Chapter 9 Logic

9.2 Deduction: Categorical Proposition

  • Mueller interview. WSJ video about the interview leaving both parties frustrated.

  • What is my take? Stephen Colbert and Christ Wallace, start from 3:30.

  • The interview by Lesko.

    • If Mueller is fired, then Trump obstructed justice.
    • Mueller is not fied.
    • Trump did not obstruct justice.
  • Another point.

    • If Trump obstructe justice, Mueller will be fired or that his investigation will be terminated.
  • Denying the antecedent.

9.3 Legitimate Inference

Decide whether the inference is legitimate.

  1. “[A]ll revolutionaries in the domain of thought, from Galileo and Columbus to Wagner and Manet, have been for a time persecuted and derided. Ergo, since the Post-Impressionists have provoked a vast amount of scornful mirth, they are necessarily great men.” [Art critic Royal Corissoz speaking ironically, “The PostImpressionist Illusion,” Century magazine, April 1913]

  2. “The chief foundations of all states, new as well as old or composite, are good laws and good arms; and as there cannot be good laws where the state is not well armed, it follows that where they are well armed they have good laws.” [Niccolo Machiavelli, The Prince]

9.4 More difficult examples

Next, a somewhat more difficult example in which the intermediate conclusions are not stated:

  • Since values are nothing more than our own evaluations of the facts, they are not objective. Knowledge of the facts is based on empirical evidence, and anything based on such evidence is objective. But evaluating facts is different from knowing them.

  • As unpleasant as the current shortage is, it means we are consuming less. And so in the long run our standard of living will rise. For in order to improve our standard of living, we must save, and in order to save, we must forgo current consumption.

9.5 Immediate Inference

First, fill out the following table.

Table 9.1: Immediate Inference Table
Standard_form Converse Obverse Contrapositive
All S are P
No S is P
Some S are P
Some S are not P
  • Catergorical proposition: an assertion about the relations among classes
  • Content: subject (S) and predicate (P)
  • Form: quantity and quality
  • Converse: switch S and P
  • Obverse: Change quality and replace predicate with complement (add non)
  • Contrapositive: Complement S and P, then swith

9.6 Immediate Inference

Table 9.2: Immediate Inference Table for Crossing-out
Standard_form Converse Obverse Contrapositive
All S are P All P are S No S is non-P All non-P are non-S
No S is P No P is S All S are non-P No non-P is non-S
Some S are P Some P are S Some S are not non-P Some non-P are non-S
Some S are not P Some P are not S Some S are non-P Some non-P are not non-S

9.7 Logical Equivalence.

  • In logic, two propositions are equivalent if the truth of each necessarily implies the truth of the other, and the falsity of each necessarily implies the falsity of the other. Logically equivalent propositions, in other words, must be either both true or both false.

9.8 Immediate Inference

Cross out cells that are not logical equivalent

Table 9.3: Immediate Inference Table for Crossing-out
Standard_form Converse Obverse Contrapositive
All S are P All P are S No S is non-P All non-P are non-S
No S is P No P is S All S are non-P No non-P is non-S
Some S are P Some P are S Some S are not non-P Some non-P are non-S
Some S are not P Some P are not S Some S are non-P Some non-P are not non-S

9.9 Venn Diagrams Explanation 1

All S are P

Figure 9.1: All S are P

9.10 Venn Diagrams Explanation 2

No S is P

Figure 9.2: No S is P

9.11 Venn Diagrams Explanation 3

Some S are P

Figure 9.3: Some S are P

9.12 Venn Diagrams Explanation 4

Some S are not P

Figure 9.4: Some S are not P

9.13 DiffIcult Cases

Note that we only need to identify cases where one is true and the other is false. Two cells in the above table could be challenging for students, labelled as A and B in the following table.

Table 9.4: Immediate Inference Table
Standard_form Converse Obverse Contrapositive
All S are P
No S is P
Some S are P B
Some S are not P A

9.13.1 Some explanation

For cell A, note that when the venn diagram is this:

All S are P

Figure 9.5: All S are P

The standard form is true, while the converse is false.

For cell B, when the venn diagram is this:

No S is P

Figure 9.6: No S is P

The standard form is false, while the contrapositive is true.

9.14 Exercises

Decide whether the inference is legitimate.

  1. “[A]ll revolutionaries in the domain of thought, from Galileo and Columbus to Wagner and Manet, have been for a time persecuted and derided. Ergo, since the Post-Impressionists have provoked a vast amount of scornful mirth, they are necessarily great men.” [Art critic Royal Corissoz speaking ironically, “The PostImpressionist Illusion,” Century magazine, April 1913]

  2. “The chief foundations of all states, new as well as old or composite, are good laws and good arms; and as there cannot be good laws where the state is not well armed, it follows that where they are well armed they have good laws.” [Niccolo Machiavelli, The Prince]

9.15 Syllogism

  • Syllogism: deductive argument with two premises and a conclusion.

  • Categorical syllogism, both of the premises and the conclusion are categorical propositions.

9.16 In-class quizzes

  1. Some international conflicts arise from just motives, but no aggressive war arises in that way. Hence some aggressive wars are not international conflicts.
  1. Some democracies are tyrannies, because any country that ignores human rights is tyrannical, and some democracies do just that.
  1. “Thus, a minimum wage is nothing more than a price control. Price controls have always been disastrous, so a minimum wage must be harmful.” [Stephan Kinsella, “Simple Economics Goes Against It,” Morning Advocate, March 25, 1989]

9.17 More terms (skip)

  • major term: predicate of the conclusion.

  • major premise: the premise in which the major term appears.

  • minor term: subject of the conclusion.

  • minor premise: the premise in which the minor term appears.

  • middle term: the term that appears in both premises and links together the major and minor terms.

9.18 Some examples

No horned animal is a carnivore.
All moose are horned animals.
No moose is a carnivore.

No privately held company is listed on stock exchanges.
Some privately held companies are large corporations.
Some large corporations are not listed in stock exchanges.

All athletes are people in good physical condition.
All fitness instructors are people in good physical condition.
All fitness instructors are athletes.

No member of the Green Party was elected to Congress.
Some candidates who favor strong environmental regulation are not members of the Green Party.
Some candidates who favor strong environmental regulation were elected to Congress.

9.19 Some examples

No horned animal is a carnivore. No M is P
All moose are horned animals. All S are M
No moose is a carnivore. No S is P

No privately held company is listed on stock exchanges. No M is P
Some privately held companies are large corporations. Some M are S
Some large corporations are not listed in stock exchanges. Some S are not P

All athletes are people in good physical condition. All P are M
All fitness instructors are people in good physical condition. All S are M
All fitness instructors are athletes. All S are P

No member of the Green Party was elected to Congress. No M is P
Some candidates who favor strong environmental regulation are not members of the Green Party. Some S are not M
Some candidates who favor strong environmental regulation were elected to Congress. Some S are P

9.20 Exercises

Determine whether the following argument is valid or not.

  1. “[A]ll revolutionaries in the domain of thought, from Galileo and Columbus to Wagner and Manet, have been for a time persecuted and derided. Ergo, since the Post-Impressionists have provoked a vast amount of scornful mirth, they are necessarily great men.” [Art critic Royal Corissoz speaking ironically, “The PostImpressionist Illusion,” Century magazine, April 1913]

  2. “The chief foundations of all states, new as well as old or composite, are good laws and good arms; and as there cannot be good laws where the state is not well armed, it follows that where they are well armed they have good laws.” [Niccolo Machiavelli, The Prince]

  3. Some international conflicts arise from just motives, but no aggressive war arises in that way. Hence some aggressive wars are not international conflicts.

  1. Some democracies are tyrannies, because any country that ignores human rights is tyrannical, and some democracies do just that.
  1. “Thus, a minimum wage is nothing more than a price control. Price controls have always been disastrous, so a minimum wage must be harmful.” [Stephan Kinsella, “Simple Economics Goes Against It,” Morning Advocate, March 25, 1989]

9.21 Acknowledgement

This section is taken from the Art of Reasoning by David Kelly.

9.22 Non-categorical syllogisms

  • Disjunctive proposition: a compound proposition that uses the connective “or.”
  • Hypothetical proposition: a compound proposition that uses the connective “if … then”

9.23 Disjunctive Syllogisms

The meeting is in room 305 or the meeting is in room 306. p or q
The meeting is not in room 305. not-p
The meeting is in room 306. q

Table 9.5: Negation Table
Quantity Proposition Negation
Universal All whales are mammals. Some whales are not mammals.
No whales are fish. Some whales are fish.
Particular Some mushrooms are hallucinogens. No mushrooms are hallucinogens.
Some cats are not pets. All cats are pets.
Singular Tom is a rich man. Tom is not a rich man.

9.24 Some rules to pay attention to.

  1. Affirming a disjunct is not a valid form of inference.

    • What makes the previous argument valid is a fact that is not actually stated and is not a matter of logical form. In particular, “or” can be used exclusively or inclusively.
  1. Denying a disjunct is valid.

The child is not a girl or the child is not a boy. p or q
The child is a girl. not-p
The child is not a boy. q

  1. not-(p and q) is always equivalent to not-p or not-q.

9.25 Hypothetical Syllogisms

First, some definition.

  • The “if” component is called the antecedent and the “then” component is the consequent.

  • Mueller interview. WSJ video about the interview leaving both parties frustrated.

  • What is my take? Stephen Colbert and Christ Wallace, start from 3:30.

  • The interview by Lesko.

    • If Mueller is fired, then Trump obstructed justice.
    • Mueller is not fired.
    • Trump did not obstruct justice.
  • Another point.

    • If Trump obstructe justice, Mueller will be fired or that his investigation will be terminated.
  • Denying the antecedent.

9.27 What are the arguments?

  • Reminder: calling the shots by Chris Wallace
  • Qualifier: not about whether it is good for the Democrats or the nation as a whole; not about whether it will be successful
  • Work in groups: what are the arguments put forth by Democrats and the GOP? Are the arguments valid?

9.28 Some examples

9.29 One example for the class

  • If Ukraine got the military aid after announcing an investigation into the Bidens, then there is quid pro quo.
  • But Ukraine got the aid without announcing an investigation.
  • Therefore, there is no quid pro quo.

9.30 In-class quizzes

  1. If the plan to build grocery stores in low-income areas is successful, then deaths from heart disease will decrease. Deaths from heart disease have decreased, so the plan to build grocery stores was successful.
  1. If salmon reliably return from the ocean to the streams in which they were spawned—and they do—then they must have some means of detecting subtle differences in the chemical composition of the water of those streams at the point where they flow into the ocean.
  1. If he had even mentioned her name, I would have hit him. But he didn’t, so I didn’t.
  1. If the errors and distortions in television news were not the result of political bias, but merely sloppy reporting, they would be randomly distributed across the political spectrum. But in fact they are all slanted in a particular direction.
  1. Under the “bubble” concept of air pollution regulation, a new factory in a given area can emit a given quan tity of pollutants into the air only if it purchases the right to emit that quantity from some other factory, and another factory is willing to sell the right only if it is willing to reduce its emissions by the same quantity. So if a new factory emits a given quantity of pollutants, some other factory is emitting less of that pollutant by the same quantity.
  1. Without some mechanism of natural selection, there would be no way for new species of organisms to arise—as they have. So clearly there is a mechanism of natural selection.

9.31 if, only if, if and only if

  • (consequent) if (antecedent)
  • (antecedent) only if (consequent)

  • A is sufficient for B = If A then B
  • A is necessary for B = If B then A

Finally,

  • p unless q = If not-q, then p
  • Without X, q = If X does not exist (occur), then q

9.32 Rules of Hypothetical Syllogism

Table 9.6: Mixed Forms of Hypothetical Syllogisms
Valid Invalid
Affirming the antecedent Affirming the consequent
Denying the consequent Denying the antecedent

9.33 Exercises

Translate into standard forms, identify any implicit premise or conclusion, identify what type of hypothetical syllogism it is, and determine whether the syllogism is valid.

  1. If the plan to build grocery stores in low-income areas is successful, then deaths from heart disease will decrease. Deaths from heart disease have decreased, so the plan to build grocery stores was successful.
  1. If salmon reliably return from the ocean to the streams in which they were spawned—and they do—then they must have some means of detecting subtle differences in the chemical composition of the water of those streams at the point where they flow into the ocean.
  1. If he had even mentioned her name, I would have hit him. But he didn’t, so I didn’t.
  1. If the errors and distortions in television news were not the result of political bias, but merely sloppy reporting, they would be randomly distributed across the political spectrum. But in fact they are all slanted in a particular direction.
  1. Under the “bubble” concept of air pollution regulation, a new factory in a given area can emit a given quan tity of pollutants into the air only if it purchases the right to emit that quantity from some other factory, and another factory is willing to sell the right only if it is willing to reduce its emissions by the same quantity. So if a new factory emits a given quantity of pollutants, some other factory is emitting less of that pollutant by the same quantity.
  1. Without some mechanism of natural selection, there would be no way for new species of organisms to arise—as they have. So clearly there is a mechanism of natural selection.

9.34 Induction

Evaluate the claims made in the following chain letter. What conclusions can you draw from the information contained in the letter? What additional information would you need to have?

“KISS SOMEONE YOU LOVE WHEN YOU RECEIVE THIS LETTER AND MAKE MAGIC.
This paper has been sent to you for good luck. The original copy is in New England. It has been around the world 9 times. The luck has now been sent to you. You will receive good luck within 4 days of receiving this letter.
Send copies to people you think need good luck. Don’t send money, as fate has no price. Do not keep this letter. It must leave your hands within 96 hours. An Air Force officer received $70,000. Joe Elliot received $42,000, and lost it because he broke the chain. While in the Philippines, Gene Welch lost his wife 6 days after receiving this letter. He failed to circulate the letter. However, before her death she had won $50,000 in a lottery. The money was transferred to him 4 days after he decided to mail out the letter.
Please send 20 copies of this letter and see what happens in 4 days. The chain comes from Venezuela and was written by Saul Anthony dé Croix, a missionary from South America. Since the copy must make a tour of the world, you must make 20 copies and send them out to your friends and associates. After a few days you will get a surprise. This is true even if you are not superstitious.
Do note the following: Constantine Dias received the chain in 1953. He asked his secretary to make 20 copies and send them out. A few days later he won a lottery of 2 million dollars. Andy Daddit, an office employee received the letter and forgot it had to leave his hands within 96 hours. He lost his job. Later, after finding the letter again, he mailed out 20 copies. A few days later he got a better job. Dalen Fairchild received this letter and not believing threw it away. Nine days later he died.
PLEASE SEND NO MONEY. PLEASE DON’T IGNORE THIS. IT WORKS!!!!!”

9.35 Induction

  • Generalization is a form of inductive inference in which we conclude that something is universally true of a class on the basis of evidence regarding a sample. To avoid the fallacy of hasty generalization, we should follow three basic rules in generalizing: (1) Use a sample that is sufficiently numerous and various; (2) look for disconfirming evidence; and (3) consider whether the conclusion is plausible in light of other knowledge we possess.

  • Logical asymmetry between positive and negative instances

  • Black swan

  • Experiments on mental abilities (innate differences or nongenetic factors)

9.36 More examples

  • In the Clinton section of Manhattan, pigeons were roosting on many apartment buildings. Residents in three buildings put plastic replicas of great horned owls on the rooftops. Result: no pigeons on those buildings. “ ‘The owls work,’ said Sarah Weinberg. . . . ‘A month ago we could not stand outside the door because of all this gook falling from the sky.’ ” [New York Times, Oct. 29, 1986]

  • “While we typically associate economic growth with technological development, organizational innovation has played an equal if not more important role since the beginning of the industrial revolution … The development of transoceanic commerce in the fifteenth century depended on the invention of the carrack, which could sail beyond coastal waters. But it also depended on the creation of the joint-stock company, by which individuals could pool their resources and share risks entailed in funding great voyages. The extension of the railroads across the continental United States in the mid-nineteenth century required large, hierarchically organized companies with geographically dispersed managers.” [Francis Fukuyama, Trust]

9.37 Mill’s Method

The following methods can provide evidence for a causal relationship between a factor a and an effect E:

  • Agreement Show that a is the only factor common to two or more cases in which E occurs. (Negative use: Show that E does not occur in one or more cases when a is present.)

  • Difference Show that E occurs when a is present and not when it is absent, all other factors being held constant. (Negative use: Show that E occurs when a is absent.)

  • Joint method Show that a is the only factor common to two or more cases in which E occurs and that E does not occur in one or more of those cases if a is removed while holding the other factors constant.

  • Concomitant variations Show that quantitative variations in a are systematically related to quantitative variations in E. (Negative use: Show that variations in a do not correlate with any change in E.)

  • Residues Show that in a given case where E occurs, the factors other than a explain only a part of E.

9.38 Acknowledgement

This section is taken from the Art of Reasoning by David Kelly.

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