Unformatted text preview: Ling 21, Lecture 4: Ling Basic Logical Concepts Basic BASIC LOGICAL CONCEPTS BASIC Task: To distinguish good arguments from bad Two questions: Are the premises true? Do the premises provide good reasons to accept Do the conclusion? the TWO ARGUMENT TYPES TWO Deductive arguments (try to) PROVE their conclusions (try PROVE Inductive arguments (try to) show that their conclusions are (try PLAUSIBLE or LIKELY PLAUSIBLE LIKELY DEDUCTIVE ARGUMENTS DEDUCTIVE Some pigs have wings. All winged things sing. Therefore, some pigs sing. Everyone has one and only one biological mother. Full sisters have the same biological mother. No one is her own biological mother. Therefore, there is no one whose biological mother Therefore, is also her sister. is EXERCISE: Solve the mysteries, CT pages 56-57. CT INDUCTIVE ARGUMENTS INDUCTIVE Every ruby discovered thus far has been red. So, probably all rubies are red. Polls show that 87% of 5-year-olds believe in the Polls tooth fairy. tooth Marta is 5 years old. Marta probably believed in the tooth fairy. Chemically, potassium chloride is very similar to Chemically, ordinary table salt (sodium chloride). ordinary Therefore, potassium chloride tastes like table salt. THE DIFFERENCE THE
Key: deductive / inductive Key: inductive If the premises are true the conclusion is If necessarily / probably true. necessarily probably The premises provide conclusive / good The good evidence for the conclusion. It is impossible / unlikely for the premises to It impossible unlikely be true and the conclusion to be false. be It is logically inconsistent / consistent to It consistent assert the premises but deny the conclusion. assert FOUR TESTS FOUR Four tests allow us to identify deductive / Four inductive arguments inductive The indicator word test The strict necessity test The common pattern test The principle of charity test INDICATOR WORD TEST INDICATOR
Deduction Certainly Definitely Absolutely Conclusively This entails that Induction Probably Likely Plausible Reasonable The odds are that CAUTION! CAUTION!
-Arguments may not contain any indicator words. Pleasure is not the same thing as happiness. The occasional self-destructive behavior of the The rich and famous confirms this too vividly. rich (Tom Morris) (Tom -Arguers may use indicator words incorrectly. (People very often overstate their cases.) -In these cases, other tests must be used to -In determine whether an argument is deductive or inductive. inductive. The Strict Necessity Test The An argument’s conclusion either follows with An strict logical necessity from its premises or it does not. does If an argument’s conclusion does follow with If does strict logical necessity from its premises, the argument should always be treated as deductive. deductive. if an arguments conclusion does not follow with strict logical necessity from its premises, the argument should normally be treated as inductive. inductive. The Strict Necessity Test The Examples: Alan is a father. Therefore Alan is a male. Jill is a six-year-old. Therefore, Jill cannot Jill run a mile in one minute flat. run COMMON PATTERN TEST COMMON Modus ponens (affirming the antecedent) antecedent If A then B. A. Therefore B. (A = antecedent; B = consequent) This is a very common pattern of deductive This reasoning. reasoning. Common Pattern Test Common Example (modus ponens) If we are in Paris, then we are in France. -------A------------------B-----------------A---------- We are in Paris. --------A----------------A-------- Therefore, we are in France. ---------B-------------------B----------- PRINCIPLE OF CHARITY TEST PRINCIPLE When interpreting an unclear argument, When always give the speaker / writer the benefit of the doubt. Fosters good will and mutual understanding in Fosters an argument. an Promotes the discovery of truth by insisting Promotes that we confront arguments that we ourselves admit to be the strongest and most plausible versions of those arguments. versions Exceptions to the Strict Necessity Test Exceptions An argument in which the conclusion does An not follow necessarily from the premises should be treated as deductive if either: should
The language or context make clear that the The arguer intended to offer a logically conclusive intended argument, but the argument is in fact not logically conclusive; logically 2. The argument has a pattern of reasoning that is The characteristically deductive, and nothing else about the argument indicated clearly that the argument is meant to be inductive. argument
1. Exceptions to the Strict Necessity Test Necessity Example Magellan’s ships sailed around the world. It Magellan’s necessarily follows, therefore, that the necessarily therefore, earth is a sphere. earth If I’m Bill Gates, them I’m mortal. I’m not Bill Gates. Therefore, I’m not mortal. SUMMARY: How to distinguish deductive from inductive arguments deductive inductive
If the conclusion follows necessarily from the premises = If deductive deductive If the conclusion does not follow necessarily from the If premises = inductive, unless inductive, Language indicates it is deductive Argument has deductive pattern of reasoning If the argument has a pattern of reasoning that is If characteristically deductive = deductive, unless deductive, Clear evidence indicates it is intended to be inductive Clear If the argument has a pattern of reasoning that is If characteristically inductive = inductive unless inductive Clear evidence indicates it is intended to be deductive If the argument contains an indicator word If indicator If still in doubt: Principle of Charity If Principle 5 COMMON DEDUCTIVE DEDUCTIVE PATTERNS PATTERNS Hypothetical syllogism Categorical syllogism Argument by elimination Argument based on mathematics Argument from definition HYPOTHETICAL SYLLOGISM HYPOTHETICAL A syllogism is a three-line argument with two syllogism premises, one of which is a conditional. premises, is a syllogism. Modes ponens Other syllogisms are: Chain arguments Modus tollens (denying the consequent) Denying the antecedent Affirming the consequent CHAIN ARGUMENT CHAIN
If A then B. If B then C. Therefore if A then C. If you are blue in the face then you are lying. If you are lying then you can’t be my friend. Therefore if you are blue in the face then you Therefore can’t be my friend. can’t MODUS TOLLENS MODUS If A then B. Not B. Therefore not A. If we’re in Sacramento, we’re in California. We’re not in California. Therefore, we’re not in Sacramento. If you love me, you’ll come with me to Tibet. You will not come with me to Tibet. Therefore you do not love me. DENYING THE ANTECEDENT* DENYING If A then B. Not A. Therefore not B. *If Payton Manning won the Superbowl then he’s a great *If athlete. athlete. Payton Manning didn’t win the Superbowl. Therefore, Payton Manning is not a great athlete. *If Jack comes to the party, Jill will leave. Jack did not come to the party. Therefore Jill did not leave. *Denying the antecedent is a fallacious deductive pattern AFFIRMING THE CONSEQUENT* AFFIRMING If A then B. B. Therefore A. If we are on Neptune then we are in the solar system. We are in the solar system. We Therefore we are on Neptune. *Affirming the consequent is a fallacious deductive *Affirming pattern pattern Exercise: Identify the argument pattern (ex. 3.2, p. Exercise: 67) 67) MODUS PONENS (affirming the antecedent): If A then antecedent): B. A. Therefore B. B. CHAIN: If A then B. If B then C. Therefore if A then C. MODUS TOLLENS: If A then B. Not B. Therefore not MODUS A. A. *DENYING THE ANTECEDENT: If A then B. Not A. If Therefore not B. Therefore *AFFIRMING THE CONSEQUENT: If A then B. B. If Therefore A. Therefore PRINCIPLE OF CHARITY PRINCIPLE Attribute an arguer the strongest argument Attribute possible. possible.
Andy told me he ate at JB’s yesterday. But JB’s was destroyed by a fire a month ago. It is certain therefore that Andy is either lying or It mistaken. mistaken. Caution: A principle of argument interpretation, interpretation not a principle of argument repair. repair CATEGORICAL SYLLOGISM CATEGORICAL A three-line argument three-line in which each statement begins with one of the words all, some, or no. some, Some pigs have wings All winged things sing. Therefore some pigs sing. ARGUMENT BY ELIMINATION ARGUMENT Rules out various logical possibilities until Rules only a single possibility remains. only Either Dutch or Jack or Celia committed the Either murder. murder. If D or J committed the murder then the weapon If was a rope. was The weapon was not a rope. Therefore neither D nor J committed the murder. Therefore C committed the murder. Therefore MATHEMATICS MATHEMATICS The conclusion depends largely or entirely The on mathematical calculation or measurement. measurement. Light travels at a rate of 186,000 miles per second. The sun is more than 94 million miles from earth. Therefore it takes more than 8 minutes for the sun’s Therefore light to reach earth. light Caution – not all arguments that make use of numbers and mathematics are deductive. numbers DEFINITION DEFINITION The conclusion follows from the definition The of some key word or phrase in the argument. argument. Josefina is a drummer. Therefore Josefina is a musician. COMMON INDUCTIVE PATTERNS PATTERNS There are 6 common inductive patterns: Inductive generalization Predictive argument Argument from authority Casual argument Statistical argument Argument from analogy A generalization attributes some generalization INDUCTIVE GENERALIZATION GENERALIZATION characteristic to all or most members of a given class. given Information about some members of the Information class is said to license the generalization. license All dinosaur bones discovered thus far All have been more than 65 million years old. have Therefore probably all dinosaur bones are Therefore more than 65 million years old. more PREDICTIVE ARGUMENT PREDICTIVE A statement about what will (likely) statement happen in the future is defended with reasons. reasons. It has rained in Vancouver every February It since records have been kept. since Therefore it will probably rain in Therefore Vancouver next February. Vancouver AUTHORITY, CAUSE, STATISTICS AUTHORITY, Argument from Authority The conclusion is supported by citing The some presumed authority or witness. some Causal Argument Asserts or denies that something is the Asserts cause of something else. cause Statistical Argument Rests on statistical evidence. ANALOGY ANALOGY Common Pattern: Common Two (or more) things are alike in one way. Therefore they are probably alike in some further way. they As a man casts off worn-out garments and puts on As others that are new, others similarly, the soul, casting off worn-out bodies, enters similarly, into others, which are new. into (Bhagavad-Gita) Exercise: Determine whether arguments are deductive or inductive (ex. 3.3.1, p. 74-76) or VALIDITY VALIDITY VALID arguments may have false premises and false conclusions! and At issue is the form. If the premises are true At form the conclusion must be true. the
All circles are squares. All squares are triangles. Therefore all circles are triangles. All fruits are vegetables. Spinach is a fruit. Therefore spinach is a vegetable. VALIDITY, CONT’D VALIDITY, It is not enough that the conclusion It happens to be true. If the conclusion doesn’t follow from the premises by strict logical necessity, a deductive argument is invalid. invalid All pigs are animals. Wilber is pink. Therefore Wilber is a pig. Exercise: What conclusions follow validly? (ex. 3.4, p. 76-77) (ex. SOUNDNESS SOUNDNESS A deductive argument is deductive sound if it is valid sound and has true premises. and A deductive argument with (at least) deductive one untrue premise, valid or invalid, iis s untrue unsound. unsound Exercise: Determine whether arguments are valid / sound (ex. 3.5 I & II, p. 84-86) are INDUCTIVE STRENGTH INDUCTIVE A ‘good’ deductive argument is valid. ‘good’ valid A ‘good’ inductive argument is ‘good’ inductive strong. strong An inductive argument is strong if the An conclusion follows probably from the premises. premises All recent US presidents have been All college graduates. It is likely that the next US president will be a college graduate. be WEAKNESS WEAKNESS An argument that is not strong is weak. An weak Most US presidents have been men. Most It is likely that the next US president will be a woman. In A weak inductive argument, the conclusion does not In not follow probably from the premises. follow I dream about monsters. You dream about monsters. Therefore everybody probably dreams about monsters. INDUCTIVE PROBABILITY INDUCTIVE The premises and conclusion do not have to be The true – The question is: true If the premises were true, would the conclusion If were follow? follow? Deductive arguments are either 100% valid or 100% invalid. 100% Inductive arguments can be somewhat strong, strong, very strong, depending on the degree of support the premises provide for the conclusion. support According the National Weather Service, there is a According 60% - 70% - 90% chance of rain today. 60% It is likely that it will rain today. INDUCTIVE ARGUMENTS INDUCTIVE A valid deductive argument with true premises is sound. valid true sound A strong inductive argument with true premises is cogent. strong true cogent An inductive argument that is either weak or has at least An one false remise is uncogent. uncogent
- No US president has been a skateboarding champion. No Therefore the next US president will probably not be a skateboarding Therefore champion. (Cogent) (Cogent) - All previous US presidents have been rocket scientists. Therefore the next US president will probably be a woman. (Uncogent) Therefore (Uncogent) - All previous U.S. Presidents have been Democrats. Therefore the next All U.S. President will be a Democrat. (Uncogent) (Uncogent) Exercise: Determine whether arguments are cogent or uncogent (ex. 3.5 III, p. 85-86) uncogent Summary of Argument Types Summary
Deductive Valid Inductive Weak (all are (all uncogent) Invalid Strong (all are (all unsound) Sound Unsound Cogent Uncogent Culminating Activity Culminating Exercise 3.5 IV, Page 86: Exercise Determine whether the arguments are deductive or inductive. If the argument is deductive, determine whether it is valid or invalid. If the argument is inductive, determine whether it is strong or weak. or COMPLEX ARGUMENTS COMPLEX Take into account only basic premises Take and conclusions. and Determine whether each conclusion is Determine arrived at deductively or inductively. Evaluate for validity or strength. Evaluate In the case of nonconvergent arguments, In if any one step is inductively weak the whole argument is weak. whole The inductive probability of a convergent The argument is as strong as its strongest branch. branch. ...
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