Part III: How to Evaluate Inductive Arguments
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Part III: How to Evaluate Inductive Arguments Week 7: Inductive Arguments Lecture 7.1: What is induction? Inductive arguments may actually be more common than the deductive ones we’ve been learning about the past few weeks. Types of inductive arguments ● Inference to the best explanation you might use to determine who committed a crime. ● Arguments from analogy ● Statistical generalizations just about all the polls in politics! ● Statistical applications how do those generalizations tell you something about a person? ● Causal reasoning ● Probability ● Decision making Inductive vs. deductive arguments How could you prove the sun will come up tomorrow morning? The sun came up the day before…etc. The sun came up the day before. The sun came up the day before. The sun came up yesterday. ∴ The sun will rise tomorrow. Or, for short: The sun came up every day before that for an awfully long time. The sun came up yesterday. ∴ The sun will rise tomorrow. ● Is the argument valid? No. It’s possible for these premises to be true and the conclusion false (say, if a meteor struck Earth and stopped it from spinning). ● Is this argument good? Yes. These premises provide good reasons for the conclusion. So we see that an argument can be good even if it’s not valid. But other arguments are no good because they’re invalid. Every sophomore is a student. Bub is not a sophomore. ∴ Bub is not a student. ● But what if Bub’s a junior or senior?
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Deductive arguments are intended to be valid, so they’re no good if they’re invalid. ● The standards are whether the argument is valid or invalid. ● Validity is allornothing no such thing as “partly valid.” ● Standard of validity is called indefeasible (or monotonic) no matter what premises you add to a valid argument, it will still be valid. If Joe is a sophomore, then Joe is a student. Joe is not a student. ∴ Joe is not a sophomore. Inductive arguments are not intended to be valid; they can still be good even when invalid. ● The standards are whether the argument is strong or weak. ● Strength comes in degrees an argument is stronger when it gives more and better reasons for the conclusion. ○ We can’t just ask if it’s strong we need to know if it’s strong enough. That depends on the context and the values at stake. When there’s a lot to lose, we demand better reasons and stronger arguments ■ Example: Sticking a piece of straw in the center works to test if cake’s done, but not a turkey you can get sicker from an uncooked turkey, so there’s more at stake. ● Standard of strength called defeasible (or nonmonotonic) adding additional premises or information can weaken or even undermine a strong argument. ○ Example: In a court scenario, a piece of new information that the suspect has an identical twin weakens eyewitnesses’ testimony that it was the suspect. Classifying an argument as inductive or deductive is important because it determines whether the argument’s good or bad. The sun came up yesterday. The sun came up every day for thousands of years. ∴ The sun will come up tomorrow. This argument isn’t valid, but it isn’t intended to be so it’s still good. Common mistakes about inductive arguments: ● Many view them as inferior to deductive arguments but it’s easy to take any inductive argument and turn it into a deductive argument. If most of Joe’s friends are seniors, then Joe is a senior. Most of Joe’s friends are seniors. ∴ Joe is a senior Adding that first conditional premise shifts doubts about the inductive argument into doubts about the premise it hasn’t really made us any more sure of Joe’s seniorness. ● Many believe induction always takes us from the particular to the general but generalization is just one kind of inductive argument.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ Inference to the best argument In taste test, this lemonade tastes awful because you put dishwashing detergent in it! ○ Argument from analogy I know because it tastes like soap. ○ Necessary condition test I didn’t spit out the first lemonade I tried. ○ Statistical generalization Did you know over 90% of those we tested agreed they preferred our lemonade to dishwashing detergent? ○ Statistical application If you try this test on your friends, they’ll agree too.
Lecture 7.2: Generalizations from samples Generalizations all around us “almost all,” “most,” “many,” “¾,” “half,” “87.2% of statistics,” etc. ● They’re useful when you need to make a decision when you’re sick, you need to know if people with the same symptoms usually go to the doctor, & if doing so would even help. ● But how do we decide which ones to believe? You can’t check them all, so you have to do a sample from which you can reach a conclusion about the larger class. Forms: ● Universal The first F is G. The second F is G. The rest of the Fs in the sample are G. ∴ All Fs are G. ● Partial The first F is G. The second F is G. The third F is not G. The fourth F is G, and so on. ∴ X% of Fs in the sample are G. ∴ X% of all Fs are G. Moving the argument from part of the class to the whole thing proves it’s inductive: Almost all bands that I know of have drummers. ∴ Almost all bands have drummers. ● Is it valid? No. Maybe I need to broaden my musical horizons. ● Is it defeasible? Yes. I’d have to change my conclusion if you gave me examples of bands without drummers. ● Does this mean it can’t be strong? No. Argument could be strengthened by making the sample larger, or weakened with a smaller sample. ● Is the argument still good? Yes, because it’s inductive you can’t criticize it for not being valid, because it doesn’t try to be. How do we tell if a generalization from a sample is strong? We have to turn to statistics.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● Some argue they’re always worse than the “damn lies” in Twain’s quote (cribbed from Disraeli) and you can’t trust ‘em at all. ● Some argue they’re infallible it’s just math, and you can’t question the numbers, right? ● But both generalizations are wrong the truth lies somewhere in the middle.
Lecture 7.3: When are generalizations strong? How can we tell when a generalization from a sample is a good argument? We’re not going to get into a lot of statistics that’d take a whole course but let’s look at some common errors that can be used to mislead.
Questions to ask about generalizations ● Are the premises false? ○ Let’s say the optimum number of chocolate chips is 1012 for a 3inch cookie, and Walter bought 10 cookies from each of 5 bakeries in his area. 80% of cookies I bought from Bakery A have 1012 chocolate chips. ∴ 80% of all cookies from Bakery A have 1012 chocolate chips. ○ Thing is, Walter is lying he didn’t buy any cookies at all! ○ An argument that’d be strong if its premises were true is no good if they aren’t. ● Are the premises unjustified? ○ It’s not enough for the premises to be true. ○ What if Walter did buy the cookies, but counted them so fast that he missed a bunch of chips? or chips were melted together and couldn’t be counted very well? or cookies from bakery A and B got mixed up? ○ Mistakes were made in the counting, so the premises are false. ● Is the sample too small? ○ Let’s say Walter is honest and counts thoroughly and carefully and is justified in believing he hasn’t made a mistake. He bought one cookie from each bakery; the Bakery A cookie had 11 chips, and the other bakeries’ had less than 10. 100% of the cookies I bought from Bakery A have 1012 chocolate chips. ∴ 100% of all cookies from Bakery A have 1012 chocolate chips. 0% of the cookies I bought from Bakery B have 1012 chips. ∴ 0% of all cookies from Bakery B have 1012 chips. ○ The problem: You can’t generalize from just one cookie. It’s not a mechanical process, so you can’t know every cookie is the same. ○ Fallacy of hasty generalization generalizing from a sample that’s too small. This fallacy is very common (ie: My neighbor’s new car broke down, so that type of car is no good). ○ We don’t need to know if the sample’s large, but if it’s large enough. In some cases, a very small sample can be big enough for the generalization.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
■ Whether the sample is large enough depends on background information and what’s at stake. ● Background info: You come across an apple tree and want to know if its apples float in the water. One apple is a big enough sample you have background info from biology that the apples on that tree are similar to each other. ● Stakes: If you want to make sure parachutes are packed properly and will work, you obviously want to check every single one. Chocolatechip cookies aren’t as big a deal. ● Is the sample biased? ○ Say Walter’s not lying, counted accurately, is justified that he counted accurately and had a big enough sample. At each of the five bakeries, he explained his survey to the person at the counter before buying 10 cookies. 80% of cookies I bought from Bakery A have 1012 chocolate chips. ∴ 80% of all cookies from Bakery A have 1012 chocolate chips. ○ The problem: The employee knows what Walter’s looking for, so he’s probably going to serve Walter the cookies with the most chips in them so Walter will continue to buy from his shop. ○ Fallacy of biased sampling the sample is not representative of the whole because it’s biased in a certain way. Seems like an obvious mistake, but it happens all the time even to some of history’s top pollsters. ■ Literary Digest poll in 1936 presidential election concluded that Alf Landon was going to beat FDR 5644. They were way off FDR won 6238. How did this happen? They sent letters to addresses pulled from the phone book but back then, few had phones, especially among poor or rural areas. Those voters favored FDR because of the New Deal and other policies to help them out. ■ Still happens today many pollsters looking to do a quick poll call people up on their phones. But there are lots of problems: ● Polls can’t be done on cellphones, which may be the only phone young people have, so they’re likely to be underrepresented. ● If those polled do have a landline, they have caller ID and may not answer if it’s a pollster low response rates. ● Some studies have found that women tend answer the phone more often than men, which skews samples. ○ What do you with a biased sample? You try to correct for that. ■ If you get a lot more people from the liberal party than last time might mean sample is skewed, or might mean more people have shifted to liberal party. Polls use different techniques to correct, which means some may reach very different results. ■ Some pollsters are dishonest and may slant their questions to get the desired result. ● Is the question slanted?
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ Question may be worded in a way that makes people feel bad about giving a certain answer. ■ “Should we kill a mouse to save a human?” → a yes indicates support of animal experimentation. ■ “Should scientists torture animals in their experiments?” → a no indicates opposition to animal experimentation. ○ Survey participants’ options may be limited. ■ Doris Day Animal League says survey found 51% “think primates are entitled to the same rights as human children.” But the survey question refers specifically to chimps, says “similar” rather than “the same,” and lists only four options: Should animals be treated (a) like property, (b) similar to children, (c) the same as adults, (d) not sure.
Lecture 7.4: Applying generalizations Application of the generalization (or statistical syllogisms) a form of argument in which we (shockingly) apply a generalization to a particular case. ● Common, as in: I almost never like horror movies. or I don’t want to invest in that restaurant 80% of restaurants fail within the first 23 years. ● Example: Almost all professors who are teaching are wearing shoes. Walter is a professor who is teaching. ∴ Walter is (probably) wearing shoes. ● Has same form as previous example, which we can explore further by substituting variables for terms in the English argument: X% of F are G. a is F. ∴ a is (probably) G. ○ F = reference class the subject in the general premise (the set of professors who are teaching) ○ G = attribute class the predicate in the general premise (the people who are wearing shoes) ○ a (usually lowercase) = the individual to whom the generalization is applied. (Walter) ○ X = quantifier percentage in generalization. (Even phrases like “almost all” are replaced with X%.) ● Note that application of a generalization moves in opposite direction from the generalization from the sample. Generalization: Application: X% of sampled F are G. X% of F are G. ∴ X% of F are G. a is F. ∴ a is (probably) G. ● Applications of generalizations are inductive and share those arguments’ features: ○ They’re invalid (it’s possible for premises to be true and conclusion false).
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ They’re defeasible (can be weakened by adding information). ○ They can vary in strength. When is an AoG considered to provide strong reasons for the conclusion? ● Premises must be true and justified. ● Percentages should be very high or low. (“99% of F are G” is stronger than “80% …”) ● No smaller reference class should conflict. ○ This is the fallacy of overlooking a conflicting reference class. 95% of F (professors) are G. a is F. ∴ a (probably) is G. BUT … 30% of F* (online professors) are G. a is F*. ∴ a (probably) is not G. ○ General rule: When you encounter conflicting reference classes, look at the smallest one it’s usually likely to give you a better estimate. ■ But keep in mind that as reference class gets narrower and narrower, you may not have enough data to support the premise of a certain percentage. ○ A medical example: 90% of people with a certain medical condition die. Bob has that medical condition. ∴ Bob is likely to die? BUT … Most people catch this illness when they’re old. Bob is quite young. Only 20% of people who get this illness when they’re young die. ∴ Bob isn’t likely to die? BUT … 80% of people who are young and have a heart condition die. Bob has a heart condition. ∴ Bob is likely to die? BUT … 30% of those given a new treatment die, even if young/heart condition. Bob lives in an area where he can get this treatment. ■ As we get more information on this illness, the chance Bob will die goes high, low, high, low … ■ So which of the generalizations should we use? The narrowest class. But the treatment’s new and probably hasn’t been tried on many young people, let alone those with heart conditions.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Lecture 7.5: Inference to the best explanation Inference to the best explanation combines qualities of both a justification and an explanation (see: Week 1) it uses an argument to justify belief in the conclusion, and yet a premise of the argument is that that conclusion explains a phenomenon you took for granted. ● This is extremely common. ○ Appears in just about every detective story. (Sherlock Holmes’ “deduction” was actually inductive reasoning.) ○ In science: What killed the dinosaurs? Best explanation is giant meteor in Yucatan Peninsula. The point of most scientific theories is to provide the best explanation of all the experimental data that has been observed by scientists. ○ In everyday lives: A drop of water hit my head that must mean the roof is leaking. ● Uses an explanation but runs in the opposite direction as a typical explanation: The conclusion does the explaining; the premises are what get explained: ○ Explanation: When there is a leak in the roof, water drops through the roof. When water comes through the roof, it drops onto whatever is in its way. I am in its way. ∴ Water drops onto me. ○ Inference to the best explanation: Water drops onto me. A leak in the roof would explain why water drops onto me. No better explanation is available. ∴ There is a leak in the roof. ● This argument isn’t valid, but it’s still good as an inductive argument. . ● Is it defeasible? Yes (by adding in information that there’s a guy with a watering can above you). ● Defeasibility does not mean it was a weak argument from the outset. ● Function is different as well: ○ In explanation, we take for granted that conclusion is true. ○ In inference to the best explanation, we argue that the conclusion is true because it’s the best explanation of the observation in the first premise.
Lecture 7.6: Which explanation is best? A good explanation: ● Starts with surprising circumstances that need to be explained. ● Involves an accurate picture of what you’re trying to explain. ● Has to be compatible with all of the facts not just the specific ones you’re trying to explain. ● Is simple and conservative. ● Is deep ● Is powerful and broad
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● Is modest Summary of the skit’s argument: 1. Observation: It’s time for class and nobody else is in the room. 2. Explanation: The hypothesis that class was canceled due to a holiday, plus accepted facts and principles, gives a suitably strong explanation of the observation in premise 1 that no one else is in the room. 3. Comparison: No other hypothesis provides an explanation nearly as good as the holiday hypothesis in premise 2. 4. Conclusion: The holiday hypothesis in premise 2 is true. ● Premise 3 is most controversial of these. To justify this, we need to compare other possible explanations (as the skit did) in a background argument.: ○ “Other students are late” hypothesis falsified by passing of time. ○ “Other students are invisible” hypothesis not conservative. ○ “Other students are undetectable” hypothesis not falsifiable. ○ “Other students are playing a joke” hypothesis not deep. ○ “Professor skipped class” hypothesis not powerful or broad. ○ “Other students are at MLK Day ceremony” not modest. ○ No other hypothesis seems plausible, therefore (premise 3 above). ● Another background argument: The holiday hypothesis … ○ Explains why nobody else is there if classes were canceled for the holiday, nobody else would be there (in that room at the usual time) ○ Is broad it explains other actual observations (for example, the whole building is empty). ○ Is powerful applies to many separate cases (for example, why students won’t be there on future holidays). ○ Is falsifiable the students might find out classes weren’t canceled for the holiday ... ○ Is not falsified … but they didn’t. ○ Is conservative does not conflict with prior, wellestablished beliefs. ○ Is deep does not depend on any assumptions that need but lack independent explanation. ○ Therefore, the holiday hypothesis, plus accepted facts and principles, gives a strong explanation of why nobody else is in the room. ● The items used in these background arguments are commonly called explanatory virtues. The more of these virtues an argument has, the stronger it is.
Lecture 7.8: Arguments from analogy Argument from analogy very closely related to inference to the best explanation; also common ● Remember that an analogy is a comparison between two things that points out similarities without being specific. ● Some might argue the lack of specificity makes them illsuited for arguments, but we use them that way all the time.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ Public policy: Phoenix is a lot like Houston in many ways. This type of transportation system worked well for Houston. So it will probably work well in Phoenix. ○ Law: Common law and other systems use precedent in deciding similar cases. ■ Useful because it’s predictable and gives flexibility. ○ Art history: You find a painting in the attic that looks like a (valuable) Cezanne but isn’t signed. How do you authenticate it? You look at a lot of works known to be by Cezanne and try to find out whether they’re similar (or get an expert to do so). This painting is like others in its colors, forms, brushwork, etc. Those other paintings were painted by Cezanne. ∴ This painting was (probably) also painted by Cezanne. ● Arguments from analogy are inductive they only try to show they’re probably true. Argument from analogy form Object A has properties P, Q, R, and so on. Objects B, C, D, and so on also have properties P, Q, R, and so on. Objects B, C, D, and so on also have property X. ∴ Object A (probably) also has property X. ● A = the subject ● B, C, D = the analogous objects ● P, Q, R = the similarities ● X = the target Standards of strength An argument from analogy is stronger when: ● the similarities are more important ● there are more similarities ● there are fewer disanalogies (there are always some) ● the analogous objects are more diverse ● the conclusion is weaker (“probably”)
Week 8: Causal Reasoning Lecture 8.1: Causal reasoning Causal judgments have to do with what caused an individual event. Supported by inductive arguments. ● Cannot be certain, because we can’t know with absolute certainty the causes of things. Causation ● Causal claims are about individual events one event causes another. ○ My phone is off. When I push this button on the bottom, it turns on.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ General rule behind that: Every time I push this button, the phone turns on. If it doesn’t hold as a general rule, it’s not a causal relation. ● Big topic for this week: How do we test general principles? ○ Which cases in the data do we need to look at, and how do we know when we have enough data to believe in a general principle? Types of general principles ● Sufficient condition ○ F is a sufficient condition for G if and only if in normal circumstances: ■ For events: Whenever event F happens, event G also happens. ■ For features: Anything that has feature F also has feature G. ● Necessary condition includes a negation on both sides. ○ F is a necessary condition for G if and only if in normal circumstances: ■ For events: Whenever event F does not happen, event G also does not happen. ■ For features: Anything that does not have feature F also does not have feature G. ● Example: ○ Being a whale is sufficient for being a mammal. ■ Anything that is a whale is a mammal. ○ Being a whale is necessary for being a sperm whale. ■ Anything that is not a whale is not a sperm whale. ● Thinking of this in propositional statements may be helpful (but it’s not quite the same because there are no quantifiers): ○ Sufficient = If F, then G ○ Necessary = If not F, then not G Causal examples: ● Whenever is a quantifier applied to a restricted domain of discourse. ● Causal cases must be understood as a general rule within an appropriate range of circumstances. ○ Striking this match is sufficient for lighting it. ■ It can’t be true that whenever the match is struck it lights it won’t if the surface is not rough. ■ This example takes for granted that this is a normal match say, you haven’t dipped it into a cup of coffee to get it wet. ○ Striking this match on a rough surface is necessary for lighting it. ■ Yeah, we could use a lit match to light this match, but … ● So we need to modify our definition to include “in normal circumstances” which tells us how “whenever” applies. Kinds of conditions:
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● Conceptual ○ Being a whale is sufficient for being mammal. (But being a whale is not necessary for being mammal.) ○ Being a whale is necessary for being a sperm whale. (But being a whale is not sufficient for being a sperm whale.) ● Causal ○ Striking this match is sufficient for lighting it. ○ Striking this match on a rough surface is necessary for lighting it. ● Moral ○ Torturing for fun is sufficient for doing wrong. ○ Doing something wrong is necessary for being punishable.
Lecture 8.2: Negative sufficient condition tests How can we tell which conditions are necessary and which are sufficient? Two tests for each type positive and negative. Negative test for sufficient condition tells you what’s not a sufficient condition. ● X is NOT a sufficient condition of Y if there is any case where X is present and Y is absent. ● A deductive argument against the generalization that whenever X is present, Y is present as well. ○ That means it’s not defeasible it’s still valid even when more information added. ○ All you need is one counterexample. ● An example (inspired by Airplane!): A group of people eat a full banquet with soup course, main course, wine, dessert. All goes well until the end; when some start to get up, they feel sick and then keel over and die. ○ We figure that they were poisoned, and that it had to have been through the food. ○ Now we look at what each particular person ate to determine which of the dishes was poisoned. (Which candidate X is sufficient for the target Y?) Diners
Soup
Main
Wine
Dessert
Result
Ann
tomato
chicken
white
pie
alive
Barney
tomato
fish
red
cake
dead (=Y)
Cathy
tomato
beef
red
ice cream
alive
○ Anything Ann or Cathy ate is not sufficient for death, since both survived. ■ Tomato soup (both ate it) ■ Chicken (Ann) and beef (Cathy) ■ White wine (Ann) ■ Red wine (Barney drank it, but so did Cathy)
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
■ Pie (Ann) ■ Ice cream (Cathy) ○ The test does not rule out things Barney ate that the others didn’t those are still potentially sufficient for death. ■ There’s no case in this data in which someone eats fish (or cake) and doesn’t die, but that doesn’t mean fish (or cake) is sufficient for death. For that, we must apply the positive sufficient condition test.
Lecture 8.3: Positive sufficient condition tests Positive test for sufficient condition tells you what you have reason to believe is a sufficient condition. ● We have good reason to believe X IS a sufficient condition of Y if all of the following conditions are met: 1. We have not found any case where X is present and Y is absent. ● Basically, it hasn’t been ruled out in the negative test. 2. We have tested a wide variety of cases, including cases where X is present and cases where Y is absent. ● Without this condition, we could reach really silly conclusions much too easily. ● An example if X is absent: a conclusion that pea soup is sufficient for death, based on the earlier data. If we didn’t have a case where someone ate pea soup, there can’t be a case where someone had pea soup and didn’t die. ● An example if Y is always present: If the only case we knew was Barney, that means everybody died and nobody lived. But if everybody died, we don’t have any cases where somebody had tomato soup (or something else) and didn’t die difficult to tell which food is the culprit. 3. If there are any other features that are never present where Y is absent, then we have tested cases where those other features are absent but Y is present. ● We need to look at a case where fish is present but cake is not, and a case where cake is present but fish is not. ● We can’t be sure from the first three cases because we don’t have that combination, so we need to do more research. ● It turns out Doug and Emily were also at the banquet: Diners
Soup
Main
Wine
Dessert
Result
Doug
tomato
beef
red
cake
alive
Emily
tomato
fish
red
pie
dead (=Y)
● Doug had cake and survived; thus, cake is not sufficient for death. ● Emily had fish and died, so fish is the only candidate on our list not ruled out as a sufficient condition for death.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
4.
● However, we don’t know whether all the different potential culprits made it onto our list, so we need one more criteria: We have tested enough cases of various kinds that are likely to include a case where X is present and Y is absent IF there is any such case. ● This cause will obviously be hard to apply it isn’t mechanical. ● We need background knowledge what kinds of things cause death/can’t cause death. ● We’ve already checked everybody’s water glasses and looked for syringe marks where someone might have injected them with something. So we have some reason to believe the sufficient condition must lie somewhere among the features we’re testing. ● This part is inductive, not deductive. ○ If something fails negative condition test, it’s valid to conclude it’s not a sufficient condition. ○ But if something passes the positive condition test, it’s not valid it’s possible that it’s not sufficient, and that argument is defeasible it could be undermined by future data. For example …
Diners
Soup
Main
Wine
Dessert
Result
Fred
leek
fish
red
cake
dead (=Y)
Gertrude
tomato
fish
white
pie
alive
Harold
tomato
beef
red
cake
alive
● Just one more case Gertrude shows that fish is not sufficient for death after all. So we’ll have to think more about what might be sufficient.
Lecture 8.4: Negative necessary condition tests These tell you what is not a necessary condition. ● X is not a necessary condition of Y if there is any case where X is absent and Y is present. Back to our example: Diners
Soup
Main
Wine
Dessert
Result
Ann
tomato
chicken
white
pie
alive
Barney
tomato
fish
red
cake
dead (=Y)
Cathy
tomato
beef
red
ice cream
alive
● From Barney’s data, we gather that neither pie nor ice cream is a necessary condition of death (whereas in the sufficient test, we got that info from Ann).
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● What might be a necessary condition? Anything Barney ate: tomato soup, fish, red wine, cake. Let’s add more data: Diners
Soup
Main
Wine
Dessert
Result
Doug
tomato
beef
red
cake
alive
Emily
tomato
fish
red
pie
dead (=Y)
● We now know cake is not necessary for death, since Doug ate it and lived. ● With Emily’s data, it now looks like fish might be a necessary condition, but we’ll need to do a positive test.
Lecture 8.5: Positive necessary condition tests These tell you what you have reason to believe is a necessary condition. ● We have good reason to believe X is a necessary condition of Y if all of the following conditions are met: 1. We have not found any case where X is absent and Y is present. (In other words, it passed the negative necessary condition test.) 2. We have tested a wide variety of cases, including cases where X is absent and Y is present. 3. If there are any other features that are never absent where Y is present, then we have tested cases where those other features are present but X is absent. Diners
Soup
Main
Wine
Dessert
Result
Fred
leek
fish
red
cake
dead (=Y)
Gertrude
tomato
fish
white
pie
alive
Harold
tomato
beef
red
cake
alive
4.
● Examples we might look at: fish/red wine/no tomato soup; fish/tomato soup/no red wine; tomato soup/red wine/no fish ● Fred tells us that tomato soup is not a necessary condition he didn’t eat tomato soup, but still died. ● The data from Gertrude and Harold rules out nothing Y is not present because they both didn’t die. ● Fish might be a necessary but not sufficient condition, given this data. ● Red wine might be necessary but is definitely not a sufficient condition ● It is NOT reasonable to conclude both fish and red wine are necessary. We have tested enough cases of various kinds that are likely to include a case where X is absent and Y is present, if there is any such case.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Lecture 8.6: Complex conditions Complex conditions conditions, features or candidates that are combinations of other conditions, features or candidates. ● Could also be a negation (“not having fish”) or a disjunction (“having fish or beef”), but here we are focusing on conjunctions.of two or more items on the banquet menu. ● Simple conditions like those previously discussed are also called atomic conditions. Negative test for conjunctive sufficient condition: (W&X) is not a sufficient condition of Y if there is any case where (W&X) is present and Y is absent. ● It’s the negative sufficient test, but with a conjunction substituted for X.(X not sufficient if there’s a case when X is present and Y is absent). Back to our example let’s look at all the data we have: ● Tomato soup + chicken? Ruled out by survivor (Ann). ● Red wine + cake? Ruled out by survivors (Doug and Harold) ● Fish + red wine? Barney, Emily and Fred, the three who died, all had both, so that conjunction might be sufficient. ● Fish + cake? Barney and Fred had both, so it might be sufficient. Negative test for conjunctive necessary condition: (W&X) is not a necessary condition of Y if there is any case where (W&X) is absent and Y is present. ● Leek soup + fish? Not necessary, since Barney and Emily ate tomato. ● Tomato soup + fish? Not necessary, since Fred ate leek. ● Fish + red wine? Might be necessary, since all three ate both, and it was also not ruled out as a sufficient condition. ● Given the data we have so far, we have at least some reason to believe fish and red wine is both necessary and sufficient for death. ● But why might fish and red wine cause death? Perhaps the chef is deeply offended that people would have the gall to drink red wine with his fish, and thus poisoned those who ordered both. Also possible that chemicals in the fish and wine interacted somehow to become a poison, but if you can rule that out from background knowledge, it’s more likely that someone in the kitchen poisoned them.
Lecture 8.7: Correlation vs. causation The tests we’ve discussed work in many cases, but not all. ● Our example took an allornothing approach you’re dead or you’re not; you ate fish or you didn’t (we don’t get into whether you ate just a little bit of fish). ● Some causal relations hold between properties that come in degrees. ○ CO2 is not sufficient for global warming there’s always some in the atmosphere, even when temperatures get cooler. So we need to consider the degree which means we can’t use sufficient or necessary tests. We can still
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
reach a conclusion; we can even figure out the mechanism (CO2 reflecting heat from sun back to Earth and trapping it in.) ○ A runner runs a 10min. mile. Why is he so slow? Maybe he’s really heavy. But weight is not an onandoff property, so it can’t be sufficient. We have a good causal hypothesis it takes more energy to move a heavier object but can’t use our tests. ○ Suppose wages go down because unemployment is high. There’s always some unemployment, so how do you get a causal claim? We have a causal story: As unemployment goes up, wages go down because people are more willing to take lowerpaying jobs. Method of concomitant variation ● Developed in 19th century by John Stuart Mill, who also created parallel variations on the necessary/sufficient tests. ● A concomitant variation is what we generally call a correlation today. 2 types: ○ Positive correlation: X and Y are positively correlated when: 1. the degree of X increases as the degree of Y increases, and 2. the degree of X decreases as the degree of Y decreases. ○ Negative correlation: X and Y are negatively correlated when: 1. the degree of X increases as the degree of Y decreases, and 2. the degree of X decreases as the degree of Y increases. ● When two features are correlated, positively or negatively, there are four possible causal relations between them: 1. A causes B may mean A causes a change in B (A → B) ● Example: Speed limits (A) are correlated with auto accidents and related deaths (B). 2. B causes A may mean B causes a change in A (B → A) ● Example: Auto accidents and related deaths (A) are correlated with fast driving (B). 3. Some third thing (C) causes both A & B ● Example: Having yellowed teeth (A) is correlated with lung cancer (B); both are caused by smoking (C). 4. The correlation is purely accidental A & B just happen to change together. ● Example: The height of my son (A) is correlated with the height of the tree outside (B); the taller one gets, the taller the other gets. (One might consider maturation or time to be C, but here it’s maturation of 2 different things, and time is an abstract concept.) ● How do you tell which possibility applies in a particular case? ○ Temporal order generally, when A causes B, A has to come first; when B comes before A, B comes first. ○ Manipulation change A to see if B changes accordingly, and vice versa. ○ Background information the correlation may be indirect and involve information not stated.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Lecture 8.8: Causal fallacies Two common fallacies: ● Confusing accidental correlation with causation (post hoc ergo propter hoc after this, therefore because of this) ○ Pushing the floor button in the elevator isn’t what set off the fire alarm that happened separately. ● Confusing cause with effect ○ A lot of schizophrenics smoke and drink heavily; that doesn’t mean alcohol and cigarettes cause schizophrenia, but that schizophrenia results in more smoking and drinking.
Lecture 9.1: Why probability matters ● It influences our beliefs inductive strength can be understood in terms of probability: An argument is strong in proportion to the probability that its conclusion is true, given its premises. ● Also influences our decisions: The probability of rain might influence whether you decide to go on a picnic, for example. ● Often people get confused about probability, in part by relying on intuition. ○ Gambler’s fallacy Rolling a pair of dice to get 3 sevens does not affect the likelihood of getting a seven on the next roll. You can’t assume the next roll won’t be a seven, nor can you assume the dice are “hot” and the next roll will be. ○ Representativeness heuristic Dealing a winning hand (four queens and an ace) is just as likely as dealing a junk hand but you perceive the winning hand as less likely because it’s more representative of what you typically experience when playing. ■ See also: “Linda the bank teller” (Kahneman & Tversky) ○ The Monty Hall problem You’re presented with three doors (a la Let’s Make a Deal) one has a car behind it; the other two have goats. If you pick door A, and Monty reveals door C had a goat behind it, should you switch to door B when offered the chance? (The answer: Yes.)
Lecture 9.2: What is probability? The language of probability ● May be expressed in a number of times it might happen out of the number of times that occur: 3 times out of 10 = 3 in 10 = 3/10 = 30% chance ● In this course, we’ll use numbers between zero and one: 0.3 probability. ○ 1 = certainly true, no chance of falsehood ○ 0 = certainly false; no chance of truth Kinds of probability
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● a priori probability: likelihoods of alternatives are assumed prior to any kind of experience. ○ flip of a fair coin = equal chance you’ll get heads or tails, so you assume 0.5 probability ○ rolling two fair dice = seven ways to get a a seven out of 36 possible outcomes (6x6), so probability is about 1/6 ● statistical (or empirical) probability: based on observed frequency; can be used when likelihoods cannot be assumed. ○ Say it’s not a fair coin you hammered an edge to bend it. It seems to come up heads far more often (but not always). Equal probability can no longer be assumes you’d need to flip it hundreds and hundreds of times to get a percentage probability. Whether a priori or statistical, probability must follow the same four general rules.
Lecture 9.3: Rules of probability - negation Rule 1 (probability of a negation): The probability that an event will not occur is one minus the probability that it will occur. ● Pr(~h) = 1Pr(h) ○ h stands for the hypothesis or event ○ ~ stands for negation ○ Pr(X) stands for the probability of X ● Why does this hold? 1 = certainty, so probabilities of h and ~h (or “not h”) must add up to 1.
Lecture 9.4: Rules of probability - conjunction Independent events: Two events are independent if and only if the occurrence of one has no effect on the probability of the other. ● Examples: Coin flips, dice rolls ● If the occurrence of one does affect probability of the other, it’s a dependent event for example, drawing a card from a deck and discarding it affects the probability of other draws from that deck. Rule 2 (probability of a conjunction) for independent events: If two events are independent, the probability of both events occurring is the product of the probability of the first event occurring times the probability of the second event occurring. ● Pr(h1&h2) = Pr(h1) x Pr(h2) ● It’s less likely that both will occur than it is for one to occur separately. Multiplying two probabilities 0. ○ A bet is unfavorable if and only if its expected value
Part III: How to Evaluate Inductive Arguments Week 7: Inductive Arguments Lecture 7.1: What is induction? Inductive arguments may actually be more common than the deductive ones we’ve been learning about the past few weeks. Types of inductive arguments ● Inference to the best explanation you might use to determine who committed a crime. ● Arguments from analogy ● Statistical generalizations just about all the polls in politics! ● Statistical applications how do those generalizations tell you something about a person? ● Causal reasoning ● Probability ● Decision making Inductive vs. deductive arguments How could you prove the sun will come up tomorrow morning? The sun came up the day before…etc. The sun came up the day before. The sun came up the day before. The sun came up yesterday. ∴ The sun will rise tomorrow. Or, for short: The sun came up every day before that for an awfully long time. The sun came up yesterday. ∴ The sun will rise tomorrow. ● Is the argument valid? No. It’s possible for these premises to be true and the conclusion false (say, if a meteor struck Earth and stopped it from spinning). ● Is this argument good? Yes. These premises provide good reasons for the conclusion. So we see that an argument can be good even if it’s not valid. But other arguments are no good because they’re invalid. Every sophomore is a student. Bub is not a sophomore. ∴ Bub is not a student. ● But what if Bub’s a junior or senior?
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Deductive arguments are intended to be valid, so they’re no good if they’re invalid. ● The standards are whether the argument is valid or invalid. ● Validity is allornothing no such thing as “partly valid.” ● Standard of validity is called indefeasible (or monotonic) no matter what premises you add to a valid argument, it will still be valid. If Joe is a sophomore, then Joe is a student. Joe is not a student. ∴ Joe is not a sophomore. Inductive arguments are not intended to be valid; they can still be good even when invalid. ● The standards are whether the argument is strong or weak. ● Strength comes in degrees an argument is stronger when it gives more and better reasons for the conclusion. ○ We can’t just ask if it’s strong we need to know if it’s strong enough. That depends on the context and the values at stake. When there’s a lot to lose, we demand better reasons and stronger arguments ■ Example: Sticking a piece of straw in the center works to test if cake’s done, but not a turkey you can get sicker from an uncooked turkey, so there’s more at stake. ● Standard of strength called defeasible (or nonmonotonic) adding additional premises or information can weaken or even undermine a strong argument. ○ Example: In a court scenario, a piece of new information that the suspect has an identical twin weakens eyewitnesses’ testimony that it was the suspect. Classifying an argument as inductive or deductive is important because it determines whether the argument’s good or bad. The sun came up yesterday. The sun came up every day for thousands of years. ∴ The sun will come up tomorrow. This argument isn’t valid, but it isn’t intended to be so it’s still good. Common mistakes about inductive arguments: ● Many view them as inferior to deductive arguments but it’s easy to take any inductive argument and turn it into a deductive argument. If most of Joe’s friends are seniors, then Joe is a senior. Most of Joe’s friends are seniors. ∴ Joe is a senior Adding that first conditional premise shifts doubts about the inductive argument into doubts about the premise it hasn’t really made us any more sure of Joe’s seniorness. ● Many believe induction always takes us from the particular to the general but generalization is just one kind of inductive argument.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ Inference to the best argument In taste test, this lemonade tastes awful because you put dishwashing detergent in it! ○ Argument from analogy I know because it tastes like soap. ○ Necessary condition test I didn’t spit out the first lemonade I tried. ○ Statistical generalization Did you know over 90% of those we tested agreed they preferred our lemonade to dishwashing detergent? ○ Statistical application If you try this test on your friends, they’ll agree too.
Lecture 7.2: Generalizations from samples Generalizations all around us “almost all,” “most,” “many,” “¾,” “half,” “87.2% of statistics,” etc. ● They’re useful when you need to make a decision when you’re sick, you need to know if people with the same symptoms usually go to the doctor, & if doing so would even help. ● But how do we decide which ones to believe? You can’t check them all, so you have to do a sample from which you can reach a conclusion about the larger class. Forms: ● Universal The first F is G. The second F is G. The rest of the Fs in the sample are G. ∴ All Fs are G. ● Partial The first F is G. The second F is G. The third F is not G. The fourth F is G, and so on. ∴ X% of Fs in the sample are G. ∴ X% of all Fs are G. Moving the argument from part of the class to the whole thing proves it’s inductive: Almost all bands that I know of have drummers. ∴ Almost all bands have drummers. ● Is it valid? No. Maybe I need to broaden my musical horizons. ● Is it defeasible? Yes. I’d have to change my conclusion if you gave me examples of bands without drummers. ● Does this mean it can’t be strong? No. Argument could be strengthened by making the sample larger, or weakened with a smaller sample. ● Is the argument still good? Yes, because it’s inductive you can’t criticize it for not being valid, because it doesn’t try to be. How do we tell if a generalization from a sample is strong? We have to turn to statistics.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● Some argue they’re always worse than the “damn lies” in Twain’s quote (cribbed from Disraeli) and you can’t trust ‘em at all. ● Some argue they’re infallible it’s just math, and you can’t question the numbers, right? ● But both generalizations are wrong the truth lies somewhere in the middle.
Lecture 7.3: When are generalizations strong? How can we tell when a generalization from a sample is a good argument? We’re not going to get into a lot of statistics that’d take a whole course but let’s look at some common errors that can be used to mislead.
Questions to ask about generalizations ● Are the premises false? ○ Let’s say the optimum number of chocolate chips is 1012 for a 3inch cookie, and Walter bought 10 cookies from each of 5 bakeries in his area. 80% of cookies I bought from Bakery A have 1012 chocolate chips. ∴ 80% of all cookies from Bakery A have 1012 chocolate chips. ○ Thing is, Walter is lying he didn’t buy any cookies at all! ○ An argument that’d be strong if its premises were true is no good if they aren’t. ● Are the premises unjustified? ○ It’s not enough for the premises to be true. ○ What if Walter did buy the cookies, but counted them so fast that he missed a bunch of chips? or chips were melted together and couldn’t be counted very well? or cookies from bakery A and B got mixed up? ○ Mistakes were made in the counting, so the premises are false. ● Is the sample too small? ○ Let’s say Walter is honest and counts thoroughly and carefully and is justified in believing he hasn’t made a mistake. He bought one cookie from each bakery; the Bakery A cookie had 11 chips, and the other bakeries’ had less than 10. 100% of the cookies I bought from Bakery A have 1012 chocolate chips. ∴ 100% of all cookies from Bakery A have 1012 chocolate chips. 0% of the cookies I bought from Bakery B have 1012 chips. ∴ 0% of all cookies from Bakery B have 1012 chips. ○ The problem: You can’t generalize from just one cookie. It’s not a mechanical process, so you can’t know every cookie is the same. ○ Fallacy of hasty generalization generalizing from a sample that’s too small. This fallacy is very common (ie: My neighbor’s new car broke down, so that type of car is no good). ○ We don’t need to know if the sample’s large, but if it’s large enough. In some cases, a very small sample can be big enough for the generalization.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
■ Whether the sample is large enough depends on background information and what’s at stake. ● Background info: You come across an apple tree and want to know if its apples float in the water. One apple is a big enough sample you have background info from biology that the apples on that tree are similar to each other. ● Stakes: If you want to make sure parachutes are packed properly and will work, you obviously want to check every single one. Chocolatechip cookies aren’t as big a deal. ● Is the sample biased? ○ Say Walter’s not lying, counted accurately, is justified that he counted accurately and had a big enough sample. At each of the five bakeries, he explained his survey to the person at the counter before buying 10 cookies. 80% of cookies I bought from Bakery A have 1012 chocolate chips. ∴ 80% of all cookies from Bakery A have 1012 chocolate chips. ○ The problem: The employee knows what Walter’s looking for, so he’s probably going to serve Walter the cookies with the most chips in them so Walter will continue to buy from his shop. ○ Fallacy of biased sampling the sample is not representative of the whole because it’s biased in a certain way. Seems like an obvious mistake, but it happens all the time even to some of history’s top pollsters. ■ Literary Digest poll in 1936 presidential election concluded that Alf Landon was going to beat FDR 5644. They were way off FDR won 6238. How did this happen? They sent letters to addresses pulled from the phone book but back then, few had phones, especially among poor or rural areas. Those voters favored FDR because of the New Deal and other policies to help them out. ■ Still happens today many pollsters looking to do a quick poll call people up on their phones. But there are lots of problems: ● Polls can’t be done on cellphones, which may be the only phone young people have, so they’re likely to be underrepresented. ● If those polled do have a landline, they have caller ID and may not answer if it’s a pollster low response rates. ● Some studies have found that women tend answer the phone more often than men, which skews samples. ○ What do you with a biased sample? You try to correct for that. ■ If you get a lot more people from the liberal party than last time might mean sample is skewed, or might mean more people have shifted to liberal party. Polls use different techniques to correct, which means some may reach very different results. ■ Some pollsters are dishonest and may slant their questions to get the desired result. ● Is the question slanted?
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ Question may be worded in a way that makes people feel bad about giving a certain answer. ■ “Should we kill a mouse to save a human?” → a yes indicates support of animal experimentation. ■ “Should scientists torture animals in their experiments?” → a no indicates opposition to animal experimentation. ○ Survey participants’ options may be limited. ■ Doris Day Animal League says survey found 51% “think primates are entitled to the same rights as human children.” But the survey question refers specifically to chimps, says “similar” rather than “the same,” and lists only four options: Should animals be treated (a) like property, (b) similar to children, (c) the same as adults, (d) not sure.
Lecture 7.4: Applying generalizations Application of the generalization (or statistical syllogisms) a form of argument in which we (shockingly) apply a generalization to a particular case. ● Common, as in: I almost never like horror movies. or I don’t want to invest in that restaurant 80% of restaurants fail within the first 23 years. ● Example: Almost all professors who are teaching are wearing shoes. Walter is a professor who is teaching. ∴ Walter is (probably) wearing shoes. ● Has same form as previous example, which we can explore further by substituting variables for terms in the English argument: X% of F are G. a is F. ∴ a is (probably) G. ○ F = reference class the subject in the general premise (the set of professors who are teaching) ○ G = attribute class the predicate in the general premise (the people who are wearing shoes) ○ a (usually lowercase) = the individual to whom the generalization is applied. (Walter) ○ X = quantifier percentage in generalization. (Even phrases like “almost all” are replaced with X%.) ● Note that application of a generalization moves in opposite direction from the generalization from the sample. Generalization: Application: X% of sampled F are G. X% of F are G. ∴ X% of F are G. a is F. ∴ a is (probably) G. ● Applications of generalizations are inductive and share those arguments’ features: ○ They’re invalid (it’s possible for premises to be true and conclusion false).
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ They’re defeasible (can be weakened by adding information). ○ They can vary in strength. When is an AoG considered to provide strong reasons for the conclusion? ● Premises must be true and justified. ● Percentages should be very high or low. (“99% of F are G” is stronger than “80% …”) ● No smaller reference class should conflict. ○ This is the fallacy of overlooking a conflicting reference class. 95% of F (professors) are G. a is F. ∴ a (probably) is G. BUT … 30% of F* (online professors) are G. a is F*. ∴ a (probably) is not G. ○ General rule: When you encounter conflicting reference classes, look at the smallest one it’s usually likely to give you a better estimate. ■ But keep in mind that as reference class gets narrower and narrower, you may not have enough data to support the premise of a certain percentage. ○ A medical example: 90% of people with a certain medical condition die. Bob has that medical condition. ∴ Bob is likely to die? BUT … Most people catch this illness when they’re old. Bob is quite young. Only 20% of people who get this illness when they’re young die. ∴ Bob isn’t likely to die? BUT … 80% of people who are young and have a heart condition die. Bob has a heart condition. ∴ Bob is likely to die? BUT … 30% of those given a new treatment die, even if young/heart condition. Bob lives in an area where he can get this treatment. ■ As we get more information on this illness, the chance Bob will die goes high, low, high, low … ■ So which of the generalizations should we use? The narrowest class. But the treatment’s new and probably hasn’t been tried on many young people, let alone those with heart conditions.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Lecture 7.5: Inference to the best explanation Inference to the best explanation combines qualities of both a justification and an explanation (see: Week 1) it uses an argument to justify belief in the conclusion, and yet a premise of the argument is that that conclusion explains a phenomenon you took for granted. ● This is extremely common. ○ Appears in just about every detective story. (Sherlock Holmes’ “deduction” was actually inductive reasoning.) ○ In science: What killed the dinosaurs? Best explanation is giant meteor in Yucatan Peninsula. The point of most scientific theories is to provide the best explanation of all the experimental data that has been observed by scientists. ○ In everyday lives: A drop of water hit my head that must mean the roof is leaking. ● Uses an explanation but runs in the opposite direction as a typical explanation: The conclusion does the explaining; the premises are what get explained: ○ Explanation: When there is a leak in the roof, water drops through the roof. When water comes through the roof, it drops onto whatever is in its way. I am in its way. ∴ Water drops onto me. ○ Inference to the best explanation: Water drops onto me. A leak in the roof would explain why water drops onto me. No better explanation is available. ∴ There is a leak in the roof. ● This argument isn’t valid, but it’s still good as an inductive argument. . ● Is it defeasible? Yes (by adding in information that there’s a guy with a watering can above you). ● Defeasibility does not mean it was a weak argument from the outset. ● Function is different as well: ○ In explanation, we take for granted that conclusion is true. ○ In inference to the best explanation, we argue that the conclusion is true because it’s the best explanation of the observation in the first premise.
Lecture 7.6: Which explanation is best? A good explanation: ● Starts with surprising circumstances that need to be explained. ● Involves an accurate picture of what you’re trying to explain. ● Has to be compatible with all of the facts not just the specific ones you’re trying to explain. ● Is simple and conservative. ● Is deep ● Is powerful and broad
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● Is modest Summary of the skit’s argument: 1. Observation: It’s time for class and nobody else is in the room. 2. Explanation: The hypothesis that class was canceled due to a holiday, plus accepted facts and principles, gives a suitably strong explanation of the observation in premise 1 that no one else is in the room. 3. Comparison: No other hypothesis provides an explanation nearly as good as the holiday hypothesis in premise 2. 4. Conclusion: The holiday hypothesis in premise 2 is true. ● Premise 3 is most controversial of these. To justify this, we need to compare other possible explanations (as the skit did) in a background argument.: ○ “Other students are late” hypothesis falsified by passing of time. ○ “Other students are invisible” hypothesis not conservative. ○ “Other students are undetectable” hypothesis not falsifiable. ○ “Other students are playing a joke” hypothesis not deep. ○ “Professor skipped class” hypothesis not powerful or broad. ○ “Other students are at MLK Day ceremony” not modest. ○ No other hypothesis seems plausible, therefore (premise 3 above). ● Another background argument: The holiday hypothesis … ○ Explains why nobody else is there if classes were canceled for the holiday, nobody else would be there (in that room at the usual time) ○ Is broad it explains other actual observations (for example, the whole building is empty). ○ Is powerful applies to many separate cases (for example, why students won’t be there on future holidays). ○ Is falsifiable the students might find out classes weren’t canceled for the holiday ... ○ Is not falsified … but they didn’t. ○ Is conservative does not conflict with prior, wellestablished beliefs. ○ Is deep does not depend on any assumptions that need but lack independent explanation. ○ Therefore, the holiday hypothesis, plus accepted facts and principles, gives a strong explanation of why nobody else is in the room. ● The items used in these background arguments are commonly called explanatory virtues. The more of these virtues an argument has, the stronger it is.
Lecture 7.8: Arguments from analogy Argument from analogy very closely related to inference to the best explanation; also common ● Remember that an analogy is a comparison between two things that points out similarities without being specific. ● Some might argue the lack of specificity makes them illsuited for arguments, but we use them that way all the time.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ Public policy: Phoenix is a lot like Houston in many ways. This type of transportation system worked well for Houston. So it will probably work well in Phoenix. ○ Law: Common law and other systems use precedent in deciding similar cases. ■ Useful because it’s predictable and gives flexibility. ○ Art history: You find a painting in the attic that looks like a (valuable) Cezanne but isn’t signed. How do you authenticate it? You look at a lot of works known to be by Cezanne and try to find out whether they’re similar (or get an expert to do so). This painting is like others in its colors, forms, brushwork, etc. Those other paintings were painted by Cezanne. ∴ This painting was (probably) also painted by Cezanne. ● Arguments from analogy are inductive they only try to show they’re probably true. Argument from analogy form Object A has properties P, Q, R, and so on. Objects B, C, D, and so on also have properties P, Q, R, and so on. Objects B, C, D, and so on also have property X. ∴ Object A (probably) also has property X. ● A = the subject ● B, C, D = the analogous objects ● P, Q, R = the similarities ● X = the target Standards of strength An argument from analogy is stronger when: ● the similarities are more important ● there are more similarities ● there are fewer disanalogies (there are always some) ● the analogous objects are more diverse ● the conclusion is weaker (“probably”)
Week 8: Causal Reasoning Lecture 8.1: Causal reasoning Causal judgments have to do with what caused an individual event. Supported by inductive arguments. ● Cannot be certain, because we can’t know with absolute certainty the causes of things. Causation ● Causal claims are about individual events one event causes another. ○ My phone is off. When I push this button on the bottom, it turns on.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
○ General rule behind that: Every time I push this button, the phone turns on. If it doesn’t hold as a general rule, it’s not a causal relation. ● Big topic for this week: How do we test general principles? ○ Which cases in the data do we need to look at, and how do we know when we have enough data to believe in a general principle? Types of general principles ● Sufficient condition ○ F is a sufficient condition for G if and only if in normal circumstances: ■ For events: Whenever event F happens, event G also happens. ■ For features: Anything that has feature F also has feature G. ● Necessary condition includes a negation on both sides. ○ F is a necessary condition for G if and only if in normal circumstances: ■ For events: Whenever event F does not happen, event G also does not happen. ■ For features: Anything that does not have feature F also does not have feature G. ● Example: ○ Being a whale is sufficient for being a mammal. ■ Anything that is a whale is a mammal. ○ Being a whale is necessary for being a sperm whale. ■ Anything that is not a whale is not a sperm whale. ● Thinking of this in propositional statements may be helpful (but it’s not quite the same because there are no quantifiers): ○ Sufficient = If F, then G ○ Necessary = If not F, then not G Causal examples: ● Whenever is a quantifier applied to a restricted domain of discourse. ● Causal cases must be understood as a general rule within an appropriate range of circumstances. ○ Striking this match is sufficient for lighting it. ■ It can’t be true that whenever the match is struck it lights it won’t if the surface is not rough. ■ This example takes for granted that this is a normal match say, you haven’t dipped it into a cup of coffee to get it wet. ○ Striking this match on a rough surface is necessary for lighting it. ■ Yeah, we could use a lit match to light this match, but … ● So we need to modify our definition to include “in normal circumstances” which tells us how “whenever” applies. Kinds of conditions:
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● Conceptual ○ Being a whale is sufficient for being mammal. (But being a whale is not necessary for being mammal.) ○ Being a whale is necessary for being a sperm whale. (But being a whale is not sufficient for being a sperm whale.) ● Causal ○ Striking this match is sufficient for lighting it. ○ Striking this match on a rough surface is necessary for lighting it. ● Moral ○ Torturing for fun is sufficient for doing wrong. ○ Doing something wrong is necessary for being punishable.
Lecture 8.2: Negative sufficient condition tests How can we tell which conditions are necessary and which are sufficient? Two tests for each type positive and negative. Negative test for sufficient condition tells you what’s not a sufficient condition. ● X is NOT a sufficient condition of Y if there is any case where X is present and Y is absent. ● A deductive argument against the generalization that whenever X is present, Y is present as well. ○ That means it’s not defeasible it’s still valid even when more information added. ○ All you need is one counterexample. ● An example (inspired by Airplane!): A group of people eat a full banquet with soup course, main course, wine, dessert. All goes well until the end; when some start to get up, they feel sick and then keel over and die. ○ We figure that they were poisoned, and that it had to have been through the food. ○ Now we look at what each particular person ate to determine which of the dishes was poisoned. (Which candidate X is sufficient for the target Y?) Diners
Soup
Main
Wine
Dessert
Result
Ann
tomato
chicken
white
pie
alive
Barney
tomato
fish
red
cake
dead (=Y)
Cathy
tomato
beef
red
ice cream
alive
○ Anything Ann or Cathy ate is not sufficient for death, since both survived. ■ Tomato soup (both ate it) ■ Chicken (Ann) and beef (Cathy) ■ White wine (Ann) ■ Red wine (Barney drank it, but so did Cathy)
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
■ Pie (Ann) ■ Ice cream (Cathy) ○ The test does not rule out things Barney ate that the others didn’t those are still potentially sufficient for death. ■ There’s no case in this data in which someone eats fish (or cake) and doesn’t die, but that doesn’t mean fish (or cake) is sufficient for death. For that, we must apply the positive sufficient condition test.
Lecture 8.3: Positive sufficient condition tests Positive test for sufficient condition tells you what you have reason to believe is a sufficient condition. ● We have good reason to believe X IS a sufficient condition of Y if all of the following conditions are met: 1. We have not found any case where X is present and Y is absent. ● Basically, it hasn’t been ruled out in the negative test. 2. We have tested a wide variety of cases, including cases where X is present and cases where Y is absent. ● Without this condition, we could reach really silly conclusions much too easily. ● An example if X is absent: a conclusion that pea soup is sufficient for death, based on the earlier data. If we didn’t have a case where someone ate pea soup, there can’t be a case where someone had pea soup and didn’t die. ● An example if Y is always present: If the only case we knew was Barney, that means everybody died and nobody lived. But if everybody died, we don’t have any cases where somebody had tomato soup (or something else) and didn’t die difficult to tell which food is the culprit. 3. If there are any other features that are never present where Y is absent, then we have tested cases where those other features are absent but Y is present. ● We need to look at a case where fish is present but cake is not, and a case where cake is present but fish is not. ● We can’t be sure from the first three cases because we don’t have that combination, so we need to do more research. ● It turns out Doug and Emily were also at the banquet: Diners
Soup
Main
Wine
Dessert
Result
Doug
tomato
beef
red
cake
alive
Emily
tomato
fish
red
pie
dead (=Y)
● Doug had cake and survived; thus, cake is not sufficient for death. ● Emily had fish and died, so fish is the only candidate on our list not ruled out as a sufficient condition for death.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
4.
● However, we don’t know whether all the different potential culprits made it onto our list, so we need one more criteria: We have tested enough cases of various kinds that are likely to include a case where X is present and Y is absent IF there is any such case. ● This cause will obviously be hard to apply it isn’t mechanical. ● We need background knowledge what kinds of things cause death/can’t cause death. ● We’ve already checked everybody’s water glasses and looked for syringe marks where someone might have injected them with something. So we have some reason to believe the sufficient condition must lie somewhere among the features we’re testing. ● This part is inductive, not deductive. ○ If something fails negative condition test, it’s valid to conclude it’s not a sufficient condition. ○ But if something passes the positive condition test, it’s not valid it’s possible that it’s not sufficient, and that argument is defeasible it could be undermined by future data. For example …
Diners
Soup
Main
Wine
Dessert
Result
Fred
leek
fish
red
cake
dead (=Y)
Gertrude
tomato
fish
white
pie
alive
Harold
tomato
beef
red
cake
alive
● Just one more case Gertrude shows that fish is not sufficient for death after all. So we’ll have to think more about what might be sufficient.
Lecture 8.4: Negative necessary condition tests These tell you what is not a necessary condition. ● X is not a necessary condition of Y if there is any case where X is absent and Y is present. Back to our example: Diners
Soup
Main
Wine
Dessert
Result
Ann
tomato
chicken
white
pie
alive
Barney
tomato
fish
red
cake
dead (=Y)
Cathy
tomato
beef
red
ice cream
alive
● From Barney’s data, we gather that neither pie nor ice cream is a necessary condition of death (whereas in the sufficient test, we got that info from Ann).
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● What might be a necessary condition? Anything Barney ate: tomato soup, fish, red wine, cake. Let’s add more data: Diners
Soup
Main
Wine
Dessert
Result
Doug
tomato
beef
red
cake
alive
Emily
tomato
fish
red
pie
dead (=Y)
● We now know cake is not necessary for death, since Doug ate it and lived. ● With Emily’s data, it now looks like fish might be a necessary condition, but we’ll need to do a positive test.
Lecture 8.5: Positive necessary condition tests These tell you what you have reason to believe is a necessary condition. ● We have good reason to believe X is a necessary condition of Y if all of the following conditions are met: 1. We have not found any case where X is absent and Y is present. (In other words, it passed the negative necessary condition test.) 2. We have tested a wide variety of cases, including cases where X is absent and Y is present. 3. If there are any other features that are never absent where Y is present, then we have tested cases where those other features are present but X is absent. Diners
Soup
Main
Wine
Dessert
Result
Fred
leek
fish
red
cake
dead (=Y)
Gertrude
tomato
fish
white
pie
alive
Harold
tomato
beef
red
cake
alive
4.
● Examples we might look at: fish/red wine/no tomato soup; fish/tomato soup/no red wine; tomato soup/red wine/no fish ● Fred tells us that tomato soup is not a necessary condition he didn’t eat tomato soup, but still died. ● The data from Gertrude and Harold rules out nothing Y is not present because they both didn’t die. ● Fish might be a necessary but not sufficient condition, given this data. ● Red wine might be necessary but is definitely not a sufficient condition ● It is NOT reasonable to conclude both fish and red wine are necessary. We have tested enough cases of various kinds that are likely to include a case where X is absent and Y is present, if there is any such case.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Lecture 8.6: Complex conditions Complex conditions conditions, features or candidates that are combinations of other conditions, features or candidates. ● Could also be a negation (“not having fish”) or a disjunction (“having fish or beef”), but here we are focusing on conjunctions.of two or more items on the banquet menu. ● Simple conditions like those previously discussed are also called atomic conditions. Negative test for conjunctive sufficient condition: (W&X) is not a sufficient condition of Y if there is any case where (W&X) is present and Y is absent. ● It’s the negative sufficient test, but with a conjunction substituted for X.(X not sufficient if there’s a case when X is present and Y is absent). Back to our example let’s look at all the data we have: ● Tomato soup + chicken? Ruled out by survivor (Ann). ● Red wine + cake? Ruled out by survivors (Doug and Harold) ● Fish + red wine? Barney, Emily and Fred, the three who died, all had both, so that conjunction might be sufficient. ● Fish + cake? Barney and Fred had both, so it might be sufficient. Negative test for conjunctive necessary condition: (W&X) is not a necessary condition of Y if there is any case where (W&X) is absent and Y is present. ● Leek soup + fish? Not necessary, since Barney and Emily ate tomato. ● Tomato soup + fish? Not necessary, since Fred ate leek. ● Fish + red wine? Might be necessary, since all three ate both, and it was also not ruled out as a sufficient condition. ● Given the data we have so far, we have at least some reason to believe fish and red wine is both necessary and sufficient for death. ● But why might fish and red wine cause death? Perhaps the chef is deeply offended that people would have the gall to drink red wine with his fish, and thus poisoned those who ordered both. Also possible that chemicals in the fish and wine interacted somehow to become a poison, but if you can rule that out from background knowledge, it’s more likely that someone in the kitchen poisoned them.
Lecture 8.7: Correlation vs. causation The tests we’ve discussed work in many cases, but not all. ● Our example took an allornothing approach you’re dead or you’re not; you ate fish or you didn’t (we don’t get into whether you ate just a little bit of fish). ● Some causal relations hold between properties that come in degrees. ○ CO2 is not sufficient for global warming there’s always some in the atmosphere, even when temperatures get cooler. So we need to consider the degree which means we can’t use sufficient or necessary tests. We can still
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
reach a conclusion; we can even figure out the mechanism (CO2 reflecting heat from sun back to Earth and trapping it in.) ○ A runner runs a 10min. mile. Why is he so slow? Maybe he’s really heavy. But weight is not an onandoff property, so it can’t be sufficient. We have a good causal hypothesis it takes more energy to move a heavier object but can’t use our tests. ○ Suppose wages go down because unemployment is high. There’s always some unemployment, so how do you get a causal claim? We have a causal story: As unemployment goes up, wages go down because people are more willing to take lowerpaying jobs. Method of concomitant variation ● Developed in 19th century by John Stuart Mill, who also created parallel variations on the necessary/sufficient tests. ● A concomitant variation is what we generally call a correlation today. 2 types: ○ Positive correlation: X and Y are positively correlated when: 1. the degree of X increases as the degree of Y increases, and 2. the degree of X decreases as the degree of Y decreases. ○ Negative correlation: X and Y are negatively correlated when: 1. the degree of X increases as the degree of Y decreases, and 2. the degree of X decreases as the degree of Y increases. ● When two features are correlated, positively or negatively, there are four possible causal relations between them: 1. A causes B may mean A causes a change in B (A → B) ● Example: Speed limits (A) are correlated with auto accidents and related deaths (B). 2. B causes A may mean B causes a change in A (B → A) ● Example: Auto accidents and related deaths (A) are correlated with fast driving (B). 3. Some third thing (C) causes both A & B ● Example: Having yellowed teeth (A) is correlated with lung cancer (B); both are caused by smoking (C). 4. The correlation is purely accidental A & B just happen to change together. ● Example: The height of my son (A) is correlated with the height of the tree outside (B); the taller one gets, the taller the other gets. (One might consider maturation or time to be C, but here it’s maturation of 2 different things, and time is an abstract concept.) ● How do you tell which possibility applies in a particular case? ○ Temporal order generally, when A causes B, A has to come first; when B comes before A, B comes first. ○ Manipulation change A to see if B changes accordingly, and vice versa. ○ Background information the correlation may be indirect and involve information not stated.
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
Lecture 8.8: Causal fallacies Two common fallacies: ● Confusing accidental correlation with causation (post hoc ergo propter hoc after this, therefore because of this) ○ Pushing the floor button in the elevator isn’t what set off the fire alarm that happened separately. ● Confusing cause with effect ○ A lot of schizophrenics smoke and drink heavily; that doesn’t mean alcohol and cigarettes cause schizophrenia, but that schizophrenia results in more smoking and drinking.
Lecture 9.1: Why probability matters ● It influences our beliefs inductive strength can be understood in terms of probability: An argument is strong in proportion to the probability that its conclusion is true, given its premises. ● Also influences our decisions: The probability of rain might influence whether you decide to go on a picnic, for example. ● Often people get confused about probability, in part by relying on intuition. ○ Gambler’s fallacy Rolling a pair of dice to get 3 sevens does not affect the likelihood of getting a seven on the next roll. You can’t assume the next roll won’t be a seven, nor can you assume the dice are “hot” and the next roll will be. ○ Representativeness heuristic Dealing a winning hand (four queens and an ace) is just as likely as dealing a junk hand but you perceive the winning hand as less likely because it’s more representative of what you typically experience when playing. ■ See also: “Linda the bank teller” (Kahneman & Tversky) ○ The Monty Hall problem You’re presented with three doors (a la Let’s Make a Deal) one has a car behind it; the other two have goats. If you pick door A, and Monty reveals door C had a goat behind it, should you switch to door B when offered the chance? (The answer: Yes.)
Lecture 9.2: What is probability? The language of probability ● May be expressed in a number of times it might happen out of the number of times that occur: 3 times out of 10 = 3 in 10 = 3/10 = 30% chance ● In this course, we’ll use numbers between zero and one: 0.3 probability. ○ 1 = certainly true, no chance of falsehood ○ 0 = certainly false; no chance of truth Kinds of probability
Lauren Phillips Think Again: How to Reason and Argue Fall 2014 Walter SinnottArmstrong, Duke University; Ram Neta, University of North Carolina Chapel Hill
● a priori probability: likelihoods of alternatives are assumed prior to any kind of experience. ○ flip of a fair coin = equal chance you’ll get heads or tails, so you assume 0.5 probability ○ rolling two fair dice = seven ways to get a a seven out of 36 possible outcomes (6x6), so probability is about 1/6 ● statistical (or empirical) probability: based on observed frequency; can be used when likelihoods cannot be assumed. ○ Say it’s not a fair coin you hammered an edge to bend it. It seems to come up heads far more often (but not always). Equal probability can no longer be assumes you’d need to flip it hundreds and hundreds of times to get a percentage probability. Whether a priori or statistical, probability must follow the same four general rules.
Lecture 9.3: Rules of probability - negation Rule 1 (probability of a negation): The probability that an event will not occur is one minus the probability that it will occur. ● Pr(~h) = 1Pr(h) ○ h stands for the hypothesis or event ○ ~ stands for negation ○ Pr(X) stands for the probability of X ● Why does this hold? 1 = certainty, so probabilities of h and ~h (or “not h”) must add up to 1.
Lecture 9.4: Rules of probability - conjunction Independent events: Two events are independent if and only if the occurrence of one has no effect on the probability of the other. ● Examples: Coin flips, dice rolls ● If the occurrence of one does affect probability of the other, it’s a dependent event for example, drawing a card from a deck and discarding it affects the probability of other draws from that deck. Rule 2 (probability of a conjunction) for independent events: If two events are independent, the probability of both events occurring is the product of the probability of the first event occurring times the probability of the second event occurring. ● Pr(h1&h2) = Pr(h1) x Pr(h2) ● It’s less likely that both will occur than it is for one to occur separately. Multiplying two probabilities 0. ○ A bet is unfavorable if and only if its expected value
