9.5: Resolving Inconsistencies
- Page ID
- 22010
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In a Peanuts comic strip, Charlie Brown says, "I tell you, Lucy, birds do fly south during the winter." Lucy responds with what she takes to be a counterexample: "Chickens are birds, aren't they? You never see a chicken flying south for the winter, do you?"
"Good grief," says Charlie Brown. This exchange contains a good example of misinterpretation resulting from ambiguity. Lucy takes Charlie Brown's claim one way, but he means it another way. In the way Lucy takes it (all birds fly south) there is a counterexample involving chickens; in the way Charlie Brown means it (many birds fly south), there is no counterexample. To avoid the misunderstanding, Charlie should revise his statement by saying what he means. The moral is that clearing up ambiguity can resolve an inconsistency.
When you are given inconsistent information, you should reject some of the information to resolve the problem and achieve consistency among the remaining pieces of information. Because you also want to find the truth, you should always reject the information that is the least well supported or the most likely to be false. However, in cases where it isn't clear what to revise, you need to search for more information (and hope that in the meantime you will not have to act on the information you have).
This apparently inconsistent sentence was published in a U.S. newspaper. Can you clear up the problem?
Instead of being arrested, as we stated, for kicking his wife down a flight of stairs and hurling a lighted kerosene lamp after her, the Reverend James P. Wellman died unmarried four years ago.
- Answer
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The newspaper was apologizing for the false report about the Reverend's violent behavior when in fact he had not behaved this way but had died four years before the report.
Sometimes a person has inconsistent moral principles that don't appear to be inconsistent. For example, suppose you, like most people, believe the moral principle that
(1) People ought to keep their promises to their family,
and also the moral principle that
(2) You shouldn't do anything that is likely to hurt innocent persons.
These don't appear to be inconsistent. Now suppose that your father insists you keep your promise to help him with his summer project. His project is, you later learn, to stop the burglaries on your family farm by booby-trapping the windows and doors of the barn. An infrared beam of light will pass by the inside of each window. If anyone forces open the window and sticks his head through, they will get a blast of birdshot in the side of the head.
Think of the innocent but curious eight-year-old girl next door who finds the window unlocked on a day when the alarm is activated. She could be hurt. This would violate principle (2) above. The very possibility of this tragic event, even if it never happens, shows that moral principle (1) is inconsistent with moral principle (2). Moral principles are supposed to cover possible situations as well as actual ones. Therefore, you are caught in an ethical dilemma. Which moral principle should be revised? One reasonable change would be to revise principle (1) in favor of (1'):
(1') People ought to keep their promises to their family unless doing so is likely to hurt innocent people.
Principles (1') and (2) are consistent. This process of resolving moral dilemmas by thinking in advance about potential situations is an important way to make moral progress, and it is the kind of thing that gets talked about in ethics classes. Attention to logical inconsistency can promote moral growth.
The notion of resolving contradictions also plays a central role in adding new information into your store of knowledge. Your goal in adopting new beliefs is always to add more while maintaining consistency. We all try to do this, but there are good ways and not so good ways to do so. Suppose, for example, that your problem is to decide whether George can swim well. If you knew that he was a lifeguard, that would be significant supporting evidence. Almost every lifeguard in the world is a good swimmer—let's say 99 out of 100 lifeguards are good swimmers. Here is a fine argument using this evidence:
George is a lifeguard.
99 out of 100 of the world's lifeguards can swim well.
So, George can swim well.
You cannot be absolutely sure of the conclusion on the basis of those two pieces of information, but you can be about 99 percent sure. It would be illogical to conclude that he cannot swim well. Now, compare that argument with this one:
Fred is a Frisian.
Frisians are poor swimmers; in fact, 8 out of 10 Frisians cannot swim well.
So, Fred cannot swim well.
You could be about 80 percent sure that Fred cannot swim well, given these two pieces of information. Both arguments are good arguments because they provide good reasons to believe their respective conclusions. You should add both conclusions into your store of information if you happen to know that the premises are true.
Now for the surprise. Suppose you acquire some new information: Fred is George. If you hold onto the conclusions from the two previous arguments, you will conclude that Fred can swim well and also can't. You can’t have that. It is time to go back and revise your store of information. How are you going to resolve your contradiction?
You should retract your belief that Fred cannot swim well. Fred is an exceptional Frisian. The best conclusion on the total evidence is that he can swim well, but now you can no longer be 99 percent sure. You need to reduce your estimate of the probability. We won’t try to figure out the new probability number.
An important moral can be drawn from our swimming story: Do not cover up counterevidence. The more evidence you pay attention to, the better position you are in to draw the best conclusions. A second moral is that belief is a matter of degree; it is not an all or nothing affair.