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Some non-proofs

15 June, 2015 - 15:10
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Of course, not all arguments are valid proofs. Identiflying invalid proofs is just as interesting as identi flying valid ones.


Homer: Ah, not a bear in sight. The Bear Patrol must be working. Lisa: That's speciousl  reasoning, Dad. Homer: Thank you, honey. Lisa: By your logic, this rock keeps tigers away. Homer: Oh? How does it work? Lisa: It doesn't work. Homer: Uh-huh. Lisa: It's just a stupid rock. Homer: Uh-huh. Lisa: But I don't see any tigers around here, do you? [pause] Homer: Lisa, I want to buy your rock! [A moment's hesitation ... and money changes hands.] 

(From The Simpsons Much Apu About Nothing .)

If Lisa isn't around, who will identify specious reasoning for us? We can certainly use her approach of finding other particular examples that follow the same argument, yet lead to a clearly erroneous conclusion.

Example 1.5

Suppose that my friend makes the following argument:


Warm cola tastes bad.



Warm salt-water tastes bad.



Therefore, mixing them together tastes bad.

“Common-sense conclusion”.
lines 1,2

I'm skeptical, so I have a sip; sure enough, the conclusion is indeed true. But is the proof correct does the "common-sense conclusion" rule actually hold? In order to refute the form of the argument, we can try similar arguments which have the same form but a false conclusion (as Lisa did).


Ice-cold coke tastes good.



Ice coffee tastes good.



Therefore, mixing them together tastes good.

“Common-sense conclusion”, lines 1 and 2.

After another unfortunate sip, I verify that this conclusion is not true, and therefore my friend's reasoning is at fault.

My friend responds by claiming that the "common-sense conclusion" is too valid; the rule is that bad-taste is preserved upon mixing, not that any taste is preserved. While I'm inclined to believe that, we realize we can still test this more refined rule: can you come up with an instance of mixing together bad-tasting things and ever getting a yummy result? (Say, salt and four, which can be mixed and baked to get delicious saltines The argument continues, about whether the form of the argument precludes baking, and so on.)

The end result (after I take some antacid) is that we have a clearer understanding of the initially vague "common-sense conclusion", and stricter rules about when it applies. Thus, refining the argument has led us to a greater understanding.

The above examples are a bit frivolous, but the procedure of looking for counterexamples applies to many real-world dilemmas. It also highlights the difference between a correct proof, and a faulty proof that might still happen to lead to a true result. (By the way, this is the exact same skill used when trying to come up with an algorithm for a problem: "well, the algorithm works for this input, but can I find a something that makes one of the steps fail?" If so, you then try refining your algorithm "well, I can add a test to take care of that problem; is that enough so that it always works?")

Exercise 1.3.1

Solve this statement for [X]: It is wrong to ban [X]. Such a ban would punish those reasonable citizens who would use [X] responsibly, while those who really want to abuse [X] will be able to get it anyway, through a black market which will only subsidize other criminal activities.

In real-world issues, there are often many subtleties, and short arguments that sound airtight might be glossing over factors which are important in practice.

Example 1.6

During daylight, there is no need to have headlights (or running lights) on: there's already plenty of light for everybody to see each other by. Even during the day, headlights slightly increase how quickly other drivers see you during (say) a routine, tenth-of-a-second glance in their mirror.

Example 1.7

When in a turn-only lane, there is absolutely no need to signal since there's only one way to turn, a signal can't communicating any information to other drivers’ Glib, but not true: Other defensive drivers presumably know you have only one legal option, but they don't know that you know that, and they are planning reactions in case you surprise them with a sudden illegal maneuver. By signaling, you give them information which helps them better plan for yet other contingencies. Furthermore, it also gives you more confidence that other drivers are expecting your turn, reducing your suspicion that they're about to pull a surprise maneuver on you. (True, these are all low-probability events which almost always turn out to be unnecessary. But avoiding accidents is all about minimizing risks for the one moment events do spiral out of control.)

Example 1.8

You'll lose weight if and only if you burn more calories than you take in. All those diet-plan books can never get around this, and all their details are pointless.

True, calorie intake and expenditure solely determine weight loss/gain. But after some thought, we can get examples where the above logic overlooks some relevant differences: If your friend told you they were switching from a diet of 2000 calories of balanced short-term and long-term energy sources (sugars, proteins, and carbs) to a diet of 2000 calories worth of Pixy Stix at breakfast plus a Flintstones multivitamin, would you be optimistic that they would have the willpower to strictly follow the new plan? The two plans are equal when counting calories, but in actuality one really is a better plan. (Even more exaggeratedly, consider a daily plan of 2000 calories of sugar while never drinking any water since water has no calories, it can't affect your calorie count, according to the above claim.)

These contrived counterexamples help illustrate that it's conceivable that there can be a difference between diet plans, so the initial claim isn't technically true.

The point illustrated is that often real-world arguments incorrectly imply that their result follows from the form of the argument, when in fact the form is not valid in the way a syllogism is. This fallacy can be illuminated by finding a different domain in which the argument fails. The practice of searching for domains which invalidate the argument can help both sides of a debate hone in on bringing the unspoken assumptions to light. The original argument, if its conclusion is indeed true, must be patched either by adding the unspoken assumptions or fixing the invalid form.

Exercise 1.3.2

Mistakes in syllogisms are hard to make: what are the only two ways to have an error in a syllogism?