If the policy is that when you get into an accident you get money... That does not tell you what happens when you do no get into an accident. You may get money, you may not. One would conclude it is undefined. You do it with a syllogism. You can deduce that you get money from the policy and the accident, but you cannot deduce that you get money from the policy and lack of accident. If doing knowledge representation, that's what I'd do. Edit: see inference rules and formal language.
This is how you conceptualize this: {A => B} (A, B) do not contradict inferring B from A. Understand inferring B from A that knowing A is true is enough to know B is true, however A being true might not be necessary for B being true. If and only if A is true and B is false, we would have found a situation where inferring B from A does not work. Meaning that we found evidence that the inference rule from A to B is wrong. If we find that A is false and B is true, that does not contradict inferring B from A. It just confirms that A being true is not necessary for B being true.
No entendí nada de lo que dijiste (del idioma) pero sí pude comprender lo demás. Ojalá mi profesor de Álgebra hubiera explicado las cosas así cuando vimos lógica.
Voy a intentar explicártelo porque es un concepto importante que se usa mucho para presentar teoremas. Muchas veces en matemática vas a encontrar que teoremas, lemas etc son enunciados usando un lenguaje formal y la implicación es esencial. La implicación lógica podría pensarse como un MODELO, una ESTRUCTURA creada por nosotros para reflejar la relación entre dos proposiciones atómicas. A implica B es equivalente a decir que si representáramos a A y B con círculos, el círculo A estaría DENTRO de B. Notá por favor, que no estoy hablando de algo en el mundo real, no me estoy refiriendo a nada en particular, simplemente estoy diciendo que el círculo A está dentro del círculo B y ahora notá lo siguiente: Si A es verdadero, quiere decir que A ocurre (por decirlo de alguna forma) entonces para representar que A ocurre los pintamos. Si ahora digo que B ocurrió también, entonces pinto a B también y se ve claramente que esto es NO CONTRADICTORIO, ES CONCEBIBLE PARA NUESTRA LÓGICA. Si pinto A y no pinto B, entonces hay una contradicción importante. Decir que B no ocurrió es equivalente a que nada de B sea pintado, pero también estoy diciendo que A ocurrió y que por lo tanto hay que pintarlo....lo cual es absurdo. Por lo tanto Verdadero implica falso es FALSO. Por último, si no pinto A y pinto B, eso es concebible, porque puedo pintar la parte de B que no es A, por lo tanto es NO CONTRADICTORIO. Por último, si no pinto A y no pinto B, nuevamente, no hay contradicción alguna, es permitido. En matemática se usa el condicional porque hay ciertos relaciones entre objetos matemáticos que siguen este comportamiento. Aplicar el condicional para el "mundo real" es un tanto problemático.
I used to write T for all the T/F problems. Then I would wait for the teacher to go over the answers, then all I needed to do is to add a - to T so it would look like F. Unfortunately I could only do that on worksheets not exams.
my problem with this video, is that he's not discussing the logic behind implication, i.e. modus ponens, necessity and sufficiency. he's just putting in math expressions and evaluating the truth value. the true importance and power of p->q is that it helps us describe the world ( if it's raining then i'll get wet), and helps us formalize logical entailment. implication is the one foundational concepts in building all of mathematics through logical entailment. it's pretty insane.
another way to see why f=>t and f=>f should be true is by motivation of the if and only if statement (iff). iff should hold true when both premises are the same truth value and iff should be false if p and q differ in truth value. p iff q is defined as p=>q and q=>p. if f=>t or f=>f were to be false, then iff would not work as we understand it to be.
Whenever I've been stuck with implication (every time), I either remind myself "Ex falso quodlibet" or I remember that if the truth table wasn't this way it would be some other logic relation like bi-implication.
r is just a parameter here. For example x^(2/3) + y^(2/3) = a^(2/3) is an asteroid with intercepts (a,0), (0,a), (-a,0), (0,-a). Anyway, A = 4*integral (r^4 - x^4)^(1/4) dx, x=0..r Dont think this will be done with elementary functions.
Hi dr Peyam! Would you mind to do a separate series or at least a video about tensors and its applications in calculus and differential geometry? I've always wanted to get to know this branch of mathematics, but unfortunately tensors weren't part of my linear algebra course and (at least for me) It's really hard to understand the core idea of it. Btw in my opinion your channel is the best youtube channel about math. Greetings from Poland!
The true nature of implication is not entailment but opposition: th-cam.com/video/supEdKORfNw/w-d-xo.html (English subtitles available) The "False imply true" problem is solved once and for all!
"Most feared statement" - pulls up Epsilon Delta definition of a limit. HAHAHAHHAHA!!! Good one Dr. Peyam. If you're up to it, a video on nested predicate quantifiers would be greatly appreciated. Thank you for explaining the truth table of the conditional through analogies. My math professor never bothered to correct the counter-intuition we naively have with F->T and just expected us to rote memorize it from textbook. Concentration was only spent on doing the exercises. Thank you for your service. Oh and just for encouragement for fellow mathematicians don't worry if you still don't completely get or like logic - there are mathematicians that hate studying logic too. It's a free country!
Hi Dr. Peyam, I was watching your video on surface integrals and their geometric interpretation, and I wanted to clarify something. Is a surface integral the volume between a surface and the surface's projection onto a 4D function (just like a line integral is the area between a curve and its projection on a surface above)? So is it actually a 3D volume/measurement?
If False is True than the consequent statment is also True, as if it's True it's cool anyway and if it's False it's also True as we've assumed. That't why the whole statment is True.
It could be also helpful to interpret (P implies Q) as (Q is not less true than P). Indeed, we are only interested if we can say P is true, once we know Q is true. So we really care about the relationship between the truth-values of P and Q, not the relationship between the statements themself. That way the truth table makes much more sense.
Dr Peyam, please, I strongly sugggest you make a video on the derivation of the Normal Distribution using this article www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf It's a beautiful derivation (none other than Gauss' derivation of the Normal PDF!) which is within the reach of anyone with minnimal knowledge in probability theory and willing to go into it. Page 104 onwards! I think everyone is going to absolutely love it! As a side note, I reckon the only additional piece of information to make the derivation fully self-contained is proof that the only function that satisfies F(kx)=kF(x) is indeed F(x)=x
"You don't get into an accident and dthe policy doesn't give you a million dollars" Would you say it is a bad policy (F)? No But *I wouldn't say it is a good policy (T) either* , because I didn't actually test the policy. That is why the thing I don't understand is why we didnt set False Implies False instead of False Implies True. Why the convention feels the opposite of common sense? It would make sense if 0->1=0 and 0->0=0
The implication is characteristic that MUST have the cause of the effect. If I fall into the pond, I am never dry.That's the most important.(If A then B . A=1, B=0 The result of implication is 0).In other cases the result of implication=1
I don't understand. I think that False as Input doesn't reveal anything about the fuction: P->Q internal state. I would prefer tri-state logic which has True,False,Undefined. To take something by 'default' is something i don't understand. Undefined for the last 2 cases.
While this is still an example of case 4 (of the truth table as both p and q are false), I think you raise an interesting point - it seems to be a special sub-case of 4 as given p (1+1=3), then q ((1+1)+1=4 i.e. p -> q is patently true ("sound"?) . This contrasts with the example given at 6:25 which in Dr Peyam's words is "complete gibberish" as there is little logical connection between p or q. Your example is a valid argument whereas Dr Peyam's is not valid (noting that either example is not sound) See en.wikipedia.org/wiki/Deductive_reasoning#Validity_and_soundness): "An argument is “valid” if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false"
If the policy is that when you get into an accident you get money... That does not tell you what happens when you do no get into an accident. You may get money, you may not. One would conclude it is undefined. You do it with a syllogism. You can deduce that you get money from the policy and the accident, but you cannot deduce that you get money from the policy and lack of accident. If doing knowledge representation, that's what I'd do.
Edit: see inference rules and formal language.
Yes, it was an if then relationship, not an if and only if relationship.
I like the insurance policy example
This is how you conceptualize this:
{A => B} (A, B) do not contradict inferring B from A. Understand inferring B from A that knowing A is true is enough to know B is true, however A being true might not be necessary for B being true.
If and only if A is true and B is false, we would have found a situation where inferring B from A does not work. Meaning that we found evidence that the inference rule from A to B is wrong.
If we find that A is false and B is true, that does not contradict inferring B from A. It just confirms that A being true is not necessary for B being true.
No entendí nada de lo que dijiste (del idioma) pero sí pude comprender lo demás. Ojalá mi profesor de Álgebra hubiera explicado las cosas así cuando vimos lógica.
Voy a intentar explicártelo porque es un concepto importante que se usa mucho para presentar teoremas. Muchas veces en matemática vas a encontrar que teoremas, lemas etc son enunciados usando un lenguaje formal y la implicación es esencial. La implicación lógica podría pensarse como un MODELO, una ESTRUCTURA creada por nosotros para reflejar la relación entre dos proposiciones atómicas. A implica B es equivalente a decir que si representáramos a A y B con círculos, el círculo A estaría DENTRO de B. Notá por favor, que no estoy hablando de algo en el mundo real, no me estoy refiriendo a nada en particular, simplemente estoy diciendo que el círculo A está dentro del círculo B y ahora notá lo siguiente: Si A es verdadero, quiere decir que A ocurre (por decirlo de alguna forma) entonces para representar que A ocurre los pintamos. Si ahora digo que B ocurrió también, entonces pinto a B también y se ve claramente que esto es NO CONTRADICTORIO, ES CONCEBIBLE PARA NUESTRA LÓGICA. Si pinto A y no pinto B, entonces hay una contradicción importante. Decir que B no ocurrió es equivalente a que nada de B sea pintado, pero también estoy diciendo que A ocurrió y que por lo tanto hay que pintarlo....lo cual es absurdo. Por lo tanto Verdadero implica falso es FALSO. Por último, si no pinto A y pinto B, eso es concebible, porque puedo pintar la parte de B que no es A, por lo tanto es NO CONTRADICTORIO. Por último, si no pinto A y no pinto B, nuevamente, no hay contradicción alguna, es permitido. En matemática se usa el condicional porque hay ciertos relaciones entre objetos matemáticos que siguen este comportamiento. Aplicar el condicional para el "mundo real" es un tanto problemático.
I used to write T for all the T/F problems. Then I would wait for the teacher to go over the answers, then all I needed to do is to add a - to T so it would look like F. Unfortunately I could only do that on worksheets not exams.
Hahaha, I hate when students write T with a - because you can’t tell if it’s T or F
top 0 professors eddie woo fears
Nice video, Dr. Peyam. Could you please upload more of these basic logic-related videos?
Sure! I don’t know if it helps, but I have a Set Theory Playlist
@@drpeyam Ok, thanks! I will check it out!
This video is out in the perfect timing
I just saw a question using F implies T for prove in today math lesson and I was confused
Thx for the video
it is true that something false can imply something true. This word "can" admits that there is another possible case, namely, F implies F.
I see, so on my next exam I should put my answers in the form p implies q and then put a wrong answer for p so I am guaranteed to get 100% correct!
my problem with this video, is that he's not discussing the logic behind implication, i.e. modus ponens, necessity and sufficiency. he's just putting in math expressions and evaluating the truth value.
the true importance and power of p->q is that it helps us describe the world ( if it's raining then i'll get wet), and helps us formalize logical entailment. implication is the one foundational concepts in building all of mathematics through logical entailment. it's pretty insane.
another way to see why f=>t and f=>f should be true is by motivation of the if and only if statement (iff). iff should hold true when both premises are the same truth value and iff should be false if p and q differ in truth value. p iff q is defined as p=>q and q=>p. if f=>t or f=>f were to be false, then iff would not work as we understand it to be.
Have you considered proving the law of non-contradiction? That it is necessarily false that statements (generally) are allowed to contradict
Why does this video is unlisted? Am I the only one who watched it? I'm special!!!
How did you find it?
Otium Abscondita ik u from fematika’s discord server
@@gregoriousmaths266 no shit Sherlock
@@OtiumAbscondita lmao
The Riemann hypothesis implies a statement about the distribution of primes.
Does the continuum hypothesis imply the Riemann hypothesis?
If the continuum hypothesis is false, then yes :P
Hey Dr.Peyam! I am kinda new to this channel...could you tell me some of the best videos of yours...
P.S. A sophomore.
Sounds funny how I ask...
Check out my top videos, those are the best :)
@@drpeyam Oh! Thank You!
False implies true is essentially true by accident.
We learned about this in my computer science class. Logic is a lot of fun!
Whenever I've been stuck with implication (every time), I either remind myself "Ex falso quodlibet" or I remember that if the truth table wasn't this way it would be some other logic relation like bi-implication.
So cool!!
How about a vid on the NOR (or NAND) operator, the only operator you'll ever need. 😋
Wonderful :)
Can you find a formula for the area inside the curve x^4+y^4=r^4 ?
But x^4+y^4r^4 in polar
What?
In polar coordinate system is x^4+y^4 different from r^4
Lida Jrs
in polar system there's no x and y
r is just a parameter here. For example x^(2/3) + y^(2/3) = a^(2/3) is an asteroid with intercepts (a,0), (0,a), (-a,0), (0,-a).
Anyway,
A = 4*integral (r^4 - x^4)^(1/4) dx, x=0..r
Dont think this will be done with elementary functions.
Hi dr Peyam! Would you mind to do a separate series or at least a video about tensors and its applications in calculus and differential geometry? I've always wanted to get to know this branch of mathematics, but unfortunately tensors weren't part of my linear algebra course and (at least for me) It's really hard to understand the core idea of it. Btw in my opinion your channel is the best youtube channel about math. Greetings from Poland!
Maybe
awesome
The true nature of implication is not entailment but opposition: th-cam.com/video/supEdKORfNw/w-d-xo.html (English subtitles available) The "False imply true" problem is solved once and for all!
Justamente estoy viendo un capítulo de introducción a la topología y este tipo de temas me dejan pensando
"Most feared statement" - pulls up Epsilon Delta definition of a limit. HAHAHAHHAHA!!! Good one Dr. Peyam. If you're up to it, a video on nested predicate quantifiers would be greatly appreciated. Thank you for explaining the truth table of the conditional through analogies. My math professor never bothered to correct the counter-intuition we naively have with F->T and just expected us to rote memorize it from textbook. Concentration was only spent on doing the exercises. Thank you for your service. Oh and just for encouragement for fellow mathematicians don't worry if you still don't completely get or like logic - there are mathematicians that hate studying logic too. It's a free country!
I love truth tables!
Me too!!!!
Love you so much! Thank‘s for all your interesting videos!
Hi Dr. Peyam,
I was watching your video on surface integrals and their geometric interpretation, and I wanted to clarify something. Is a surface integral the volume between a surface and the surface's projection onto a 4D function (just like a line integral is the area between a curve and its projection on a surface above)? So is it actually a 3D volume/measurement?
That’s precisely what it is, and it’s a 3D volume measurement
@@drpeyam Thank you! :)
True always implies true says nothing about what False is allowed to imply. It's a simple spell, but quite unbreakable!
Ex Falso Sequitur Quodlibet!
Well done
If False is True than the consequent statment is also True, as if it's True it's cool anyway and if it's False it's also True as we've assumed. That't why the whole statment is True.
You can deduce anything from a contradiction, c'mon!
Ex falso quodlibet. (Scholastic logic)
It could be also helpful to interpret (P implies Q) as (Q is not less true than P). Indeed, we are only interested if we can say P is true, once we know Q is true. So we really care about the relationship between the truth-values of P and Q, not the relationship between the statements themself. That way the truth table makes much more sense.
True
Hi Doctor excuse me where are you come from? Are you Iranian?
Yeah
So, in JavaScript terms, it's really *falsey!*
Dr Peyam, please, I strongly sugggest you make a video on the derivation of the Normal Distribution using this article www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf It's a beautiful derivation (none other than Gauss' derivation of the Normal PDF!) which is within the reach of anyone with minnimal knowledge in probability theory and willing to go into it. Page 104 onwards! I think everyone is going to absolutely love it! As a side note, I reckon the only additional piece of information to make the derivation fully self-contained is proof that the only function that satisfies F(kx)=kF(x) is indeed F(x)=x
(a implies b) being true, and a being false and b being true, means that b being true does not force a being true. I.e a is not affected by b.
I like that!!!
... vacuously true.
(Just completing the title haha)
"You don't get into an accident and dthe policy doesn't give you a million dollars"
Would you say it is a bad policy (F)? No
But *I wouldn't say it is a good policy (T) either* , because I didn't actually test the policy.
That is why the thing I don't understand is why we didnt set False Implies False instead of False Implies True. Why the convention feels the opposite of common sense? It would make sense if 0->1=0 and 0->0=0
More math like this!
Amazing!
If I fall into the pond, I am wet.
And if I don't, I still have the right to be wet. :-)
The implication is characteristic that MUST have the cause of the effect. If I fall into the pond, I am never dry.That's the most important.(If A then B . A=1, B=0 The result of implication is 0).In other cases the result of implication=1
You got wet from a bathtub instead lol.
Para un futuro video PQ
Logic
Claro, demostrar un teorema del tipo P->Q, es demostrar que el razonamiento P->Q es una tautología, es decir que siempre es verdadero
I don't understand. I think that False as Input doesn't reveal anything about the fuction: P->Q internal state. I would prefer tri-state logic which has True,False,Undefined. To take something by 'default' is something i don't understand. Undefined for the last 2 cases.
How about 1+1 = 3 -> (1+1)+1 = 4
I think you missing this nice example
While this is still an example of case 4 (of the truth table as both p and q are false), I think you raise an interesting point - it seems to be a special sub-case of 4 as given p (1+1=3), then q ((1+1)+1=4 i.e. p -> q is patently true ("sound"?) . This contrasts with the example given at 6:25 which in Dr Peyam's words is "complete gibberish" as there is little logical connection between p or q. Your example is a valid argument whereas Dr Peyam's is not valid (noting that either example is not sound)
See en.wikipedia.org/wiki/Deductive_reasoning#Validity_and_soundness):
"An argument is “valid” if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false"
Ex falso quodlibet
"False Implies True is..." False does not imply anything.