Reasoning

## Deductive

Given that "all men are mortal" and "Socrates is a man" then an example of deduction is "Socrates is a mortal".

Given that "if *A* is true then by the laws of cause-and-effect *B*, *C*, and *D* will necesarily be true", an example of deduction would be "*A* is true therefore we can deduce that *B*, *C*, and *D* are true".

## Inductive

From Wikipedia:Inductive reasoning:

Given that "if *A* is true then by the laws of cause-and-effect *B*, *C*, and *D* will necesarily be true", an example of induction would be "*B*, *C*, and *D* are observed to be true therefore *A* may be true". *A* is a reasonable explanation for *B*, *C*, and *D* being true.

For example:

A large enough asteroid impact would create a very large crater and cause a severe impact winter that could drive the non-avian dinosaurs to extinction.
We observe that there is a very large crater in the Gulf of Mexico dating to very near the time of the extinction of the non-avian dinosaurs
Therefore it is possible that this impact could explain why the non-avian dinosaurs became extinct.
Note however that this is not necessarily the case. Other events also coincide with the extinction of the non-avian dinosaurs. For example, the Deccan Traps in India.

Attempting to use inductive reasoning in a situation where the laws of cause and effect are not understood results in an error. (However, Bayesian probability may still be applicable)

A classical example of an *incorrect* inductive argument was presented by John Vickers:

All of the swans we have seen are white.
Therefore, we *know* that all swans are white.
The correct conclusion would be: we expect that the next swan we see will be white.

## Fuzzy logic

If I pull a rondom coin out of my pocket and flip it what is the probability it will land heads? It is tempting to say 50/50 but the truth is that the probability is unknown. The coin might have 2 heads or 1 head or 0 heads.

So what do you use when the probability is unknown? You use Bayesian probability. The Bayesian probability that the coin will land heads is 50/50. But if I flip it 10 times and get heads each time then the Bayesian probability will change. Bayes' theorem shows how it will change after each flip.

Given that "I think all men are mortal" and "I think that Socrates is a man" then an example of fuzzy logic is "I think that Socrates is a mortal".

From Wikipedia:Bayesian inference:

Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1?

Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. Let

H

1

{\displaystyle H\_{1}}

 correspond to bowl #1, and

H

2

{\displaystyle H\_{2}}

 to bowl #2.
It is given that the bowls are identical from Fred's point of view, thus

P
(

H

1

# )
P
(

H

2

)

{\displaystyle P(H\_{1})=P(H\_{2})}

, and the two must add up to 1, so both are equal to 0.5.
The event

E

{\displaystyle E}

 is the observation of a plain cookie. From the contents of the bowls, we know that

P
(
E
∣

H

1

# )
30

/

# 40
0.75

{\displaystyle P(E\mid H\_{1})=30/40=0.75}

 and

P
(
E
∣

H

2

# )
20

/

# 40
0.5.

{\displaystyle P(E\mid H\_{2})=20/40=0.5.}

 Bayes' formula then yields

P
(

H

1

∣
E
)

=

P
(
E
∣

H

1

)

P
(

H

1

)

P
(
E
∣

H

1

)

P
(

H

1

)

P
(
E
∣

H

2

)

P
(

H

2

)

=

0.75
×
0.5

0.75
×
0.5
0.5
×
0.5

=
0.6

{\displaystyle {\begin{aligned}P(H\_{1}\mid E)&={\frac {P(E\mid H\_{1})\,P(H\_{1})}{P(E\mid H\_{1})\,P(H\_{1})\;+\;P(E\mid H\_{2})\,P(H\_{2})}}\\\\\ &={\frac {0.75\times 0.5}{0.75\times 0.5+0.5\times 0.5}}\\\\\ &=0.6\end{aligned}}}

Before we observed the cookie, the probability we assigned for Fred having chosen bowl #1 was the prior probability,

P
(

H

1

)

{\displaystyle P(H\_{1})}

, which was 0.5. After observing the cookie, we must revise the probability to

P
(

H

1

∣
E
)

{\displaystyle P(H\_{1}\mid E)}

, which is 0.6.

## Base rate fallacy

From Wikipedia:Base rate fallacy:

In a city of 1 million inhabitants let there be 100 terrorists and 999,900 non-terrorists. To simplify the example, it is assumed that all people present in the city are inhabitants. Thus, the base rate probability of a randomly selected inhabitant of the city being a terrorist is 0.0001, and the base rate probability of that same inhabitant being a non-terrorist is 0.9999. In an attempt to catch the terrorists, the city installs an alarm system with a surveillance camera and automatic facial recognition software.

The software has two failure rates of 1%:

* The false negative rate: If the camera scans a terrorist, a bell will ring 99% of the time, and it will fail to ring 1% of the time.
* The false positive rate: If the camera scans a non-terrorist, a bell will not ring 99% of the time, but it will ring 1% of the time.

Suppose now that an inhabitant triggers the alarm. What is the chance that the person is a terrorist? In other words, what is P(T | B), the probability that a terrorist has been detected given the ringing of the bell? Someone making the 'base rate fallacy' would infer that there is a 99% chance that the detected person is a terrorist. Although the inference seems to make sense, it is actually bad reasoning, and a calculation below will show that the chances he/she is a terrorist are actually near 1%, not near 99%.

The fallacy arises from confusing the natures of two different failure rates. The 'number of non-bells per 100 terrorists' and the 'number of non-terrorists per 100 bells' are unrelated quantities. One does not necessarily equal the other, and they don't even have to be almost equal. To show this, consider what happens if an identical alarm system were set up in a second city with no terrorists at all. As in the first city, the alarm sounds for 1 out of every 100 non-terrorist inhabitants detected, but unlike in the first city, the alarm never sounds for a terrorist. Therefore, 100% of all occasions of the alarm sounding are for non-terrorists, but a false negative rate cannot even be calculated. The 'number of non-terrorists per 100 bells' in that city is 100, yet P(T | B) = 0%. There is zero chance that a terrorist has been detected given the ringing of the bell.

Imagine that the first city's entire population of one million people pass in front of the camera. About 99 of the 100 terrorists will trigger the alarm—and so will about 9,999 of the 999,900 non-terrorists. Therefore, about 10,098 people will trigger the alarm, among which about 99 will be terrorists. So, the probability that a person triggering the alarm actually is a terrorist, is only about 99 in 10,098, which is less than 1%, and very, very far below our initial guess of 99%.

The base rate fallacy is so misleading in this example because there are many more non-terrorists than terrorists, and the number of false positives (non-terrorists scanned as terrorists) is so much larger than the true positives (the real number of terrorists).

## Always know

If I pull a coin out of my pocket and flip it what are the odds it will come up heads?

Since there are only 2 possibilities (heads and tails) then the bayesian probability is exactly 50%. But thats not the same as the probability.

The probability is unknown because you dont know how many heads it has.

* If its a normal coin then the odds are 50%.
* If its two headed then the probability is 100%.
* If its two tailed then the probability is 0%.

If you dont know the certainty then you can still know the probability.

If you dont know the probability then you can still know the Bayesian probability.

You can always know.

You can know that you know.

You can know that you think.

But you should never under any circumstances "think that you know" (i.e. believe).

## Assumptions

The brain makes many hidden assumptions that we dont even realize it is making.