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
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 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.
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
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 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 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!
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.
@@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.
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!
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.)
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'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.
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.
@@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.
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.
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 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.
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!
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.
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😊.
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.)
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.
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)
@@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.
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.
@@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?')
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.
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)
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
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. 🙏
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 👍
'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
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
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.
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.
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! :)
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
This video is very nice for beginners, I'll probably recommend this video to students at the beginning of a Math proving class so they know what they'll be getting into.
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)
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.😉
26:37 1. true (either have to be true) 2. true (same as last) 3. true (not positives are negative, but the sign could also be ) 4. false (no prime is even since even is divisible by 2) 5. false (**BOTH** **MUST** be 0, and Ay e R is not only 0) 6. false (same as last) 7. true (the only number squared that isnt itself is its negative) 8. true (both x and y must be in Q for the x+y to be in Q) (EDIT: false, irrational+irrational can equal a rational, e.g. π+(1-π)=1) 9. true (you can always fit a real number between 2 other) 10. false (no one REAL number squared is < 0)
OMG you're so helpful your video is really interesting and informative. Also love the jokes ,it break the tense that built with in me every time the problem and the materials getting difficult. Ty kind sir. You're Video is Amazing all love from me you sure put a lot of care into it♥🥰.
Can you do a video explaining all 16 diagrams of VENN-diagrams, in relation to Binary numbers and logical reasoning. So, I would like to entirely understand why the Venn diagrams are the best way to use for logical reasoning. Thank you very much in advance.
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.
I used to work with programmable logic controllers (PLC) which use "ladder logic" essentially a programming language based on relay logic. I got quite good at reading the logic and found that i could easily translate it to English in order to make sense of it to laypersons who weren't familiar with it. The plc took "inputs" that could be discreet switches or analog sensors and then based on the logic would set "outputs" that could be motor starters for examples. I thought of it as basically conditions and outcomes, if the conditions are true then make this outcome true.
@@logickedmazimoon6001 You might know this then? I was wondering is there an actual logic gate that is capable of carrying out that Implies operation? From what I remember physical NOT gates and OR gates exist in real life as does a physical NAND gate (you add a NOT gate at the output to get the AND operations just as you would add a NOT gate to the output of an OR gate to get the NOR operations). Also there's the XOR (Exclusive Or) gate which gives you a 1 (or True) at the output only when both the inputs are different from one another, so again you can just add a NOT gate at the output and you will get those IF and only IF operations like in the video which only give you a True when the inputs are the same, ie either both True or both False. But the Implies operation? I can't remember any that will perform that operation. Maybe a combination of others will?
Generally I love how clear your explanations are, however in this instance, I think you missed a trick by not including : (such that). I found your list of patreon suggestions at the end very hard to parse, as commas were used for both their usual separation of statements, and to mean 'such that'.
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?
My attempt at the 10 questions: 1. True, all integers are either odd or even. 2. False, 0 is a rational that is neither positive or negative. 3. False, again 0 is not positive and also not negative. 4. True, 2 is prime and even. 5. False, there is no number that is 0 when anything is added to it. 6. True, y=-x 7. True, I can't think of any counterexamples. 8. False, there can exists two non-rational numbers that add to a rational number, (pi)+(1-pi)=1 9. True, you can take the average of x and y to get z. 10. True, x=0.
Nice job! 👍 I'd just like to add the following note: Just because you are unaware of any counterexamples doesn't immediately imply that the statement must be true. Though, luckily for you, #7 can be proven to be True because if x² = y² then x² - y² = 0. Applying the difference of squares and solving for the cases where x ≠ y gives us the desired x = -y. Hope this helps, and stay curious! :)
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
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
For those who want an easy translation, the intro statement says “for all x that exist in the real numbers, so long as x does not equal zero, then there exists a number y which when multiplied to create xy, you get 1.”
Wait did the joke at 6:21 just go over my head, or does UK English keyboard layout actually have the logical negation symbol on it? I get the feeling I missed the joke but it would be cool if that symbol was actually standard on a keyboard
I didn't realise this was only a UK thing! But yeah our key below Esc and to the left of 1 as the ¬ symbol on it. As well as ` which only TeX-savvy people use. And also ¦ which... your guess is as good as mine.
@@AnotherRoof Oh interesting! Yeah, on an US English keyboard layout, that key has ` (backtick) also, but the other symbol on it, which you reach with shift, is ~ (tilde). But you say there's a third symbol there? I recognize that broken-vertical-bar but the US layout only has the regular vertical bar | on the same key as the backslash (under backspace)
@@Scum42 Yeah I realise that now -- we also have the vertical bar on the backslash key but it's left of Z. I do remember some of the differences now because I once bought a US-layout keyboard but I like our big, tall 'enter' key, But I never realised ¬ was a UK thing. Don't know why -- literally no idea what it's used for outside of formal logic!
For all the programmers watching this video marvelling at the similarity with various programming languages, that is no accident. Programming languages were basically conceived as a way to implement first order logic.
I keep thinking that implication is just an artifact of the premise that truth conditions can only be either True or False. Functionally, A -> B is equivalent to ¬A V B, which is easier to interpret. Though the arrow makes for a nice shorthand. Considering the alternatives, saying an implication is false when the condition in not met just gives you conjunction. Any other alternative gives you the identities of the inputs. But realistically, I think what people would _want_ to say is that an implication whose condition isn't met is Unknown "U", rather than True "T". That's just not an option, though. Say one wanted to represent A -> B with sets. They could draw A as a set subsumed by B, but that carries the subtle additional idea that A exists. It's the same discrepancy with "The king of France is bald." Does that mean that the king of France _exists_ and is bald, or only the implicational relationship? Similarly, What is the probability of B given A, if the probability of A is 0? Freaky stuff...
Wait, "0 is part of N" depends on the speaker? I thought the entire point of having separate names for the whole numbers (W) and the natural numbers (N) was so that one of them, W, would include 0 and the other wouldn't!
But the whole numbers is just another name for the integers? More seriously, "Whole Numbers", "Counting Numbers" and "Natural Numbers" all get used in attempts to distinguish between positive and non-negative integers, but there isn't agreement on which name should be used for which set (and, as I mentioned, "whole numbers" is also used for the integers). The general solution is that most times you're not really interested in whether you start with 0, 1, or 42; if you're talking about natural numbers, you're more interested in what happens as they get bigger, not what happens with the special cases at the start. If 0 works differently, then you may need to clarify whether or not you're including 0, but most of the time that's not the interesting bit.
@@rmsgrey Some authors do specify the naturals as 1, 2, 3, ... and the whole numbers as 0, 1, 2, ... Like I said in my video, there's really no consensus!
In set theory, it really makes sense to include 0 in the set of natural numbers because this allows you to construct 1,2,3,... and so on. In other cases like defining the harmonic series, it makes sense to exclude 0 if you want to call the harmonic series "the sum of the reciprocals of the natural numbers". We are just stuck with competing jargon because of the history of math because the language didn't evolve with high-use, distinct names for {0,1,2,...} and {1,2,3,...}.
@@AnotherRoof Yeah, if you look long enough, you can find people who state with great authority that X is the name for the positive integers and Y is the name for the non-negative integers for any of the 6 possible ways of picking distinct X and Y from {"natural numbers", "whole numbers", "counting numbers"}. Which does invite speculation about the sort of mind that thinks people should count starting with 0, but that's another issue. The fact that "whole numbers" also gets used for the integers as a whole only adds to the potential for confusion. The whole thing is reminiscent of XKCD strip 927
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.
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.
I m very sad how Computer Science student go away from mathematics logic by spend time to languages and hardware ; and they don’t realize that Computer Science is created by Mathematicians.
The formal-logic _or_ rarely lines up with how it's used in plain English. Think "cake or death." If someone asks you if you'd like coffee or tea, they're not asking if you'd like both and they don't want to hear about it if you do; what they're saying is that you may have one of the two, now declare which one of the two you're choosing to have. And the idea that you don't have a preference for one over the other won't stand to scrutiny, unless you're going to sit unable to choose like Buridan's Ass. The exclusive-or is what aligns with intuition.
