The real trick by which you tend to be deceived in Zeno's paradox is not being prevalently talked. Let us learn how to resist any trick.
Topics
About: truth
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Zeno's Paradox
- 2: The Prevalent Explanation
- 3: Why Did You Not Bother to Do the Calculation?
- 4: The Real Trick
- 5: We Should Learn from the Example, How to Resist Any Trick
- 6: Logically Speaking, It Is Simple
- 7: Any Trick Is About Attention
- 8: Still We Tend to Fall into Traps, so We Should Beware of Usual Types of Traps
- 8-1: Concept Replacement
- 8-2: Some Traps Concerning Quantification
- 8-3: Unbalanced Citation
- 8-4: Inadvertently Accepting an Implicit Premise
- 8-4-1: Faulty Dichotomy (or Multichotomy)
- 8-4-2: Faulty Existence
- 8-4-3: Some Widely-Assumed, Selfish, Ludicrous, Implicit Propositions
- 8-5: Illegitimately Generalizing a Proposition
- 8-6: Rushingly Accepting a Spurious Deduction
- 9: In Any Case, Your Laziness, a.k.a. Labor-Saving Is Being Taken Advantage of
- 10: The Countermeasures
- 10-1: The Principle: State Explicitly and Check Diligently
- 10-2: A Technique: Rephrase the Proposition
- 10-3: A Technique: Thinking of Extreme Cases
- 10-4: A Technique: Making a Wise, Typically Whole and Static Visualization
Starting Context
- The reader knows the background of this site.
Target Context
- The reader will understand the real trick of Zeno's paradox and how to resist any trick.
Orientation
There is an article on harmful rhetoric.
Main Body
Stage DirectionHere is Special-Student-7 in a room in an old rather isolated house surrounded by some mountains in Japan.
1: Zeno's Paradox
Special-Student-7-Hypothesizer
Zeno's paradox claims like this.
Running Achilles will never catch up a turtle that is crawling ahead of him, because when Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles reaches the new location of the turtle, the turtle will be already ahead of the location; when Achilles ~, and so on, which will never end.
Special-Student-7-Rebutter
Ah.
Special-Student-7-Hypothesizer
"Ah" what?
Special-Student-7-Rebutter
Were you taken in by that?
Special-Student-7-Hypothesizer
I was not exactly taken in, because it is obviously wrong: Achilles can undoubtedly catch up the turtle. However, I must confess that it took certain time until I understood the trick.
2: The Prevalent Explanation
Special-Student-7-Hypothesizer
The prevalent explanation is, in short, that the time taken in Achilles's catching up the turtle converges.
In long, for the initial distance between Achilles and the turtle, l, the velocity of Achilles, \(v_A\), and the velocity of the turtle, \(v_t\), the time taken in Achilles's reaching the present turtle location is \(t_0 = v_A^{-1} l\), the distance covered by the turtle in \(t_0\) is \(v_t t_0\); the time taken in Achilles's reaching the new turtle location is \(t_1 = v_A^{-1} v_t t_0 = v_A^{-1} v_t v_A^{-1} l = l v_A^{-1} v_A^{-1} v_t\), the distance covered by the turtle in \(t_1\) is \(v_t t_1\); generally, \(t_i = l v_A^{-1} v_A^{-i} v_t^{i}\); so, the time taken in Achilles's catching up the turtle is \(t = \sum_{i = 0, 1, ..., \infty} t_i = l v_A^{-1} (1 + v_A^{-1} v_t + v_A^{-2} v_t^2 + . . . ) = l v_A^{-1} (1 - v_A^{-1} v_t)^{-1}\), which is not the infinity.
For example, supposing that \(l = 1\), \(v_A = 1\), and \(v_t = 0.1\), \(t = 1 + 0.1 + 0.01 + . . . = 1.11 . . . \), which obviously does not go over even 1.2.
3: Why Did You Not Bother to Do the Calculation?
Special-Student-7-Hypothesizer
In fact, you did not need to know the concept of convergence, in order to break the paradox.
You could just do the calculation, then, you would have noticed that the time did not go to the infinity.
Why did you not just bother? . . . You should repent.
4: The Real Trick
Special-Student-7-Hypothesizer
Anyway, why was the paradox so successful?
The reason why you were tricked is not that you did not know that an infinite series did not necessarily go to the infinity, I guess.
The reason is that you confused 2 times: the time taken in saying "when Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles ~" and the time taken in Achilles's catching up the turtle.
What Zeno really proved is just that you will never finish saying "when Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles ~", not that Achilles will never catch up the turtle.
To think in one's right mind, it is kid's stuff.
It takes only several seconds to say "Ten years have passed", but it takes ten years for ten years to pass; the 2 times are obviously different.
Special-Student-7-Rebutter
Well, people may not have exactly confused the 2 times, but at least seem to have had a misconception that there was a kind of relationship between the 2 times in which if one of the 2 times was the infinity, also the other would be the infinity.
Special-Student-7-Hypothesizer
It is obviously a misconception: it takes only several seconds to say "An infinite time has passed", but it takes an infinite time for an infinite time to pass.
If they had not exactly confused the 2 times, at least they had "identified" the 2 times with respect to being infinite or non-infinite.
Anyway, as a proof of that the confusion or identification of the 2 times is the real reason, this "paradox" will not be as successful as Zeno's: "In order for Achilles to measure the distance from him to the turtle (which is not moving in this case), Achilles has to go to the half point, measuring the distance; Achilles has to go to the half point of the rest distance, measuring and adding the distance; Achilles ~; and so on. So, the distance is the infinity.".
Special-Student-7-Rebutter
As many people would not be tricked by it as by Zeno's.
Special-Student-7-Hypothesizer
If the issue was about convergence of infinite series, you would be tricked by it likewise by Zeno's.
You would not be tricked by it because you would not confuse (or identify) the time taken in saying "Achilles has to go to the half point measuring the distance; Achilles ~" with the distance.
Special-Student-7-Rebutter
You mean, people may be likely to confuse or identify a time with another time, but they are far less likely to confuse or identify a time with a distance.
Special-Student-7-Hypothesizer
I suspect also that Zeno's paradox was so successful because it was about time; I mean, people will less easily confuse or identify a distance with another distance.
Special-Student-7-Rebutter
It may be so; people are rather insensitive to time.
5: We Should Learn from the Example, How to Resist Any Trick
Special-Student-7-Hypothesizer
Now we know how we were tricked by Zeno's paradox.
But it is a waste of opportunity to stop there: we should learn how to reject any trick, with the example as a teaching material.
6: Logically Speaking, It Is Simple
Special-Student-7-Hypothesizer
In order to reject any trick, logically speaking, it is simple: accept only true propositions and take only valid deductions. Then, you cannot be deceived by any trick.
Special-Student-7-Rebutter
Certainly, but it will be "easy to be said but hard to be done".
Special-Student-7-Hypothesizer
I guess so, so, we have to adopt some practical countermeasures.
7: Any Trick Is About Attention
Special-Student-7-Hypothesizer
As a grounding, we should know that any trick is about attention.
The trickster diverts your attention from the trick; very often, the trick itself is not so complex to be difficult to be penetrated, once a due attention is directed to it; it is that the trick is hidden from your attention.
Special-Student-7-Rebutter
The trickster may be you, in fact.
Special-Student-7-Hypothesizer
Yes. It is called 'self-deception', which may be intentional as a rationalization or unintentional as an innocent mistake.
Anyway, the countermeasure should be to distribute your attention evenly, especially to things being neglected by you.
Special-Student-7-Rebutter
You cannot catch out the trick by very closely watching what the magician is inviting you to watch, because that is exactly the magician's ploy.
Special-Student-7-Hypothesizer
In the case of Zeno's paradox, you are invited into the details of "when Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles ~", and you are impressed by the impeccability and the cleverness of the details, with your attention being fixated on the presented details.
Inundating your attention with some innocent details is a usual ploy: as your attention capacity is limited, occupying your attention with the details prevents your attention from noticing the trick.
Of course, it is not that details are bad or unnecessary; the trick can be revealed by checking the details around the trick, but your attentions is being kept far away to attend other details.
Special-Student-7-Rebutter
Sure; we should distribute our attention evenly, but that is very difficult to be really executed.
Special-Student-7-Hypothesizer
Certainly, it will be impossible to be executed perfectly. But we should be able to execute it better.
Special-Student-7-Rebutter
Even only being aware that any trick is about attention will make a difference, but being equipped with some tactics will be better.
Special-Student-7-Hypothesizer
That is the reason why this article does not end here.
8: Still We Tend to Fall into Traps, so We Should Beware of Usual Types of Traps
Special-Student-7-Hypothesizer
While the trap can be different for each trick, there are some usual types of traps, and knowing and bewaring of such types will be helpful.
8-1: Concept Replacement
Special-Student-7-Hypothesizer
A usual type of traps is to surreptitiously replace a concept with another concept.
That means, we had been talking about a concept, but without noticed, we have been tricked to be talking about another concept.
In the case of Zeno's paradox, the paradox was really talking about 'the time taken in your saying "When Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles ~"', but the conclusion is about 'the time taken in Achilles's catching up the turtle'.
Special-Student-7-Rebutter
As we talked before, such replacements seem to be performed frequently via careless identifications of the concepts instead of via mere confusions of the concepts.
Special-Student-7-Hypothesizer
"identification" means that although the 2 concepts are known to not be exactly the same, we consent to talk in terms of the alternative concept, supposing that it will not make any difference for the immediate argument; a very dangerous move, I say.
You know, it is hazardous to casually identify concepts, because usually, you cannot be sure that the concepts are interchangeable for the sake of the argument, until the argument has been completed.
Special-Student-7-Rebutter
If you consent to a concepts identification, you will have to be always watching whether the identification is still valid in the argument, but you will not probably do that, because than doing such a thing, doing without the identification will be far easier.
Special-Student-7-Hypothesizer
So, it will be a sound principle that you do not consent to the identification in the first place.
Anyway, the hotbed of concepts confusions is to use the same term for multiple concepts.
While the term is brought forth in front of your attention for each appearance, as the relation between the term and the concept associated with is not stable, you can be easily tricked to switch to be talking about another concept.
The concept associated with for the term is believed to be determined by the context, but the context does not consist only of the explicit concept but also of the implicit context like the person's knowledge and expectation.
Special-Student-7-Rebutter
"expectation" is especially a vital factor: if the person is interested in a concept, he or she will easily associate the term with the concept.
Special-Student-7-Hypothesizer
So, as the trickster leads your interest, your association is manipulated.
Using the same term for multiple concepts includes omitting due qualifications: when 2 concepts have to be differentiated by some qualifications, if the qualifications are stripped off, the 2 concepts are being represented by the same term.
In the case of Zeno's paradox, saying just "the time" blurs the distinction between the concept of 'the time taken in your saying "When Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles ~"' and the concept of 'the time taken in Achilles's catching up the turtle', and "the time" tends to be mistakenly associated with the latter, which you are interested in.
Special-Student-7-Rebutter
The paradox does not even use the term, 'time'.
Special-Student-7-Hypothesizer
It uses "never", which is inevitably about time, but anyway, "never" itself is also ambiguous, because "never" what? You "never" finish saying "When Achilles reaches the present location of the turtle, ~"?, Achilles "never" catch up the turtle?, or what?
Using "never" is logically the same with using "the time taken is more than the infinity", but it seems to make you more vulnerable.
Special-Student-7-Rebutter
Why is that?
Special-Student-7-Hypothesizer
Well, because "never" is harder to be qualified, I guess.
Special-Student-7-Rebutter
What do you mean by "harder to be qualified"?
Special-Student-7-Hypothesizer
When you are told "never", you will usually not ask "Which never?".
Special-Student-7-Rebutter
Is "Which never?" grammatically valid?
Special-Student-7-Hypothesizer
It is not, which is the point: you are discouraged against asking the due question because it is grammatically invalid.
Special-Student-7-Rebutter
So, should we prefer nouns over adverbs?
Special-Student-7-Hypothesizer
I think so. Look, you may think that you can choose any expression if it is logically the same, but the choice of expression makes a difference because the issue is attention.
When 'time' is used, you are more likely to ask "Which time?" or "Time taken in what?" to be more deliberately expressed.
Special-Student-7-Rebutter
It is quite likely.
Special-Student-7-Hypothesizer
And while "never" means nothing but 'the time taken is more than the infinity' and you have to confirm that the time is really more than the infinity in order to be able to claim "never", when "never" is used, you are more likely to do without really calculating the time.
In the case of Zeno's paradox, the time taken in your saying "When Achilles reaches the present location of the turtle, the turtle will be already ahead of the location; when Achilles ~" is certainly infinite without any minute calculation, but by doing the calculation, you are likely to notice that you are not talking about the time taken in Achilles's catching up the turtle.
Generally speaking, doing appropriate quantification is helpful.
8-2: Some Traps Concerning Quantification
Special-Student-7-Rebutter
"appropriate quantification" will be helpful, but we should be aware that sometimes, quantification is the trap.
Special-Student-7-Hypothesizer
I did not say that any quantification was good; any inappropriate quantification is more harmful than non-quantification.
Special-Student-7-Rebutter
What is "inappropriate quantification"?
Special-Student-7-Hypothesizer
Any inappropriate quantification is to quantify something with an inappropriate measure for the purpose.
For example, humans may be quantified by weight, but weight may not be the right measure for them to be evaluated with, unless they are sumo wrestlers or being sold as meat.
Special-Student-7-Rebutter
Especially, statistics can be traps for many people.
Special-Student-7-Hypothesizer
A typical trap of statistics is the biased population: by tampering the population deliberately or inadvertently, the statistic can be significantly swayed.
It is important to always check the population when faced with any statistic.
Special-Student-7-Rebutter
There is also the matter of accuracy of quantification.
Special-Student-7-Hypothesizer
Yes. The number may be just fairly wrong, while some people tend to suppose that the number should be accurate just because it is a number,
8-3: Unbalanced Citation
Special-Student-7-Hypothesizer
This is a plain example that any trick is about attention.
You cite only what are convenient for you, for example, examples that conform to your claim or some facts that you have exerted much effort, and you insist that your claim is true or you deserve specific (additional) reward for the effort, while there are really also examples that refute your claim or you have already reaped the reward of the effort enough.
Special-Student-7-Rebutter
Those both examples are quite prevalent among Earthians.
Special-Student-7-Hypothesizer
While the former example is explained in an another article, let us talk about the latter example.
After you have taken the effort and have reaped the reward explicitly or implicitly, you tend to keep bringing up only the effort and demanding more reward again and again, saying "Have you forgotten my effort? You, ungrateful creature!"
Special-Student-7-Rebutter
It is more often "implicitly", which is the reason why "Nothing is more costly than being given free.". As we have not decisively settled the transaction, the "giver" endlessly keeps demanding more.
Special-Student-7-Hypothesizer
In the 1st place, while the "giver" is assuming that his or her effort should be rewarded by me, I am not obliged to pay for his or her self-satisfying effort even if I feel it appropriate to pay only as much as the benefit I have received as the result of the effort if any.
Special-Student-7-Rebutter
If you want a reward from your deed, you have to obtain an agreement that you be rewarded how before you do the deed; otherwise, you have no right to demand any reward afterward.
8-4: Inadvertently Accepting an Implicit Premise
Special-Student-7-Hypothesizer
Another usual type of traps is to inadvertently accept an implicit premise.
Special-Student-7-Rebutter
Ah, a tricky one.
Special-Student-7-Hypothesizer
Yes, because it is very difficult to completely reject the trap.
Again, it is about attention: without directing your attention to the implicit premise, you cannot check the premise, but directing your attention to the implicit premise means making the implicit premise explicit, so, you have to make all the premises explicit, which is practically very difficult.
Special-Student-7-Rebutter
Again, it is about making it better, not about making it complete; a detrimental attitude is to despair that you cannot make it complete, so you just give it up completely.
Special-Student-7-Hypothesizer
When you accept an implicit premise, you do so not necessarily because the premise seems obvious, but because you do not direct your attention to it. If you ask yourself whether you should accept the premise, you may rather swiftly reject it.
And a typical case seems to be that you fleetingly wonder about an implicit premise, but let the trickster go ahead, thinking that you can go back to it later, but forget about it, with your attention being distracted to other things by the trickster.
Special-Student-7-Rebutter
It seems to be a prevalent vice to be too confident of your ability to command your attention. People seem to tend to assume that they can easily command their attention at their will because it is their own attention, but it is not so.
Special-Student-7-Hypothesizer
There are some usual types of implicit premises and knowing them will be helpful for noticing faulty implicit premises.
8-4-1: Faulty Dichotomy (or Multichotomy)
Special-Student-7-Hypothesizer
A typical type of faulty implicit premises is faulty dichotomy (or multichotomy).
It is, "(A or B.) Not A. So B." where "(~)" means that it is implicitly supposed or inconspicuously expressed as not to draw attention.
Special-Student-7-Rebutter
But is it really "A or B."? Is it not a possibility of C, D, or whatever?
Special-Student-7-Hypothesizer
If we are asked so, we are likely to rather promptly notice that there are other possibilities, but we have to ask ourselves so.
Special-Student-7-Rebutter
It may be "(A, B, or C)" in fact.
Special-Student-7-Hypothesizer
Yes: that is the reason why I said "multichotomy", although I cannot find such a word in any Biasian dictionary, I do not understand why.
Anyway, faulty dichotomy is prevalent also in self-deception: "(I should get that woman or I should die.) I cannot get that woman. So, I should die.". . . . That is a pretty too-simplistic dichotomy, I say!
Special-Student-7-Rebutter
It is prevalently seen phenomena that people are fixated on choosing only from some fixed ways; you should ask "Is there not another way?".
8-4-2: Faulty Existence
Special-Student-7-Hypothesizer
Usually, the argument is constructed based on the presupposed existence of something.
Special-Student-7-Rebutter
Yes. When a proposition is presupposed, the entities that appear in the proposition are presupposed to exist.
Special-Student-7-Hypothesizer
It seems more difficult to doubt the existences of the entities than to doubt just the proposition.
Special-Student-7-Rebutter
It is more difficult because the existences are implicitly presupposed.
Special-Student-7-Hypothesizer
Yes. When a proposition is "A is B.", it is really "(A exists.) (B exists.) A is B, by the way."
We cannot check implicit presuppositions unless we make them explicit.
I notice that a type of entities that tend to be faultily presupposed to exist is ability.
Special-Student-7-Rebutter
Besides being implicitly presupposed, the existence of an ability is frequently difficult to be refuted.
Special-Student-7-Hypothesizer
That is because we cannot refute the existence of the ability just because the ability is not exerted this time or even has never been exerted.
In order to exactly refute the existence of the ability, we will have to prove that the ability is physically impossible, which is frequently difficult, especially for human abilities.
In the ideal of so-called "democracy", majority are expected to respect interests of minority, but it is preposterous to presuppose the existence of that ability in majority.
Majority have the power to decide things as to maximize only their profits, and can they refrain from doing so, in order to respect interests of minority? . . . It is nobleness, and I know that there exists nobleness,even among Biasians, but I know that nobleness is very rare, found in only a tiny percentage of people. It is preposterous to assume such nobleness in majority.
Special-Student-7-Rebutter
That is a reasonable estimation, but is not an exact proof.
Special-Student-7-Hypothesizer
So, the preposterous ideal of "democracy" keeps being preserved.
Special-Student-7-Rebutter
It should not really be about the exact refutation of the existence of the ability, but about unwiseness of relying on such an at-least-unlikely and never-demonstrated ability, even if it exists.
Special-Student-7-Hypothesizer
It is a widely seen harm that people just wish the existence of an ability and pretend as though the ability was being exerted, which is exactly how majoritarianism is being defended.
Special-Student-7-Rebutter
As I said, whether the ability exists or not, as far as the ability is not being exerted in fact, at least reliably, we should take measures based on the assumption that the ability probably will not be exerted.
8-4-3: Some Widely-Assumed, Selfish, Ludicrous, Implicit Propositions
Special-Student-7-Hypothesizer
There are some selfish implicit propositions, which are ludicrous, but are widely-assumed nevertheless.
For example, "What I believe to be true is true.".
Special-Student-7-Rebutter
Ah, I see many arguments in which a party claims something to be true just because the party believes it to be true.
Special-Student-7-Hypothesizer
Or "What I wish to be true is true."
Special-Student-7-Rebutter
More ludicrous, but not particularly less seen.
Special-Student-7-Hypothesizer
Sometimes, it takes a form like "If it is true, it will be troublesome, so, it is not true.".
Special-Student-7-Rebutter
What a selfish assumption!
Special-Student-7-Hypothesizer
And there are some anthropocentric propositions adopted by even many educated people on the Bias plant, like "What can't be measured by humans don't exist.", "What can't be known to humans don't exist.", "What cannot be defined by humans don't exist.", "What cannot be done calculation on by humans don't exist.".
8-5: Illegitimately Generalizing a Proposition
Special-Student-7-Hypothesizer
Another usual type of traps is to illegitimately generalize a proposition.
Special-Student-7-Rebutter
Is it not just an induction?
Special-Student-7-Hypothesizer
Any induction is fine only if it is presented as a hypothesis, and it requires being examined whether it conforms to all the known cases, even if it is not logically proved.
"to illegitimately generalize a proposition" means to claim the generalization to be a truth without any legitimate proof, usually without even mentioning the existence of the generalization.
Special-Student-7-Rebutter
If the proposition is presented explicitly, the trap seems rather easy to detect, if we are rigidly logically-minded.
Special-Student-7-Hypothesizer
I guess so, but anyway, the practice of this type of traps is prevalent, and most Biasians are not rigidly logically-minded.
Special-Student-7-Rebutter
If the proposition is implicit, even a rigidly logically-minded person can be tricked.
Special-Student-7-Hypothesizer
Yes. Being implicit means that our attention is not given at it, and what is not being attended to can be easily tampered, a general mechanism of tricks.
Even if we are rather rigidly logically-minded, we sometimes could be tricked, so, we should remind ourselves the basics of propositions.
We should always regard any proposition as an inseparable combination of a set of some preconditions and a conclusion, like "If ~, then ~.".
Special-Student-7-Rebutter
Is there always a precondition?
Special-Student-7-Hypothesizer
There might not be in rare cases, but we can think that there is the always-true precondition even in those cases.
My point is that we should always assume that there is the set of preconditions, because you tend to forget it and forgetting it is detrimental.
Again, it is about attention: it is important to not let your attention absorbed solely into the conclusion with the set of preconditions left out of the scope of your attention.
"to illegitimately generalize a proposition" means to abuse the conclusion with the set of preconditions loosened or even completely forgotten.
It is different from accepting implicit propositions, because the preconditions are not particularly being accepted, but the necessity for the preconditions is being forgotten.
8-6: Rushingly Accepting a Spurious Deduction
Special-Student-7-Hypothesizer
Another usual type of traps is to rushingly accept a spurious deduction.
Special-Student-7-Rebutter
You mean, the deduction is just wrong?
Special-Student-7-Hypothesizer
Yes. You just did not check the deduction enough or even at all.
Special-Student-7-Rebutter
Is it not just that you were just poor-minded?
Special-Student-7-Hypothesizer
Well, you may say so . . ., but there is a usual tactic that entices you to skip checking the deduction.
Special-Student-7-Rebutter
What is that?
Special-Student-7-Hypothesizer
The usual tactic shows a meticulous and correct proof of a proposition and says like "Also this is true while I entrust you to prove it.".
Special-Student-7-Rebutter
And you do not bother to prove it, probably.
Special-Student-7-Hypothesizer
As the trickster has done the previous proof impeccably, you believe that the new proposition should have been proved by the trickster as well.
Special-Student-7-Rebutter
If the trickster really says "I entrust you to prove it" explicitly, he or she is rather conscientious, but usually, the trickster does not, and you just fill the gap.
Special-Student-7-Hypothesizer
The same trick is prevalently used in story-telling: the author shows a meticulous description of a part of an imaginary world, and the reader believes that the world should be filled with details all over, while the world is really almost empty, of course.
Special-Student-7-Rebutter
A TV show shows a few rooms in a spaceship, and viewers imagine the spaceship full of rooms.
Special-Student-7-Hypothesizer
The tactic can be more nasty than you imagine, because human brains are wired to unconsciously interpolate wishful imaginations to fill gaps, but knowing the existence of the tactic will be helpful.
9: In Any Case, Your Laziness, a.k.a. Labor-Saving Is Being Taken Advantage of
Special-Student-7-Hypothesizer
While there are various types of traps, you fall into one because of your laziness.
In the case of concept replacement, as you grudge the effort of using a strict terminology (in which, each term corresponds to a definite concept), you allow a term to mean multiple concepts, so, you allow a concept to be surreptitiously replaced with another concept in virtue of being represented by the same term.
In the case of implicit premise acceptance, as you grudge the effort of making all the premises explicit, you allow the spurious premise to slip into the argument.
In the case of proposition generalization, as you grudge the effort of checking the fulfillment of all the preconditions each time, you allow the conclusion to be abused.
In the case of spurious deduction acceptance, as you grudge the effort of strictly checking the deduction, you allow the deduction to go through.
Special-Student-7-Rebutter
Labor-saving is fine while it is skipping doing unnecessary things, but most people cannot refrain from also skipping doing necessary things.
Special-Student-7-Hypothesizer
Wanting to save one's energy is a major animal instinct, but containing such urge should be a major human distinction.
10: The Countermeasures
10-1: The Principle: State Explicitly and Check Diligently
Special-Student-7-Hypothesizer
To sum up, the principle for the countermeasures is to state any proposition explicitly and check any deduction diligently.
Special-Student-7-Rebutter
A principle which most Earthians cannot or will not follow, I guess.
Special-Student-7-Hypothesizer
Which is the case?
Special-Student-7-Rebutter
I do not know whether their will is too weak to resist the animal instinct or their will is adamant that they should be allowed to follow the animal instinct.
Special-Student-7-Hypothesizer
I find that a useful tactic to counter an animal instinct by human will is to form a habit, which can be formed by human will.
If you think that you can invoke the principle only when it is required, probably, you will not notice the occasion when it is required.
Special-Student-7-Rebutter
But as you cannot practically always fully employ the principle, some dynamics should be required.
Special-Student-7-Hypothesizer
Of course. I think that there is a difference between that you basically employ the principle, but loosen the principle occasionally and that you basically does not care the principle, but intent (but probably fail) to employ the principle occasionally.
Well, while the principle should be enough if executed perfectly, I admit that it is hard to be executed perfectly, so, let us know some effective techniques.
10-2: A Technique: Rephrase the Proposition
Special-Student-7-Hypothesizer
You may snub rephrasing, regarding it as a waste, because the new phrase should mean nothing new, but it often makes a critical difference.
If you have managed to rephrase "Achilles never catch up the turtle." to "The time taken in Achilles's catching up the turtle is more than infinity.", you will be already very near breaking the trick.
Special-Student-7-Rebutter
Is there some guidance for what should be the new expression?
Special-Student-7-Hypothesizer
Expressing concepts, at least core concepts, in nouns will be beneficial.
That is because that makes the concepts more easy to be scrutinized.
Special-Student-7-Rebutter
Especially, nominalization seems to encourage quantification.
Special-Student-7-Hypothesizer
Yes. Being just "fast" is just a matter of impression, and introducing 'speed' encourages us to ask "What speed?", "What km/h"?
Special-Student-7-Rebutter
We should know that human impressions are hugely unreliable.
Special-Student-7-Hypothesizer
I can think of some other types of rephrasing like negation-affirmation changes and subject-object changes.
Special-Student-7-Rebutter
Are "subject-object changes" passive-active changes?
Special-Student-7-Hypothesizer
Not necessarily: "A hit B." can become not only "B is hit by A." but also "B got a punch from A." or "B enjoyed a punch by A.".
Special-Student-7-Rebutter
Ah, B may be a masochist.
Special-Student-7-Hypothesizer
It is one of our purposes to escape prejudice like "B must be regrettably hit.", which "A hit B." should not necessarily connote.
10-3: A Technique: Thinking of Extreme Cases
Special-Student-7-Hypothesizer
A useful check for any proposition is to think of extreme cases.
That is because a proposition tends to show gapes in extreme cases.
Special-Student-7-Rebutter
Many Earthians seem to try to get away with the proposition by claiming that such gapes are just exceptions.
Special-Student-7-Hypothesizer
I say, any proposition with any exception is meaningless, because if there is an exception, any concerned case can be an exception, so, nothing can be claimed by the proposition.
Such a selfish ruse as opportunistically dismissing only inconvenient cases as exceptions is never allowed.
If there are some exceptions, the exceptions must have been incorporated into the proposition as preconditions; that is what you must do.
For example, if you support the proposition, "People must obey any evil law.", you should think about a case that a law orders you to kill your baby, for example.
Special-Student-7-Rebutter
Why does the law order that?
Special-Student-7-Hypothesizer
Who knows? It is an evil law; perhaps, because the baby is not eugenic, is immature, or has a little crooked little toe.
Anyway, will you obey the law? If not, you must not support the proposition.
Special-Student-7-Rebutter
While there are some people who nonchalantly claim that you must lawfully try to repeal the law instead of disobeying the law, . . .
Special-Student-7-Hypothesizer
you are ordered to kill your baby right now.
You are not allowed to claim that this case is an exception; if this case is your exception, any case can be someone's exception.
Special-Student-7-Rebutter
I do not guess that many can really support that proposition.
Special-Student-7-Hypothesizer
Brandishing such a rough proposition is irresponsible; what we are supposed to do is to argue on what conditions any law must be obeyed.
Anyway, if you are familiar with software engineering, the check is like a borderline test, which is efficient in finding out bugs.
10-4: A Technique: Making a Wise, Typically Whole and Static Visualization
Special-Student-7-Hypothesizer
Making a visualization sounds a trite technique, but it should be useful if done wisely.
I say "wisely" because 'what kind of visualization?' is the issue.
For example, a vivid picture of an instant at which Achilles is perspiring on his forehead with his teeth clenched is a kind of visualization, but is not particularly useful in breaking the paradox.
Special-Student-7-Rebutter
We are talking about a graph, not about a picture, at least in that case.
Special-Student-7-Hypothesizer
But in another case, a picture may be useful.
So, just saying "do a visualization" is easy, but choosing a right kind of visualization is not necessarily easy.
Special-Student-7-Rebutter
Are there some criteria for being a right kind of visualization?
Special-Student-7-Hypothesizer
Being whole and static, perhaps.
Special-Student-7-Rebutter
What does that mean? The picture of perspiring Achilles is static.
Special-Student-7-Hypothesizer
But is not whole; I mean, it is only of an instant (a part), not of the whole of the concerned phenomenon.
If you gather up such pictures and make them an animation, it is a kind of visualization, but is not static.
Special-Student-7-Rebutter
Is animation bad?
Special-Student-7-Hypothesizer
I am not particularly an enemy of animation, but you tend to be easily deceived by an animation.
For example, in a baseball animation, a pitcher pitches, the ball comes, the batter begin to remember old days in which he was ridiculed by fellow children as a hacker and he practiced swings in front of a red giant setting sun or under a starry sky and he was still ridiculed and he practiced more and . . ., before the ball reaches him in zero point what second, and you accept such absurdity and even shed tear.
Special-Student-7-Rebutter
Humans are fairly blurry about time.
Special-Student-7-Hypothesizer
So, it had better be static in order for you not to be deceived.
A typical wise visualization is the world lines of the concerned objects.
If you had just drawn the world lines of Achilles and the turtle, you would have easily seen what the paradox was saying: it was saying of increasingly short line segments whose time coordinates did not reach the catch-up time at all, to say nothing of infinity.
