HOW TO SOLVE PARADOXES
2.1 Introduction: Solutions as Reeducations ofIntuition
Imagine that you are a contestant on a game show. The host shows you three closed doors and tells you that there is a prize of a brand new car behind one door and goatsbehind each of the other two doors. - P43
To see how this is so, imagine that you are now thegameyou can show host and know where the car is. If the contes-tant picks the correct door on the first try, thenreveal either of the two other doors to the contestant andthen make the offer to switch. But, what is the chance that this will happen? - P44
This thought experiment, sometimes called the Monty Hall paradox (figure 2) after a famous game showhost, nicely illustrates how some of the weaker paradoxes are solved (Clark 2007). - P44
By showing that what we expected to be the case turns out to not be thecase, we have "reeducated" our intuitions about probabil-ity. In the Monty Hall paradox, we did this by looking atthe game show host‘s choices in the situation and show-ing that he is more likely to have revealed the only door with the goat behind it, leaving the remaining door the more likely winner and switching the better choice. - P46
Another way to reeducate our intuition about a part of the paradox is to show that the very notion that leads to the paradox is contradictory. - P46
(전략).
By reeducating our intuitions about the barber andour chances on the game show, we find a way out of the paradoxes. Earlier, you learned that a useful way to think about intuitions or "what we would say" in a given situation is using degrees of belief and is sometimes called subjective probability. - P47
(전략). Because subjective probability involveswhat a rational thinker believes, solutions attempt to provide rational grounds for lowering one‘s degrees of belief about the parts of the paradox. This "reeducation" of our intuitions, in which we send our intuitions back to school,
as it were, can be done in numerous ways. - P47
Table 3 catalogs different strategies for solving paradoxes. I use the term solution-type, by which I mean a strategy for analyzing paradoxes. - P49
Let‘s examine each of these strategies for solving para-doxes in more detail. You may suspect, along the way, that this taxonomy, or catalog, of solution-types has an element of artificiality. If so, you are correct. Some solutions occupy border regions between types. - P50
Although detailed analyses of all the logical systems available for solving paradoxes is too big a task for any one book, it is helpful to look at some of the more promising systems and how they provide solutions to some of themore troubling paradoxes. - P50
Fuzzy logic has uses in a wide range of areas: for example, the way patterns such as signatures are recognized by computers. And decision theory has importantconsequences for actions taken by governments, researchers, and others. - P51
2.2 Solution-Type 1: The Preemptive Strike, or Questioning the Paradoxical Entity
(전략).
Also, consider detective Flint and her dilemma of pre-venting a future explosion from happening. She was givenimages that, it was assumed, were accurate images of the future. But if she prevents the future explosion, how could the pictures have been accurate? One way to deal with this paradoxical situation is to deny that there can be accurate pictures of the future. - P52
A variety of this type of solution applies to paradoxesdealing with abstract concepts like statements and sets,
and argues that the proposed abstract entity is eithermeaningless or self-contradictory. - P52
2.2.1 Example of a Preemptive Strike against a Paradox:Zermelo-Fraenkel Set Theory‘s Solution to Russell‘s Paradox
(전략).
The "Rule V" that Frege mentioned in his response to Russell was a basic principle of traditional "naive" set theory, a theory of sets that defines set informally. The principle is known as the unrestricted comprehension principle. - P53
(전략). It is the empty set. As long as there is a property, there is a set that corresponds to it. This statement is the unrestricted comprehension axiom, and Frege‘s Rule V.
The contradiction that prompted Frege‘s response to Russell was the paradoxical set R. With the paradoxicalset R, the set that corresponds to the property of being
"a set of all sets that do not contain themselves as members," a problem arises. - P54
Russell‘s paradoxical set R
R: set of all sets that do not contain themselves at members - P55
Why not say here that something is defective about R?
That would be good, but the problem with this straightforward preemptive strike against the paradox is that there is nothing, in principle, wrong with sets that contain other sets as members. - P55
Zermelo-Fraenkel set theory (ZF), which is still standard today¹, provides a way out of Russell‘s paradox. Thebasic strategy is to get rid of the unrestricted comprehen-sion axiom, which holds that for any property there is a setof all things that satisfy that property. - P57
2 How to Solve Paradoxes
1. Usually, another axiom is added that wasn‘t original to ZF－the axiom ofchoice－and the theory is abbreviated ZFC. - P213
ZF assumes that for any set and any definable property, thereis a subset of all the elements of the given set that satisfy that property. Under this approach, you don‘t start with aproperty and then say that there is a set that corresponds to it. - P57
According to ZF, R cannot be constructed. Norcan a set of all sets, for that matter, because you are using abottom-up approach by constructing sets and saying whatsubsets of that given set satisfy a property. - P57
Zermelo-Fraenkel set theory provides a "preemptive strike" against Russell‘s paradox in that it gives rules for the construction of sets that precludes the problematic "set of all sets that do not contain themselves as members." - P57
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