@@bobmarley9905 It can not be that the premise holds but the result does not, that's an interpretation of the de Morgan'd formula. Also that from a wrong premise you can only derive a wrong conclusion.
From 'Material Implication' and 'Contraposition', start with (-P v Q) (P => Q) -P). I take this to mean that (-P v Q) (-Q => -P). But I don't think this holds...keep reading below. P = I'm in the inner sanctum Q = I'm in the Castle Hence, P => Q -Q => -P. If I'm in the inner sanctum, I'm in the castle; if I'm not in the castle, I'm not in the inner sanctum. Yes? But then how can it follow that P=> Q -P v Q as well? I'm not in the inner sanctum or I'm in the castle. Going the other way around: P = I am hungry Q = I eat Hence, P => Q -P v Q. If I am hungry, then I eat; I cannot be hungry and not eat, though I may eat even though I'm not hungry. Yes? But then how can it follow that P => Q -Q => -P as well? I don't eat if I'm not hungry, even though -P v Q states that I may eat even if I'm not hungry. Sure enough, you run it through the truth table and--so long as you take vacuously true statements to be true true statements, they are equivalent. But, if I'm reasoning clearly, they can't be equivalent. Something that occurs to me, though I haven't thought it all the way the through, is that -Q => P seems to be descriptive (this is how it is) whereas -P v Q seems prescriptive (I'm asserting this). Where am I going wrong?
How does material implication hold up? From what I understand P => Q means: if P, then Q. With that being said, how can one infer that if P and Q are false, then P=>Q is true?
Assume the following model. p = I am Miley Cyrus and q = I am ugly. Let's also assume the proposition p -> q (If I am Miley Cyrus, then I am ugly). Now if p is false, I wouldn't be miley cyrus. Therefore it doesn't even matter what q is, because I never lied to you, since I was never Miley Cyrus to begin with. This is called vacuous truth, because if you dont meet the condition of p then you never lied. An implication only becomes false once you lie, which only happens if p is true and q is false.
+Jason Papoutsis If you know the basics about programming, you can see this "vacuous truth" as an If-Then-Else block with an empty Else statement. The code will still run; it just won't do anything when the If condition (P) is not met, because in that case, it will simply ignore everything under Then (Q). In other words, even if you have faulty code under Then (Q is false), the program will never crash (implication is false) unless the If condition is met (P is true).
I don't think this applies to english well: P: Today is monday Q: Tomorrow is tuesday P=>Q: If today's monday, then tomorrow's tuesday. Three things possible: 1. Today is monday and tomorrow is tuesday. (ok) 2. Today is not monday and tomorrow is tuesday (no way) 3. Today is not monday and tomorrow is not tuesday (ok as well) As you can see, only two things are possible. Anyone wants to explain?
If someone were to say to you "If I am hungry, then I eat" and then proceeds to start eating, and when you assume that they are hungry, they say ""no I am not". Would it be coherent for you to then say his claim was true, and him eating really did follow from implication of him being hungry?
@Anarchy made a crucial point. I just wanted to elaborate so that it might help someone. Row 3 of the truth-table for the conditional says the conditional P --> Q is true when P is false and Q is true. P --> Q is logically equivalent to ~P v Q. So: ~(F) v T T v T T Row 4 of the truth-table for the conditional says that the conditional P --> Q is true when P is false and Q is false. Again, P --> Q is logically equivalent to ~P v Q. So: ~(F) v F T v F T There's a really helpful and short paper (only six pages!) by Todd M. Furman called "Making Sense of the Truth Table for Conditional Statements" that explains what I outline above and more in plain idiomatic english. Hope any of that helps. And if it's wrong, please do comment further. I certainly don't want to mislead anyone!
These videos are extremely helpful. However, I think they will be even clearer and easier to follow if you point to the elements in rows or columns you're referring to with a pen/cursor instead of naming them.
Until you come to the ~P v Q column it is not clear if this is ~(P v Q) or the equivalent of Q v ~P The same goes for the the english which would be much clearer if you added the word either before the or, and even more so if you would change the order and have the eating before the no hunger.
"I am not hungry or I eat" is not the same as first possible sentence 1. "I am hungry and I eat" according to truth table. Can someone explain this to me?
"I am hungry and I eat" is just representing one situation in the truth table where p and q are both true. You should compare "I am not hungry or I eat" and "If I am hungry then I eat."
I'm not a logician so excuse my possibly strange questions. FIRST, Why is there a truth value to P->Q when P is false? How does truth transfer when P is denied in the first place? What is the "motivation" in doing it that way? SECOND, How easily do you transition between the argumentation of natural language and formal logic? You give examples in natural language but few people outside of logicians can intuit what you've stated here. I'm wondering whether you, being both a logician and a natural language speaker, can sense a difference and whether this difference is due to the distinct rules of both. THIRD, What knowledge can logic produce? For instance, you say that ~P v Q is the same as P->Q . Is this knowledge? What is it supposed to be? I mean, how does it fit into the analyses of the real world when we're trying to sort stuff out? (Recall that you gave examples regarding being hungry and eating.)
1) He talked about that in a previous video, this is called "vacuous truth". For example, if you have the statement, all cellphones in the room are off, this is true when there are cellphones in the room and they are off, it is false if there are cellphones in the room but they are on, and it is true if there are no cellphones in the room. Which could be represented by: C: There are cellphones in the room O: The cellphones are off C=>O
2) Natural language can be very dependant on meaning and is can be very vague, while logic is very specific. So it can be difficult to translate from Natural language to logic, but very easy in reverse
3) This is often used in proving statements and arguments, and is used in many fields, like lawyers, computer science, philosophy, etc. Although in real world situations it can tell you the same conclusion in a different way.
You need to handle the following cases: 1. premise = 1, deduction = 1 2. premise = 0, deduction = 1 3. premise = 1, deduction = 0 4. premise = 0, deduction = 0 Then 1. is valid, from true you derive true. 2. When the premise is false, it has no bearing on the truth value of the result, else there must be a connection between them. This is the vacuous truth. 3. From a true statement you can never derive a false one. 4. You derive false from false, and that is valid. As you see, the material implication does not check for the semantic consistency of the implication, but that should be handled by the formal system, then, when formalized, the material implication assigns the correct truth values under assumed semantic pattern matching based on whether the preconditions or results are consistent. Look into model theory and interpretations, offers more flexibility than classical logic.
@@Wabbelpaddel It's too late now, but if you read comment carefully carefully, then you'd realize I was looking for real life relevance. Not interested in a discussion unless you say something extraordinarily interesting. I'm likely to erase my original comment.
Confused. What if, in natural language “if P then Q” stands for “if a dog, then a mammal”, . And this is equivalent to “either not a dog or a mammal”? I guess there are so many non-dog mammals that if feels this concept could not be relied on well for natural language. What am I missing? Thanks for any possible help
The confusion spring from considering these two statements the same. 'If a dog, then a mammal' and ' either not a dog or mammal' are logically equivalent, that is, their truth values are the same in any specific case, which does not have anything to do with their meaning and usage in the natural languages. They are just equivalent logically.
As far as I can tell, "material implication" just means that you can replace a conditional statement (A => B) with a disjunction (~A v B). I think that converting from one form to another can help simplify things while working on a proof, and I am betting that William might address that in upcoming videos.
Don't mind this comment, it's a bookmark just for me: "IF I AM HUNGRY, THEN I EAT." (Implication) 1. I AM HUNGRY AND I EAT: (P = T THEN Q = T) = T 2. I AM NOT HUNGRY AND I EAT: (P = F THEN Q = T) = T (vacuously true) 3. I AM NOT HUNGRY AND I DON'T EAT: (P = F THEN Q = F) = T (vacuously true) I AM HUNGRY AND NOT EAT: (P = T THEN Q = F ) = F ----------------------------------------------------------------------------------------------------------------------------------- "I AM NOT HUNGRY OR I EAT." (Disjunction) 1. I AM HUNGRY AND I EAT: (P = F OR Q = T) = T 2. I AM NOT HUNGRY AND I EAT: (P = T OR Q = T) = T 3. I AM NOT HUNGRY AND I DON'T EAT: (P = T OR Q = F) = T I AM HUNGRY AND NOT EAT: (P = F OR Q = F ) = F
Note that according to DeMorgan's law, (~P v Q) is the same as ~(P ∧ ~Q), which makes much more intuitive sense when you compare it to P -> Q.
could u explain why?
@@bobmarley9905 It can not be that the premise holds but the result does not, that's an interpretation of the de Morgan'd formula.
Also that from a wrong premise you can only derive a wrong conclusion.
I've been watching so many videos about implication, and I still don't understand it.
😭
why do you need p=>Q when you have ~P v Q?
Material implication? More like "it's knowledge you're spittin!" Thanks for making and posting such informative videos.
From 'Material Implication' and 'Contraposition', start with (-P v Q) (P => Q) -P). I take this to mean that (-P v Q) (-Q => -P). But I don't think this holds...keep reading below.
P = I'm in the inner sanctum
Q = I'm in the Castle
Hence, P => Q -Q => -P. If I'm in the inner sanctum, I'm in the castle; if I'm not in the castle, I'm not in the inner sanctum. Yes?
But then how can it follow that P=> Q -P v Q as well? I'm not in the inner sanctum or I'm in the castle.
Going the other way around:
P = I am hungry
Q = I eat
Hence, P => Q -P v Q. If I am hungry, then I eat; I cannot be hungry and not eat, though I may eat even though I'm not hungry. Yes?
But then how can it follow that P => Q -Q => -P as well? I don't eat if I'm not hungry, even though -P v Q states that I may eat even if I'm not hungry.
Sure enough, you run it through the truth table and--so long as you take vacuously true statements to be true true statements, they are equivalent. But, if I'm reasoning clearly, they can't be equivalent.
Something that occurs to me, though I haven't thought it all the way the through, is that -Q => P seems to be descriptive (this is how it is) whereas -P v Q seems prescriptive (I'm asserting this).
Where am I going wrong?
How does material implication hold up? From what I understand
P => Q means: if P, then Q. With that being said, how can one infer that if P and Q are false, then P=>Q is true?
Assume the following model. p = I am Miley Cyrus and q = I am ugly. Let's also assume the proposition p -> q (If I am Miley Cyrus, then I am ugly). Now if p is false, I wouldn't be miley cyrus. Therefore it doesn't even matter what q is, because I never lied to you, since I was never Miley Cyrus to begin with. This is called vacuous truth, because if you dont meet the condition of p then you never lied. An implication only becomes false once you lie, which only happens if p is true and q is false.
+Jason Papoutsis If you know the basics about programming, you can see this "vacuous truth" as an If-Then-Else block with an empty Else statement. The code will still run; it just won't do anything when the If condition (P) is not met, because in that case, it will simply ignore everything under Then (Q).
In other words, even if you have faulty code under Then (Q is false), the program will never crash (implication is false) unless the If condition is met (P is true).
This is the best explanation I’ve found so far.
Because from a false premise you can only derive a false statement (exclusion of the middle), hence the derivation is correct, by the truth values.
Logic exploring the boundaries of language; noice!
I don't think this applies to english well:
P: Today is monday
Q: Tomorrow is tuesday
P=>Q: If today's monday, then tomorrow's tuesday.
Three things possible:
1. Today is monday and tomorrow is tuesday. (ok)
2. Today is not monday and tomorrow is tuesday (no way)
3. Today is not monday and tomorrow is not tuesday (ok as well)
As you can see, only two things are possible.
Anyone wants to explain?
If someone were to say to you "If I am hungry, then I eat" and then proceeds to start eating, and when you assume that they are hungry, they say ""no I am not". Would it be coherent for you to then say his claim was true, and him eating really did follow from implication of him being hungry?
@Anarchy made a crucial point. I just wanted to elaborate so that it might help someone.
Row 3 of the truth-table for the conditional says the conditional P --> Q is true when P is false and Q is true.
P --> Q is logically equivalent to ~P v Q. So:
~(F) v T
T v T
T
Row 4 of the truth-table for the conditional says that the conditional P --> Q is true when P is false and Q is false.
Again, P --> Q is logically equivalent to ~P v Q. So:
~(F) v F
T v F
T
There's a really helpful and short paper (only six pages!) by Todd M. Furman called "Making Sense of the Truth Table for Conditional Statements" that explains what I outline above and more in plain idiomatic english.
Hope any of that helps. And if it's wrong, please do comment further. I certainly don't want to mislead anyone!
These videos are extremely helpful. However, I think they will be even clearer and easier to follow if you point to the elements in rows or columns you're referring to with a pen/cursor instead of naming them.
Until you come to the ~P v Q column it is not clear if this is ~(P v Q) or the equivalent of Q v ~P
The same goes for the the english which would be much clearer if you added the word either before the or,
and even more so if you would change the order and have the eating before the no hunger.
"I am not hungry or I eat" is not the same as first possible sentence 1. "I am hungry and I eat" according to truth table. Can someone explain this to me?
"I am hungry and I eat" is just representing one situation in the truth table where p and q are both true.
You should compare "I am not hungry or I eat" and "If I am hungry then I eat."
I'm not a logician so excuse my possibly strange questions.
FIRST, Why is there a truth value to P->Q when P is false? How does truth transfer when P is denied in the first place? What is the "motivation" in doing it that way?
SECOND, How easily do you transition between the argumentation of natural language and formal logic? You give examples in natural language but few people outside of logicians can intuit what you've stated here. I'm wondering whether you, being both a logician and a natural language speaker, can sense a difference and whether this difference is due to the distinct rules of both.
THIRD, What knowledge can logic produce? For instance, you say that ~P v Q is the same as P->Q . Is this knowledge? What is it supposed to be? I mean, how does it fit into the analyses of the real world when we're trying to sort stuff out? (Recall that you gave examples regarding being hungry and eating.)
1) He talked about that in a previous video, this is called "vacuous truth".
For example, if you have the statement, all cellphones in the room are off, this is true when there are cellphones in the room and they are off, it is false if there are cellphones in the room but they are on, and it is true if there are no cellphones in the room.
Which could be represented by:
C: There are cellphones in the room
O: The cellphones are off
C=>O
2) Natural language can be very dependant on meaning and is can be very vague, while logic is very specific. So it can be difficult to translate from Natural language to logic, but very easy in reverse
3) This is often used in proving statements and arguments, and is used in many fields, like lawyers, computer science, philosophy, etc. Although in real world situations it can tell you the same conclusion in a different way.
You need to handle the following cases:
1. premise = 1, deduction = 1
2. premise = 0, deduction = 1
3. premise = 1, deduction = 0
4. premise = 0, deduction = 0
Then 1. is valid, from true you derive true.
2. When the premise is false, it has no bearing on the truth value of the result, else there must be a connection between them. This is the vacuous truth.
3. From a true statement you can never derive a false one.
4. You derive false from false, and that is valid.
As you see, the material implication does not check for the semantic consistency of the implication, but that should be handled by the formal system, then, when formalized, the material implication assigns the correct truth values under assumed semantic pattern matching based on whether the preconditions or results are consistent.
Look into model theory and interpretations, offers more flexibility than classical logic.
@@Wabbelpaddel It's too late now, but if you read comment carefully carefully, then you'd realize I was looking for real life relevance. Not interested in a discussion unless you say something extraordinarily interesting. I'm likely to erase my original comment.
This video was extremely helpful for me.
THANK YOU SO SO SO SO MUUUUUUUCH !!!!!!!!!!!
Confused. What if, in natural language “if P then Q” stands for “if a dog, then a mammal”, . And this is equivalent to “either not a dog or a mammal”? I guess there are so many non-dog mammals that if feels this concept could not be relied on well for natural language. What am I missing? Thanks for any possible help
The confusion spring from considering these two statements the same. 'If a dog, then a mammal' and ' either not a dog or mammal' are logically equivalent, that is, their truth values are the same in any specific case, which does not have anything to do with their meaning and usage in the natural languages. They are just equivalent logically.
I need to understand this but I can't. If anyone can help, please contact me.
As far as I can tell, "material implication" just means that you can replace a conditional statement (A => B) with a disjunction (~A v B). I think that converting from one form to another can help simplify things while working on a proof, and I am betting that William might address that in upcoming videos.
This is amazing! Simple but profound!
How do i apply this in real life?
Man what is this,
I know it makes sense I know how it makes sense but I don't know WHY it makes sense
Thank you!
Blurd can't see
Don't mind this comment, it's a bookmark just for me:
"IF I AM HUNGRY, THEN I EAT." (Implication)
1. I AM HUNGRY AND I EAT: (P = T THEN Q = T) = T
2. I AM NOT HUNGRY AND I EAT: (P = F THEN Q = T) = T (vacuously true)
3. I AM NOT HUNGRY AND I DON'T EAT: (P = F THEN Q = F) = T (vacuously true)
I AM HUNGRY AND NOT EAT: (P = T THEN Q = F ) = F
-----------------------------------------------------------------------------------------------------------------------------------
"I AM NOT HUNGRY OR I EAT." (Disjunction)
1. I AM HUNGRY AND I EAT: (P = F OR Q = T) = T
2. I AM NOT HUNGRY AND I EAT: (P = T OR Q = T) = T
3. I AM NOT HUNGRY AND I DON'T EAT: (P = T OR Q = F) = T
I AM HUNGRY AND NOT EAT: (P = F OR Q = F ) = F