第二天的课程是basic relations！

mike1999

第一天的课程：http://forum.chasedream.com/dispbbs.asp?boardID=24&ID=6014
第七课：Introduction to Conjunctions and Disjunctions
("And" and "Or")

The simplest deductions are those involving two or more things connected by "and" (a conjunction) or "or" (a disjunction), and are governed by the following rules:


"And" (and "but"): Affirm all, negate one.
To affirm an "and" claim, all parts of the "and" must be affirmed as true.
To negate an "and" claim, at least one part of the "and" must be negated as false.
"Or": Affirm one, negate all.
To affirm an "or" claim, at least one part of the "or" must be affirmed as true.
To negate an "or" claim, all parts of the "or" must be negated as false.


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Example 1. Consider the claim, "Pat and Juan have arrived." If that claim is affirmed as true, we can conclude that both Pat and Juan have arrived, because all parts must be affirmed. If, however, the claim is false, then we can conclude that at least one of the terms must be negated: either Pat has not arrived, or Juan has not arrived, or neither has arrived.

Example 2. Consider the claim, "Farida or Marcia has won the race." If true, then (because at least one part of the "or" must be affirmed) one of the following must be true: Farida has won, Marcia has won, or they both have won. If false, then neither Farida nor Marcia have won.

These rules always apply, even when the deductions are complicated by more elements ("Farida, Marcia, Pat, and Juan"), the use of negatives ("Farida and not Marcia"), or some combination of these. So, to affirm the claim "Farida and Pat but not Marcia or Juan have finished," we would employ the following steps:

To negate the "or" ("not Marcia or Juan"), we would negate both parts, concluding that Marcia has not finished and Juan has not finished.
To affirm the "and" ("Farida and Pat"), we would affirm both parts, concluding that Farida has finished and Pat has finished.
To affirm the "but" (which operates logically as an "and"), we would affirm all parts, concluding that Farida has finished, Pat has finished, Marcia has not finished, and Juan has not finished. All these must be true in order for the claim "Farida and Pat but not Marcia or Juan have finished" to be true.


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Inclusive and Exclusive "Or"
Sometimes, "or" is used in an exclusive sense. For example, you might read on a menu, "Soup or salad comes with the dinner." This means, "soup or salad, but not both," because the menu is describing what is included with the price of the meal. However, if there is no contextual reason to think otherwise, assume every "or" is inclusive--that is, "A or B or both." The difference between the inclusive and exclusive "or," then, has to do with cases in which "both" are true. Since "Soup or salad comes with the dinner" is exclusive, it places "soup and salad" outside the range of things that are included in the price of the meal.

Now suppose an advisor tells you that you can take English 7 or History 60 to satisfy a critical thinking requirement. Though, in this case, it's clear that you don't need to take both, that "or" is still inclusive, because if you did take both, you would still be satisfying the requirement: English 7 or History 60 or both satisfy the requirement.

As a result, the use of "and/or" is unnecessarily confusing and should be avoided, since "or" by itself, in the absence of any exclusionary language or context, means the same thing. Using "A or B or both" makes the possibilities clearer but, in most cases, a simple "or" will suffice.

[此贴子已经被作者于2003-6-10 16:22:06编辑过]

mike1999

第八课    Introduction to Options


Critical thinking concerns the processes by which we make decisions, and the most basic decisions are made between two choices. We can better understand the consequences of the choices we make, and learn to make better choices, by employing the concepts of critical thinking.

Making a choice involves the apparently simple operation of either affirming one possibility (saying "yes" to it), or denying another (saying "no" to it). Creating a string of such choices, however, can quickly get complicated, and such a binary string ("yes-yes-no-no-no-yes") is actually the basis for computerized computations, where the options are usually expressed as zeros and ones ("110001"). If we go a little further, and establish relations between the choices using conjunctions and disjunctions, we have created a Boolean string: "a and b or (c and not d)."

Let's say you are visiting Avshi, who offers you the use of a car.

Avshi might ask, "Do you want the red car or not?" In this case, you have been given two options: red or not red. Driving the blue car is part of "not red," but so is declining both cars. If you are seen driving "red" (that is, when "red" is true), the implication is that you have not chosen "not red," and if you are seen driving anything but "red," the implication is that you have not chosen "red." When you have only two options, both of which cannot be true and both of which cannot be false, you are faced with contradictions, or contradictory choices.
Avshi might also ask, "Do you want the red car or the blue car first?" Since "first" makes it clear that the "or" here is exclusive (for more on this, see the "And" and "Or" Introduction), you have been given three options: "red," "blue," and "neither." In this case, if you are seen driving "red" (that is, if "red" is true), the implication is that you have not chosen "blue" or "neither." But the implication of "not red" isn't as clear: we can't conclude that you have chosen "blue" because "neither" is also an option. When you have two options ("red" and "blue"), and a third option that neither of the first two are true ("neither"), you are faced with contraries, or contrary choices.
And someone else might ask, "Which car or cars are you going to drive while you are here?" Now you have four different options: "red," "blue," "both," and "neither." In this case, if you are seen driving "red," we can't determine whether you might also drive "blue" at another time, because "both" is an option; and if you aren't driving "red," we can't conclude whether you have chosen "blue," because "neither" is an option. When you have two options ("red' and "blue"), and two more options (that "both" of the first two are true, and that "neither" of the first two is true), then you are faced with open or unrestricted choices.
As you can see, those three questions have very different implications. We can summarize the three different possibilities for an option between two things as follows:

Contradictions: both cannot be true, and both cannot be false.
Contraries: both cannot be true, but both can be false.
Choices: both can be true, and both can be false.
In confronting options, then, your job is first to determine whether you are dealing with a contradiction, a contrary, or an open choice, and then to understand the consequences of that.

Example 1. My options are A or B, and I choose "not B." What can you conclude about "A"?

If A and B are contradictories, you can conclude that "A" is true.
If A and B are contraries or open choices, you cannot conclude anything, because either "A" is true or "neither A nor B" is true.
Example 2. My options are A or B. I have chosen "A." You can only conclude that "not B" is true if you know that A and B are either contradictory or contrary.

Example 3. My options are A or B. I have chosen "not A." You can only conclude that "B" is true if A and B are contradictory. If they are contrary or open choices, "neither" is a possibility.


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Note: In common speech, words paired as "opposites" are sometimes contradictories and sometimes contraries. Often, this is determined by the context. "Night" and "day," for example, may be understood as contradictory if "night" is the time between sunset and sunrise, and "day" between sunrise and sunset. On the other hand, if "twilight" is recognized as a time that is neither "night" nor "day," then they are only contrary. We usually accept "male" and "female" as contradictory for humans, but contraries or just choices for other kinds of animals, like snails, some of which are asexual or hermaphroditic. In fact, there is always a range of definitions, depending on the context the terms are used: for gender identities, from physical appearance to genetic composition; for "night" and "day," from the common to the meteorological and astronomical. The only way to be sure that two terms are contradictory, therefore, is to use the "A and not-A" format. Thus, "night" and "not-night" are certainly contradictory, whatever "night" and "day" may be. (And even here, common usage may undermine the meanings. Many people, for example, assume that not everyone falls into the categories of the "haves" and the "have nots.")