 In the previous video we introduced the notions of logical inference and talked about a valid argument that is an argument for which if all of the premises are true then the conclusion has to necessarily be true. Now not all arguments have to be valid and therefore if you have an argument that is invalid then we can't necessarily guarantee the truthfulness of the conclusion even if the premises are true. This is an example of what we call a logical fallacy. A logical fallacy is whenever we can't guarantee the conclusion of an argument based upon the premises of the argument. If a argument is invalid then it is a logical fallacy and so in this video I actually want to present the two most common logical fallacies that come from invalid arguments that is the truthfulness of the premise doesn't guarantee the truthfulness of the conclusion. So let's give an example of such a one. Consider the following argument. If I have the new calling plan then I can text my friends for free. Now again if anyone is watching this video and you're like wait was there ever a time you couldn't text your friends for free? Well yes back in the Stone Age I have to apologize here when you know cell phones first came out they didn't necessarily have free texting this was sort of an add-on feature are they tech they cost you they charge you by the text I know we live like animals back then not free you know texting wasn't necessarily included in the plan but let's let's pretend we live in those dark ages right now. If you have the new calling plan then I can text my friends for free and then the second statement here in your argument is I can text my friends for free and therefore I have the new calling plan okay let's look at the structure of this argument here this first one is conditional so let's break it up into two primitive statements let's call the first one I have the new calling plan statement p and let's take the second statement I can text my friends for free as statement q and so then that first statement if p then q we can write in the following form okay then you look at the second statement right here I can text my friends for free that's just statement q and then therefore you get p right there uh because the last statement I have the new calling plan so the argument made here is p implies q q therefore p is this a valid argument does the truth value of the premises guarantee the truth of the conclusion well like we consider in the previous video we can prove an argument is valid or in this case invalid from a truth table we construct a truth table where we consider all the possibilities of all the truth values of the primitives in this case p and q so we have four rows for those possibilities then we're going to list every single premise so we and we're doing in order p implies q and q and then the conclusion p now what we have to do is we have to look for every row for which all the premises are true you'll notice that in the first row p implies q and q are true statements this is followed up by a true p that's that's great and dandy that does by itself does not say that the argument is valid when you look at the second row you'll see that both premises are false so we can ignore this row entirely it has no bearing on our on our arguments validity when you look at the third row we'll actually jump to the fourth row for a second you'll notice that one of the premises is true the other one's false we ignore that row entirely okay it's the third row that's going to be of consequence here you'll notice that in the third row both premises are true so true and true but on the other hand the conclusion is false so in this case this has been evidence that the argument is invalid now sure there was a row where the conclusion was true when the premises were true but there's also more importantly a row for which the premises are true but the conclusion is false what this tells us is that the truth of the premises does not guarantee the truth of the conclusion the conclusion could be true or it could be false we cannot guarantee its truth and therefore this is an invalid argument and when you look at the argument itself you perhaps can see it oh sure if you have the new plan then you can text for free you can text for free therefore you have the new plan no no no i guess there could be other reasons you could text for free perhaps there's old plans that um there are old plans that you could text for free then you have one of those just because the new plan gives you free text he doesn't mean that no other plan gives you free text and there could be another explanation that doesn't guarantee that you have the new plan and so this argument we see right here which is definitely invalid is an example of what we call the fallacy of converse and this is perhaps the most common logical fallacy that people ever make the fallacy of converse where we have some established implication p implies q now we know by the law of detachment if p is true then q is true but the converse which is the reverse of that is not necessarily true um a statement and a conditional and its converse are not logically equivalent the conditional could be true when its converse is false or vice versa and so if we had the converse p implies q and we know q then we would get p right if the converse is true then this is just the law of detachment this is a valid argument but because the converse is not logically equivalent to the conditional we don't have it and therefore we don't know if it's true or not and therefore we could get disagreement here we don't we can't guarantee it's true so you have to watch out for the fallacy of converse the fallacy here is you're assuming that a conditional is equivalent to its converse which it's not we don't necessarily have implications in both directions there could be other other exhalations that would explain why p why q is true without it being p now certainly if p is true then q would have to be true but just because q is true doesn't mean it was because of p there could have been some other reason for doing that so this is one of the most dangerous most common types of logical fallacies and so considering example of such a thing uh determine whether the following argument is valid or not if i win the lottery then i'll take my family on a vacation to hawaii i took my family on a vacation to hawaii therefore i won the lottery looking at this argument again if p is winning the lottery and if q is going to hawaii then the first statement could be written as p implies q the second statement i took my family on a vacation to hawaii that's q and therefore i won the lottery this is an example of the fallacy of converse and therefore this is an invalid argument there could be other reasons why i took my family on a vacation to hawaii maybe i didn't win the lottery maybe my great-aunt ruth died and left me a lot of money i don't even know who great-aunt ruth is but i appreciate the check in which case then i celebrate her death by taking my family to hawaii and maybe i'm not celebrating her death i'm just in celebrating the her inheritance what have you but whatever the reason there could be other reasons to explain the vacation to hawaii that don't necessarily come from winning the lottery again the fallacy of converse is essentially assuming a conditional statement p implies q is logically equivalent to its converse q implies p that is not the case and we have seen that previously and so as i end this lecture i do want to mention something related to the converse of the fallacy of converse which is known as the fallacy of inverse which has the the following structure p implies q not p therefore not q the argument structure you can kind of see is the following right p implies q we know that now if if the premise doesn't hold for the conditional then you're saying oh the conclusion doesn't hold either but that's not true either much like the previous example right that if i say that if i win the lottery then i'll take my family to hawaii i didn't win the lottery therefore i didn't take my family to hawaii that's the exact same fallacy as before because if dear great aunt ruth passes away i could still take my family to hawaii even though i didn't win the lottery there could be other reasons to explain it now we've learned before that when you have a conditional statement this is equivalent to its contrapositive for which the contrapositive of a statement is not p implies not sorry not q implies not p there's also the converse of a statement which is q implies p this is equal to the inverse of a statement which is not p implies not q so as we've talked about these different logical structures so there's the valid argument the law of detachment that basically is saying that if p implies q and p happens then q happens that's a valid argument the law of contraposition is basically using the fact that conditionals and their contrapositive are logically equivalent to each other so that's also a valid argument conversely in this video we saw that the the fallacy of converse supposing that a converse of a statement is equivalence to its conditional that's an invalid argument because those things are not logically equivalent and then similarly here the fallacy of inverse is supposing that an inverse of a conditional is logically equivalent which is also false so you must be you must be cautious of logical fallacies because if you use a logical fallacy in a proof that makes your proof invalid right you know if you have a brownie and there's a little bit of feces in that just a little bit don't eat any of the brownie right none of the brownies acceptable because it's contaminated by the logical fallacy so we have to be very cautious to avoid these in the future we'll talk about some more about more logical fallacies in the next lecture but that does bring us to the end of lecture 12 for right now thanks for watching if you learned anything in these videos please like them subscribe to the channel to see more videos like this in the future and as always if you have any questions please post them in the comments below and i'll be glad to answer them as soon as i can