A star (*) indicates that there are exercises covering this section and previous unmarked sections.

Transcription

1 1 An Introduction To Reasoning Some Everyday Reasoning 1 Introduction 2 Reasoning Based On Properties 3 Part-Whole Relationships 4 Reasoning With Relations 5 The Tricky Verb 'To Be' 6 Reasoning With Categorical Generalizations 7 Mixing General & Specific Propositions 8 Reasoning With Conditionals 9 Elimination 10 Induction 11 Fallacious Reasoning & Degree Of Difficulty 12 Concluding Remarks & Summary* A star (*) indicates that there are exercises covering this section and previous unmarked sections. This piece in relation to others: This is an introductory chapter covering various forms of everyday reasoning. It can be skipped entirely but it shows that reasoning is a familiar activity and many of the types of reasoning introduced here re-appear elsewhere in the book. Cathal Woods

2 2 Some Everyday Reasoning 1 Introduction Reasoning is a familiar activity. You reason hundreds if not thousands of times every day. This introductory chapter begins the book by looking at the kinds of reasoning you do, and typically do well, on an everyday basis. These kinds of reasoning are simple enough that you do them in real time without needing a pencil and paper. Some of them have more complicated versions or related forms of reasoning (both good and bad), and these are covered in more detail elsewhere in the book. There are lots of names of types of reasoning in this chapter, but the names aren't terribly important. What's important here is that you are able, with a little thought, to recognize how each kind of reasoning works. Once you see the features that allow each kind of reasoning to work, and how they are different in each case, you can come up with your own names if you like. 2 Reasoning Based On Properties 1. Entities have various properties. By keeping track of these properties you draw various conclusions. You can compare the same entity with itself at different times, and say whether it has changed or remained the same, whether with respect to place (i.e. it moves or stays still) or any other quality, or with respect to its being one thing or another (it changes what it is, it lives or dies, it persists or falls apart). For example: Henry was in the kitchen earlier. Now he is in the living-room. So, Henry has moved. Henry's skin was pale before his holiday to Majorca. When he came back it was brown. So, Henry got a tan. These inferences work by comparing properties of Henry at different times. The first involves his location at two different times, the second involves the darkness of his skin at two different times. 2. Based on a comparison of the properties of two (different) entities, you can make a judgment that they are different, or are the same. For example: This mourning dove has a dark spot, that looks permanent, on its left leg. The one last week did not have a mark. So, this bird is not the same as that bird.

3 3 In order to conclude that two things are different, compare the properties of the two different entities, either at the same time or at different times (if you can be sure that the second thing is not just a changed version of the first one). In the inference above, the mark on the first bird was thought to be permanent, and so the second bird, with a different mark, must be a different bird. 3. By comparing the properties of entities on a single scale you can infer that they are have the property in the same or different amount or extent. For example, with respect to height, you can use information about two entities to conclude that one is taller than the other or the same in height; with respect to time, you can use information about two entities to conclude that one happens earlier than other or at the same time; with respect to number or amount, you can compare how many are in one group and how many in another and conclude that one is more numerous (has a greater number) than the other or that they are equal; and so on. Here is an example involving speeds: Bill ran the race in 10.5 seconds. Henry ran the race in 11.8 seconds. So, Bill was faster (by 1.3 seconds) than Henry. 3 Part-Whole Relationships 1. Another type of relationship you have a decent intuitive grasp of is the relationship of parts to wholes. The properties of the parts sometimes transfer and often do not transfer to the whole (and are sometimes tricky). And the same is true of the relationship between the properties of the whole and the parts. Consider the following: Jim (a dog) has a white paw. So, Jim is white. Jack's hand smells of the garlic he has been chopping. So, Jack smells of garlic. FC Milan is a great (football) team. So, each player on the team is great. Does Jim's white paw mean that he is white? It's not clear. Judging the quality of the first inference is difficult because it depends on how you construe the word "is" in the conclusion. Imprecision is a BIG and CONSTANT problem in the study of reasoning and is discussed in Real-World Reasoning. The second example seems (do you agree? ) more straightforward: if Jack's hand smells of garlic then Jack smells of garlic, unless, again, we want to be very careful in distinguishing what it is for Jack to smell, rather

4 4 than his hand. The third inference is an example of division, which is a kind of part-whole reasoning. Division says that all of the parts have the property that the whole has. Sometimes reasoning by division is appropriate and sometime it is not; it depends on the type of thing and the parts. The opposite form of reasoning is collection, which says that the whole has a property that all of the parts have. The first argument is not an instance of collection, since it only talks about one part of Jim, namely his paw. The general lesson here is that whenever you see an argument involving a thing (a whole) and its parts, think carefully about the relationship between them. An inference is poor or fallacious or bad when the reasons on offer, even assuming they are true, don't give you strong reason to accept the conclusion. 4 Reasoning With Relations 1. Some propositions express how two entities are related. Some examples are "Jack is not the same (person) as Gill.", "Jack is opposite Gill.", "Jack is taller than Gill.". example: 2. Some relations between entities are symmetrical and some are not. For Jack is next to Gill. So, Gill is next to Jack. The relation "is next to" is symmetrical; Jack cannot be next to Gill without Gill being next to Jack. "Is behind", however, is not symmetrical: Jack is behind Gill. So, Gill is behind Jack. The relationship between an entity and its properties is obviously not symmetrical; the terms in the proposition cannot be reversed. For example: Jack is (has the property of being) tall. So, tall is Jack. 3. A very familiar form of reasoning involves making a chain. Chain reasoning can be applied to three entities and the relationships between them so that the first and third can be linked. Two types of basic relationship that are easily formed into chains are their relative location in space (expressed with words such as "next to", "behind"

5 and so on) and in time (expressed with words such as "before", "after" and so on). For example: Jim (the dog) is in his kennel. Jim's kennel is in the back garden. So, Jim is in the back garden. "Jim's kennel" acts as a link so that "Jim" and "the back garden" can be joined. This passage uses the relationship "is in" in all three propositions. However, just using the same relationship throughout is not a guarantee that the reasoning will be good. Consider the following pair: Seattle is west of Chicago. Chicago is west of New York. So, Seattle is west of New York. Seattle is close to Eugene. Eugene is close to San Francisco. So, Seattle is close to San Francisco. Both of these examples involve the same relationship throughout: in the first inference, the relationship used is 'west of' and 'Chicago' acts as a link; in the second, the relationship is 'close to' and 'Eugene' acts as the link. (All three are cities on west coast of the U.S.) However, the relation 'west of' is transitive, while 'close to' is not in fact (or, is not necessarily) transitive, and a speaker who tries to form a chain using 'close to' is reasoning poorly. 4. From your competence in English, you understand the logic of thousands of relational properties and verbs. For example, you can also easily spot that there's something wrong with the following inference: Smith is Jones' friend. Jones is Henry's friend. So, Smith is Henry's friend. You know that the reasoning is suspect and the conclusion could well be false because you know that 'is a friend of' is not (necessarily) a transitive relationship. 5. Now consider an example of chain reasoning there is a link, a shared middle term which does not involve the same relationship throughout: Jim (the dog) is in his kennel. Jim's kennel is green. So, Jim is green. The shared item is again Jim's kennel. But the reasoning in this example is poor because the relationship "is in" cannot reliably be linked with "is" (in the sense of "has 5

6 the property"). A correct inference would be to the conclusion that Jim is in something green. 6 5 The Tricky Verb 'To Be' 1. Let's pause for a moment to reflect on the tricky nature of the verb "to be" in English. The example of Jim in his kennel, which is green, raises an important point. The word "is" (and its cognates) in English has many different meanings and is used in many different types of predication. Consider the following inference: Smith is white. White is a color. So, Smith is a color. If this example gives you pause, it is probably because you are trying to work out what meaning of the word "is" will make the best sense of the inference. The first proposition uses "white" as a property, while the second uses "white" as a general term. The two types of predication are different, even though the same word "is" appears in both of them. And the conclusion too is open to different interpretations, depending on how you understand the "is". In order for this inference to make sense, you might need to employ three different meanings of "is": Smith is white. White is a color. Smith is a color. Jack has the property white. Anything white is colored. Smith is colored (i.e. has some color). You intuitively feel the difference between different kinds of "is" propositions, but it can take some work to articulate exactly what kind of relationship is being mentioned in each proposition. There are different types of predication, many of which English expresses with the same word "is". Here are only some of the different types of predication using "is": Jack is. Jack is white. Jack is a human. Jack is Captain Arnold Jack exists. Jack has the property white. Jack belongs to the class human. Jack and Captain Arnold are identical. 6 Reasoning With Categorical Generalizations

7 1. Above, we saw chain reasoning applied to relationships (especially spatial and temporal relationships) between entities,. Chain reasoning can also be applied to relationships between classes (also called categories). Categorical generalizations are propositions stating what proportion of the members of one class are also members of another. Here are some examples: Anything blue is colored. (Every member of the class of blue things is also a member of the class of colored things.) No human is greater than 10 feet tall. (No member of the class of human things is also a member of the class of things greater than 10 feet tall.) Most dogs have tails. (Most of the members of the class of dogs are members of the class of things with tails.) Some dogs are brown. (Most of the members of the class of dogs are members of the class of things that are brown.) The first two examples are universal generalizations; they use words like "all" or "any" and "no" or "none". Another way you can think of universal categorical generalizations (though it rarely occurs in regular speech) is in terms of a conditional with a variable "x". For example, "Anything blue is colored." can be thought of as "If x is blue, then x is colored.". "x" stands for any specific entity. The third and fourth examples are not universal. They use "most" and "some" as quantities. 2. Generalizations, both universal and non-universal, can be linked together in a chain. For example: All mammals are animals. All humans are mammals. So, all humans are animals. All fish live in water. Some pets are fish. So, some pets live in water. The first example involves three "all" propositions. The second example involves one "all" and two "some" propositions. They are both types of chain reasoning using classes or categories. P&C-Categorical takes a brief look at chain reasoning involving classes. 7 7 Mixing General & Specific Propositions

8 1. A familiar type of reasoning mixes categorical generalizations and propositions about specific entities. For example: All humans are mammals. Jack is a human. So, Jack is a mammal. Here, Jack is a specific member of the category 'human', and since the generalization tells you that if some thing is a human, it is a mammal, you can infer that Jack is also a mammal. This kind of reasoning is called instantiation (or instantiation syllogism or quasi-syllogism). One familiar use of instantiation is reasoning with cause-and-effect generalizations. You are constantly making instantiations inferences about what is happening or going to happen, based on a generalization about what typically happens in these circumstances. Or to put it simply, a prediction is a kind of instantiation. For example: There is a heat wave over most of Europe. Heat waves cause deaths. So, some (heat-related) deaths are occurring. Here, the generalization in the second sentence might have been written as "If there is a heat wave, then deaths occur." or "Any time there's a heat wave, people die.". Since there is a current, specific, heat wave, there must be some people dying. Instantiation is discussed in Inductive & Scientific Reasoning. 8 Reasoning With Conditionals 1. Chain reasoning can also be applied to (specific) states of affairs or events. States of affairs are expressed in propositions such as "Jack is at the store." and "The Chargers won by 30 runs.". For chain reasoning, we need three states and the pairs to be connected using the "If then " construction. We saw in section 6 that a generalization can be expressed using "if x is then x is " but the conditionals we're talking about involve propositions about some specific thing and not the members of a class in general. Here are some examples of conditionals linking specific propositions: If it continues to rain, (then) the river will burst its banks. If the Bulls win tonight, (then) they will go through to the finals. If you don't get off my land right now, (then) I'll call the police. 8

9 9 If Bill is in the library, (then) Henry is too. When filled in, if-then propositions are also called conditionals. The proposition that goes with the "if" in a conditional is called the "antecedent" and the proposition that goes with the "then" is called the "consequent". Each of the conditionals above involves particular states or event: this current rainfall; the Bull's winning tonight's game; the departure of a particular person from the speaker's land; and so on. Note that the "then" is often omitted in English, and that there are other ways of expressing a conditional in English without using "if then " or "if, " at all. For example, the first sentence above ("If it continues to rain, the river will burst its banks.") could be expressed as "Any more rain and the river will burst its banks.". (Various different ways of expressing a conditional are covered in P&C 1.5.) 2. Now that we have conditionals connecting two states, we can build a chain of conditionals. For example: If Jack arrives before noon, we will eat lunch at home. If we eat lunch at home, we will save time. So, if Jack arrives before noon, we will save time. In this example, eating lunch at home is the event that provides the linking state which joins Jack's arrival with saving time. It is the consequent of one conditional ("If Jack arrives before noon, we will eat lunch at home.") and the antecedent of the other (If we eat lunch at home, we will save time."). Note that conditionals do not say how the world is; they only say what would be true if something else were true. Thus the conclusion of this chain inference says only that if Jack arrives before noon, we will save time; it does not say that we will save time. Chain reasoning applied to conditionals is covered in both P&C-Big and P&C- Derivation. 3. Besides chain reasoning, another very familiar kind of reasoning involves "if then " propositions (conditionals). If you believe (or grant) that a conditional is true and also believe that the antecedent (the "if" part) is true, you can infer that the consequent (the "then" part) is also true. Here is an example: It is seven o'clock. If it is seven o'clock, the cafeteria is closed. So, the cafeteria is closed.

10 10 One name for this kind of reasoning involving a conditional about specific events and then asserting the antecedent is matching the antecedent (also asserting the antecedent or modus ponens). This type of reasoning is discussed in P&C-Big 8 and P&C-Derivation. 9 Elimination 1. A final very common form of reasoning concerns situations with a limited range of options, followed by all but one of the possibilities being eliminated. Here's an example: You think to yourself: I'll get either the stir-fry or else the stuffed ravioli. The waiter then says: Unfortunately, there is no ravioli this evening. You think: So, I'll get the stir-fry. This kind of reasoning is called elimination inference or argument by elimination or disjunctive syllogism. Elimination is discussed in P&C. 10 Induction 1. How you come to know propositions about categories of things or about cause-and-effect relationships is another kind of reasoning, one based on generalizing from your experience of specific things. (You can of course be told a generalization by someone else. But they, or somebody further back in time, used their experience to generate the generalization.) The process of generating quantified categorical propositions from repeated experience of specific things is called induction or generalization. Induction is not a reasoning process that is easy for humans to do. Or rather, it is not one that is easy for humans to do well. Humans perform generalizations very quickly, but (as is discussed in the I&S chapter on Induction) they often do so too quickly. 11 Fallacious Reasoning & Degree Of Difficulty 1. Many of these types of reasoning are so familiar that you can also easily tell when reasoning involving them goes wrong. For example, you can easily spot that there's something wrong with the following attempt at transitivity:

11 11 Jack is standing next to (i.e. shoulder to shoulder with) Gill. Gill is next to Henry. So, Jack is next to Henry. You can spot the fallacious reasoning because you know that 'next to' (at least in the sense of "shoulder to shoulder with", doesn't transfer. Now compare that with this piece of reasoning: Jack is to the right of Gill. Gill is to the right of Henry. So, Jack is to the right of Henry. Unlike 'next to', 'to the right of' is transitive, and so this is good reasoning. We will introduce some additional terminology later, but for the moment we will simply say that reasoning is either good or bad (also "poor" and "fallacious"). By "good", we mean that, assuming the reasons offered are true, the conclusion would have to be true or be very likely to be true. By "bad", we mean that even assuming the reasons offered are true, the conclusion is false or not very likely to be true. 2. Telling the difference between good and bad reasoning gets more difficult as the passages get more complex. The types of simple reasoning we have seen so far can quickly become difficult. First look at this passage, involving the relative location of three cities in Ireland: Belfast is north of Newry, and Newry is north of Dublin. So, Belfast is north of Dublin. No problem. You can do this in your head. But notice that it gets a little trickier if we change the order of the information around. Consider: Newry is north of Dublin. Belfast is north of Newry. So, Belfast is north of Dublin. This is a little trickier because there's a longer gap between the two mentions of Newry. Your mind has to go back and get the first piece of information about Newry (that it is north of Dublin) once it hears Newry mentioned the second time, in relation to Belfast. We prefer the reasoning to proceed step by step, with an obvious connection at each step. Now, let's also increase the number of items involved (and change the example): Jack is left of Gill. Henry is left of Smith. Jones is left of Jack. Gill is left of Henry. So, Jack is left of Smith.

12 12 The relationship here is a fairly simple one, 'left of', but the number of people involved, along with the non-sequential ordering, makes this difficult enough that you probably need to move very slowly, constructing a mental model of the information in your mind, or perhaps on paper. The pattern of reasoning itself, however, is still familiar. 12 Concluding Remarks & Summary* 1. As you can see from the brief survey, reasoning is part of every human's everyday life. Everyone makes basic inferences about the identity and location of objects, and about various different relationship between things (including between things and their parts), and is able to link together information in chains, to apply general propositions to specific cases and to argue by elimination. 2. In the remainder of this book we take these basic modes of reasoning and expand them up to and somewhat beyond the point of most people's intuitive reasoning. Some of the reasoning discussed in this book is very simple and the rest you are capable of performing even if it takes not only a pencil and paper but some learning, practice and concentration. 3. The summary on the next page provides a brief description and an example of each of the different types of reasoning you have seen in this chapter.

13 Change/Rest of Entity 'Properties of one entity are same/different at different times. So, entity is same/ different.' Jack was in front of the TV when I came in. He still there now. So, he hasn't moved. Same/Different Entities 'Properties of different entities are same/different. So, the entities are the same/ different.' Jack has dark hair. The thief had light hair. So, Jack is not the thief. Comparative Properties 'Properties (of different entities) have certain values and so are same/different.' A tumbler is 3.5" high. A tall-boy is 6". So, the tall-boy is taller. Part-Whole 'A part has a certain property. So, the whole does.' (Or vice versa.) My arm is sore. So, I am sore. (If about the relationship between all parts and the whole, or vice versa, these are called Collection or Division.) Symmetricality Can the relationship be reversed? Usually between entities. Jack is sitting next to Gill. So, Gill is sitting next to Jack. Chain Reasoning involving Entities: Jack is taller than Gill, who is taller than Jim. So, Jack is taller than Jim. (Transitive if same relationship used (successfully) throughout.) Classes: Anything with kidneys has a liver. All humans have kidneys. So, all humans have a liver. States/Events (in specific propositions): If the Chargers win, they are champions. If the Chargers are champions, I'll lose my bet. So, if the Chargers win, I'll lose my bet. Matching the Antecedent A conditional plus the antecedent gives the antecedent. If the Chargers win, they are champions. The Chargers win. So, the Chargers are champions. Instantiation A generalization is applied to a specific entity. Dogs love hot dogs. Jim is a dog. So, he loves hot dogs. Elimination 'One or other. Not the one. So, the other.' Jack will go to see either Snakes On A Plane or Mission Impossible. Snakes On A Plane is sold out. So, he will see Mission Impossible. Exercise Set (1) on Some Everyday Reasoning (1.1-12) 13

14 For each passage, (i) Identify the type(s) of reasoning involved: Change/Rest of entity; Same/Different entities; Comparison of entities on a scale; (state with respect to what they are being compared: height, time, place, etc.); Part-whole (including collection and division); Symmetricality; Chain reasoning (involving: entities (possibly transitive), classes, (specific) propositions); Matching the antecedent; Instantiation; Elimination. (ii) Say whether the reasoning is good or bad. If bad, explain why. Sample Politicians get a generous allowance for transportation costs. Enda Kenny is a politician. So, he gets a generous transportation allowance. Instantiation (1) The meeting is at two p.m. The lecture is at three p.m. So, the meeting is first. 14 (2) Jack's book is in his room. His room is on the second floor. So, his book is on the second floor. (3) Jack is 5' 7" and Gill is 5' 5". So, Jack is taller than Gill. (4) Jack ate the left-over slice of pizza from the fridge. So, the pizza is no longer in the fridge. (5) In a house with a dog. There are feces on the kitchen floor. If there are feces on the floor, a dog pooped on the floor. So, a dog pooped. (6) Jack is Henry's cousin. Cousins have parents who are either brothers or sisters. So, their fathers are brothers.

15 15 (7) Jack is sitting next to Gill. Gill is a person wearing a hat. So, Jack is sitting next to a person wearing a hat. (8) The hall's capacity, for fire safety purposes, is 400 people, but we have admitted 423 people. The hall is over-full. (9) Smith is Jones' boss. And Henry is Smith's boss. So, Henry is Jones' boss. (10) I was talking about John with the glasses and short hair, but Jack was talking about John with the long hair and no glasses. So, we were talking about two different Johns. (11) It generally takes half an hour to get to the school. I have to be there by nine. So, I'll leave shortly before eight thirty. (12) Henry is worse than Bill at basketball, but Henry is better than Smith. So, Bill is better than Smith.

16 16 Answers to Exercise Set (1) on Some Everyday Reasoning - Even Numbers (2) Jack's book is in his room. His room is on the second floor. So, his book is on the second floor. Chain w/ entities; not transitive - two different spatial relationships; "Jack's room" as the link (4) Jack ate the left-over slice of pizza from the fridge. So, the pizza is no longer in the fridge. Change (of place) Note that "the" is used in both propositions (6) Jack is Henry's cousin. Cousins have parents who are either brothers or sisters. So, their fathers are brothers. Instantiation (8) The hall's capacity, for fire safety purposes, is 400 people, but we have admitted 423 people. The hall is over-full. Comparison in number. The number of people in the hall is greater than it should be. (10) I was talking about John with the glasses and short hair, but Jack was talking about John with the long hair and no glasses. So, we were talking about two different Johns. Comparison of properties leading Though glasses can be taken off. to a same/different conclusion. (12) Henry is worse than Bill at basketball, but Henry is better than Smith. So, Bill is better than Smith. Chain. (Though "better" might be equivocated upon.)

17 17 Exercise Set (2) on Some Everyday Reasoning (1.1-12) For each passage, (i) Identify the type(s) of reasoning involved: Change/Rest of entity; Same/Different entities; Comparison of entities on a scale; (state with respect to what they are being compared: height, time, place, etc.); Part-whole (including collection and division); Symmetricality; Chain reasoning (involving: entities (possibly transitive), classes, (specific) propositions); Matching the antecedent; Instantiation; Elimination. (ii) Say whether the reasoning is good or bad. If bad, explain why. (1) Dark clouds are gathering. Dark clouds usually bring rain. So, it will rain soon. (2) Jack is next to Gill. So, Gill is next to Jack. (3) Gill likes Jack. Jack likes Rocky Road ice-cream. So, Gill likes Rocky Road icecream. (4) The Virginia Symphony is not a great orchestra. So, none of its players are great. (5) The fire bell is ringing. If the fire bell is ringing, someone broke an alarm. So, someone broke an alarm.

18 (6) Jim (the dog) went outside, through the back door, into the enclosed back garden. I've been watching the door and he hasn't come back through it. So, he is still outside. 18 (7) I'll sign up for either a pottery course or a painting course. (Later:) The pottery course is full. So, I guess I'll take the painting course. (8) Jack is close to the pond. The pond is close to the playground. So, Jack is close to the playground. (9) Cherry blossom tress bloom in spring. The cherry blossom trees are blossoming. So, it is Spring. (10) Extra strength Tanqueray (gin) contains alcohol. Alcohol causes intoxication. So, Tanqueray causes intoxication. (11) There's always a good deal of truth to a cliché. The saying "One man's meat is another man's poison" is a cliché. So, there's a good deal of truth to the saying "One man's meat is another man's poison".

19 (12) At a square, four person, table, Jack is sitting across from Gill, and Henry is on his (Jack's) right. Gill is on Bill's left. So, the order of people is Henry, Bill, Gill, Jack. 19

20 Bonus Exercise Draw a diagram (a family tree) to illustrate the song "I'm My Own Grandpa" by Dwight Latham and Moe Jaffe. Watch/listen at It sounds funny, I know, but it really is so! Oh, I'm my own grandpa! CHORUS I'm my own grandpa! I'm my own grandpa! It sounds funny, I know, but it really is so! Oh, I'm my own grandpa! Now, many, many years ago when I was just twenty-three, I was married to a widow who was pretty as could be. This widow had a grown-up daughter who had hair of red My father fell in love with her and soon they too were wed. This made my dad my son-in-law and changed my very life. My daughter was my mother 'cause she was my father's wife. To complicate the matter, even though it brought me joy, I soon became the father of a bouncing baby boy. My little baby then became a brother-in-law to Dad. And so became my uncle though it made me very sad. For if he was my uncle then that also made him brother Of the widow's grown-up daughter, who, of course, was my stepmother. CHORUS My father's wife then had a son who kept them on the run. And he became my grandchild for he was my daughter's son. My wife is now my mother's mother and it makes me blue Because, although she is my wife, she's my grandmother too. CHORUS Now if my wife is my grandmother, then I'm her grandchild, And every time that I think of it, it nearly drives me wild. For now I have become the strangest case you ever saw As husband of my grandmother, I am my own grandpa! 20

21 CHORUS 21

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