[Repost] What is Code? by Paul Ford

I was recently assigned to read this entire article/essay/magazine in a class of mine and was blown away by how comprehensive it is. If you guys are just beginning to dip your toes into IT or software development, then this is a must-read!

To read the article:


Symbolic Logic: 10 Replacement Rules

This is part two of our little intro to Symbolic Logic. We’re going to expand our repertoire of rules we can employ in our proofs. These rules are all about putting logic statements into an alternative form. A lot of these rules will be familiar as they’re used in mathematics. One of the differences between these rules and the basic 8 is that these are reversible, hence the :: symbol to denote a two-way operation. The format of this post is going to be similar to the last one; the rules will be listed first, then some simple examples and then a couple of practice problems on the following pages.

1. Double Negation (D.N)

p :: ~~p

2. Commutation (Comm.)

p V q :: q V p

p • q :: q • p

3. Association (Assoc.)

[(p V (q V r)] :: [(p V q) V r)]

*applies to AND operators the same way*

4. Duplication (Dup.)

p :: p V p

5. DeMorgan’s Law (DeM.)

~(p V q) :: ~p • ~q

This law describes how a negation gets distributed into a parenthesised statement. It negates the two variables and switches the operator from an AND to an OR or vice versa. It only works on AND and OR operators though so if you have a (BI)CONDITIONAL operator inside the parenthesis, you’ll need to use one of the later replacement rules to make them into one.

6. Biconditional Exchange (B.E.)

(p ≡ q) :: [(p ⊃ q) • (q ⊃ p)]

7. Contraposition (Contra.)

(p ⊃ q) :: (~q ⊃ ~p)

8. Conditional Exchange (C.E.)

p ⊃ q :: (~p V q)

9. Exportation (Exp.)

[(p • q) ⊃ r] :: [(p ⊃ (q ⊃ r)]

10. Distribution (Dist.)

[p • (q V r)] :: (p • q) V (p • r)

[p V (q • r)] :: (p V q) • (p V r)


Alright y’all, on to the practice problems.

Symbolic Logic: 8 Basic Inference Rules

Hi y’all! So, if you’re computer science majors/philosophy major/etc., you probably have to take this class in college. I love this stuff because it’s very procedural and the proofs they give for you to solve are like puzzles and puzzles are super fun. Today, we’re gonna look at the 8 basic rules and then we’ll look at the replacement rules and more. I’m going to assume that y’all know the basic structure of sentential logic including operators and truth tables. Let’s get started.


A proof is a procedure which is supposed to derive the desired conclusion from a set of premises. To do this, the proof has to be set up in a certain way. First, all lines of a proof must be numbered. The premises make up the first lines of the proof along with the desired conclusion. Then, all subsequent derivations from the premises are listed below with the justification for each step listed along the right side, noting which rule was used and what lines of the proof were referenced. Here’s an example:

  1. Premise Pr. (for premise)
  2. Premise Pr. /:. Conclusion
  3. Derivation (Name of Rule) 1, 2 (lines 1 & 2 referenced)

For our purposes on this page, the visualisations for each of the rules below will not be written in this vertical fashion as they are cumbersome to format in the WordPress editor so it’ll be horizontal.

Let’s get into the rules and then work on some examples which will be on page 2.

1. Simplification (Simp.)

p • q /:. p


p • q /:. q

If there is a conjunction, then both conjuncts can be individually represented as being true by themselves.

2. Conjunction (Conj.)

p; q /:. p • q

If two variables are true, then they can be joined in a conjunction.

3. Addition (Add.)

p /:. p V q

This rule is incredibly powerful as it allows you to introduce new elements into a disjunction as long as we have one of its disjuncts as true.

4. Disjunctive Syllogism (D.S.)

p V q; ~q /:.p


p V q; ~p /:. q

If one of the disjuncts is stated to be false, then the remaining disjunct is true.

5. Modus Tollens (M.T)

p q; ~q /:. ~p

If the consequent of a conditional is false, then its antecedent is also false.

6. Modus Ponens (M.P.)

p q; p /:. q

If the antecedent of a conditional is true, then the consequent is also true.

7. Hypothetical Syllogism (H.S.)

p q; q r /:. p r

If the antecedent of a conditional leads to an antecedent of another conditional, then you can infer that the first antecedent leads to the consequent of the second conditional.

8. Rule of Dilemma/ Constructive Dilemma (C.D)

p V q; p s; q r /:. s V r

If either of the antecedents of two conditionals is true, then either of their consequents must also be true.

Alright, off to examples! We’re gonna start off easy and build onto harder and longer proofs.

Alright, now on to real problems; I have included two. I’ll give you guys the premises and the desired conclusion and I’ll post answers on how to derive the conclusion on the page after that. On to the next page!

[Repost] Don’t Feed the Trolls, and Other Hideous Lies

As the internet population grows and the influence of the internet over people grows as a result, the internet becomes an increasingly accessible tool to spread one’s views and attitudes. Trolls in recent years have received increasing coverage as their numbers grew and their tactics more malicious. Discussions on how to combat them have popped up, out of which, the phrase “Don’t feed the trolls” came from.  But, how does this strategy actually work out and how can these social parasites be cut from their host?

A Twitter follower reminded me of a line in the famous parable from Bion of Borysthenes: “Boys throw stones at frogs in fun, but the frogs do not die in fun, but in earnest.” Defenders of trolling insist it’s all just a joke, but if trolling is inherently designed to get a rise out of someone, then that’s what it really is. In many cases, it is designed to look and feel indistinguishable from a genuine attack. Whether you believe what you are saying or not is often immaterial because the impact is the same — and you are responsible for it, regardless of how funny you think it is. It is a lesson kids learn time and time again on the playground, and yet, it is ridiculously difficult for people to accept the same basic notion in online culture, no matter their age. Why is that so? Because those are the social norms that develop when you create a culture where everything is supposed to be a joke.

For the whole article, click here.

AP Lang Essay Prep: How Your Essay Should Look

Welcome y’all.

So, as you all should know, there are three essays that you need to write for AP Lang.

  1. Synthesis
  2. Argumentative
  3. Rhetorical Analysis



If you’ve taken an AP history class, then you are in for an easy ride.

Synthesis is exactly like the DBQ except a whole lot easier because you’re only required to use three documents instead of the usual six. So your layout would look something similar to the DBQ, which is something like this:

  1. Intro (+2 points for contextualization)
  2. Thesis (+1 point for thesis/argument)
  3. Body
    1. Introduce/cite document (ex: [Doc A]) after summarizing
    2. HIPPO* (+1 point for HIPPO; required 4 out of 6 documents for full credit)
    3. Relate back to how document proves thesis (+1 point for cohesive argument)
    4. Repeat for however many docs you have
  4. Outside Knowledge (+1 point for knowledge not mentioned in text helping argument)
  5. Conclusion
    1. Restate thesis
    2. Restate each body paragraph into one sentence each

Except your synthesis should look something like this: (Keep in mind that there is no rubric for this so your essay is based on your rhetoric and organization more than a DBQ would be.)

  1. Intro
  2. Thesis
  3. Body
    1. Introduce/cite document
    2. Analyze and explain (kinda like HIPPO)
    3. Relate back to how document proves thesis
    4. Repeat for only 3 documents!!
  4.  Conclusion
    1. Restate thesis
    2. Restate each body paragraph into one sentence each
    3. End conclusion with broad and deep message that resonates throughout the audience

* Historical context, Intended audience, Purpose, Point of view, Outside information



Again, for argumentative essay, it’s exactly like the LEQ for AP US. You’re legit pulling knowledge out of your butt to argue for a claim. This is similar to synthesis except for the fact that you don’t have documents to facilitate your claim. This can be good or bad. Good because you’re not required to waste time on finding the documents that help your argument. Bad because you have to come up with your own evidence. There’s less structure for an argumentative essay, but the basic outline goes like this:

  1. Intro
  2. Thesis
  3. Body
    1. Clearly state the purpose of this body paragraph
    2. Analyze evidence you have provided
    3. Relate back to how evidence proves thesis
    4. Repeat for however much evidence you want to include, but remember the time limit!
  4. Conclusion
    1. Restate thesis
    2. Restate each body paragraph into one sentence each
    3. End conclusion with broad and deep message that resonates throughout the audience

Because an argumentative essay prompt can go in infinite directions based off of the evidence that you provide, this is a little vague, but an argumentative essay can easily be the most difficult one or a breeze depending on the prompt and how you want to structure your essay.


Rhetorical Analysis

This one is, in my opinion, the hardest one because I don’t have experience with this as I do with the other two in different classes. I ONLY FOUND OUT MUCH LATER THAT THIS IS YOUR USUAL SAT ESSAY. Done SAT before? Then no problem. Haven’t done SAT yet? You need this. My suggestion is that you should separate the analysis into beginning, middle, and end. What does the author do at the beginning of the passage to be persuasive? How does the author sound convincing in the middle of the passage? In what way does the author conclude his/her piece that’ll make the audience see his/her perspective? I usually outline my essay like this:

  1. Intro (SOAPS* it: introduce the document that you are analyzing)
  2. Thesis
  3. Body (analyze the most important persuasive tactics used—ask yourself, does that negligible metaphor used to persuade me more than the loaded words used to elicit some sort of feeling from me? Remember, you are on a time limit. Prioritize which rhetoric you want to go in-depth with.)
    1. Beginning
    2. Middle
    3. End
  4. Conclusion
    1. Restate thesis
    2. Restate each body paragraph into one sentence each
    3. End conclusion with broad and deep message that resonates throughout the audience

You can draw multiple parallels between many of the essays that I mentioned. Here are some pointers to always, always, ALWAYS remember when writing your essay.

  • You want to be flowery with your language a little. Show off. Throw a few words like “juxtaposition” or “plethora” or “effervescent” here and there. Utilize that vocabulary.
  • But don’t be too verbose. Do not overstep your boundaries. While you want to sound sophisticated, you also want to be succinct. Passive tone, being verbs, bland adjectives—all unnecessary. (NO “to be able to”!!)
  • Power of three. Reread the point above. “Passive tone, being verbs, bland adjectives”, it all just sounds so right. When listing items, don’t list two or four. Listing three gives a sort of unknown power that makes your writing sound… correct. There’s some sort of science behind this, I just don’t think it’s been figured out yet. This is an actual thing, though. Search it up.
  • Scaling along with being too wordy, try to avoid using the words “logos”, “pathos”, and “ethos” in your essay. While you are trying to show how an author uses these three points to persuade their audience, you can easily say how the author does so without explicitly saying it. Your grader will know what you’re talking about. Spend time analyzing what the author is doing rather than summarizing what the author is doing using these three terms. Show, not tell.
  • While these points seem counterintuitive and contradictory to each other, more practice will allow you to find that happy medium that will achieve all these points in no time! You just have to believe!!

*Speaker, Occasion, Audience, Purpose, Subject

Thanks for taking your time to read my lecture to you. Good luck to you all on the exam!

[Repost] The Essence of Calculus | A Series

Hello! I’m aware that I may be a little late since midterms are already over but I have “refound” a channel that I had previously discovered that was a big help to me when it comes to understanding what calculus is, how certain theories and formulas we’re taught in class came to be and what practical problems those formulas and theories were made to solve. This is what your calculus teacher didn’t teach you.

The channel, called 3Blue1Brown, is a channel in the same vein as Numberphile where it tackles mathematical questions but in a very visually stimulating format. For anyone taking calculus (and especially those who’re taking AP Calculus AB/BC), this will help fill in some of the holes in your knowledge on why some things are the way they are.

Different mathematical theories and formulas weren’t invented in a vacuum; each theory and formula was made to solve a problem. So, when learning about math, you shouldn’t learn it in a vacuum either.

Here’s the first episode of a series on the channel, The Essence of Calculus:


Happy learning!

“Are You a Human?” How Captcha and ReCaptcha Works



Captcha stands for Completely Automated Public Turing test to tell Computers and Humans Apart. It is basically what it sounds like it is. It’s a test to see if the user is human or not. It prevents spam and automated data downloading. It uses simple tests that computers can’t solve like the distorted letters test or it may ask you to point out the street signs in a picture. Since both of the tests require the ability to analyse and comprehend contextual patterns,

There’s not much to explain for captchas. Let’s move on.

Here’s the wiki page for this.


reCaptcha is the same thing as Captcha except it has a double purpose of digitising books. There are several ongoing projects that are trying to convert books to digital text. They use a software program to scan the book and convert it to digital text. However, they can’t be quite sure that the program works correctly so they employ the help of humans. They show two words that can’t be read by computer programs. The first word is confirmed to say what it’s supposed to say. The second word is an unconfirmed word. If the person gets the first word right, the system assumes that they’ll get the second word right as well. When enough people have entered the same result for the second word, then that second word then becomes confirmed.