Final Exam Phil 1102 Chapters 13, 14, 5, 6, 7
Ordinary Language Arguments: Reducing the Number of Terms in an Argument
*But first: add appropriate quantifiers when needed (all, no, some) Ex: Whales are mammals. Some sea-creatures are whales. Some sea-creatures are mammals. V.S. All whales are mammals. Some sea-creatures are whales. Some sea-creatures are mammals.
Translations and the Main Operator
*Locating the main operator helps to translate sentences and place parentheses accurately
Ordinary Language Arguments Reducing the Number of Terms in an Argument:
*Substitute class terms and copula Ex: Some non-citizens pay taxes ->>> Some non-citizens are taxpayers
Ordinary Language Arguments Reducing the Number of Terms in an Argument:
*Substitute complements for terms Ex: All knives are sharp objects. Some knives are illegal items. Some legal items are dull objects.
Ordinary Language Arguments Reducing the Number of Terms in an Argument:
*Use conversion, obversion, contraposition, to SIMPLIFY propositions Ex: All knives are sharp objects. Some knives are non-legal items. ------------------------ Some legal items are non-sharp objects. Change: All knives are sharp objects. Some knives are not legal items. Some legal items are not sharp objects.
Ordinary Language Arguments: Reducing the Number of Terms in an Argument
*eliminate needless words All managers are college graduates. Some [of the] managers are lazy people. Some lazy people are college graduates. [delete]
Ordinary Language Arguments Reducing the Number of Terms in an Argument:
*substitute words that are synonyms All rich people are materialistic individuals. No materialistic individuals are altruists. No altruists are wealthy people.
The Modern Square of Opposition
- P: the complement of class P (i.e. the class containing everything that is not contained in class P) The square of opposition is a chart that was introduced within classical (categorical) logic to represent the logical relationships holding between certain propositions in virtue of their form. A and O propositions are contradictory; E and I are contradictory Contradictories have opposite values for quantity and quality. A: Universal Affirmative contractive to... O: Particular Negative E: Universal Negative contractive to... I: Particular Affirmative
Chapter 6: Categorical Syllogisms
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The Need for a Fair Test
-A good test of a hypothesis involves making a prediction -The truth or falsity of the prediction provides evidence to refute (disconfirm) or support (confirm) the hypothesis -A good prediction helps judge the strength of a causal argument.
The Square of Opposition shows further relationships, some resulting in so-called immediate arguments (i.e. arguments with only one premise and one conclusion)
-Conversion -Obversion -Contraposition
Confirmation Bias: Problems
-False beliefs may be compounded by ignoring evidence to for what would be a correct belief. • Poor decisions may be made based on biased interpretation of evidence. • Sexist, racist, homophobic, gender-identity related, ethnic, ableist, ..., discriminatory beliefs and practices may go uncorrected. • Biased false or selective memory can have grave effects in the court or law, when a case rests on witness reports: witnesses report what they remember, but that may be biased in various and very problematic ways.
The best hypothesis should provide a causal explanation based on...
-Historical data -Contemporary knowledge -Strong probability
Conditional Statements: Distinguishing if from only if:
-If typically precedes the antecedent of a conditional statement. - Only if precedes the consequent of a conditional statement. The light is on only if the neighbors are home. L (horseshoe) N If the neighbors are home, then the light is on. N (horseshoe) L Only if the light is on will the neighbors be home. N (horseshoe) L
Diagramming UNIVERSAL propositions in a syllogism
-TWO areas must now be shaded instead of one
Truth Functions 1) Truth value
1) every statement is either true or false, the 2 truth values. Each statement has exactly one truth value.
Determining Causality (Explanation of the five criteria)
1. A correlation is required, but cannot in itself establish a causal relationship. 2 & 3. The time lag/ spatial distance between the cause must be considered (the longer the time lag/greater the spacial distance, the more situation can be interrupted by other variables). 4. Requires that "X caused Y" be backed by: (1) X was sufficient to bring about Y; and (2) X was necessary for Y (without X, Y would NOT have occurred). 5. Ruling out plausible alternative explanations means that the evidence both REFUTES rival claims and CONFIRMS the HYPOTHESIS.
A categorical syllogisms that meets three requirements
1. All three statements must be standard-form categorical propositions. 2. The two occurrences of each term must be identical and have the same sense. 3. The major premise must occur first, the minor premise second, and the conclusion last.
Confirmation Bias
1. Discrimination: Sexists beliefs may be "confirmed" by citing one mistake a female coworker made, and ignoring all the evidence of women outperforming they male coworkers. Analogously for racist, homophobic, gender, ethnic, ableist, etc. discrimination. 2. Conspiracy: Conspiracy theorists seek confirming data, ignore evidence to the contrary, and interpret data in a biased way ("Of course scientists argue that climate change is real: they are part of the conspiracy!")
Requirements for a Fair Test of a Causal Hypothesis
1. The prediction should be verifiable. 2. The prediction should not be trivial. 3. The prediction SHOULD have a logical connection to the hypothesis.
Introduction to Categorical Logic- Aristotle
4th century B.C. he developed his logic as a foundation for science His system of logic was based on CLASSIFICATION: what is the relationship between one class of objects and another?
Biconditional
A compound statement consisting of TWO CONDITIONALS- one indicated by the word 'if' and the other indicated by the phrase 'only if.' *The triple bar symbol is used to translate a biconditional statement Ex: You get ice cream if, and only if, you eat your spinach. (I [triple bar] S) Ex: IF you eat your spinach, then you get ice cream, and you get ice cream ONLY IF you eat your spinach. (S [horseshoe]) [dot] (I [horseshoe] S)
Conjunction
A compound statement with two statements (conjuncts) connected by the dot symbol (·) (·) translates and, but, still, moreover, while, however, also, although, yet, nevertheless, whereas, etc. Honesty is the best policy, and lying is for scoundrels. H · L Honesty is the best policy; moreover, lying is for scoundrels. H · L Frank and Ernest teach music. F · E (Frank teaches music and Ernest teaches music.)
Disjunction
A compound statement with two statements (disjuncts) connected by the wedge (v) (v) translates or, unless, otherwise, and either ... or Paris is the city of lights or Big Ben is in London. P v B She is either a Pisces or a Scorpio. P v S (She is a Pisces or she is a Scorpio.) Unless it rains today, we will go swimming. R v S
Categorical Syllogisms
A deductive argument constructed entirely of categorical propositions with exactly two premises and a conclusion All comedians are shy people. Some comedians are good actors ------------------------- Some good actors are shy people Contains three terms, each mentioned exactly twice -Minor term (S of conclusion) (ex: good actors) -Major term (P of conclusion) (ex: shy people) -Middle term (M) (ex: comedians) *The major premise must occur first, the minor premise second, and the conclusion last.
Categorical Syllogisms
A deductive argument constructed entirely of categorical propositions with exactly two premises and a conclusion. Ex: All comedians are shy people. Some comedians are good actors. Some good actors are shy people. Minor term: (S of conclusion) ex: good actors Major term: (P of conclusion) shy people Middle term: (M) comedians *occurs in premises
Conditional Statements
A dog is not dangerous IF it has been well-trained. IF a dog has been well-trained, then it is not dangerous. *E: No well-trained dogs are dangerous animals.
Class
A group of objects.
Standard Deviation
A measure of the amount of DIVERSITY in a set of numerical values. -Although the numerical value are spread throughout the curve (the distribution), the majority of the values are clustered around the mean. *The area from -1 SD to 1 SD contains roughly 68% of the values *A bell curve has a few values at the extremes (-2SD to +2SD): 95% and (-3SD to +3SD): 99.5%
Implied Quantifiers
A professor is a human being. A: All professors are human beings. A professor is not a machine. E: No professor are machine. A professor won the Nobel Prize. I: Some professors are winners of the Nobel Prize.
Existential (existence) Import (significance/meaning)
A proposition has existential import if it presupposes (assumes) the existence of certain kind of objects. *Compare: All cats are mammals. (True since cats exist?) All unicorns are mammals. (False since unicorns do not exist?) Problem: All unicorns are mammals. (False (?)) Some unicorns are not mammals. (???) Problem: All female US-presidents are female. (True) Some female US-presidents are female. (There are none! (Yet.))
Representative sample
A sample that accurately reflects the characteristics of the population as a whole
Causal network
A set of conditions that bring about an effect -Normal state: The historical INFORMATION regarding an object. -Abnormal state: A DRASTIC CHANGE in the normal state regarding an object -Precipitating cause: The object or event DIRECTLY INVOLVED in bringing about an effect -Remote cause: Something that is CONNECTED to the precipitating cause by a chain of events
Sorites
A special type of enthymeme that is a chain of arguments; its missing parts are intermediate conclusions, each becoming a premise in the next link in the chain
Mean
A statistical average that is determined by adding the numerical values in the data concerning the examined objects, then dividing the number of objects that were measured
Mode
A statistical average that is determined by locating the value that occurs most
Median
A statistical average that is determined by locating the value that separates the entire set of data in half
Sample
A subset of a population
Syllogisms
Adding a circle for M, we have eight areas.
Syllogisms
Adding a circle for M, we have eight areas. syllogism= drawing a conclusion with two premises, each sharing a middle term
Universal Affirmative
All S are P. An A- proposition affirming that every member of the subject class is a member of the predicate class. ALL (every) members of S is a member of P. Ex: All cell phones are expensive toys. S= cell phones P= expensive toys
Distribution: A-propositions
All S are P. Quantity: universal Quality: Affirmative Term Distributed: Subject Ex: All cats are mammals. *Subject "cats" is distributed; the proposition asserts that every cat is a mammal. * Predicate "mammals" is undistributed; the proposition does not make an assertion about every mammal.
"It Is False That...."
All professional athletes are people who use steroids. Negation: It is false that every professional athlete uses steroids. Contradictory: Some professional athletes are not people who use steroids.
Modern Interpretation
All unicorns are mammals = "If a thing is a unicorn, then it is a mammal." (Gottlob Frege)
Chapter 7: Propositional Logic
Analyzes sentences containing: and, not, or, if, and, only if *provides precise definitions for these logical words -statements rather than classes -not restricted to A, E, I, O propositions (it contains an unlimited number of complex statements)
"The Only" (not at beginning of sentence)
Android phones are the only phones imported by their company. All phones imported by her company are Android phones.
Population
Any group of objects, not just human populations
Science and Superstition
Applying the criteria for testing hypotheses can help expose FALSE BELIEFS. Evolutionary theory: -Highly testable -Repeatedly tested -Successful and fruitful in explaining a large part of the world Creationism or "intelligent design" (the belief that the universe and living organisms originate from specific acts of divine creation, as in the biblical account, rather than by natural processes such as evolution) -Does not generate testable hypotheses -Is not fruitful in explaining anything new about the physical world -Does not meet the basic criteria for scientific theory
Enthymemes
Arguments with missing premises, missing conclusions, or both
Obversion (A, E, I, O)
CHANGE the QUALITY of the proposition then replace the predicate term with its COMPLEMENT CAN be (A, E, I, O) Obvertend A: All carrots are vegetables. Obverse: No carrots are vegetables. Obvertend A: All S are P. Obverse E: No S are non-P. Obvertend I: Some S are P. Obverse O: Some S are not non-P. Obvertend E: No S are P. Obverse A: All S are non-P. Obvertend O: Some S are not P. Obverse I: Some S are non-P.
Chapter 5
Categorical Logic
Chapter 6
Categorical Syllogisms
Chapter 14
Causality
The Modern Square of Opposition
Contradictories have OPPOSITE values for quantity and quality. Examples: A and O I and E
Check Your Understanding 5D: Some martial arts teachers are not tournament winners.
Conversion: Some tournament winners are not martial arts teachers. (switch S and P) Obverse: Some martial arts teachers are non-tournament winners. (change the quality and replace predicate with its complement) Contrapositive: Some non- tournament winners are not non martial arts teachers. (Exchange subject with predicate; then replace each term with complements) *Conversion is not valid for O- propositions
Propositions Requiring Two Translations
Everyone but gamblers sleep well at night. No gamblers are people who sleep well at night, and all non-gamblers are people who sleep well at night.
Check Your Understanding 13A
Ex: Why Women Choose Manly Men By FREAKONOMICS, April 7, 2010 Researchers at the University of Aberdeen's Face Research Laboratory have some news of interest for particularly "masculine"-looking men. Almost 4,800 women from around the world logged on to the organization's online lab over the past year, viewed 20 pairs of male faces (similar, but one slightly more masculine than the other), and expressed their preferences. The Wall Street Journal reports on what the researchers found: "They could predict how masculine a woman likes her men based on her nation's World Health Organization statistics for mortality rates, life expectancy and the impact of communicable disease." In short, women in less healthy countries preferred more masculine men, perhaps for their evolutionary advantages (testosterone is linked to health). So if you're blessed or burdened with a short, broad face and a strong jawline, you might want to think about moving to Argentina.
The Shifting Base Fallacy
Ex: 50% pay cut (100 X 50%) $100->>> $50 Then next week an increase 50 X 50% = $75 (lower salary than original) Always important to know the base rate, especially in relation to increase and decline
Post Hoc Fallacy: Pure chance
Ex: During the last two months our football team was defeated in every game. In each game, the cheerleaders wore blue ribbons in their hair. Therefore, to avoid any further defeats, the cheerleaders should get rid of their blue ribbons. ****A correlation can exist between two types of events by pure chance. Ex: Correlation by pure chance are exploited by companies for advertisement purposes with very convincing, but misleading effects. Our toothpaste is more effective against cavities than our main competitor's. Tests have shown that among a group of randomly chosen people, twice as many people had cavities who use our main competitor's toothpaste compared to people using our toothpaste. Pick small sample size (say, 10 people), do the study 100 times, discard the 99 studies that do not have the desired outcome.
Post Hoc Fallacy: Common Cause
Ex: Peter wakes up with a fever one morning. Around noon, he finds red spots on his skin. He concludes that his high body temperature caused the red spots. ***The red spots are not caused by the fever, but rather both the fever and the spots have a common cause: some infection. The will be a correlation between two events, A and B, if both events are caused by the same third event, C. A C B Recall the example of the drop in atmospheric pressure causing both a storm and the barometer to drop (from chapter 4).
Conversion is NOT valid for A- or O- proposition!!!
Example: Convertend A: All carrots are vegetables. Converse: All vegetables are carrots. Convertend O: Some vegetables are not carrots. Converse: Some carrots are not vegetables. *These are not valid immediate arguments
Contraposition is NOT VALID for I- and E- propositions...
Example: I: Some humans are non-registered to vote. Contrapositive: Some registered voters are non-humans. E: No gorillas are lions. Contrapositive: No non-lions are gorillas. ***These are NOT VALID immediate arguments
Limitation of Mill's Methods
FALSE CAUSE FALLACIES -can occur if Mill's methods are applied simplistically Mill's methods help reveal a NECESSARY ingredient in causation, BUT do NOT PROVIDE sufficient EVIDENCE of causation Understanding the HOW and WHY of causal functions goes BEYOND identifying cause-effect relationships to the development of THEORIES and HYPOTHESES- the basis of scientific reasoning
Categorical Propositions
Four standard forms of categorical propositions A: All S are P. I: Some S are P. E: No S are P. O: Some S are not P. A...f..f...I...r...m..o (affirmo, I affirm) n...E...g...O (nego, I deny)
Main Operator
Has as its range the LARGEST component or components in a compound statement Either one of the four operators that go between statements or else the negation operator *There can be only ONE main operator in a compound statement
Confirmation bias is a general phenomenon...
Humans just are constituted to behave in this way. Like before: the only thing we can do is know about this bias, be conscious of its pervasiveness, and to pay very close attention not to fall prey to it. Problems arise when we think that we are immune to this bias. Alas, we are not. Nobody is immune to confirmation bias! The only thing we can (and should) do is to examine critical our own beliefs when we are confronted with evidence that contradicts it, and remember that we are all subject to this bias.
The following case has been cited in support of the hypothesis that pyramids have special powers. A young woman who was having difficulty with her complexion was told to keep a pitcher of water under a pyramid and then wash her face in that water, with only the mildest soap, once in the morning and once in the evening. She was also told to put nothing else on her face, no creams or medications of any kind and no makeup. Although she has been in the habit of using makeup, she agreed to the experiment. Within two weeks there was a clearly noticeable improvement in her complexion.
Hypotheses: Her face is sensitive to products. Background knowledge: She shows bad complexion on her face and she uses make-up... Prediction: Without the use of make-up and by washing her face with a mild soap her face's complex will clear up.
What type of proposition? Some political documentaries are shows deserving of acclaim.
I- proposition Subject: political documentaries Predicate: shows deserving of acclaim
Transposition (change the order)
If a dog is NOT obedient, then it has NOT been well-trained. If a dog has been well-trained, then it is obedient. All well-trained dogs are obedient.
Conditional Statements: Revisit necessary and sufficient conditions
If in doubt which way around the conditional goes, ask yourself: what is claimed to be necessary to happen if the other thing does—or, equivalently, what is claimed to be sufficient to be the case, to be guaranteed that the other occurs? Ex: If the light is on [sufficient condition], then the neighbors are home [necessary condition]. L (horseshoe) N Ex: The neighbors will be home only if the light is on. The light's being on is necessary for the neighbors' being home. N (horseshoe) L
Immediate Arguments
If something is in Area 1, then it is an S and a non-P. If something is in Area 2, then it is both an S and a P. If something is in Area 3, then it is a P and a non-S. If something is in Area 4, then it is both a non-S and a non-P.
Diagramming in the Modern Interpretation: Immediate Arguments
If something is in Area 1, then it is an S and a non-P. If something is in Area 2, then it is both an S and a P. If something is in Area 3, then it is a P and a non-S. If something is in Area 4, then it is both a non-S and a non-P.
Joint Method of Agreement and Difference
If two or more instances of an event have only ONE thing in COMMON, while the instances in which it does not occur all share the ABSENCE of that thing, then the item is likely cause ->one commonality ex: all John, Robert, Christina had an appetizer and got food poisoning (Have effect in common) ->one absence ex: Kristin did not have an appetizer and didn't get food poisoning (Does not have effect) Assessment: The conclusion allows us to assert that the cause was PROBABLY both a SUFFICIENT and NECESSARY CONDITION for the effect.
Contradictories
In categorical logic, contradictories are pairs of propositions in which one is the NEGATION of the other (one is true & the other false) A- and O- propositions ->All zoos are places where animals are treated humanely. ->Some zoos are not places where animals are treated humanely. E- and I- propositions -> No zoos are places where animals are treated humanely. -> Some zoos are places where animals are treated humanely.
Translating Ordinary Language into Categorical Propositions
Missing Plural Nouns Nonstandard Verbs Singular Propositions Adverbs and Pronouns Implied Quantifiers Nonstandard Quantifiers Conditional Statements Transposition Exclusive Propositions "The Only" Propositions Requiring Two Translations
The Base Rate Fallacy
More worrisome examples of faulty reasoning involving an incorrect base for the calculation of a certain percentage are gathered under the label "base rate fallacy": again, an incorrect base rate is assumed when calculating, or, more often, estimating a percentage.
Singular Propositions
My car is in Joe's garage for repairs. All things identical to my car are things in Joe's garage for repairs.
Universal Negative
No S are P. An E- proposition affirming that no members of the subject class are members of the predicate class. NO cell phones are expensive toys. (No members of S are members of P. )
Distribution: E- propositions
No S are P. Quantity: universal Quality: Negative Term Distributed: subject and predicate Ex: No turtles are mammals. *Both subject and predicate are distributed; the proposition asserts that not even one turtle is a mammal and not even one mammal is a turtle.
Ex: A dolphin is not an animal that enjoys captivity.
No dolphins are animals that enjoy captivity.
Determining Causality
None of the five criteria alone is sufficient to establish a cause-effect relationship; answers to all five together ESTABLISH GROUND for a cause-effect relationship 1. There should be a correlation between the cause and the effect. 2. The cause should precede the effect. 3. The cause should be in the proximity of the effect. 4. A set of necessary and sufficient conditions should exist. 5. Alternative explanations should be ruled out.
What if the Results are Skewed?
Not all data are quantifiable; measuring devices may not provide "objective" results Ex: Original grade scores could be reinterpreted with a curve: (Statistical analysis can lead to new questions, new disagreements, new interpretations)
Nonstandard Quantifiers
Not every investment baker is a crook. O: Some investment bankers are not crooks.
Controlled Experiments
One in which multiple experimental setups differ by only ONE VARIABLE, thereby helping to uncover CAUSAL relationships Experimental group: The group that GETS the variable being tested. Control group: The group in which the variable being tested is WITHHELD
Verifiable Predictions
One where the prediction, if it is TRUE, must include an OBSERVABLE EVENT. Ex: Everyone in the house got sick for no apparent reason. Hypothesis 1: A high degree of radioactivity in the house is causing the illness. Prediction: Use a Geiger counter to find high readings of radioactivity. Hypothesis 2: A disease-causing ghost is haunting this house. Prediction: Is there a method to obtain observable evidence?
Exclusive Propositions (only, none but, solely, alone, none except)
Only persons with tickets can enter the arena. If a person does NOT have a ticket, then that person cannot enter the arena. If a person can enter the arena, then that person has a ticket. ***All persons who can enter the arena are persons who have tickets.
Well-Formed Formulas Rule #4
Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement
Well-Formed Formulas
Precise RECURSIVE (repetition) definition of well-formed formula (WFF): Atomic sentences: A, B, C, D, E, ... Contains a: 1) Base clause Ex: Every atomic sentence is a WFF. 2) Recursive clauses ex: a, b, c, d (see examples) 3) Exclusion clause ex: Nothing else is a WFF.
INFERENCE to the Best EXPLANATION:
Reasoning from the premise that a hypothesis would explain certain facts to the conclusion that the HYPOTHESIS is the best EXPLANATION for those facts
Nontrivial Predictions
Requires reference to BACKGROUND KNOWLEDGE (everything we know to be true) Halley's hypothesis: Comets reappear in regular cycles. Background knowledge: Newtonian Theory Prediction: A comet will reappear in 1758 in a precise location. *The prediction is SPECIFIC and unlikely to be true without this HYPOTHESIS.
Well-Formed Formulas Rule #2
Rule 2: The tilde, goes in front of the statement it is meant to negate. WFFs: ~(P v Q) [horseshoe] ~R (S · P) v ~(~Q · S) Not WFFs: P~ (P v Q)~ ~(C · P)~
Conditional Statements: Mnemonic for necessary and sufficient conditions: If [Sufficient condition], then [Necessary condition]
S U N U= horseshoe S (sufficient condition) then (necessary condition)
Answer to Check your Understanding 13A
Sample: 4,800 women from around the world Population: All women. Sample size: This is a very large sample, but we need to know more about how the women were selected. Potential Bias: The sample excludes women who do not have access to the Internet and this organization's online lab. Also, only 20 pairs of male faces were viewed and compared, allowing for the possibility that including additional male faces, fitting alternative definitions of "masculine," might have led to different results. Randomness: This is not a random sample, because not every woman had an equal chance of getting into the sample. This reduces the likelihood that the sample is representative of the population.
Hypothesis Testing, Experiments, and Predictions
Scientific theories must: -stand up to severe and repeated testing of HYPOTHESES -provide coherent and effective EXPLANATIONS -provide correct PREDICTIONS -help to DISCOVER new facts about the world
Distribution: O- propositions
Some S are NOT P. Quantity: Particular Quality: Negative Term Distributed: ???n/a Ex: Some cars are not fuel-efficient vehicles. Subject "cars" is undistributed; the proposition does not make an assertion about every car. Predicate: n/a
Particular Affirmative
Some S are P. An I- proposition affirming that at least one member of the subject class is a member of the predicate class. SOME cell phones are expensive toys. AT LEAST ONE (some) of S is a member of P.
Distribution: I- propositions
Some S are P. Quantity: Particular Quality: Affirmative Term Distributed: no distribution Ex: Some students are sophomores. *Both subject and predicate are undistributed; the proposition does not make an assertion about every student or every sophomore.
Particular Negative
Some S are not P. An O- proposition affirming that at least one member of the subject class is NOT a member of the predicate class. AT LEAST ONE MEMBER of S is NOT a member of P. Some cell phones are not expensive toys.
Determining Validity
Some designer drugs are addictive chemical substances. Some D are A. Some illegal drugs are not designer drugs. Some I are not D. Some illegal drugs are not addictive chemical substances. Some I are not A. Is conclusion true? Some I are not A (NO: X could be in Area 6) 4. Syllogism is therefore INVALID.
Missing Plural Nouns
Some political parties are disorganized. Some political parties are disorganized groups.
Negation
Statements in which the word 'not' and the phrase 'it is not the case that' are used to deny the statement that is contained in them, using a tilde Ex: Today is Monday M Ex: Today is NOT Monday. ~ M Ex: Gold is selling at $1000 an ounce. G Ex: It is not the case that gold is selling at $1000 an ounce. ~ G
Chapter 13
Statistics
The Misuse of Statistics
Statistics can mislead either intentionally or unintentionally Quantitative terms can be interpreted differently than those containing qualitative terms The latest album by the Green Biscuits is number one in sales this year. (quantitative) The latest album by the Green Biscuits is the best album this year. (qualitative)
The Allure of Superstition
Superstitious beliefs claim causal relationships between events where NONE EXIST -Predictions cannot transfer truth value back to the hypothesis if they are not the product of a repeatable experiment -The scientific process is PARTIAL and TENTATIVE, with evidence accumulating over a long time; mistakes are revealed by the results of repeated experiments
Complement
The class of objects that do not belong to a given class.
Method of Agreement
The method that looks at two or more instances of an event to see what they have in common Ex: Three people get food poisoning in a restaurant -> finding commonality (i.e. appetizer) Criticism: While the method may help narrow the search, it does NOT provide conclusive proof that we have found the cause. It can offer only PARTIAL, tentative inductive evidence.
Method of Concomitant Variations
The method that looks for TWO factors that VARY together. Correlation: A correspondence between two sets of objects, events, or sets of data. Isolate one variable (speed), keep others constant: *RESULT: speed and MPG vary together (are correlated) Criticism: Every cause-effect relationship is a case of correlation, but NOT EVERY correlation is a cause-effect relationship *CORRELATION DOES NOT EQUAL CAUSATION
Method of Difference
The method that looks for what all the instances do NOT have in common -> difference Who did not get food poisoning? (the effect) Kristin- only one that did not each French Fries Criticism: The method identifies a SUFFICIENT CONDITION as the probable cause, but still does NOT provide conclusive proof.
INFERENCE to the Best EXPLANATION: Abduction
The process that occurs when we infer EXPLANATIONS FOR certain FACTS.
Conjunction Fallacy
The reason to think that (b) is more likely seems to be the relatively high probability that Linda is an active feminist and the relatively low probability that she is a bank teller (given what we know about her) Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more likely? a) Linda is a bank teller. b) Linda is a bank teller and is active in the feminist movement. Most people answer (b). In fact, (a) is more likely know about her).
Copula
The terms ARE and ARE NOT. (connecting word)
Quantifiers
The terms: ALL, NO, and SOME
Ex: The Base Rate Fallacy
The test for a certain disease delivers "false positives" in 5% of the cases, that is, for 5% of those tested, the test will say that they have the disease even though the person tested does not in fact have that disease. The test is 100% accurate for people who actually do have the disease. At a certain time (during some outbreak) 1 in 1,000 people have the disease. Tim has a positive test (i.e., the test says, he has the disease). How likely is it that Tim is in fact ill? Surveys suggest that many people think it is as high as 95%, the vast majority thinks it's very high (over 60%). In fact, it's about 2%. 999 out of a 1,000 people do not have the disease; 5% of these are false positives, i.e., around 50. But only 1 in 1,000 actually has the disease. So, the base rate is about 51, with a chance of 1 in about 51 (= about 2%) of being ill.
Well-Formed Formulas Rule #3
The tilde cannot, by itself, go between two statements *Not a WFF WFF: P v ~Q Not a WFF: P ~ Q
Conditional Statements
The word 'if' typically precedes the antecedent of a conditional statement The horseshoe is used to translate conditional statements. If you wash the car, then you can go to the movies. W (horseshoe) M You can go to the movies, if you wash the car. W (horseshoe) M Every time P, then Q. Given that P, then Q. Each time P, then Q. Provided that P, then Q. Any time P, then Q. P implies Q
Post Hoc Fallacy
There are several possibilities how an event A could be correlated with an event B without A causing B or B causing A. The two most important ones (for our purpose here) are: • Pure chance • Common cause
Connecting the Hypothesis and Prediction
There must be a direct connection between the hypothesis and the prediction in order to transfer the TRUTH VALUE from the prediction back to the hypothesis. -Halley noticed a pattern: a comet appeared in a specific place in the sky in 1682, 1606, and 1530. -Halley conjectured that they were instances of the same comet (his hypothesis) -From this data and background knowledge (Newtonian Theory), Halley calculated the next return of the comet, leading to a specific prediction.
The post hoc fallacy again
There was- in fact- a strong positive correlation between the salaries of Presbyterian Ministers and the price of Rum in Havana in the 1950s: whenever one went up (or down), so did the other. Should we conclude that rising prices of Rum in Havana caused the salaries of the Ministers to go up? The other way around? Surely not!
Nonstandard verbs
Trespassers WILL BE prosecuted All trespassers ARE people who will face prosecution.
Diagramming particular propositions in a syllogism
Two areas must now considered instead of one
Conversion, Obversion, and Contraposition Revisited
Venn diagrams illustrate validity and invalidity of immediate arguments Know how to make diagrams for each Draw diagrams for each
Modern Position on Existential Import
We do NOT want to rely on knowledge about what exists to decide whether an argument is valid. -> just live with trivially (easy proof that shows truth) true A- propositions
Confirmation Bias
We have a cognitive bias for accepting information we believe already, and for rejecting data that contradicts what we believe (The more emotionally involved we are, the stronger this bias) This can take different forms: - We may SEARCH for confirmation for our beliefs, and ignoring evidence to the contrary - We may INTERPRET available information in such a way that is confirms our beliefs, even if the information may be better interpreted differently. - We may REMEMBER events, people, ..., selectively and even incorrectly in such a way that it confirms our beliefs.
Compound Statements Rule #1
Well-formed formulas (WFFs): Statement forms that are grammatically correct. Rule 1: The dot, wedge, horseshoe, and triple bar symbols must go between two statements (either simple or compound)
Distributed
When a categorical proposition asserts something definite about EVERY member of a class.
Undistributed
When a proposition does NOT assert something definite about every member of a class.
Random sample
Where every member of the population has an equal chance of getting into the sample
Quality
Whether a categorical proposition is AFFIRMATIVE or NEGATIVE. Some cats ARE mammals. ->AFFIRMATIVE Some mammals are NOT cats. -> NEGATIVE
Quantity
Whether a categorical proposition is UNIVERSAL or PARTICULAR. Ex: ALL cats are mammals. -> UNIVERSAL Ex: SOME cats are mammals. -> PARTICULAR
Modern Square of Opposition
___ A: SP = 0 (Every S is P) *A is contradictory to... O: SP not equal to 0 (Some S are not P) [contradictories have opposite values for quantity and quality] *A contraries (opposite) with E (E: No S is P ) [contraries are opposite] E: SP= 0 (No S is P) E contradictory to... I: SP= not equal to 0 (Some S are P) *NO S IS P (E) is CONTRADICTORY for SOME S IS P (I).
Graphs and Pictograms
can be a useful and instructive way of representing statistical data But they also present an "opportunity" to MISREPRESENT the statistical data How the data is represented graphically is always a choice: some ways to do so can be significantly misleading Ex: Money bags for bar graphs ->What is representative here is only the height of the money bags. But the larger bag is also wider than the smaller one. Moreover, the pictures represent three-dimensional objects, with a disproportional increase in volume.
George Boole
connected logic to algebra E: No S are P written SP = 0 (the intersection of S and P is zero) I: Some S are P written SP ≠ 0 (at least one member is in both S and P)
John Venn
developed diagrams for modern logic after him: Venn Diagrams
The larger the size of the standard deviation, the larger the diversity; the smaller the size of the standard deviation, the smaller the .....?
diversity! If the range of scores for each SD is small, the sides of the curve slope down sharply from the top, indicating the small diversity in the group.
Compound Statement
has at least one simple statement as a component H and K: Hamlet is a tragedy AND Kung Fu Panda is a comedy.
Contraposition
of categorical positions, give rise to valid immediate arguments for ALL and ONLY A- and O- propositions: Contraposition (A,O): Exchange subject with predicate; then replace each with term complements. A: All S are P. Contrapositive: All non-P are non-S.
Conversion
of categorical propositions give rise to valid immediate arguments for ALL and ONLY: E- and I- propositions: Conversion (E, I): Interchange the subject and predicate terms. Convertend: E: No S are P. Converse: E: No P are S. Convertend I: Some S are P. Converse: I: Some P are S.
Spurious correlations
post hoc fallacy ex: Apple iPhone sales correlate with Suicides by handgun
John Stuart Mill (1806-73) Mill's Methods
presented five methods for making causal inductive arguments in his System of Logic: 1. Method of agreement 2. Method of difference 3. Joint method of agreement and difference 4. Method of residues 5. Method of concomitant variations
Categorical Propositions
relates two class of objects Subject term (S) Predicate term (P) All STAND-UP COMEDIANS are WITTY PERSONS.
Cause-effect
relationship can involve many factors
Venn Diagrams for Categorical Propositions
shaded= excludes/none x= shows overlap (see powerpoint) A: All S are P (Because ALL S are P, unshaded is all of P, including the shared region, shaded [excluded region] is S) E: No S are P (Because NO S are P, the shaded region lies in the shared region, since it is excluded) I: Some S are P. (Since SOME S are P, this means at least one S are P so t show the overlap the shared region has an "x." O: Some S are NOT P. (Since SOME S are NOT P, it is impossible for the overlap to be in region P or the shared region so the "x" lies in the S side of the Venn diagram.
Logical Operators
symbols used in translations of ordinary language statements Ex: tilde, dot, wedge, horseshoe, triple bar
Simple statement
the basic component of propositional logic (also called atomic sentences) H: Hamlet is a tragedy
Method of Residues
the method that subtracts form a complex set of events those parts that already have known causes. -Subtract what is known from past experience *Whatever REMAINS is the most likely cause of the remaining effect. Ex: Rash-> macadamia nuts
Truth Function
the truth value of a compound proposition is determined by the truth values of its components and by the logical operators. We say the logical operators determine the truth value of the statement truth functionality; or that the logical operators denote truth functions
