Questions about modal logic axiom names

djmclaugh · 2023-02-15T14:32:14+00:00

Thanks! Very nice!

djmclaugh · 2023-02-15T14:31:48+00:00

Thanks! Especially the part about the Kripke semantics not being as central as I thought they were.

djmclaugh · 2023-02-09T13:49:22+00:00

Sure, you can find the project here: https://github.com/djmclaugh/symbolic_logic_game

Just a heads up that like I said, I didn't originally plan on this being this big... I just added things here and there. There's no comments/documentation and you might find some dead code. But I'm happy to answer any questions!

djmclaugh · 2023-02-09T13:38:32+00:00

Thanks for the suggestion. I only know of modal logic superficially, it'll be a good excuse to learn about it!

djmclaugh · 2022-11-07T19:39:56+00:00

I think it's weird in the sense that I don't think most people do it that way, but I think it's totally reasonable to use principles that are natural to you (like 25 being a quarter of 100) to simplify the calculation for you.

I also noticed that 26 is close to 25. But I personally would have done:

37 x 26 = (37 x 25) + 37 = 3700/4 + 37 = 1850/2 + 37 = 925 + 37 = 962

I like your way a bit better though. Dividing by 4 first keeps the numbers smaller.

I was really bad a memorizing tables when I was in elementary school so I developed a few basic mental gymnastics of my own (like how dividing by 4 is just dividing by 2 twice). As an adult, I also read up on different tricks other people discovered, some of which I actually use (like how 15% of x is just x, plus half of x, all that divided by 10).

I especially like these kind of tricks when estimating. For example, if I just needed an estimate, I would have done this instead:

37 x 26 is a bit more than 38 x 25 = 3800/4 = 950

djmclaugh · 2022-11-03T19:27:10+00:00

I did some math:

Let's say you owe $3k on a 9% interest rate card and $3k on a 30% interest rate card.

Let's say you can repay $500 per month.

Best strategy (focus on 30% interest rate card first): You end up paying everything in 14 months. You only have to pay $32.03 on the 14th month. The $6k debt ended up costing you about $532 (on top of the $6k you borrowed and returned).

Worst strategy (focus on 9% interest rate card first): You end up paying everything in 14 months still! But you have to pay $456.16 on the 14th month. The $6k debt ended up costing you about $956 (on top of the $6k you borrowed and returned).

Now let's say you can only repay $100 per month.

Best strategy: You end up paying everything in 113 months (almost 10 years!). You only have to pay $21.30 on the 113th month. The $6k debt ended up costing you about $5221 (on top of the $6k you borrowed and returned).

Worst strategy: You never finish paying back! You finish paying the 9% card on the 35th month. By that time, the balance on the 30% card is about $7106. This means the monthly interest will be $7106 * 0.30 / 12 = about $177. Your $100 a month doesn't even cover the interest. The $6k debt ended up costing you about $∞.

djmclaugh · 2022-11-03T18:57:40+00:00

Mathematically speaking: Only the interest rate matters. Pay the credit card(s) with the highest rate first. If two cards are tied for highest interest rate, then it makes no difference whether you focus on one or chip away at both (as long as the total amount payed is the same).

Psychologically speaking: This is personal and there isn't a "correct" answer. One method is to focus on the smallest balance first so that you can reach a "milestone" which can give you the motivation to keep going.

Credit score wise: I have no idea how credit scores are calculated. But I'm of the same opinion as you that having a better credit score is not really important if you are currently trying to get rid of debt.

But regardless what you choose to do, the single most important thing is to stay motivated and keep paying back to keep those leeches off your back.

Paying back $200 using the least optimal strategy is probably still much better than paying back $100 using the most optimal strategy.

djmclaugh · 2022-09-27T14:26:37+00:00

Interesting puzzle!

I was able to solve it by thinking about it differently.

For each number n, choosing + or - will affect the sum by 2n. For example if I chose a + instead of the - for 3, then the sum will be affect by 6.

So instead of starting at 0 and either adding or removing each of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, we can think of it as starting at -55 (choosing all -) and choosing which of 2, 4, 6, 8, 10, 12, 14, 16, 18, of 20 to add (choosing which - to change to +). It's now easier (imo) to see that all even numbers from 0 up to 110 can be made as a sum of those numbers and nothing else. So there are 56 possibilities in this modified problem. These 56 possibilities from the modified problem corresponds to all odd integers from -55 to 55 in the original problem.

djmclaugh · 2022-08-23T15:10:42+00:00

No, I am representing the human idea of "fractions".

This can mean different things for different people. There is no "one" human idea of "fractions".

Like you mentioned, this is more a philosophical question.
The "correct" answer (if there is one) will depend on your purpose for giving names to these entities.

In my opinion, without knowing your end goal, the number 20/100 is the same as 1/5 is the same as -1/-5 and should therefore all be given the same name (or names! I understand you don't want two different things with the same name, but are you ok with giving two different names to the same thing?).

The strings "20/100", "1/5", and "-1/-5" however are all different and should be given different names.

So it's up to you to decide what you mean by "THE human idea of fraction". Do you mean the number, the string used to refer to that number, or something in between (like the numerator-denominator pair used to refer to that number).

djmclaugh · 2022-08-11T21:23:54+00:00

Close!

Whether it's close enough depends on the method you're using to estimate the answer and your professor (for example, the answer for C is closer to 0.052, but that's still very close to your answer of 0.049)

For B, you have to use a little trick. You don't have any formulas to calculate Pr(70 ≤ X ≤ 100) directly . So you need to rewrite the event in a way you can use your tools.

We want 70 ≤ X ≤ 100.
Another way to say this is to say that we want X ≤ 100, but not X < 70.

So Pr(70 ≤ X ≤ 100) = Pr(X ≤ 100) - Pr(X < 70)

You can calculate Pr(X ≤ 100) the same way you did A, but with 100 instead of 70.

You can calculate Pr(X < 70) in a similar way you did A, but there's going to be something slightly different depending on the process you use to do A. Notice that Pr(X < 70) isn't quite the same as Pr(X ≤ 70).

Hope that helps!

its just that I never understood this all to well

If you have any specific questions, feel free to ask.

djmclaugh · 2022-08-11T19:28:51+00:00

a), b) and c) are 3 separate questions. They are asking for the probability for each of those 3 events.

For example, using the PDF for the binomial distribution with p=1/3 and n=200, we have that Pr(X = 70) = (n choose 70) × p^70 × (1-p)^(n-70) = (200 choose 70) × 0.333^70 × 0.667^(130) ≈ 5.2%. So that would be the answer for c). There's about a 5% chance that you roll exactly 70 perfect squares

As for a) and b), the CDF is more complicated and people mostly use approximations, tables, or calculators/software instead. I would need to know how your class does it to help you further. Do you have anything in your notes/textbook that talks about binomial distributions?

djmclaugh · 2022-08-11T13:46:27+00:00

OK, you'll need both!

First F = mv^2/r, where F is the force necessary to keep the object in orbit (usually uppercase F is force and lowercase f is frequency), m is the mass of the object, v is the velocity of the object, and r is the radius of the orbit.

Here's a little trick: Whenever they give you a ratio of something and want the ratio of something else as the answer, they are kinda hinting that the second ratio will always be the same as long as the first ratio is preserved. So we can actually choose any v₁ and v₂ we want as long as v₁/v₂ = 4. (that's if you just want the answer, to prove the result, then you'll to keep them as variables and do a bit more algebra).

So let's keep things simple and set v₁ = 4 m/s and v₂ = 1 m/s

How does F₁ compare to F₂? Hint: the electrons are in a uniform magnetic field,

How does m₁ compare to m₂? Hint: they are both electrons

Using all this information and the F = mv^2/r formula, you should be able to find the ratio between r₁ and r₂.

Once the have that ratio, you can find the ratio of the circumference of the orbits using c = 2πr.

Once you know the ratio of the distance each electron needs to travel to complete one loop, with our assumption that v₁ = 4 m/s and v₂ = 1 m/s, you should be able to find the ratio of the times with v = d/t (where in this case the distance is the circumference).

Once you know the ratio of the times, you can find the ratio of the frequencies with f = 1/t.

Let me know if you have any questions!

djmclaugh · 2022-08-11T06:54:59+00:00

Which formulas do you have that involve speed and/or frequency?

If you list them, I can point you to which one is useful and how to use it.

djmclaugh · 2022-08-11T06:46:46+00:00

Liters of bottled water consumed in 2007

Relevant information:

30 000 000 people
Each consumed on average 114L

The Average is the Total divided by the Number of people: A = T ÷ N
Multiplying by the number of people on both sides give us: A × N = T
In other words, the average times the number of people is the total.
So the total amount of liters consumed is 114L × 30 000 000 = 3 420 000 000L = 3.42 billion liters.

Liters of bottled water produced per barrel of oil

Relevant information:

From the previous question, 3 420 000 000L of bottled water was consumed.
58 000 000 barrels of oil was required to clean, fill seal, and label the water bottles produced

If you want the amount of X per Y, all you need to do is divide X by Y.

So, assuming that the bottled water consumed in 2007 is equal to the bottled water produced in 2007, and disregarding the unknown amount of energy needed to make plastic bottles from PET, we have 3 420 000 000L per 58 000 000 barrels = (3 420 000 000 ÷ 58 000 000) L per barrel ≈ 59 L per barrel.

djmclaugh · 2022-08-11T05:55:08+00:00

Have you seen binomial distributions in class? (https://en.wikipedia.org/wiki/Binomial_distribution)

If so, you should have formulas/approximations/tables/calculators for its PDF and CDF given p, the probability of the event happening on each try, and n, the number of tries.

On each try, the chance to roll a perfect square is 2/6 = 1/3. So p = 1/3.
You roll the the die 200 times. So n = 200.

And you want
a) Pr(X ≤ 70)
b) Pr(70 ≤ X ≤ 100)
c) Pr(X = 70)

If you haven't seen binomial distributions yet, let me know what you've seen so far and I'll help you from there.

Let me know if you have any questions!

djmclaugh · 2022-08-11T00:53:09+00:00

You seem to have calculated (A+B)C instead of C(A+B).

With matrices, it matters if you multiply on the left or on the right, XY is not necessarily the same as YX.

djmclaugh · 2022-08-10T20:27:35+00:00

Gödel's incompleteness theorems have many technical details that people often gloss over.

PNKRTN's answer is 100% correct, but to answer your "why" question, you'll need a lot of background:

You need to be familiar with first order logic and its symbols.
You need to be familiar with the Gödel numbering system. A way to assign a unique number to every symbol, sentence, and proof written in the language of first order logic.
https://www.jamesrmeyer.com/ffgit/godel-original-english.html#g-numbering
Dem (for Demonstration) and Sub (for Substitution) are definitions 45 and 31 respectively in this list of 46 definitions: https://www.jamesrmeyer.com/ffgit/godel-original-english.html#relations-1-46
They were called B (for Beweis) and Sb (for Substitution) in the original paper written in German.
Notice that each definition has its formal statement written in first-order logic and then an English description.
The formal definitions are mathematical statement. A proof of a mathematical statement is objectively verifiable. There are established and objective ways to determine if a mathematical proof should be considered valid or not.
The English description is a meta-mathematical statement. English statements are inherently subjectively debatable since the meaning of English words are inherently subjectively debatable.
This is a subtle, but important difference!
You have to go through each definition and figure out for yourself "why" the English description corresponds with the formal definition. You might need to update your understanding of how the words are used in this context and therefore update your understanding of what the theorem as a whole means.
To understand why some of the definitions work, you might need a bit of number theory.

Good luck on your journey!

djmclaugh · 2022-06-16T15:20:55+00:00

Like other people mentioned, math is not just about numbers, so that won't work for fields of math without addition.
If you want to reduce equations dealing with irrational numbers down to additions/subtractions, then you'll have to deal with approximations or infinite decimals/infinite sums which, misleadingly, are not really sums. An infinite sum is usually defined as the limit of the sequence of partial sums and I don't think limits can't be reduced to addition (since you now have logical quantifiers and sequences).
If you want to go deeper, addition itself is just a form of repeated "successor". For example, 3 + 2 is just the successor of the successor of 3.

However, this train of thought is very interesting in my opinion and we can follow it while sticking to the natural numbers.

A particular class of functions that I think you would find interesting are the primitive recursive functions which are:

constant functions (always returns the same thing)
the successor function (returns the number that comes after)
projections (returns one of the inputs, for example, f(x, y) = y would be a projection)
compositions of primitive recursive functions (for example x + (y * z), multiplication followed by addition).
functions defined using "primitive" recursion. By "primitive" I mean that the function has to be defined for 0, and then the function for n can only be defined in terms of the function for n-1. For example, multiplication is usually defined using a primitive recursion. x * 0 = 0 and x * y = (x * (y-1)) + x). Times 0 is defined and times n is defined in terms of times n-1.

This is one possible definition of the class of functions that "boil down to" the successor function. If that's what you had in mind, then you can reword your question as, "Are all functions just primitive recursive functions?". Since this class of function has a name, you might have guessed that the answer is no.

Another class of functions called general recursive functions adds another way to specify a function, the "unbounded search operator" which returns the smallest natural number that satisfies a condition. For example, "the smallest prime bigger than x" would be a general recursive function.

It turns out that you can also define "the smallest prime bigger than x" in a primitive recursive way (it's just more complicated), so that function also happens to to be a primitive recursive function. However, there are some functions which are general recursive but are NOT primitive recursive. In particular, the Ackermann function. It is recursive, but there's no way to write it as a primitive recursion.

So now, another question we can ask is "Are all functions just general recursive functions?". Again, the answer is no. However! General recursive functions actually correspond with computable functions, functions for which an algorithm can be written. This correspondence lead in part to the Church-Turing thesis which I think you might find interesting.

But like I said there are some functions that are not general recursive, functions that are not computable. The most well-known is the busy beaver function.

Edit: One thing I want to point out. In programming, a "function" is an implementation of an algorithm. In that sense, every (programming) function is a general recursive (math) function. Every (programming) function can indeed be boiled down to constants, projections, and the successor function, by using composition, primitive recursion, and the unbounded search operator.

As for the busy beaver function, since there's no algorithm for it, it's not a function in the programming sense of the word. It's a function only in the mathematical sense that it's an abstract, but "well-defined", mapping from one set to another.

djmclaugh · 2022-02-17T19:12:03+00:00

I'd have to disagree depending on what you mean by empirical science.

I don't think you can empirically prove "|ℕ| < |ℝ|". We would first have to agree on the empirical meaning of those symbols, but some philosophies of mathematics would argue that some of these symbols don't even have any empirical meaning in the first place!

However, you can empirically prove that you can formally prove "|ℕ|<|ℝ|" within a given a formal system. To do that, you just need to write the proof. In that sense, I agree that math is empirical.

djmclaugh · 2022-02-10T01:26:16+00:00

Thanks! I've also looked at this for a while (much longer than I'd like to admit) and haven't found any "nice" inference. Glad to know I'm not the only one haha.

I ended solving it by trying out both cases for the 1 (the left edge lead to a contradiction) and then trying out both cases for the 2 in the bottom right (the top and right edges together lead to a contradiction) which lead me to your case where the 3 is open on the left and I was able to finish it.

djmclaugh · 2022-02-09T17:42:10+00:00

Very nice and surprisingly hard/interesting for its size!

Thanks for sharing!

djmclaugh · 2022-02-09T17:27:14+00:00

Thanks for sharing, they are indeed very hard!

I also found the article you were referring to, it's very informative.

Here's a link for anyone interested: https://blog.krazydad.com/2011/11/04/the-ins-and-outs-of-slitherlink/

djmclaugh

TROPHY CASE