In set theory implication can be understood as inclusion. P implies Q translates to set theory as P is within Q, graphically speaking we have two circles, P's area is smaller than Q's and P is INSIDE Q. Then it's easy to see why the truth table is the way it is. If P is false, Q might still have happened. If n is a primer number bigger than two then n is odd. If it's false that n is prime, n can still be odd, since 9 is bigger than two and it's odd.
@@Eugene-rq8kr Not necessarily; if p is false then we can't argue that p -> q is false because we're out of the statement (not false = true) or at least that's my limited understanding
This video provides the definition, a few examples, but it fails to provide any insight or intuition about why a false antecedent leads to a true implication.
I think it's purely definitional and there is no real logical explanation for why this operator is the way it is. You can come up with multiple valid english translations for how implication works in scenarios that are already consistent with real-world logic, but such description will always fail in scenarios that are already illogical in a real-world sense. Take the statement: "if 2+2=10, then 2+2=100". This is a true logical statement, but there is no intuitive, real-world reason for why this should be the case. That's just how implication works. The fact that it's called "implication" is just an unfortunate coincidence. If you absolutely need a language-consistent way of thinking about it, try this: Take the statement "P implies Q". If you can 100% prove that this statement is false (i.e. P is true but Q, for whatever reason, is still false), then the statement is false. For all other cases, the statement is true because *you can't show that the statement is necessarily false*.
@@ancienthero2876 If you absolutely need a language-consistent way of thinking about it, try this: Take the statement "P implies Q". If you can 100% prove that this statement is false (i.e. P is true but Q, for whatever reason, is still false), then the statement is false. For all other cases, the statement is true because *you can't show that the statement is necessarily false* GREAT ANSWER.
The dog mammal example made me say "oh my god" out loud, because it finally clicked in my brain how implies works. The reason FF =T, is because the statement is correct still, if you pet is a dog, then it's a mammal, well its not a dog, and since you took the dog part of the question away, it's really doesn't matter if it's a mammal or not
Thank you. My lecturer did a poor job of explaining this in Computer Science. It doesn't help that he has a thick accent, and slightly broken English. This helps a great deal, thanks again.
So basically all we really need to remember is: if the antecedent is true and the consequent is false then the statement is false. In all other situtations the statement is true? Great.
If anyone is still struggling with this, I think I've figured it out. Let p = "The shape is a Square" Let q = "The shape is a Rectangle" IF "The shape is a Square" (T) THEN "The shape is a Rectangle" (T) = True (squares are indeed rectangles) IF "The shape is a Square" (T) THEN "The shape is not a Rectangle" (F) = False (all squares are rectangles) IF "The shape is not a Square" (F) THEN "The shape is a Rectangle" (T) = True (not all rectangles are squares) IF "The shape is not a Square" (F) THEN "The shape is not a Rectangle" (T) = True (the shape is neither a square nor a rectangle) Let me know if I made a mistake or anything. I think the "if then" way of stating it implies causality, which is where many people (myself included) get confused.
The logicality of the statement in terms of the english language is irrelevant to this, only the truth value is important. The example I used is just an illustration of p and q, in that q is a necessary condition for p, but p is a sufficient condition for q. Think of it like this: if the "event" of q "occurs" then the event of p could occur but it could also not occur. if the event of q does not occur, however, the event of p will not occur under any circumstances.
this is why when q is false and p is false, the truth value of the statement is true, because without q, p cannot occur. when q is false and p is true, the truth value of the statement is completely false. That scenario is impossible because the only way for p to "occur" is if q occurs.
"Everyday" meaning of implication. For an easier understanding of the meaning of direct implication and memorizing its truth table, an everyday model can be useful: A is the boss. He can order "work" (1) or say "do what you want" (0). B - subordinate. He can work (1) or mess around (0). In this case, the implication is nothing more than the obedience of the subordinate to the superior. According to the truth table, it is easy to check that there is no obedience only when the boss orders to work (1), and the subordinate is idle(0).
Let's imagine that I have a set of four cards laid on the table, each of which shows a certain color on one face, and shows a certain number on its opposite face. And I state that "In this set, if a card shows an even number on one face, then its opposite face is red". In real life, this statement makes sense only when there is at least one element in the set that satisfies the first condition, and it is true only when each card that satisfies the first condition also satisfies the second condition. On the other hand, in logic, this statement can make sense even if there is no element that satisfies the first condition and it is true only when each card verifies any of the following clauses: a) The first condition is true and the second condition is true, b) the first condition is false and the second condition is true, and c) the first condition is false and the second condition is false. Because in that way we guarantee that there are no cards that contradict the implication. So, in logic, this statement means "There are no cards that verify the first condition but not the second" (In this case, we do not need any card to fulfill the first condition for this statement to make sense.) Furthermore, if there exists at least one card that satisfies the first condition, then by guaranteeing the logical implication, we guarantee that that or those cards also satisfy the second statement. That is, we’re guaranteeing that each card that satisfies the first condition also satisfies the second condition. So we can say that, in this context, the logical implication and the real life implication actually mean exactly the same thing when there exists at least one card that satisfies the first condition. PD: Question for you, ¿would they mean the same thing if there were no cards that satisfy the first condition?
But how can False (p) with True (q) be always implied to be True? Suppose I use False (p) with True (q) using this implication - "If your pet is a bird, then your pet is a mammal." Isn't this implication false here?
the statement we're evaluating is: If the moon is made of green cheese, then the sun flashes purple on Saturdays. This is true, because the moon is not made of green cheese, therefore, we don't care if the sun flashes purple or not, the implication is true. We don't care about the two statements themselves, we care if one implies the other. If the first statement is shown to be false, then it implies that anything else is true by convention.
@@keokawasaki7833 If you pass your exam your mother says she will buy you a car. Only way this promise can be false is if you pass the exam and she doesn't buy the car.
The only thing I can think of is: if your pet is a dog -> then your pet is a mammal. It would be logically impossible to have a pet that is a dog and not a mammal (therefore Dog (True) - > mammal (False), is always false. What do you think?
![[LIVE] : ONE ลุมพินี 78 | คู่เอก "ปกรณ์ vs ฟาบิโอ"](http://i.ytimg.com/vi/LSOmgl0Th2c/mqdefault.jpg)
but WHY? my problem is understanding why it is the way it is
I feel you ^
i feel ya
yep lmao same
have anybody find out why?
That's what I was asking myself as well ... Explain to me where is this used in a real-life scenario
In set theory implication can be understood as inclusion. P implies Q translates to set theory as P is within Q, graphically speaking we have two circles, P's area is smaller than Q's and P is INSIDE Q. Then it's easy to see why the truth table is the way it is. If P is false, Q might still have happened. If n is a primer number bigger than two then n is odd. If it's false that n is prime, n can still be odd, since 9 is bigger than two and it's odd.
VERY VERY INTERESTING PERSPECTIVE
Why is " might still have happend" defined as true? It looks like false is a better option. Just think back: "might still have happend..."
@@Eugene-rq8kr Not necessarily; if p is false then we can't argue that p -> q is false because we're out of the statement (not false = true)
or at least that's my limited understanding
This video provides the definition, a few examples, but it fails to provide any insight or intuition about why a false antecedent leads to a true implication.
I think it's purely definitional and there is no real logical explanation for why this operator is the way it is. You can come up with multiple valid english translations for how implication works in scenarios that are already consistent with real-world logic, but such description will always fail in scenarios that are already illogical in a real-world sense. Take the statement: "if 2+2=10, then 2+2=100". This is a true logical statement, but there is no intuitive, real-world reason for why this should be the case. That's just how implication works. The fact that it's called "implication" is just an unfortunate coincidence.
If you absolutely need a language-consistent way of thinking about it, try this: Take the statement "P implies Q". If you can 100% prove that this statement is false (i.e. P is true but Q, for whatever reason, is still false), then the statement is false. For all other cases, the statement is true because *you can't show that the statement is necessarily false*.
@@ancienthero2876 If you absolutely need a language-consistent way of thinking about it, try this: Take the statement "P implies Q". If you can 100% prove that this statement is false (i.e. P is true but Q, for whatever reason, is still false), then the statement is false. For all other cases, the statement is true because *you can't show that the statement is necessarily false*
GREAT ANSWER.
The dog mammal example made me say "oh my god" out loud, because it finally clicked in my brain how implies works. The reason FF =T, is because the statement is correct still, if you pet is a dog, then it's a mammal, well its not a dog, and since you took the dog part of the question away, it's really doesn't matter if it's a mammal or not
Legendary video. Very clear way of explaining, thanks!
Thank you. My lecturer did a poor job of explaining this in Computer Science. It doesn't help that he has a thick accent, and slightly broken English.
This helps a great deal, thanks again.
I know them feels... its like its part of the job description.
same here. My lecturer even completely skipped the explanation and asked us to "just remember"
So basically all we really need to remember is: if the antecedent is true and the consequent is false then the statement is false. In all other situtations the statement is true? Great.
If anyone is still struggling with this, I think I've figured it out.
Let p = "The shape is a Square"
Let q = "The shape is a Rectangle"
IF "The shape is a Square" (T) THEN "The shape is a Rectangle" (T) = True (squares are indeed rectangles)
IF "The shape is a Square" (T) THEN "The shape is not a Rectangle" (F) = False (all squares are rectangles)
IF "The shape is not a Square" (F) THEN "The shape is a Rectangle" (T) = True (not all rectangles are squares)
IF "The shape is not a Square" (F) THEN "The shape is not a Rectangle" (T) = True (the shape is neither a square nor a rectangle)
Let me know if I made a mistake or anything. I think the "if then" way of stating it implies causality, which is where many people (myself included) get confused.
for the last one I meant "The shape is not a Rectangle" (F)
cause square is already one case of rectangle, it's like NEWYORK implies US. but what if p is more general and q is more specific???
The logicality of the statement in terms of the english language is irrelevant to this, only the truth value is important. The example I used is just an illustration of p and q, in that q is a necessary condition for p, but p is a sufficient condition for q. Think of it like this: if the "event" of q "occurs" then the event of p could occur but it could also not occur. if the event of q does not occur, however, the event of p will not occur under any circumstances.
this is why when q is false and p is false, the truth value of the statement is true, because without q, p cannot occur. when q is false and p is true, the truth value of the statement is completely false. That scenario is impossible because the only way for p to "occur" is if q occurs.
That helped me, thanks.
Wow.... A really good video....
Last example was a good one.... :)
"Everyday" meaning of implication. For an easier understanding of the meaning of direct implication and memorizing its truth table, an everyday model can be useful:
A is the boss. He can order "work" (1) or say "do what you want" (0).
B - subordinate. He can work (1) or mess around (0).
In this case, the implication is nothing more than the obedience of the subordinate to the superior. According to the truth table, it is easy to check that there is no obedience only when the boss orders to work (1), and the subordinate is idle(0).
What is the programmatical equivalence to logical implications? I really can't understand it this way.
Let's imagine that I have a set of four cards laid on the table, each of which shows a certain color on one face, and shows a certain number on its opposite face. And I state that "In this set, if a card shows an even number on one face, then its opposite face is red".
In real life, this statement makes sense only when there is at least one element in the set that satisfies the first condition, and it is true only when each card that satisfies the first condition also satisfies the second condition.
On the other hand, in logic, this statement can make sense even if there is no element that satisfies the first condition and it is true only when each card verifies any of the following clauses: a) The first condition is true and the second condition is true, b) the first condition is false and the second condition is true, and c) the first condition is false and the second condition is false. Because in that way we guarantee that there are no cards that contradict the implication. So, in logic, this statement means "There are no cards that verify the first condition but not the second" (In this case, we do not need any card to fulfill the first condition for this statement to make sense.)
Furthermore, if there exists at least one card that satisfies the first condition, then by guaranteeing the logical implication, we guarantee that that or those cards also satisfy the second statement. That is, we’re guaranteeing that each card that satisfies the first condition also satisfies the second condition.
So we can say that, in this context, the logical implication and the real life implication actually mean exactly the same thing when there exists at least one card that satisfies the first condition.
PD: Question for you, ¿would they mean the same thing if there were no cards that satisfy the first condition?
But how can False (p) with True (q) be always implied to be True?
Suppose I use False (p) with True (q) using this implication - "If your pet is a bird, then your pet is a mammal." Isn't this implication false here?
Thank you very much.
But the sun does not flash purple on Saturdays so the consequent is false no? Have I missed a memo? How can that 'always be true'?
the statement we're evaluating is: If the moon is made of green cheese, then the sun flashes purple on Saturdays.
This is true, because the moon is not made of green cheese, therefore, we don't care if the sun flashes purple or not, the implication is true. We don't care about the two statements themselves, we care if one implies the other. If the first statement is shown to be false, then it implies that anything else is true by convention.
Thank you. Very helpful!
Great video. thank you!
You...... I just can’t believe what you did
I want an example for when the antecedent is true and the consequent is false then the implication is false..
if your pet is a dog fish(type of small shark), it's a dog, but it's not a mammal. Then, if it is a dog then it's a mammal, is not true.
No-one truly understands this operator
I think the same.
I loved it
Can you just explain why "if we don't even care about the consequence value" then the whole statement is defined to be true rather than false?
A better name would be 'the promise gate'
what in the flying fuck led you to that implication
@@keokawasaki7833 If you pass your exam your mother says she will buy you a car. Only way this promise can be false is if you pass the exam and she doesn't buy the car.
@@timseytiger9280 yo thanks! 😘
The Professor sounds like Hyman Roth from Godfather!
Thank you!
Clearly explained🤗
but sir if my pet is a bird then my pet is a mammale
I DONT SEE WHY THE IMPLICATION IS FALSE IF THE ANTECEDENT IS TRUE & THE CONSQUENT IS FALSE SINCE THERE IS NO CAUSALITY????????
The only thing I can think of is: if your pet is a dog -> then your pet is a mammal. It would be logically impossible to have a pet that is a dog and not a mammal (therefore Dog (True) - > mammal (False), is always false. What do you think?