This is Introduction to Logic

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Class

Logic is the study of arguments and inferences as such it is an indispensable tool in all reasoning. The course has both formal and informal aspects. This is good news. The informal and formal sections make it so there's something that everyone can do.

Arguments have premises and conclusions

  • A premise supports or gives warrant for a conclusion
  • Arguments have conclusions

Arguments can be deductive or inductive

  • Valid arguments establish their conclusions with certainty given their premises are true. Validity is a formal notion, and thus valid arguments do not have to be sound.
  • Validity applies to arguments not premises.

Premises

  • Premises are propositions.
  • They can be true or false
  • They can be expressed in sentences of a natural language.

Soundness

  • Soundness is a property of valid arguments that have true premises.

Argument

  • The moon is made of cheese
  • Cheese is a dairy product.
  • Therefore the moon is made of a dairy product.

Inductive arguments

These support their conclusions with evidence of various kinds. Science is largely inductive. Inductive arguments do not give us certainty about the conclusion, but most of the reasoning that we do is inductive.

Terms

To reach precision, we must have precise meanings for our terms. If they are not precise, it is difficult to assign a truth value.

Lexical Definitions

  • These explain current usage
    • They must be neither too broad nor too narrow
    • Avoid Circularity.
    • BUT lexical definitions match the vagueness of our ordinary usage :(
    • Matches Emotional Tone
    • Includes essential to the term

Stipulative Definition

  • Clarifies

Analytic and Synthetic Propositions

  • Analytic statements are self-contradictory to deny
  • Synthetic statements are contingent and you actually learn something from the statement, so to speak. The truth does not follow from the meaning alone.

Using Neutral Terms

  1. filthy rich
  2. heroic
  3. an extremist
  4. a bastard

Analytic versus Synthetic

  1. All triangles have three angles.
  2. 2+2=4.
  3. Combining two drops of mercury with two other drops results in one big drop.
  4. There are ants that have established a system of slavery.

Analytic versus Synthetic

  1. All triangles have three angles.
    • Analytic + A priori
  2. 2+2=4.
    • Analytic & A priori
  3. Combining two drops of mercury with two other drops results in one big drop.
    • Synthetic + A posteriori
  4. There are ants that have established a system of slavery.
    • Synthetic + A posteriori

Analytic versus Synthetic

  1. Either some ants are parasitic or else none are.
  2. No three-year-old is an adult.
  3. No three-year-old understands symbolic logic.
  4. Water boils at 90C on that 10,000-foot mountain.
  5. Water boils at 100C at sea level.

Analytic versus Synthetic

  1. Either some ants are parasitic or else none are.
    • Analytic + A priori
  2. No three-year-old is an adult.
    • Analytic + A priori
  3. No three-year-old understands symbolic logic.
  • Synthetic + A posteriori
  1. Water boils at 90C on that 10,000-foot mountain.
    • Synthetic + A posteriori
  2. Water boils at 100C at sea level.
    • Analytic + + A priori

Analytic versus Synthetic

  1. No uncle who has never married is an only child.
  2. All swans are white.
  3. Every material body is spatially located and has spatial dimensions. > + Analytic
  4. Every material body has weight.
  5. The sum of the angles of a Euclidian triangle equals 180o.

Analytic versus Synthetic

  1. No uncle who has never married is an only child.
    • Analytic + A priori
  2. All swans are white.
    • Synthetic + A posteriori
  3. Every material body is spatially located and has spatial dimensions.
    • Analytic + A priori
  4. Every material body has weight.
    • Synthetic + A posteriori
  5. The sum of the angles of a Euclidian triangle equals 180o.
    • Analytic + A priori

Analytic versus Synthetic

  1. If all Parisians are French and all French are European, then all Parisians are European.
  2. Every event has a cause.
  3. Every effect has a cause.

A priori and A posteriori

  1. If all Parisians are French and all French are European, then all Parisians are European.
    • Analytic and A priori
  2. Every event has a cause.
    • Synthetic and A posteriori
  3. Every effect has a cause.
    • Analytic + A priori

Understanding Arguments

  • Arguments are everywhere.
  • They are sometimes difficult to distinguish from explanations
  • They can be diagrammed to capture the logical linkages between pre mises
    • These linkages can be complex.

Rules for Diagramming

  • Conclusions are below premises
  • Premises are joined to their conclusions via arrowed lines
  • Some premises support a conclusion jointly
    • We represent this by bracketing premises together
  • Some premises suport other premises (sub-conclusions)

Finding the Conclusion

The [ Detroit] Pistons did not lose because of the lack of ability. They are an all- around better team. They lost because of the law of averages. They will beat the [ San Antonio] Spurs every two times out of three. When you examine the NBA finals [ of 2005], that is exactly how they lost the seventh ( last game) because that would have been three out of three. The Spurs will beat the Pistons one out of three. It just so happens that, that one time was the final game, because the Pistons had already won two in a row.

— Maurice Williams, “ Law of Averages Worked Against Detroit Pistons,” The Ann Arbor ( Michigan) News, 8 July 2005

Finding the Conclusion

The [ Detroit] Pistons did not lose because of the lack of ability. They are an all-around better team. They lost because of the law of averages. They will beat the [ San Antonio] Spurs every two times out of three. When you examine the NBA finals [ of 2005], that is exactly how they lost the seventh ( last game) because that would have been three out of three. The Spurs will beat the Pistons one out of three. It just so happens that, that one time was the final game, because the Pistons had already won two in a row.

— Maurice Williams, “ Law of Averages Worked Against Detroit Pistons,” The Ann Arbor ( Michigan) News, 8 July 2005

College are not learn students well

Hundreds of thousands of recent college graduates today cannot express themselves with the written word. Why? Because universities have shortchanged them, offering strange literary theories, Marxism, feminism, deconstruction, and other oddities in the guise of writing courses.

— Stanley Ridgeley, “ College Students Can’t Write?” National Review Online, 19 February 2003

College are not learn students well

Hundreds of thousands of recent college graduates today cannot express themselves with the written word. Why? Because universities have shortchanged them, offering strange literary theories, Marxism, feminism, deconstruction, and other oddities in the guise of writing courses.

— Stanley Ridgeley, “ College Students Can’t Write?” National Review Online, 19 February 2003

Nicholas Kristof equates the hunting of whales by Eskimos with the whaling habits of Japanese, Norwegians, and Icelanders. The harsh environment of the Inupiat [ Eskimos] dictates their diet, so not even the most rabid antiwhaling activist can deny their inalienable right to survive. The Japanese and the European whale- hunting countries can choose the food they consume; they have no need to eat whales. It is not hypocritical to give a pass to the relatively primitive society of the Inu-piat to hunt a strictly controlled number of whales for survival while chastising the modern societies that continue to hunt these magnificent mammals for no good reason.

— Joseph Turner, “ Their Whale Meat, and Our Piety,” The New York Times, 18 September 2003

Separate the important parts

Premise: Japanese and European whale-hunting countries have no need to eat whales; they can choose their diets.

Premise: Eskimos live in an environment so harsh that their survival obliges them to eat whales; they have no choice in dietary matters.

Conclusion: Permitting primitive Eskimos to kill some whales for survival, while at the same time demanding that modern societies cease to hunt whales, is fair and reasonable, not hypocritical.

Where's the conclusion

If you marry without love, it does not mean you will not later come to love the person you marry. And if you marry the person you love, it does not mean that you will always love that person or have a successful marriage. The divorce rate is very low in many countries that have prearranged marriage. The divorce rate is very high in countries where people base their marriage decisions on love.

— Alex Hammoud, “ I Take This Man, for Richer Only,” The New York Times, 18 February 2000

The Conclusion is unstated

We ought not suppose that romantic love is a necessary precondition of successful marriage.

An Argument to Diagram

The dominant characteristic of sprawl is that each component of a community—housing, shopping centers, office parks, and civic institutions—is segregated, physically separated from the others, causing the residents of suburbia to spend an inordinate amount of time and money moving from one place to the next. And since nearly everyone drives alone, even a sparsely populated area can generate the traffic of a much larger traditional town.

Numbered

  1. The dominant characteristic of sprawl is that each component of a community—housing, shopping centers, office parks, and civic institutions—is segregated, physically separated from the others,
  2. causing the residents of suburbia to spend an inordinate amount of time and money moving from one place to the next.
  3. And since nearly everyone drives alone,
  4. even a sparsely populated area can generate the traffic of a much larger traditional town.

Diagrammed

LBJ

  1. At his best, Lyndon Johnson was one of the greatest of all American presidents.
  2. He did more for racial justice than any president since Abraham Lincoln.
  3. He built more social protections than anyone since Franklin Roosevelt.
  4. He was probably the greatest legislative politician in American history.
  5. He was also one of the most ambitious idealists.
  6. Johnson sought power to use it to accomplish great things.

Find the conclusion

  • This can be tough
  • And up for interpretations
  • Was LBJ a great prez because he sought power to achieve great things?
  • 1 seems like a more interesting conclusion in the overall context

Diagrammed

Marriage (# this yourself)

Married people are healthier and more economically stable than single people, and children of married people do better on a variety of indicators. Marriage is thus a socially responsible act. There ought to be some way of spreading the principle of support for marriage throughout the tax code.

— Anya Bernstein, “ Marriage, Fairness and Taxes,” The New York Times, 15 February 2000

Marriage

Blair on the Euro

“The argument is simple. We are part of Europe. It affects us directly and deeply. Therefore we should exercise leadership in order to change Europe in the direction we want.”

— Reported by Alan Cowell in the The New York Times, 9 December 2001

Diagrammed

Mo Money, Mo Problems

No government can ever guarantee that the small investor has an equal chance of winning. It is beyond dishonest to pretend that rules can be written to prevent future financial scandals. No set of regulations can insure fairness and transparency in the [ securities] markets.

— Lester Thurow, “Government Can’t Make the Market Fair,” The New York Times, 23 July 2002

Diagrammed (Is this right?)

MORE DIAGRAMMING

[1]People and governments want to talk, talk, talk about racism and other forms of intolerance;[2] we are obsessed with racial and ethnic issues.[3] But we come to these issues wearing earplugs and blinders,[4] and in a state of denial that absolves us of complicity in any of these hateful matters.[5] Thus, the other guy is always wrong.

— Bob Herbert, “ Doomed to Irrelevance,” The New York Times, 6 November 2001

Racism

Drawing it

Revolution!

[1]To hasten the social revolution in England is the most important object of the International Workingman’s Association.[2] The sole means of hastening it is to make Ireland independent.[3] Hence the task of the “ International” is everywhere to put the conflict between England and Ireland in the foreground,[4] and everywhere to side openly with Ireland.

Wealth Gap!

[1]Why decry the wealth gap? [2]First, inequality is correlated with political instability.[3] Second, inequality is correlated with violent crime.[4] Third, economic inequality is correlated with reduced life expectancy.[5] A fourth reason? Simple justice.[6] There is no moral justification for chief executives being paid hundreds of times more than ordinary employees.

— Richard Hutchinsons, “When the Rich Get Even Richer,” The New York Times, 26 January 2000

I'll have to draw this one.

Genes

[1]Genes and proteins are discovered, not invented. [2] Inventions are patentable, discoveries are not.[3] Thus, protein patents are intrinsically flawed.

Our Groveriest President

[1]Since Grover Cleveland has a terrific public record, but a blemished private life, and [2]since his opponent, James G. Blaine, has a storybook private life but a checkered public record,[3] why not put both where they perform best—[4] return Blaine to private life, keep Cleveland in public life.

Drawing it on Board

More Whaling

Ultimately, whaling’s demise in Japan may have little to do with how majestic, smart, or endangered the mammals are, but a good deal to do with simple economics. A Japanese newspaper conducted a survey in Japan re-garding the consumption of whale meat, and reported that of all the thou-sands of respondents, only 4 percent said that they actually ate whale meat at least sometimes. The newspaper then wrote this: “ A growing number of Japanese don’t want to eat whale meat. And if they won’t eat it, they won’t buy it. And if they won’t buy it, say goodbye to Japanese whaling.”

— Reported in Asahi Shimbun, April 2002

Stop breaking us, Socrates

In the Crito, Socrates imagines the laws speaking to him:

He who disobeys us is, as we maintain, thrice wrong; first, because in disobey-ing us he is disobeying his parents; secondly, because we are the authors of his education; thirdly, because he has made an agreement with us that he will duly obey our commands.

-Plato, Crito

What is it good for?

"Wars don’t solve problems; they create them,” said an October 8 letter about Iraq. World War II solved problems called Nazi Germany and militaristic Japan, and created alliances with the nations we crushed. The Revolution-ary War solved the problem of taxation without representation, and created the United States of America. The Persian Gulf War solved the problem of the Iraqi invasion of Kuwait. The Civil War solved the problem of slavery. These wars created a better world.

— Keith Kraska, “Necessary Wars,” The New York Times, 15 October 2002

I will Draw it

It's a hit.

[1]The reality is that money talks.[2] Court officers, judges and juries treat private lawyers and their clients differently from those who cannot pay for representation.[3] Just as better- dressed diners get prime tables at a restaurant, human nature dictates better results for those who appear to have money.

— Desiree Buenzle, “ Free Counsel and Fairness,” The New York Times, 15 January 2007

Diagrammed

HARD!

[1] It is very unlikely that research using animals will be unnecessary or poorly done. [2]Before an experiment using a vertebrate animal is carried out, the protocol for that experiment must be reviewed by an institutional committee that includes a veterinarian and a member of the public,[3] and during the research the animal’s health and care are monitored regularly.[4] Researchers need healthy animals for study in science and medicine,[5] because unhealthy animals could lead to erroneous results.[6] This is a powerful incentive for scientists to make certain that any animals they use are healthy and well nourished.[7] Furthermore, research involving animals is expensive,[8] and because funding is limited in science,[9] only high- quality research is able to compete effectively for support.

Diagrammed

Big Bang

[1]The Big Bang theory is crumbling.[2] According to orthodox wisdom, the cosmos began with the Big Bang. [3] The problem is that astronomers have confirmed by observation the existence of huge conglomerations of galaxies that are simply too big to have been formed in a mere 20 billion years.[4] Studies based on new data collected by satellite, and backed up by earlier ground surveys, show that galaxies are clustered into vast ribbons that stretch billions of light years, and are separated by voids hundreds of millions of light years across.[5] Because galaxies are observed to travel at only a small fraction of the speed of light,[6] mathematics shows that such large clumps of matter must have taken at least one hundred billion years to come together. [7] Structures as big as those now seen can’t be made in 20 billion years.

Diagrammed

Diagramming Practice

  1. Nuclear weapons raise novel moral issues.
  2. Nuclear weapons have new and undreamed-of long-term effects.
  3. The radioactive fallout pollutes the environment and alters human genes.
  4. A nuclear war could destroy human civilization in its entirety.
  5. In the case of nuclear war, the dust caused by the explosions would prevent the sun’s rays from reaching the earth’s surface.
  6. A nuclear war would result in a drastic lowering of the earth’s temperature.
  7. A nuclear war would result in a “nuclear winter.”
  8. No human or human group has the right to gamble with the very climate on which life itself is based.

Teletubbies

  1. Po cannot come to the party.
  2. Po's scooter is broken.
  3. Dipsy cannot come to the party.
  4. Dipsy has to pick up his new hat.
  5. I did not invite the other teletubbies.
  • [Conclusion] No teletubby will come up to the party.

My wife is unfashionable.

  1. Marriage is becoming unfashionable.] [2. Divorce rate is at an all time high], and [3. cohabitation is increasingly presented in a positive manner in the media]. [4. Movies are full of characters who live together and unwilling to commit to a lifelong partnership]. [5. Even newspaper columnists recommend people to live together for an extended period before marriage in order to test their compatibility.]

Damn right.

[1. All university students should study critical thinking.] After all, [2. critical thinking is necessary for surviving in the new economy] as [3. we need to adapt to rapid changes, and make critical use of information in making decisions.] Also, [4. critical thinking can help us reflect on our values and purposes in life.] Finally, [5. critical thinking helps us improve our study skills.]

Sunshine

[1]Solar-powered cars can never be anything but experimental devices.[2] Solar power is too weak to power even a mini- car for daily use. [3] The solar power entering the atmosphere is about 1 kilowatt per square yard.[4] Because of the scattering in the atmosphere,[5] and because the sun shines half a day on the average at any place on earth, [6]average solar power received is 1/ 6 kilowatt, or 4 kilowatt hours a day.[7] Tests on full- size cars indicate that 300,000 watt hours are required in a battery for an electric car to perform marginally satisfactorily.[8] So, 40 square yards of cells would be needed to charge the car batteries, about the size of the roof of a tractor-trailer.

(1.1) If x and y are identical, then x cannot exist without y; but if x and y are not identical, then x can exist without y (axioms of identity). (1.2) Therefore, if we are identical to our bodies, then we cannot exist without them; but if we are not identical to our bodies, we can exist without them.(1) Therefore, if we can exist without our bodies, then we cannot be bodies.(2.1) If it is conceivable that x can exist without y, then it is possible that x can exist without y. (2.2) It is conceivable that we could exist without bodies (including brains).(2) Therefore, we can exist without bodies. (3)Therefore, we cannot be bodies.

Solar Diagrams

Journal of Medicine

Diagrammed

You should not take illegal drugs. 1 They can kill you. 2 If you overdose, you can die. 3 If you share a needle, you could get AIDS. 4 If you get AIDS, then you die. 5

1.We should never take our friends for granted. 2.True friends are there when we need them. 3.They suffer with us when we fail, and they are happy when we succeed.

Democracy

[1]Democratic laws generally tend to promote the welfare of the greatest possible number;[2] for they emanate from the majority of the citizens, who are subject to error, but who cannot have an interest opposed to their own advantage.[3] The laws of an aristocracy tend, on the contrary, to concentrate wealth and power in the hands of the minority;[4] because an aristocracy, by its very nature, constitutes a minority.[5] It may therefore be asserted, as a general proposition, that the purpose of a democracy in its legislation is more useful to humanity than that of an aristocracy.

— Alexis de Tocqueville, Democracy in America, 1835

Elementary

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