Understanding Thought part 2

Idea: Truth Projection

Let us start wth an example:

1. You have 2 apples and 3 pears
2. Someone asks you "How many apples do you have?"
3. You are only allowed to give a number in your answer.
4. What will you respond?

My answer here would be "3 apples" I think. Why?

Let us first agree on some things. If the rule in 3 wasn't there, you'd say "I have only 2 apples, but I do have 3 pears also". Perhaps someone is hungry, baking an apple cake, making apple juice or something else. Here it would be interesting to know about the pears.

The other thing we should easily agree on is that some answers are plain wrong: If you answer "1" or "0" that is (for all possible intents) wrong. If you answer "6 apples", "7 apples", or more, that is also wrong (for all possible intents of the question).

So why "3 apples"? I can picture a range of different scenarios (intents of the question). If the person asking only could eat apples and was hungry (think Death Note), then you would be slightly wrong. The same goes if the one posing the question was in the store and could easily have bought 1 more apple.

What if the one who asked wanted to make apple juice, and he thinks that pears are good too. Well, then 2 apples and 3 pears would make a lot more apple/pear juice than would 2 apples (150% more). On the other hand, 2 apples and 3 pears would make a bit more juice than 3 apples (67%).

In day-to-day life, this doesn't happen with apples and pears, as it is simple to give all the information. But you can never give all possible information, so we try to say what's relevant, and to not say stuff we think is irrelevant

This phenomenon appears clearly in the teaching of mathematics and physics.

Mathematics example: We normally teach that all (positive) functions can be integrated. Some teachers tell the students that there are some (positive) functions that cannot be integrated, but you would never teach what kind of functions these are, so in essence you say "Any function you will ever encounter is integrable". Then when you study some more you learn that there are lots of functions that cannot be integrated. Then you learn that these can actually be integrated, we just have to expand our method of integration a bit. Then you learn that even with this new kind of integration there are some super-strange functions that cannot be integrated.

Physics example: When you learn about electrons on a small scale (quantum mechanics), you learn that they work the way waves on the water works. You can have one big "up" wave meeting a big "down" wards wave, and cancel out. When a wave hits a small opening, it expands out after this opening. But electrons do not work as waves on the ocean; they can be collapsed, they have more directions than just "up" and "down" (they have a kind of imaginary "left" and "right" also).

Often the whole point of being pedagogical is to be able to give the best possible Projection of what you considers to be True, down onto the 'plane' of the listeners understanding.

Understanding Thought part 1

- Preliminaries

There's nothing like reading a book every day for two weeks, so I'm a bit behind. I have a dozen half-finished blog posts that, for some reason, never seems to get past the initial stage into something half-finished (i.e. ready to post).

This series/collection I will start on now is a rethinking of how I blog. I will take one big issue and go with it for several months. I would very much like to be able to start with the beginning and just give a clear and concise presentation, but this is impossible as I don't know this stuff. What I will do is present my finding when researching the research and ideas on the subject.

My goal is to post at least once in each calendar month. The purpose is to understand how we think, how the processes of thinking works, what we do differently, and the difference between intuitive and analytic processes.

Before we start this adventure we need to agree on some things and some definitions of words.

I read an Internet-poll today that asked "Do you believe in God" with alternatives "Yes" and "No". I could write a book about how limited this model of yes/no is, but let's instead look at what I will use as definitions (with minimum time in parenthesis):

A Thought – A stray thought, random combination of ideas or words (2 seconds)

An Idea – An insight that you think may be fruitful to investigate (10 minutes)

Theory – An understanding of the world that you consider to be important and true, and from which you can draw conclusions about actions and results (descriptive and/or normative) (1 hour)

Active Belief – Something you, personally, use to decide what you do, which actions to take. A theory where you follow the conclusions.

I understand some natural progress between these four: Though -> Idea -> Theory -> Active Belief.

So let me try to describe the differences here. You are not allowed to believe in a though or an idea, they are independent objects of study. You are allowed to 'have faith in an idea', think it is a very good idea, when working with it, but it is not something ready for a true/false discussion yet. You can believe/unbelieve in a theory (and, of course, in an active belief). I will use the word unbelieve when you believe the opposite of the statement. This gives us 3 modes: belief, uncertainty, unbelief.

What should the possible answers to "Do you believe in God" be using these words? (After any clarifications you may need; it is, after all, a rather ambiguous question.)

- I am uncertain (I think this was in the original poll)

- In Theory Yes

- In Theory No

- In Active Belief Yes

- In Active Belief No

I am uncertain is typical agnostic. In Theory No is typical 'I don't see any reason to believe it', but someone who is too lazy to withdraw from the state church; someone who does nothing about it. In Active Belief No is a typical atheist, someone who tries to remove the state church, someone who argues that there should be no mandatory religion course in school (or if it is, then it should contain humanism and other big religions equally). These people are certain that there is no God, so why should we spend time on Him/Her/They?

Most people I know who call themselves christian are 'In Theory Yes', they say "we believe", go to church, marries in church, and does everything religious that is considered normal in the culture/society. If I am of this group, and I think abortion is OK (within the set limits), and one night God comes to me in a dream and says "Abortion is wrong" (and I have a religious experience), what will I do? Nothing/Ask for proof of His existence. So this belief is not active, I do not take actions based on it. This group writes God with capital G, and Him with capital H, no because they are afraid to be disrespectful to God, but to other religious people. One of the things that makes this group uncomfortably misunderstood by atheists (Active Belief No) is that they tend to believe strongly things that seem very contradictory. For example, they know that monotheism was invented about 3000-5000 years ago, that there are a thousand different gods (and it is improbable that you should choose the one in your state religion as The God), that the earth is round and goes around the sun, that the exodus from Egypt is more of a folk tale than significant history, that life on earth is a million years old.

Last, but not least, we have the 'In Active Belief Yes'. They can start an argument by "The bible says ..." or "The pope said, ex cathedra, that ...", they can spend resources converting others to the faith, and they can feel bad when their children are not properly religious because (in many religions) they will end up in (some sort of) hell (eternally?).

First we make the decisions, then we make the reasons. I don't know what this is called, but it is a very strong fallacy. One of the modes in which I think is deciding on an option, then writing down any arguments I can come up with. Then deciding on the other alternative, and writing down any arguments for that. Then I try to read the arguments with an open mind. This method is a result of how our brains (at least mine) work – it is a lot easier to come up with arguments after taking a standpoint. I really need a good name for this fallacy; the fact that the reasons you present are not the reasons that made you decide.

Example: Let's say I drive at a speed of 110 on the highway with speed limit 100. Perhaps that is because it is unsafe to drive at a speed of 120? But more commonly it is because you get an expensive speeding-ticket. 

The most important thing here is to stop lying to yourself. Later you can consider telling the truth (surprisingly often this is embarrassing, or you'll come across as very frank). I will do my best to note when I make this fallacy.

Other stuff:

- I will use a numbering for later reference.

- Note that I have allowed room for nonscientific theories with these assumptions. You could make up untestable hypotheses, like "praying only helps the faithful" etc. Even though you can't do a proper double blind, there is usually some way to do a statistical test if you allow for weaker conditions.

- Surprise is a good thing when we search the scientific literature.

Infinity in an Hour

- How it Feels to study Pure Mathematics

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

- William Blake

I have a master in pure mathematics (it also includes a substantial amount of applied mathematics and physics). People often wonder what is it that we do? And they visualize the most complicated mathematics they know, and they try to take it to the next level mentally. But it is very hard to try to understand something you know nothing of. People who only have a minimum of mathematical knowledge guess "Oh, so you multiply really large numbers", those who had math in high school/college guess "Oh, so you differentiate and integrate difficult functions?", those who have a university education where they needed advanced mathematics guess "Oh, so you solve hard differential equations on difficult grids/spaces?". Today I will endeavor to give a picture of how I feel when doing pure math.

For inspiration, I recommend the following videos. After we have seen them, we will try to do some of this mathematics. Note that we mention Cantor several times.

http://video.google.com/videoplay?docid=-8492625684649921614&hl=en

(Part 2 if you're interested)

http://video.google.com/videoplay?docid=-1663091361786740235

Pure mathematics is that of unlimited abstraction and precision. We define everything as precisely as humanly possible, and we try to make everything into abstract concepts. Understanding something in pure mathematics is often like walking a tightrope over an abyss, no room for small deviations or imperfect intuition.

A lot of the problems I think about when working, you would need 4 years of university education just to be able to ask. But now I will use some examples that you need almost no mathematics education to understand.

First we will look at sizes of infinity (this is very close to the paradoxes talked about in the videos above). To talk about size, we have to define it, and size of what?

Definition A set contains elements, and an element is contained in a set.

Example {1,2,3,5} is a set containing the elements 1, 2, 3, and 5.

Example 2 {a, b, car, boat, 99} is a set containing the elements a, b, car, boat and 99.

Definition 3 The size of a set is the number of elements it contains

Example 4 The size of {1,2,3,5} is 4

Example 5 The size of {a, b, car, boat, 99} is 5

Using these definitions, we could have defined a set to be infinite size if it has no finite size. But to be able to speak of different infinity sizes, we have to use a more fine tuned definition.

Definition 6 Two sets are equal if there is a bijection between them

Definition 7 A bijection is a function which maps all the elements of a set to the elements of another set, and where no two elements are mapped to the same.

To understand this last definition, you should see some examples. So look at wikipedias page bijection.

For more information, see function and Bijection-Injection-Surjection.

Now we can state the first question

Question 1 Are there as many even (positive) numbers as there are (positive) numbers in total?

Answer 1 (with proof) Yes. Let E={2, 4, 6, 8, 10, ...} be the set of even numbers, and let N={1, 2, 3, 4, 5, ...} be the set of all numbers. Then we can construct a function, f, from E to N by f(x)=x/2. This function is a bijection. Hence the size of E is the same as the size of N.

How can this be? Clearly the set N contains the set E, and then some, how can they be equal? Well we just proved that they were. The only thing we can conclude form the fact that N contains E is that the size of E is smaller or EQUAL to the size of N.

What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite.

Calculations with infinite

If you use a modern computer program, it often has inf (infinite) as a kind of number. It will normally give the following computations:

inf + 1000 = inf

inf – 1700 = inf

inf + inf = inf

inf*1000 = inf

inf*inf+inf^inf = inf

1/inf = 0

1/0 = inf

inf-inf = NAN

inf/inf = NAN

(-1)^inf = NAN

Here NAN means Not A Number. That is because you are not allowed to do these operations, the result is undefinable (if you chose a definition you would end up with a contradiction). The problem with inf-inf is that we don't know which inf is "largest". In some applications you may get the answer 0, in others 31, in yet others you may get inf.

Countable and Uncountable infinity

The size of the set Z (all finite numers), or the set N (all positive finite numbers), is called countable infinity (the sizes of Z and N are equal). It is easy to prove that the set of all fractions, the rational numbers Q, also has countable size (see http://www.homeschoolmath.net/teaching/rational-numbers-countable.php).

But what about the set R of real numbers (all numbers, including pi and the square root of 2, but not including imaginary numbers), is this also the same size as Z and N and Q? No. This proof is quite deep, and took me several days to understand (several years ago). If you want a challenge, see wikipedia's page on Cantors diagonal argument.

The Length of the Rationals (the set of fractions Q)

There is another very much used notion of size. This is what we use for integration, and to avoid confusing it with the size of sets from before, we call this new thing for length.

Definition The length of an interval on the real line, is the right endpoint minus the left endpoint.

Example We write [-3,7] for all the numbers between -3 and 7 including -3 and 7. The length of this interval is 7-(-3) = 10.

We can generalize this concept of length to other sets than intervals, for example to the union of intervals. Not surprisingly we get:

Theorem If one set is contained in an interval, the length of the set is smaller or equal to the length of the interval.

Supertheorem: The size of Q is countably infinite, but the length of Q is 0 (on the real line).

Proof: We have already seen that the size of Q is countably infinite. What about its length? Well, write Q as a sequence Q={q1, q2, q3, q4, ...} where all qi are fractions. Let K>0 be any arbitrary number (for example 0.000000001). Then the first fraction, q1, is contained in an interval of length K, namely [q1-K/2, q1+K/2]. The second fraction is contained in an interval of half that length (namely [q2-K/4, q2+K/4]). The third fraction q3 is contained in an interval with half that length again. Let us sum up this:

q1 contained in an interval of length K

q2 contained in an interval of length K/2

q3 contained in an interval of length K/4

q4 contained in an interval of length K/8

q5 contained in an interval of length K/16

...

So Q must be contained in the union, which will be an set with size smaller than (smaller because some of the intervals may overlap):

K + K/2 + K/4 + K/8 + K/16 + ... = 2K

How to calculate this? This is what we call a geometric series.

What do we now know? Q is contained in something of length 2K (you can choose any K>0). Hence the length of Q is smaller or equal to 2K (you can choose any K>0). The only possibility is that the length of Q is 0. QED.

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Comment by Nok er Nok:

I'll keep beating the same horse as always, I guess. We have spent some time discussing these questions and generally agree, but I can't help but object to the following paragraph. I have mostly the same knee-jerk reaction to it as the Newcomb's paradox, Hangman's paradox, etcetera. On the whole, I'll even illustrate my point with an XKCD strip (gasp!),  http://xkcd.com/169/   , though I'm not sure if the miscommunication is intentional or not.

"What we just did is known as Hilbert'sparadox of the Grand Hotel, with Infinitely many new guests. You should look at the link. This is NOT a paradox, this is a well established mathematical fact. What seems to be a paradox is only because our intuition is not used to the concept infinite."

This is most certainly a paradox. The mathematical formalism you describe and which mathematicians use is consistent and useful, yes, that's not what I'm trying to deny. Using that formalism to claim there are -as many- even numbers as natural numbers, however, is dishonest; using layman's terms in that manner -creates- the paradox. Nobody protests against the bijection-wizardry which is firmly belonging to alternate math-dimension. They do, however, have issues with mathematicians redifining common English words to mean something entirely different, and then marveling at how counterintuitive it is - particularly when they a few moments later set the trap by mixing real world example, impossible mathematical constructs, normal English and indistinguishable mathematical definitions.


Let's deconstruct Hilbert's Hotel to start with, and let's pretend we are not familiar with the mathematical formalism for dealing with infinities. Hilbert then claims the following:

 - hotel with infinite number of rooms
 - infinite number of guests at the hotel
 - such that all rooms are occupied, each by a single guest
which is all fine and dandy, but sets up for the counterintuitive conclusion
 - the hotel can still house another group of guests, exactly as large as the number which currently occupies every single room

Now, if you look at that without consulting your English<->Mathematics dictionary, you will surely conclude that this is perfect nonsense. Somebody is pulling a cheap parlour trick on you, one of these words have to mean something else that is appears. You can easily construct equally (more?) valid arguments than Hilbert presents, to show that the conclusion is impossible. For instance, it is easy to visualise that no room will be left unoccupied after you swap any two guests between their respective rooms, any number of times, and even though Hilbert does this an infinite number of times, this shouldn't change anything.


The parlour trick here is, of course, that occupied does not mean occupied at all, it has to do with bijections, and infinity does not mean a number you can increment arbitrarily many times, it is instead some mathematical construct with such properties that it cannot possibly have anything to do with any actual hotel. You might say the point of Hilbert's Hotel is that infinity cannot be treated as just any large number, you claim that our intuitions are not prepared to deal with infinity, but I strongly disagree. Hilbert's Hotel only shows that the mathematical lingo he ends up translating to 'occupied' and 'infinity' has nothing to do with a normal understanding of these words.

Taken as a story to accompany the mathematical formalism, to illustrate how it handles infinities, cardinality and size as something to do with bijections, it does an okay job. Without that context, it is not the slightest bit clever or enlightening, but just a load of gibberish. Hilbert's Hotel says -nothing- -whatsoever- about how hotels of arbitrary size work; it intentionally mixes mathematical formalism with a real world example which it then -fails to describe-!


Of course, all of this comes from the same sort of people who with a straight face will call f(x) = constant  an increasing function - and a decreasing one, at the same time. Nevermind that increasing is a code word for non-decreasing, which you cannot know without consulting your Google-translate English<->Mathspeak, or being familiar with the tradition of inclusive definitions in mathematics.

I think inclusive definitions are useful, and I'm not quite decided on whether using somewhat familiar but inaccurate and misleading terms is better than inventing new, arbitrary ones. However, I'm certainly not going to give mathematicians any credit for clever paradoxes which does nothing but illustrate that the mathematicians themselevs do not understand that their redefined words cannot be inserted into common English prose without appropriate and careful translation.


As an endnote, I feel fairly certain that it would be very possible to develop a formalism in which the natural numbers, the even numbers, the prime numbers and so on and so forth were -not- the same size. Of course, these alternate defintions would not develop fruitfully into integration and cardinality, like the current one does, but this alternate mathematics would be able to present the exact same Hilbert Hotel and the exact opposite conclusion; the hotel -cannot- accomodate even one more guest, much less another infinity of them. Or perhaps they would balk the moment you suggested that every one of the infinite number of rooms is currently occupied. And if this is true, it should be all the more obvious why Hilbert's Hotel is, indeed, a paradox of sorts.  

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Morality from Evolution

-Prisoners dilemma, tit for tat, and cheating.

Today I want to look at how the (Darwinian) Theory of Evolution can model morality from purely egoistic assumptions.

Let us quickly recap the assumptions of Darwin:

1. There are more offspring than can survive (there would be exponential growth if everyone survived) (survive means: have offspring, not survive forever).
2. There are differences between offspring.
3. These differences are heritable (if your father is taller than average, we expect you to be – this only needs to be statistically true)

The conclusion then follows:

Those traits we see after many generations are the traits that is well suited to survive the environment (survive still means: have offspring). Note also that you cannot develop a complicated trait without there being several stages of beneficial traits leading up to this complicated trait.

In morality we have a principle known as "tit for tat". Tit for tat is: "Do unto another as another has done unto you". If Sam helps you, then the next time you help Sam. If Sam steals from you, then you steal from Sam.

We will discuss how this principle (tit for tat) can follow from the Theory of Evolution and purely egoistic assumptions. First let us define our environment/setup/problem, what is known as the iterated prisoner's dilemma.

The prisoner's dilemma has its name for a reason, but for reasons of clarity of exposition I will present it differently. It goes as follows. You and Joe find some food in the jungle (think about our ancestral environment). You and Joe get two options: Fight or Cooperate. You can choose different actions. Results are as follows:

1. Both cooperate: Both get 3 (units of) food
2. Both fight: Both get 1 food
3. A fights while B cooperates: A gets 4 food, B gets 0 food.

Note how this is supposed to model how food is ruined (the prey escapes etc) when someone fights (in case 2 we loose 4 food, in case 3 we loose 2 food). We assume that the choice you and Joe picks are independent (you cannot wait and see what he chooses).

How is the prisoner's dilemma soled by two egoistical entities? If you and Joe are egoistic you will always choose to fight. Does this make any sense when both could have more food if both of you cooperated? Yes, as the choice is independent. No matter which choice Joe picks, it's always better for you to fight than cooperate. In that way outcome 2) is recognized as a stable Nash equilibrium (anyone seen the film "A beautiful mind"?).

Before we go on to iterate this problem, I want to take a small digression. How can we model the solution of a single prisoner's dilemma? Well, we can have utilitarianism (everyone's welfare is important) instead of egoism, but that was not our assumptions. One common solution is "kin selection". If Joe is your brother, then his survival will bring your genes (statistically about half of them) down the line. So his survival is half as important as yours: He getting 4 food has value 2 to your genes, but you getting 1 food each has value 1.5. This makes cooperating always better for your genes. (We just assumed probability of survival is linearly dependent on amount of food.) But what about your friend, they are not of your blood, can you still have kin selection? Before modern transportation it was highly likely that your children would have kids with the children of one of your friends some day. This might (with a slightly different setting, and some more assumptions) make cooperating with your friends a good strategy. From now on we assume that the two '

'prisoners' are complete strangers with nothing in common.

What its iterated prisoner's dilemma? Well, you go hunting with Joe every week for a few years. Now, every week you make a new choice, independent of Joe, whether you should fight or cooperate. But now you are allowed to remember everything that has happened up to this point. Now you are allowed to choose "Because Joe did X last time, I will do X now", which we called the tit for tat strategy. NB: you always start by cooperating. What is special about this strategy? It is essentially unbeatable. Let us disregard small deviations (like tit for tat with forgiveness or extra randomness). What does unbeatable mean? If you have a population of 100 people, each with their own strategy; some of them with the tit for tat strategy, some of them with completely different strategies then no one will get more food than you if everyone plays the iterated prisoners dilemma against everyone else (and you are using tit for tat). (This also relies on the assumption that the world is not dominated by a big number of tit for tat haters whose strategy is discovering the 'tit for tat strategy users' and killing them.)

Now the Theory of Evolution concludes that after many generations, everyone will have something close to the tit for tat strategy. Yes, this is dependent on even more assumptions. Maybe we should model this on the computer one of these days?

Positive Morality

Are we fundamentally cooperative or egoistic?

It would be easy to argue that everything that everyone does is based on pure egoism. I disagree with that viewpoint, but let me present it.

If Eve does something that benefit her and nobody else (or has a negative impact on others), we call the action egoistic. If Eve is kind to a friend in a selfless way, it is because she wants something in return. The notion of reciprocity. If she needs help at some later time, this friend will help her out (she assumes). It can actually be proven that reciprocity is the best strategy (tit for tat) in a 'game' supposed to model real life. Let us come up with a situation where no reciprocity is expected.

If you find a drunk man lying in the gutters, and help him get a taxi home, would you expect reciprocity? Let us assume no. But then your genes are. Through millennia of evolution your genes have found the perfect way for you to behave; group-orientation, giving, reciprocity, caring, emotions, family values, etc. But in the end all your behaviours are designed for one purpose only: You (or your genes at any rate).

Let us return to my viewpoint.

Survival of the fittest. In accordance with Darwin's theory of evolution, the strongest (those most fit to survive the world and bring offspring) survive (or they genes do anyway). This is easy to believe (note how a lot of people that are pro-Darwinistic consider the theory of gravity to be just a theory, but the theory of Evolution to be given as an Axiom of The World).

Few people would argue against that kindness and selflessness are important concepts that we use to model our world. So in everyday life these models are true in some sense. Some may argue that the more fundamental model of Evolution is more true (since it is in a sense more fine-grained; this is the typical Reductionist view), but I'd say we have another way to choose what to consider 'the most true'.

What you believe changes who you are. If you disagree, for example if you do not believe in free will and purpose, reading this has no value anyway (you're just doing it as a consequence of random or deterministic happenstance). The question is: Do you want to live in an egoistic world, where you can always just interpret everything as egoism? Or do you want to live in a kind world, and learn time and again that the world is not so kind? Or do you want to live somewhere in between, thinking the best of people, that is, the best that your experience permits?

Question anyone: If I give you a present (and conscious reciprocity is not in the picture), is there any experiment that would give a different result whether it was fundamentally an egoistic or a selfless act?

Sometimes the purpose can be one thing, and the important sub goal something else. This is how I view free will vs. Darwinism. Free will is the goal, but survivability is an important bi-product.

The reason I blogged about this today was essentially

http://www.youtube.com/watch?v=GcJxRqTs5nk

To sum up:

Return favours: Conscious egoism,

Empathy: Unconscious egoism, or something else?

Is giving away something with absolutely no future gain a "bad" Darwinian side effect of the powers of empathy, or is it a sign of our "true" nature?

Is this really a question for science or a question of some other kind?

Solving Conundrums 3

- Solutions to Solving Conundrums 1 (and 2)

The Second Conundrum

"Which way would you tell me to go if I were to ask?" (Then take that way)

If it's the one who always lie, he would tell you to go the wrong way, but as he is lying he has to lie to the question by telling you the right way.

The Fourth Conundrum

The solutions to this depends on what you mean by surprised. In practical terms, what the death-sentenced king does is to believe with all his mind that he will be killed tomorrow (he believes this every day). Tuesday morning he is not killed, but does he consider himself to be surprised?

There are three ways to define surprised (that I can think of). The first is "I am surprised whenever I am wrong". Using this interpretation, where is he wrong in his analysis? He must decide in advance on one day when he thinks he will be executed. If he decides 'I think I'll be executed on Tuesday', then he is surprised when he is executed on Friday (or, rather, when he is not executed on Tuesday).

The second interpretation is "I am surprised whenever I have a false negative, that is, when I predict that I will not be executed but I am." So if you have a false positive (you think you'll be executed but you're not) you don't count as surprised. Then the king is correct, his analysis is good, it is impossible to surprise him. (This goes against the way I told the story, but there exist different versions.)

The third interpretation is a bit more complex. If it rains/not rains tomorrow, are you surprised? Not too much, because you don't have a strong prediction. But if a volcano erupts in your neighborhood, then you are surprised. If someone asked you beforehand you'd say that the chance of a volcano erupting so near you was almost 0, or just say 0. The interpretation is: "You are surprised whenever something occurs that you assigned less than 10% probability" (I chose 10% for convenience, you can substitute it with any number less than 50%). Every night the king can assign 50% probability to being executed the next morning. Then he is never surprised.

The Third Conundrum

When making statements in logic you can make almost any statement you can dream up. Almost. You are not allowed to define a logical variable by 'P:(not P)'. The easiest way to make sure your statements are definable is to only use logical variables that you have already defined, together with logical operations (and, or, not, etc.).

Let us define the logical variables:

P: not Q

Q: not P

This is impossible in the same way as

P: not Q

is impossible (you can't know if P is true or false, it can't be true or false, as you have not even defined Q).

If you'd rather think of this as a satisfiability question (see the last solution on this post) for 'P equivalent (not P)', then the answer is "no, there is no truth values satisfying this expression".

The First Conundrum

Using the clarified version:

If A) is correct then D) is also correct, and then A) is not correct as there are two correct solutions.

If B) is correct, then B) is false by its own statement.

If C) is correct, then C) is false by its own statement.

If D) is correct then A) is also correct and then D) is not correct.

So every answer gives a contradiction, hence none of them are correct. But isn't then alternative C) correct, since no of the alternatives are correct?

The resolution here comes from thinking about satisfiability. If you have a logical system you can give a number of variables A, B, C, ..., and a logical 'equation' (statement), for example

(A and B) or (C and (not A) and (not D) or (A equivalent B),

and ask the question "Is there a set of values for the variables that make the equation true?". The answer can be

"yes; A=true, B=false, ...", or

"no; there is no such choice of values".

Finally, the answer becomes "The question you posed has no correct answer". On this higher level, where you defined the question, you can say that there is no answer that can be correct (as all of them leads to contradictions).

The question cannot be satisfied. It is not so that 0 of the alternatives are correct on the level of the question. On the level above, it is so.

If you disagree, I would recommend a book on mathematical logic, and one on set theory (not naive set theory, but the serious kind), at least read the wikipedia articles

http://en.wikipedia.org/wiki/Satisfiability

http://en.wikipedia.org/wiki/Propositional_satisfiability

And to see how complicated stuff becomes when someone tells you everything, see

http://en.wikipedia.org/wiki/Russels_paradox

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

Solving Conundrums Part 2

- Hints (including clarifications) to Solving Conundrums Part 1

The First Conundrum

Restatement of the problem:

How many of the alternatives in this question is/are correct?

A) 1 of the alternatives

B) 2 of the alternatives

C) 0 of the alternatives

D) 1 of the alternatives

The Second Conundrum

This solution is supposed to only use a question referring twice to the brother who is alive. This one is connected to the third conundrum.

The Third Conundrum

This question is related to the post on Limitations of Logic

The Fourth Conundrum

The wikipedia-page on this (unexpected hanging paradox) is the worst I have seen. It is a lot simpler than that. Start with defining surprised.

If you think something (H) will happen (or not), there are four possible outcomes with respect to your information:

You said H would happen, and it does -> you're right

You said H would not happen, and it does not -> you're right

You said H would happen, but it does not happen -> False Positive

You said H would not happen, but it does happen -> False Negative

Are you surprised when you do a false negative, or are you surprised whenever you do either a false negative or a false positive?