# grist

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  1.Some things that people say are true. Some things that people say are false. There are even things that people say that are neither true nor false, but that in the future will turn out to be one or the other.

retve:
  1.Some things that people say are true. Some things that people say are false. There are even things that people say that are neither true nor false, but that in the future will turn out to be one or the other might not believe how much a single person under stress can lift, but even so, their top performance will be well below that required to life an airliner.

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  There are lots of statements that would require investigation to determine their truth or falsity. When people try to reason using those sentences, the have to start out by saying something like, “First, let us suppose that Sally Forth can actually lift that airplane.

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  Then they might say, “Let’s also think about what will happen on the assumption that Sally Forth cannot lift that airplane.”  

retvh:
  We are all familiar with sentences such as, “I think that the teacher must have heard the projector fall. Let’s get out of here.” We suppose that the teacher has heard the noise and plan accordingly. 


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  2. When we try to think ahead, we often use suppositions about what might be true or might become true. “If he turns out to have Ebola, then we must quarantine him.” In logic people are often dealing with situations that have not yet been made  clear.

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  People often explore abstract proposition. They say something like, “Supposing she is right about that, then what must we do?” We don’t have to be faced with a wildfire to take precautions against one. People make guesses or suppositions about what might happen, and then they try to plan ahead just in case.

retvf:
  If students in primary school broke something in the activity room while the teacher was away, how would they act if they needed to hide the broken pieces, but they thought they heard someone coming down the hallway?


retve: Let us look more carefully at if-then sentences to discover how we judge them true or false.
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  3. I didn’t believe my grandmother when she claimed that if I ate mulberrries from her tree I would get intestinal worms, so I ate lots of her mulberries, and I never got sick. Do you think she was right? (See answer at bottom of next screen for the answer.

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  We are all familiar with sentences such as, "I think the teacher must have heard the projector fall. Let's get out of here." We suppose that the teacher has heard the noise and plan accordingly.


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  4. Surely she was wrong. Lots of people eat mulberries even if they are not as tasty as some other kinds of fruits and berries. Nobody seems to get sick.

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  We can use a formal device called a truth table to discover what the facts tell us about whether an if-then prediction, or other use for it, should be judged true or false.


retvf: The "conditional" or "if-then" statement in this case can be symbolized by “E ➔ S”.
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  5. We. can make a diagram to help with our inquiry into my grandmother’s idea about the danger of mulberries. We will use capital letters to represent each little sentence in the prediction. 

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  “You Eat mulberries” will be symbolized by “E” for “eat.” “You get Sick” will be symbolized  by “S” for “sick.” 

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  Under “E” we will write “t” to indicate cases we study where somebody eats mulberries. Under “S” we will write “t” for the cases we investigate where somebody gets sick. Places where nobody eats mulberries and where nobody becomesr ill will be marked with an "f."

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  Then under “E ➔ S” we will write “t” for the possiblecases where Grandma was right, and we will write “f” for the possible cases where she was wrong. For the times when both “E” and “S” are judged to be true, what do we write for “E ➔ S” ?  See the diagram to ther right.

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  Surely my grandmother was wrong. Lots of people eat mulberries even if they are not as tasty as some other kinds of fruits and berries. Nobody seems to get sick.


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  6. Grandma said that if you ate mulberries you would get sick. You ate mulberries despite her warning, and you then got sick. So it looks to me like Grandma was definitely not proven to be wrong. Mark "E➔S" with a “t.”


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  7.  Suppose that you ate mulberries, but then you didn’t get sick. How would you mark “E”? How would you mark “S”? Then how would you mark "E➔S"?  See the bottom of next screen for the answer. Check your own work.


retve: Write your answer down and then check it against the next step.
retvf: Check your work against the answer ar rhe bottom of the next page.
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retvn: Mark E(ating) true, and mark getting S(ick)false.. Mark E➔S false.
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  8. Suppose that it was not true that you ate mulberries, but you did get sick? Does that make Grandma wrong?


retve: Write your answer down somewhere and check the bottom of the next screen.
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  9. Supposea your dentist told you not to eat candy because if you do eat candy then you would get cavities. So you didn't eat any candy but you still got cavities. Did your dentist mislead you? 

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  Your grandmother didn't say what would happen if you did not eat mulberries. Mark E ➔ S (not disproven =) true.


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retvn: Your dentist did not deceive you. Eating candy is not the only way you can get tooth decay.
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  10. Just as your dentist did not deceive you, your grandma never claimed that you would only get sick if you ate mulberries. All she claimed was that if you did eat mulberries you would get sick. So “E ➔ S” has not been disproven. Mark it “t” on the truth table. See the bottom line on the truth table.


retve: Write down your answer and then check it against the information at the bottom of the next screen.
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  11. In logic, as in science, when people are not speaking with complete accuracy, what do people mean when they say that something is "true"?


retvf: Write out your own answer somewhere, and then see the next screen..
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retvm: It's a little like the idea of "innocent until proven guilty."
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  12. There is another possibility. We are trying to understand what real-world results would be relevant to Grandma’s claim that if you eat mulberries you will get sick.

retve:
    What if you do not eat mulberries and you do not get sick? We would have to mark both E and S as false. How should we mark  “E ➔ S?

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  A sentence is said to be "true" as long as it has not been demonstrated to be false. If I say that all Martians are violet, nobody can say for sure that I am wrong until somebody finds a Martian of some other color. If some people in 16th century England claimed that all swans are white, nobody could prove them wrong until an explorer sailed to Australia and found their first black swan. When something is well attested we casually say it is "true."


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retvg: So, when E and S are both not true, what does that mean about “E ➔ S”?
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  13. If you think about it as seeing whether Grandma was right, then you can easily go astray. You can eat mulberries but get sick because of some other factor. So proving an “if-then” statement true in some absolute way is impossible.

retve:
  But if you find one case where the person does not get sick after eating mulberries, then you know that  “E ➔ S” is false. 


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  14. What we have just discussed shows that when neither part of the if-then sentence is true, i.e., when there is no information indicating that the whole sentence  can be proven false, we have to mark it true — but we also need to remember that “true” only means “has not yet been disproven.”

retve:
  A couple of irrelevancies can't tell you anything about the true subject of inquiry. Suppose somebody said, "If humans establish a successful colony on Mars, then the human species will be better protected against extinction," and somebody says, "Humans cannot establish a colony on the surface of Venus, and, besides, the UN Secretariat says that the human species will live forever." 

retvf:
  People have pretty good reason to believe that the surface of Venus would be a bad place to start a colony. On top of that, the UN Secretariat is a sort of governmental organization, and it is definitely not qualified to make true prophecies. So why would we let the improbable ideas of one person or group make us discount the word of others?

retvg:
  Nobody knows for sure what would happen if humans eventually try to establish a colon on Mars. We will have to wait to see how things really turn out. But exgtrapolating from what we do know about planets, environments, and survival, this idea doesn't seem to be impossible to realize..


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  15. Let’s look at a new kind of proposition, the “if and only if” sentence. For instance, “You will graduate if and only if you complete 120 semester credit hours.” If somebody makes that claim, under what conditions will they be shown to be saying something true, and under what conditions will they be shown to be saying something false. 

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  (Remember that "true" just means something like "innocent until proven guilty," i.e., "not yet shown to be false."

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  According to that standard, if you get 120 semester credit hours, you count as having graduated, and if you have graduated you must have earned 120 semester credit hours.


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  16. How would you write the truth table for “G if and only if C”  (G ↔︎ C)?  Write down the top line, and then go to the next step to compare your answer with what is written there.


retvd: 18. Here is what the top two lines of the truth table should look like.
retve: Do you see any conflict that would make it wrong to mark “G ↔︎ C” as true?
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  You predict a situation when both little sentences (clauses, to give them their proper name) have to go together, and you have encountered such a situation so your prediction has not been shown to be wrong.


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  19. How about when somebody graduates without getting 120 semester credit hours? Would that make “G ↔︎ C” true or false? See the next step for discussion after you have written your answer down.


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  20. A statement such as “G ↔︎ C” can be said beforehand as a rule for how people should be handled. The school says you have to get so many points toward graduation, or you will not graduate.

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  If in a real situation somebody has fewer than 120 semester credit hours, the school would say that the other way of stating the rule is that if you do not have at least 120 hours then you do not graduate.

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  When  people outside the system see somebody with 120 credit hours or more not graduate, they must then conclude their hypothesis is false. 

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  Depending on circumstances, either somebody outside the school system has tried to figure out the rule but has somehow made a mistake, or else somebody inside the school system has failed to follow their own rules.

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  Note that you can take this "if and only if" sentence as an outside observer's guess about the school's rule, or you  can take it as a rule that the school should live up to.It depends on your point of view.


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  21. Sometimes “G ↔︎ C” is made from the outside, For instance, somebody observes the school’s processes and make an inductive observation. As they see it, obtaining 120 semester credit hours amounts to graduating.

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  When they see instances where people get 120 points graduate, and people who fail to get 120 points do not graduate, they say their hypothesis was supported. The more positive instances they see, the better supported they will believe their hypothesis to be.


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  22. Whether it is a rule of the college, or it is a hypothesis about how the college runs things, what can we say when somebody does not graduate and also does not get at least 120 semester credit hours?


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  23. If it’s a rule of the college and the person does not graduate and also does not get enough semester credit hours to graduate, there is nothing that says the rule has been broken. 

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  So, looking at things that way would you mark the fourth line of the truth table as true or as false?


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  24. How about the situation in which an external observer, a scientist, observes the school and predicts that “G ↔︎ C”  is true in those situations where students who do not get 120 credit hours in fact never get to graduate?


retve: Here is the truth table as you will find it in all the standard textbooks:
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  25. If you have any doubt that the fourth line under “G ↔︎ C” should be marked true, then go to the Internet and find the explanations of the many people who write on the subject of logic.


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  26. Here is another kind of logic problem. Kong An-guo was a descendant of Confucius who lived from 156 to 74 BC. He was an official who lived near a tributary of the Yellow River.

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  In the city he governed there were people who took the most beautiful maiden they could find every year and sacrificed her to the god of that river. Kong An-guo told them they should stop. “Why should we stop?” they asked. “We have always sacrificed a maiden to the god of the river."

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  Did Kong An-guo think those priests had good reason for making yearly sacrifices of virgins to the god of the river? Can you restate their rationalization for continuing to make these sacrifices?

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  "Kong" rhymes with the German family name Jung, "An" is pronounced something like "ahn" as in the German word "Autobahn." "Guo" is something like "gwoh." 


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  27. Kong An-guo did not argue with them directly. He went to the river at the time they were going to load the maiden onto a boat and send it out into the river where it would sink. He told them the maiden was not beautiful enough.

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  He ordered them to find a more beautiful maiden. Nothing happened for a long time. Kong An-guo put the chief priest into the boat and ordered him to meet with the god of the river and apologize for being late with the sacrifice.

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  Nothing happened for a long time. Kong An-guo said that the chief priest ought to have returned by then, so he had the second highest priest get into another sinkable boat and go out to find why the god of the river had not responded, and to reassure the god that they were still industriously looking for a suitable girl to sacrifice.


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  28.A while later,  Kong An-guo looked around and discovered that all the other priests had gone home, so he went home too.

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  The rationale offered by the priests for doing a sacrifice again that year was that they had always made a sacrifice to the river god once a year at this time. Centuries earlier, Confucius had advised his students to respect (be wary of) spirits and stay away from them. So Kong  An-guo probably thought that this sacrifice had been wrong the first time, and doing it time after time did not make it better. It was still wrong and should be stopped.


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  29. So far we have learned how to construct two truth tables. Please write out the two from memory and then check the next two steps to be sure that you know how to make them correctly.


