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December 31st, 2012 (Permalink)

Headline

Source: Jay Leno, Jay Leno's Headlines, Books I, II, III (1992), p. 39

December 29th, 2012 (Permalink)

Blurb Watch

In previous "Blurb Watch"s I've always picked on the blurbs for new movies, which are often taken out of context from reviews. However, new books often have blurbs on their jackets and in ads published in periodicals. One difference between the two types of blurb is that books often have blurbs from other writers, rather than from reviews. For some reason it's rare, though not unheard of, for a film to be blurbed by another filmmaker. Blurbs from other authors are, of course, less likely to be quoted out of context than those taken from reviews, and even less likely to be checkable since there's no review to read.

However, as an article in England's The Independent newspaper explains, such solicited blurbs have their own problems:

When browsing the shelves, an endorsement from an author you admire can help make that purchase a bit easier. But dig deeper, and glowing recommendations become harder to trust.
The latest paperback edition of Lee Child's Killing Floor, the popular thriller writer's debut novel, comes complete with some glowing praise from Stephen King…. "All are ripping yarns," the cover says, "but since this is the first, it seems the logical place to start." The back-slapping fest between Child and King is well-documented, and Child once described King as "America's greatest living novelist".
A Trick I Learned From Dead Men, by former actress Kitty Aldridge, was praised by Liz Jensen. … Both writers share the agent Clare Alexander.
"Salley Vickers sees with a clear eye and writes with a light hand and she knows how the world works. She's a presence worth cherishing," says Phillip Pullman… of Vickers's The Cleaner of Chartres. Just two years ago, Vickers described Pullman as a "supreme storyteller" in a review of his The Good Man Jesus and the Scoundrel Christ.

Source: Nick Clark, "Why blurbs remain important in the digital age", The Independent, 12/30/2012

Fallacy: Appeal to Misleading Authority

December 28th, 2012 (Permalink)

New Book: Bad Pharma

Ben Goldacre, author of the "Bad Science" column and subsequent book, has a new one titled Bad Pharma: How Drug Companies Mislead Doctors and Harm Patients. It won't be out until February fifth of next year, but you can preorder copies from Amazon now. However, you can read the Introduction and at least part of the first chapter using Amazon's "Look Inside!" feature. The first chapter discusses publication bias, or "the file drawer effect", which we've discussed here previously (see the Resources, below).

Resources:

Publication Bias, 5/6/2006
Worse Science, 9/20/2008

December 7th, 2012 (Permalink)

How to Prove that God Exists, Part Three

Meg and Jack continue their coffeehouse discussion of the existence of God.

"If you have doubts about that one, there's another way to prove that God exists, Jack," Meg said. "Everything's the same as itself, right?"

"Of course," Jack answered, writing a formula on the napkin:

∀x(x = x)

"Instantiate it to 'g'," Meg said, adding another formula beneath the first:

g = g

"So that says God is the same as Himself. Now, existentially generalize on 'g'." So, Jack added the following formula:

∃x(x = x)

"No, Jack, that says that something is the same as itself, which is true but not what we want. Just generalize on the first variable." Picking up the pencil, Meg rubbed out the last "x" and wrote in a "g".

∃x(x = g)

"Can you do that?" asked Jack, "Can you generalize on just one variable?"

"Sure, it doesn't break any rule. Anyway, all we've done is go from God is the same as Himself to something is the same as God, which is valid. But, as you translated it previously, that last statement says that God exists. We've just proven the existence of God! Now, do you believe?"

Meg smiled triumphantly at Jack, who just stared down at the napkin with a puzzled frown and didn't answer.

Should Jack be convinced by Meg's proofs? If not, where did Meg go wrong? Do Meg's arguments commit a logical fallacy and, if so, which one?

Click on the link below for the answer!

Answer

December 6th, 2012 (Permalink)

How to Prove that God Exists, Part Two

Note: The argument between Meg and Jack gets a little technical from this point on. Picking up where we left off, Jack objects:

"There's something sketchy about that argument, Meg."

"But you can prove it using the symbolic logic we've been learning in class," Meg replied. "How would you symbolize your belief that God doesn't exist?"

"Let's see. I remember we learned in class how to symbolize that Socrates exists. We'll let 's' stand for Socrates." Jack grabs a napkin and a pencil and writes the following formula:

∃x(x = s)

Meg looks over his shoulder and says: "Wait. Doesn't that say that there's something that's the same as Socrates?"

"Yeah, but that's the same as saying that he exists, 'cause there's only one thing the same as Socrates; that's Socrates."

"I see, so replace s with g for God."

Meg rubs out the "s" and writes "g" in its place, so that it now looks like this:

∃x(x = g)

"So, that says that God exists," Jack adds, "and all we have to do now is negate it." Jack made a squiggly mark in front of the formula, so that it looked like the following:

∼∃x(x = g)

Jack showed the napkin to Meg, who nodded, and Jack asked: "So, how am I contradicting myself?"

"Well, do you remember the quantifier-negation rules?"

"Vaguely."

"If nothing goes faster than light, what's that equivalent to in terms of a universal quantifier?"

"Don't tell me. Let me think a minute. Oh, yeah! Everything doesn't go faster than light." Jack pulled the napkin towards him and wrote another formula below the first:

∀x∼(x = g)

"So," Jack continued, "that says that everything is different than God. In other words, God doesn't exist. I don't see anything wrong with that. That's what I believe!"

"Well, but do you remember how to do universal instantiation?" Meg asked.

"Of course, that's easy!"

"Then instantiate with 'g'."

Jack wrote a third formula on the napkin:

∼(g = g)

"That says," Meg continued, "God isn't the same as God, but that's false!"

"We must've made a mistake somewhere!" Jack insisted.

Is Jack right? If so, where did they go wrong? Stay tuned for Part Three, when all will be revealed.

December 5th, 2012 (Permalink)

How to Prove that God Exists, Part One

Meg and Jack are two university students taking their first course in logic. They started out helping each other with their logic homework, but that has led Meg to take a romantic interest in Jack. However, Jack is an atheist, which is a big drawback for Meg, who believes that God exists. Knowing that Jack is highly logical, Meg sets out to convert him to theism over coffee at the campus coffeehouse.

"You believe that God doesn't exist, don't you, Jack?" Meg asked.

"That's right", Jack answered.

"Which means that you believe the statement 'God doesn't exist' is true."

"Of course, Meg."

"But if it's true, it's got to be meaningful."

"No doubt, but where are you going with this?"

"Be patient, Jack! Now, if the statement 'God doesn't exist' is meaningful, then the word 'God' has to mean something."

"Yes, but so what?"

"So, you've contradicted yourself!"

"What? How so?"

"If the name 'God' is meaningful, then there must be something that 'God' refers to, and that contradicts your claim that God doesn't exist."

What, if anything, is wrong with Meg's argument? Does it commit a fallacy? If so, which one?

Stay tuned for Part Two, when things start to get hairy.

December 4th, 2012 (Permalink)

Check it Out

USA Today has one of the best articles I've read in the popular press on how laypeople should evaluate news reports on health research. Short of studying research design, this is probably the best that you can do to avoid being misled by health news. Don't just read the whole thing; learn it!

Source: Liv Osby, "Research studies often leave consumers confused", USA Today, 12/2/2012

December 1st, 2012 (Permalink)

A Fourth Puzzling List

Suppose that you find a piece of paper with the following list of sentences printed on one side:

At least one of the statements on this list is false.
At least two of the statements on this list are false.
At least three of the statements on this list are false.
At least four of the statements on this list are false.
At least five of the statements on this list are false.
At least six of the statements on…

At this point, the piece of paper has been torn across and the bottom is missing. Let's assume that the list goes on in this fashion, but that you can't tell exactly how many sentences are on the list. However, let's also assume that there are an even number of sentences on the list. Call the last sentence on the list "2n". So, 2n would read:

2n. At least 2n of the statements on this list are false.

Also, let's assume that each of the 2n sentences on this list are either true or false. Can you determine how many of them are false? How many sentences, if any, are true? Which sentences are true and which false?

Solution

Resources:

Answer to "How to Prove that God Exists" (12/9/2012): Jack was right to be skeptical. Let's deal with the informal argument from Part One, first: it's not technical but, as a result, it's a little hard to pin down.

First of all, even if we can't be sure exactly what's wrong with Meg's argument, we can see that something is wrong in the following way: suppose that we substitute "Santa Claus" for "God" throughout. Then, Jack believes that Santa doesn't exist, which means that he believes the statement "Santa Claus doesn't exist" is true, and so the sentence must be meaningful, and for the sentence to be meaningful the name "Santa Claus" has to be meaningful. So far, so good. But if "Santa" is meaningful, then there must be something for it to refer to. Therefore, Santa Claus exists! Q.E.D.

What's wrong here is in the step from "Santa" is meaningful to "Santa" refers to something, as names can be meaningful without referring to anything. Otherwise, we could prove that nonexistent things exist just by naming them. Moreover, to assume that "Santa" or "God" refers to something is to assume that it exists. So, Meg's argument assumes what it sets out to prove, namely, that God exists, which means that it commits the informal logical fallacy of begging the question.

Now, to turn to the two formal proofs given by Meg in parts Two and Three: both proofs go wrong at the point when Meg symbolizes the word "God" as "g". In the type of symbolic logic that most students study in college, it is assumed that all names must refer. So, to symbolize the word "God" as a name is to assume that it has a reference, and thus that God exists. So, the proofs also beg the question of God's existence, just as the informal argument did; otherwise, they are formally valid. In other words, in standard logic anything that has been given a name can be proven to exist. Thus, in translating English sentences that talk about nonexistent things―such as Santa Claus―or fictional things―such as Meg or Jack!―or about controversial things―such as God―it is a mistake to translate such words as logical names.

Now, there are logical languages that allow for names that do not refer to anything, but they are more complicated than the standard logic usually taught in university classes, which is why Meg and Jack were not familiar with them. Of course, natural languages, such as English, do allow for names that fail to refer, but standard logic does not parallel natural languages in this respect.

Source: The informal argument is a reconstruction of an argument that I heard in high school from one of my teachers―yikes!―and it subsequently occurred to me that it could be formalized in standard logic in ways that might fool a novice. Usually, I prefer examples to be documented, which is why I presented these arguments in the form of a fictional conversation.

Fallacy: Begging the Question

Solution to a Fourth Puzzling List: Assuming that all of the sentences on the list are either true or false, the first n sentences will be true and the last n will be false.

This puzzle is a generalization of the previous puzzling list (see the Resources, above): as we saw in the previous puzzle, if any sentence on the list is true then all previous sentences will also be true. That is, if the sentence "at least n of the statements on this list are false" is true, then "at least m of the statements on this list are false" will also be true, for any m<n. In other words, if at least two sentences are false, then at least one is false; if at least three sentences are false, then at least two are false; if at least four sentences are false, then at least three are false, etc.

Similarly, if any sentence on the list is false, all later sentences on the list will also be false. Suppose that the mth sentence on the list is false, that is, "at least m of the statements on this list are false" is false. Then, it's not the case that at least m of the sentences on the list are false, which means that less than m sentences on the list are false. However, if less than m are false, then less than m+1 are false, less than m+2 are false, etc. In other words, if less than 2 sentences on the list are false, then less than 3 are false; if less than 3 are false, then less than 4 are false; if less than 4 are false, then less than 5 are false, etc.

Now, consider the nth sentence on the list: since we assumed that all of the sentences on the list are true or false, sentence n is true or false. So, there are two possibilities to consider:

Sentence n is false: Then, less than n sentences on the list are false. However, n itself and all later sentences on the list are false, which means that more than n sentences on the list are false, which is impossible. Therefore, sentence n must be true.
Sentence n is true: Then, all prior sentences on the list are also true. Moreover, there must be at least n false sentences on the list, since that's what n says. So, n+1 through 2n are all false.

Notice that, since we didn't specify what number n is, this holds true for any finite list of similar sentences with an even number of members: the first half of any such list will be true and the second half false.

Source: Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments (1986). The puzzle was suggested by one on page 70.