Requiem aeternam

dona eis, Domine, et lux perpetua luceat eis. Animae eorum et omnium animae fidelium defunctorum per misericordiam Dei requiescant in pace.

Name that work

The objects in space and time, on the contrary, exist in the phenomena. As is proven in the ontological manuals, it must not be supposed that, in reference to ends, space can thereby determine in its totality time. As is evident upon close examination, the phenomena are the clue to the discovery of, however, our faculties. By means of analytic unity, we can deduce that, so regarded, the objects in space and time have nothing to do with the transcendental unity of apperception, yet the paralogisms constitute the whole content of, for these reasons, our a priori knowledge. Because of the relation between the thing in itself and the paralogisms of natural reason, it is not at all certain that the transcendental unity of apperception (and it is obvious that this is true) stands in need of the never-ending regress in the series of empirical conditions; on the other hand, the Antinomies (and there can be no doubt that this is the case) are the clue to the discovery of our ideas. Let us apply this to the Ideal of pure reason.

This discussion cracks me up

What about the system that takes as an axiom that there are self-evident truths? And why, oh why, don’t they teach them philosophy in these schools? They could start with Aristotle’s Organon.

Biased use of the word “bias”

Psychologists have discovered that humans in general – and children in particular – exhibit three innate biases:

• Essentialist bias: all natural kinds have and immutable essence
• Teleological bias: what they see must be purposeful and goal-orientated
• Intentionality bias: actions and outcomes must be the work of an intentional agent

These biases are actually useful for children to make predictions in the world and are defaults that adults revert to at times.

 

Source.

Busy

Next to no free time around here, especially for the next couple of weeks. Prayers for a special intention would be appreciated. I need the intercession of people holier than myself; random blog readers qualify.

Mathematical puzzle

Last post on probability for now. This one doesn’t involve propositions or questions of ignorance vs. knowledge, just straightforward mathematics. The point is to illustrate how counterintuitive conditional probabilities can be.

Suppose in a given population there are an equal number of males and females. This population is subject to a disease, which I’ll call Bayesitis. 1 out of 10,000 males has Bayesitis, while 1 out of 12,500 females has it. There is a test for Bayesitis. If a person has the disease, the test returns a positive result 98% of the time. If a person does not have the disease, the test returns a negative result 99% of the time. The test results are indifferent to the sex of the person. UPDATE: It can be assumed that the probabilities work exactly; that is, the size of the population is a common multiple of 10,000 and 12,500, and exactly 1 out of 10,000 males and 1 out of 12,500 females has the disease.

A member of the population is selected at random and tested for Bayesitis. The result is positive.

1) What is the probability that the person has the disease?

2) What is the probability that the person is male?

Answers should be given to at least five decimal places.

Propositional evidence part III: Applying the lesson learned

This post is a thought experiment. As is evident from the number of comments on the first Bayes’s theorem post below, it has sparked quite a debate. I’d like to step into my opponent’s shoes for a moment and think about this debate from the point of view that I’m seeing presented.

<tongue in cheek>

Suppose I give a demonstration of a certain fact. Then suppose some person unknown to me comes along and asserts that the fact is false. What am I to make of this?

First, I consider my own judgment of the probability of the truth of my demonstration. I judge it to be at least as certain as, say, the argument by which Rutherford proved the existence of the proton, and additionally strongly confirmed by personal experience, which I don’t have regarding the existence of the proton.

Second, I think about this in relation to someone’s contradiction of the position. Clearly, a person’s contradiction of such a highly probable fact significantly raises the probability that the person for whatever reason is prone to making assertions opposite what is actually the case. Ultimately, I judge this to be the most likely hypothesis.

Third, I reason as follows: “A person who I have reason to believe has a probability greater than 0.5 of making assertions opposite what is actually the case has just asserted not-X. Clearly, this increases the probability of X.” Consequently, I judge that this assertion actually confirms my original belief, so I am now more sure than ever that it is true.

Further, this hypothesis allows me to make predictions about the person’s future behavior, namely making similar assertions about related issues. When what actually obtains turns out to be exactly what my hypothesis predicts, my hypothesis is confirmed even more strongly, along with my original position. Interestingly, there is nothing the person can do about this—all further arguments against my position simply serve to further confirm it.

</tongue in cheek>

 

Yes, that was tongue in cheek. However, it is precisely the sort of reasoning that the opposing position must ultimately lead to. In fact, it is worse, since it implies that any time anyone disagrees with me, I should view their position as increasing the probability of my own view rather than theirs. For even if I judge the person to be more probably correct than me on the general subject in question, I cannot judge them to be more probably correct with regard to the exact proposition being disputed; since if I thought this I could not disagree about it.

 

<tongue in cheek>

Now, I predict responses to this telling me why I’m wrong, that the opposing position does not imply these consequences. This will, of course, further confirm my hypothesis, causing me to become more certain of the truth of my position.

</tongue in cheek>

Propositional evidence part II: Raspberries

In the combox of the last post, we have an interesting example in which it seems that some random person’s assertion in fact constitutes zero evidence. I’d like to analyze that example in more detail.

Here’s the scenario. You’re sitting in front of your computer. Some random guy instant messages you, saying, “You’re eating raspberries.” Does this claim constitute evidence that you are, in fact, eating raspberries? It may intuitively appear that it does not.

Consider the possible scenarios:

A. The assertion that you are eating raspberries (I’ll call this the IM) is uninfluenced by the state of your eating.

B. The IM is influenced by the state of your eating.

Let H be the hypothesis that you are eating raspberries. Bayes’ theorem gives us:

P(A)+P(B)=1

R (the evidential ratio, P(IM|H)/P(IM|~H) ) = [(P(IM)*P(A))+(P(IM|H)*P(B))] / [(P(IM)*P(A))+(P(IM|~H)*P(B))]

The formula breaks down like this because on possibility A, P(IM|H) = P(IM); that is, if we assume that the IM is uninfluenced by the state of your eating, then it is equally likely regardless of that state.

Scenario A obtains with high probability, say 0.90. The exact value doesn’t matter, it can be as high as we want as long as it is not 1, which would imply that it is absolutely impossible that the sender of the IM could have any knowledge of whether you’re eating—obviously there is some chance of this, however small it may be.

Notice, though, that the probability of A is irrelevant, because it occurs in both the numerator and the denominator of the fraction, and the only thing that matters is the ratio. Thus, in order for the IM to not affect our judgment of P(H), it is necessary that given scenario B

P(IM|H)=P(IM|~H)

That is, the sender must be as likely to send the IM knowing that you are eating raspberries as he is likely to send the IM knowing that you are not eating raspberries. But to send an IM asserting that you are eating raspberries, knowing that you are not, is lying. Thus, claiming that the IM does not increase P(H) is equivalent to asserting that the sender is as likely to lie about what he believes as tell the truth about what he believes. Clearly this is hardly ever the case, and certainly would be irrational to assume of an anonymous person. Therefore, the IM increases the probability that you are eating raspberries.

A pre-emptive response to objections: “lie about what he believes” is not the same as “be in error about whether you are eating raspberries.” If you think it is, you didn’t understand the argument.

Second pre-emptive response: If you want to object that the assertion can be influenced by the state in some other way than the sender knowing the state, you need to describe the precise mode of influence.

Postscript: I haven’t assigned numbers, but in reality such an IM would constitute quite strong evidence, not very slight evidence. This is because of the improbability of someone sending such a message in the first place.

Bayes’ theorem and propositional evidence

Suppose I want to evaluate the probability, given the state of my knowledge, of a certain proposition X. I say that learning the additional fact that some random guy entirely unknown to me, John Smith, asserts that X is true should increase my assessment of the probability that X is true.

Reductio: We know from Bayes’ Theorem that the probability of a hypothesis, given certain evidence, equals the probability of the evidence given the hypothesis, times the probability of the hypothesis simply, divided by the probability of the evidence simply. In other words,

1. P(h | e) = ( P(e | h) x P(h) ) / P(e)

2. Let the hypothesis be that X is true, and the evidence that John Smith says that X is true. Let us further assume that Smith’s assertion does not change the probability of the hypothesis. In other words,

3. P(h | e) = P(h)

4. From 1 & 3, P(h) = ( P(e | h) x P(h) ) / P(e)

5. Dividing both sides by P(h), we get

6. 1 = P(e | h) / P(e)

7. So, P(e) = P(e | h). That is, the probability of the evidence equals the probability of the evidence given the hypothesis.

9. From 2 & 7, the probability of John Smith saying that X is true equals the probability of Smith saying that X is true given that X is true. In other words, Smith is equally likely to assert X whether or not X is true.

10. Since we know nothing about Smith, we must judge him evidentially as a random person, and the same about the proposition since its content was unstated.

11. Thus, on average, a given random person has the same probability of asserting a given random fact regardless of whether it is true or false.

12. Now, with regard to observable facts, no one holds (11). For instance, no one would say that the probability of a random person’s asserting “the sky is blue” is independent of the sky’s actually being blue.

13. So in order for (11) to be true on average, the class of unobservable facts must have a negative correlation between truth and assertion. First, this is highly dubious in itself. Second, it would not be sufficient even if true, simply because most things people say are about observable facts.

14. Therefore, the second assumption in (2) is false.

This can also be shown from a related probability formula. The probability of A given B equals the probability of both A and B, divided by the probability of B. That is,

P(A | B) = P(A & B) / P (B)

But if P(A | B) = P(A), then

P(A) x P(B) = P(A & B)

But the probability of two events both occurring equals the product of the two independent probabilities when there is no causal relationship between the two, for instance two separate coin tosses. So this would imply that there is no causal relationship between the truth of things and what people say about them, which again is impossible. Thus, it is necessary to say that the mere assertion of a fact constitutes evidence for the truth of that fact.

One last argument: there are many things that we hold simply on the basis of someone’s assertion. We would not do this unless we thought that the assertion was evidence for its own truth.

Self-referential multiple choice quiz

The solution is unique, and can be found by progressive elimination of impossible answer choices (or by trial and error). Post the solution in the combox.

 

1) The number of questions whose answer is a vowel is:

a) 4

b) 1

c) 3

d) 5

e) 2

 

 

2) The answer to question 5 is:

a) B

b) C

c) D

d) E

e) A

 

 

3) The most number of times the same answer occurs on this test is:

a) 2

b) 1

c) 4

d) 3

e) None of the above

 

 

4) The first question whose answer is B is:

a) 3

b) 4

c) 5

d) 1

e) 2

 

 

5) The only adjacent pairs of questions whose answers are also adjacent in the alphabet are:

a) 1 & 2

b) 1 & 2 and 3 & 4

c) 4 & 5

d) 2 & 3 and 3 & 4

e) 3 & 4