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孟子的辩论技巧英文回答：As a skilled debater, Mencius employed a diverse arrayof persuasive techniques to effectively convey his arguments and dismantle opposing viewpoints. These included:Logical Reasoning: Mencius constructed compelling arguments based on sound logic and clear reasoning. By meticulously dissecting his opponents' claims andidentifying inconsistencies, he adeptly exposed their flaws and undermined their credibility.Inductive Reasoning: He drew persuasive inferencesfrom specific observations and examples. By presenting a series of carefully selected cases that supported his assertions, Mencius gradually built a strong case for his position, making it difficult for his opponents to refute.Analogies and Metaphors: Mencius frequently employedanalogies and metaphors to illustrate complex concepts and make his arguments more accessible and relatable. By comparing abstract ideas to familiar objects or experiences, he effectively appealed to his audience's intuition and common sense.Emotional Appeals: Recognizing the power of emotionsin persuasion, Mencius skillfully incorporated emotional appeals into his arguments. By evoking feelings of empathy, sympathy, or indignation, he engaged his audience on a deeper level, bridging the gap between logic and passion.Rhetorical Questions: Mencius posed thought-provoking rhetorical questions to challenge his opponents' assumptions and sow doubt in their minds. By compellingthem to question their own beliefs, he subtly weakenedtheir arguments and fostered a sense of uncertainty.Audience Engagement: Mencius actively engaged with his audience, using humor, wit, and personal anecdotes to connect with them on a human level. By establishing arapport with his listeners, he increased theirreceptiveness to his ideas and created a more persuasive atmosphere.For instance, in his debate with King Xuan of Qi, Mencius employed his mastery of logical reasoning to expose the king's misguided views on human nature. By skillfully dismantling the king's flawed arguments, Mencius demonstrated the inherent goodness of human beings and the importance of benevolent governance.中文回答：作为一名高超的辩论家，孟子运用了多种多样的说服技巧，来有效地传达自己的论点，并击破对方的观点。
高二英语哲学讨论单选题30题1. In the philosophical debate, the term "metaphysics" refers to the study of _____.A. the nature of realityB. human behaviorC. social systemsD. language structure答案：A。
“metaphysics”意为形而上学，主要研究现实的本质。
选项B“human behavior”指人类行为；选项C“social systems”指社会系统；选项D“language structure”指语言结构。
在哲学讨论中，“metaphysics”通常指对现实本质的研究，故选A。
2. When discussing philosophy, the phrase "epistemology" is concerned with _____.A. moral valuesB. knowledge and beliefC. artistic expressionD. economic systems答案：B。
“epistemology”指认识论，主要涉及知识和信念。
选项A“moral values”是道德价值观；选项C“artistic expression”是艺术表达；选项D“economic systems”是经济系统。
哲学中“epistemology”侧重于知识和信念方面，所以选B。
3. In a philosophical context, "ontology" is the branch of philosophythat examines _____.A. beauty and aestheticsB. the nature of beingC. political theoriesD. logical reasoning答案：B。
In the realm of English composition,writing an essay that generously offers valuable advice requires a thoughtful approach and a clear structure.Heres a detailed guide on how to craft such an essay:1.Introduction:Begin with a captivating opening that introduces the topic and sets the tone for the advice you will provide.You may want to use a quote,a rhetorical question, or a short anecdote to engage the reader.2.Statement of Purpose:Clearly state the purpose of your essay.This is where you outline the main advice or suggestions you will be offering.3.Body Paragraphs:Develop your body paragraphs around the main points of advice. Each paragraph should focus on a single piece of advice,starting with a topic sentence followed by explanations,examples,or evidence to support your point.Paragraph1:Offer the first piece of advice.Explain why it is important and how it can be beneficial.Provide a reallife example or a hypothetical scenario to illustrate your point.Paragraph2:Present the second piece of advice.Discuss its relevance and e statistics,research findings,or expert opinions to back up your suggestion.Paragraph3:Continue with additional advice.Ensure that each piece of advice is distinct and contributes to the overall message of your essay.4.Counterarguments:If applicable,address potential counterarguments or concerns that readers might have regarding your advice.This shows that you have considered different perspectives and strengthens your argument.5.Conclusion:Summarize the main points of your essay and reiterate the importance of your advice.End with a strong closing statement that leaves a lasting impression on the reader.6.Style and Tone:Use a formal and respectful tone throughout your essay.Be concise and clear in your language to ensure your advice is easily understood.7.Proofreading:Before submitting your essay,proofread it for grammatical errors,clarity, and coherence.Make sure your advice is presented logically and flows well from one point to the next.8.Citations:If you have used any sources to support your advice,ensure you cite themproperly to avoid plagiarism and to give credit to the original authors.Heres a brief example to illustrate the structure:Title:The Art of Effective Time ManagementIntroduction:In todays fastpaced world,time is a precious commodity.This essay aims to provide practical advice on mastering the art of time management.Statement of Purpose:The following advice will guide readers through the process of organizing their schedules, prioritizing tasks,and avoiding common timewasting pitfalls.Body Paragraphs:Firstly,setting clear goals is crucial for effective time management.By knowing what you aim to achieve,you can allocate time wisely.Secondly,prioritizing tasks based on importance and urgency can significantly improve productivity.The Eisenhower Matrix is a useful tool for this purpose.Lastly,taking regular breaks can actually enhance focus and efficiency.The Pomodoro Technique is one method that incorporates short breaks into a work routine.Counterarguments:While some may argue that strict time management can lead to stress,the advice provided encourages a balanced approach that includes rest and flexibility.Conclusion:In conclusion,by implementing the advice offered in this essay,individuals can take control of their time,leading to increased productivity and a more balanced life.Remember,the key to a successful advice essay is to provide actionable,relevant,and practical tips that readers can apply to their own lives.。
旧有建筑的处理方式——六种观点直到近几年来，于建筑改造，界内流行的原则是旧建筑和新建筑必须明确区分开来。
在那些将新旧建筑对立起来的案例中可以明显感受到这一点：它们通保留原有建筑和新建筑产生对比，并以一条狭窄的交接缝将新旧区分。
与此同时，改造建筑的方式逐渐多元化：从保留旧有建筑特定时期的气质（例如柏林新博物馆），到尝试在原有建筑的设计概念下进行更新与深化，最后到实验性的改造方式。
在下文中，六位涉足该领域的著名建筑师，深入地阐述了他们在处理旧建筑时的态度。
一对现存肌理的阅读现代建筑将建筑师的形象转变为独立设计者，他们对整个设计过程拥有完全的控制权。
而在别人设计的建筑上继续工作，是和这一形象相矛盾的---正如人们很难认同一位修改别人的音乐、绘画、文学或电影作品的艺术家。
或许这就是为何旧建筑都被取而代之，没有历史积累的现代主义仍视历史建筑为边缘作品的原因。
因此，在二十世纪上半年，当代思潮鲜有体现对旧有营造物的关怀。
直到二十世纪末，欧洲建筑界对于上世纪的建筑的态度逐渐改变——这是一种回应，针对的是发生在欧洲城市中那些不可逆的改造行为。
在那里，老建筑不幸地被新建筑野蛮取代。
自1980年后，旧建筑的更新改造作为城市可持续发展的手段，人们逐渐意识到其重要性。
无论是单纯的建筑保护还是实验性改造，一些近年来完成的重要改造项目都开始关注建筑的过去，而这一点正是被现代建筑所忽视的。
一位改造建筑师，必然会面临旧建筑在不断改变的问题。
当推进一个改造项目时，无论该建筑的年龄是长是短，我们产生一种奇怪的错觉：这个建筑似乎拥有自我改变的能力，他们的存在是基于时间和空间以外的某种东西，因此我们的任务最终仅仅是揭示这座建筑的内在密码。
我们喜欢这个想法，那就是每一座建筑都可以表明自己该如何被对待，我也们必须具备解读这些指示的能力，因为它会告诉我们要如何去延伸、包围、抽离、隐藏或拆分这座建筑。
更新设计或适应性改造也许就是破解原来设计者隐藏的意图，就像阅读一本重写本一样阅读建筑，去揭示那最难以破译的原始篇章。
The Greeks Assumed That the Structure of LanguageIntroductionLanguage is a fundamental aspect of human communication and plays a significant role in shaping our thoughts and ideas. The Greeks, renowned for their contributions to philosophy and literature, also pondered over the nature and structure of language. This article aims to delve intothe Greek assumptions regarding the structure of language, exploringtheir theories and implications.Origins of Greek Linguistic ThoughtThe Greek fascination with language can be traced back to prominent philosophers such as Plato and Aristotle. Plato believed that language was not a mere tool for communication but a reflection of the ultimate reality. According to him, words and their meanings were not arbitrarybut had a deeper connection to the essence of objects or concepts. Aristotle, on the other hand, studied language from a more empirical perspective, focusing on its function and structure.Greek Assumptions about Language StructureThe Greeks made several assumptions about the structure of language,which had a profound impact on subsequent linguistic thought. These assumptions include:1. Words Reflect RealityThe Greeks assumed that words had an inherent connection to the objectsor concepts they represented. They believed that through language, individuals could access and understand the true nature of reality. This assumption laid the foundation for the philosophical concept of “logos,” which refers to the relationship between words and reality.2. Language Is Composed of Basic ElementsThe Greeks recognized that language could be broken down into smaller units with distinctive meanings. They postulated that these basic elements, known as morphemes, combined to form words. This assumption paved the way for the development of morphological analysis in linguistics, which studies the internal structure of words.3. Syntax and Grammar Govern LanguageAncient Greek philosophers acknowledged the importance of syntax and grammar in organizing and conveying meaning. They recognized that language followed specific rules and structures that determined the relationships between words in a sentence. This assumption laid the groundwork for syntactical analysis, which explores the arrangement of words and phrases in a sentence.4. Language Is InnateThe Greeks assumed that the ability to acquire and understand language was innate to humans. They believed that language proficiency stemmed from natural predispositions rather than external influences. This assumption aligns with modern theories of language acquisition, such as Noam Chomsk y’s concept of a Universal Grammar.Implications of Greek Linguistic ThoughtThe Greek assumptions about language structure had far-reaching implications for various disciplines, including linguistics, philosophy, and literature. Some of these implications are:1. Language as a Mirror of RealityThe concept of language reflecting reality influenced subsequent philosophical and metaphysical thought. It prompted thinkers to explore the relationship between language, perception, and knowledge. This exploration ultimately shaped diverse philosophical schools, such as phenomenology and hermeneutics.2. Development of Linguistic AnalysisThe Greek assumptions regarding the composition of language elements and the importance of syntax and grammar laid the groundwork for linguistic analysis. These assumptions influenced the development of structural linguistics, generative grammar, and other linguistic theories that investigate the form and function of language.3. Influence on Literary StylesGreek linguistic thought permeated literary works, influencing writing styles and literary devices. Writers began incorporating rhetorical techniques, such as metaphors and analogies, to convey deeper meanings and evoke emotional responses. These techniques shaped the foundations of poetry, prose, and dramatic literature.4. Evolution of Language EducationThe Greek assumptions about language being innate and governed by rules contributed to the development of language education methodologies. They inspired instructional approaches that emphasize the systematic teaching of grammar, syntax, and vocabulary. These approaches continue to influence language teaching methodologies worldwide.ConclusionThe Greeks’ assumptions about the structure of language have left an indelible mark on human understanding and exploration of linguistic phenomena. Their belief that language reflects reality, the recognition of basic language elements, the importance of syntax and grammar, and the innate nature of language have shaped various disciplines. From philosophy to linguistics, and literature to education, the Greek assumptions continue to shape our understanding and appreciation of language.。
In the vast expanse of history,there have been numerous figures whose wisdom and foresight have illuminated the path for future generations.One such luminary is Ji Gangming,whose contributions and influence have spanned across the four corners of the world.This essay aims to explore the life and legacy of Ji Gangming,reflecting on his profound impact on society and the lessons we can learn from his example.Ji Gangming was born in an era marked by significant changes and challenges.His early life was characterized by a thirst for knowledge and a desire to understand the world around him.He was deeply influenced by the cultural and intellectual currents of his time, which inspired him to pursue a life of learning and inquiry.As he matured,Ji Gangmings intellectual pursuits led him to develop a unique perspective on the world.He believed in the power of education to transform individuals and societies,and he dedicated his life to the pursuit of knowledge and the dissemination of ideas.His teachings emphasized the importance of critical thinking,openmindedness, and a commitment to truth and justice.One of Ji Gangmings most significant contributions was his development of a comprehensive educational system that sought to promote intellectual growth and moral development.He believed that education should not be limited to the acquisition of factual knowledge but should also foster the development of character and the cultivation of virtues such as empathy,compassion,and a sense of social responsibility.Ji Gangmings educational philosophy was grounded in the belief that every individual has the potential to contribute positively to society.He advocated for an inclusive and egalitarian approach to education,arguing that access to learning opportunities should not be determined by ones social status or economic background.This vision of a more equitable and just society was a driving force behind his efforts to reform the educational system and expand access to education for all.In addition to his work in education,Ji Gangming was also a prolific writer and thinker. His writings covered a wide range of topics,from philosophy and ethics to politics and social issues.His ideas were characterized by a deep respect for human dignity and a commitment to the principles of justice,equality,and freedom.One of the most enduring aspects of Ji Gangmings legacy is his emphasis on the importance of dialogue and the exchange of ideas.He believed that true understanding and progress could only be achieved through open and respectful engagement with diverse perspectives.This commitment to dialogue and the pursuit of truth has inspired countless individuals and continues to resonate in todays world,where the need foropenmindedness and mutual respect is more important than ever.In conclusion,Ji Gangmings life and work serve as a shining example of the power of ideas and the transformative potential of education.His commitment to intellectual growth,moral development,and social justice has left an indelible mark on the world and continues to inspire generations of thinkers and doers.As we reflect on his legacy,we are reminded of the importance of pursuing knowledge,fostering openmindedness,and working towards a more just and equitable society.。
关于实验是检验真理的唯一标准英语作文全文共3篇示例，供读者参考篇1Experiment: The Only Yardstick for Measuring TruthTruth, that elusive and coveted prize that humanity has chased after for millennia. We've constructed elaborate philosophies, devised ingenious thought experiments, and spent countless hours pondering and debating what constitutes truth and how to discern it from fiction. Yet, amid this intellectual odyssey, one approach has emerged as the undisputed champion, a beacon of light cutting through the fog of speculation and conjecture – the scientific experiment.As a student, I've been taught to revere the sanctity of the scientific method, to view it as the ultimate arbiter of truth in a world often clouded by biases, assumptions, and unfounded beliefs. Through rigorous experimentation, we can strip away the veneers of preconceived notions and subject our hypotheses to the unforgiving crucible of empirical evidence.The strength of the experiment lies in its objectivity and replicability. It transcends the limitations of individualperspectives, cultural biases, and ideological leanings, offering a universal language that any rational mind can comprehend. When conducted with precision and adherence to established protocols, an experiment becomes a testament to the pursuit of truth, a beacon guiding us through the labyrinth of uncertainty.Consider the countless breakthroughs and paradigm shifts that have reshaped our understanding of the world, from Galileo's revolutionary observations of the heavens to the groundbreaking experiments of Marie Curie that unveiled the mysteries of radioactivity. Each of these monumental discoveries was forged not in the realm of abstract theorizing but through meticulous experimentation, where hypotheses were put to the ultimate test, and nature itself was allowed to speak its truth.The beauty of the experiment lies in its ability to challenge our preconceptions and shatter long-held beliefs. It acts as a bulwark against the insidious influence of dogma, forcing us to confront reality head-on and embrace the uncomfortable truths that may contradict our cherished notions. The annals of science are replete with examples of experiments that have upended conventional wisdom, from the earth's revolution around the sun to the counterintuitive principles of quantum mechanics.Moreover, the experiment fosters a culture of intellectual humility, a recognition that our understanding of the universe is ever-evolving and subject to constant refinement. It reminds us that truth is not a static entity to be grasped once and for all but a dynamic pursuit, a journey of continuous exploration and discovery. Through experimentation, we acknowledge the limitations of our current knowledge and remain open to the possibility of revising our beliefs in the face of new evidence.Yet, the power of the experiment extends far beyond the realms of natural sciences. In the social sciences, carefully designed experiments have illuminated the intricate workings of human behavior, shedding light on topics as diverse as decision-making, social dynamics, and cognitive biases. By isolating and manipulating variables in controlled environments, researchers can tease apart the complex tapestry of human interactions, uncovering truths that would otherwise remain obscured by the noise of everyday life.Even in the abstract domains of mathematics and logic, the experiment plays a crucial role. Through the construction of formal systems and the derivation of theorems, mathematicians and logicians engage in a form of intellectual experimentation, subjecting their axioms and conjectures to the rigors of logicalscrutiny. The truth of a mathematical statement is not determined by mere assertion but by its ability to withstand the relentless probing of logical deduction and proof.Of course, the experiment is not without its limitations. It is a tool, and like any tool, it can be misused or misinterpreted. Flawed experimental designs, measurement errors, and selective reporting of results can lead us astray, obscuring the truth rather than revealing it. This is why the scientific community places such emphasis on rigorous peer review, replication studies, and a commitment to transparency and integrity in the experimental process.Furthermore, there are realms of inquiry where the experiment may not be applicable or practical, such as in the study of historical events or in the exploration of certain metaphysical and philosophical questions. In these domains, we must rely on other modes of inquiry, such as textual analysis, logical argumentation, and reasoned discourse, while maintaining a healthy skepticism and a willingness to revise our beliefs in the face of new evidence.Yet, despite these caveats, the experiment remains the gold standard for testing truth, a beacon that guides us through the murky waters of uncertainty and conjecture. It is a testament tothe human spirit's insatiable curiosity and our relentless pursuit of knowledge, a pursuit that has yielded countless wonders and revelations about the universe we inhabit.As a student, I have been indelibly shaped by this reverence for the experiment and the scientific method. It has instilled in me a deep appreciation for the power of evidence, a respect for the rigor of the scientific process, and a commitment to intellectual honesty. It has taught me to question assumptions, to embrace uncertainty, and to remain open to revising my beliefs in the face of compelling evidence.More importantly, the experiment has imbued me with a sense of wonder and awe at the grandeur of the universe and the boundless potential of human inquiry. Each time a hypothesis is tested, a new door is opened, revealing glimpses of truth that were previously obscured. It is a journey of endless discovery, where each answer begets a multitude of new questions, propelling us ever forward in our quest for understanding.In a world often beset by dogmatism, misinformation, and the allure of convenient fictions, the experiment stands as a beacon of hope, a reminder that truth is not a matter of opinion or belief but a pursuit rooted in evidence and reason. It is a call to embrace intellectual humility, to shed our preconceptions,and to fearlessly confront the unknown, armed with the tools of scientific inquiry and a steadfast commitment to uncovering the truths that lie beyond the veil of our limited perceptions.So, as I embark on my academic and professional journey, I carry with me this unwavering conviction: the experiment is not merely a tool for testing truth but a way of life, a embodiment of the human spirit's insatiable thirst for knowledge and understanding. It is a torch that illuminates the path forward, guiding us towards a future where truth reigns supreme, and the boundaries of our understanding are continually pushed ever outward, into the vast expanse of the unknown.篇2Experimentation: The Sole Criterion of Truth?As a student grappling with the complexities of epistemology – the study of knowledge and its acquisition – I find myself drawn to the notion that experimentation is the sole criterion of truth. This assertion challenges the traditional methods of acquiring knowledge and raises pertinent questions about the nature of truth itself. In this essay, I will delve into the merits and limitations of this stance, drawing upon philosophicalinsights and empirical evidence to present a comprehensive analysis.The proposition that experimentation is the sole arbiter of truth finds its roots in the empirical tradition, which emerged during the Scientific Revolution of the 16th and 17th centuries. Thinkers such as Francis Bacon and René Descartes advocated for a systematic and methodical approach to understanding the natural world, rejecting the authority of ancient texts and embracing the power of observation and experimentation.Proponents of this view assert that truth can only be established through controlled, replicable experiments that test hypotheses against empirical data. This approach places a premium on objectivity, rigorous methodology, and the ability to reproduce results. By subjecting our assumptions to the scrutiny of empirical inquiry, we can weed out unfounded beliefs and superstitions, allowing us to uncover the underlying principles that govern the universe.The success of the scientific method in unveiling the mysteries of the natural world lends credence to this perspective. Through experimentation, we have unraveled the intricacies of physics, chemistry, biology, and myriad other disciplines, enabling technological advancements that have transformed ourlives. The theories and laws derived from empirical investigations have withstood the test of time, serving as the bedrock of our understanding of the universe.Moreover, the reliance on experimentation fosters a spirit of skepticism and critical thinking, which are essential for the pursuit of truth. By constantly challenging our assumptions and subjecting them to empirical verification, we safeguard against the pitfalls of dogmatism and blind acceptance of authority. This approach encourages intellectual humility, as even the most well-established theories must be continuously scrutinized and refined in the face of new evidence.However, it would be remiss to adopt an unwavering stance on experimentation as the sole criterion of truth without acknowledging its limitations and the existence of other legitimate modes of inquiry. While experimentation excels in the realm of the natural sciences, it may fall short in addressing questions of ethics, aesthetics, and metaphysics, which often defy empirical verification.For instance, how can we experimentally determine the inherent value of human life or the moral implications of our actions? The realm of ethics and morality is rooted in philosophical reasoning, cultural traditions, and subjectiveexperiences, which may not lend themselves readily to experimental methodologies. Similarly, our appreciation of art and beauty, while grounded in neural and psychological processes, transcends mere empirical analysis and involves subjective interpretations shaped by individual experiences and cultural contexts.Furthermore, the pursuit of truth is not solely confined to the observable and measurable aspects of reality. Metaphysical inquiries into the nature of existence, consciousness, and the fundamental constituents of the universe often engage with realms that lie beyond the reach of direct experimentation. While empirical evidence can inform and constrain our metaphysical theories, the ultimate truths about the origin and essence of reality may elude the confines of the experimental method.It is also important to acknowledge the inherent limitations of experimentation itself. Despite our best efforts to maintain objectivity and rigor, our experiments are subject to the constraints of our current technological capabilities, theoretical frameworks, and human biases. The history of science is replete with instances where flawed experimental designs, faulty data analysis, or cognitive biases led to erroneous conclusions that were later overturned by more rigorous investigations.Moreover, the reductionist approach inherent in experimentation may fail to capture the holistic and emergent properties of complex systems, leading to an incomplete understanding of the phenomena under study. The interplay of multiple factors, non-linear dynamics, and the inherent unpredictability of certain systems may defy the controlled conditions and simplifying assumptions of experiments, necessitating the integration of alternative modes of inquiry.In light of these considerations, a more nuanced perspective emerges: while experimentation is an indispensable tool in our quest for truth, it should not be regarded as the sole criterion. Instead, we must embrace a pluralistic approach that recognizes the complementary roles of various modes of inquiry, each contributing to our understanding of the world in unique and invaluable ways.Philosophical reasoning, introspection, and subjective experiences offer insights into the realms of ethics, aesthetics, and consciousness, domains that may elude the grasp of empirical investigation. Cultural traditions and indigenous ways of knowing can provide alternative perspectives and enrich our understanding of the human experience. Mathematical and logical reasoning can unveil truths about abstract concepts andformal systems, transcending the boundaries of the physical world.Ultimately, the pursuit of truth is a multifaceted endeavor that requires a synthesis of diverse modes of inquiry, each illuminating different facets of reality. Experimentation remains a pivotal component of this pursuit, providing a rigorous and systematic method for testing hypotheses and uncovering the underlying principles that govern the natural world. However, it is not the sole criterion of truth, but rather a powerful tool that must be wielded in conjunction with other modes of inquiry to achieve a more comprehensive and holistic understanding of the world we inhabit.As students and seekers of knowledge, our task is to cultivate a spirit of intellectual humility, recognizing the limitations of any single approach while embracing the richness and diversity of human inquiry. By integrating the insights gleaned from experimentation with those derived from philosophical, cultural, and subjective modes of understanding, we can navigate the complexities of truth with greater wisdom and depth, ultimately enriching our collective knowledge and enhancing our ability to comprehend the mysteries that surround us.篇3Experiment as the Sole Criterion of TruthThe quest for truth and knowledge has been an enduring pursuit throughout human history. As we navigate the complexities of the natural world, we are confronted with numerous assertions, theories, and beliefs that compete for our acceptance. In this landscape, the question arises: How can we discern truth from falsehood? Is there a universal standard by which we can evaluate the validity of claims? Many philosophers and scientists have grappled with this fundamental inquiry, and one perspective that has gained significant traction is the notion that experiment is the sole criterion of truth.At first glance, this proposition may seem overly simplistic or even radical. After all, the realm of human knowledge encompasses a vast array of disciplines, from the abstract realms of mathematics and philosophy to the tangible domains of the natural sciences. How can a single standard encompass such diversity? However, upon closer examination, the argument for experiment as the ultimate arbiter of truth holds considerable weight.The essence of this perspective lies in the recognition that empirical evidence, derived from carefully controlled and replicable experiments, provides the most reliable foundation for establishing objective truth. Unlike mere speculation, anecdotal accounts, or subjective interpretations, experiments offer a systematic and rigorous approach to testing hypotheses and uncovering the fundamental principles that govern the universe.One of the strongest arguments in favor of this view is the remarkable success of the scientific method, which relies heavily on experimentation. Throughout history, countless discoveries and technological advancements have been made possible through the application of experimental techniques. From the groundbreaking work of pioneers like Galileo and Newton to the cutting-edge research in fields like particle physics and molecular biology, experiments have consistently yielded insights that have reshaped our understanding of the world.Moreover, the power of experimentation lies in its ability to challenge and refine existing theories. By subjecting hypotheses to rigorous testing and scrutiny, experiments can either confirm or refute proposed explanations. This process of continuous questioning and verification is essential for advancing ourknowledge and ensuring that our beliefs align with empirical reality.Critics of this perspective may argue that not all domains of knowledge are amenable to experimental investigation. For instance, how can one conduct experiments to explore abstract philosophical concepts or subjective experiences? While this objection holds some merit, it is important to recognize that even in these realms, the principles of empiricism and verifiability remain paramount. Philosophical arguments and theories that cannot be subjected to any form of empirical scrutiny or logical analysis run the risk of becoming mere speculation or dogma.Furthermore, the notion of experiment as the sole criterion of truth does not necessarily preclude other forms of inquiry or knowledge acquisition. Rather, it suggests that any claim, whether derived from reason, intuition, or revelation, must ultimately be subjected to the litmus test of empirical verification through experimentation. This process may involve indirect methods, such as the analysis of observable phenomena or the construction of logical arguments based on empirical premises.Another compelling argument in favor of this perspective is the inherent objectivity and universality of experimental results. Unlike subjective interpretations or culturally specific beliefs,well-designed experiments transcend personal biases and can be replicated and verified by researchers across different geographical and cultural contexts. This universality of empirical evidence fosters a shared understanding of the natural world and promotes scientific collaboration on a global scale.However, it is crucial to acknowledge the limitations and potential pitfalls associated with experimental research. Experiments can be influenced by a variety of factors, including flawed experimental designs, measurement errors, and unconscious biases. Additionally, the interpretation of experimental results may be subject to varying theoretical frameworks or philosophical assumptions. These challenges underscore the importance of rigorous peer review, replication studies, and a commitment to continually refining experimental methodologies.Despite these limitations, the weight of evidence supporting the primacy of experimentation as the ultimate arbiter of truth is overwhelming. From the remarkable achievements of modern science to the consistent ability of experiments to challenge and revise longstanding beliefs, the empirical approach has proven itself as the most reliable path to uncovering objective truth.In conclusion, the proposition that experiment is the sole criterion of truth represents a powerful and compelling perspective. While acknowledging the limitations and potential objections, the overwhelming success of the scientific method and the inherent objectivity of empirical evidence strongly support this view. As we continue to explore the mysteries of the universe and seek to expand the boundaries of human knowledge, the principles of experimentation and empirical verification must remain at the forefront of our endeavors. Only through a steadfast commitment to empiricism and a willingness to subject our beliefs to rigorous testing can we hope to uncover the deepest truths of the natural world.。
辩论的技巧英文回答：As a seasoned debater, I've honed my skills through countless rounds of heated exchanges and persuasive arguments. Here are some of the key techniques that have helped me succeed in the art of debate:1. Research and Preparation:Thorough preparation is paramount. Delve deep into the topic, gathering credible evidence and building a solid foundation for your arguments. Anticipate potential counterarguments and prepare rebuttals in advance.Example: In a debate on climate change, I spent hours reading scientific papers, analyzing data, and consulting experts to build a comprehensive understanding of the issue.2. Logical Reasoning and Argumentation:Present well-reasoned arguments based on evidence and sound logic. Avoid emotional appeals and fallacies. Structure your points clearly, using transitions to guide your audience.Example: When arguing that the government should invest in renewable energy, I presented data on the environmental benefits, cost-effectiveness, and job creation potential of such investments.3. Effective Communication:Articulate your ideas clearly and concisely. Use vivid language, compelling examples, and relatable analogies to engage your audience. Adapt your communication style tosuit the context and audience.Example: In a debate targeted at a general audience, I used simple language and concrete examples to explain the complexities of monetary policy.4. Refutation and Counterarguments:Anticipate and address opposing viewpoints respectfully. Use evidence and logic to dismantle weak arguments and effectively counter your opponents' claims.Example: When my opponent argued that a proposed tax increase would stifle economic growth, I presented evidence showing that similar tax increases in other countries hadled to increased investment and job creation.5. Rebuttals and Closing Arguments:Prepare strong rebuttals to your opponents' arguments and use them to reinforce your own positions. Conclude your speech with a powerful summary of your main points and a compelling call to action.Example: In the closing remarks of a debate on immigration, I emphasized the positive contributions of immigrants to society and urged the audience to support policies that foster a welcoming and inclusive environment.6. Respect and Etiquette:Engage in respectful discourse, even when debating contentious issues. Avoid personal attacks and maintain a professional demeanor throughout the debate.Example: After a particularly heated exchange, I made a point of thanking my opponent for their insights and acknowledging the importance of diverse perspectives.7. Practice and Feedback:Regular practice is crucial. Join a debate team or engage in mock debates with friends or colleagues. Seek feedback from experienced debaters to improve your skills.Example: I participated in countless debate tournaments and sought mentorship from renowned coaches to refine my techniques and develop a strong debating style.中文回答：作为一名经验丰富的辩手，我通过无数轮激烈的交流和有力论证磨练了自己的技能。
The Guide to Critical ThinkingCritical thinking is an essential skill that is highly valued by employers and society at large. It allows individuals to assess information objectively, analyze complex problems, and make informed decisions. In this guide, we will explore what critical thinking is, why it is important, and some techniques for developing and improving this skill.What is Critical Thinking?Critical thinking is the ability to objectively assess information, arguments, and evidence in order to form reasoned judgments and make informed decisions. It involves being able to identify assumptions, evaluate arguments, and recognize different perspectives on a given issue. Critical thinking is not simply about being skeptical or contrarian, but about being able to analyze information in a systematic and logical way.Why is Critical Thinking Important?The ability to think critically is becoming increasingly important in our complex and rapidly changing world. With the abundance of information available through the internet and other sources, it is essential for individuals to be able to assess information reliably and make informed decisions. Critical thinking also allows individuals to identify biases and assumptions, evaluate arguments, and recognize when information is incomplete or unreliable.From an organizational perspective, critical thinking is valuable because it allows individuals to solve complex problems, make better decisions, and adapt to changing circumstances. Companies are looking for employees who are capable of analyzing information, identifying trends, and making strategic recommendations. In short, critical thinking is a fundamental skill for success in the modern world.Techniques for Developing and Improving Critical Thinking SkillsWhile some people may be naturally inclined to think critically, it is a skill that can be developed and improved through practice and effort. Here are some techniques for improving your critical thinking skills:1. Ask QuestionsOne of the most basic techniques for developing critical thinking is to ask questions. When presented with a statement or argument, ask yourself questions such as:•What evidence supports this claim?•What assumptions is this argument based on?•Are there alternative explanations for this phenomenon?•What are the implications of this argument?By asking questions, you can begin to identify underlying assumptions and evaluate the strength of an argument.2. Evaluate EvidenceAnother important aspect of critical thinking is the ability to evaluate evidence. When presented with information, you should ask yourself questions such as: •What is the source of this information?•Is the evidence reliable?•What is the context of this information?•Is there any contradictory evidence?By evaluating evidence, you can avoid being misled by unreliable or biased information.3. Consider Different PerspectivesA key aspect of critical thinking is the ability to consider different perspectives. This involves being able to recognize and evaluate arguments from different viewpoints. You should ask yourself questions such as:•What are the underlying assumptions of this argument?•What are the strengths and weaknesses of this argument?•Are there any alternative perspectives on this issue?Considering different perspectives can help you develop a more comprehensive and nuanced understanding of a given issue.4. Practice Analyzing ArgumentsAnalyzing arguments is one of the core skills of critical thinking. When presented with an argument, you should identify the premises and conclusion, and evaluate the logic of the argument. You can practice this skill by reading articles or essays and analyzing the arguments presented.5. Be CuriousFinally, one of the best ways to develop critical thinking skills is to be curious. Ask questions, seek out new information, and challenge your assumptions. By being curious, you can develop a deeper understanding of the world around you and become a more informed and engaged citizen.ConclusionCritical thinking is an essential skill for success in the modern world. By developing and improving your critical thinking skills, you can analyze information more effectively, make better decisions, and adapt to changing circumstances. The techniques outlined in this guide provide a starting point for developing this important skill. Remember to ask questions, evaluate evidence, consider different perspectives, practice analyzing arguments, and be curious. With practice and effort, you can become a more effective critical thinker and achieve greater success in your personal and professional life.。
英语四级范文的分论点Captivating the essence of effective communication, the art of crafting an English Level Four essay hinges on the strategic deployment of well-structured arguments. The core of such a composition lies in the meticulous arrangement of its main points, which not only serve to support the central thesis but also to engage the reader in a compelling narrative. Each sub-point acts as a pillar, reinforcing the overarching message with clarity and precision. To achieve this, one must delve into the intricacies of the topic, identifying the most persuasive angles that resonate with the audience. The first sub-point should be a natural extension of the introduction, seamlessly leading the reader into the body of the argument. It is here that the writer begins to unravel the complexities of the subject matter, providing concrete examples and evidence to bolster the claim. Moving forward, subsequent points should build upon the foundation laid by the first, each offering a fresh perspective or a deeper insight into the issue at hand. The transition between points must be smooth and logical, ensuring that the reader remains engaged and follows the flow of thought without confusion. Finally, the conclusion should not only summarize the main points but also leave a lasting impression, encapsulating the significance of the argument and its implications for the broader discourse. In essence, the art of writing a compelling English Level Four essay is akin to weaving a tapestry of ideas, where each thread contributes to the overall design and beauty of the final piece.。
a rXiv:q uant-ph/0112178v318D ec23On a Supposed Conceptual Inadequacy of the Shannon Information in Quantum Mechanics C.G.Timpson ∗The Queen’s College,Oxford,OX14AW,UK 18December 2002Abstract Recently,Brukner and Zeilinger (2001)have claimed that the Shannon information is not well defined as a measure of information in quantum mechanics,adducing arguments that seek to show that it is inextrica-bly tied to classical notions of measurement.It is shown here that these arguments do not succeed:the Shannon information does not have prob-lematic ties to classical concepts.In a further argument,Brukner and Zeilinger compare the Shannon information unfavourably to their pre-ferred information measure,I ( p ),with regard to the definition of a notion of ‘total information content’.This argument is found unconvincing and the relationship between individual measures of information and notions of ‘total information content’investigated.We close by considering the prospects of Zeilinger’s Foundational Principle as a foundational principle for quantum mechanics 1Introduction What role the concept of information might have to play in the foundations of quantum mechanics is a question that has recently excited renewed interest(Fuchs 2000,2002;Mermin 2001;Wheeler 1990).Zeilinger,for example,has put forward an information-theoretic principle which he suggests might serve as a foundational principle for quantum mechanics (Zeilinger 1999),(see Appendix).As a part of this project,Brukner and Zeilinger (2001)have criticised Shannon’s (1948)measure of information,the quantity fundamental to the discussion of information in both classical and quantum information theory.They claim that the Shannon information is not appropriate as a measure of information in the quantum context and have proposed in its stead their own preferred quantity and a notion of ‘total information content’associated with it,which latter is supposed to supplant the von Neumann entropy (Brukner and Zeilinger 1999,2000a,2000b).The main aim in Brukner and Zeilinger(2001)is to establish that the Shan-non information is intimately tied to classical notions,in particular,to the pre-conceptions of classical measurement,and that in consequence it cannot serve as a measure of information in the quantum context.They seek to establish this in two ways.First,by arguing that the Shannon measure only makes sense when we can take there to be a pre-existing sequence of bit values in a message we are decoding,which is not the case in general for measurements on quantum systems(consider measurements on qubits in a basis different from their eigen-basis);and second,by suggesting that Shannon’s famous third postulate,the postulate that secures the uniqueness of the form of the Shannon information measure(Shannon,1948)and has been seen by many as a necessary axiom for a measure of information,is motivated by classical preconceptions and does not apply in general in quantum mechanics where we must consider non-commuting observables.These two arguments do not succeed in showing that the Shannon informa-tion is‘intimately tied to the notion of systems carrying properties prior to and independent of observation’(Brukner and Zeilinger2000b:1),however.Thefirst is based on too narrow a conception of the meaning of the Shannon information and the second,primarily,on a misreading of what is known as the‘grouping axiom’.We shall see that the Shannon information is perfectly well defined and appropriate as a measure of information in the quantum context as well as in the classical.We will begin by reviewing some of the different ways in which the Shannon information may be understood(Section2),before examining this pair of arguments and seeing where they go wrong(Section3).Brukner and Zeilinger have a further argument against the Shannon infor-mation(Section4).They suggest it is inadequate because it cannot be used to define an acceptable notion of‘total information content’.Equally,they insist, the von Neumann entropy cannot be a measure of information content for a quantum system because it has no general relation to information gain from the measurements that we might perform on a system,save in the case of mea-surement in the basis in which the density matrix is diagonal.By contrast,for a particular set of measurements,their preferred information measure sums to a unitarily invariant quantity that they interpret as‘information content’,this being one of their primary reasons for adopting this specific measure.This prop-erty will be seen to have a simple geometric explanation in the Hilbert-Schmidt representation of density operators however,rather than being of any great in-formation theoretic significance;and thisfinal argument found unpersuasive,as the proposed constraint on any information measure regarding the definition of‘total information content’seems unreasonable.Part of the problem is that information content,total or otherwise,is not a univocal concept and we need to be careful to specify precisely what we might mean by it in any given context.22Interpretation of the Shannon information The technical concept of information relevant to our discussion,the Shannoninformation,finds its home in the context of communication theory.We are concerned with a notion of quantity of information;and the notion of quantityof information is cashed out in terms of the resources required to transmit messages(which is,note,a very limited sense of quantity).We shall highlighttwo main ways in which the Shannon information may be understood,thefirstof which rests explicitly on Shannon’s1948noiseless coding theorem.2.1The communication channelIt is instructive to begin with a quotation of Shannon’s:The fundamental problem of communication is that of reproducingat one point either exactly or approximately a message selected atanother point.Frequently these messages have meaning...These se-mantic aspects of communication are irrelevant to the engineeringproblem.(Shannon1948:31)The communication system consists of an information source,a transmitter orencoder,a(possibly noisy)channel,and a receiver(decoder).It must be able to deal with any possible message produced(a string of symbols selected in thesource,or some varying waveform),hence it is quite irrelevant whether what is actually transmitted has any meaning or not.It is crucial to realise that‘information’in Shannon’s theory is not associatedwith individual messages,but rather characterises the source of the messages. The point of characterising the source is to discover what capacity is requiredin a communications channel to transmit all the messages the source produces;and it is for this that the concept of the Shannon information is introduced. The idea is that the statistical nature of a source can be used to reduce thecapacity of channel required to transmit the messages it produces(we shall restrict ourselves to the case of discrete messages for simplicity).Consider an ensemble X of letters{x1,x2,...,x n}occurring with proba-bilities p i.This ensemble is our source,from which messages of N letters are drawn.We are concerned with messages of very large N.For such messages,we know that typical sequences of letters will contain Np i of letter x i,Np j ofx j and so on.The number of distinct typical sequences of letters is then given byN!is the Shannon information(logarithms are to base2tofix the units of infor-mation as binary bits).Now as N→∞,the probability of an atypical sequence appearing becomes negligible and we are left with only2NH(X)equiprobable typical sequences which need ever be considered as possible messages.We can thus replace each typical sequence with a binary code number of NH(X)bits and send that to the receiver rather than the original message of N letters(N log n bits).The message has been compressed from N letters to NH(X)bits(≤N log n bits).Shannon’s noiseless coding theorem,of which this is a rough sketch,states that this represents the optimal compression(Shannon1948).The Shannon information is,then,appropriately called a measure of information because it represents the maximum amount that messages consisting of letters drawn from an ensemble X can be compressed.One may also make the derivative statement that the information per letter in a message is H(X)bits,which is equal to the information of the source. But‘derivative’is an important qualification:we can only consider a letter x i drawn from an ensemble X to have associated with it the information H(X)if we consider it to be a member of a typical sequence of N letters,where N is large,drawn from the source.Note also that we must strenuously resist any temptation to conclude that because the Shannon information tells us the maximum amount a message drawn from an ensemble can be compressed,that it therefore tells us the ir-reducible meaning content of the message,specified in bits,which somehow possess their own intrinsic meaning.This idea rests on a failure to distinguish between a code,which has no concern with meaning,and a language,which does(cf.Timpson(2000),Chpt.5).2.2Information and UncertaintyAnother way of thinking about the Shannon information is as a measure of the amount of information that we expect to gain on performing a probabilistic ex-periment.The Shannon measure is a measure of the uncertainty of a probability distribution as well as serving as a measure of information.A measure of un-certainty is a quantitative measure of the lack of concentration of a probability distribution;this is called an uncertainty because it measures our uncertainty about what the outcome of an experiment completely described by the prob-ability distribution in question will be.Uffink(1990)provides an axiomatic characterisation of measures of uncertainty,deriving a general class of measures of which the Shannon information is one(see also Maassen and Uffink1989).Imagine a truly random probabilistic experiment described by a probabil-ity distribution p={p1,...,p n}.The intuitive link between uncertainty and information is that the greater the uncertainty of this distribution,the more we stand to gain from learning the outcome of the experiment.In the case of the Shannon information,this notion of how much we gain can be made more precise.4Some care is required when we ask‘how much do we know about the out-come?’for a probabilistic experiment.In a certain sense,the shape of the probability distribution might provide no information about what an individualoutcome will actually be,as any of the outcomes assigned non-zero probabilitycan occur.However,we can use the probability distribution to put a value on any given outcome.If it is a likely one,then it will be no surprise if it occursso of little value;if an unlikely one,it is a surprise,hence of higher value.A nice measure for the value of the occurrence of outcome i is−log p i,a decreas-ing function of the probability of the outcome.We may call this the‘surprise’information associated with outcome i;it measures the value of what we learn from the experiment given that we know the probability distribution for theoutcomes.If the information that we would gain if outcome i were to occur is−log p i, then before the experiment,the amount of information we expect to gain is givenby the expectation value of the‘surprise’information, i p i(−log p i);and this, of course,is just the Shannon information H of the probability distribution p.Hence the Shannon information tells us our expected information gain.More generally,a crude sketch of how the relationship between uncertainty and expected information gain might be cashed out for the whole class of mea-sures of uncertainty may be given as follows.What we know given the proba-bility distribution for an experiment is that if the experiment is repeated very many times then the sequence of outcomes we attain will be one of the typical sequences.How much we learn from actually performing the experiments and acquiring one of those sequences,then,will depend on the number of typical se-quences;the more there are,the more we stand to gain.Thus for a quantitative measure of how much information we gain from the sequence of experiments we could just count the number of typical sequences(which would give us NH,the Shannon information of the sequence),or we could choose any suitably behaved function(e.g.continuous,invariant under relabelling of the outcome probabili-ties)that increases as the number of typical sequences increases.This property will follow from Schur concavity,which is the key requirement on Uffink’s gen-eral class of uncertainty measures U r( p)(for details of the property of Schur concavity,see Uffink(1990),Nielsen(2001)and Section4.1below).NU r( p) then,can be understood as a measure of the amount of information we gain from a long series N of experiments;we get the average or expected informa-tion per measurement by dividing by N,but note that this quantity only makes sense if we consider the individual measurement as part of a long sequence of measurements.A precisely similar story can be told for a measure of‘how much we know’given a probability distribution.This will be the inverse of an uncertainty:we want a measure of the concentration of a probability distribution;the more concentrated,the more we know about what the outcome will be(which just means,the better we can predict the outcome).A function that decreases as the number of typical sequences increases(Schur convexity)will give our quantitative measure of how much we know about what the outcome of a long run of experiments will be:the more typical sequences there are the less we5know.We are again talking of a long run of experiments,so‘how much we know’for a single experiment will only make sense as an average value,when the single experiment is considered as a member of a long sequence.So note again that to say we have a certain amount of information(knowledge)about what the outcome of an experiment will be is not to claim that we have partial knowledge of some predetermined fact about the outcome of an experiment.2.3The minimum number of questions needed to specifya sequenceThefinal common interpretation of the Shannon information is as the minimum average number of binary questions needed to specify a sequence drawn from an ensemble(Uffink1990;Ash1965),although this appears not to provide an interpretation of the Shannon information actually independent of the previous two.Imagine that a long sequence N of letters is drawn from the ensemble X,or that N independent experiments whose possible outcomes have probabilities p i are performed,but the list of outcomes is kept from us.Our task is to determine what the sequence is by asking questions to which the guardian of the sequence can only answer‘yes’or‘no’;and we choose to do so in such a manner as to minimize the average number of questions needed.We need to be concerned with the average number to rule out lucky guesses identifying the sequence.If we are trying to minimize the average number of questions,it is evident that the best questioning strategy will be one that attempts to rule out half the possibilities with each question,for then whatever the answer turns out to be,we still get the maximum value from each question.Given the probability distribution,we may attempt to implement this strategy by dividing the possible outcomes of each individual experiment into classes of equal probability,and then asking whether or not the outcome lies in one of these classes.We then try and repeat this process,dividing the remaining set of possible outcomes into two sets of equal probabilities,and so on.It is in general not possible to proceed in this manner,dividing afinite set of possible outcomes into two sets of equal probabilities,and it can be shown that in consequence the average number of questions required if we ask about each individual experiment in isolation is greater than or equal to H(X).However,if we consider the N repeated experiments,where N tends to infinity,and consider asking joint questions about what the outcomes of the independent experiments were,we can always divide the classes of possibilities of(joint)outcomes in the required way.Now we already know that for large N,there are2NH(X)typical sequences,so given that we can strike out half the possible sequences with each question,the minimum average number of questions needed to identify the sequence is NH(X).(These last results are again essentially the noiseless coding theorem.)It is not immediately obvious,however,why the minimum average number of questions needed to specify a sequence should be related to the notion of information.(Again,the tendency to think of bits and binary questions as irreducible meaning elements is to be resisted.)It seems,in fact that this is6either just another way of talking about the maximum amount that messages drawn from a given ensemble can be compressed,in which case we are back to the interpretation of the Shannon information in terms of the noiseless coding theorem,or it is providing a particular way of characterising how much we stand to gain from learning a typical sequence,and we return to an interpretation in terms of our expected information gain.3Two arguments against the Shannon informa-tionWith this preamble behind us,we may turn to thefirst of the arguments against the Shannon information.3.1Are pre-existing bit-values required?Since the quantity− i p i log p i is meaningful for any(discrete)probability distribution(and can be generalised for continuous distributions),Brukner and Zeilinger’s argument must be that when we have probabilities arising from mea-surements on quantum systems,− i p i log p i does not correspond to a concept of information.Their argument concerns measurements on systems that are all prepared in a given state|ψ ,where|ψ may not be an eigenstate of the observable we are measuring.The probability distribution p={p1,...,p n}for measurement outcomes will be given by p i=T r(|ψ ψ|P i),where P i are the op-erators corresponding to different measurement outcomes(projection operators in the spectral decomposition of the observable,for projective measurements).Brukner and Zeilinger suggest that the Shannon information has no meaning in the quantum case,because the concept lacks an‘operational definition’in terms of the number of binary questions needed to specify an actual concrete sequence of outcomes.In general in a sequence of measurements on quantum systems,we cannot consider there to be a pre-existing sequence of possessed values,at least if we accept the orthodox eigenvalue-eigenstate link for the ascription of definite values(see e.g.Bub(1997))1,and this rules out,they insist,interpreting the Shannon measure as an amount of information: The nonexistence of well-defined bit values prior to and indepen-dent of observation suggests that the Shannon measure,as definedby the number of binary questions needed to determine the partic-ular observed sequence0’s and1’s,becomes problematic and evenuntenable in defining our uncertainty as given before the measure-ments are performed.(Brukner and Zeilinger2001:1)...No definite outcomes exist before measurements are performed andtherefore the number of different possible sequences of outcomes doesnot characterize our uncertainty about the individual system beforemeasurements are performed.(Brukner and Zeilinger2001:3)These two statements should immediately worry us,however.Recall the key points of the interpretation of the Shannon information:given a long message (a long run of experiments),we know that it will be one of the typical sequences that is instantiated.Given p,we can say what the typical sequences will be, how many there are,and hence the number of bits(NH(X))needed to spec-ify them,independent of whether or not there is a pre-existing sequence of bit values.It is irrelevant whether there already is some concrete sequence of bits or not;all possible sequences that will be produced will require the same num-ber of bits to specify them as any sequence produced will always be one of the typical sequences.It clearly makes no difference to this whether the probability distribution is given classically or comes from the trace rule.Also,the number of different possible sequences does indeed tell us about our uncertainty before measurement:what we know is that one of the typical sequences will be instan-tiated,what we are ignorant of is which one it will be,and we can put a measure on how ignorant we are simply by counting the number of different possibili-ties.Brukner and Zeilinger’s attempted distinction between uncertainty before and after measurement is not to the point,the uncertainty is a function of the probability distribution and this is perfectly well defined before measurement2.Brukner and Zeilinger have assumed that it is a necessary and sufficient condition to understand H as a measure of information that there exists some concrete string of N values,for then and only then can we talk of the minimum number of binary questions needed to specify the string.But as we have now seen,it is not a necessary condition that there exist such a sequence of outcomes.We are not in any case forced to assume that H is about the number of questions needed to specify a sequence in order to understand it as a measure of information;we also have the interpretations in terms of the maximum amount a message drawn from an ensemble described by the probability distribution p can be compressed,and as the expected information gain on measurement. (And as we have seen,one of these two interpretations must in fact be prior.) Furthermore,the absence of a pre-existing string need not even be a problem for the minimum average questions interpretation—we can ask about the minimum average number of questions that would be required if we were to have a sequence drawn from the ensemble.So again,the pre-existence of a definite string of values is not a necessary condition.It is not a sufficient condition either,because,faced with a string of Ndefinite outcomes,in order to interpret NH as the minimum average number of questions needed to specify the sequence,we need to know that we in fact have a typical sequence,that is,we need to imagine an ensemble of such typical sequences and furthermore,to assume that the relative frequencies of each of the outcomes in our actual string is representative of the probabilities of each of the outcomes in the notional ensemble from which the sequence is drawn. If we do not make this assumption,then the minimum number of questions needed to specify the state of the sequence must be N—we cannot imagine that the statistical nature of the source from which the sequence is notionally drawn allows us to compress the message.So even in the classical case,the concrete sequence on its own is not enough and we need to consider an ensemble, either of typical sequences or an ensemble from which the concrete sequence is drawn.In this respect the quantum and classical cases are completely on a par.The same assumption needs to be made in both cases,namely,that the probability distribution p,either known in advance,or derived from observed relative frequencies,correctly describes the probabilities of the different possible outcomes.The fact that no determinate sequence of outcomes exists before measurement does not pose any problems for the Shannon information in the quantum context.Reiterating their requirements for a satisfactory notion of information,Brukner and Zeilinger say:We require that the information gain be directly based on the ob-served probabilities,(and not,for example,on the precise sequenceof individual outcomes observed on which Shannon’s measure of in-formation is based).(Brukner and Zeilinger2000b:1)But as we have seen,it is false that the Shannon measure must be based on a precise sequence of outcomes(this is not a necessary condition)and the Shannon measure already is and must be based on the observed probabilities(a sequence of individual outcomes on its own is not sufficient).There is,however,a difference between the quantum and classical cases that Brukner and Zeilinger may be attempting to capture.Suppose we have a sequence of N qubits that has actually been used to encode some informa-tion,that is,the sequence of qubits is a channel to which we have connected a classical information source.For simplicity,imagine that we have coded in orthogonal states.Then the state of the sequence of qubits will be a prod-uct of|0 ’s and|1 ’s and for measurements in the encoding basis,the sequence will have a Shannon information equal to NH(A)where H(A)is the infor-mation of the classical source.If we do not measure in the encoding basis, however,the sequence of0’s and1’s we get as our outcomes will differ from the values originally encoded and the Shannon information of the resulting se-quence will be greater than that of the original3.We have introduced some‘noise’by measuring in the wrong basis.The way we describe this sort of sit-uation,though(Schumacher1995),is to use the Shannon mutual information H(A:B)=H(A)−H(A|B),where B denotes the outcome of measurement ofthe chosen observable(outcomes b i with probabilities p(b i))and the‘conditionalentropy’H(A|B)= n i=1p(b i)H(p(a1|b i),...,p(a m|b i)),characterises the noise we have introduced by measuring in the wrong basis.H(B)is the information(per letter)of the sequence that we are left with after measurement,H(A:B) tells us the amount of information that we have actually managed to transmit down our channel,i.e.the amount(per letter)that can be decoded when we measure in the wrong basis.3.2The grouping axiomThefirst argument has not revealed any difficulties for the Shannon information in the quantum context,so let us now turn to the second.In his original paper,Shannon put forward three properties as reasonable requirements on a measure of uncertainty and showed that the only function satisfying these requirements has the form H=−K i p i log p i.4 Thefirst two requirements are that H should be continuous in the p i and that for equiprobable events(p i=1/n),H should be a monotonic increasing function of n.The third requirement is the strongest and the most important in the uniqueness proof.It states that if a choice is broken down into two successive choices,the original H should be a weighted sum of the individual values of H.The meaning of this rather non-intuitive constraint is usually demonstrated with an example(see Fig.1).A precise statement of Shannon’s third requirement(one that includes also the second requirement as a special case)is due to Faddeev(1957)and is often known as the Faddeev grouping axiom:Grouping Axiom1(Faddeev)For every n≥2H(p1,p2,...,p n−1,q1,q2)=H(p1,...,p n−1,p n)+p n H(q1p n)(2)where p n=q1+q2.The form of the Shannon information follows uniquely from requiring H(p,1−p) to be continuous for0≤p≤1and positive for at least one value of p,permu-tation invariance of H with respect to relabelling of the p i,and the grouping axiom.2,16}can be considered asgiven for the three outcomes directly,or we could considerfirst a choice of two equiprobable events,followed by a second choice of two events with probabilities23,conditional on the second,say,of thefirst two events occurring,a‘decom-position’of a single choice into two successive choices,the latter of which will only be made half the time.Shannon’s third requirement says that the uncer-tainty in p will be given by H(13,12,12H(23):the uncertainty ofthe overall choice is equal to the uncertainty of thefirst stage of the choice,plusthe uncertainty of the second choice weighted by its probability of occurrence.‘Grouping axiom’is an appropriate name.As it is standardly understood(see e.g.Ash(1965),Uffink(1990),Jaynes(1957)),we consider that instead of giving the probabilities p1,...,p n of the outcomes x1,...,x n of a probabilis-tic experiment directly,we may imagine grouping the outcomes into compositeevents(whose probabilities will be given by the sum of the probabilities of their respective component events),and then specifying the probabilities of the out-come events conditional on the occurrence of the composite events to whichthey belong;this way of specifying the probabilistic experiment being precisely equivalent to thefirst.So we might group thefirst k events together into anevent A,which would have a probability p(A)= k i=1p i,and the remaining n−k into an event B of probability p(B)= n i=k+1p i;and then give the con-ditional probabilities of the events x1,...,x k conditional on composite event Aoccurring,(p1/p(A)),...,(p k/p(A)),and similarly the conditional probabilities for the events x k+1,...,x n conditional on event B.The grouping axiom then concerns how the uncertainty measures should be related for these different de-scriptions of the same probabilistic experiment.It says that our uncertainty about which event will occur should be equal to our uncertainty about which group it will belong to plus the expected value of the uncertainty that would remain if we were to know which group it belonged to(this expected value be-ing the weighted sum of the uncertainties of the conditional distributions,with weights given by the probability of the outcome lying within a given group).So in particular,let us imagine an experiment with n+1outcomes which welabel a1,a2,...,a n−1,b1,b2,having probabilities p1,...,p n−1,q1,q2respectively.We can define an event a n=b1∪b2,b1∩b2=∅,which would have probability p n=q1+q2and the probabilities for b1and b2conditional on a n occurring willthen be q1p n respectively.Grouping Axiom1says that the uncertainty in theoccurrence of events a1,a2,...,a n−1,b1,b2is equal to the uncertainty for the11。
