How to Read Logic

Symbolic logic looks intimidating, combining familiar symbols like equality and inclusion with lesser-known backwards E’s and upside down A’s. But with a bit of guidance, anyone can understand the meaning of these symbols and interpret logical statements.

Check out my series on building numbers from the ground up:

TIMESTAMPS

00:00 - Intro
03:07 - Or, And, Not
06:28 - Implication
16:39 - Quantifiers
26:26 - Outro

INVESTIGATORS

ftfftttftf is not the slug you are looking for.

CORRECTIONS

*Propositions vs predicates: So that I didn’t overwhelm the viewer I stuck to just using “proposition” throughout. I know this isn’t strictly correct as many of the statements involving variables are actually prediates.

**For some reason when recording I had it in my head that ‘n’ was a British thing when it is widely used throughout the Anglosphere and beyond.

***Slip of the tongue that kind of undermines my point — the converse of Legrange’s Theorem would be “H is a subset with cardiality dividing |G| ⇒ H is a subgroup of G."

****Another slip of the tongue that undermines the point — we are showing that whenever x is NOT zero, it has a reciprocal y=1/x.

CREDITS

All music by Danjel Zambo.

[NStripleseven]

Mathematicians really like their flipped/rotated letters. Upside-down V, rotated/flipped L, flipped A, flipped E

[skylardeslypere]

The way my teacher explained the implication and its truth table is as follows.

Suppose I say "If I win the lottery, I will buy you a house!"
Logically, this is saying P => Q where P is "I win the lottery" and Q is "I buy you a house".

Now think about the following: in which cases are you satisfied?

When P is true and Q is true, then I kept my promise and you're happy. So T => T is T
When P is true but Q is false, then I broke my promise. I won the lottery, but didn't buy you a house. You're angry, sad, dissapointed. So T => F is F
When P is false, I haven't really made any promises. I never said what I'd do if I did NOT win the lottery. So, if I still buy you a house, you're definitely going to be happy, but even if I don't, you won't be mad because I didn't win the lottery. Hence, F=>T and F=>F are both T.

[ShevkoMore]

When I was learning this, the hard part for me to digest was that implication is a logical operator. For a while I was thinking about it like "this symbol is used as equals AND an operation?!".
Nowadays, whenever I try using any complex logic with my friends I always say something like "We'll assume it's true, because it doesn't matter if it isn't", because it feels like that mindset is what "non-math" people struggle with, and many "math" people take for granted

[pandabeargogo]

Having studied Maths at uni, I saw this thumbnail and thought 'YEAH?? OBVIOUSLY??' Then I actually watched the video and it's a really good video explaining the basics. Nice!

[titou]

I loved the joke about "THERE EXISTS" as someone did the same for the factorial in my class some days ago 😂
More seriously, I really enjoy your videos, they're very recognizable because of their graphic identity and the music behind, and your way to show examples to be very clear, to stick little good-looking papers and to write on a black board, it's very pleasant! I particularly loved your series about foundations of numbers, but this video about logic was very good as well and I appreciated it!
Continue like that!

[thinker]

Having never studied this kind of math yet having years of experience in the field of programming, it's incredibly interesting seeing how a lot of concepts in both fields are equatable.

[zenithparsec]

Be careful when translating natural languages into logic: people often can be tricky.
"Everybody loves somebody" seems to have an obvious meaning. But it could either mean "There exists one person that every person loves" or "Every person has (at least) one person that they love."

(I find it fun to deliberately misinterpret ambiguous sentences.)

[IronFairy]

Hey! You're tricking me into studying maths by making interesting and well explained videos! Not cool!

please keep making them thank you

[GreatCollapsingHrung]

Great video! I seriously went from seeing an incomprehensible mess to thinking “well yeah, obviously”. I’m a fan of math, but never felt I had enough talent to go get an advanced degree in it. But your videos make these esoteric sounding ideas easy to grasp. I would love it if you covered Gödel’s incompleteness theorem at some point. Love your work!

[AnotherCroydonWanabi]

Where was this guy when I was doing my maths degree?!? Really clearly explained

[gabington]

I'm about 6 minutes into this video and I can't unsee the similarities(atleast so far) with logical math, and programming operators. I think I might be able to understand this.

[NonTwinBrothers]

Unrelated but I always watch youtube videos with auto-generated captions on, and I'm continually impressed at how far its evolved.
Especially at 2:22

[wiggles]

I'll just add this because it's something that really clarified the existential and universal quantifiers for me: the existential quantifier creates a giant OR statement, and the universal quantifier creates a giant AND statement. For example, let the universe of discourse be {0, 1, 2, 3, 4, 5}. Then:

For all x: (x > 3) <=> 0>3 and 1>3 and 2>3 and 3>3 and 4>3 and 5>3 <=> false.

There exists x: (x > 3) <=> 0>3 or 1>3 or 2>3 or 3>3 or 4>3 or 5>3 <=> true.

[beldraith]

Never understood those weird math symbols but this video really helped.
Also as a Software Developer I can find many similarites in the language of maths and code.

[Fun_maths]

That "Is it a boy or a girl? Yes" joke reminded me how in the national math team training camps we do this joke all the time, some asks for example "Wait so is lunch now or do we have a lecture?" and someone else responds "Yes", man that does not get old

[bennettpalmer]

I think your reciprocal proposition is actually a really strong argument for why implication is the way that it is.
For all real numbers x, (if x is not 0, then there exists a real number y such that xy=1)

We really want the implication to be true for all x for our universal quantifier, but there is a value of x where the first statement of the implication is false. The thing we're trying to prove doesn't really care what happens when x=0, but we still need the implication as a whole to be true for all x, including 0, the thing we were trying to exclude. So, we just define that case to be true no matter what, because it means we don't have to worry about it breaking our quantifier.

[thedarkspeedninjashadittsux]

I was always curious about logic notation. Now I won't have to be haunted with pages and pages of unknown symbols when I choose to study this subject. Very good video!

[Noh_Mercy]

Wow I finally get the implication part. Looking at 11:06,
If P is a circle inside Q in this 'space of all possibilities'
then you can point you finger at any point on that space and say:
Point x is inside P and Q
Point x is outside of P but inside of Q
Point x is outside of P and outside of Q
But you cant point to a space that is inside of P and outside of Q. Thats why that is F, its an impossible state.
In the case of "greg is a cat->greg is a mammal" the only contradiction is where greg is a cat and not a mammal. No other scenario is contradictictory.

[dojelnotmyrealname]

A major factor in the confusion of the or statement is the implied use in natural language of "or" as "exclusively or". With the drinks example, I don't know if I'd be happy being served two hot drinks. Those kinda have a time limit.

[videogamefan]

22:28 another way to prove that one false would be, since x is an element of the real numbers, and y covers all the real numbers, y also covers x, and x is never less than itself