INDUCTIVE REASONING IN THE JUSTIFICATION OF THE RESULT OF ADDING TWO EVEN NUMBERS

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2022年考研考博-考博英语-中国社会科学院考试全真模拟易错、难点剖析AB卷(带答案)试题号：100

2022年考研考博-考博英语-中国社会科学院考试全真模拟易错、难点剖析AB卷（带答案）一.综合题(共15题)1.单选题Russia’s economy — until recently one of the fastest growing in Europe — is in dire straits. Traditional industries such as steel are hurting badly. The decade-long consumer boom has turned into a slump as unemployment soars. The government has cash to spend after years of sensible budget policies, but the central bank will be forced to keep interest rates high as long as inflation is stuck in double digits and trust in the ruble remains shaky.The reversal in Russia’s economic fortunes is particularly pain ful. Since 1998 — the year of Russia’s last financial crisis —the economy has expanded eight-fold. As oil prices rocketed, so did the country’s self-confidence. Not content with presiding over the economic boom, the President Vladimir Putin vowed to resto re his country’s great power status. Talks about a partnership with the West gave way to belligerent statements about a new Cold War. In the summer of 2008, Russian tanks trundled into Georgia. In early 2009, a dispute with neighboring Ukraine led Russia to cut off gas flows, leaving people in some European Union countries freezing and factories idle. Most Europeans want to see Russia stable and well-off. But they also believe that the economic crisis might bring opportunities for a political rapprochement. Some hope that the recession might just make the Russian leadership a little more humble or at least trigger reforms that would make it easier for the E. U. to strengthen trade and investment links.But while Russia’s relations with the U. S. have been th awing since Barack Obama took over the White House, E. U.-Russia relations remain frosty. Talks about a new bilateral treaty on political and economic cooperation have made little headway. Hopes for a free trade agreement between Brussels and Moscow have withered after Russia put its application for membership in the World Trade Organization on ice. E. U. -Russia energy cooperation remains stuck, which increases the risk of yet another gas crisis. Europeans have responded to Moscow’s ideas about constructin g a “new European security architecture” with a distinct lack of enthusiasm. Most importantly, perhaps, Russia is incensed about E. U. efforts to draw the countries that lie between the E. U. and Russia closer into its orbit. Russia has traditionally regarded Ukraine, Belarus, Moldova and other former Soviet states along its border as its “privileged sphere of influence”,in the words of President Dmitry Medvedev. The E. U.’s new “Eastern Partnership” initiative, launched in May, offers these countries econo mic integration and stronger political ties. Although the E. U. has shied away from talking about the prospect of membership,however distant, it hopes to help its eastern neighbors to become richer, more stable and more democratic. This would leave them better equipped to withstand Russian meddling and bullying.Moscow is particularly unhappy about the E. U.’s offer to include Belarus — traditionally a staunch Moscow ally —in the Eastern Partnership, albeit on the condition that Minsk improve its shoddy human-rights record. When the E. U. recently offered a multibillion-dollar loan to help modernize the Ukrainian pipeline system — conduit for 70% of Russian gas sales to Europe 一 Russian leaders were furious. Moscow has also tried to foil European attempts to build stronger energy links with Azerbaijan. Potential for conflict exists in Georgia, where E. U. observers are the only ones left after Russia force Organization for Security and Cooperation in Europe and United Nations’ monitors to leave Abkhazia and South Ossetia. Wary of ceding influence to Europe, the Russians have stepped up efforts to maintain their traditional fighting weight in the region. They have given large loans to neighbors hit by the economic crisis and sought to strengthen regional security and economic organizations that tie their neighbors closer to Moscow. They have also taken a more hands-on approach to “frozen conflicts” in Moldova and the Caucasus to keep neighboring governments on their toes.E. U. officials like to insist that its eastern policy does not clash with Russian interests in their common neighborhood. They have asked Russia to take part in some regional initiatives such as an effort to strengthen energy security. So far, though, Russia has refused to play ball. But the E. U. cannot simply pull back and allow Russia to dominate Eastern Europe. It must stick firmly to its objective of helping its neighbors to decide their own destiny. If Europe is to remain credible, there is no other course worth pursuing.1.Which of the following titles would best describe this article?2.Which of the following words best describes the tone of the passage?3.For the author, which of the following adjectives best describes President Putin’s attitude?4.What does the phrase “on their toes” m ean?5.For the author, which of the following should be considered a top priority to ensure peace and stability in Eastern Europe?问题1选项A.Europe and Russia’s Continental Rift.B.Russia’s Geopolitical Role.C.Financial and Economic Mayhem in Russia.D.Crisis Averted between Russia and E .U.问题2选项A.Argumentative.B.Satirical.C.Objective.D.Critical.问题3选项A.Diplomatic.B.Pugnacious.C.Pusillanimous.D.Infantile.问题4选项A.To render neighboring governments impotent.B.To weaken the resolve of the neighboring governments.C.To keep the neighboring governments on a state of constant alert.D.To gain the support of the neighboring governments.问题5选项A.The E.U. should acknowledge Russia’s pan European initiatives.B.Russia’s new se curity and energy initiatives will foster pan European cooperation.C.Russia must agree to promote bilateral, multilateral and regional economic cooperation.D.E.U. should acknowledge Russia’s pan European initiatives.B.Russia’s new security and energy initiatives will foster pan European cooperation.C.Russia must agree to promote bilateral, multilateral and regional economic cooperation.D.E.U. should acknowledge Russia’s pan European initiatives.B.Russia’s new security and energy initiatives will foster pan European cooperation.C.Russia must agree to promote bilateral, multilateral and regional economic cooperation.D.E.U. should acknowledge Russia’s pan European initiatives.B. Russia’s new security and energy initiatives will foster pan European cooperation.C. Russia must agree to promote bilateral, multilateral and regional economic cooperation.D.E.U. should acknowledge Russia’s pan European initiatives.B. Russia’s new security and energy initiatives will foster pan European cooperation.C. Russia must agree to promote bilateral, multilateral and regional economic cooperation.D. Bilateral contacts between Russia and individual E.U. member states reinforce rather than undermine common E .U. objectives.【答案】第1题:A第2题:A第3题:B第4题:C第5题:C【解析】第1题:1.主旨大意题。

考研作文英语真题范文

考研作文英语真题范文The Importance of Critical Thinking in the Modern AgeIntroduction:In today's fast-paced and information-driven world, critical thinking skills have become more crucial than ever. This essay aims to explore the significance of critical thinking and its impact on personal and professional development, as well as its contribution to society as a whole.Development of Critical Thinking Skills:Critical thinking involves actively and skillfully analyzing, evaluating, and synthesizing information to make reasoned judgments and decisions. It goes beyond memorization of facts and encourages individuals to think independently, question assumptions, and challenge existing knowledge. The development of critical thinking skills should start at an early age and be nurtured throughout one's educational journey.Benefits of Critical Thinking in Personal Life:1. Problem-solving: Critical thinking equips individuals with the ability to identify and solve problems efficiently. It enables them to approach challenges with logical reasoning and creativity, finding innovative solutions that benefit their personal lives and the lives of others.2. Effective communication: Critical thinking encourages the development of strong communication skills. Individuals can express their thoughts and ideas clearly, while also actively listening to others and appreciating diverse perspectives. This fosters better relationships, moreconstructive conversations, and a deeper understanding of different cultures and backgrounds.3. Decision-making: With strong critical thinking skills, individuals are better equipped to make informed decisions. They can consider various factors, weigh different options, and anticipate potential outcomes. This leads to more effective decision-making in all aspects of life, from career choices to everyday choices.Importance of Critical Thinking in Professional Life:1. Workplace problem-solving: Critical thinking is highly valued in the professional world. It enables individuals to identify and solve complex problems, which is essential in a constantly evolving and competitive job market.2. Adaptability: Critical thinking promotes adaptability in the face of change. It allows individuals to quickly analyze new situations, process information, and adjust their strategies accordingly. This is particularly important in industries that are rapidly evolving, such as technology and healthcare.3. Innovation and creativity: Critical thinking encourages individuals to think outside the box, fostering innovation and creativity. This is especially valuable in industries that require fresh ideas and solutions to stay ahead.Contribution of Critical Thinking to Society:1. Informed citizenship: Critical thinking skills enable individuals to be active and informed citizens. They can evaluate political decisions, mediamessages, and social issues critically, ensuring that they make informed choices and contribute to a well-functioning society.2. Scientific progress: Critical thinking is essential in scientific research and discovery. It allows scientists to question existing theories, design experiments, and make groundbreaking discoveries that advance human knowledge and improve our lives.3. Problem-solving on a global scale: Addressing global issues, such as climate change, poverty, and healthcare, requires critical thinking on a societal level. It enables individuals and organizations to collaborate, analyze problems, and develop sustainable solutions that benefit all.Conclusion:In conclusion, critical thinking plays a vital role in personal and professional development, as well as in advancing society as a whole. Its benefits in problem-solving, decision-making, and communication make it an invaluable skill in the modern age. It is crucial that educators and institutions prioritize the development of critical thinking skills to equip individuals with the tools they need to thrive in an ever-changing world.。

INDUCTIVE REASONING IN MATHEMATICS

INDUCTIVE REASON I NG I N MATHEMAT CSF. Malloy Brown a:Department o f A r t i f i c i a l In t e l l i g e n c eU n i v e r s i t y o f EdinburghScotlandA b s t r a c tWe i n v e s t i g a t e s e v e r a l methods of i n d u c t i v ereasoning i n the domain o f d i f f e r e n c e e q u a t i o n s ,i n c l u d i n g the method o f g e n e r a l i z a t i o n w i t h b e l i e f s , the method of successive r e f i n e m e n t , and temporal methods based on comparisons w i t h p r e v i o u s l y solved problems.1. I n t r o d u c t i o nWe begin a study of i n d u c t i v e reasoning inmathematics. I n d u c t i v e reasoning in mathematicsd i f fe r s from i n d u c t i v e reasoning i n the e m p i r i c a lsciences i n t h a t t h e r e i s a n u l t i m a t e t e s t althoughnot n e c e s s a r i l y a d e c i s i o n procedure, which can beused to determine what is a c o r r e c t i n d u c t i o n.Namely, t h a t i f what i s induced i s t r u e , t h a t i s :if it can be deduced by one's deductive systemthen i t must b e c o r r e c t. Because o f t h i s r e l a t i o n s h i p between i n d u c t i o n and deduction one of ourgoals i s t o c l a s s i f y what instances o f reasoningare i n d u c t i v e and what are d e d u c t i v e.Our main goal however, is to create a taxonomy of f e a s i b l e i n d u c t i v e methods. Such a goal is not u n r e l a t e d to the work of the mathematician: G. Polya,[1967, 1968A, 1968B], i t i s j u s t more d e t a i l e d. I n p a r t i c u l a r , we t r y toc a r r y out the a n a l y s i s of each method to a le v e l d e t a i l e d enough so as to beprogrammed. A l s o , where necessary, we r e l a t e thei n d u c t i v e methods to the l e v e l and c a p a b i l i t i e s ofcontemporary deductive systems. Due to t h i s r e quirement f o r d e t a i l we s h a l l r e s t r i c t our a t t e n t i o n t o a p a r t i c u l a r , but s i g n i f i c a n t domain.1.1 Our Problem DomainWe consider the problem of t r y i n g to f i n d byi n d u c t i v e reasoning a closed-form s o l u t i o n , t h a t is an a l g e b r a i c s o l u t i o n , to a r e c u r s i v e f u n c t i o n.That i s , from a set K of equations of the f o r m *:ψk (n ,f n ,f (n-h) ,f (n-2h) . . .) =0 or r a t h e r :fn = ψk (n ,f (n -h ),f (n -2h ),...) we wish to f i n d an equation of the form: Vn fn - øn where f does n o t occur in ø. For example, given the r e c u r s i v e equation f o r the Fibonacci f u n c t i o n : F(n+2) = F(n+1)+Fn F l = 1 Fo = 0 We would l i k e to f i n d a theorem of the form: Vn Fn = øm where øn is a sentence c o n s t r u c t e d from a l g e b r a i c symbols such as numerals, p l u s (+), times (*), power (), minus (-), d i v i s i o n (/), l o g a r i t h m (I n ), * Our convention f o r p a r e n t h e s i z i n g complex subexpressions i s t o p l a c e the l e f t parenthesis bef o r e the f u n c t i o n symbol as is done in EVAL-LISP. Thus we w r i t e (F n) and (a + b) not F(n) and (a) + (b ). S o c i a l i z e d S 8 Sten-Ake Tarnlund Dept. of I n f o r m a t i o n Processing & C omputer Science U n i v e r s i t y o f Stockholm, Royal I n s t i t u t e of Technology, Sweden. sine (sin) and cosine (c o s ).Or as another example, given the r e c u r s i v eequations f o r the minimum number of moves t h a t must be made in the Tower of Hanoi puzzle of n d i s c s (a d e s c r i p t i o n of t h i s puzzle may be found in Luger [19761.)H(n+1) = 2(Hn)+l H O = 0we would l i k e to f i n d a closed form s o l u t i o n : Vn Hn = øn.1.2 A d e f i n i t i o n of "I n d u c t i v e Reasoning" O f course i f we, o r r a t h e r our program a l ready "knew" a closed form s o l u t i o n øn f o r a r e c u r s i v e f u n c t i o n such as F or H then we would not want to c a l l the method by which t h a t øn was p r o duced: "i n d u c t i v e r e a s o n i n g ". The problem then of d e f i n i n g j u s t which methods of producing øn areexamples of i n d u c t i v e r e a s o n i n g , and which are not - resides in the question as to what it means f o r a system to "know" something. For example if the system had;Vn Hn = øne x p l i c i t l y s t o r e d as an axiom we would c l e a r l y say t h a t the system knew t h a t Vn Hn = øn. F u r t h e r more, if the system could apply a sequence of items (i.e. axioms, lemmas, axiom schemas and lemma schemas, w r i t t e n say in LISP) which transformed Hn i n t o øn, then if the system knew t h a t each a p p l i c a t i o n of an item in t h a t sequence produced a c o r r e c t r e s u l t , then again we would say t h a t the system knew t h a t (Vn Hn = øn) . So f a r t h i s is an omnipotent sense of knowledge: We know whatever is d e d u c i b l e. We would l i k e to modify i t b y the f u r t h e r requirement o f f e a s i b i l i t y. That i s , any sequence of a p p l i c a t i o n of items must not be so long as to exhaust the system's resources o r our p a t i e n c e. I n regard t o f e a s i b i l i t y , i t i s p o i n t e d out t h a t we d o n 't s t r i c t l y speaking have to "know" t h a t each a p p l i c a t i o n of an item is c o r r e c t , (f o r a f t e r a l l , p r o o f s from axioms o n l y are very l o n g ), but r a t h e r a l l t h a t i s needed i s the p o s s i b i l i t y o f "knowing" t h a t our items are c o r r e c tin the sense of an e x t e n s i b l e deductive system (Brown [1976A]), coupled w i t h the "b e l i e f "* t h a t our items are c o r r e c t in at l e a s t the sense of never having been r e f u t e d (i.e. shown to be i n c o r r e c t ) (Popper[1968]) . In summary then we w i l l c a l l any method of producing a closed-form s o l u t i o n øn, which is not "known" to be the c l o s e d -f o r m s o l u t i o n , i n d u c t i v e reasoning. For example, enumeration o f a l l possi b l e a l g e b r a i c f u n c t i o n s or random guessing wouldcount as i n d u c t i v e reasoning (a l b e i t poor i n d u c t i v e * We speak of "knowledge" as t r u e "b e l i e f ". The p o i n t is t h a t although we hope t h a t our deductive system is sound, t h e r e does not seem to be anyway t o a b s o l u t e l y guarantee t h i s f a c t.s t e m s -4: Brown4reasoning) because as items of a deductive system they would q u i c k l y be r e f u t e d by producing f a l s e r e s u l t s.2. Elementary I n d u c t i v e MethodsWe f i r s t d e s c r i b e two basic methods of i n d u c t i v e r e a s o n i n g , and then compare them.2.1 The Method of G e n e r a l i z a t i o n w i t h B e l i e f sOne popular theory of i n d u c t i v e reasoning (Meltzer [1969], P l o t k i n [1969, 1971],Popplestone [1969], Feldman[1969], Hardy[1976]) suggests t h a t i t i s b a s i c a l l y g e n e r a l i z a t i o n.* C onsider f o r example the Tower of Hanoi problem. Using the r e c u r s i v e equations we can deduce closed-form s o l u t i o n s f o r numerical instances o f Hn. H O = 0HI = HO+1 = 2.04-1 = 1 H2 - 2H1+1 = 2.1+1 = 3 H3 - 2H2+1 = 2.3+1 = 7 H4 = 2H3+1 =2.7+1-15 H5 - 2H4+1 = 2.15+1 = 31 H6 - 2H5+1 = 2.31+1 = 63I suspect the reader has already induced a c l o s e d -form s o l u t i o n f o r Hn, b u t l e t us i n v e s t i g a t e how t h i s might be done. FromH0=0 Hl=1 H2=3 H3=7 H4=15 H5=31 H6=63 we wish to f i n d * *to the elements of each sequence. For example,in the case of Hn we might s u b t r a c t from each element of the b e l i e f sequence, the corresponding element generated by Hn, o b t a i n i n gThis would give us the know-and v i a our b e l i e f , we induce t h a t : Hn = 2n - 1which i n f a c t i s t r u e.There i s , however, another i n d u c t i v e method which, o t h e r than f o r p o s s i b l y the t r i g g e r i n g o f the b e l i e f t h a t the closed form s o l u t i o n might i n volve an e x p e n e n t i a l f u n c t i o n of base 2, does not r e a l l y need to produce those instances of the Hn f u n c t i o n. Nor does it need to go through any g e n e r a l i z a t i o n s t e p s. We c a l l t h i s method, the Method of Successive Refinement.2.2 The .Method of Successive RefinementThe basic idea of the Method of Success Refinement is to make an i n i t i a l (i n c o r r e c t ) guess as to what is the closed form s o l u t i o n , l e t a theorem prover t r y to prove t h i s guess, and when i t f a i l s t o prove i t , g o back and t r y t o modify each branch of the p r o t o c o l on which n had been produced. For example, l e t us suppose t h a t we b e l i e v e a s o l u t i o n to Hn might i n v o l v e an exponenti a l f u n c t i o n to a power of two:(B e l i e f Hn i n v o l v e s 2n )I n o t h e r words, l e t u s f i r s t form the hypothesis t h a t ;Hn - 2nand see what a deductive system w i l l do to t h i s e x p r e s s i o n. I n d u c t i n g on n we g e t *:Already we are in t r o u b l e as the deductive system shows t h a t our hypothesis is f a l s e , on the base case. Can we modify our hypothesis to induce a b e t t e r one? Analysing our p r o o f we see t h a t if the expression 0=1 were replaced by 0 = 1 - 1, then o and not n would be produced. This means t h a t HO ■ 1 would have to be HO = 1 - 1,HO = 2 would have to be HO = 2 - 1, and Hn = 2n would have to be hn = 2n - 1.Giving our new hypothesis to a deductive system we g e t :* ■ is our symbol f o r t r u e ; a is our symbol f o rf a l s e , and >- is our symbol f o r i m p l i c a t i o n. The l e f t branch i s the base o f the i n d u c t i o n , and the r igh t branchi s the i n d u c t i o n s t e p.S p e c i a l i z e d Systems-U: BrownF i r s t l e t u s note t h a t a l e a s t general g e n e r a l -i x a t i o n method such as P l o t k i n [1969,1971] does not work because the l e a s t general g e n e r a l i z a t i o n : Furthermore,even if we i n c l u d e an a l g e b r a i c matching f a c i l i t y s i m i l a r t o t h a t used i n Feldman [1969] and Hardy [I 976]***w h i c h allows us to t r y to r e l a t e the arguments o f the H n f u n c t i o n t o i t s v a l ues by a d d i n g ,m u l t i p l y i n g ,s u b t r a c t i n g ,a n d d i v i d i n g , we s t i l l would not o b t a i n a s o l u t i o n. For example, adding one to the sequence of values of Hn g i v e s : HO+1=1 Hl+1=2 H2+l=4 H3+l=8 H4+l=16 H5+l=32 H6+1=64But s t i l l the remaining l e a s t general g e n e r a l i z a t i o n i s f a l s e.So how can t h e closed form s o l u t i o n be induced?Let us suppose t h a t the i n d u c t i v e system has a v a i l able a b e l i e f t h a t any f u n c t i o n which produces the sequence o f values 1, 2, 4, 8, 16, 32, f o r the arguments O, 1, 2, 3, 4, 5, is probably equal to the e x p o n e n t i a l f u n c t i o n of base 2:(B e l i e f 1, 2, 4, 8, 16, 32 is probably 2n )Then given t h i s b e l i e f a system might t r y to compare t h e sequence 1, 2, 4, 8, 16, 32 to the s e quence of values g i v e n by the r e c u r s i v e f u n c t i o n by a p p l y i n g v a r i o u s a l g e b r a i c o p e r a t i o n s p a i r w i s e * We consider analogy to be an e s s e n t i a l component of the method of g e n e r a l i z a t i o n as here d e s c r i b e d , and hence do not d e f i n e a separate: method of analogy.** V i s our symbol f o r a l l .*** In our case, however, we work d i r e c t l y on the numbers themselves r a t h e r than a p p l y i n g a l g e b r a i c o p e r a t i o n s to the number of occurrences of symbols such a s : the successor symbol.which we deduce to be t r u e.We see then t h a t we can o b t a i n a proof by modi f y i n g a previous unsuccessful attempt at p r o v i n ga theorem. This leads n a t u r a l l y to the questionas to what is the space of a l l such m o d i f i c a t i o n s? Although, p o t e n t i a l l y t h i s space may be q u i t e l a r g e, we w i l l o n l y consider two simple m o d i f i c a t i o n s:1) a d d i t i o n of a constant to an equation in ordert o make i t t r u e2) m u l t i p l i c a t i o n of a constant to an a d d i t i v e p a r to f a n equation i n order t o make i t t r u e.We have j u s t seen an example of (1) in o b t a i n i n g as o l u t i o n to the Hanoi f u n c t i o n. An example of(2) a p p l i e d to the f a l s e equation1-/5 would b e t o m u l t i p l y /5 b y 1//5and an example of (2) a p p l i e d to an a d d i t i v e p a r tof the f a l s e equation0 - 1 + /5would be to m u l t i p l y the a d d i t i v e p a r t 1 by - 1//5 Note t h a t m o d i f i c a t i o n s are only made to ther i g h t hand side of an equation. The reason f o rt h i s i s t h a t t h a t side contains the closed-forms o l u t i o n whereas the other side merely containsthe r e c u r s i v e f u n c t i o n.The reader w i l l see in the method of successi v e refinement an echo of M e l t z e r's [1969] hypot h e s i s t h a t i n d u c t i v e reasoning i s inverse deduct i o n. But in our case, t h i s is not so much a semantical hypothesis: if A->B then A may be i nduced from B, as it is a n o t i o n of search s t r a t e g y in the sense t h a t if B is deduced from A by a p p l yi n g the i t e m* A -> B to A, then A is induced fromB by a p p l y i n g the item in the opposite d i r e c t i o n.We have seen how the closed-form s o l u t i o n f o r Hn may be obtained from the b e l i e f t h a t the s o l ut i o n i n v o l v e s an exponential f u n c t i o n of base two. Note t h a t t h i s b e l i e f i s stronger than what i s necessary, and t h a t the b e l i e f t h a t some a r b i t r a r y exponential f u n c t i o n a is i n v o l v e d s u f f i c e s: That i s, i f 2 i s replaced b y a i n the previous p r o t o c o l s,e s s e n t i a l l y the same r e s u l t w i l l be o b t a i n e d.2.3 C omparison of the two MethodsSo f a r, we have found t h a t simple g e n e r a l i z at i o n methods are not s u f f i c i e n t and t h a t s o p h i s t icated b e l i e f s seem to be used in i n d u c t i v e reasoni n g. We have cast doubt on the suggestion t h a ti n d u c t i v e reasoning i s o n l y g e n e r a l i z a t i o n b y des c r i b i n g another technique, the method of successi v e refinement, which seems to be more powerful (because it needs a weaker b e l i e f to produce thes o l u t i o n).One advantage of the g e n e r a l i z a t i o n method is the p o s s i b i l i t y o f t r i g g e r i n g the 2n b e l i e f b y a n algebraic matching of the 1, 2, 4, 8, 16, 32, sequence to the sequence generated by the Hanoif u n c t i o n: Hn. A second advantage i s t h a t i t i s*<-> i s our symbol f o r i f f (i f and o n l y i f). q u i t e easy to e x p l a i n how the b e l i e f, t h a t a f u n ct i o n whose i n i t i a l instances are 1, 2, 4, 8, 16,32, is the exponential of base 2:(B e l i e f 1, 2, 4, 8, 16, 32 is probably 2n) nwas produced by simply i n s t a n t i a t i n g n in 2 to successively 0, 1, 2, 3, 4, 5 whereas it does not seem q u i t e as easy to e x p l a i n the p r o d u c t i o n ofthe i n i t i a l hypothesis t h a t Hn i n v o l v e d some exp o n e n t i a l f u n c t i o n:(B e l i e f Hn i n v o l v e s a n)Are we f o r example to b e l i e v e t h a t the closed-forms o l u t i o n of every r e c u r s i v e f u n c t i o n i n v o l v e s an exponential f u n c t i o n or what?Although i t i s easier t o e x p l a i n the o r i g i n sof a b e l i e f of the form:(B e l i e f 1, 2, 4, 8, 16, 32 is probably 2n)than of the form:(B e l i e f Hn i n v o l v e s a )we note t h a t if a r e c u r s i v e f u n c t i o n is not a c t ua l l y equal to a simple a l g eb r a ic f u n c t i o n such as2 then it is q u i t e improbable t h a t the necessaryb e l i e f of t h i s f i r s t form could be a v a i l a b l e to be used in o b t a i n i n g the closed-form s o l u t i o n. Wew i l l now give an example of such a f u n c t i o n and suggest t h a t the closed-form s o l u t i o n s of most r ec u r s i v e f u n c t i o n s are o f t h i s c h a r a c t e r. This example i s i n t e r e s t i n g because i t supports the viewt h a t i n d u c t i v e reasoning based o n l y on the methodo f g e n e r a l i z a t i o n even w i t h s o p h i s t i c a t e d b e l i e f s,is not s u f f i c i e n t, and t h a t other methods are also needed.Our example is the Fibonacci f u n c t i o n. Using i t s r e c u r s i v e equations we deduce:FO = 0F1 = 1F2 = F l + FO = 1 + 0 = 1F3 = F2 + F l = 1 + 1 = 2F4 = F3 + F2 = 2 + 1 = 3F5 = F4 + F3 - 3 + 2 = 5F6 = F5 + F4 = 5 + 3 = 8F7 = F6 + F5 = 8 + 5 = 13Thus fromF0=0 F l=l F2-1 F3=2 F4=3 F5=5 F6=8 F7=13we wish to f i n d:Fn = ?But there is no simple a l g e b r a i c f u n c t i o n which produces a n i n i t i a l sequence anything l i k e 0, 1,1,2, 3, 5, 8, 13. In other words there could notbe any b e l i e f of the form:(B e l i e f a1....a m is probably a )which would be h e l p f u l in s o l v i n g t h i s problem.For a f t e r a l l we cannot expect to have b e l i e f sabout every a l g e b r a i c f u n c t i o n in our system; o n l ythe simple ones. We can see t h a t the closed-forms o l u t i o n i s not simple, b y a c t u a l l y e x h i b i t i n g i t. Concurrently we w i l l take the o p p o r t u n i t y to showhow t h i s closed-form s o l u t i o n could be obtainedby the method of successive refinement using theb e l i e f t h a t the s o l u t i o n i n v o l v e s two exponentialf u n c t i o n s:(B e l i e f Fn i n v o l v e s a , b )where a # 0 can b # 0.From t h i s b e l i e f we form an i n i t i a l hypothesist h a t:Fn = a n + b nand see what our deductive system w i l l do to t h i s expression. I n d u c t i n g on n t w i c e, because FnS p e c i a l i z e d Systems-4: Brown846does not recurse on Fn+1, we o b t a i n ;Working upwa rds we see t h a t the f i v e previous l i n e s in the p r o o f must ha ve been something l i k e :The i n d u c t i o n two ba se ca ses a re t h e s i s to induce a step seems to be t r u e , but the f a l s e. C an we modify our hypo-b e t t e r one?Analysing the Fl ba se ca se we see t h a t if the in (1 = /5) h d been d i v i d e d by √ 5 then ■ would have been produced. This means t h a t the basecase would have had to have looked something l i k e : F l =a F O +((b -a )//5)b F l =a -0+((b -a )/5)-l F l -0+(b -a )/√5 F l =(b -a )//5 1= (b-a )/√51= ((l -√5)/2-(l +√5)/2)/√5 1= 2√5/2√5 1-1D* In contempora ry deductive systems (Brown [1976B, 1977], Boyer [1975]) the e q u a l i t y or u n i f i c a t i o n items would end up r e p l a c i n g a l l occurrences of sa y "a " by b -1. A system used in successive r e f i n e ment should not do t h i s beca use if Fn = a n + b n is Fn+2=aFn+l+(Fn+l-aFn)band since the t h i r d l i n e i s i d e n t i c a l t o our p r e v ious t h i r d l i n e we see t h a t the i n d u c t i o n step branch o f our proof w i l l s t i l l r e s u l t i n ■.Of course we d o n 't a c t u a l l y have to analyse our proofsteps downward, as we could simply apply our deductive system to such a step and see what it does. For example, to see t h a tFn+1 = aFn +((b -a )// 5)b n r e s u l t s in = a l l we need do is to apply our deductive system to i t .We have now produced a m o d i f i c a t i o n which r e s u l t s in a on both the Fl base, and i n d u c t i o n step branches of our p r o o f. Only the FO base step r e mains. The FO base branch r e s u l t e d in o, so we know t h a t some s o r t of m o d i f i c a t i o n is what was needed to make the FO base branch r e s u l t in ■:We f i r s t check to see if the m o d i f i c a t i o n induced from the Fl base branch w i l l s u f f i c e to make t h i s branch r e s u l t in :f a l s e a might not equal b - 1, and then we would not want t o propagate t h i s f a l s i t y i n t o the branch of the proof which is the i n d u c t i o n step which may a f t e r a l l be t r u e as in t h i s example. The reason we p r e f e r to have n derived on the branch of the p r o o f which is the i n d u c t i o n base r a t h e r than the i n d u c t i o n step i s t h a t i t i s a simpler branch and w i l l b e easier t o analyse why i t produced n , i n order to induce a hypothesis.Special fzed Systems-4: Brown847i s t r u e.Replacing a and b by the a l g e b r a i c terms t h a t we have found them to be in our p r o o f we f i n d t h a t the closed form s o l u t i o n f o r the Fibonacci f u n c t i o n i s :So f a r we have been able to induce a new hypothesii which w i l l probably make the Fl-base branch of the proof r e s u l t in . But, does t h i s change the r e s u l t of we had p r e v i o u s l y obtained in the i n d u c t i o n step branch of the proof? Since i n d u c t -Thus we see t h a t although it is p l a u s i b l e t h a t wemight be able to induce t h i s s o l u t i o n by the i n d u c t i ve method of successive refinement, i t i s notvery p l a u s i b l e t h a t we could have induced t h i ss o l u t i o n by the g e n e r a l i z a t i o n method. The reason f o r t h i s , as w i l l be r e c a l l e d , is because the gene r a l i z a t i o n method would r e q u i r e p r i o r b e l i ef st h a t :And it simply does not seem p l a u s i b l e t h a t suchb e l i e f s would be a v a i l a b l e.2.4 The Supplementary Method of E x i s t e n t i a lFunctionsWe have one f i n a l p o i n t to make about theFibonacci example: I f we had a s l i g h t l y mores o p h i s t i c a t e d b e l i e f as to what the closed-forms o l u t i o n might be, then our deductive system wouldbe able to o b t a i n the s o l u t i o n in a very d i r e c tmanner. Let us suppose t h a t we have the b e l i e ft h a t the closed-form s o l u t i o n o f the Fibonacci i n volves a l i n e a r combination of two non-zero f u n c t i o n s :(B e l i e f Fn i n v o l v e s α a n + βb n )Then the hypothesis i s : = αa n +βbn A s o l u t i o n is obtained by i n d u c t i n g twice on n:This technique of using e x i s t e n t i a l f u n c t i o n s has been used by B i b e l [1976](i n our case c o n s t a n t s : a,b,α,β).Because i t i s s o powerful t h a t w e s h a l l r a i s e i t t o the s t a t u s o f a t h i r d i n d u c t i v e method which we s h a l l c a l l : The supplementary method of e x i s t e n t i a l f u n c t i o n s. I t should b e noted t h a t t h i s technique is no s u b s t i t u t e f o r the more gene r a l method of successive refinement, f o r i t s power depends on knowledge possessed by the deductive system. In the p r o t o c o l s using the method of e x i s t e n t i a l f u n c t i o n s we have assumed the a b i l i t y to solve equations of the form: f - 0f o r a n a r b i t r a r y e x i s t e n t i a l c o n s t a n t. This knowledge was not assumed when not using thi smethod. We s h a l l see more of t h i s method in sect i o n 3.2.5 Summary In summary then we have described two basici n d u c t i v e methods and one supplementary method; and have shown t h a t the e f f e c t i v e n e s s of each method depends on the a v a i l a b i l i t y of p a r t i c u l a r b e l i e f s. We have suggested t h a t the b e l i e f s needed by the g e n e r a l i z a t i o n method would o n l y be a v a i l a b l e if the r e c u r s i v e f u n c t i o n is equal to some very simple a l g e b r a i c f u n c t i o n , such as is the case w i t h the Tower of Hanoi f u n c t i o n. For most other r e c u r s i v e f u n c t i o n s such as the Fibonacci f u n c t i o n s we have suggested t h a t the b e l i e f s needed by the g e n e r a l i z a t i o n method would not be a v a i l a b l e. In s e c t i o n 3 we describe how the b e l i e f s needed by the method of successive refinement might be a u t o m a t i c a l l y produced. 3. Temporal In d u c t i v e MethodsWe now describe methods of i n d u c t i v e reasoningbased on i n f o r m a t i o n obtained from the s o l u t i o n s of p r e v i o u s l y solved problems. Such methods are c a l l ed temporal i n d u c t i v e methods. There are two temporal i n d u c t i v e methods, d i s t i n g u i s h e d by thetype of i n f o r m a t i o n on which they are based. The f i r s t temporal method is based s o l e l y on i n f o r m a t i o n obtained from the theorem which expresses a p r e v i o u s l y solved problem, whereas the second temporal method is based on i n f o r m a t i o n obtained from the p r o o f of a p r e v i o u s l y solved problem.R e f e r r i n g back to s e c t i o n 2.5 we s h a l l see in s e c t i o n 3.2 how the second temporal method may be a p p l i e d to producing the k i n d of b e l i e f s needed by the method of successive refinement.3.1 The Temporal Method based on TheoremsThe basic idea of the temporal method based on theorems is f o r any given problem F to t r y to f i n d a s i m i l a r problem H which has already been solved, and then to use something s i m i l a r to the s o l u t i o n f o r H as a hypothesis f o r the s o l u t i o n of F.For example l e t H be the Tower of Hanoi p r o b lem whose s o l u t i o n we discovered in Section 2, and l e t F be the Fibonacci f u n c t i o n whose s o l u t i o n we are t r y i n g to f i n d. The theorem which expresses the s o l u t i o n to the Hanoi problem is t h e n : (Vn H (n +l )=2(H n )+l A H O 0) -> Hn=2n -1and the (as y e t incomplete) theorem which expressesS p e c i a l i z e d Systems-4: Brown848We see t h a t t h e r e is no reasonable analogy t h a t we can make which w i l l help us to solve F. For ex-ample, if we form the analogy t h a t m is n - 1 from (Hn+1 Fm+2), then we might conjecture t h a t : Fm=2m-1 -1which is f a l s e. In any case, so much syntax has been l e f t unexplained by t h i s analogy (f o r example the 2. , +1, HO = 0, +Fm, and Fl = 1) t h a t we can have no confidence in t h i s conjecture anyway.Furthermore even if our g e n e r a l i z a t i o n system were s o p h i s t i c a t e d enough to r e w r i t e the H theorem as the e q u i v a l e n t theorem:(VnH(n+2)=2H(n+l)+l 'H l «l ^H O =0) + Hn=2n -1 (VroF(m+2)-F(ra+l)+Fm / F l -1 > FO=0) + Fm=?The c l o s e s t analogy would s t i l l not e x p l a i n the 2* , +1, and +Fm.The problem here is t h a t H and F are so d i s -s i m i l a r t h a t no immediate analogy can be made. 3.2 Temporal Method based on ProofsThe basic idea of the temporal method based on p r o o f s is as f o l l o w s :1) F i r s t using the temporal method based on t h e o r -ems f i n d a problem H which is s i m i l a r to the p r o b -lem F you are t r y i n g to solve and conjecture a s o l u t i o n f o r F.2) Using the deductive system produce the proof of H and a p r o t o c o l of F w i t h t h a t hypothesis.3) F i n a l l y compare the proof w i t h the p r o t o c o l to f i n d out why no s o l u t i o n was o b t a i n e d , and form a new hypothesis.Suppose, now t h a t we are t r y i n g to f i n d a s o l u t i o n to the Fibonacci f u n c t i o n F, given t h a t we already know the s o l u t i o n to the Tower of Hanoi f u n c t i o n H. We s h a l l use the p r o o f of the s o l u t i o n of H which i s given a s described i n s e c t i o n 2.2,s t a r t i n g w i t h : 3a Vn Hn = a n - 1The hypothesis f o r the s o l u t i o n of F is 3 a Vn Fn - a n -l(We could throw in a few e x i s t e n t i a l constants, f o r example: Fn = a.a n -3, but it won't make any d i f -ference a t t h i s stage.)Using our deductive system we o b t a i n the f o l -lowing p r o t o c o l :was f i r s t a p p l i e d. We then see t h a t the steps in the base case of proof and p r o t o c o l were the same, and t h a t the f i r s t few steps on the i n d u c t i o n step branch of the p r o o f and p r o t o c o l were the same. In f a c t we f i n d t h a t they d i f f e r e x a c t l y where H(n+1) is replaced by 2(Hn)+l H(n+l)=a(Hn)+a-lThis then is the problem. On F the theorem prover inducts because it c a n 't r e c u r s e , because F recurses on n+2 not n +1, whereas H immediately recurses to o b t a i n a s o l u t i o n.So granted t h a t F recurses on n+2 and t h a t the theorem proven inducts twice the q u e s t i o n , i s : why is not a s o l u t i o n obtained a f t e r the second induc-t i o n ? Let us compare the steps beginning w i t h the second i n d u c t i o n in F to the steps in Hn:3a Vn Hn = a n - 1 3a Vn. F(n+1) = aFn+a - 1We see t h a t Hn involves some term "a " exponentiated by the i n d u c t i o n v a r i a b l e n. Comparing Hn to Fn+1 t h i s leads us to suspect t h a t Fn+1 might also i n -volve some term exponentiated by the i n d u c t i o n term n +1. Thus we would hypothesize t h a t Fn+1 i n v o l v e s some term b or r a t h e r than Fn involves some term b . The reader should bear in mind t h a t the hypothesis is not a f o r t h i s a would clash w i t h the bound v a r i a b l e a already occuring in the above Fibonacci expression. The p o i n t is t h a t the "a " in the Hanoi expression and "a " in the Fibonacci expression are two d i s t i n c t bound v a r i a b l e s ,w h i c h do not n e c e s s a r i l y r e f e r to the same number.There are two other i n d i c a t i o n s t h a t Fn i n -volves a term of the form b . I n d u c t i n g on n ineach ca^se we get r e s p e c t i v e l y the base cases:Note t h a t in the p r o o f of H a skolem f u n c t i o n d i s -appears by having a replaced by 1 whereas in the p r o t o c o l of F no skolem f u n c t i o n disappears int h i s manner. Here then is a second i n d i c a t i o n t h a t a term of the form b or r a t h e r b is needed in the Fibonacci p r o t o c o l.F i n a l l y , by comparing the i n d u c t i o n steps of H and F we note t h a t in H the e q u a l i t y item r e -placed a by an e x p r e s s i o n , whereas in F it merely replaced "a " by an expression. Here then is a t h i r d i n d i c a t i o n t h a t Fn i n v o l v e s an expression of the form b (where b is not n e c e s s a r i l y equal to a ).We now compare t h i s p r o t o c o l w i t h the proof of Hn. We see t h a t in both cases the i n d u c t i o n p r i n c i p l eThus, f o r three reasons we are lead to the b e l i e f t h a t Fn i n v o l v e d some term b where b is not neces-s a r i l y a. We now go back and modify our o r i g i n a lS p e c i a l i z e d S ystems-4: Brown849。

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Deductive & InductiveApproachesReasoning andResearch MethodsWANG YanUIBEReasoning Methods●The main division between forms of reasoning made in philosophy is between deduction and induction.●Formal logic has been described as 'the science of deduction'.●The study of inductive reasoning is generally carried out within the field known as informal logic or critical thinking.Two basic categories of human reasoning●Deduction and induction refer to two distinctlogical processes. Deduction begins with the general and ends with the specific, whileinduction moves from the specific to thegeneral.●Deduction:reasoning from general premises,which are known or presumed to be known, to more specific, certain conclusions.●Induction:reasoning from specific cases tomore general, but uncertain, conclusions.DeductiveGeneralization/Rule →Specific Examples/ActivitiesInductiveSpecific Examples/Activities →Generalization/Rule Inductive and Deductive Reasoning ●Deductive reasoning is a movement from generalization to specific.●Inductive reasoning is a movement from specific instances to a general law or rule.Deductive vs. Inductive Reasoning Deduction ●commonly associated with “formal logic .”●involves reasoning from known premises, or premises presumed to be true, to a certain conclusion.●the conclusions reached are certain, inevitable, inescapable.Induction●commonly known as “informal logic ,” or “everyday argument”●involves drawinguncertain inferences, based on probabilistic reasoning.●the conclusions reached are probable, reasonable, plausible, believable.Deductive vs. Inductive Reasoning Deduction ●It is the form or structure of a deductive argument that determines its validity ●The fundamental property of a valid, deductive argument is that if the premises are true, then the conclusion necessarily follows.●The conclusion is said to be “entailed” in, or contained in, the premises.Induction●By contrast, the form or structure of an inductiveargument has little to do withits perceived believability or credibility, apart from making the argument seem clearer or more well-organized.●The receiver (or a 3rd party) determines the worth of an inductive argument.Inductive or deductive reasoning?● A sample of fifty motoristswho were stopped by theCHP at a sobrietycheckpoint on a Saturdayat midnight revealed thatone in four drivers wereeither uninsured,intoxicated, or both. Thus,if you get involved in anaccident on the freewaythere is a 25% chance theother motorist will be drunk or uninsured.●The Law of the Sea treatystates that any vesselbeyond a 12 mile limit is ininternational waters. Thetreaty also states that anyvessel in internationalwaters cannot be legallystopped or boarded.Therefore, when the U.S.Coast Guard interceptsboats coming from Cuba orHaiti more than 12 milesfrom the U.S. coast, it isviolating the Law of the Sea.Deductive and Inductive Arguments An argument consists of one or morepremises and one conclusion.● A deductive argument is an argument such that thepremises provide complete support for the conclusion,i.e., the conclusion is taken to necessarily follow fromthe premises●An inductive argument is an argument such that thepremises provide some degree of support (less than complete support) for the conclusion, i.e., theconclusion is taken to probably but not necessarilyfollow from the premisesDeductive and Inductive ArgumentsExample of Deduction ●Major premise:Alltortoises arevegetarians.●Minor premise:Bessieis a tortoise.●Conclusion:Therefore,Bessie is a vegetarian Example of Induction●Lily and Davidgraduated from CentralCollege. They are goodelectrical engineers.Most electrical engineer majors from CentralCollege are well trainedin their field.Exercise:Deductive or Inductive Arguments?●Tom:I’ve noticed previously that every time I kick a ball up, it comes back down, so I guess the next time when I kick it up, it will come back down, too.●Jack:That’s Newton’s Law. Everything that goes up must come down. And so, if you kick the ball up, it must come down.11Deductive Arguments●For a deductive argument, if all its premises are true, its conclusion is necessarily true (or it is logicallyimpossible for the conclusion to be false.)–i.e., the truth of premises guarantees the truth of conclusion.●e.g.:1.Cats are mammals.2.Mammals are animals.3.Therefore, Cats are animals.12Deductive Arguments ●When we talk about deductive arguments, we have already presupposed that the arguments are successful or valid deductive arguments.The conclusion of a valid argument is called a valid conclusion.●For an unsuccessful deductive argument (the premises are intended to guarantee the conclusion but fail to do so), we call it an invalid argument.● A deductive argument may be valid or invalid, there is nothing in between.13Deductive Arguments●Whether a deductive argument is valid or invalid depends on its form or structure.●The above-mentioned argument is valid because it has this valid form–All A are B.–All B are C.–Therefore, All A are C.●Any argument having that form will also be a valid argument.14Deductive Arguments●A valid argument may have false conclusion if it has false premises.●e.g.:–All cows are animals.–All animals are carnivores.–Therefore, all cows are carnivores.15Deductive Arguments ●In order to guarantee the truth of conclusion, we have to make sure all the premises are true.●When all the premises of a valid argument are true, the argument is called a “sound argument”.●And the conclusion of a sound argument is called a sound conclusion.16Inductive Argument●The following is a typical inductive argument:Swan1 is white.Swan2 is white.Swan3 is white.…Swan n is white._______________All swans are white.17Inductive Argument●The main difference between deductive arguments and inductive arguments is that, for the latter, if all its premises are true, its conclusion is likely to be true but still possible to be false–i.e., the truth of its premises makes it reasonable to hold that the conclusion is true but the content of the premises does not include (imply) thecontent of the conclusion. 18Inductive Argument ●We call a good inductive argument a strong argument, a bad inductive argument a weak argument.●Whether an inductive argument is strong or weak depends on its content, not on its structure.●Example of strong induction :–Every day to date, the law of gravity has held.Therefore, the law of gravity will hold tomorrow.●Example of Weak induction:–Many speeding tickets are given to teenagers.Therefore, all teenagers drive fast.19Inductive Argument●Even for a strong argument, if its premises are false, we still have no reasons to believe in the conclusion.●If all the premises of a strong argument are true, the argument is called a “cogentargument.”●The conclusion is called a “cogentconclusion.” 20Deductive and Inductive Arguments ●Sometimes it is difficult to conduct both deductive and inductive reasoning together.Consider thefollowing question.●The following is a description of Linda:–Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeplyconcerned with issues of discrimination and social justice,and also participated in many demonstrations.●Which of the following has a higher probability?(a) Linda works in a bank.(b) Linda works in a bank and as a volunteer in the Red Cross. Deduction vs. Induction ●Deductive reasoning is either “valid” or “invalid.” A deductive argument can’t be “sort of” valid.●If the reasoning employed in an argument is valid and the argument’s premises are true, then the argument is said to be sound .valid reasoning + true premises = sound argument ●Inductive reasoning enjoys a wide range of probability; it can beplausible, possible, reasonable, credible, etc.●The inferences drawn may be placed on a continuum ranging from cogent at one end to fallacious at the other.fallacious cogentHow to Organizean Argumentative Essay●There are two very common ways of developing anargumentative essay: One is to declare the writer’sposition upfront and then use reasons and evidenceto support it. The other method is to explore thesubject with the reader, presenting the evidence andreasons before coming to a conclusion.●The first method, which proceeds from a generalclaim before moving on to specific details, is calledthe deductive method. The second method, whichmoves from the specific details before ending in ageneral claim, is called the inductive method ofdevelopment.Body paragraph development●What is a body paragraph?–Body paragraphs are all theparagraphs between the introductionand the conclusion.–Body paragraphs support and proveyour thesis.Sentences in a body paragraph:e.g. an argumentative paragraph●A topic sentence is the first sentence in a body paragraph.●A support sentence gives a reason in support of the paragraph’s topic sentence.●A proof sentence proves a support sentence by providing a detail or quotation from a source.●A concluding sentence refers back to the topic, provides a logical closing, and may provide a transition to the next body paragraph.The McParagraph sentences Topic sentenceSupport sentence 1Proof sentence 1Support sentence 2Proof sentence 2Support sentence 3Proof sentence 3Concluding sentenceTopic Sentence●Topic sentences state the main idea of theparagraph.●The rest of the paragraph must expand on,describe, or prove what the topic sentencestates in some way.●A good topic sentence makes a point andsuggests the logical structure of the rest of the paragraph.Exercise 1:Which are good topic sentences?1.Texas has 267,000 square miles.2.Texas is so big that you can find many thingsto do.3.There are several ways of accurately tellinghow old fossils are.4.The animal dies and sinks to the sea floor.Exercise 2:Identify the topic, support, proof,and concluding sentences in thefollowing body paragraph.The political success of Lincoln’s speech -the last speech in a series sponsored by the Young Men’s Central Republican Union of New York that winter (Holtzer13) -had something to do with timing and luck. A sizable number of Republican leaders were worried that the front-running candidate, New York Senator William Henry Seward, was perceived by the Northern electorate as too close to the unpopular abolitionist movement (Holtzer32). “Lincoln’s best ally in the winter of 1860 was his lack of association with the abolitionists in the mind of New Yorkers,” according to Holtzer(32). Republicans were worried also that Seward has little appeal in the West (Illinois, Ohio, etc.) (Burris 126). Burris asserts that “Indiana and Illinois Republicans perceived Seward as an Eastern liberal” (127). Lincoln also benefited from the political machinations of the speech series’s sponsors. The Young Republicans planned the speech series ostensibly to introduce alternative candidates to Seward, but the real motivation of the group's leader, James A. Briggs, was to damage Seward enough to promote his favorite alternative, Ohio governor Salmon P. Chase (Holtzer34). The Republican party’s soul-searching and the secret motivations of the series sponsors gave Lincoln the opening he needed.Example of a Deductive Essay●Look at the outline of a deductive essay:Should cloning be encouraged?●The first and last paragraphs are given in full. Thethesis and the concluding statements are highlightedfor your attention.●After reading the outline, answer the questions below:1.Where is the thesis statement?2.What do the subsequent paragraphs do after the thesisstatement?3.How does the essay end?4.What pattern of development can you find in the aboveexample?Example of an Inductive Essay●Look at the example of an inductive essay:Is cloning a technology which brings us benefits?●After reading the outline, answer the questionsbelow:1.Where is the thesis statement?2.What do the subsequent paragraphs do after theintroduction?3.How does the essay end?4.What pattern of development can you find in theabove example?What to Watch Out For in Planning the Pattern of Essay Development●You should first decide whether to adopt a deductive or inductivepattern of presenting the arguments in the essay.–The deductive method is to present the Writer’s Position first before proceeding to prove it.–The inductive method, on the other hand, is to give the reasons first before drawing a conclusion, which is the Writer’s Position.● A common problem is not to adopt either a deductive orinductive approach–The writer does not develop the essay in a straight line, but deviates from the Writer’s Position stated either at the beginning or at the end of theessay.–The English reader generally looks for either a deductive or inductive development, any deviation from a one-line pattern makes it veryconfusing for the reader.●To be effective, all paragraphs should be logically related to theWriter’s Position.Examples of Problematic Essays●Look at the two examples of problematicessays.●Find out the problems.●Try to improve the writing.Quiz:●Answer the questions about deduction and induction.●Test your knowledge and skill of argumentative essay development. Research Methods●In research, we often refer to the two broad methods of reasoning as deductive and inductive approaches.Research TypesDeductive Approach InductiveApproach Deductive Research Approach ●Deductive reasoning works from the more general to the more specific.●Sometimes this is informally called a “top -down” approach.●Conclusion follows logicallyfrom premises (availablefacts)Theory Hypothesis Observation Confirmation WaterfallInductive Research Approach●Inductive reasoning worksthe other way, movingfrom specific observations to broader generalizations and theories.●Informally, we sometimescall this a “bottom up”approach.●Conclusion is likely basedon premises.●Involves a degree ofuncertaintyTheoryTentativeHypothesisPattern ObservationHillClimbingDeductive approach vs. Inductive approachTheory HypothesisObservationConfirmationTheoryTentativeHypothesisPattern ObservationDistinctions between Quantitative and Qualitative MethodsConcepts usually associated with quantitative methodsType of reasoningDeductionObjectivityCausationType of questionPre-specifiedOutcome-orientedType of analysisNumerical estimationStatistical inference Concepts usually associated with qualitative methodsType of reasoningInductionSubjectivityMeaningType of questionOpen-endedProcess-orientedType of analysisNarrative descriptionConstant-comparisonAssignment●Select one of your research questions, and make it a topic for an argumentative essay.●Write an outline for the topic (including the topic sentence of each paragraph), developing your essay in a deductive way.●Write an outline for the topic (including the topic sentence of each paragraph), developing your essay in an inductive way.。

Inductive Reasoning and Chance Discovery

Inductive Reasoning and Chance Discovery* AHMED Y.TAWFIKSchool of Computer Science,University of Windsor,Windsor,Canada ON N9B3P4;E-mail: atawﬁk@uwindsor.caAbstract.This paper argues that chance(risk or opportunity)discovery is challenging,from a reasoning point of view,because it represents a dilemma for inductive reasoning.Chance discovery shares many features with the grue paradox.Consequently,Bayesian approaches represent a potential solution.The Bayesian solution evaluates alternative models generated using a temporal logic planner to manage the chance.Surprise indices are used in monitoring the conformity of the real world and the assessed probabilities.Game theoretic approaches are proposed to deal with multi-agent interaction in chance management.Key words:Bayesian conﬁrmation,chance discovery,inductive reasoning1.IntroductionA chance(opportunity or risk)can be characterized as a high impact event, situation or change.Typically,these situations are rare but their eﬀects (payoﬀor loss)are so signiﬁcant that it is advantageous to discover them as early as possible to try to avert the risks and exploit the opportunities (Ohsawa,2001).The term chance discovery has been coined to refer to the process of discovering such situations using automated reasoning.From a reasoning perspective,chance discovery diﬀers from knowledge discovery. Knowledge discovery extracts common patterns from data while chance discovery predicts future outcomes.For example,forecasting the market potential for a new product represents a form of chance discovery.To illustrate how diﬃcult and how far oﬀhuman may be in discovering chance,consider the following examples:In1943,Thomas Watson,then chairman of IBM Corporation,predicted a world market for aboutﬁve computers.In1970,Ken Olsen,founder of Digital Equipment Corporation is reported to have said that no one needed to have a personal computer at home.In both cases,it was diﬃcult to assess the opportunities because traditionally,forecasting has relied on extrapolation.Extrapolation is a form of inductive reasoning that assumes that current trends would carry on into the future.Clearly,this approach does not work well with new types of products.This problem is not unique to extrapolation as a procedure but it is *The author would like to thank the reviewers for their helpful comments.This work is supported by a grant from the Natural Sciences and Engineering Research Council,Canada.Minds and Machines14:441–451,2004.Ó2004Kluwer Academic Publishers.Printed in the Netherlands.inherently a problem in inductive reasoning that is closely related to Good-man’s new riddle of induction (Goodman,1955).However,solving the problem of induction does not completely solve the chance discovery problem.The representational challenge is also as signiﬁ-cant.Generally,our knowledge representations suﬀer from functional ﬁxa-tion,thus,hiding potential opportunities and risks.Thomas Watson’s forecast implicitly assumes some function for computers that is rather lim-ited.Certainly,in 1943,the range of computer applications imagined was rather limited.The same is true for personal computers in the 1970’s.This work argues that ﬁnding the proper knowledge representation is of great importance for chance discovery.Knowledge representation and reasoning frameworks typically favor ‘the normal’and the ‘common’to the ‘rare’or ‘exceptional’.Consequently,conventional frameworks will likely miss rare situations.Moreover,for these rare situations,it is necessary to distinguish cases that represent opportunities or risks from other rare changes.To identify these situations a decision theoretic approach for assessing such rare situations is needed.The paper is organized as follows:Section 2shows that chance discovery is a practical example of grue.As such,proposed solutions to the grue paradox are surveyed for clues that may help with chance discovery.The Bayesian approach seems to hold some promise.Section 3discusses the use of entropy maximization to come up with probabilities.Section 4presents a technique for managing chance.Section 5suggests the need for chance monitoring.Section 6discusses the impact of intent on probability and utility assessment.2.Goodman’s Riddle of InductionThe color of an emerald is grue (green then blue)if it is and has always been observed green until some future time (say year 2222)when it will turn blue.This notion presents a paradox to inductive reasoning because our obser-vations support the statement that emeralds are green as well as the claim that they are grue (Goodman,1955).This paradox,ﬁrst proposed in 1955,has inspired arguments about the validity of induction.The essence of the problem lies in the inductive temporal uniformity assumption that implies that the future will look like the past.Many have contended that a correct solution would justify preferring green emeralds to grue ones.In the context of chance discovery,the correct solution would be one that minimizes grue predictions (i.e.by maximizing temporal uniformity)without missing any cases of grue (i.e.when temporal uniformity does not apply).It may be necessary to ﬁrst demonstrate that certain properties are really grue (Akeroyd,1991).The statement that gold is soluble is grue (or falue –for false then true).For a long time in history,observations supported the AHMED Y.TAWFIK 442INDUCTIVE REASONING AND CHANCE DISCOVERY443 notion that gold cannot be dissolved until the invention of regal water. Scientiﬁc discoveries have generally challenged human conception of the universe that has proved to be grue in some ways.Grue phenomena challenge the notion of temporal uniformity.In this treatment,a phenomenon is grue if it involves an unexpected change.For example,the discovery of a treatment for a previously untreatable disease is grue.Similarly,we consider the eruption of a volcano or a strong earthquake in a historically stable area as grue phenomena.These phenomena present a new challenge to automated reasoning.Therefore,chance discovery is to some extent about discovering when our experiences mislead us.It is about identifying situations when grue is true.Favoring green over grue for its simplicity is of limited relevance for two reasons.First,the deﬁnitions of green and blue in terms of grue and bleen (blue then green)are as simple as those of grue and bleen in terms of green and blue.In other words,measuring complexity by comparing message length can be an artifact of the representation.Therefore,trying to minimize message length does not necessarily resolve the problem.Minimum message length induction approximates a full Bayesian inference over the entire hypothesis space(Solomonoﬀ,1999).A better approximation is obtained if we use more terms corresponding to short codes.In chance discovery, whether the objective is to identify a risk or an opportunity,it is necessary to consider the more complex scenarios as long as they are possible no matter what measure of complexity is appropriate.Notions of simplicity such as Ockham’s Razor would always miss some chances.Preferring persistence(as in green)to change(as in grue)corresponds to the common sense law of inertia(McCarthy and Hayes,1969).The com-monsense law of inertia can be considered as a nonmonotonic circum-scriptive assumption that minimizes change(Shanahan,1997).This approach does not capture correctly many practical situations involving indigenous change or partially observable systems(Dean and Kanazawa,1989).More-over,chances will necessarily be missed.The commonsense law of inertia would prefer to assume that a volcano will not erupt,that a new product will not sell and in general that diﬀerent new conditions will not occur.Bayesian conﬁrmation theory evaluates the probabilities of green and grue at future times based on prior and conditional probabilities incorporating evidence and background information(Horwich,1982).As such the degree of belief in a particular outcome can be calculated provided an accurate theory exists to assess the probabilities and the causal dependencies.As such, Bayesian conﬁrmation theory extends the use of probabilities as a measure of belief beyond frequency interpretation to include other interpretations such as subjective probabilities and propensities.The Bayesian approach is to explicitly list all possibilities including all alternative models(all possible worlds),assess priors and conditionalprobabilities,and calculate posterior probabilities given all available obser-vations for the diﬀerent models under consideration.Therefore,the proba-bility of a statement (or conclusion)S given some evidence E is given byP ðS j E Þ¼P ðS ÞP ðE j S ÞP jP ðE j h j ÞP ðh j Þwhere the h 0j s represent all possible models consistent with E .As chance discovery is the other side of the coin,a Bayesian approach might help.Consider for example that S represents the occurrence of a strong earthquake,and that E consists of a history of seismic activity in the region including small earthquakes.A number of competing theories h i ’s are con-sistent with observations but diﬀer in future predictions.The probability of a strong earthquake can be derived provided some prior and conditional probabilities.Typically,the number of possible theories or models can be very large.This large number of possible models in any practical situation presents a challenge to Bayesian chance discovery.If Thomas Watson were to apply the above approach to assess the market for computers,he would have to consider a myriad of possible models including the one that actually happened.In hindsight,we know that the computer market became strong because computer prices went steadily down,performance increased exponentially and applications have been developed to fulﬁll a wide range of needs.This particular scenario was not very likely in 1943.Having been involved in the eﬀorts to build some of the very ﬁrst computers,Thomas Watson might have not seen the information revolution coming.Considering the complexity of using the early machines,their poor reliability,high prices,and limited performance as evidence,he predicted a very limited market for these machines.However,it is fair to assume that the need for computation in domains such as accounting,engineering design,banking,and planning,was evident even then.There may have been some speculations (theories)about ways to reduce prices and increase performance.Therefore,to discover a chance,it is important to incorporate relevant evidence within possible models.This requirement adds additional challenges to the challenge of coming up with prior and conditional probabilities usually encountered in Bayesian approaches.3.Finding the ProbabilitiesA knowledge representation suitable for chance discovery has to be able to concisely encode a possibly very large number of models (possible worlds).To achieve this representation eﬃciency,models can be grouped in AHMED Y.TAWFIK 444INDUCTIVE REASONING AND CHANCE DISCOVERY445 equivalence classes or categories such that a group of models belonging to the same class K share the same PðS;E j h kÞfor all h k in K.The formation of these equivalence classes can be based on the structural similarities of the models or on the propensity of the models to support particular evidence(Bacchus et al.,1996).For example,the probability of increased sales of computers in the future may be the same according to a theory projecting future educa-tional applications and another projecting more business applications.In this case,both theories belong to the same equivalence class.In Equation1above,PðS j EÞis inversely proportional to R i PðE j h iÞPðh iÞ. The other terms in the expression do not depend on the particular model,and are constants for any particular combination of any statement S and evidence E.However,the choice of priors Pðh iÞand conditional probability distri-butions PðE j h iÞhas to reﬂect the background knowledge or the lack thereof (ignorance).Choosing the values that maximize entropy reﬂects ignorance (Jaynes,1968).This information theoretic approach to the determination of priors has an advantage over other subjective approaches in the case of ignorance.However,entropy maximization,like many other probabilistic inference procedures,is representation dependent(Halpern and Kollar, 2004).It appears that all non-trivial probabilistic inference procedures are representation dependent to some extent.The entropy is given byXPðh iÞlog Pðh iÞEntropy¼ÀiIn the context of chance discovery,the determination of conditional proba-bilities for PðE j h iÞhas to rely on a background theory.Edis(2000)suggests that,in the absence of any background theory,the evidence E supports all competing models equally as long as they are consistent with it.In other words,if we do not have a background theories allowing us to prefer the model of green emeralds,all observations of green emeralds also support the model of grue emeralds to the same extent.This results in equal weights for all alternatives.For example,in the absence of any information to guide the assessment of probabilities,scenarios representing computers becoming more expensive are as likely as scenarios representing computers getting cheaper and so on.4.Model FormationThe treatment so far assumed the availability of three elements:a chance statement S,some related evidence E,and a set of models f h1;h2;...;h N g.All three elements are hardly ever readily available in a chance discovery context.Formally,the chance discovery problem can be represented by a Kripke structure(Kripke,1963).The structure M¼ðW;U;p;RÞrepresents a chancediscovery Kripke structure.W denotes a set of worlds.Each world is de-scribed using truth assignment p deﬁned for a set of propositions U .An accessibility relation R determines the set of worlds reachable from a par-ticular world.Each world w occurs with a probability l ðw Þ.The probability of a proposition /is given by P ðu Þ¼X w j¼ul ðw ÞA chronicle C is a path between a start world w 0and a ﬁnal world w f such that for any two consecutive worlds along the path w i and w j ,w i follows w j if and only if w j 2R ðw i Þ.The evidence set E is a subset of U such that E holds in a temporally constrained set of worlds in all chronicles.Depending on the nature of E and W ,temporal constraints may be ordering constraints over points and intervals or clock constraints.The chance states S constitute another subset of U such that a utility function U when applied to s 2S in some world(s)w 2W results in a signiﬁcant chance.In chance discovery,unlike in decision theory,utilities are assessed for worlds irrespective of their probabilities in order to detect rare chances.Each accessibility relation r 2R between a pair of worlds encodes a set of assumptions,actions,or events.A model h is the conjunction of all the assumptions actions or events along a chronicle c 2C .This deductive approach to chance discovery has some complexity and feasibility limitations.In realistic domains,it is diﬃcult to encode all possible combinations of events,actions,and assumptions as well as all their conse-quences.Moreover,as chances are rare,a chance discovery procedure that expands all future worlds would waste a tremendous amount of computa-tional resources,seldom discovering chances.However,this last observation suggests that backward chaining is a more eﬃcient solution if there is a proper characterization of risks and opportunities.Accordingly,the chance discovery process proceeds from a chance statement S .Similarly,McBurney and Parsons (2001)start with a statement and proceed with building a chance discovery dialogue between collaborative agents.The Bayesian analogue to the backward reasoning approach is to consider the probability of the model given the evidence E and the chance S .P ðh j S ;E Þ¼P ðS ;E j h ÞP ðh ÞP ðS ;E ÞThe purpose of the above equation is to measure if there is a model that explains both E and S well.The model h that maximizes the probability in Equation (4)above is the model (sequence of actions)to follow or prevent most depending on our interpretation of S as opportunity or risk respectively.Thus far,the development does not provide any insights into how to enumerate the models compatible with given S and E .Assuming that the AHMED Y.TAWFIK 446INDUCTIVE REASONING AND CHANCE DISCOVERY447 number of models satisfying the constraints imposed by S and E isﬁnite,the list h1;h2;...;h N represents these models(or equivalence classes of such models).Recent advances in planning(Bacchus and Kabanza,2000),allow us to build eﬃcient backward chaining planners that guide their search by incorporating temporal domain dependent constraints.Each plan thus gen-erated corresponds to a model.The use of planning for model formation leaves one last challenge:how to enumerate possible actions,events,and assumptions?This is a knowledge representation issue.The challenge stems from the fact that many threats and opportunities result from unusual or innovative use or interaction of previ-ously deﬁned rgely,our knowledge representations suﬀer from functionalﬁxation and it is imperative that the actions,events,and assumptions available be as general as possible.One way to achieve this generality is using object hierarchies and generic actions.For example,a saltshaker ought to be deﬁned as a rigid container that has holes and that may contain salt.Actions that may involve the saltshaker include those applicable to rigid objects such as using it to drive a nail,as well as actions for containers(e.g.ﬁlling it)and those speciﬁc to saltshakers(e.g.pouring salt).This approach may discover many common situations such as crossing a street or driving to work as risks and other daily occurrences may be labeled as opportunities.In risk and opportunity discovery,we have to assume that such common risks and opportunities have already been addressed and the focus is on discovering unusual and rare risks and opportunities.Here,the expedient of setting a threshold on PðSÞto qualify as a chance avoids this problem at the risk of missing some chances.5.Chance MonitoringAs the Bayesian approach to chance discovery relies on subjective probability assessment,it is important to monitor the chance exploitation plan execution to verify that these assessments conform to reality.Moreover,our chance discovery algorithm may still miss some chances because of an inadequate representation or limited observations.Both of these concerns can be ad-dressed using chance monitoring.Given some observations during the exe-cution of the chance exploitation plan,how to identify new chances that may come about or detect deviations preventing proper chance exploitation?For example,how to tell if a particular technology is no longer a good investment opportunity or if another new one is promising?Chance monitoring relies on surprise measures to detect deviations be-tween expected behavior and observed behavior.From a Bayesian perspec-tive,all alternative scenarios must be considered and the probabilities are used to determine potential risks and opportunities.This approach ismethodologically sound but of limited practical value because it is nearly impossible to enumerate all alternatives in a given situation.A considerable simpliﬁcation results from enumerating some common subset of alternative scenarios and using a surprise measure to detect other situations.Surprise measures reﬂect the degree of incompatibility of observed data and enu-merated models (Bayarri and Berger,1997).A particular observation is surprising if its probability is small in com-parison with the probability of other possible results (Weaver,1948).The occurrence of an event such as ‘someone won the lottery’is not surprising despite its small probability because all other alternatives are equally un-likely.However,the person who wins the lottery would be surprised because the alternative event (i.e.not winning)is much more probable.Weaver (1948)uses the ratio of the expected probability value to the probability of the observed event as a surprise measure.Surprise ¼P iP 2ðx i ÞP ðx Obs ÞThe numeric value of this surprise index is less than 1as long as the more likely event takes place.It gets higher the less likely the event is compared to the alternatives.Other surprise indices that diﬀer in their generality,math-ematical properties,and ease of use have been proposed 1including loga-rithmic forms (Good,1983).In the context of chance monitoring,the probabilities used for evaluating the surprise index are model probabilities.The frequent occurrence of sur-prising events signals model inadequacy.By adding the new surprising observations as evidence and revisiting the model selection stage,it is possible to adjust the probabilities as well as choosing a better plan.6.Intent-Based Chance ManagementThe chance discovery software agent relies on utilities to assess the desir-ability of a situation.These utilities express a form of intent.The chance discovery process typically involves interactions between many intelligent entities with converging or diverging intentions.These intentions guide re-sponses to challenges and opportunities forcing a deﬁned structure when a random response would be expected.For example,knowing that a business aims at making proﬁts implies that it will not try to exploit a chance in such a way to maximize its losses.Moreover,chronicles inconsistent with the intention are highly unlikely.The bias introduced by intentions can be very signiﬁcant.For example,Schelling (1960)reports that a group of individuals AHMED Y.TAWFIK 448INDUCTIVE REASONING AND CHANCE DISCOVERY449 asked to choose head or tail with the intent of matching the choice of another unknown person,have predominately(86%)chosen head.A preponderance of participants(90%),asked to divide100with a rival to increase the chance of a match,have divided the sum equally.These observations clearly suggest that intentions can strongly bias otherwise random choices.Incorporating motives and intentions within the proposed framework would improve the probability assessments.By ascribing motives to agents,the behavior of these agents becomes more deterministic.Multi-agent systems that exploit this phenomenon,also known as focal points,can achieve a certain level of coordination without communications(Fenster et al.,1995).The perception of value depends on the intentions of the agent.Some agent’s trash could be another’s treasure.The utilities or values assignments depend to a great extent on the intent of an agent.In a chance management process involving humans and intelligent agents,a proper formulation would require a game theoretic framework to account for multi-agent conﬂict, coordination,and cooperation.The interaction between two agents may involve strict competition,strict cooperation,or a combination of cooper-ation,coordination,and a degree of conﬂict.The latter results in a non-zero sum game(Schelling,1960).In such situations,proper chance management requires building on common interests and resolving conﬂicts.Humans tend to accept conﬂict resolution compatible with the notion of focal points. These focal points are zones within a solution space that possess some specially appealing features like uniqueness,symmetry,simplicity,or pre-cedence.Formally,a game theoretic(Osborne and Rubinstein,1994)formulation would describes the interaction of a set of agents with a Markov environment in which they all receive some payoﬀs for reaching their intended goals.The game consists of a tuple h S;p;G;T;r i where S is a discrete state space that corresponds roughly to the set of world states in Section4;p is a probability distribution over the initial state;G is a collection of agents,each described by deﬁning three sets:A,O,B where A is a discrete action space,O is a discrete observation space,and B is a set of mappings from the observation space to a probability distribution denoting the world state corresponding to an observation;T is a mapping from states of the environment and actions of the agents to probability distributions over states of the environment.;and r: SÂA G!R is the payoﬀfunction,where A G is the joint action space of the agents.Upon observing an observation o,an agent would try to deduce the corresponding world state,and act according to a strategy.The objective of each agent’s strategy is to maximize its reward.Algorithm for optimal and suboptimal strategy development has been proposed(Boutilier,1999; Peshkin et al.,2000).While,this game theoretic formulation is very similar to the techniques described in previous sections,it is a necessary extension to account for multi-agency in chance discovery contexts.7.Conclusion The chance discovery process described here relies on a background theory to build a plan for managing and exploiting chances.By combining abductive and deductive reasoning,the Bayesian treatment of chance discovery com-plements the Bayesian solution to the Goodman’s riddle of induction.The role of probability in chance management is fundamental.Attempts to reap the rewards of discovery plans are not likely to be productive unless the chance exploitation plan can reasonably improve the probability of oppor-tunities and reduce that of risks.A game theoretic approach may be neces-sary to exploit chances in a multi-agent environment.Note1Consult Bayarri and Berger (1997)for a survey.ReferencesBrown,J.S.and Burton,R.R.(1978),‘Diagnostic Models for Procedural Bugs in Basic Mathematical Skills’,Cognitive Science 2(2),pp.155–192.Akeroyd,A.(1991),‘A Practical Example of Grue’,British Journal for the Philosophy of Science 42(4),pp.535–539.Bacchus,F.Grove,A.,Halpern,J.and Koller,D.(1996),‘From Statistical Knowledge to Degrees of Belief’,Artiﬁcial Intelligence 87,pp.75–143.Bacchus,F.and Kabanza,F.(2000),‘Using Temporal Logics to Express Search Control Knowledge for Planning’,Artiﬁcial Intelligence 116(1–2),pp.123–191.Bayarri,M.and Berger,J.(1997),‘Measures of Surprise in Bayesian Analysis’,Duke Uni-versity Institute of Statistics and Decision Sciences Working Paper No.97-46,Durham,North Carolina.Boutilier,C.(1999),‘Sequential Optimality and Coordination in Multiagent Systems’,in Proceedings of the Sixteenth International Joint Conference on Artiﬁcial Intelligence .Dean,T.and Kanazawa,K.(1989),‘A Model for Reasoning about Persistence and Causa-tion’,Computational Intelligence 5(3),pp.142–150.Edis,T.(2000),‘Resolving Goodman’s Paradox:How to Defuse Inductive Skepticism’,Unpublished Manuscript./$edis/.Fenster,M.,Kraus,S.and Rusenschein,J.(1995),‘Coordination without Communications:Experimental Validation of Focal Point Techniques’,in Proceedings of the First Interna-tional Conference on Multi-Agent Systems (ICMAS-95),San Francisco,CA.Good,I.J.(1983),Good Thinking:The Foundation of Probability and its Applications ,Min-neapolis:University of Minnesota Press.Halpern,J.and Koller,D.(2004),‘Representation Dependence in Probabilistic Inference’,Journal of Artiﬁcial Intelligence Research 21,pp.319–356.Horwich,P.(1982),Probability and Evidence ,Cambridge:Cambridge University Press.Jaynes,E.T.(1968),‘Prior Probabilities’,IEEE Transactions on System Science and Cyber-netics 4,pp.227–241.AHMED Y.TAWFIK450INDUCTIVE REASONING AND CHANCE DISCOVERY451 Kripke,S.(1963),‘Semantical Analysis of Modal Logic I:Normal Modal Propositional Calculi’,Zeitschrift f.Math.Logik und Grunlagen d.Math.9,pp.67–96.McBurney,P.(2001),‘Review of:First International Workshop on Chance Discovery’, Knowledge Engineering Review16(2),pp.215–218.McBurney,P.and Parsons,S.(2001),‘Chance Discovery Using Dialectical Argumentation’,in Y.Ohsawa,ed.,Proceedings of the First International Workshop on Chance Discovery, Matsue,Japan,pp.37–45.McCarthy,J.and Hayes,P.(1969),‘Some Philosophical Problems from the Standpoint of Artiﬁcial Intelligence’,Machine Intelligence4,pp.463–502.Goodman,N.(1955),‘The New Riddle of Induction’,Fact,Fiction and Forecast,Cambridge, MA:Harvard University Press.Ohsawa,Y.(ed.)(2001),Proceedings of the First International Workshop on Chance Discovery Matsue,Japan:Japanese Society for Artiﬁcial Intelligence.Osborne,M.and Rubinstein,A.(1994),A Course in Game Theory Cambridge,MA:MIT Press.Peshkin,L.,Kim,K.,Meuleau,M.and Kaelbling,L.(2000),‘Learning to Cooperate via Policy Search’,Proceedings of the Sixteenth Conference on Uncertainty in Artiﬁcial Intel-ligence.Prendinger,H.and Ishizuka,M.(2001),‘Some Methodological Considerations on Chance Discovery’,in Y.Ohsawa,ed.,Proceedings of the First International Workshop on Chance Discovery,Matsue,Japan,pp.1–4.Schelling,T.(1960),A Strategy of Conﬂict,Cambridge,MA:Harvard University Press. Shanahan,M.(1997),Solving the Frame Problem,Cambridge,MA:MIT Press.Solomonoﬀ,R.(1999),‘Two Kinds of Probabilistic Induction’,The Computer Journal42(4), pp.251–259.Weaver,W.(1948),‘Probability,Rarity,Interest and Surprise’,Scientiﬁc Monthly67,pp.390–392.。

归纳总结能力英文

归纳总结能力英文The Ability of Inductive Reasoning and SummarizationIn the fast-paced world we live in, the ability to analyze information, make connections, and summarize key points has become increasingly important. Inductive reasoning and summarization skills are crucial in various domains, including education, research, and decision-making. This article explores the significance of developing and enhancing these abilities and provides useful strategies for doing so.1. Understanding Inductive ReasoningInductive reasoning is the process of deriving general principles from specific observations or examples. It involves looking for patterns, making educated guesses, and drawing conclusions based on limited evidence. This type of reasoning is often used in scientific research, problem-solving, and drawing accurate predictions.1.1 Why is Inductive Reasoning Important?Inductive reasoning helps us make sense of complex information and make informed decisions. By identifying patterns and establishing connections between data points, we can develop a deeper understanding of the world around us. This ability is especially valuable in scientific fields, where researchers constantly gather data and draw generalizations to advance knowledge.1.2 Strategies for Enhancing Inductive Reasoning Skills- Engage in puzzles and brainteasers: These activities force you to think creatively, look for patterns, and make logical connections. Riddles, Sudoku, and crossword puzzles are great examples.- Analyze case studies: Practice drawing general conclusions from specific examples. Analyzing case studies in various fields, such as business or psychology, will help you hone your inductive reasoning skills.- Learn from experience: Reflect on past experiences and consider how they can be applied to present situations. Look for recurring patterns and develop a broader understanding based on these observations.- Stay curious: Ask questions, seek new perspectives, and challenge assumptions. Curiosity fuels inductive reasoning by encouraging exploration and discovery.2. Mastering the Art of SummarizationSummarization is the process of condensing information into concise, coherent, and meaningful summaries. It requires identifying key points, eliminating irrelevant details, and presenting information in a clear and organized manner. Effective summarization skills are beneficial in academic settings, professional environments, and everyday communication.2.1 The Importance of SummarizationIn today's information overload, being able to extract essential information and communicate it efficiently is vital. Summarization enables efficient learning, effective communication, and concise reporting. It helps us identify the main ideas in a text and retain important information.Additionally, summaries are valuable tools for transferring knowledge and sharing key insights with others.2.2 Strategies for Enhancing Summarization Skills- Read actively: Engage with the material by highlighting key points, taking notes, and summarizing main ideas in your own words. Organize the information hierarchically, with the most important points at the top.- Practice paraphrasing: Take a paragraph or an article and rewrite it in a more concise form, while preserving the original meaning. This exercise helps develop your ability to extract essential information and present it in a succinct manner.- Use graphic organizers: Visual tools such as mind maps, concept maps, or flowcharts can help you organize information and identify relationships between ideas. These aids facilitate summarization by providing a clear structure for presenting key points.- Seek feedback: Share your summaries with others and ask for feedback. This will help you identify areas for improvement and refine your summarization skills.ConclusionThe ability to reason inductively and summarize effectively is vital in a world overflowing with information. By honing these skills, we can better understand complex concepts, make informed decisions, and communicate more efficiently. Whether in academic, professional, or everyday life, developing and enhancing our inductive reasoning and summarization abilities will undoubtedly contribute to our success. So, embark on thejourney of cultivating these skills and witness the positive impact they can have on your personal and professional growth.。

【必刷题】2024高一英语上册完形填空逻辑推理专项专题训练(含答案)

【必刷题】2024高一英语上册完形填空逻辑推理专项专题训练（含答案）试题部分一、选择题：1. In the first paragraph, the author mainly wants totell us that ______.A. logic reasoning is important in daily lifeB. logic reasoning is widely used in various fieldsC. logic reasoning can help us solve plex problemsD. logic reasoning is a skill that everyone should master2. According to the passage, which of the following is NOT a type of logic reasoning?A. Deductive reasoningB. Inductive reasoningC. Abductive reasoningD. Emotional reasoning3. When it es to solving a math problem, which type of reasoning is usually used?A. Deductive reasoningB. Inductive reasoningC. Abductive reasoningD. Analogical reasoning4. The passage mainly discusses ______.A. the importance of logic reasoningB. the types of logic reasoningC. how to apply logic reasoning in daily lifeD. the benefits of logic reasoning5. Which of the following sentences best describes the author's attitude towards logic reasoning?A. PraisefulB. CriticalC. IndifferentD. Objective6. In the second paragraph, the author mentions "For example, if all cats have four legs and Whiskers has four legs, then Whiskers is a cat." This is an example of ______.A. deductive reasoningB. inductive reasoningC. abductive reasoningD. analogical reasoning7. When using logic reasoning, which of the following is the most important?A. Making assumptionsB. Gathering evidenceC. Analyzing dataD. Drawing conclusions8. According to the passage, logic reasoning can help us ______.A. make better decisionsB. improve our memoryC. bee more imaginativeD. municate more effectively9. Which of the following is NOT mentioned as a benefit of logic reasoning in the passage?A. Improving critical thinking skillsB. Enhancing problemsolving abilitiesC. Boosting creativityD. Developing munication skills10. In the last paragraph, the author suggests that we should ______.A. learn different types of logic reasoningB. practice logic reasoning in daily lifeC. read more books about logic reasoningD. attend logic reasoning training courses二、判断题：1. Logic reasoning is only used in academic fields. ( )2. Deductive reasoning always starts with specific information and leads to general conclusions. ( )3. Inductive reasoning is based on patterns and observations. ( )4. Abductive reasoning is a bination of deductive and inductive reasoning. ( )5. Emotional reasoning is a type of logic reasoning. ( )三、填空题：1. Logic reasoning is a process of ________ that involves the use of rational thought to deduce a conclusion from a set of premises.2. The two main types of reasoning are ________ reasoning and ________ reasoning.3. In a syllogism, the "major premise" is the statement that ________, while the "minor premise" is the statementthat ________.4. An example of inductive reasoning is when we observe specific instances and ________ a general principle.5. A logical fallacy occurs when the argument's logic is ________ and leads to a false or incorrect conclusion.6. The process of using known facts to reach a specific conclusion is known as ________ reasoning.7. When we use analogical reasoning, we pare two ________ to infer something about the ________.8. To strengthen a logical argument, it is important to provide ________ and ________ evidence.9. Critical thinking is a skill that involves analyzing, evaluating, and ________ information.10. One of the benefits of logic reasoning is that ithelps improve our ________ thinking skills.11. In a logical argument, the "conclusion" is the statement that is ________ to be true based on the premises.12. A valid argument is one where if the premises are true, then the ________ must also be true.13. The structure of a logical argument includes premises, ________, and logical ________.14. ________ reasoning is often used in mathematics and formal logic to prove theorems.15. An example of circular reasoning is when theconclusion is ________ in one of the premises.16. ________ reasoning is used to make predictions based on limited data or observations.17. The process of elimination is a strategy often usedin ________ reasoning to narrow down possible answers.18. A counterargument can be used to ________ thevalidity of an opposing viewpoint.19. ________ reasoning is based on the idea that if two things are similar in some respects, they will be similar in other respects as well.20. To avoid logical fallacies, it is important to________ assumptions and biases that could weaken the argument.四、简答题：1. Explain the difference between deductive and inductive reasoning.2. How does critical thinking relate to logic reasoning?3. What is a logical fallacy, and how can it affect an argument?4. Describe the steps involved in constructing a valid deductive argument.5. Give an example of how inductive reasoning is used in everyday life.6. What is the role of evidence in logic reasoning?7. How can analogical reasoning be used to solve problems?8. Explain the concept of a syllogism and provide an example.9. Why is it important to avoid emotional reasoning in logical discussions?10. Discuss the importance of logic reasoning in academic and professional settings.本套试题答案如下一、选择题：1. B2. D3. A4. A5. D6. A7. D8. A9. C10. B二、判断题：1. ×2. ×3. √4. ×5. ×三、填空题：1. inference2. deductive, inductive3. makes a general statement, makes a specific statement4. infer5. flawed6. deductive7. cases, case in question8. relevant, sufficient9. synthesizing10. critical11. claimed12. conclusion13. conclusion, connectors14. Deductive15. restated16. Inductive17. deductive18. challenge19. Analogical20. identify and eliminate四、简答题：1. Deductive reasoning starts with a general principle and moves to a specific conclusion, while inductive reasoning starts with specific observations and moves to a general conclusion.2. Critical thinking involves the evaluation and analysis of arguments, which is essential in logic reasoning to ensure that the conclusions are valid.3. A logical fallacy is an error in reasoning that undermines the argument's validity; it can lead to false conclusions and weaken the argument's persuasive power.4. Identifying the major premise, stating the minor premise, and drawing the conclusion based on the relationship between the two premises.5. Observing that the sky is clear and predicting that it will not rain, based on past experiences.6. Evidence supports the premises and makes the argument more convincing and reliable.7. By identifying a known similarity between two situations and using that similarity to infer a similarity in another respect.8. A syllogism is a form of deductive reasoning that consists of a major premise, a minor premise, and a conclusion. Example: All mammals are warmblooded (major premise), a whale is a mammal (minor premise), therefore a whale is warmblooded (conclusion).9. Emotional reasoning can cloud judgment and lead to biased or illogical conclusions.10. Logic reasoning is essential for making wellreasoned decisions, solving plex problems, and effectively municating ideas in academic and professional contexts.。

Thinking of argument

Thousands of children are needlessly denying ቤተ መጻሕፍቲ ባይዱhemselves milk and other common foods because of allergies they don’t have, a research team at the National Jewish Hospital and Research Center has concluded.
The research, if its findings are confirmed by further testing, could mean that many “allergy sufferers” can enjoy more palatable and nourishing diets.
—“Doctors Find Two-Thirds of ‘Allergic’ Kids Aren’t,”
Atlanta Constitution, March 19, 1979, p. 13A
Induction. Induction is thinking that arrives at a general principle from particular facts or experiences.
But the most significant result of the study so far is the revelation that possibly only one-third of children thought to have food allergies actually have such allergies.
All flight attendants know how to swim. Ralph knows how to swim. Therefore Ralph is a flight attendant.

推理英文作文技巧

推理英文作文技巧1. Deductive reasoning is a valuable skill that allows us to draw logical conclusions based on available information. It involves making inferences and using evidence to support our claims. By honing this skill, we can become more effective problem solvers and critical thinkers.2. When engaging in deductive reasoning, it is important to carefully examine the evidence at hand. This may include analyzing data, evaluating the credibility of sources, and considering alternative explanations. By thoroughly assessing the information, we can make informed judgments and avoid jumping to hasty conclusions.3. Another key aspect of deductive reasoning is identifying patterns and connections. By recognizing similarities or trends, we can make predictions or draw conclusions about related situations. This ability to see the bigger picture allows us to make logical leaps andsolve complex problems.4. In addition to analyzing evidence and identifying patterns, deductive reasoning also involves considering counterarguments. By acknowledging opposing viewpoints, we can strengthen our own arguments and address potential weaknesses. This critical evaluation of different perspectives helps us to make more well-rounded and persuasive claims.5. Deductive reasoning is not limited to academic or professional settings. It is a skill that can be applied to everyday life as well. For example, when faced with a decision, we can use deductive reasoning to weigh the pros and cons, consider potential outcomes, and make a well-informed choice.6. Developing strong deductive reasoning skills takes practice. Engaging in activities such as puzzles, riddles, and logical reasoning exercises can help to sharpen our ability to think critically and draw logical conclusions. Additionally, seeking out diverse perspectives and engagingin thoughtful discussions can further enhance our deductive reasoning abilities.7. In conclusion, deductive reasoning is a valuableskill that allows us to draw logical conclusions based on available evidence. By carefully analyzing information, identifying patterns, considering counterarguments, and applying this skill in various aspects of life, we can become more effective problem solvers and critical thinkers.。

2022年同等学力英语作文

2022年同等学力英语作文：探索终身学习之路In the fast-paced and constantly evolving era of 2022, the concept of equivalent academic qualifications has gained significant traction, especially in the realm of lifelong learning. This paradigm shift recognizes the value of continuous education beyond traditional academic boundaries, enabling individuals to enhance their skills and knowledge throughout their lifetimes. The essence of equivalent qualifications lies in the belief that learning is not confined to specific institutions or periods but is a never-ending journey of personal growth and intellectual enrichment.In this context, the significance of English essay writing within the framework of equivalent qualifications becomes apparent. English, as a global language, is not only a means of communication but also a gateway to a vast array of knowledge and cultural exchange. Through essay writing, individuals can express their thoughts, analyze complex issues, and develop critical thinking skills. It is a powerful tool that fosters creativity, logical reasoning,and the ability to communicate effectively in a globalized world.The challenge, however, lies in the fact that equivalent qualifications often require individuals to demonstrate their proficiency in English essay writing without necessarily having undergone formal academic training. This demands a high level of self-directed learning, research capabilities, and writing skills. It is a test of resilience and adaptability, as learners must navigate through various resources, understand the nuances of academic writing, and craft coherent and convincing arguments.To succeed in this endeavor, several strategies can be employed. Firstly, establishing a solid foundation in English language proficiency is crucial. This involves enhancing vocabulary, grammar, and comprehension skills through regular practice and exposure to authentic materials. Secondly, developing a structured approach to essay writing is essential. This includes planning the outline, conducting thorough research, organizing ideas logically, and expressing them clearly and concisely.Additionally, reading widely and critically can help individuals broaden their perspectives, deepen their understanding of complex topics, and learn from the writing styles and techniques of established authors.Moreover, the utilization of technology and online resources can significantly enhance the learning experience. Platforms such as online libraries, educational websites, and interactive learning tools provide access to vast repositories of information and facilitate collaborationand feedback. These resources can be leveraged to enhance research capabilities, writing skills, and overall understanding of the subject matter.In conclusion, the pursuit of equivalent qualifications through English essay writing represents a journey of intellectual exploration and personal growth. It demands a commitment to continuous learning, adaptability to new challenges, and the ability to harness the power of language to express ideas and understand the world. By embarking on this path, individuals can not only enhance their professional prospects but also enrich their liveswith knowledge, understanding, and the joy of discovery.**2022年同等学力英语作文：探索终身学习之路** 在2022年这个快节奏且不断变革的时代，同等学力概念逐渐受到广泛重视，尤其是在终身学习领域。

关于表述理由的四级英语作文

关于表述理由的四级英语作文Forming a convincing argument requires careful consideration of various factors. 表达理由需要仔细考虑各种因素。

First and foremost, it is crucial to support your argument with solid evidence. 首先，用可靠的证据支持你的论点是至关重要的。

This could include statistics, research findings, expert opinions, and real-life examples. 这可能包括统计数据、研究结果、专家意见和实例。

By presenting compelling evidence, you can strengthen your argument and make it more persuasive. 通过呈现令人信服的证据，你可以加强你的论点并使其更有说服力。

In addition to evidence, providing logical reasoning is also essential in forming a coherent argument. 除了证据，提供逻辑推理也是形成连贯论证的重要因素。

By carefully laying out the logical steps that lead to your conclusion, you can help your audience follow your train of thought. 通过仔细阐述导致你结论的逻辑步骤，你可以帮助听众理解你的思路。

Logical reasoning involves connecting the dots between your evidence and your main point, forming a clear and structured argument. 逻辑推理涉及将证据与主要观点联系起来，形成一个清晰而结构化的论据。

Inductive-Reasoning-Test1

Inductive Reasoning Test 1 - SolutionsQuestion 1Solution: EExplanation:In this question, there are two rules to follow.The first rule is that the spiral rotates 90° clockwise each time. Following this rule, the next diagram in the sequence should have the end of the spiral facing north. Therefore, the correct answer could be B or E.The other rule is that the arrow alternates from pointing vertically upwards to vertically downwards. Following this rule, the next diagram in the sequence must have the arrow pointing vertically downwards. Therefore, the correct answer is E.Question 2Solution: AExplanation:In this question, there are four rows of squares, one with four squares, one with three squares, one with two squares and one with a single square.There are also two rules to follow.The first rule is that the rows of squares move one place down vertically each time, with the bottom row moving to the top. Following this rule, the next diagram in the sequence should have two squares in the top row, four in the next, one in the next and three squares in the bottom row. Therefore, the correct answer could be A or E.The other rule is that the row containing a single square moves one place to the right each time, but the other rows do not move horizontally. Following this rule, the next diagram in the sequence must have the single square in the second column from the left. Therefore, the correct answer is A.Question 3Solution: DExplanation:In this question, there is a pattern of black, white and grey squares.The rule in this case is that all the squares in the pattern move successively one square to the left and then in the next step they move one square up.The correct answer is, therefore, D.Question 4Solution: BExplanation:In this question, there are two rules to follow.The first rule is that the pattern of three circles rotates through 90° clockwise each time. Following this rule, the next diagram in the sequence should have circles in the top right, bottom right and bottom left corners. Therefore, the correct answer could be B or D.The other rule is that the number of black circles increases from 1 to 2 to 3, then back to 1 again. Following this rule, the next diagram in the sequence must have 3 black circles. Therefore, the correct answer is B.Question 5Solution: CExplanation:In this question, there are two solid lines and one dotted line. The dotted line doesn’t move. One solid line rotates 90° anticlockwise every time and the other solid line rotates 45° anticlockwise every time. Sometimes the solid lines cover and obscure the dotted line (as in the second diagram) or cover each other (as in the fifth diagram).Following this rule, one solid line that was pointing towards the west in the fifth diagram will turn 90° anticlockwise and will point towards the south in the next diagram and the other solid line that was also pointing towards the west in the fifth diagram will turn 45°anticlockwise and will point towards the south-west in the next diagram. Therefore, the correct answer is C.Question 6Solution: DExplanation:In this question, there is a triangle, a square and a circle which each follow a rule.The triangle alternates between the top left corner and the bottom right corner and is coloured white twice before changing colour, then black twice before changing colour again. Following this rule, the next diagram in the sequence should have a white triangle in the bottom right corner. Therefore, the correct answer could be B or D.The square moves clockwise around the eight main compass points and is coloured black twice before changing colour, then white twice before changing colour again. Following this rule, the next diagram in the sequence should have a black square in the top right corner (north-east) position. Therefore, the correct answer could be A, D or E.The circle stays in the same position and alternates between black and white. Following this rule, the next diagram in the sequence should have a white circle.Therefore, the correct answer is D.Question 7Solution: CExplanation:In this question, there are three rules to follow.The first rule is that the arrow rotates through 90° clockwise each time. Following this rule, the next diagram in the sequence should have the arrow pointing east-west. Therefore, the correct answer could be A, B, C or D.The second rule is that the number of arrowheads increases by 1 each time, from 1 to 2 to 3, then back to 1. Following this rule, the next diagram in the sequence must have 3 arrowheads. Therefore, the correct answer could be A or C.The third rule is that the arrowheads alternately point to and away from the centre of the square. Following this rule, the next diagram in the sequence should have the arrowheads pointing away from the centre. Therefore, the correct answer is C.Question 8Solution: EExplanation:In this question, the black square follows a path of a knight’s moves in chess – alternately its moves are:•two squares down and one square to the right,•then one square down and two squares to the right.When it reaches the right edge of the grid, it continues from the left edge. When it reaches the bottom edge of the grid, it continues from the top edge.Following this rule, from the fifth diagram the black square must move two squares down and one square to the right. The correct answer is, therefore, E.Question 9Solution: AExplanation:In this question, there are two rules to follow.The first rule is that the diagonal of the hexagon rotates through 60° anticlockwise each time. Following this rule, the next diagram in the sequence should have the diagonal drawn from the bottom left corner to the top right corner. Therefore, the correct answer could be A, C or E.The other rule is that the black circle moves clockwise around the vertices of the hexagon, moving one place, then two places, then three places etc. Following this rule, in the next diagram in the sequence the black circle should have moved another five places clockwise, which takes it to the vertex of the hexagon at the extreme right. Therefore, the correct answer is A.Question 10Solution: BExplanation:In this question, there are two rules to follow. There are eight different patterns around the outside eight squares of the design.The first rule is that the eight patterns rotate anticlockwise around the outside of the design. The first time, they all rotate one place anticlockwise, the second time two places, the third time three places and so on. Following this rule, the next diagram in the sequence could be any one of A, B, C or D.The other rule is that patterns with contrasting colours – black and white, or grey and white, alternate from one diagram to the next e.g. the pattern with two stripes - white on the left and grey on the right - changes to grey on the left and white on the right in the next diagram. The patterns with diagonal and vertical lines do not change. Following this rule, the next diagram in the sequence must be B.。

Deductive Reasoning

Deductive Reasoning
1
Deductive Reasoning
Is a process of thinking in a logical way, where conclusion are drawn from the information given, for example: if there are clouds in the sky
Premises Conclusion Observation
So Jenna must have completed her general psychology requirements.
Valid Conclusion
If a student is in cognitive psychology, Then she must have completed her general psychology requirements. David has NOT completed his general psychology requirements.
Valid
If an officer is a general, then he was a captain. Gerald is NOT a captain, So he NOT a general.
3
Invalid Conclusions
Affirming If p is true then q is true the q is true consequent p is true Invalid
Cognitive Functions
Research has shown that conditional reasoning and syllogisms tasks both affect the same cognitive functions (Gilhooly, 1996; Klauer et al., 2000; Matlin 1994; Stanovich, 1999). How?

归纳思想总结英语翻译

归纳思想总结英语翻译Inductive Reasoning Summary:Inductive reasoning is a form of logical thinking that involves making generalizations or predictions based on specific observations or patterns. It is one of the two main types of reasoning, the other being deductive reasoning.In inductive reasoning, the conclusions or predictions are not guaranteed to be true, but they are considered to be probable or likely based on the evidence available. This is because inductive reasoning moves from specific instances to general principles.The process of inductive reasoning involves several steps. First, specific observations or instances are gathered and examined. These observations can be qualitative or quantitative in nature, and they can be collected through various methods such as experiments, surveys, or observations in real-life situations.After gathering the observations, patterns or trends are identified. These patterns can be similarities or commonalities between the different instances. It is important to note that inductive reasoning does not involve proving a universal truth, but rather finding trends or probabilities based on the available evidence.Once patterns are identified, a hypothesis or general principle is formulated. This hypothesis is a tentative explanation or prediction that is based on the observed patterns. It can be thought of as an educated guess about the general principle behind the specific instances.The next step involves testing the hypothesis through additional observations or experiments. This is done to gather more evidence and determine whether the hypothesis holds true in other instances or situations. The more observations and evidence that support the hypothesis, the more confident one can be in its validity.If the hypothesis is supported by the additional observations, it can be considered as a general principle or theory. However, it is important to remember that inductive reasoning does not provide absolute certainty. The conclusions are based on probabilities and is subject to revision if new evidence or observations are found.One common example of inductive reasoning is the prediction of weather. Meteorologists collect data on current weather conditions, such as temperature, wind direction, and cloud cover. They then analyze these observations to identify patterns and trends. Based on these patterns, they can make predictions about future weather conditions.Inductive reasoning is also widely used in scientific research. Scientists gather observations or data from experiments or observations and analyze them to identify patterns or trends. Based on these patterns, they can formulate hypotheses or theories that explain the observed phenomena.In conclusion, inductive reasoning is a logical thinking process that involves making generalizations or predictions based on specific observations or patterns. It is a valuable tool in many fields, including science, research, and everyday decision-making. Whileit does not guarantee absolute truth, it provides a way to make educated guesses and predictions based on the available evidence.。

《国际课程学习技能》期末试题(14土建、电气等)

I.Please list two examples with inductive reasoning (归纳推理) and deductive reasoning(演绎推理) respectively. （分别各举一例说明归纳推理和演绎推理）Inductive reasoningPremise: So far, according to experience, every time an object is released from certain height close to the ground, it fell downward.Conclusion: Every time an object is released from certain height close to the ground, it always falls downward, whether it is in the past, now, or in the future.Deductive reasoningPremise: If someone was born in Beijing, then he/she was born in China.Conclusion: if someone was not born in China, then he/she was not born in Beijing.II.Please list specific examples, which suffers flawed reasoning of hasty generalizations, circular reasoning, and biased generalizations.(请列举出分别犯有“以偏概全”、“循环论证”、和“偏见性概括”的具体事例)(Note: Flawed reasoning(无效推理) includes hasty generalizations(以偏概全), circular reasoning（循环论证）, and biased generalizations(偏见性概括) .A hasty generalization is a conclusion that is based on too little evidence. Circular reasoning is often of the form: "A is true because B is true; B is true because A is true." If the conclusion is based on evidence from biased sources, then the generalization is biased.)Flawed reasoning(无效推理)A: This egg tastes disgusting.B: Y ou're eating these eggs grow up, you have the power to say what it does not taste the egg?includes hasty generalizations(以偏概全)"A hasty generalization is a broad claim based on too-limited evidence. It is unethical to assert a broad claim when you have only anecdotal or isolated evidence or instances. Consider two examples of hasty generalizations based on inadequate data:- Three congressional representatives have had affairs. Therefore, members of Congress are adulterers.- An environmental group illegally blocked loggers and workers at a nuclear plant. Therefore, environmentalists are radicals who take the law into their own hands.In each case, the conclusion is based on limited evidence. In each case the conclusion is hasty and fallacious."(Julia T. Wood, Communication in Our Lives, 6th Ed. Wadsworth, 2012)circular reasoning（循环论证）A: the bible is right.B: how do you know the bible written is right?A: because the bible was written by god, and god cannot lie and not to make mistakes.B: how do you know?A: because the bible so write.biased generalizations(偏见性概括)Heroine Elizabeth was born in a small landowning family, for children of the rich Darcy loves. Darcy despite the gap between the door first and wealth, to propose to her, but it was rejected. Elizabeth for his misunderstanding and prejudice is one reason, but the main thing is that she hated his arrogance. Because of this arrogant Darcy actually reflect differences in status, as long as the existence of such arrogance, it is impossible to have common thoughts and feelings between him and Elizabeth, it is impossible to have an ideal marriage. After Elizabeth Darcy acted personally observed a series of actions and, in particular, changes in the past that saw him conceited manner, eliminating misunderstanding and prejudice against him, which he concluded with a happy marriage.III. Compare the features of descriptive writing and critical writing.IV. Fifteen years ago, Omega University implemented a new procedure that encouraged students to evaluate the teaching effectiveness of all their professors. Since that time, Omega professors have begun to assign higher grades in their classes, and overall student grade averages at Omega have risen by 30 percent. Potential employers, looking at this dramatic rise in grades, believe that grades at Omega are inflated and do not accurately reflect student achievement; as a result, Omega graduates have not been as successful at getting jobs as have graduates from nearby Alpha University. To enable its graduates to secure better jobs, Omega University should terminate student evaluation of professors.(翻译：15年前，Omega大学实施了一项新措施，鼓励学生对所有教授的教学效果进行评价。

InductiveReasoningExamples

InductiveReasoningExamples Having a tough time trying to make sense of inductive reasoning? Here's a brief write-up, which will put forth some examples of the same and make it easier for you to understand. In psychology, inductive reasoning or 'induction' is defined as reasoning based on detailed facts and general principles, which are eventually used to reach a specific conclusion. It is one of the two types of reasoning; deductive reasoning being the other type. Also known as inductive logic or the bottom-up approach, induction is basically a type of reasoning wherein the chances of the conclusion being false are significant even when all thepremises, on which the conclusion is based, are true.As opposed to deductive reasoning, which goes from general to specific, inductive reasoning goes from specific to general. In other words, it begins with a specific argument and arrives at a general logical conclusion. At times, induction is termed as strong, or weak, on the basis of the credibility of the argument put forth.Example of Strong Inductive ReasoningAll the tigers observed in a particular region have black stripes on orange fur. Therefore all the tigers native to this region have black stripes on orange fur.Even though all the tigers that were observed in this region sported black stripes on orange fur, the existence of a white tiger, i.e., black stripes on white fur, cannot be ruled out. Based on this, one can assume that the conclusion mentioned in this example is not certain. But then, the chances of coming across a white tiger are actually very rare, and that in itself makes this statement a good example of strong induction. In other words, a strong induction is the one wherein the conclusion is backed by the premises to a significant extent. Example of Weak Inductive ReasoningJoe always jumps the red light. Therefore everybody jumps the red light.Unlike strong induction, in weak induction, the conclusion is not linked to the premises. Concluding that everybody jumps the red light just because one person does, is not an exercise of logical thinking. Simply put, weak induction is one which is backed by a faulty logic. CategoriesInductive reasoning is further categorized into different types, i.e., inductive generalization, simple induction, causal inference, argument from analogy, and statistical syllogism. Given below are some examples, which will make you familiar with these types of inductive reasoning.All observed people are right-handed, therefore all the people are right-handed. (Inductive generalization)All the dogs that have been observed, can bark, therefore all the dogs can bark. (Simple induction)Joe leaves home at 08:30 in the morning and arrives late for work, based on which he concludes that he will be late for work every time he leaves at 08:30. (Causal inference)John and Joe are friends. John likes to sing, write and read. Joe likes to sing and write. Therefore one assumes that Joe also likes to read. (Argument from analogy)John plays as a pitcher for his team. All pitchers pitch at an average speed of 90 MPH, therefore John pitches at an average speed of 90 MPH as well. (Statistical syllogism)More ExamplesThe relationship between the premises and proposition forms the base of any inductive reasoning argument. Going through some examples of this form of reasoning will help you get a better understanding of the concept.Every time John eats shrimp, he gets cramps, and therefore he assumes that he gets cramps because he eats shrimp.John is an amazing athlete. So John's son too will go on to become an amazing athlete.When chimpanzees are exposed to rage, they tend to become violent. Humans are similar to chimpanzees, and therefore they tend to get violent when exposed to rage.The woman in the neighboring apartment has a shrill voice. I can hear a shrill voice from outside. There is a high probability that the woman in the neighboring apartment is shouting.All the dogs which were subjected to routine diagnosis had fleas, so one concludes that all the dogs have fleas.The Philadelphia Falcons have won their last four matches in a one-sided contest, and therefore their fans conclude that the Falcons will win their fifth match as well.Every time you get a call from some unknown number, you find a telemarketer on the other side of the line. It makes you conclude that if it's an unknown call, it is most likely to be a telemarketer.You see a dog chasing a cat in your neighborhood a couple of times, and start believing that the two animals cannot be kept in one house.A few episodes of a particular sitcom make you laugh, and you conclude that the said sitcom is very funny.100 pens are kept in front of you. On checking the first 10 pens, you note that 5 had black ink and 5 had blue ink, and therefore you conclude that half of the 100 pens are black and half are blue.To induce is to "bring about", and inductive reasoning is all about arriving at a conclusion on the basis of principle facts which guide you towards it. Comparing these examples of inductive reasoning with those of deductive reasoning will give you a better idea about the difference between the two. While we may not realize it, we resort to inductive reasoning for numerous day-to-day activities in our life. In fact, there are certain circumstances wherein you are left with no other option, but to rely on this form of reasoning -- even when you think it's unreliable.。

Legal Reasoning

Legal ReasoningInductive and Deductive pattern in argument Inductive and deductive refer to two distinct logical processes.Inductive reasoning is a movement from a specific examples or activities to generalization or rule.Deductive reasoning is a movement from a generalization to specific.(一)Inductive ReasoningInductive reasoning is reasoning in which the premises seek to supply strong evidence for (not absolute proof of) the truth of the conclusion. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.The philosophical definition of inductive reasoning is more nuanced than simple progression from particular/individual instances to broader generalizations. Rather, the premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it. In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms, discussed below).An example of an inductive argument:90% of biological life forms that we know of depend on liquid water to exist.Therefore, if we discover a new biological life form it will probably depend on liquid water to exist.This argument could have been made every time a new biological life form was found, and would have been correct every time; however, it is still possible that in the future a biological life form not requiring water could be discovered.As a result, the argument may be stated less formally as:All biological life forms that we know of depend on liquid water to exist.All biological life probably depends on liquid water to exist.The forms of inductive reasoning1.1 GeneralizationA generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.The proportion Q of the sample has attribute A.Therefore:The proportion Q of the population has attribute A.ExampleThere are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. A good inductive generalization would be that there are 15 black, and five white, balls in the urn.How much the premises support the conclusion depends upon (a) the number in the sample group, (b) the number in the population, and (c) the degree to which the sample represents the population (which may be achieved by taking a random sample). The hasty generalization and the biased sample are generalization fallacies.1.2 Statistical syllogismMain article: Statistical syllogismA statistical syllogism proceeds from a generalization to a conclusion about an individual.A proportion Q of population P has attribute A.An individual X is a member of P.Therefore:There is a probability which corresponds to Q that X has A.The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".1.3 Simple inductionSimple induction proceeds from a premise about a sample group to a conclusion about another individual.Proportion Q of the known instances of population P has attribute A.Individual I is another member of P.Therefore:There is a probability corresponding to Q that I has A.This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.1.4 Argument from analogyMain article: Argument from analogyThe process of analogical inference involves noting the shared properties of two or more things, and from this basis inferring that they also share some further property:P and Q are similar in respect to properties a, b, and c.Object P has been observed to have further property x.Therefore, Q probably has property x also.Analogical reasoning is very frequent in common sense, science, philosophy and the humanities, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning.1.5 Causal inferenceA causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.1.6 PredictionA prediction draws a conclusion about a future individual from a past sample.Proportion Q of observed members of group G have had attribute A.Therefore:There is a probability corresponding to Q that other members of group G will have attribute A when next observed.(二)Deductive ReasoningDeductive reasoning, also deductive logic, logical deduction or, informally, top-down" logic, is the process of reasoning from one or morestatements (premises) to reach a logically certain conclusion. It differs from inductive reasoning or abductive reasoning.Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from, i.e., there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used inmathematical proofs – mathematical inductionis actually a form of deductive reasoning.For example:Major premise: All men are mortal.Minor premise: Socrates is a man.Conclusion: Socrates is mortal.Rule: To be enforceable, the contract for the sale of goods for a price of $500 or more must be in writing.Facts: The oral contract for the sale of these goods was not for $500 or more.Legal conclusion: The contract in this case is enforceable.Summary:Derives its conclusion by reasoning from the major premise to the minor premise. Its essence is the syllogism.Legal problem solving in the civil law tradition is a kind of deductive reasoning, which is based on rules the contents of which are posited prior to the problems to which they must be applied.For legal deductive reasoning, one must first establish the Rule. This may be from the constitution, a statute, an administrative regulation, treaty, executive order, or the other sources of law. Second, find the facts. Third, conclude.The forms of deductive reasoning1.1 The law of detachmentThe law of detachment is the first form of deductive reasoning. A single conditional statement is made, and a hypothesis (P) is stated. The conclusion (Q) is then deduced from the statement and the hypothesis. The most basic form is listed below:1.P → Q (conditional statement)2.P (hypothesis stated)3.Q (conclusion deduced)In deductive reasoning, we can conclude Q from P by using the law of detachment.[3] However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no definitive conclusion.The following is an example of an argument using the law of detachment in the form of an if-then statement:1.If an angle satisfies 90° < A < 180°, then A is an obtuse angle.2. A = 120°.3. A is an obtuse angle.Since the measurement of angle A is greater than 90°and less than 180°, we can deduce that A is an obtuse angle. If however, we are given the conclusion that A is an obtuse angle we cannot deduce the premise that A = 120°.1.2 The law of syllogismThe law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:1.P → Q2.Q → R3.Therefore, P → R.The following is an example:1.If Larry is sick, then he will be absent.2.If Larry is absent, then he will miss his classwork.3.Therefore, if Larry is sick, then he will miss his classwork.We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the second statement. We also allow that this could be a false statement. This is an example of the Transitive Property in mathematics. The Transitive Property is sometimes phrased in this form:1. A = B.2. B = C.3.Therefore, A = C.1.3 The law of contrapositiveThe law of contrapositive states that, in a conditional, if the conclusion is false, then the hypothesis must be false also. The general form is the following:1.P → Q.2.~Q.3.Therefore, we can conclude ~P.The following are examples:1.If it is raining, then there are clouds in the sky.2.There are no clouds in the sky.3.Thus, it is not raining.。

英语作文推理

英语作文推理Title: The Power of Deductive Reasoning。

Deductive reasoning is a fundamental aspect of critical thinking, allowing individuals to draw logical conclusions based on premises and evidence. In this essay, we will explore the significance of deductive reasoning, its applications in various fields, and its impact on problem-solving.To begin with, deductive reasoning involves deriving specific conclusions from general principles. It follows a top-down approach, where one starts with a general premise and then applies it to reach a specific conclusion. This method is commonly used in mathematics, science, philosophy, and even everyday decision-making.In mathematics, deductive reasoning plays a crucialrole in proving theorems and solving mathematical problems. Mathematicians use axioms and logical rules to deduce newtruths, building upon existing knowledge to expand mathematical theory. For example, in geometry, the process of deducing properties of shapes and angles relies heavily on deductive reasoning.Similarly, in science, deductive reasoning is essential for forming hypotheses and making predictions. Scientists formulate general theories based on observations and empirical evidence, and then use deductive reasoning to derive specific hypotheses that can be tested through experimentation. This process lies at the heart of the scientific method, allowing researchers to systematically explore and understand the natural world.In philosophy, deductive reasoning is employed to construct logical arguments and analyze concepts. Philosophers use deductive logic to evaluate the validity of arguments and assess the soundness of philosophical theories. Through careful reasoning and logical analysis, they aim to uncover truths about existence, morality, and the nature of reality.Moreover, deductive reasoning extends beyond academic disciplines and finds practical applications in variousreal-world scenarios. In law, for instance, legal professionals use deductive reasoning to analyze evidence, construct legal arguments, and reach verdicts in court cases. Lawyers build their cases by applying legal principles to specific facts, systematically deducing the guilt or innocence of the accused.In medicine, doctors utilize deductive reasoning to diagnose illnesses and develop treatment plans. By gathering patient information, conducting tests, and applying medical knowledge, physicians deduce the underlying causes of symptoms and prescribe appropriate interventions. Deductive reasoning helps healthcare professionals make informed decisions and provide effective care to patients.Furthermore, deductive reasoning is valuable in everyday problem-solving and decision-making. Whether it's planning a budget, organizing tasks, or solving puzzles, individuals rely on deductive logic to analyze situations,identify patterns, and draw conclusions. By employing deductive reasoning skills, people can navigate complex challenges and make rational choices in their personal and professional lives.In conclusion, deductive reasoning is a powerful cognitive tool that enables individuals to draw logical conclusions based on premises and evidence. From mathematics and science to law and everyday life, deductive reasoning plays a crucial role in problem-solving and decision-making processes. By understanding and applying the principles of deductive logic, we can enhance our critical thinking abilities and approach challenges with clarity and rationality.。

对外经贸大学跨文化交际导论期末考试题

对外经贸大学跨文化交际导论期末考试题对外经济贸易大学《跨文化交际（英）》期末考试I.True-False: Decide whether each of the following statements is tme orfalse. Write T for "true” and F for “false”.（每题 1 分? 共20 分）1.The tenn “intercultural coniniumcation M was fiist used by Geert Hofstede ill 1959.2? Hall defines culture as the "software of the mind" that guides us ill our daily interactions ?3.In most of Afiica, Argentina and Peru, putting one^s index finger to his temple means'You are crazy/4.Stereotyping is a complex form of categorization that mentally organizes yourexperiences and guides your behavior toward a paiticular group of people?5.\alues aie social principles, goals, or standards accepted by persons in a cultme?They are the imiennost “skin of the oniony6.People from some cultures may lower their gaze to convey respect, whereas this may betuiderstood as evading or even insulting in other cultiues?7.Unbuttoning one^s coat is a sign of openness, fi iendliness or willingness to reach anagieement.8.In order for iiitercultiiral negotiation to be successful, the parties must providefor a wiii-lose situation.9.Edward Hall's theory states that the fdur levels embody the total concept of culturelike an onion 一symbols, heroes, rituals, and values?10. Successfill intercultiual business conununication involves knowing the ethnocentrismsof persons in other cultaes. Understanding the mindsets of both oneself and the person of another culture will result in more efficient communication.11? Etlmocentrism is the belief that somebody else，s cultural background、including ways of analyzing problenis, values, beliefs, language, and verbal and non verbalcommunication, is better than our own?12? People ill the United States place a greater emphasis on liistory and do not like change as compared with people of Asian and Latin cultiwes.13.When dealing with German business people, you should avoid jokes and other forms ofhumor during the actual business sessions.14.In the business ciicle, American business people use fii st names immediately.15? Companies should avoid sending female employees to the Middle East, as in Aiab coimtries men may refiise to work with women.16.In Southeast Asia, you should avoid presenting your business card with your right hand17.Wlien accepting a business card, Gennan business people carefiilly look at the card、observe the title and organizatioiL acknowledge with a nod that they have digested the information, and perhaps make arelevant comment or ask a polite question.18.The OK sign may be iiiteipreted as asking for money by Japanese business people.19.Nonverbal communication is important to the study of intercultmal communicationbecause a great deal of nonverbal behavior speaks a universal language.20.In short，iiitercultiiral communication competence requiies sufficient awarenessknowledge, motivations, and skills. Each of these components alone is sufficient to achieve intercultmal conmiunication conopetence?II.Tdmskdion: Translate the following Chinese terms into English and English terms into Chinese.（每题 1 分?共20 分）1. stereotypes3.etlinocentrism5.high-context culture 7.speech act9.vocal qualifiers 2.paralangiiage4.masculinity6.monochronic time 8.conversation taboos 10.power distance12.偏见14.不确定性回避16.语用错误18.归纳法20.礼仪与礼节III. Multiple Choice: Choose the ONE appropriate answer.（每题 1 分，共20分）1? Understanding another cultuie ____________ .a. enables businesspeople to know why foreign associates believe and act as theydob ? is best achieved tlirough “do's and don'ts^ listsc. is important for businesspeople because they can appear to be better informedd. isn't necessary for businesspeople2. Non-liiiear languages ________ .a. are object orientedb ? see time as a continuum of present, past and fiitiirec.are circular, tradition oriented and subjective d. lead to short-range planning in business practices3? Which statement about values is incorivct?a. \alues are social principles, goals, or standards accepted by persons in aculture.b. \alues are learned by contacts with family members, teachers, and religiousleaders.c. "Values will be influenced by what is seen on television or read innewspapers.d. People in various cultiues have basically similar values.4. People from cultiues that follow the monoclironic time system tend toa.do one thing at a time. b. be committed to people.11.译码 13.文化震惊 15.概念意义 17.礼貌原则 19.空间语言c?boiTOW and lend things often.d. build lifetime relationsliips.5? Which statement regarding haptics is incorrect?a.Ill Thailand, it is offensive to touch the head.b.Japan is considered a "don't touch11 culture.c.Greece is considered a ”touch” culture?d.Ill Latin American comitries, touching between men is miacceptable.6. The opinion that everyone has a position and clearly defined privileges is _________ ?a. a view of hierarchical stmcture of social relationshipb. a view of group orientation stmctiue of social relationsliipc. a view of individual orientation stnictiire of social relationshipd.none of the above7? General guidelines to follow when conversing with someone from another cultiu e include all of the following except:a.politics is a safe topic in most cultures.b.avoid telling jokes.c.avoid personal questions.d.keep the conversation positive.8.Wliich statement best describes an incorrect handshake?a.Ill the U.S., a handshake should be firm.b? An Asian handshake is usually gentle.c.Germans repeat a bnisque handshake upon anival and departure.d. A British handshake is film and repeated frequently.9.Which statement refeniiig to thouglit patterns is inconvct?/doc/fb7560339.html,ns typically use theinductive method of reasoning.b? Thouglit patterns impact oral conmiunication.c.Wlien using the deductive method of reasoning, one starts with the facts andgoes to generalizations.d.Recognizing different thouglit patterns is important in negotiation withdifferent cultxues.10.Which statement is incoiTect?a.Costly business blunders are often the result of a lack of knowledge ofanother cultured nonverbal conmiimication patterns.b.Processes of reasoning and problem solving are the same in all cultures?c.Attitudes toward time and use of space convey nonverbal messages inintercultiiral encounters.d.When in another culture、an appropriate caution would be to watch thebehavior of the persons you are talking with and match their style.11 ? Language is important because it ________a.helps us shape concepts, contiols how we think, and controls how we perceiveothers.b.allows us to be understood by foreigners?c.is determined by colonialism?d.is stable, easily understood, and free of diversity.12.Which of the following comitries uses liigh-context language?a.Canadab.Germanyc.Japand.United States13.Slang is generally _______a.imderstood by everyone.b.spoken by the masses.c.easily translated./doc/fb7560339.html,ed by subgroups?14.Nonverbal conmiunication does not include __________a.chromatics.b.clironemics.c.haptics.d.semantics.15.Dominance, harmony and subjugation are all value orientations that conespond to。

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INDUCTIVE REASONING IN THE JUSTIFICATION OF THE RESULT OF ADDING TWO EVEN NUMBERSConsuelo Cañadas Santiago Universidad de Zaragoza, SpainEncarnación Castro Martínez Universidad de Granada, SpainIn this paper we present an analysis of the inductive reasoning of twelve secondary students in a mathematical problem-solving context. Students were proposed to justify what is the result of adding two even numbers. Starting from the theoretical framework, which is based on Pólya’s stages of inductive reasoning, and our empirical work, we created a category system that allowed us to make a qualitative data analysis. We show in this paper some of the results obtained in a previous study. Proof appears as a real problem in different educational levels. On one hand, although pre-service teachers are accustomed to do many formal proofs, they have difficulties in proof teaching (Cañadas, Nieto & Pizarro, 2001). On the other hand, secondary students do not make as much progress as they are supposed to do. One possible reason lies on the fact that they cannot suddenly acquire the necessary reasoning skills for developing formal proof. They need a period of time to transform their daily reasoning into formal one (Jones, 1996). Some studies show that primary and secondary students are able to formulate conjectures, examine and justify them if they start working from particular cases (Lampert, 1990; Healy & Hoyles, 1998). These actions are part of inductive reasoning process. We are interested in analyzing this reasoning as a process instead that a result.This paper consists of four main parts. First, we present the theoretical framework of the study reported. Second, we show the methodology used to analyze inductive reasoning process. Third, we present some sample data and finally, we discuss some results of the study.THEORETICAL FRAMEWORKProof and reasoningMany studies on mathematical proof use rigor as a criterion for establishing a rank between different types of proofs. The sort of reasoning and the type of language used determine rigor. Proofs where inductive reasoning is predominant must be in a lower level that proofs which involve deductive reasoning. In this sense, if we find a proof in which drawings or concrete numbers are involved and where inductive reasoning is predominant, it will be considered an informal proof. On the other hand, as much deductive reasoning and algebraic language involve in a proof, more formal it will be considered (Cañadas, Castro & Gómez, 2002).Due to the aforementioned relation between a proof and the reasoning involved, we will consider proof as a formal part of the reasoning process. Reasoning is to givereasons for explaining a fact. A chain of reasoning with some rigor characteristics (which includes sort of reasoning and language) leads to a formal proof. Mathematical induction is a formal proof based more on deductive than on inductive reasoning, so, it is not considered part of inductive reasoning. We have to mention that some processes of inductive reasoning finish with mathematical induction but it does not always occur. The task proposed to the students in this study cannot be justified by mathematical induction.Inductive reasoningInductive reasoning and deductive reasoning are the two traditional types of reasoning considered. Inductive reasoning is the natural reasoning that allows us to get the scientific knowledge (Pólya, 1967). Neubert & Binko (1992) connect inductive reasoning in Mathematics with patterns recognizing and its application to Numbers Theory. Pólya indicates inductive reasoning in mathematics teaching as a method to discover properties from phenomena and to find the regularities in a logical way.Inductive reasoning in Mathematics Education is the reasoning process that begins with particular cases and gets the generalization from these cases. Pólya (1967) indicates four steps for a correct process of inductive reasoning: experiences with particular cases, conjecture formulation, conjecture proof and verification with new particular cases. Based on these steps and taking into account our previous study (Cañadas, 2002), we will consider the following actions related to the justification of a statement where inductive reasoning appears:− Observation of particular cases. The starting point is the experiences with particular cases of the problem set out. We observe if the students used particular cases spontaneously, which types of particular cases they used and how many.−Organization of particular cases. It can be used some strategies to systematize and facilitate work with particular cases. The most common strategy used is the organization of particular cases by data lists or tables (Allen, 2001; Grupo PI, 2002). −Search and prediction of Patterns. Pattern is an important notion in Mathematics based on the idea of repeated and regular situation. Find patterns and use them is an important strategy in mathematical problem solving (Steen, 1988; Stacey, K. & Groves, 1989). Mathematics is considered by Keith (1994) as the patterns science.−Conjectures formulation. A conjecture is a statement based on empirical facts, which has not been validated. In this report we seek that students formulate their own conjectures.−Conjectures validation. The conjecture can be true for some particular cases. We are sure about the truth of such conjecture for those specific cases but not for other ones.−Conjectures generalization. Based on the knowing that a conjecture is true for some particular cases, we hypothesize that the conjecture is also true for moreparticular cases apart from the first ones. Generalization is one of the actions considered by Pólya as one of the basis of inductive reasoning.−General conjectures justification. The first step in the way to confirm or reject a general conjecture is validating it with particular cases. These cases never draw us to confirm the veracity of the general conjecture. It is necessary to give reasons that explain the conjecture with the intention of convincing another person to justify the generalization. We look for a fair examination of the conjecture and, if it is necessary, we will do a formal proof as the latest justification that guarantee the veracity of such conjecture.METHODOLOGYTo observe the students working, we chose a methodology based on individual interviews. The interviewer (represented by “I” in collection and data analysis) was one of the researchers. She had an interview plan which allowed her to guide students by questions so that we can observe their reasoning in getting the problem solution (Cohen & Manion, 1990).Due to curriculum indications and our research objective, this study involved secondary students. We interviewed twelve Spanish students (six girls and six boys) from the four years that comprise Secondary Education in Spain. We interviewed three students from each year. Academic results were the main criterion to choose the three students from each year in order to have a wide variety of answer. For presenting and analyzing the data, we symbolize the students as 1, 2, 3 and 4 depending on the year s/he belongs to. We assigned A, B or C for high, medium or low academic results. For example, 3A is a third year student whose academic results are higher than her/his classmates.In Mathematics Education, problem solving is a highly formative task due to the knowledge, skills and reasoning that it arises (Segovia & Rico, 2001). Spanish curriculum recognizes problem solving as one of the main objectives in secondary mathematics because it is supposed to develop reasoning abilities and provide typical attitudes and habits of mathematical work (Boletín Oficial del Estado, 2003). In this report, problem solving is used to arise the reasoning of secondary students.The task we proposed to these students was to justify the result obtained when adding two even numbers. The main aspect was the justifications of their reasoning, paying more attention to the process instead of the result. This task was adequate because it came up actions related to inductive reasoning (generalization, patterns…) and it refers to Numbers Theory. Moreover, we took into account that the students belonged to an educational level where they are supposed to be able to solve the task.DATA COLLECTION AND ANALYSISWe collected data in three ways: the interview was recorded in an audio tape, we gave worksheets to the students so that they could write their work if they wanted andthe interviewer took notes during and after each interview about relevant research aspects impossible to be registered in the tape.We completed the interviewer’s notes with the students’ worksheets. We transcribedthe tapes and introduce them, together with the interviewer notes, in a qualitative data analysis program, Nud*ist revision (N4). This program allowed us to see the data in a structured way and to discover details, patterns and relations that would be more complicated by hand.In the following table we show the category system used in this study. The categories emerged from the actions mentioned in the theoretical framework. For almost all the categories, we considered subcategories which emerged from other studies (Goetting, 1995; Edwards, 1999; Miyakawa, 2002).CATEGORIES SUBCATEGORIESObservation of particular cases SpontaneityProposed by interviewer Number of particular cases Sort of particular casesOrganization of particular cases Systematic way TablesSearch and prediction of patternsConjectures formulation Use of school knowledge Conjectures validationConjectures generalization Characterization of even numbersGeneral conjectures justification Justification necessity Based on particular cases General caseKind of languageRESULTSAll the students gave an answer equivalent to “the result of adding two even numbers is another even number”. We based on the aforementioned category system to present the results. All the subcategories did not appear in the analysis of this task. We summarize some of the main results in two tables and then we will make some comments in relation with the determinated categories. The first table refers to particular cases and the second one shows students’ advance forward the general case.Table 1. Particular cases ObservationSpont 2 is oneaddendOrganization Patterns1C X X1B X X1A X X2C X2B X2A3C X X3B X X3A X4C X X4B X X4A X- Observation of particular cases. Just two students (1B and 2B) mentioned particular cases before giving the correct answer. All the students stated that the result of adding two even numbers is another even number. As we can see in the table, six students turned to particular cases in a spontaneous way, without any interviewer’s indication. Finally, when they were required to justify their conjectures, all the students used specific numbers, although in different ways. 1A, 2A, 3A and 4A used particular cases just as support of their reasoning because they look for the justification in the general case. The rest of them used particular cases when they try to justify their conjectures.All the students used 2 as a highlighted number in their justification. Four of them (1C, 1B, 4C and 4B) considered 2 as one of the addends in all their sums. All the students mentioned in their reasoning small and close numbers. No students mentioned even numbers higher than 18 as addends. They tried to work with higher numbers when the interviewer guided them.I: ok… and what happens for example, with 1784 and 2320?- Organization of particular cases. There were no students who organized particular cases. Just two of the students (2A and 3B) used written language for particular cases, listing them.- Search and prediction of patterns. The interviewer guided the students’ work with particular cases so that they could find regular situation for even numbers. Half of the students (1C, 1A, 3C, 3B, 3A and 4A) noticed that even numbers ended by an even digit, which is the most frequent characterization used by the students. We found among them the students who finally justify correctly their conjecture.- Conjectures formulation. All the students formulated the right conjecture. They were convinced that the sum of two even numbers is another even number.In this study, the inductive process before the conjecture formulation is short. One reason for this is that all the students seemed to previously know the result of the proposed task. Table 2. General caseGeneralJustification Characterization LanguageTermination Divisib ParticularcasesNatural Written Algeb 1C X X1B1A X X2C X2B X X X2A X X X3C X X3B X X X3A X X X4C X4B X X4A X- Conjectures validation. 2B, 2C, 3C, 3A based their justification on particular cases and they did not make progress to general justification.I: imagine that you have to convince me that the result is always an even number. 2C: Then... we had to... maybe... in a practical way, hadn ’t we? For example, tell me an even number.I: ok, one thousand and seven hundred.2C: One thousand and seven hundred… and another even number, for example four. You see? Then, I add both numbers and the result is an even number.- Conjectures generalization. All the students except 2C based their reasoning on even numbers characterizations. 3A noticed the common termination of even numbers but did not advance in this sense. Four students (2B, 2A, 4C and 4B) detected this pattern of even numbers. Then, they used the characterization of even numbers as divisors or multiples of number 2. 2B, 2A and 4B turned to algebraic language to express this characterization and write 2x to represent an even number. The difficulties in the task appeared when they were required to justify this conjecture. After getting the general conjecture, two students (4C and 4A) proved it with new particular cases but they did not justify it.- General conjectures justification. Three students (1A, 2C and 4B) recognized the necessity of justifying their result by their own. The rest of them saw the result as anevident statement obtained from particular cases, without any additional justification to be convinced of its truth.Some of the students used the termination characterization (1C, 1A, 3B, 3C and 4A) of even numbers. They based their justifications on even numbers smaller than 10 and they argued that all the sums made with numbers whose termination was 0, 2, 4, 6 or 8, gave as result another number that ended in 0, 2, 4, 5, or 8. This means that the result was another even number.2B, 2A, 3A, 4C and 4B mentioned divisibility criterion. 2A, 4C and 4B used this criterion and the common factor concept. Just one of them (2A) justified her conjecture with algebraic language:2x (represents) an even number2y (represents) another even number2x + 2y = 2 (x + y) and this is another even number because it appears multiplied by 2. Another of these students (4B) made the right justification spoken:Because all the numbers can be decomposed in a number multiplied by two. Then, for adding two even numbers, you can get the number two as a common factor and you have the number two multiplied by a number and then, this is an even number too.Finally, seven students justified the general conjecture correctly using natural, written or algebraic language (1C, 1A, 2B, 2A, 3B, 4B and 4A).DISCUSIONSWe made a categories and subcategories system basing on the theoretical framework and previous studies related to inductive reasoning but due to specific characteristics of this report, some of the considered subcategories did not come into view in the data analysis.Inductive reasoning appeared in an implicit or explicit way in all the students. Students turned to particular cases when they try to justify their conjecture, so, we can conclude that inductive reasoning appears naturally on these educational levels. Although all of the students stated directly the result of adding two even numbers, the justification for the general case was considered difficult by them. Many students considered evident the conjecture from particular cases and did not think of the necessity of a general conjecture justification to validate their statements. This was an obstacle because it did not allow them to make progress in their reasoning forward the generalization.The seven students who justified the general conjecture found a mathematical pattern from particular cases obtained from the characterization used in their justifications. This confirms that searching pattern is a relevant step in inductive reasoning process and an important strategy in mathematical problem solving.We did not notice significant differences among students’ reasoning of different educational levels. It happened in the same way with students with different academicresults belonging to the same year. We just detected some differences in the way they expressed their argumentations. First year students used natural language and not used neither written nor algebraic language.ReferencesAllen, L. G. (2001). Teaching Mathematical Induction. An Alternative Approach.Mathematics Teacher, 94, 500-504.Bolletín Oficial del Estado. (2003). Real Decreto, 832/2003.Cañadas, C., Castro, E. & Gómez, P. (2002). Didactical reflections about some proofs of the Pythagorean proposition. En A. Cockburn & E. Nardi (Eds.). Proceedings of the 26th International Group for the Psychology of Mathematics Education, England, 177-184. Cañadas, M. C. (2002). Razonamiento inductivo puesto de manifiesto por alumnos de secundaria. Granada: Universidad de Granada.Cañadas, M. C., Nieto, E. & Pizarro, A. (2001). El valor de la demostración en la educación secundaria. En J. Berenguer, B. Cobo y J. M. Navas (Eds.). Proceedings of Investigación en el aula de matemáticas: retos de la educación matemática del siglo XXI, España, 167-172.Cohen, L. & Manion, L. (1990). Métodos de investigación educativa. Madrid: Editorial La Muralla.Edwards, L. D. (1999). Odds and Evens: Mathematical Reasoning and Informal Proof among High School Students. Journal of Mathematical Behavior, 17, 4, 489-504. Goetting, M. (1995). The college student´s understanding of mathematical proof. Doctoral Disseration. Philosophy Department of University of Maryland.Healy, L. & Hoyles, C. (1998). Justifying and Proving in School Mathematics. (Technical Report on The Nationwide Survey. Institute of Education, University of London, February, 1998.Jones, L. (1996). A Developmental approach to proof. Proceedings of the 8th International Congress on Mathematical Education, 109-154.Keith. D. (1994). Mathematics. The Science of Patterns. Scientific American Library: New York.Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-36.Miyakawa, T. (2002). Relation between proof and conception: the case of proof for the sum of two even numbers. Proceedings of the 26th International Group for the Psychology of Mathematics Education, 3, 353-360.Neubert, G. A. & Binko, J. B. (1992). Inductive reasoning in the secondary classroom.Washington D.C.: National Education Association.Pólya, G. (1966). Matemáticas y razonamiento plausible. Madrid: Tecnos.Pólya, G. (1967). Le découverte des mathématiques. París: DUNOD.Segovia, I. y Rico, L. (2001). Unidades didácticas. Organizadores. En E. Castro (Ed.), Didáctica de la Matemática en la Educación Primaria (pp. 83-104). Madrid: Síntesis. Stacey, K. & Groves, S. (1999). Resolver problemas: estrategias. Madrid: Narcea.Steen, L. (1988). The Science of Patterns, Science, 611-616.。