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BASIC LOGICAL REASONING


BASIC MATHEMATICAL REASONING


AND


THE PHYSICAL SCIENCES

by

David R. Burgess

And

Jeremiah Dolan


PREFACE

This course provides a way for non-science freshman students to learn basic reasoning skills and appreciate their use in science. The approach strengthens the "liberal arts" notion of education by integrating logic, basic mathematics and some concepts from physical science into one course of study. In addition, there is practice at synthesizing ideas and writing. The process and "everyday" nature of science is emphasized. The general philosophy is to use physical science as a medium to help students develop logical reasoning, critical thinking and problem solving abilities.

The intent of this book is to provide a basic framework for a more detailed discussion among students, facilitated by an instructor. The "answers" are not found in these pages, but there is some basic background material that can be used as a reference.

Further explanation of this material, practice quizzes, and other details specific to PHY101 are found on the Introduction to Physical Sciences homepage which can be accessed through the Chemistry/Physics Department Access Page (http://www.rivier.edu/chemistry).

The homepage also has information about concept outlines, writing assignments, and oral presentations.

This class will not cover very much formal physical science. It will concentrate on basic skills and the processes that scientists use. Physical science will be used as a medium to develop logical reasoning and basic math skills. This is intended as a course that integrates and provides opportunities to develop general skills that can be used throughout life.

We will begin by looking at the atom so that an understanding of charge can be developed. Basic logic will be used to investigate the implications that arise from the commonly known concept that like charges repel and unlike charges attract. This will lead to some elementary chemistry and physics that can be applied to the world around us and help us to see the need to understand some basic science when dealing with the environmental concerns of today. After this you will be in a position to investigate one of the most important concepts in chemistry, the concept of intermolecular forces. Some elementary classical physics will then be investigated. The idea of forces causing motion and opposing forces balancing will be ongoing concepts. Conservation of energy, specifically dealing with kinetic and potential energy will be studied in more detail.

This book is intended as a supplement to class discussion and activities that will cover this material. It is not intended to encompass all details about these topics and, furthermore, other topics will also be part of the course.

Copyright © 2006-2015

Dr. David R. Burgess

All Rights Reserved
TABLE OF CONTENTS

PREFACE                                                                                                                 2

                                                                                                                                        

I. INTRODUCTION                                                                                             5

                                                                                                                                        

A. Counter-Intuitive Situations                                                                                       5

B. Modeling                                                                                                                     6

C. A Good Learner                                                                                                          6

D. The Nature of Science                                                                                                6

                                                                                                                                        

II. BASIC LOGICAL REASONING                                                             9

                                                                                                                                        

A. Logic and Reasoning                                                                                                  9

B. Argument and Logical Form                                                                                      11

C. Other Types of Argument                                                                                           12

D. Complex Arguments                                                                                                   15

E. Truth and Falsity of Premises                                                                                     16

Exercises                                                                                                                          17

                                                                                                                                        

III. BASIC MATHEMATICAL REASONING                                        19

                                                                                                                                        

A. Arithmetic and Basic Algebra                                                                                    19

B. Reporting Numbers                                                                                                     20

C. Mathematical Modeling                                                                                              21

Exercises                                                                                                                          24

                                                                                                                                        

IV. WAVES AND ATOMIC THEORY                                                       25

                                                                                                                                        

A. Waves                                                                                                                         25

B. Elements and Atomic Structure                                                                                  25

C. Atomic Structure, Line Spectra, and Light Absorption                                             26

                                                                                                                                        

V. INTERMOLECULAR FORCES                                                              27

                                                                                                                                        

A. Charge, Elements, and Compounds                                                                           27

B. Nonpolar and Polar Molecules                                                                                    28

C. Molecular Interactions                                                                                                28

Exercises                                                                                                                          28


VI. BASIC PHYSICS                                                                                           29

                                                                                                                                        

A. Kinetic Energy, Potential Energy, and Conservation of Energy                                29

B. Momentum and Conservation of Momentum                                                            30

C. Two Dimensional Physics                                                                                           31

D. Forces and Equilibrium                                                                                               31

Exercises                                                                                                                          32

                                                                                                                                        

                                                                                                                                        

Appendix A: MODELING AND SCIENCE                                              33

                                                                                                                                        

Appendix B: MODEL OF A GOOD LEARNER, Daniel K. Apple35

                                                                                                                                        

Appendix C: CRITICAL THINKING SKILLS, CONCEPTS, AND TERMINOLOGY FROM THE LOGIC COURSE, J. Dolan                                                   36

                                                                                                                                        

                                                                                                                                        


I. Introduction

A. Counter-Intuitive Situations

One reason the world is such a fascinating place is that something is always surprising us. This gives variety and excitement to our lives. But why are we surprised? Maybe it is due to experience. Gravity never changes, yet hot air balloons rise, airplanes take off, rockets propel men and women into space, and space probes leave the earth and travel beyond the limits of our solar system. These are not surprising to anyone that lives in our day, but can you imagine how surprised people who lived 1000 years ago would have been to find out that someone walked on the moon? How would they have explained such a feat? It would not have made any sense according to their explanation of the world, but in our day it can be completely explained by the "laws of physics." An event that goes against what we think should happen is said to be counter-intuitive.

Here is one for you to try. After Thanksgiving dinner (or at some other convenient time) connect the tines of two forks together, put one end of a toothpick through the tines, and place the other end of the toothpick on a drinking glass. If done correctly the forks will "hang in mid-air!" Figure 1.1 is a model of the situation.

Isn't that fun! Some people think physics is hard because it is counter-intuitive, but the counter-intuitive situation is what makes physics interesting. One of our jobs is to understand the counter-intuitive situation. How do you go about understanding something that seems impossible?


B. Modeling

One way to try and make sense out of the insensible is to model it. Figure 1.1 is a model. It has been constructed from experience and can be used by others to duplicate the situation. Is this model useful? Can you make a better model? Is there another kind of model that would help us understand this situation better? Here is a picture of this situation. Is it a better model?


Modeling is central to a scientist's activities. It might be said that a scientist's goal is to correctly model observable events. The models can take several forms. This book will emphasize logical reasoning models as taught in beginning philosophy courses (see chapter two) and basic mathematical reasoning models as taught in Junior High and High Schools (see chapter three). Building models, using logical reasoning and mathematical reasoning, should become a skill that is applied to all problem solving, whether in science or any other discipline. Appendix Ahas a summary of some of the kinds of models that might be used in science.

C. A Good Learner

In order to understand counter-intuitive situations it is important to be a good learner. One important concept that must be learned is that scientists are ordinary people who use ordinary methods to solve problems. They must often learn new ideas and use those ideas in solving the problem at hand. It is important to improve learning skills. Appendix B is a model of a good learner. Good learners seem to have the characteristics listed in Appendix B. These are characteristics that should be developed through the study of science. Study Appendix B and decide what you think are the three most important characteristics for a good learner. Use the activities suggested in this book to develop those characteristics.

D. The Nature of Science

Science is inherently experimental and, as such, inherently has uncertainty associated with it. Scientists model physical observations using the best tools available. They are constantly trying to refine their understanding of how nature works. They are in a never-ending quest for truth. Consider the following quotes from Daniel Dennett in his article Postmodernism and Truth (http://www.butterfliesandwheels.com/articleprint.php?num=13).

"Then we invented measuring, and arithmetic, and maps, and writing. These communicative and recording innovations come with a built-in ideal: truth. The point of asking questions is to find true answers; the point of measuring is to measure accurately; the point of making maps is to find your way to your destination."

"We human beings use our communicative skills not just for truth-telling, but also for promise-making, threatening, bargaining, story-telling, entertaining, mystifying, inducing hypnotic trances, and just plain kidding around, but prince of these activities is truth-telling, and for this activity we have invented ever better tools. Alongside our tools for agriculture, building, warfare, and transportation, we have created a technology of truth: science. Try to draw a straight line, or a circle, "freehand." Unless you have considerable artistic talent, the result will not be impressive. With a straight edge and a compass, on the other hand, you can practically eliminate the sources of human variability and get a nice clean, objective result, the same every time."

"What inspires faith in arithmetic is the fact that hundreds of scribblers, working independently on the same problem, will all arrive at the same answer (except for those negligible few whose errors can be found and identified to the mutual satisfaction of all). This unrivalled objectivity is also found in geometry and the other branches of mathematics, which since antiquity have been the very model of certain knowledge set against the world of flux and controversy."

"Yes, but science almost never looks as uncontroversial, as cut-and-dried, as arithmetic. Indeed rival scientific factions often engage in propaganda battles as ferocious as anything to be found in politics, or even in religious conflict. The fury with which the defenders of scientific orthodoxy often defend their doctrines against the heretics is probably unmatched in other arenas of human rhetorical combat. These competitions for allegiance--and, of course, funding--are designed to capture attention, and being well-designed, they typically succeed. This has the side effect that the warfare on the cutting edge of any science draws attention away from the huge uncontested background, the dull metal heft of the axe that gives the cutting edge its power. What goes without saying, during these heated disagreements, is an organized, encyclopedic collection of agreed-upon, humdrum scientific fact."

"The methods of science aren't foolproof, but they are indefinitely perfectible. Just as important: there is a tradition of criticism that enforces improvement whenever and wherever flaws are discovered. The methods of science, like everything else under the sun, are themselves objects of scientific scrutiny, as method becomes methodology, the analysis of methods. Methodology in turn falls under the gaze of epistemology, the investigation of investigation itself--nothing is off limits to scientific questioning."

The "technology of truth" called science often seems contradictory. It is inherently uncertain, but everyone can get the same result. It is always changing, but there is a "humdrum of scientific fact" that is agreed upon by all scientists. Maybe some explanations are in order.

Are There Facts in Science?

In science the words hypothesis, theory, law, and fact are often used. What is the difference between a hypothesis and a theory? Or between a theory and a law? Or between a law and a fact?

Here are some definitions that may be used.

  • Hypothesis - An idea that could explain an interesting observation. It is limited in scope and is an educated guess, meaning that there is reasoning that supports the idea. A guess without any reason for the guess is just a guess, not a hypothesis!
  • Theory - A theory is broader in scope, has experimental support, is coherent, and is internally consistent. Experimental support means that experimental observations are consistent with the theory and experimentation bears out the truthfulness of predictions made by the theory. To be coherent and internally consistent means that all aspects of the theory itself are consistent. One part of the theory doesn't contradict another part of the theory. In science we are interested in theories that explain how something works or how an observation came to be that way.
  • Law - Laws can be thought of as theories that are so well established that they seem to always be true under the specified conditions. 's laws are about force and motion as applied to large objects. If a very small particle is traveling very fast (close to the speed of light) 's laws no longer hold true. This idea that there are conditions that must be met in order for the law to be true is often overlooked.
  • Fact - Facts can be thought of as pieces of experimental data that are so well established that they seem to always be true under the specified conditions. The boiling point of water is 100oC (a fact), but it is only true at one atmosphere of pressure. Science depends on the idea that the "facts" of science can be obtained by anyone anywhere as long as the conditions are the same. I emphasize that every fact will have a set of conditions attached to it.
  • Data - Data comes from confirmed observations. A confirmed observation is a fact, it has been experimentally verified and is reproducible under the specified conditions. One piece of data is a fact.

Fundamental Principles

There is another category that is sometimes called fundamental principles. These fundamental principles have held true throughout all of the history of humankind. The laws of thermodynamics are in this category. Conservation laws are also in this category. One expression of the conservation laws is that you can't get something from nothing. That seems to be a principle that is always true.

These fundamental principles hold a special place when evaluating an idea as a scientific theory. No idea will have very much merit if it violates one of the fundamental principles that have been established over time.


II. Basic Logical Reasoning

". . . each chief step in science has been a lesson in logic."

A quote from Charles Sanders Pierce

A. Logic and Reasoning

One of the most important aspects of a scientist's work is the use of argument, presenting evidence to support conclusions. Scientists must argue logically in order to organize their observations into general, scientifically useful statements. They must argue logically in order to effectively support their theories. They must also argue logically in order to have their theories accepted by other members of the scientific community. Information, however accurate, which is not formulated completely and organized logically will be quickly dismissed. It is, therefore, important for the scientist, and the student of science, to be able to recognize arguments, understand how arguments are structured, and be able to determine whether arguments are effective or not.

It is also important to realize that the use of argument is not by any means confined to the scientific context. In fact, arguments are exchanged in all domains of life: in the various academic disciplines, in the workplace, in the law courts, in ordinary, daily conversation. Whenever we want to prove a point, to convince someone of an opinion held, and we provide information to support that point or opinion, then we are engaged in argument. In this sense, an argument is not a fight or disagreement, but rather an effort to reason with someone. In every argument information already known or agreed upon is used to prove a point not yet known or accepted. In an argument the known information is expressed as premise statement(s), and the point to be proved is the conclusion statement.

Some examples of arguments are now given to illustrate both how arguments work and the diversity of contexts in which they can occur:

1. The first argument is taken from The Trial, by Franz Kafka.

Someone must have been telling lies about Joseph K., for without having done anything wrong he was arrested one fine morning.

In this argument from the novel by Kafka the speaker is drawing the conclusion that "someone must have been telling lies about Joseph K." As readers, we would assume, or know from the context in the novel, that the speaker is aware of Joseph K.'s arrest and also aware that Joseph K. had done nothing wrong. We would then see that, in light of this premise information, the speaker concludes that Joseph K. was a victim of lying.

2. The second argument is from an article by A. M. Turing, "Computing Machinery and Intelligence," Mind, Vol. 59, 1950.

Thinking is a function of man's immortal soul. God has given an immortal soul to every man and woman, but not to any other animal or to machines. Hence no animal or machine can think.

In this argument A. M. Turing, the pioneering giant in the field of computers and artificial intelligence, is exploring the position that computers can not think. In order to make a case for that position he is appealing, in his premises, to information he believes his audience might well already accept about the nature of the human soul and the absence of a soul in machines and other animals. Notice how this argument is building. The first premise connects thinking with having an immortal soul. The second statement actually makes two points. It links an immortal soul with man and woman. It also excludes an immortal soul from animals and machines. This premise information leads then to the conclusion statement. In other words, Since thinking requires an immortal soul and animals and machines have no soul, we are led to conclude that "no animal or machine can think."

3. The third argument is from William Hochkammer, "The Capital Punishment Controversy," Journal of Criminal Law, Criminology, and Political Science, Vol. 60, No. 3, 1969.

But since the death penalty is in fact imposed for only those capital crimes which shock the public, where guilt is clear, and in light of the existing safeguards of appellate review and the possibility of commutation, execution of the innocent is unlikely.

In this journal article Hochkammer is arguing, from information about how the death penalty is actually applied, to the conclusion that execution of the innocent is unlikely. Several points are included in the premise material: That the death penalty is actually imposed (1) only for capital crimes, (2) only for those capital crimes that shock the public, (3) only in cases where guilt is clear, and (4) only in the context of the existing legal safeguards of appellate review and the possibility of commutation. From this premise information Hochkammer draws the conclusion that "execution of the innocent is unlikely."

4. The last example is from Robert Heilbroner, "Reflections: Boom or Crash," The New Yorker, .

Human activity, especially in the industrialized regions where its effect on nature is most concentrated, is at the verge of creating violent and irreversible effects on the planet. The immense magnitude of technological assault on the environment is indicated by the ongoing debates over the possibility of carbon dioxide creating a "greenhouse effect" that would alter the temperature of the entire earth; the possibility that massed industrial heat could change the patterns of air circulation and of precipitation on a continental scale; the possibility that the release of chemical waste might contaminate the ground water of a large region; and the possibility that the volume of nuclear wastes might constitute a hazard for an entire city or state.

In this argument economist Robert Heilbroner is drawing on four major points of information to establish and support his conclusion, which is expressed in the first statement. His premises are based on observations about (1) the possibility of the "greenhouse effect," (2) potential changes in the patterns of air circulation and precipitation, (3) the possibility of widespread chemical contamination of ground water, and (4) the potential dangers of nuclear wastes. In light of these serious dangers from industrial activity, Heilbroner asks us to conclude that we are "at the verge of creating violent and irreversible effects on the planet."

B. Argument and Logical Form

Arguments can be looked at in various ways. We can notice their content, what the statements are about. We can also pay attention to their form or structure, how they are set up and the relationships among the various statements. We can try to determine how well the premise statement(s) support the conclusion statement. And we can ask whether the premise statements are true or false. In fact, for a critical interpretation and evaluation of any particular argument we must do all of these.

Questions about how well the premises support the conclusion are often questions primarily about the form or structure of the argument. Thus, it is important to be able to recognize and work with different argument forms.

As an example take the following argument, drawn from Meno, a dialogue by the ancient Greek philosopher, Plato:

If virtue is a kind of knowledge, then it can be taught. But virtue is a kind of knowledge. Therefore, virtue can be taught.

This is a type of argument that has a very specific logical structure. Its form or structure can be clearly seen by re-writing the argument in standard form, identifying the premises (labeled "P") and conclusion (labeled "C") in the following manner:

P1 --- If virtue is a kind of knowledge, then it can be taught.

P2 --- virtue is a kind of knowledge.

C --- virtue can be taught.

Perhaps you can already begin to see the logical form or structure of this argument. The first premise in this argument is a particular kind of compound statement called a conditional statement. It has two parts: the antecedent, which comes after the word "if" and expresses a condition, and the consequent, after the word "then," which expresses what will be true, if the condition expressed in the antecedent is met. In this argument the second premise affirms that the antecedent condition has been met. From these two premise statements I may now infer or draw the conclusion that virtue can be taught. The following analysis and labeling of the argument highlights these points.

                                    [ antecedent ]                         [ consequent ]

P1 --- If virtue is a kind of knowledge, then it can be taught.

P2 --- virtue is a kind of knowledge. (affirms the antecedent)

C --- virtue can be taught. (affirms the consequent)

This argument is valid. That is, the premise statements successfully support the conclusion statement to such an extent that if these two premise statements are true, then the conclusion statement has to be true. The argument has very strong reasoning.

What is interesting and important about this example, however, over and above what it seeks to prove about the teachability of virtue, is its logical form or structure. It is worth noting that any argument, about any subject matter, provided it has the same logical form or structure, will also be a valid argument.

Consider the following argument:

If I travel for one hour at fifty-five miles per hour, then I will travel fifty-five miles. I did travel for one hour at fifty-five miles per hour. Thus, I traveled fifty-five miles.

In standard form this will be as follows:

                                    [ antecedent ]                                                       [ consequent ]

P1 -- If I travel for one hour at fifty-five miles per hour, then I will travel fifty-five miles.

P2 -- I did travel for one hour at fifty-five miles per hour. (affirms the antecedent)

C -- I traveled fifty-five miles. (affirms the consequent)

This argument, and any argument having the same logical form, will be a valid argument, with very strong reasoning between premise(s) and conclusion.

Conditional arguments are only one of many different types of argument that can be analyzed and evaluated from a consideration of their logical form. It is highly useful to become acquainted with the different forms of argument, not only the forms that are valid, but also the forms that are invalid.

C. Other Types of Argument

Some arguments work or fail to work, not merely because of their logical form, but due to other kinds of relations among premise(s) and conclusion.

Consider the following argument from an article in the May 1992 edition of Discover Magazine, titled "Ruffled Feathers" by Carl Zimmer, which is based on analogy, similarity and dissimilarity:

"In 1983 Chatterjee found some bones, . . . [As he pieced them together] he began to notice some odd things. The shoulder bone, for instance, was much longer than those of most dinosaurs but a lot like those of modern birds. And the neck vertebrae had a saddle shape, one peculiar for dinosaurs but normal for birds --- it makes their necks flexible.

"As Chatterjee assembled the shattered bits of skull, he found something that for him was even more striking. Behind the eye of a dinosaur are two holes in its skull, divided by a bony strut. In the course of developing a more flexible jaw, birds have lost this strut. 'I noticed there was just one hole,' says Chatterjee. 'This is the most distinctive feature of the bird skull.' He believed he could even see little knobs in the arms of the skeleton where feathers would have been rooted. By the end of 1985 he thought there was a pretty good chance that he had actually found a bird."

Put into standard form, this argument looks like this:

P1  The shoulder bone was much longer than those of most dinosaurs, but a lot like those of modern birds.

P2  The neck vertebrae had a saddle shape, one peculiar for dinosaurs, but normal for birds (making their necks flexible).

P3  In the skull, behind the eye, there was just one hole. Behind the eye of a dinosaur, there are two holes in the skull, divided by a bony strut. In the course of developing a more flexible jaw, birds have lost this strut; the presence of a single hole being the most distinctive feature of the bird skull.

P4  Chatterjee also believed that he could see little knobs in the arms of the skeleton where feathers would have been rooted.

C   The bones Chatterjee found were not those of a dinosaur, but rather those of a bird.

This argument combines the use of analogy and disanalogy, similarity and dissimilarity, to draw a conclusion about the bones that were discovered, in order to determine to what kind of animal they belonged.

A strong literal analogy, sufficient for drawing a conclusion, exists between two or more things when they share a number of important and relevant characteristics in common. In the above example, the identified skeletal structures (length of the shoulder bone, shape of the neck vertebrae, etc.) are important and relevant characteristics for identifying and distinguishing different types of animals. Since observation of the bones revealed several (in this case four) relevant and important points of similarity to the modern bird, there is good reason to conclude that the bones were those of a bird. There is, of course, no way of knowing with certainty that this is the case, yet the strong analogy makes the conclusion very likely to be true. This kind of argument is based on a logical principle called the "principle of analogy": Things observed to have a number of closely similar properties in respects relevant to the comparison, most likely have other, unobserved properties in common as well.

The argument presented above also establishes a strong disanalogy between the bones found and those of dinosaurs. Once again the comparison is made in terms of important, relevant characteristics for identifying and distinguishing animals. In this case, the number of relevant dissimilarities establishes a disanalogy, supporting the other part of the conclusion, i.e. that the bones were not those of a dinosaur. The logical idea here is that things observed to have a number of dissimilar properties in respects relevant to the comparison are more likely to be dissimilar in other respects as well.

The following argument is a causal argument. The conclusion is that the diet of polished, white rice is what caused the chickens to develop polyneuritis and die.

Eijkman fed a group of chickens exclusively on white rice. They all developed polyneuritis and died. He fed another group of foul unpolished rice. Not a single one of them contracted the disease. Then he gathered up the polishings from rice and fed them to other polyneuritic chickens, and in a short time the birds recovered. He had accurately traced the cause of polyneuritis to a faulty diet. For the first time in history, he had produced a food deficiency disease experimentally, and had actually cured it. It was a fine piece of work and resulted in some immediate remedial measures.

            ------ Bernard Jaffe, Outposts of Science (1935)

This argument could be written in the following way:

P1   Eijkman fed a group of chickens exclusively on white rice.

P2   They all developed polyneuritis and died.

P3   He fed another group of foul unpolished rice.

P4   Not a single one of them contracted the disease.

P5   He gathered up the polishings and fed them to other polyneuritic chickens.

P6   In a short time the birds recovered.

C   The diet of polished, white rice is what caused the chickens to develop polyneuritis and die.

D. Complex Arguments

Some arguments are simple inferences, in which any number of premises are offered to support one overall conclusion. All the arguments used above are simple inferences.

Other arguments have a complex structure. In these arguments one or more premises are offered to support a conclusion statement, which is itself then used as a premise to support a further conclusion. In these cases an argument is a chain of reasoning. Each link of the chain is a simple inference (simple argument), but, when strung together, they make a complex argument chain. Such an argument has what are called basic premises (BP), an intermediate conclusion (IC), which is also called a non-basic premise (NBP) in light of the fact that it functions as the conclusion of one inference, but then also as a premise for the next inference, and one final, overall conclusion (C).

Consider the Rutherford scattering experiment, which gives rise to a complex argument. During the time of the raisin pudding model of the atom was popular. In this model very small electrons were "sprinkled" around in a positive charge that had no region of concentrated mass. This was like raisins (electrons) in a pudding (the positive charge). knew about relatively large alpha particles and decided to test the raisin pudding model. He shot alpha particles at gold atoms and watched where the alpha particles went. He thought they should go straight through because, according to the raisin pudding model, there was no concentrated mass to deflect them. knew that usually the only way for an object to be deflected is for it to collide with a larger object. The electrons were known to be too small to deflect the larger alpha particles. To his surprise the alpha particles were deflected in all directions, some of them even bouncing back toward the source of alpha particles! What argument might write?

(1) BP --------- An alpha particle is deflected from its path only as a result of a collision with a solid object larger than itself.

(2) BP -------- Alpha particles shot into a gold atom are deflected.

(3) IC/NBP --- Gold atoms contain particles larger than alpha particles.

(4) BP --------- Electrons are much smaller than alpha particles.

(5) C --------- Gold atoms must have another particle besides the electron.

In this argument statements #1 and #2 are basic premises supporting the conclusion in statement #3. Statement #3, however, is not the final, overall conclusion. That is why it is labeled as an intermediate conclusion (IC). Statement #3 is also functioning as a premise, working with statement #4, to support the overall conclusion of the argument, statement #5. That is why statement #3 is also labeled as a non-basic premise (NBP). It is functioning as the conclusion of the first inference (a simple argument) and as a premise in the second inference. Any argument with an IC/NBP is called a complex argument in order to recognize its structure as a chain of reasoning, consisting of two or more inferences interconnected in this fashion.

E. Truth and Falsity of Premises

As suggested above, the overall purpose of an argument is to prove that its conclusion is true. In order for an argument to be successful at this, it must meet two criteria:

1. the reasoning or logical connection between premise(s) and conclusion must be sufficiently strong;

2. the premise(s) must be true.

Only if both criteria are satisfied does the argument give us good reason to believe that the conclusion is true. When both criteria are satisfied the argument leads to the truthfulness of the conclusion and it is said to be a sound argument.

Consider the following argument:

P1 --- If an acid is placed in water, then the concentration of the solution is increased.

P2 --- HCl is an acid.

C --- If HCl is placed in water, then the concentration of the solution will be increased.

This is a valid argument. Its premises support the conclusion such that if the premises are true, the conclusion will have to be true. The reasoning from premises to conclusion is very strong. However, the argument fails overall to prove that the conclusion is true because a premise, the first premise, is false.

The following argument is a fully successful argument because its reasoning is strong and all its premises are true. This argument does lead to the truthfulness of the conclusion.

P1 --- If an acid is placed in water then the H+ concentration of the solution is increased.

P2 --- HCl is an acid.

C --- If HCl is placed in water then the H+ concentration of the solution will be increased.

The principal job of logic is to help us understand under what conditions the reasoning of arguments is strong or weak, whether the premise statements do or do not support the conclusion. Logic does not help us to know whether, for example, the above statements about acids are true or false. One of the objectives of this course is to build up a store of knowledge as a basis for determining the truth or falsity of such statements. A second, and more important, objective of this course is to provide you with some general concepts and principles and to give you practice with the process of science. This should allow you to look at the premises (sometimes called theories, laws, hypotheses, etc.) in a more sophisticated way, in a way that allows you to better gage the truthfulness of those premises. It should by now be clear why both logical reasoning and truthful statements (or at least statements that agree with all of the available information from experiments, theoretical considerations, etc.) are necessary for the ongoing work of science.

In summary it should be noted how each piece of evidence (each premise) is directly related to a conclusion in every argument. This structure of substantiating a conclusion statement (hypothesis, opinion, etc.) is critical to scientists. In addition, the statements used as premises must agree with currently held concepts and principles (they must be "true").

Appendix C clarifies and expands upon the material presented in this chapter by summarizing the critical thinking skills, concepts, and terminology from the logic course.

Exercises

1. Analyze the following argument (identify the premises and the conclusion). Notice how the structure of the argument can be seen, even if the truthfulness of the premises is unknown or even if the language is unfamiliar.

A body on which a freely swinging pendulum of fixed length has periods of oscillation which decrease slightly with increasing latitude from the equator to both poles is an oblate spheroid slightly flattened at the poles.

But the earth is a body on which a freely swinging pendulum of fixed length has periods of oscillation which decrease slightly with increasing latitude from the equator to both poles.

Therefore the earth is an oblate spheroid slightly flattened at the poles.

(W. A. Wallace, Einstein, Galileo, and Aquinas: Three Views of Scientific Method)

2. Analyze the following argument (identify the premises and the conclusion).

In spite of the popularity of the finite world picture, however, it is open to a devastating objection. In being finite the world must have a limiting boundary, such as Aristotle's outermost sphere. That is impossible, because a boundary can only separate one part of space from another. This objection was put forward by the Greeks, reappeared in the scientific skepticism of the early Renaissance and probably occurs to any schoolchild who thinks about it today. If one accepts the objection, one must conclude that the universe is infinite.

(J. J. Callahan, "The Curvature of Space in a Finite Universe," Scientific American, August 1976)

3. Analyze the following argument (identify the premises and the conclusion). What kind of an argument is this? This argument may have been convincing in the 19th century, but is it convincing today? Explain. Hint: Are there really inhabitants on mars?

The planet Mars possesses an atmosphere, with clouds and mist closely resembling our own; It has seas distinguished from the land by a greenish color, and polar regions covered with snow. The red colour of our sunrises and sunsets. So much is similar in the surface of mars and the surface of the Earth that we readily argue there must be inhabitants there as here.

            ( Jevons, 19th century logician)

4. Explain why the following argument is invalid. The argument is presented in its original form and in standard form.

Total pacificism might be a good principle if everyone were to follow it. But not everyone is, so it isn't.

            (Gilbert Harmon, The Nature of Morality)

                                    [ antecedent ]                          [ consequent ]

P1    If everyone were to follow it, then pacificism might be a good principle.

P2    Not everyone is following it. (deny antecedent)

C      Pacificism is not a good principle. (deny consequent)

5. Does the following argument lead to the truthfulness of the conclusion? Explain.

P1     If the sun is shining, then the house becomes warm.

P2     The house becomes warm.

C      The sun is shining.

6. Find three short science articles (or paragraphs from longer articles) in newspapers or magazines that are arguments or explanations (see Appendix C) and write them in standard form. Please hand in a copy of the article with the analysis of the article.

7. Choose one of the characteristics of a good learner in Appendix B and write a conditional argument that has the following conclusion:

"Studying this book has helped me become a better learner"

8. Identify a scientific "fact" and write an argument to support the position that the "fact" is indeed true. Please refer to the Rutherford scattering experiment, which eventually resulted in the conclusion (or "fact") that protons exist, as an example of what is wanted in this problem.


III. Basic Mathematical Reasoning

A. Arithmetic and Basic Algebra

Basic mathematical reasoning is essential to everyone's life. It is used to determine where to buy gas, which shirt to buy, how much to tip the waitress, etc. Even though everyone has learned the basic mathematical reasoning skills needed to do basic science, it might be a good idea to review some of them. This chapter will review some of the specifics needed to understand basic science, but it is assumed that the rules of arithmetic are known and can be readily applied.

Some very basic arithmetic is required. As an example, consider the density of an object, which is a function of mass and volume. Being a function of mass and volume means that the density depends on mass and volume. Specifically the density is found by dividing the mass by the volume. If the letter d stands for density, the letter m stands for mass, and the letter V stands for volume, then the functional relationship could be written d = m/V. The mass, m, and the volume, V, are said to be variables of density. They can change, but knowing their values allows calculation of density, d.

If m = 2 and V = 4, then d = m/V = 2/4 = 0.5

When doing physical calculations it is important to express the units being used. Throughout this book units will always be used, even though they are not discussed in any detail. One common unit of mass is the kilogram which is abbreviated kg. A common unit for volume is the liter, abbreviated L. One common unit for density, therefore, is kg/L. The above calculation is more appropriately given as

If m = 2 kg and V = 4 L, then d = m/V = 2 kg/4L = 0.5 kg/L

The idea of using a symbol for a variable should already be familiar. The words we use to communicate with each other are just symbols (cow is a symbol for a four legged animal that produces milk). It should also be clear that the context of the symbol is important and that the same symbol could mean two different things (turn right, you are right). An example of that in science is that g could mean grams (a unit of mass) or it could mean the acceleration due to gravity (a number, usually about 9.8 meters per second squared or 9.8 m/s2). The context will tell. If an object has a mass of 2.4 g, you would assume it means 2.4 grams. If, on the other hand, the symbol g is found in a mathematical equation, like PE = mgh (potential energy equals mass times g times height), you would assume it to be the constant 9.8 m/s2.

So far this example has only dealt with arithmetic and calculating density given mass and volume. Sometimes density and volume might be known and it is necessary to calculate mass. This is possible by rearranging the formula d = m/V as m = d*V. There are several ways to explain how this rearrangement is done, but it is expected that one of them is already familiar to the reader. The relationship to find volume would be V = m/d.


B. Reporting Numbers

Any time a number is reported there should be some measure of how “accurate” it is. There should be an indication about how confident we are that the number is correct. A common practice today is for pollsters to provide a “margin of error” along with those numbers. The margin of error is reporting how confident they are that the result is meaningful. If a report says that 52% of the American people agree on an issue, but that it has a margin of error of 4%, it would suggest that the nation was pretty evenly divided. The people who reported the data think that the actual number could be anywhere from 56% down to 48%. If the margin of error was 10%, we would have less confidence in the reported number. The greater the margin of error, the less we believe the number.

In science every number should have an indication of the uncertainty in that number. Every number should have something like a margin of error. There are many ways to determine the uncertainty in a number. Scientists use instruments to get numbers. Each instrument has some uncertainty associated with it. A thermometer may have lines representing 0.5 oC between each line. You could tell that a temperature was between 21.0 and 21.5, but you would have to guess if it was 21.2, 21.3, etc. The uncertainty would be in the first decimal place. You might think that the reading should be 21.3 and that you are confident it is between 21.1 and 21.5. You could then report the number as 21.3 ± 0.2. The “plus or minus” would tell anyone who saw the number how much confidence you have in the number. In this case it says that the actual temperature is likely between 21.3 + 0.2 = 21.5 and 21.3 – 0.2 = 21.1. If you had a better thermometer, you could report more decimal places and the “plus or minus” would be smaller.

There are several mathematical ways to get the uncertainty. You may have heard of the standard deviation. It could be used to get an uncertainty number (a “plus or minus” number). A maximum differential error analysis could be performed to get the number for the uncertainty. In this class we will use a simple way of getting an uncertainty number using what is called the range for a set of data.

Consider the following set of data to illustrate the language and process of reporting numbers to be used in this class:

6.14, 6.03, 5.96, 6.11, 5.93

The sum of the numbers is 30.17 and the average is 6.034 (the sum divided by the number of data points). The median of a set of data is the data point that has exactly the same number of data points greater than itself as it has data points less than itself. In our case the median would be 6.03 since there are two numbers greater than 6.03 and two numbers less than 6.03.

The range is defined as the largest number minus the smallest number. In our case it would be 6.14 – 5.93 = 0.21. The range is how far it is between the high and the low.

- Фонд электронных границ замучил неприкосновенностью частной жизни и переписки. Сьюзан хмыкнула. Этот фонд, всемирная коалиция пользователей компьютеров, развернул мощное движение в защиту гражданских свобод, прежде всего свободы слова в Интернете, разъясняя людям реальности и опасности жизни в электронном мире.

Фонд постоянно выступал против того, что именовалось им «оруэлловскими средствами подслушивания, имеющимися в распоряжении правительственных агентств», прежде всего АНБ.

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