I had a quick mnemonic of sorts to remember the truth table for ‘If.. Then’ statements that I thought I’d share in case anyone would benefit from it. It’s basically the ‘Poison Ivy’ mnemonic. The way we think about it is let P: You touch poison ivy Q: You get a rash P => Q T , T -> If you touch poison ivy => you get a rash. This is always true. T , F -> If you touch poison ivy => you don’t get a rash. This is never true. It’s assumed that you always get a rash when you touch poison ivy, so this combination is false. F , T -> If you don’t touch poison ivy => you get a rash. This is possible. You can get a rash from other things. So this is true. F , F -> If you don’t touch poison ivy => you don’t get a rash. This makes sense. If you don’t touch something, it’s impossible to get a rash from it, so this is true as well. Making all true except the (T,F) combination :-) This is my second time going through this lecture series, the first time was for my midterm and now it’s to revise for my finals.I did really, really, well on my midterm because of this series, I’m super grateful to you for making it :-) Many thanks!
For me it is only mnemonic. I do understand the mathematical harmony of equivalence, but the problem is, that I don’t really view the „if and only then” as a simple mathematical equivalence, but rather my mind imagines it as a sort of „hard” implication. I know this is academically wrong, I have been taught this and i remember this, but outside of class, I still have a rebellious responce to such system. I understand that a true statement derived from false premise remains to be true after all, but cmon. Yes, „my hair is not my hair” therefore „tommorow has to be a next day after today” has the right conclusion because of its definitional identity, but since it is derived from a premise that has to be false by the same law (ignoring the absurdity of why would someone think that the coming of a new day would have anything to do with a hair that isn’t what it is), should the truthfulness not be considered likewise a conjunction? It just feels very wrong that coming to the right conclusion through utter nonsense is still logically considered correct. The equivalent of that would be to solve a mathematical problem on a test by using a completely false method, making an error in calculatation and coming up with the correct number for the question by pure random chance… Would You give a point for that?
I used to have a problem with this idea of assignment of truth values to what appear to be trivial cases. I put this down to my electrical engineering education in which emphasis was always on analysing definitive states with logic gates. The interpretation that best helped me was to consider what 'truth' was being tested as whether, in the IF conditional, the THEN promise, was or wasn't being kept. After this, and now, it seems silly to observe the so called 'vacuously' true assignments as anything other than totally necessary and valid !
I think a more intuitive way to explain this concept is with a real life example. For example, if we have M meaning murderer, C meaning criminal. Firstly, if you are M, you are C, make sense, it's true. Secondly, if you are M, you are not C, doesn't make sense, false. Thirdly, if you are not M, yet you are C, it could happen because you could be doing other crimes, so we have to let it be true. Lastly, if you are not M, and not C, it could also happen, you're simply not a criminal, so we have to let it be true. Basically, if it's possible, it has to be true. If it's impossible, it has to be false. I think this is intuitive.
Your example describes the definition well, but it does not _justify_ the definition, which was the point of this video. Justification of the definition is necessary because the definition itself is counterintuitive/unsupported in the case of false antecedents. If it is the case that "if M is true then C is true", then like you said, when M is false it's _possible_ that the conditional (and thus the consequent) could be true. *But* it's also _equally_ as possible that the conditional (and thus the consequent) could be false. We can't know either way, because what's implied _when M is true_ is entirely irrelevant to what's implied when M is false. Case in point, if I said "I know that [you're not a murder], but I also know that [if you _were_ a murder then you would be a criminal], therefore [you _must_ be a criminal]", you would almost certainly disagree, because that's actually not intuitively coherent at all. (Yet it's exactly what "vacuous truths" would imply, thus the dilemma.) In actuality there's no way I could know whether you're a criminal or not based only on knowing that you're _not_ a murderer, because with "if _M is true_ then C is true" we're given _zero_ information about what happens when M is false; we only know what happens when it's true. Therefore, since saying "it could be true, therefore it must be true" makes exactly as much sense as saying "it could be false, therefore it must be false" (which is to say, no sense), further justification for choosing one designation over the other is warranted (which, in this case, is specifically the impact of each designation on the coherence of biconditionals).
Ive just watched 17 Material Implication. Vacuous truth, to me, seems more sensible in terms of this rather than the bicondition argument. These little lectures are really good. Keep up the good work!
Not sure if I said this before, but the "vacuous-truth" concept should make more sense when you put it into the context of a computer program or a game. For example, the statement "if you get hit, you take damage" can only be falsified when a hit does 0 damage, since there can be other sources of damage, like poison and hazards. Similarly, the statement "if you take damage, you are hit" can only be falsified when the damage is from other sources, since attacks can miss or do 0 damage. Therefore, "you take damage if and only if you get hit" is proven when you get hit and take damage (T/T) or when an attack misses and you take no damage (F/F), and falsified when there are 0-damage hits (T/F) or other sources of damage (F/T). To understand why "vacuous falsity" doesn't make sense, simply read the statement "if you get hit, you take damage" as "all hits do damage." If the statement were vacuously false when you aren't hit, it would imply that you might not take damage when you do get hit, i.e. there can be 0-damage hits (assuming F condition = F statement). But since the statement is true when you get hit and take damage (T condition + T result = T statement), there's an obvious contradiction.
Can you make this argument with the empty room, all mobile phones in that room are on, example? Because as a programmer i cant wrap my head around it. If i don't initialize a phone object and check it for a boolean it will return an undefined not a true.
@@hermaeusmora424 Think of there being a variable X for how many phones are in the room, and a variable Y for how many phones are both in the room and turned on. In the case of there being no phones in the room, X = Y because both are 0.
@@hermaeusmora424 "If i don't initialize a phone object and check it for a boolean it will return an undefined not a true." 100%. That's why this idea is bullshit. You have no outcome for when P is not true.
Hi William, the truth table of conditional statement shows that when P is false, the vacuously true statements are required to satisfy the PQ logic. However, if the proof is simply a matter of putting in the T and F in the truth table to satisfy some logic equivalence, it seems by having p=f, q=t as vacuously false, while maintaining p=f, q=f as vscuously true also satisfy the logic requirement. I also verified it with the contraposition of p=>q. That said, were there more solid proof of why vacuously true is the way it is ?
It seems easier to understand if-then this way: Reject P And Not Q, everything else is ok. Example, P is "A person born in US", Q is "the person is US citizen". It can be understood as, it is not acceptable (or should be rejected) if a person is born in US, but Not a US citizen, everything else is ok.
The last sentence in this paragraph in another document helped me nail the 'Vascoulsy True" issue! - Conditional Statements In conditional statements, "If p then q" is denoted symbolically by "p q"; p is called the hypothesis and q is called the conclusion. For instance, consider the two following statements: If Sally passes the exam, then she will get the job. If 144 is divisible by 12, 144 is divisible by 3. Let p stand for the statements "Sally passes the exam" and "144 is divisible by 12". Let q stand for the statements "Sally will get the job" and "144 is divisible by 3". The hypothesis in the first statement is "144 is divisible by 12", and the conclusion is "144 is divisible by 3". The second statement states that Sally will get the job if a certain condition (passing the exam) is met; it says nothing about what will happen if the condition is not met. If the condition is not met, the truth of the conclusion cannot be determined; the conditional statement is therefore considered to be vacuously true, or true by default. - "it says nothing about what will happen if the condition is not met" --!!!
** if you couldn't understand something and automatically assumed it was false you would never reapproach that issue so would never solve the problem. Unless you assume its true. (until proven false) ** Kinda goes against intuition?
"if Sally passes the exam, then she will get the job." So then we know for a fact (F,T) can not allow the statement to be true, as you would be claiming that sally got the job, without passing the exam, meaning that the claim of passing the exam => getting the job, must be false. They must be independent events if I can get the job without the exam. (T,F) is false because it allows one to pass the exam without reaping its implication, which would make said implication, not an implication. Why does (F,T) not follow this same logic? Where not passing the exam, still allows you to reap the implications of passing the exam. Does that not mean passing the exam failing the exam, as both have the same implication, getting the job. The implication will then look like this (PvQ)=>R, where P = passing Q = failing and R = getting the job
It's 7 years late. Idk if you'll read this. I don't think it matters much but it would be awesome. If could've real sentence based example of this lecture.
Ok so I get why its import for them to be truth for the bi-conditional cause its a matching game, but I don't understand what the meaningful interpretation of the vacuously true statements are. Like going back to the conditional lecture: If you have finished your homework, then you can play video games, so in that the 3rd row would be where you haven't finished your homework but you can still play video games?
Vacuously true statements often lose coherent meaning precisely because of the vacuousness. For your example, let H you finished your homework and V represent you can play video games. So H => V. Suppose ~H. Then there is no direct relationship between the homework not being done and being able to play video games. You may very well be allowed to play video games--perhaps you need your math tutor to finish your homework assignment, and he isn't there currently. Or you might not. So, really, the implication is asking you to think about a counterfactual world. If you really had finished your homework, would you be allowed to play video games? If the answer is yes, the implication is still fine. If the answer is no, the implication is false, since it claimed that fulfilling H means V is fulfilled as well.
There is another reason for those vacuously true statements to be true. Suppose they were false. P=>Q would be logical equivalent to Q=>P would be logical equivalent to PQ would be logical equivalent to P^Q, which doesn't make any sense at all.
I don't think they make sense if they're true or false. I think they should be undefined. That's a more conservative approach anyway. I mean, you can say if George Washington is alive in 2024 then grass is green. And you can also say if George Washington is not alive in 2024 then grass is not green. Those are contradictory statements, and since the premise is false it allows contradictions, which I can never see as being a good thing.
I cant buy the whole argument for vacuously true. Though it makes sense in the last row. However 'if false then false' makes sense without any need to appeal to vacuity. I still dont comprehend why its not permissible to take 'if false then true' as a false statement.
heavymedley Think of it in cause and affect. For example "If it is my birthday, then I will eat cake." The argument shows that I will eat cake on my birthday, but it doesn't say that I won't eat cake if it isn't my birthday. I still may or may not eat cake any other day, but if it is my birthday, I will DEFINITELY eat cake.
@@headlesskoala How do we turn your example around? "If I eat cake, it is my birthday". So if I don't eat cake, it is my birth; and if I don't eat cake, it is not my birthday, are these vacuously true?
If if->then truth table only has True in p=true and q=true, then it is the same as p AND q, which is not what conditional or implication statement is intended for.
I looked this up because I don't accept the concept of vacuous truth at all. When given a statement: "If P then Q" The truth of Q is dependent on P being true such that when P is true, Q. In computer programming, Q will be an action carried out subject to the entry condition P being met. But you have no contingency for P being false. So if your system has only one condition, which is Q when P, then you have no condition for not P. Therefore, when the test condition is false, then neither truth nor falsehood can be determined. Let's put the same facts a different way: If you have a logic system which you can query, and in your system you have the one statement "If P then Q", and you then query the system with "When P, Q?", then it's true. If you query "When P, not Q?", the response is false. But, if you query ""When not P", because you have no case in the system for not P it is undecidable. Meanwhile, there isn't really any valid "truth table" for this kind of state expression like there is for conjunction and distinction. Realistically, it's a contingency listing.
Yeah, the words "if" and "then" are a little confusing. Think of it as "P is a sufficient but not necessary condition for Q" and "Q is a necessary but not sufficient condition for P".
@@Nicoder6884 Why? You have given no explanation for why anybody would think of it like that. When given the statement "If P then Q". You know P is sufficient for Q but you Don't know if it's necessary. Q Might be true if, and only if, P is true. You have no statement to the contrary so you don't know if it's necessary or not. Also, given the true statement "If P then Q", the only truth you can derive from querying "Q" is that when Q is false P must also be false so your second statement is wrong. If P then Q then rearranging for Q you get "Not Q is a sufficient condition for not P" and that's all you have.
@@vapourmile I don't think you understand the meanings of sufficient and necessary. P is not necessary for Q, but Q is a necessary consequence of Q being true.
This argument only holds based on practical everyday use. However everybody feels that it is a bit funny. You could argue that columns 3 and beyond only ask if the implication are "not falsified". But being "not falsified" is not the same as being true. Mathmatical conjectures are not necessarily true. They are basically undetermined until proven or disproven.
x implies y and y implies z ,y is false what can be concluded ?? You people never discuss such condition. What you do is just the common knowledge . If you are very aware of teaching then reply me this problem i mention above then i would be very happy. I hope you will do this. I wait for explaination .
What if you make vacuous statements the empty set, because it’s undefined? T empty empty empty sounds good to me. Empty equals empty is undefined, hence empty. intersection between false and empty is still empty.
Surly the only information we have from the expression, P => Q, P => Q[(T,T) = T], (The truth values within the brackets are pre-supposed on the basis the original P,Q will always be assumed true) Is that, we are to assume that P brings about, or is sufficient, for Q, and therefore are connected on that basis. Any other information is an extrapolation from said basis. Such include, P => Q(F,F) = T, Where we are to extrapolate, if P and Q are connected, on the basis that P is to bring about Q, then it must be true that when P does not happen, so too will Q not happen. From this, we can draw the equivalence, [P => Q(F,F)] [-P => -Q(T,T)] = T And, [P => Q(T,T)] [-P => -Q(F,F)] = T We can use these two equivalences to define the parameters in which, P implies Q, P => Q, where, we can only logically assume that P implies Q, IF, ([P => Q(T,T)] [-P => -Q(F,F)]) => ([P => Q(F,F)] [-P => -Q(T,T)]) = T, THEN, P => Q Conclusion: The following truth values, (F,T) and (T,F), must be false for all cases of implicit conditions, as the disjunction of the truth values, where one can happen, irrespective of the other, is contradictory to the nature of implication, where a logical connection is being established, and an event Q, follows from the truth of an event P.🤔
Thank you. This explains it - the biconditional statement (as a conjunction of the two conditionals) would not be able to recover the fact that “if P is false, Q is false - and vice versa” is a true statement. Right? Thanks
I know this is a very old video. But could you repost with either written directions or verbal corrections? You really confuse me when you start getting inconsistent with rows when you refer to a row as 4th row and then go back and say 5th row. Overall great job and very useful videos but you lose me a little in this one with inconsistency
The best example is the theorem that a empty set is a subset of all sets. This means that IF x belongs to empty set THEN it belong to any set. The antecedent is always false since nothing belongs to the empty set, so the proposition is true.
@@pingdingdongpong The antecedent does not claim that there is something in the empty set, just that if there were, it would be a part of any set, which is true based on, empty set => a member of all sets. I ask, how can you formulate (F,T)? Do you claim that, not belonging to the empty set => it belongs to any set?
@@TheMorhaGroup Empty set is a subset of all sets mathematically means: for all x, x in empty set implies that x is also in whatever set. For all x, x in empty set is false for all x by definition, therefore the antecedent is always false, hence the implication is always true.
@@pingdingdongpong ""IF x belongs to an [viz] empty set THEN it belongs to any set" does not mean by any necessity, that x has to belong in the empty set, it can mean, x, if it where to be in the empty set, would then also be a part of any set. The statement can be speaking on the quality of the empty set alone, and that it has the quality of being a part of any set, and if it were to have an element, that element in this case being x, the quality of being in any set, would be extended to that element.
I have just watched this again and it gets even worse when it comes to the fourth column. First off, it's Querying the relationship between Q and P "If Q is true is P true?". That is a completely different thing from testing the logical Assertion "When Q is true, P is true". Also, the question he is trying to answer is "Why are they true? Why aren't they false?". It's a false dichotomy which proves the invalidity of bivalent logics. In the first isn't when P is not true, it automatically doesn't matter what Q is, because you have no implication statement for when P is not true.
Assumption that P=>Q when P is false has the same truth value irrespective of Q. Perfectly consistent to have it true when Q false but false when Q is true.
I think William was trying to show how defining them to be false lead to something that didn't make sense, and so he "corrected" the definition back to the usual value of true.
I think William was just trying to show that if they were defined to be false, it would lead to something that seemed a bit off. But once he reverted them to true values, things made more sense. Namely, p q should be true if both p and q are false, since as William noted, the biconditional in logic is analogous to the equal sign in math.
Logic really seams unlogical. P -> Q. If P and Q are true, then the whole expression P->Q is true. What if, say, P is 'the Earth is round', and Q is 'Michael want to eat a fruit cake". P and Q are true, so, "If the Earth is round, then Michael want to eat a fruit cake". What it even means for it expression to be true? P and Q in this case totally unrelated.
Logic reasoning is about accepting or rejecting the state of one or more(composite) statements, the state being binary, namely true or false, black or white, 1 or 0..... It is up to the creators of the statement to accept or reject the statement, they dont need to sound related to anyone else.
We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg. If 3 squared = 27,then 2+2=5. Can we check if it is indeed true that 3 squared equals 27 then 2+2 is not 5. No. Because 3 squared equals 27 is non-existent. Or false. So we can't check if the statement is false. Hence it must be true.
@@MathCuriousity not really,unless you can show it's false,its true....if it is false,why can't one show it?...thst's what i get from this topic.it's a loopish one really.
@@MathCuriousity note,in my example,the statement is not that 2+2=5 is true...in the example i am saying the whole satement, if 3 squared =27,then 2+2=5 is true as vacuous truth or false implies truth or whatever you call it.
@@rishabhnarula1999 I get what you are saying - but when we dig down - what we realize is vacuous truths are conventional so when evaluating p implies q for when it’s true based on the true or false states of p and q, then when we get to the two situations where p are false ( FT and FF), we basically default to it being vacuously true with those state combinations because we cannot falsify something that doesn’t exist ie we cannot falsify p implies q if p doesn’t exist which it doesn’t if we are saying p is false.
in fact that's not at all accurate because you are saying that if you ever stop watching TH-cam videos then pigs will suddenly fly, and that's simply not the case. The antecedent has to be always false and the consequent can either be true or false in order for the conditional to be vacuously true. "The day string cheese is made out of yarn, is the day that cheese will stop being orange"
@@StaticBlaster No, the statement I wrote is equivalent to: IF pigs can fly, THEN I will stop watching TH-cam. In this case the antecedent is always FALSE. Having the antecedent at the end of the original sentence can be confusing, but that's just the English language for ya!
Is there are difference between a "truth table" and "the method of assigning values"? I thought they were the same and now I'm studying and I'm not sure
This makes absolutely no sense in real life. Lets take 2 statements, "if i eat a mango, i will die", "if i eat a mango, i will live." Here p is "if i eat a mango" and q1 is "i will live" and q2 is "i will die". In other words q1 is not-q2. If i don't eat a mango, i.e, p is false, then both statements have to be true, vacuously. But both statements are literally the opposite of each other and cannot be simultaneously true. This is just ridiculous and illogical.
The if-then statement has its logic based on the entire evaluation of the if and then. It seems you were only looking at the "then" individually. To counter an example, in the case of P OR Q, if P is true, Q could be false or true, the result still true, but one would not argue that Q and Not Q are the same yet they are opposite, what do you think ? On the other hand, if vacuously true were to be changed to vacuously false, the if-then statement becomes identical to the AND statement, as only when both p and q are true the statement is evaluated true. That mustnt be the reason why if-then statement was created.
I had a quick mnemonic of sorts to remember the truth table for ‘If.. Then’ statements that I thought I’d share in case anyone would benefit from it.
It’s basically the ‘Poison Ivy’ mnemonic.
The way we think about it is let
P: You touch poison ivy
Q: You get a rash
P => Q
T , T -> If you touch poison ivy => you get a rash. This is always true.
T , F -> If you touch poison ivy => you don’t get a rash. This is never true. It’s assumed that you always get a rash when you touch poison ivy, so this combination is false.
F , T -> If you don’t touch poison ivy => you get a rash. This is possible. You can get a rash from other things. So this is true.
F , F -> If you don’t touch poison ivy => you don’t get a rash. This makes sense. If you don’t touch something, it’s impossible to get a rash from it, so this is true as well.
Making all true except the (T,F) combination :-)
This is my second time going through this lecture series, the first time was for my midterm and now it’s to revise for my finals.I did really, really, well on my midterm because of this series, I’m super grateful to you for making it :-)
Many thanks!
this isn't really a mnemonic it's more of a really good example that fits implication very well, thanks a lot man
Freaking finally! An example and explanation that helped me make sense of this!
For me it is only mnemonic. I do understand the mathematical harmony of equivalence, but the problem is, that I don’t really view the „if and only then” as a simple mathematical equivalence, but rather my mind imagines it as a sort of „hard” implication.
I know this is academically wrong, I have been taught this and i remember this, but outside of class, I still have a rebellious responce to such system.
I understand that a true statement derived from false premise remains to be true after all, but cmon.
Yes, „my hair is not my hair” therefore „tommorow has to be a next day after today” has the right conclusion because of its definitional identity, but since it is derived from a premise that has to be false by the same law (ignoring the absurdity of why would someone think that the coming of a new day would have anything to do with a hair that isn’t what it is), should the truthfulness not be considered likewise a conjunction?
It just feels very wrong that coming to the right conclusion through utter nonsense is still logically considered correct.
The equivalent of that would be to solve a mathematical problem on a test by using a completely false method, making an error in calculatation and coming up with the correct number for the question by pure random chance…
Would You give a point for that?
Thank you
This mnemonic helped me make sense of truth tables for "if...then" statements. Thank you!
I used to have a problem with this idea of assignment of truth values to what appear to be trivial cases. I put this down to my electrical engineering education in which emphasis was always on analysing definitive states with logic gates. The interpretation that best helped me was to consider what 'truth' was being tested as whether, in the IF conditional, the THEN promise, was or wasn't being kept. After this, and now, it seems silly to observe the so called 'vacuously' true assignments as anything other than totally necessary and valid !
I think a more intuitive way to explain this concept is with a real life example. For example, if we have M meaning murderer, C meaning criminal. Firstly, if you are M, you are C, make sense, it's true. Secondly, if you are M, you are not C, doesn't make sense, false. Thirdly, if you are not M, yet you are C, it could happen because you could be doing other crimes, so we have to let it be true. Lastly, if you are not M, and not C, it could also happen, you're simply not a criminal, so we have to let it be true.
Basically, if it's possible, it has to be true. If it's impossible, it has to be false. I think this is intuitive.
Your example describes the definition well, but it does not _justify_ the definition, which was the point of this video. Justification of the definition is necessary because the definition itself is counterintuitive/unsupported in the case of false antecedents. If it is the case that "if M is true then C is true", then like you said, when M is false it's _possible_ that the conditional (and thus the consequent) could be true. *But* it's also _equally_ as possible that the conditional (and thus the consequent) could be false. We can't know either way, because what's implied _when M is true_ is entirely irrelevant to what's implied when M is false. Case in point, if I said "I know that [you're not a murder], but I also know that [if you _were_ a murder then you would be a criminal], therefore [you _must_ be a criminal]", you would almost certainly disagree, because that's actually not intuitively coherent at all. (Yet it's exactly what "vacuous truths" would imply, thus the dilemma.) In actuality there's no way I could know whether you're a criminal or not based only on knowing that you're _not_ a murderer, because with "if _M is true_ then C is true" we're given _zero_ information about what happens when M is false; we only know what happens when it's true. Therefore, since saying "it could be true, therefore it must be true" makes exactly as much sense as saying "it could be false, therefore it must be false" (which is to say, no sense), further justification for choosing one designation over the other is warranted (which, in this case, is specifically the impact of each designation on the coherence of biconditionals).
Ive just watched 17 Material Implication. Vacuous truth, to me, seems more sensible in terms of this rather than the bicondition argument. These little lectures are really good. Keep up the good work!
It would be helpful to point out that this is the only truth table assignment for p->q for pq to have the desired outcome.
I've learned more in 3 of your videos than all my readings and in class lectures 😒 Thank you!!
I'm glad we both found William's videos! How did the rest of your class and the final go?
thank you so much for these videos.. i have trouble understanding my symbolic logic lectures at uni and this broke it down slowly for me!
How'd the rest of your class go?
Not sure if I said this before, but the "vacuous-truth" concept should make more sense when you put it into the context of a computer program or a game.
For example, the statement "if you get hit, you take damage" can only be falsified when a hit does 0 damage, since there can be other sources of damage, like poison and hazards.
Similarly, the statement "if you take damage, you are hit" can only be falsified when the damage is from other sources, since attacks can miss or do 0 damage.
Therefore, "you take damage if and only if you get hit" is proven when you get hit and take damage (T/T) or when an attack misses and you take no damage (F/F), and falsified when there are 0-damage hits (T/F) or other sources of damage (F/T).
To understand why "vacuous falsity" doesn't make sense, simply read the statement "if you get hit, you take damage" as "all hits do damage."
If the statement were vacuously false when you aren't hit, it would imply that you might not take damage when you do get hit, i.e. there can be 0-damage hits (assuming F condition = F statement). But since the statement is true when you get hit and take damage (T condition + T result = T statement), there's an obvious contradiction.
Can you make this argument with the empty room, all mobile phones in that room are on, example? Because as a programmer i cant wrap my head around it. If i don't initialize a phone object and check it for a boolean it will return an undefined not a true.
@@hermaeusmora424 Think of there being a variable X for how many phones are in the room, and a variable Y for how many phones are both in the room and turned on. In the case of there being no phones in the room, X = Y because both are 0.
@@Nicoder6884
LOL What rubbish.
@@hermaeusmora424
"If i don't initialize a phone object and check it for a boolean it will return an undefined not a true."
100%. That's why this idea is bullshit. You have no outcome for when P is not true.
Hi William, the truth table of conditional statement shows that when P is false, the vacuously true statements are required to satisfy the PQ logic. However, if the proof is simply a matter of putting in the T and F in the truth table to satisfy some logic equivalence, it seems by having p=f, q=t as vacuously false, while maintaining p=f, q=f as vscuously true also satisfy the logic requirement. I also verified it with the contraposition of p=>q. That said, were there more solid proof of why vacuously true is the way it is ?
It seems easier to understand if-then this way:
Reject P And Not Q, everything else is ok.
Example, P is "A person born in US", Q is "the person is US citizen".
It can be understood as, it is not acceptable (or should be rejected) if a person is born in US, but Not a US citizen, everything else is ok.
The last sentence in this paragraph in another document helped me nail the 'Vascoulsy True" issue!
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Conditional Statements
In conditional statements, "If p then q" is denoted symbolically by "p q"; p is called the hypothesis and q is called the conclusion. For instance, consider the two following statements:
If Sally passes the exam, then she will get the job.
If 144 is divisible by 12, 144 is divisible by 3.
Let p stand for the statements "Sally passes the exam" and "144 is divisible by 12".
Let q stand for the statements "Sally will get the job" and "144 is divisible by 3".
The hypothesis in the first statement is "144 is divisible by 12", and the conclusion is "144 is divisible by 3".
The second statement states that Sally will get the job if a certain condition (passing the exam) is met; it says nothing about what will happen if the condition is not met. If the condition is not met, the truth of the conclusion cannot be determined; the conditional statement is therefore considered to be vacuously true, or true by default.
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"it says nothing about what will happen if the condition is not met" --!!!
** if you couldn't understand something and automatically assumed it was false you would never reapproach that issue so would never solve the problem. Unless you assume its true. (until proven false) ** Kinda goes against intuition?
"if Sally passes the exam, then she will get the job." So then we know for a fact (F,T) can not allow the statement to be true, as you would be claiming that sally got the job, without passing the exam, meaning that the claim of passing the exam => getting the job, must be false. They must be independent events if I can get the job without the exam. (T,F) is false because it allows one to pass the exam without reaping its implication, which would make said implication, not an implication. Why does (F,T) not follow this same logic? Where not passing the exam, still allows you to reap the implications of passing the exam. Does that not mean passing the exam failing the exam, as both have the same implication, getting the job. The implication will then look like this (PvQ)=>R, where P = passing Q = failing and R = getting the job
It's 7 years late.
Idk if you'll read this.
I don't think it matters much but it would be awesome.
If could've real sentence based example of this lecture.
Ok so I get why its import for them to be truth for the bi-conditional cause its a matching game, but I don't understand what the meaningful interpretation of the vacuously true statements are. Like going back to the conditional lecture: If you have finished your homework, then you can play video games, so in that the 3rd row would be where you haven't finished your homework but you can still play video games?
Vacuously true statements often lose coherent meaning precisely because of the vacuousness.
For your example, let H you finished your homework and V represent you can play video games. So H => V. Suppose ~H. Then there is no direct relationship between the homework not being done and being able to play video games. You may very well be allowed to play video games--perhaps you need your math tutor to finish your homework assignment, and he isn't there currently. Or you might not.
So, really, the implication is asking you to think about a counterfactual world. If you really had finished your homework, would you be allowed to play video games? If the answer is yes, the implication is still fine. If the answer is no, the implication is false, since it claimed that fulfilling H means V is fulfilled as well.
There is another reason for those vacuously true statements to be true. Suppose they were false. P=>Q would be logical equivalent to Q=>P would be logical equivalent to PQ would be logical equivalent to P^Q, which doesn't make any sense at all.
Yep, that makes my head hurt.
@@Gametheory101, 7:42 Why did you change the 2nd and 3rd rows form being F to being T?
@@Gametheory101, hi, by the way.
I don't think they make sense if they're true or false. I think they should be undefined. That's a more conservative approach anyway. I mean, you can say if George Washington is alive in 2024 then grass is green. And you can also say if George Washington is not alive in 2024 then grass is not green. Those are contradictory statements, and since the premise is false it allows contradictions, which I can never see as being a good thing.
I cant buy the whole argument for vacuously true. Though it makes sense in the last row. However 'if false then false' makes sense without any need to appeal to vacuity. I still dont comprehend why its not permissible to take 'if false then true' as a false statement.
heavymedley Think of it in cause and affect. For example "If it is my birthday, then I will eat cake." The argument shows that I will eat cake on my birthday, but it doesn't say that I won't eat cake if it isn't my birthday. I still may or may not eat cake any other day, but if it is my birthday, I will DEFINITELY eat cake.
You can make countless examples to remember the tables. such as "If you get A, I will buy you chocolate" but it's not logical to memorize them.
@@headlesskoala How do we turn your example around? "If I eat cake, it is my birthday". So if I don't eat cake, it is my birth; and if I don't eat cake, it is not my birthday, are these vacuously true?
If if->then truth table only has True in p=true and q=true, then it is the same as p AND q, which is not what conditional or implication statement is intended for.
I looked this up because I don't accept the concept of vacuous truth at all.
When given a statement:
"If P then Q"
The truth of Q is dependent on P being true such that when P is true, Q. In computer programming, Q will be an action carried out subject to the entry condition P being met.
But you have no contingency for P being false. So if your system has only one condition, which is Q when P, then you have no condition for not P.
Therefore, when the test condition is false, then neither truth nor falsehood can be determined.
Let's put the same facts a different way:
If you have a logic system which you can query, and in your system you have the one statement "If P then Q", and you then query the system with "When P, Q?", then it's true. If you query "When P, not Q?", the response is false.
But, if you query ""When not P", because you have no case in the system for not P it is undecidable.
Meanwhile, there isn't really any valid "truth table" for this kind of state expression like there is for conjunction and distinction. Realistically, it's a contingency listing.
Yeah, the words "if" and "then" are a little confusing. Think of it as "P is a sufficient but not necessary condition for Q" and "Q is a necessary but not sufficient condition for P".
How can q b necessary but not sufficient?!
@@Nicoder6884
Why? You have given no explanation for why anybody would think of it like that.
When given the statement "If P then Q". You know P is sufficient for Q but you Don't know if it's necessary. Q Might be true if, and only if, P is true. You have no statement to the contrary so you don't know if it's necessary or not.
Also, given the true statement "If P then Q", the only truth you can derive from querying "Q" is that when Q is false P must also be false so your second statement is wrong. If P then Q then rearranging for Q you get "Not Q is a sufficient condition for not P" and that's all you have.
@@MathCuriousity Being a mammal is necessary but not sufficient for being a dog
@@vapourmile I don't think you understand the meanings of sufficient and necessary. P is not necessary for Q, but Q is a necessary consequence of Q being true.
This argument only holds based on practical everyday use. However everybody feels that it is a bit funny. You could argue that columns 3 and beyond only ask if the implication are "not falsified". But being "not falsified" is not the same as being true. Mathmatical conjectures are not necessarily true. They are basically undetermined until proven or disproven.
x implies y and y implies z ,y is false what can be concluded ??
You people never discuss such condition. What you do is just the common knowledge . If you are very aware of teaching then reply me this problem i mention above then i would be very happy. I hope you will do this. I wait for explaination .
What if you make vacuous statements the empty set, because it’s undefined? T empty empty empty sounds good to me. Empty equals empty is undefined, hence empty. intersection between false and empty is still empty.
Surly the only information we have from the expression,
P => Q,
P => Q[(T,T) = T],
(The truth values within the brackets are pre-supposed on the basis the original P,Q will always be assumed true)
Is that, we are to assume that P brings about, or is sufficient, for Q, and therefore are connected on that basis. Any other information is an extrapolation from said basis. Such include,
P => Q(F,F) = T,
Where we are to extrapolate, if P and Q are connected, on the basis that P is to bring about Q, then it must be true that when P does not happen, so too will Q not happen. From this, we can draw the equivalence,
[P => Q(F,F)] [-P => -Q(T,T)] = T
And,
[P => Q(T,T)] [-P => -Q(F,F)] = T
We can use these two equivalences to define the parameters in which, P implies Q, P => Q, where, we can only logically assume that P implies Q,
IF,
([P => Q(T,T)] [-P => -Q(F,F)]) => ([P => Q(F,F)] [-P => -Q(T,T)]) = T,
THEN,
P => Q
Conclusion:
The following truth values, (F,T) and (T,F), must be false for all cases of implicit conditions, as the disjunction of the truth values, where one can happen, irrespective of the other, is contradictory to the nature of implication, where a logical connection is being established, and an event Q, follows from the truth of an event P.🤔
Thank you. This explains it - the biconditional statement (as a conjunction of the two conditionals) would not be able to recover the fact that “if P is false, Q is false - and vice versa” is a true statement. Right? Thanks
I know this is a very old video. But could you repost with either written directions or verbal corrections? You really confuse me when you start getting inconsistent with rows when you refer to a row as 4th row and then go back and say 5th row.
Overall great job and very useful videos but you lose me a little in this one with inconsistency
Can you give a few more concrete examples of "vacuously true" proposition please?
The best example is the theorem that a empty set is a subset of all sets. This means that IF x belongs to empty set THEN it belong to any set. The antecedent is always false since nothing belongs to the empty set, so the proposition is true.
@@pingdingdongpong The antecedent does not claim that there is something in the empty set, just that if there were, it would be a part of any set, which is true based on, empty set => a member of all sets. I ask, how can you formulate (F,T)? Do you claim that, not belonging to the empty set => it belongs to any set?
@@TheMorhaGroup Empty set is a subset of all sets mathematically means: for all x, x in empty set implies that x is also in whatever set. For all x, x in empty set is false for all x by definition, therefore the antecedent is always false, hence the implication is always true.
@@pingdingdongpong ""IF x belongs to an [viz] empty set THEN it belongs to any set" does not mean by any necessity, that x has to belong in the empty set, it can mean, x, if it where to be in the empty set, would then also be a part of any set. The statement can be speaking on the quality of the empty set alone, and that it has the quality of being a part of any set, and if it were to have an element, that element in this case being x, the quality of being in any set, would be extended to that element.
This is way easier for me to understand in terms of "q is necessary but not sufficient for p".
I have just watched this again and it gets even worse when it comes to the fourth column. First off, it's Querying the relationship between Q and P "If Q is true is P true?". That is a completely different thing from testing the logical Assertion "When Q is true, P is true".
Also, the question he is trying to answer is "Why are they true? Why aren't they false?". It's a false dichotomy which proves the invalidity of bivalent logics. In the first isn't when P is not true, it automatically doesn't matter what Q is, because you have no implication statement for when P is not true.
Assumption that P=>Q when P is false has the same truth value irrespective of Q. Perfectly consistent to have it true when Q false but false when Q is true.
oh my God what a Life saver
7:42 Why did you change the 2nd and 3rd rows form being F to being T?
I think William was trying to show how defining them to be false lead to something that didn't make sense, and so he "corrected" the definition back to the usual value of true.
I’m having trouble understanding why he switched the Falses at the end to Truths
I think William was just trying to show that if they were defined to be false, it would lead to something that seemed a bit off. But once he reverted them to true values, things made more sense. Namely, p q should be true if both p and q are false, since as William noted, the biconditional in logic is analogous to the equal sign in math.
I met your brother Cocker the other day; nice fellow. I think he prefers Richard these days though.
I wish you'd use a mouse to point at what you're talking about, thanks though
Why don't we just change row 4th rather than the whole table?
I should have paid you my tuition fee rather than paying it to my University course
Logic really seams unlogical. P -> Q. If P and Q are true, then the whole expression P->Q is true. What if, say, P is 'the Earth is round', and Q is 'Michael want to eat a fruit cake". P and Q are true, so, "If the Earth is round, then Michael want to eat a fruit cake". What it even means for it expression to be true? P and Q in this case totally unrelated.
The simple sentences must be related though.
Logic reasoning is about accepting or rejecting the state of one or more(composite) statements, the state being binary, namely true or false, black or white, 1 or 0..... It is up to the creators of the statement to accept or reject the statement, they dont need to sound related to anyone else.
We say that an implication p --> q is vaccuously true if p is false.
Since now it's impossible to have p true and q false.
That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent.
So we take the implication as true.
For eg. If 3 squared = 27,then 2+2=5.
Can we check if it is indeed true that 3 squared equals 27 then 2+2 is not 5.
No.
Because 3 squared equals 27 is non-existent. Or false.
So we can't check if the statement is false.
Hence it must be true.
“We can’t check if it’s false hence it’s true” lmao that’s some horrible logic!
@@MathCuriousity not really,unless you can show it's false,its true....if it is false,why can't one show it?...thst's what i get from this topic.it's a loopish one really.
@@MathCuriousity note,in my example,the statement is not that 2+2=5 is true...in the example i am saying the whole satement, if 3 squared =27,then 2+2=5 is true as vacuous truth or false implies truth or whatever you call it.
@@rishabhnarula1999 I get what you are saying - but when we dig down - what we realize is vacuous truths are conventional so when evaluating p implies q for when it’s true based on the true or false states of p and q, then when we get to the two situations where p are false ( FT and FF), we basically default to it being vacuously true with those state combinations because we cannot falsify something that doesn’t exist ie we cannot falsify p implies q if p doesn’t exist which it doesn’t if we are saying p is false.
I think that a good example of vacuously true statement in English would be :
"I will stop watching TH-cam videos when pigs can fly."
"every element of the empty set is equivalent to a zebra" would be a more accurate vacuously true statement.
in fact that's not at all accurate because you are saying that if you ever stop watching TH-cam videos then pigs will suddenly fly, and that's simply not the case. The antecedent has to be always false and the consequent can either be true or false in order for the conditional to be vacuously true. "The day string cheese is made out of yarn, is the day that cheese will stop being orange"
@@StaticBlaster No, the statement I wrote is equivalent to: IF pigs can fly, THEN I will stop watching TH-cam. In this case the antecedent is always FALSE.
Having the antecedent at the end of the original sentence can be confusing, but that's just the English language for ya!
Vacuous? More like "making us" super smart; thanks for sharing all of your knowledge William!
Is there are difference between a "truth table" and "the method of assigning values"? I thought they were the same and now I'm studying and I'm not sure
You just taught 15k+ to count ____s and bend ___ ______
thank you so much. i will subsribe
Isn't this material condition? rather than implication?
This makes absolutely no sense in real life. Lets take 2 statements, "if i eat a mango, i will die", "if i eat a mango, i will live."
Here p is "if i eat a mango" and q1 is "i will live" and q2 is "i will die". In other words q1 is not-q2.
If i don't eat a mango, i.e, p is false, then both statements have to be true, vacuously. But both statements are literally the opposite of each other and cannot be simultaneously true. This is just ridiculous and illogical.
The if-then statement has its logic based on the entire evaluation of the if and then. It seems you were only looking at the "then" individually.
To counter an example, in the case of P OR Q, if P is true, Q could be false or true, the result still true, but one would not argue that Q and Not Q are the same yet they are opposite, what do you think ?
On the other hand, if vacuously true were to be changed to vacuously false, the if-then statement becomes identical to the AND statement, as only when both p and q are true the statement is evaluated true. That mustnt be the reason why if-then statement was created.
Nice.
😱😭
hmmm..
(P Q) (P XNOR Q)