbelief16. in despair17. mental arithmetic18. modestly educated：slightly educated19. be upset by/about: = be worried / unhappy about20. good-natured: good-tempered21. be ignorant of/about22. ignorance23. a tiny slice of the problem24. perform the calculations25. sth be only a memory among sb26. be nonexistent27. a chilling (=discouraging) indifference (= a lack of interest)28. sum up: summarize29. compete with=compete against (sb for sth)30. financial, material, and human capital31. human capital: people’s knowledge and skills32. accumulate: collect33. ancestor=forefather←→posterity/offspring= descendant34. at some point soon用不了多久35. intellectual laziness 懒于用脑36. affect: have an influence on37. idle: lazy38. labor force: work force39. crash 坠落: fall down40. jam: get stuck41. break down: ①stop working ②fall/collapse42. have a humble suggestion 有一愚见43. dramatize: describe vividly; give a vivid description of44. feel bitter: feel painful45. headphones: earphones46. portable47. CD: compact disk48. sth put sb to sleep49. search for: look for50. get a job as a clerk51. with steady wages52. feed himself53. allow no leisure for such luxuries as…使某人没有闲暇去享受诸如...的奢侈的事情54. His only hope lies in his children.55. fast-forward56. a steel mill: a steel factory57. subway: the Underground/ the Tube (BrE)58. well off=rich59. a determined foe：a strong-minded enemy60. scared=frightened61. be scared of; too scared to do; be scared that;62. A scare B63. live by cf. live on64. peace and plenty (alliteration押头韵)65. safe and sound66. healthy and wealthy (rhyme尾韵)67. all day and all night= day and night68. offer services to sb69. long since cf. long ago70. complex: complicated71. manual work: physical labor72. discipline: training73. saving and discipline74. discipline sb75. earn an adequate living wage76. slum: ghetto77. plumbing cf. plumber78. trash piles79. trash；rubbish; garbage; junk80. in a word: in short81. much as: nearly as82. be befriended by: make friends with83. decline ①decrease ②refuse politely84. acquire(=get) financial capital85. rise from the ashes: rebuild itself after the devastation of WWII86. swear: promise seriously87. miracle: wonder: phenomenon88. sth catch one’s eye: sth arouse one’s attention89. intellectual faculties=abilities90. contemptible: deserving contempt 令人可鄙的91. cf. contemptuous: showing, feeling or expressing contempt 表示鄙视的92. stand in the doorway93. have a day off94. do the deal95. take an equal slice of the profit96. wash clothes by hand97. in my humble opinion98. appoint an assistant99. reduce the cost of production100. to his delight101. switch to sth102. fine arts103. make a good lawyer104. The story makes interesting reading.105. make it106. my sense of self107. take various measures to encourage local economic development 108. develop a sense of responsibility109. narrow the wide gap between A and B110. first-hand knowledge111. build up a large vocabulary112. gain a good command of English113. a likeable girl 讨人喜欢的女孩114. be content to do115. be unconcerned by：be indifferent about116. fine-looking: handsome117. call the fire department and ambulance118. put sth in a different perspective119. change one’s mind120. ditch: throw away sth in the ditch121. sigh inwardly 暗自叹气122. mentally battle with sth: 苦苦的心算123. look longingly at sth 眼巴巴地望着124. off-limits adj. 禁止进入的→可望不可及的125. tutor a student126. allow unlimited access to sth 可无限制的使用sth127. attend a maths seminar128. casually state sth随口说129. education establishment 教育机构130. call for tutoring in algebra and geometry131. in defense of 辩解132. rationally: reasonably133. proceed from known information to desired information 根据已知信息找到所需信息134. be competent with sth/in doing sth135. get tied up with tedious computations为繁琐的运算所拖累136. addition, subtraction, multiplication and division137. a word problem 应用题138. cut down his work: stop his work139. reflect on: think over140. repetitious: repeated141. sth fall to such a low priority in education circles 在教育界变得如此不受重视142. much less=let alone143. Math is as much about knowing why the rules work as knowing what the rules are.数学要讲有哪些规则，更要讲这些规则为什么成立。

离散数学英中名词对照表

离散数学英中名词对照表

离散数学英中名词对照表英文Abel categoryAbel group (commutative group) Abel semigroup Abelian groupabsorption property accessibility relation acyclicaddition principleadequate set of connectives adjacentadjacent matrixadjugateadjunctionaffine planealgebraic closed field algebraic element algebraic extensionalphabetalternating groupannihilatorantecedentanti symmetryanti-isomorphismarc setargumentarityarrangement problem associateassociativeassociative algebraassociatorasymmetricatomatomic formulaaugmenting pigeon hole principle augmenting path automorphism automorphism group of graph auxiliary symbol A 离散数学英文—中文名词axiom of choiceaxiom of equalityaxiom of extensionalityaxiom of infinityaxiom of pairsaxiom of regularityaxiom of replacement for the formulaaxiom of the empty setaxiom of unionB balanced imcomplete block designbarber paradoxbase (base 2 exponential function)base (logarithm function to the base 2)Bell numberBernoulli numberBerry paradoxbiconditionalbijection (one-to-one correspondence)bi-mdulebinary relationbinary operationbinary symmetric channel (BSC)binary treebinomial coefficientbinomial theorembinomial transform bipartite graphblockblockblock codeblock designBondy theoremBoolean algebra Boolean expression Boolean functionBoole homomorophism Boole latticeBoolean matrixBoolean productbound occurrencebound variablebounded latticeBruijn theorem Burnside lemmaC cagecancellation property canonical epimorphism Cantor conjecture Cantor diagonal method Cantor paradoxcapacitycardinal number cardinalityCartesion product of graph Catalan numbercatenationCayley graphCayley theoremceiling functioncell (block)centercertain eventchain (walk) characteristic function characteristic of ring characteristic polynomial check digitsChinese postman problem chromatic number chromatic polynomial circuitcirculant graph circumferenceclassclassical completeness classical consistent cliqueclique numberclose with respect to closed termclosureclosure of graphcode elementcode lengthcode wordcoefficientcoimageco-kernalcoloringcoloring problemcombinationcombination numbercombination with repetationcommon divisorcommon factorcommutativecommutative diagramcommutative ringcommutative seimgroupcomparablecompatible withcomplementcomplement elementcomplement of B with respect to A complementary relation complemented latticecomplete bipartite graphcomplete graphcomplete k-partite graphcomplete latticecomplete matchcomplete n-treecompositecomposite operationcomposition (molecular proposition) composition of graph (lexicographic product) compound statementconcatenation (juxtaposition) concatenation graphconditional statement (implication) congruence relationcongruent toconjectureconjunctive normal form connected component connective connectivityconnectivity relation consecutively consequence (conclusion) conservation of flow consistent (non-contradiction) constructive proofcontain (in)contingencycontinuumcontraction of graph contradiction contravariant functor contrapositiveconversecoproductcorankcorresponding universal map countable (uncountable) countably infinite set counter examplecountingcovariant functorcoveringcovering numbercrossing number of graph cosetcotreecutcut edgecut vertexcyclecycle basiscycle matrixcycle rankcycle spacecycle vectorcyclic groupcyclic indexcyclic permutation cyclic semigroupD De Morgan's law decision procedure decoding table deduction theorem degreedegree sequence derivation algebra Descartes product descendant designated truth value deterministic diagonal functor diagonal matrix diameterdigraphdilemmadirect consequence direct limitdirect sumdirected by inclutiondisconnecteddiscrete Fourier transform discrete graph (null graph) disjoint setdisjunctiondisjunctive normal form disjunctive syllogism distancedistance transitive graph distinguished element distributivedistributive lattice divisibilitydivision subringdivison ringdivisor (factor) dodecahedrondomaindual categorydual formdual graphdual principledual statementdummy variableE eccentricityedge chromatic number edge coloringedge connectivityedge coveringedge covering numberedge cutedge setedge-independence number eigenvalue of graph element (entry) elementary divisor ideal elementary product elementary sumempty graphempty relationempty set endomorphismendpointentry (element) enumeration function epimorphismequipotentequivalenceequivalent category equivalent class equivalent matrix equivalent object equivalent relationerror functionerror patternEuclid algorithmEuclid domainEuler characteristicEuler circuitEuler functionEuler graphEuler numberEuler pathEuler polyhedron formula Euler tourEuler traileven permutationeventeverywhere defined excess capacity existence proof existential generalization existential quantification existential quantifier existential specification explicitextended Fibonacci number extended Lucas number extensionextension field extension graphexterior algebraF facefactorfactorablefactotialfactorizationfaithful (full) functor Ferrers graphFibonacci numberfieldfilterfinite dimensional associative division algebra finite extensionfinite field (Galois field )finite groupfinite setfinitely generated modulefirst order theory with equalityfive-color theoremfive-time-repetitionfixed pointfloor functionflowforestforgetful functorfour-color theorem (conjecture)F-reduced productfree elementfree monoidfree occurrencefree R-modulefree variablefree-Ω-algebrafull n-treefunction schemeG Galileo paradoxGauss coefficientGBN (G?del-Bernays-von Neumann system) GCD (Greatest Common Divisor) generalized Petersen graphgenerating functiongenerating proceduregeneratorgenerator matrixgeneric elementgenusgirthG?del completeness theoremgolden section numbergraceful graphgraceful tree conjecturegraphgraph of first class for edge coloring graph of second class for edge coloring graph rankgraph sequencegreatest common factorgreatest elementgreedy algorithmGrelling paradoxGr?tzsch graphgroupgroup codegroup of graphgrowth of functionHajós conjectureHamilton cycleHamilton graphHamilton pathHarary graphhash functionHasse diagramHeawood graphheightHerschel graphhom functorhomemorphism homomorphism homomorphism image homomorphism of graph hyperoctahedronhypothelical syllogism hypothesis (premise)idealidempotentidentityidentity functionidentity natural transformation imageimbeddingimmediate predcessor immediate successorimpossible eventincidentincident axiomincident matrixinclusion and exclusion principle inclusion relationindegreeindependentindependent number independent setindependent transcendental element indexindirected method H Iindividual variableinduced subgraphinfinite extensioninfinite groupinfinite setinitial endpointinitial objectinjectioninjection functorinjective (one to one mapping) inner faceinner neighbour setinorder searchintegral domainintegral subdomaininternal direct sum intersectionintersection of graph intersection operation intervalinvariant factorinvariant factor idealinverseinverse limitinverse morphisminverse natural transformation inverse operationinverse relationinversioninvertableinvolution property irreflexiveisolated vertexisomorphic categoryisomorphismisomorphism of graphjoinjoin of graphJ Jordan algebraJordan product (anti-commutator)Jordan sieve formulaj-skewjuxtapositionk-chromatic graphk-connected graphk-critical graphk-edge chromatic graphk-edge-connected graphk-edge-critical graph Kanaugh mapkernelKirkman schoolgirl problem Klein 4 groupKonisberge Brudge problem Kruskal's algorithm Kuratowski theoremlabeled graphLah numberLatin rectangleLatin squarelatticelattice homomorphismlawLCM (Least Common Multiple) leader cosetleast elementleafleast upper boundleft (right) identityleft (right) invertible element left (right) moduleleft (right) zeroleft (right) zero divisorleft adjoint functorleft cancellableleft cosetlengthlexicographic orderlLie algebraline- grouplinear array (list)linear graphlinear order (total order)K Llinear order set (chain)logical connective logical followlogically equivanlent logically implies logically valid loopLucas numbermagicmany valued proposition logic map coloring problem matchingmathematical structure matrix representation maximal element maximal idealmaximal outerplanar graph maximal planar graph maximum flow maximum matching maxtermmaxterm normal form (conjunctive normal form)McGee graph meetMenger theorem Meredith graph message word mini term minimal -connected graph minimal polynomial minimal spanning tree Minimanoff paradox minimum distance Minkowski summinterm (fundamental conjunctive form)minterm normal form (disjunctive normal form)M?bius function M?bius ladder M?bius transform (inversion)modal logic modelmodule homomorphismMkmoduler latticemodulusmodus ponensmodus tollensmodule isomorphismmonic morphismmonoidmonomorphismmorphism (arrow)M?bius functionM?bius ladderM?bius transform (inversion)multigraphmultinomial coefficientmultinomial expansion theoremmultiple-error-correcting codemultiplication principlemutually exclusivemultiplication tablemutually orthogonal Latin squareN n-ary operationn-ary productn-ary tree (n-tree)n-tuplenatural deduction systemnatural homomorphismnatural isomorphismnatural transformationnearest neighbernegationneighbour setnext state transition functionnon-associative algebranon-standard logicNorlund formulanormal formnormal modelnormal subgroup (invariant subgroup)n-relationnull graph (discrete graph)null objectnullary operationobjectodd permutationoffspringone to oneone-to-one correspondence (bijection) onto optimal solutionorbitorderorder (lower order,same order) order ideal order relationordered pairOre conditionorientationorthogonal Latin square orthogonal layoutoutarcoutdegreeouter faceouter neighbourouterneighbour setouterplanar graphpancycle graphparallelismparallelism classparentparity-check codeparity-check equationparity-check machineparity-check matrixpartial functionpartial ordering (partial relation) partial order relation partial order set (poset)partitionpartition number of integerpartition number of setPascal formulapathperfect code O Pperfect t-error-correcting code perfect graph permutationpermutation grouppermutation with repetation Petersen graphp-graphPierce arrowpigeonhole principleplanar graphplane graphPolish formPólya theorempolynomailpolynomial codepolynomial representation polynomial ring positional treepossible worldpostorder searchpower functorpower of graphpower setpredicateprenex normal formpreorder searchpre-ordered setprimary cycle modulePRIM's algorithmprimeprime fieldprime to each otherprimitive connectiveprimitive elementprimitive polynomialprincipal idealprincipal ideal domainprinciple of dualityprinciple of mathematical induction principle of redundancy probabilisticprobability (theory)productproduct categoryproduct partial orderproduct-sum formproof (deduction)proof by contraditionproper coloringproper factorproper filterproper subgroupproperly inclusive relationproposition (statement)propositional constantpropositional formula (well-formed formula,wff) propositional functionpropositional variablepseudocodepullbackpushoutquantification theoryquantifierquasi order relationquaternionquotient (difference) algebraquotient algebraquotient field (field of fraction)quotient groupquotient modulequotient ring (difference ring , residue ring) quotient set Ramsey graph Ramsey number Ramsey theorem rangerankreachability reconstruction conjecture recursive redundant digits reflexiveregular expression regular graph R Qregular representationrelation matrixrelative setremainderreplacement theoremrepresentationrepresentation functorrestricted proposition formrestrictionretractionreverse Polish formRichard paradoxright adjoint functorright cancellableright factorright zero divisonringring of endomorphismring with unity elementR-linear independencerooted treeroot fieldrule of inferenceRussell paradoxS sample spacesatisfiablesaturatedscopesearchingsectionself-complement graphsemantical completenesssemantical consistentsemigroupseparable elementseparable extensionsequencesequentsequentialSheffer strokesiblingssimple algebraic extensionsimple cyclesimple extensionsimple graphsimple pathsimple proposition (atomic proposition) simple transcental extension simplicationsinkslopesmall categorysmallest element Socrates argument soundness (validity) theorem sourcespanning subgraph spanning treespectra of graphspetral radiussplitting fieldsquare matrixstandard modelstandard monomil statement (proposition) Steiner tripleStirling numberStirling transformstrong induction subalgebrasubcategorysubdirect product subdivison of graph subfieldsubformulasubdivision of graph subgraphsubgroupsub-modulesubmonoidsublatticesubrelationsubringsub-semigroup subscript。

WhatisLogic（什么是逻辑）

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;。

高级英语第二册paraphrase原句+译句(可编辑修改word版)

高级英语第二册paraphrase原句+译句(可编辑修改word版)

lesson 21.The burying –ground is merely a huge waste of hummocky earth,like a derelict building-lot.The burying-ground is nothing more than a huge piece of wasteland full of mounds of earth looking like a deserted and abandoned piece of land on which a building was going to be put up.2.All colonial empires are in reality founded upon that fact.All the imperialists build up their empires by treating the people in the colonies like animals (by not treating the people in the colonies as human beings).3.They rise out of the earth,they sweat and starve for a few years,and then they sink back into the nameless mounds of the graveyard.They are born. Then for a few years they work, toil and starve. Finally they die and are buried in graves without a name.4.A carpenter sits crosslegged at a prehistoric lathe,turning chair-legs at lightning speed.Sitting with his legs crossed and using a very old-fashioned lathe, a carpenter quickly gives a round shape to the chair-legs he is making.5.Instantly,from the dark holes all round,there was a frenzied rush of Jews.Immediately from their dark hole-like cells everywhere a great number of Jews rushed out wildly excited.6.…every one of them looks on a cigarette as a more or less impossible luxury.Every one of these poor Jews looked on the cigarette as a piece of luxury which they could not possibly afford.7.Still,a white skin is always fairly conspicuous.However, a white-skinned European is always quite noticeable.8.In a tropical landscape one’s eye takes in everything except the human beings.If you take a look at the natural scenery in a tropical region, you see everything but the human beings.9.No one would think of running cheap trips to the Distressed Areas.No one would think of organizing cheap trips for the tourists to visit the poor slum areas (for these trips would not be interesting)．10.…for nine-tenths f the people the reality of life is an endless,back-breakiing struggle to wringa little food out of an eroded sold.life is very hard for ninety percent of the people．With hard backbreaking toil they can produce a little food on the poor soil．11.She accepted her status as an old woman,that is to say as a beast of burden.She took it for granted that as an old woman she was the lowest in the community，that。