assignment #016 ÿwåõºéËE„Z†l}ɆרyO®Ãäã Liberal Arts Mathematics I 07-10-2006
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15:18:01 query 3.5.6 all dogs love to bury bones. Archie doesn't. Therefore Archie isn't a dog .
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RESPONSE --> Valid. Dogs that bury bones and Archie are two completely different circles, so Archie can't be a dog.
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15:19:34 ** You would put all dogs in a circle, and this circle would be inside another circle consisting of all things that love to bury bones. Archie is outside this bone-burying circle and since this circle contains all dogs Archie isn't a dog. This makes the argument valid. The x for Archie has to go outside the outer circle, so it has to be outside the inner circle. Thus the x can't be in the inner circle, and Archie therefore can't be a dog. The conclusion can't be contradicted. COMMON ERROR WITH INSTRUCTOR RESPONSE: I put 'all dogs like to bury bones' in one circle and 'archie likes to bury bones' in another. INSTRUCTOR RESPONSE: You don't want to use a single circle to represent a compound statement. 'All dogs like to bury bones' and 'Archie likes to bury bones' are compound statements. SIMILAR ERROR: in one circle ,I put all dogs love to bury bones, inthe other circle I put Archie, so I knew that Archie wasn't a dog, so the statement is valid . INSTRUCTOR COMMENT: See previous comment.
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RESPONSE --> Ok. I need to keep all compound statements within one circle. I used two cirlces.
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15:20:26 query 3.5.20 all chickens have a beak. All hens are chickens. Therefore all hens have beaks.
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RESPONSE --> Valid. The big circle is chickens with beaks. The smaller circle inside the big circle is hens. Therefore, all hens have beaks.
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15:22:37 ** You need to show the relationship between chickens and things with breaks. You would make a circle for everything with a beak and another circle for chickens. Since all chickens have beaks the chicken circle has to be inside the 'beaked' circle. Then you have hens. They are all chickens so the hen circle is inside the chicken circle. Since the chicken circle is already inside the beaked circle the hen circle (inside the chicken circle) is also inside the beaked circle, and you conclude that all hens have beaks. COMMON ERROR WITH INSTRUCTOR COMMENT: In the outer circle, I put chickens with beaks. Inside that circle, I made another circle for hens are chickens. INSTRUCTOR COMMENT: 'hens are chickens' is a statement, not a thing. The circles have to be defined by things. SIMILAR ERROR WITH COMMENT: Two circles: large circle of hens are chickens and a smalled circle within of hens have beaks. Valid INSTRUCTOR COMMENT: You don't put propositions into circle (e.g., 'hens are chickens' isn't a circle). You put sets of things into circles (e.g., a circle for hens and a circle for chickens, with the hens circle inside the chickens circle). **
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RESPONSE --> I am getting these correct, yet not in the correct way. For this problem, I needed to add another circle and separate the chickens and the beaks.
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15:24:02 When the diagram is drawn according to the premises, is it or is it not possible for the diagram to be drawn so that it contradicts the conclusion? If it is possible describe how.
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RESPONSE --> I am sure that the diagram can be drawn to be contradicting. I just don't know how.
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15:25:14 ** The circle for hens must be inside the circle for chickens, which is inside the circle for beaked creatures. Therefore the circle for hens must be inside the circle for beaked creatures. No other way to draw it consistent with the conditions. **
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RESPONSE --> So, for this problem, there is no way else to draw the diagram. I thought the previous question was a hypothetical question.
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15:31:20 3.5 27 all drivers contribute. All contributors make life a little worse. Some people in the suburbs make life a little worse. Therefore some people who contribute live in the suburbs.
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RESPONSE --> Invalid. ""Some people who contribute to air pollution live in a suburb."" This does not necessarily mean that all people within the circle of air pollution live in a suburb or that all people who live in a suburb pollute the air.
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15:34:24 ** BRIEF SOLUTION: Drivers circle inside contributors circle. Contributors circle inside make-life-worse circle. Suburbs circle overlaps make-worse circle but the degree of overlap is not specified, and the circle doesn't necessarily extend all the way into the contributors circle. So the picture can be drawn to contradict the conclusion without contradicting the given premises. WITH A LITTLE MORE DETAIL: You would have circles for drivers, contributors, people who make life worse and people in the suburbs. All drivers contribute so the drivers circle would be inside the contributors circle. All contributors make life worse so their circle would be inside the 'people who make life worse' circle. The people-in-the-suburbs circle has to be inside the make-life-worse circle. The make-life-worse circle contains the other two circles, and there is a place in that circle for the people-in-suburbs circle such that the people-in-suburbs circle doesn't intersect any of the other circles (put it in the part of the make-life-worse circle that doesn't contain either of the other two circles). So it's possible to put all people in the suburbs outside of the 'contribute' circle. COMMON SENSE: In commonsense terms, which you should relate to the picture, nothing is said that forces all people in the suburbs into the drivers circle or the contributors circle, since the statement is that SOME people in the suburbs make life worse. There might be a little old lady in the suburbs who doesn't drive and does nothing but make life better--this is not contradicted by the 'some people in suburbs make things worse' statement. Thus the argument is invalid. COMMON ERROR WITH COMMENTARY: all drivers contribute goes in one circle. All contributors make life a little worse goes in another. Some people in the suburbs make life a little worse goes in a third. INSTRUCTOR COMMENT Again classes of things, not relationships, go into the circles. Everything you listed here is a relationship, not a class of objects. **
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RESPONSE --> I drew my picture the way that you described. I just didn't explain it very well. I understand what you put as the explanation.
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