How to Read Logic
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- เผยแพร่เมื่อ 2 ก.ค. 2024
- PATREON: / anotherroof
CHANNEL: / anotherroof
WEBSITE: anotherroof.top
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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:
• Mathematics from the G...
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.
Octopus: www.freepik.com/free-vector/d... Image by brgfx on Freepik
Platypus: www.freepik.com/free-vector/p... Image by brgfx on Freepik
House: www.freepik.com/free-photo/ho... Image by senivpetroon Freepik - บันเทิง
Mathematicians really like their flipped/rotated letters. Upside-down V, rotated/flipped L, flipped A, flipped E
Mathematicians are running out of symbols and must recycle them
Isn't flipped L called Gamma Γ?
and a whole set of fancy superscript subscript fraktur serif sans serif letters that online "font generators" use to their advantage
I don't know about the A and the E, but upside down V is the very real capital Lambda from the greek Alphabet while the upside down L is the capital Gamma from the greek Alphabet. We just like greek symbols for the most part :p
I imagine that this comes from early printing - it’s much easier to rotate a block than carve a new symbol
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.
That's a nice explanation
That's similar to how I explain it to my students. Suppose I claim "If you do your homework, I'll give you an A". In which situation could you claim I lied? Only in the situation where you did your homework but I didn't give you an A.
In constructive logic, "not P" is actually expressed as "P implies falsehood", i.e., "if P, then pigs can fly."
@@markuspfeifer8473 Huh, interesting! If you know that Q is false, then P=>Q is indeed logically equivalent with ¬P
I still think my "tennis on a cylinder" idea fits better.. only one way fails to cross an imaginary line, the one where the two guys are actually playing back and forth normally, instead of around the world (which in all other cases will cross the line)
And T throws rights and F throws left. F left, because it looks more like an L with a beard.
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
The most helpfull comment I came across and it actually helped me finally tackle this topic.
I think the hard part is that thing we don't typically think of as true-or-false statements are so in logic/math, because everything is. Like if you write x=3, you are actually saying "the statement x = 3 is true", except the whole "is true" part is implicit. However, this is fine because setting x to a value is an intuitive enough concept you can just think of it that way. It gets weirder though with things like implications, where we especially don't think of those as true or false. Without a background in math, I feel like a lot of people would be confused if you asked "if x then y, is this true"? Once you get a grasp on the implicit things, it starts to make more sense
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.
All programming logic is inherited from traditional formal logic so there are tons of similarities!
as far as i know, boolean algebra and [0th-order] propositional logic are equivalent
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!
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!
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.
Off topic but ur protogen sona is so damn cute
How have you worked in programming for years and yet not come across prop-logic?
cute sona
@@mbdxgdb2 bro he just said " a lot of concepts in both fields are equatable.". Which mean he might have studied it in programming but not through math.
@@farhanaditya2647 Nah - you’re taught the maths before you’re taught to program if you’ve “studied it”.
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.)
For all person in the set of all humans there exist another person in the set of all humans in which person loves another person.
"Every hour, some person in new york is getting run over in traffic." (What a rough time that somebody has! Getting run down every hour)
That reminds me of the Beatles song "All you need is love". People tend to hear that philosophically as something like "the only thing that any person actually needs in life is to be loved", whereas I believe the song was actually intended to mock consumerism and greed and actually meant something closer to "you already own literally everything material and/or of financial value, and now the only thing you are still lacking is love".
@@Repsack2 The Chuckle Brothers used to end their live shows by asking people to drive carefully, saying...
Paul: On your way home please take care, as statistics show that a man gets knocked down every other night of the season
Barry: Yeah, and he's getting really fed up of it now!
I wonder whether the existential symbol contains the uniqueness symbol
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
Wow, I actually can't believe that!
@@AnotherRoof they've come a long way in 10 years 😄
Hey! You're tricking me into studying maths by making interesting and well explained videos! Not cool!
please keep making them thank you
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!
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.
That's a nice way of thinking about it!
Wait, there are some people that don't think of them like this?
@@quantumgaming9180 I think it was at least a year or multiple years between the time I was introduced to the quantifiers and the time I found out they were equivalent to AND statements or OR statements.
In fact there are alternative symbols for quantifiers: ⋀ and ⋁. In the same way ∏ and Σ mean product and sum over elements in a set, ⋀ and ⋁ mean conjunction and disjunction over all elements in the set.
so does that mean the Unique quantifier makes an XOR statement?
Where was this guy when I was doing my maths degree?!? Really clearly explained
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.
Isn't there already a 'such that' symbol?
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.
When I was in undergrad, all CS majors had to double in something else and most chose math.
@dootie8285 Underrated comment!
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.
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!
Fantastic video! I'm super curious to see where this series goes next!
Thank you very much for this! Stumbled upon this randomly and I always wanted to know yet i never had the ambition to really look it up
These videos are really interesting; I'm curious to see what you do next.
0:10 definition of the multiplicative inverse. Booya.
At the beginning of the video this stuff was just gibberish but after 30 min I could actually read and understand what you showed in the beginning, without you needing to explain it. You're a wonderful teacher.
Absolutely love your videos. I did know most of this, learned through some very basic math courses at uni doing CS, but it's always nice with a refresher and maybe seeing things in a different light. I didn't know the uniqueness symbol, that was quite nice. However, I'm wondering about your use of comma, wouldn't a : be more correct?
Great video and such a good explanation. This will help me to understand some of my subjects on engineering
Great! That's exactly what I'm learning and being tested on right now
Love these videos! Keep 'em comin'!
omg thanks for explaining this in such an understandable way... a lot of these ideas I already kinda knew from functional programming concepts (the "for all" and "there exists" seemed very familiar like the .all() and .any() methods for iterators in Rust) but I had no clue how people describe them in math terms. Especially that "implies" part was sorta tricky, but that circle diagram was pretty helpful.
if statement {
assert!(implication);
}
This is such a beautiful video. Thanks so much for making it. I really appreciate it dude ❤
I was always ... curious about symbolic logic, thanks for clarifying!
I've always wondered how to do this but never looked into it... might as well start here!
Thanks, I've never seen a satisfactory explaination for the truth table of implication before!
"by short I mean there's four videos in the series, it is three and a half hours long, but, you know"
I really like your jokes!
What a wonderful video. Subscribed.
My cousin used to always answer questions like that. “Do you want to watch Frozen or Moana”
“Yes.”
“Would you like water or milk”
“Yes. Yes I would”
Very clear, thanks !
Very good video!
I think that the solution to the exercise at the end of the video is this
1: true (ironically this is the only one i'm unsure about) 2: false (because of 0) 3: false (because of 0) 4: true (because there's 2) 5: false (there isn't a value that works for every y) 6: true (for every x there is that works) 7: true (i think this doesn't need explanation) 8: false (because there are numbers that aren't elements of Q and their sum is an element of Q: π and -π if you sum them you get 0 which is an element of Q 9: true (because R is dense) 10: true (the only value is 0)
Could you pls clarify no.5 a bit more
@@ziadhossamelden9241 there isn’t a value that added to any y equals 0
The proportion says that a value that works with any of the real numbers but there isn’t because for examples for 3 only -3 works such as 3+(-3)=0 but it doesn’t work for -4
I was looking for this comment, I have a question, tho:
1: Does this mean we all think that 0 is even? It follows the pattern, but it's just kind of weird lol.
And I got 8 and 10 wrong. Both were obviously you're answer after thinking about them further. I didn't think about transcendentals for 8, and I didn't think about how the uniqueness of 0 is the special characteristic that makes statement 10 true.
@@kindlinEvery even number "x" is a multiple of 2, which means you can write it as x=2k where k is an integer. 0 is obviously an integer and 2*0=0 => 0 is an even number.
I believe 10 is false. Edit: I was wrong
Great video, you're a really effective communicator!
I do really love your videos, I hope that you upload more videos like this touching math from first principles ❤
This is an amazing basis for logic, solving, and all the frontier forms of programming/gaming
Awesome explanation, thank you!
In language, we tend to use "or" to mean "xor" or "exclusive or". This version is true when one of the inputs is true, but not both. This is why the "yes" to "or" questions play as jokes.
Actually, if I say 'x or y' without context, it's ambiguous. 'either x or y' is xor and 'x and/or y' is 'or'
@@Anonymous-df8it
No, I would have to say, “either x or y, but not both,” not just “either x or y.”
However, we often do say “or” when we mean “xor.”
Do you want steak or chicken? Yes, both, please.
@@bethhentges a) Why wouldn't "either x or y" be sufficient? b) "However, we often do say “or” when we mean “xor.”" Your example question isn't meant to be taken literally; even if you interpret the or as xor, you still don't get the intended meaning (see 'is it a boy or a girl?')
@@Anonymous-df8it
Could be intersex! Both.
So glad to hear I'm not the only one who remembers the AND symbol as the n in fish n' chips (it's also how i remember the difference between union and intersection in set theory)
Definitely gave me a clearer perspective on why implication works the way it does. Thank you!
I'm familiar with basic logic but I watched the video anyway because I really like the way you explain it. I would love to see a follow up on higher order logic.
man I love all these videos so much, can't wait to see what's next in store, I'm super curious
The way you explained this topic was funny 😂 and I liked it
Thank you for making this hard topic look simple and interesting
I always loved logic notation because in a certain way I'm lazy to write in paper, so I answered these on questions on college and the math teachers and myself loved.
Really really great video! Very clear and helpful 😊
0:40, well it might not have been geared for me, but somehow I had never come across that use of “!” to mean “a unique” before, so I learned something! :)
I find the visual representation very helpful!
Dear another roof,
You may discourage me all you want,
in no way possible will that ever stop
me from consuming your carefully
crafted content.
I would read this “for all x in R it is true that there exists a y in R such that x times y is equal to 1”
I have almost finished my CS degree and never understood the implication and this guy managed to explain it to me, what a guy, implication is one of the most important "tools" in math an it made me always feel insecure because i've never understood it correctly, thank you very much
haven't finished the video yet but im trying to apply what i know so far by trying to define the XOR operation:
R XOR Q = (R∨Q)∧￢(R∧Q)
reasoning: in XOR, one of R and Q has to be true (the first term) and they cannot be both true (the second)
watching more, i guess you could also define:
R XOR Q = ￢(R⇔Q)
I dont have this keys on the keyboard so I use ! for not, & for and, | for or as you'l see in programming
the first way is the formal along with (!R&Q) | (R&!Q)
the second way is just a reflaction of the fact that "if and only if" means they are the same which is what the xnor gate checks, and xor is not xnor
but xor also have a diffrent symbol which is a + in a circle
At first,i used to find logic math/logic philosophy hard because i thought that it was for prodigies or geniuses,well basically,i'm good at math,but i was not that good at logic math,i only knew the element of,and the sets,but because of you, I'm starting to love logic math,and it got easier for me,and I'm starting to get hooked up with it,thank you😊.
Weirdly enough, a few days ago, I was thinking in bed "How would I introduce the concepts of Boolean logic to a middle school/high school class?" (I am totally serious) Your video tracked almost precisely with the way I would have laid it out (I didn't go into quantifiers, but I did cover a few things like DeMorgan's Laws, and various alternate notations). otoh, you taught me something I did not know (or at least remember?), the uniqueness quantifier ∃!
Turning the lights on by flipping the wall switch, and then turning them off.
1+1=0
Compare two circuits: one in series, one in parallel. Then put a switch in each circuit.
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.
Thank you so much for teaching
Lovely video! My exposure to logic has been in computer programming, really neat to see the parallels!!
edit: my logic answers
1) If x is in the set of Integers Z, the Odd set holds x or the Even set holds x.
This is True (assuming 0 has parity)
2) If x is in the set of Real Numbers R, the Positive set holds x or the Negative set holds x.
False (assuming 0 is Real and Unsigned)
3) For every value x contained in the set of Real Numbers R, if Positive doesn't hold x then Negative does hold x
False (assuming 0 is Real and Unsigned)
4) There exists some value x in the set of Natural Numbers N where x is prime or x is even
True (This will work for any prime or even number)
5) There exists some value x in the set of Real Numbers R, which for every value y in the set of Real Numbers x+y=0
False (by contradiction: x:4+ y:3 ≠ 0)
6) For every value x contained in the set of Real Numbers R, there exists some value y contained by R in which x+y=0
True (Any number's opposite added to the same number will yield 0)
7) For any two values x,y contained by the set of Real Numbers R, if x ≠ y AND the square of x is the square of y, that x = -y
True (if x and y are not equal but their squares are the same, then the magnitude of x and y must be identical. X^2 = Y^2, X = ±Y)
8) For any two values x,y contained by the set of Real Numbers R, that if x and y are both contained in the set of Determinate Fractions Q, the sum of x and y must also be contained in the set of Determinate Fractions Q (and vice versa)
True (the sum of two fractional numbers will never yield a non-determinate fractional number, and the addends of a fractional number will always be two determinate fractional numbers--otherwise the definition of a detemrinate fractional number breaks)
9) For any two values x,y contained by the set of Real Numbers R, if x < y, then there exists some number z contained by the set of Real Numbers R that falls between x and y.
True (Real Numbers allows non-wholes, and a non-whole number can __always__ be subtracted or added to. An easy way to guarantee this is by picking the minimum place value of x and y together, making it one order of magnitude smaller, and adding a single unit of that place value to x)
10) There exists a unique value x contained in the set of Real Numbers R, that with any value y contained in the set of Real Numbers R, wherein if y > 0, the square of x will be less than y
False (by contradiction: More than one value. x:2^2 < y:5 and x:2^2 < y:6, therefore x:2 is not unique)
Super super fun brain teasers!!!!! My favorite was number 8 and I do hope I'm correct on these. Thanks for a fantastic video.
edit edit: somebody added an irrational number [(π) that can be a determinate fraction] to itself on #8 and proved me wrong. Cheers!!!
For #4, you used the wrong connective; properly it should be "There exists a natural number x such that x is prime and x is even." Which is true, x=2.
For #5, you're correct, but you your proof is not a proof by contradiction, and isn't sufficient to prove the statement true or false, because a claim is being made about a property of all real numbers. A proper proof by contradiction here would be something like y = 1-x, x + (1-x) = 1, 1!= 0.
For #8, even considering your edit, the rationale is wrong. Q is the set of rational numbers; π is not a rational number, nor is π + π, so plugging it in for x or y creates a vacuous statement and doesn't prove anything. A better example would the counterexample x=π, y = 1-π. x+y=1, which is in Q, but neither (x in Q and y in Q) does not hold, so the biconditional is not satisfied and the statement is false.
For #9, you're correct, but I think a better explanation is that it's possible to define z such that the value of z always falls between x and y; the simplest example I can think of is z = (x+y)/2.
For #10, you're misinterpreting the meaning of ∀. "∀y ∈ R" means that the proposition must be true for all values of y, not for any single value of y. The statement is true; x=0 is the unique value whose square is smaller than any positive number.
I think the best way to think of ∀ and ∃ is that in both cases, you must consider every possible value of the variable. For ∀x, the predicate must be true for every possible value of x, but for ∃y, you only have to prove that out of every single possible value of y, the predicate is true for at least one.
I would try to avoid using "any" in phrases like "for any value" because that usage is ambiguous; "for any value" could mean that we should be able to plug any conceivable value in and the statement is true, or it could mean that we want it to be true given at of the set. For example "x+1=2" is true for "any" real number because it's true for 1, but it's not true for "any" real number because it's not true for 2.
Also for #8, another way to prove it false is that if you let (x ∧ y) = 1/2, then x + y = 1 which is not within the set of rational numbers. Likely, if x + y ∈ Q, then it doesn’t mean that (x ∧ y) ∈ Q because you can let x = 1 and y = 1/2 which means x + y = 3/2, and even tho the sum is rational, its components x and y are not. 🙏
@@zaydsalcedo3009 All integers are rational numbers. For example, 2 can be written as 2/1
Amazing channel
Lovely channel and awesome video :)
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
1. True. All integers are either odd or even.
This is a direct consequence of the Theorem of Euclidean Division, which states:
For every pair of integers m,n, there exists a unique pair of integers q,r, with r < n, such that m = qn + r.
In this case, n = 2; therefore, all integers can be expressed as either 2n or 2n + 1.
2. False. There exists a real number, namely 0, such that it is neither positive nor negative. This follows axiomatically from the fact that the set of real numbers is an ordered field.
3. False. There exists a real number, namely 0, such that 0 is not positive, but 0 is not negative. This is equivalent to the previous proposition.
4. True. There exists a natural number, namely 2, such that 2 is prime and also even.
2 is prime beacuse it cannot be expressed as a product of two smaller natural numbers. Being the first natural number greater than 1, the only possible "product of two smaller natural numbers" is 1x1 = 1, not 2.
2 is even because it can be expressed as 2 = 2x1, that is, 2 = 2n for n=1.
5. False. There exists NO real number which has the property of "destroying all numbers" through addition, that is, that the result of it added to any number will always result in 0.
To prove this, suppose, by contradiction, that such a number x exists. That is, x + y = 0, for any real number y. Then, take y + 1: x + (y + 1) = (x + y) + 1 = 0 + 1 = 1 =/= 0, which is a contradiction.
(On the other hand, there exists such a number for multiplication: 0 "destroys all numbers" through multiplication since y.0 = 0, for any real y.)
6. True. For every real number x, there exists a real number y, called the "additive inverse" of x, with the property that x + y = 0. This number is y = -x. This is a property that defines the set of the real numbers as a field.
7. True. To prove this, consider the second member of the "and" relation: x² = y². By subtracting y², we have x² - y² = 0. Factoring, we have (x - y)(x + y) = 0. A product of two real numbers is zero if, and only if, one of the numbers is zero. Therefore, either x - y = 0, which would mean x = y (not allowed by our premise), or x + y = 0, which would mean x = -y. Therefore, the implication holds.
8. False.
P: x is rational and y is rational.
Q: (x+y) is rational.
Q does not imply P: this means that, if (x+y) is rational, then x and y need not be both rational.
In fact, for x = √2 and y = -√2, we have (x+y) = √2 - √2 = 0 rational, but neither x nor y is rational.
9. True.
In fact, we can take z = (x+y)/2 which has the required property:
z - x = (x+y)/2 - x = (y-x)/2 which is positive when x < y, meaning x < z.
y - z = y - (x+y)/2 = (y-x)/2 which is positive when x < y, meaning z < y.
10. True. That unique number is 0.
It is true that 0 has the required property, since for y > 0, 0² = 0 < y.
The proof that 0 is the unique number with this property is as follows:
Suppose, by contradiction, that another number x =/= 0 has the same property.
Then, x² > 0, which implies x²/2 > 0. Take y = x²/2. y > 0 but x² > y = x²/2, which is a contradiction.
(This proof requires the knowledge of the fact that: x² = 0 iff x = 0, x² > 0 otherwise.)
I'm not convinced by 7. What if x and y are both 0?
Oh, it's in the proposition that x and y are unequal.
Beautiful list! I also have a question about 7, but for a different reason. I believe that for all real numbers x,y, if x = -y, then x ≠ y and x² = y² is also true, meaning it has a two way relationship. Is it a problem if one way implication is stated for a two way relationship like that? After typing it, my gut is saying that (a b) ==> (a ==> b) (I hope I did that right 😅), but I am unsure. Do you know the answer to this?
I also need to ask if 0 counts as an even number for the response to question 1. I believe it is a counterexample to the statement.
'so long and/or farewell'
since 'and v or' has the same truth table as or, and equivalent statement should be
'so long or farewell'
ps not sure if my language is mathematically rigorous or not. if error lemme know tq tq
This video is delightful. Mathematics would probably be more palatable to a general audience, if children were given a helpful introduction to logic in the early days. Most of the benefit of mathematics in real life is logic. Many people who struggle with math either fail to see its relevance or fail to grasp the basic logic underpinning the statements.
"Well, who doesn't like pi?"
*vihart has entered the chat*
nice one
I know us French people tend to make everything different when it comes to maths, but I just wanted to tell you that for us, 0 is always a natural number and 1 is never prime, except if you wanna define it otherwise but I've never seen anyone do so yet
1 is not prime in America either
For some reason in the USA, in K-12 ed, and the first two years of college, we make a distinction between natural numbers (positive integers) and whole numbers (non-negative integers).
Then once you are in your third yr at college and start group theory/abstract algebra, then we change the definition of natural number to include zero.
In the USA, the number written
-3 is “negative three,” NOT “minus three.” The word “minus” should be used only for the operation of subtraction. In everyday life, we often hear “minus” used incorrectly as “negative.”
Also, in the USA -3 is an integer, but it’s not a whole number, because the whole numbers are the non-negative integers only.
I tell my students that definitions develop over time. They start as a general description, and they get more precise as the object becomes more understood. Along the way, “edge cases” are sometimes included and other times not. It’s important to know what those edge cases are so that when you engage with a new person/course/text, you will know you need to agree as to whether or not the definition is inclusive of the edge case or not.
For the purpose of the new discussion we need to know:
Is zero a natural number?
Can a line be parallel to itself?
Is a rectangle a trapezoid?
When we say suppose a and b are two _____ , are we allowing them to be the same _____ , or are we assuming they are distinct?
Regardless of which choice we make, we need to keep that in mind as we go forward in the statements of new theorems and definitions.
The way I would explain implication is basically that, because of the law of the excluded middle, P implies Q still has to have a truth value when P is false. Either way could work, but saying implication is true when P is false has less weird implications. It really should be "indeterminate", but that's not an option.
This is the part of maths that actually feels like learning a language, and being able to just translate it into an English sentence is very satisfying
A good short recap on my first weeks lectures as a maths undergrad.
he'll yeah brother💪😤
(you should do one on group theory)
He just did!
Excelente video!
Ive already done my final exam in calculus, linear algebra and discrete maths. But I wish I know about this channel sooner. 😹
This guy is good at explaining for sure
Thanks!
Thanks alot ❤
I'm a tenth grader and I understand the notation. Before I actually watch the video, here is how I would say that example statement:
"For every real x that is nonzero, there exists a real y such that xy=1"
I thank my math teacher for teaching at above grade level, and we also strangely learned formal logic and its notation in philosophy classes 👍
25:09 i wasn't expecting that. good job. i laughed.
WOW ANOTHER VIDEO I'm exited! I very love logical and other CS stuff
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
I first learned about formal logics in high school, then mathematics and programming in college, and logic in phd, now I think the best way to describe both at the same time is type theory.
What is type theory? Care to give an example of why it's helpful for formal logic?
Your video is perfect! Don't listen to these robot comments. Thank you for your time and effort! I am learning a lot! You are appreciated!
thanks for this - there are so many papers i read where i think “i should have done a math degree”
great video
For all values of x in the set of real numbers, a value of x not equal to 0 implies that there exists a y value in the set of real numbers, specifically one that multiples by that x value to equal 1
Thanks, do you have more math logic videos ?
xor is missing => outrageous imperfection!))
Also it's a bit confusing that in propositions "=" may be false unlike in equasions
As the relatively fames phrase goes "One man's modus ponens is another man's modus tollens, another man's disjunctive syllogism or another man's indirect proof.".
Since these four logical expressions are logically equavalent to each other:
[A⇒B] ≡ [¬B⇒¬A] ≡ [¬A∨B] ≡ ¬[A∧¬B]
the corresponding syllogisms are then also logically equivalent to each other:
(P1) A⇒B
(P2) A
(C1) B [from P1 and P2 by modus ponens]
(P3) ¬B⇒¬A [from P1 by contraposition]
(C2) B [from P2 and P3 by modus ponens]
(P4) ¬A∨B [from P1 by material implication]
(C3) B [from P2 and P4 by disjunctive syllogism]
(P5) ¬[A∧¬B] [from P4 by de Morgan's law]
(P6.0) ¬B [indirect proof assumption]
(P6.1) A∧¬B [from P2 and P6.0 by conjunction introduction]
(P6.2) [A∧¬B]∧¬[A∧¬B] [from P5 and P6.1 by conjunction introduction]
(C4) B [from P2, P5 and P6.0-P6.2 by indirect proof]
So I guess, that these are all from the same partition of syllogisms, which we might call the simplest valid derivation from two premises.
Just Another Roof for you.😉
Doesn't it just say multiplicitive inverses exist for all nonzero real numbers?
I looked at it for half a minute and then said, well yeah, obviously.
I already know this stuff, but your videos are good and maybe I'll learn something. I'll watch the vid anyway if ony for the algorithm.
Edit: I did actually learn something. I didn't know about the "there exists a unique" notation
I always remember and in logic as simplify version of &.
"Be or not to be" - bv~b - tautology
Laughing at how when I saw this first I paused the video, read it, and said “oh yeah, obviously” 😂
I laughed at the birth joke.
There's something weirdly attractive about some dude successfully teaching me mathematical bureaucratese under an hour.
You are my life saveer ❤
5:35 me talking to half of my friends
24:08 wouldn't the opposite of the inside statement also be true?
if there's a real number y such that xy = 1, then x must not be 0.
Wouldn't that be an "if, and only if" statement then?
17:08 So an unsigned integer, got it
Using mathematical symbolism and logic can provide a powerful bridge to connect theological/metaphysical concepts with scientific/physical descriptions in a rigorous way. Instead of relying solely on binary true/false valuations, engaging non-contradictory/contradictory modes of reasoning could be highly fruitful.
Here are some thoughts on how we could apply this approach:
1. Multi-valued and Fuzzy Logics
Rather than classical bivalent logic, we could explore multi-valued algebraic logic systems that allow for more nuanced truth valuations beyond just 0 and 1. This could capture theological notions of paradox, ineffability, and transcendent reality that goes beyond strict binarization.
Fuzzy logics which admit truth values in the continuous range [0,1] could model metaphysical concepts that are irreducibly vague or context-dependent. Non-contradictory/contradictory could then be represented by sub-ranges of the multi-valued domain.
2. Paraconsistent Logics
Paraconsistent logical systems are designed to deal with contradictions in a controlled, discriminating way rather than just admitting logical explosion. This could allow rigorously reasoning about metaphysical statements that are paradoxical or logically inconsistent from a classical perspective.
Non-explosive paraconsistent frameworks like relevance logic could formalize theological ideas involving prescribed inconsistencies or contradictories without trivializing the entire system. Non-contradictory and contradictory conditions could be encoded precisely.
3. Modal Logics and Intuitionistic Systems
Modal logics explicitly capture notions of necessity, possibility, and ontological modalities. We could use graded/fuzzy modal systems to represent transcendent, ineffable realities beyond typical ontological constraints.
Intuitionistic logics based on constructive reasoning avoid strict bivalence and the principle of excluded middle. This could model metaphysical concepts that are not straightforwardly decidable in a binary fashion.
4. Substructural Logics and Resource Semantics
Substructural logics like linear logic impose resource-consciousness by controlling structural rules like weakening and contraction. This limited, pay-as-you-go approach could capture theological ideas of existential scarcity, ontological austerity, and irreducible indeterminacies.
Phase semantics and resource models in these logics could provide novel metaphysical interpretations and construct ontological stances beyond strictly bivalent modes.
5. Topological Semantics and Cohesion
Cohesive topological models using homotopy theory and algebraic topological semantics could provide a powerful geometric metaphor for non-contradictory/contradictory conditions in terms of intrinsic continuities, boundaries, and points of inflection.
This could unify metaphysical and scientific descriptions by embedding them in a common topological setting where contradictions are smoothly navigable via continuous pathways rather than pure bivalence.
By leveraging the immense richness of mathematical logic and non-classical reasoning frameworks, we could indeed use symbolic representations to bridge theological abstractions and physical observations in a philosophically robust yet scientifically grounded manner.
The non-contradictory/contradictory mode could become a new conceptual lens, expanding rigid true/false binaries into a continuum of coherence where metaphysics and science fluidly intersect. I'm happy to further explore concrete examples of how to apply these ideas to specific theological/metaphysical notions and their scientific counterparts.