 We have now, in this video, obtained our first primary goal with our study of logic, and that is to get to this point of logical inference. What does this word inference mean, particularly in the case of logic? An inference would be a truth that we know is to be true, that is, we infer it to be true because it's a consequence of things we already know to be true. And so in logic, you can think of it over here. There's some type of collection of known truth. Things that we know to be true, like statement P is true, statement Q is true. We also can take things we know to not be true, like let's say that R is a false statement, we know that. Well, if R is a false statement, then not R is a true statement. And so basically, if we know a statement is true or false, then it's known truth, okay? Being false isn't such a big deal because we can always do its negation. But then there might be some other statement over here like S, where this is something we don't know. It's unknown whether statement S is true or false. But because S might be related to statements P, Q, and R, is it possible that we could use the laws of logic, logical inference, then to decide because P is true, Q is true, and R is false? Because we know those things, then maybe we can then infer that S is likewise a true statement or maybe we discover that it's a false statement, it doesn't make much of a difference there. But by using inference, we can then discover new truth. And as we constructed truth tables previously in this lecture series, we would do that. It's like, okay, if I know P to be true and I know Q to be true, then I would know that P implies Q is also true, right? If you use those truth values, we can then determine the truth of other related statements. And so logical inference is then this process of which we can take known truth and then infer new truth that is truth of other logical statements. Now how does one actually do that? Well, like I said, you could draw truth tables and you could read off rows of truth tables. And that can get technical, mechanical, and very sterile as an argumentation method. And so instead, we aren't really writing truth tables again, it sort of is not the best form. But also as our statements get more complicated and more complicated, a truth table is more of an obstacle than a tool in that situation. And so now we want to focus on the idea of a logical argument. An argument is actually a series of statements. And all of the statements in this argument are called premises. These are known truth statements. But again, they could be false in case we know their negation is true, not a big deal there. So these are all known statements, but then there's also a single final statement which we call the conclusion, for which the conclusion we then think of as the statement that was inferred to be true. And so an argument might look something like the following. We have our first statement P1, our second statement P2, our third statement P3, all the way down to our nth statement PN. And then there's some type of concluding statement, which we're going to call it Q. And so this is the structure of a logical argument. These first statements, like we said before, are called the premises. So you have several premises, each one is a premise. And then the final statement is the conclusion. Oftentimes a line is drawn to separate the premises from the conclusion so that you can see it in that fashion. And then also this symbol right here, this represents therefore an abbreviation of that. And what you want to then think of as the premises are those statements which are known to be true. So by previous proof or for whatever reasons, we know P1 is true, P2 is true, P3 is true, PN is true. And then the conclusion is then a new truth that we infer from the premises. Because the premises are true, then the conclusion is true as well. That is what a logical argument is. But be aware that by definition, an argument is just a sequence of statements. Now I could list a bunch of statements and that doesn't necessarily make it a useful argument. I mean think of the following argument. The sky is blue, today is a Tuesday, I am an Aquarius and therefore the moon landing was faked. Right? I mean that is an argument. Right? I listed three premises and then I stated a conclusion. I usually even the word therefore suggest that the truthfulness of the first three statements that implies the last one for which I'll be aware the first three statements I've listed are all true statements. But the last statement could be true, it could be false maybe but certainly the truthfulness of the first three statements didn't seem to have any consequence on the truthfulness of the last statement. I mean so this important when we talk about arguments that we consider only valid arguments. An argument is valid if whenever all the premises are true then the conclusion must also be true. That is the truth value of the conclusion follows because the premises are true. If you don't have a valid argument it is then invalid and is basically useless from a logical perspective. As we thought of truth tables for conditional statements like P implies Q, notice we use the word premise and conclusion when we talk about these things as well. It's very possible that your truth value, your premise could be true but your conclusion could also be true. Right? That's what we're looking for right here. It could be false. It could be completely bonkers. You also have things like vacuously true statements. We won't get into those right now in this video here but just because the premise is true doesn't necessarily mean the conclusion is true. Now if you have a valid argument then the conclusion must be true because the premises are true. Another way of stating this is the following. If you take all of these premises P i and look at their conjunction so we take P 1 and P 2 and P 3 and P 4 and P 5 all the way up to and P n. If you take that as the hypothesis of a conditional statement and then you put as the conclusion the conclusion of your argument then this is a valid argument. Let me let me say it this way. This is a valid argument. The premises and the conclusions if and only if this logical statement right here is a totality because this being a totality totality is the same thing as saying that this will be true exactly when all of these things are true. Now again it could be paused for the conclusion because it could be true even if the premises are false but that's not the argument. The argument says that because these are true then this has to be true as well. So I want to give us some examples of valid arguments because these are the types of arguments we would use in writing a proof. Invalid arguments will never discover new truth. There's no guarantee. A valid argument guarantees the conclusion is true whenever the hypotheses are true. So the most probably the most popular and most used of all valid arguments is known as the law of detachment are in schools of logic. This is often referred to as modem ponens if you want to if you want to look at Latin phrases and things like that the law of detachment and this is this argument we have seen before. Suppose it is true that the conditional statement P implies Q so that suppose that conditional is a true statement and then likewise suppose that is true that the hypothesis of the conditional statement is true then this implies that the conclusion Q is likewise true so if P implies Q and P then Q also has to be a true statement and this gives us the law of detachment modem ponens and we can actually prove that this is a valid argument using a truth table so what we do is we draw a truth table that involves every statement in the argument including all of the premises and the conclusion. Now some of the arguments are just primitive so we have P and Q like so and then we should also include P implies Q in here and then for the sake of it I also included the statements here as well because the idea is you have all your helping columns as necessary but then you're going to have a column for each statement in the argument typically going in the order that they appear so the first statement and the second one in the conclusion. Okay so what we have to do is look for those rows in the truth table for which all of the premises are true and that only happens in one which is the first row you'll notice that in this row only are the two premises true because of the second row the first premise is false and in the last two rows the second premise is false we only have to look at the rows that have true premises all of the premises are true and then you look at the conclusion in that situation and that situation the conclusion is likewise true this is evidence that we have a valid argument because every row in the truth table for which all the premises are true the conclusion is likewise true therefore what this tells us is that whenever the premises are true the conclusion has to be true as well giving us this most important valid argument the so-called law of detachment and so let's look at an example of what an argument might look like in plain English even though I still have placed an illogical structure here so consider the following if you subscribe to the most popular Netflix DVD rental plan then you have unlimited rental rentals per month now if some of you are wondering like does Netflix rent DVDs I thought this was a streaming platform I should remember mentioned that back in the day of you know the when Netflix was a thing when it was a brand new thing I should say the Netflix started off as a DVD rental website so as opposed to like blockbuster video which you're like what's blockbuster video the idea is you could go online to rent DVDs hence the name Netflix but then of course Netflix started this concept of streaming in which case it became so popular that their DVD plan is virtually non-existence but I do believe at the timing of this video that you still have DVD plans that you can rent maybe I'm wrong about that one but let's the current the current the current events don't really matter here with the logical structure let's just suppose that if you subscribe to the most popular DVD rental plan then that means you get unlimited rentals per month as opposed to some of their other plans where you get two rentals a month or something like that okay the unlimited plan was definitely very popular then we could say something like the following you do not have unlimited rentals per month therefore you do not have the most popular Netflix DVD rental plan so if we look at the argument and play here this does feel like a conditional statement because I see the words if and then okay and so let's say that you subscribe to the most popular Netflix plan we'll call that P and then you have unlimited rentals per month we'll call that Q this first statement is then just P implies Q okay when you look at the second statement though you do not have unlimited rentals per month that would be the statement not Q so this is not the law of detachment this seems to be a different argument that's being made here and then when you look at the conclusion you do not have the most popular Netflix DVD plan that's a not P right there source our argument when we simplified is the following P implies Q not Q therefore not P now if you look at at the surface of this argument here we've put the argument in symbolic form now when you look at the argument it does seem to make sense right because if if P was true then that would apply Q is true basically by the law of detachment that we saw beforehand but Q's not true so that would seem like oh it has to be the case that P is not Q sorry P is not true we could try to make that argument there but one has to be careful are we are we actually gathering the truth that's in play here are we placing human bias or misunderstanding onto it here so let's construct a truth table to determine the validity of this argument here well our arguments involve the statements P implies Q not Q and not P so examining the table here we're gonna look for all those rows where the premises are true well you're not true in the second row for the first premise but when you look at the second premise you're not true and those ones so it turns out we only have to look at the fourth column this is the only place where all of the premises are simultaneously true they have to all be true together or we don't worry about it in which case in the fourth row they're true but what about the conclusion the conclusion is likewise true so because every row for which all the premises are true the conclusion is likewise true that makes this a valid argument the truth value of the conclusion does follow as a logical inference of the truth of the premises now what we have here this argument structure it's similar to the law of detachment but it it is a different argument and let's get a different name we refer to this as the law of contra position or again if you like the logical terms in the logical circles this gets the name modus tollens as opposed to the modus potens we saw previously now these two argument structures we see are valid and very very important and we will see more about them in the future in the future when we learn about the proof pattern of direct proof this is basically utilizing the argument structure of mode impotence that is the law of detachment and then we'll also talk about contra positive proofs this uses the logical argument structure of the law of contra position so that's why I mentioned these two first we're going to see these ones a lot going forward I also want to discuss in this video some other valid logical arguments but I'm going to leave it up to you to prove the validity of these arguments so this next argument is referred to as the law of syllogism for which it's a valid argument and the argument goes in the following way p implies q q implies r therefore p implies r on the surface it seems like a reasonable argument it is in fact valid how would you prove that to prove the law of syllogism you can construct a truth table you would construct a truth table which has in it among other things p implies q you would have q implies r and then finally you would have p implies r and then you would search every time that the premises are true you would then check that the conclusion is likewise true anytime that one of the premises is false I don't care about that row okay we just check that every time the both premises are true that the conclusion is likewise true that would make it a valid argument so I'll let you come and complete the table there and argue that the law of syllogism is a valid argument another argumentation type that is very very useful is what's called a disjunctive syllogism sometimes called the law of elimination or just elimination for short for which elimination the elimination arguments the following p or q so I did so the statement that disjunction hence why we call this a disjunctive syllogism syllogism is again another fancy word you use in logic as like a synonym for arguments right so a disjunctive argument a disjunctive syllogism that's what we mean here so it involves a disjunction p or q all right then your second premise is that not q and then you conclude therefore p so when you look at this is like okay how is a or statement true well one of these things has to be true at least one of the statements is true well if q is not true then that implies that p is q p is true excuse me and so the disjunctive syllogism is basically what's going to lead to a proof technique we discussed we discussed later in our lecture series known as proof by contradiction because the most fundamental of all contradictions is that a statement is true or or not that statement is true okay and so you basically look at not not p and therefore it's got to be p it seems a little weird when you see it that way but yes this is the basics these are basic argumentation structures we're going to use in the future nearly every argumentation style any type of proof pattern we develop in this lecture series is based upon one of these fundamental valid arguments we've talked about right now let's look at an example of such a thing let's look at the following argument if I win the lottery then I'll take my family on a vacation to Hawaii I did not take my family on a vacation to Hawaii therefore I did not win the lottery if we look at the structure of this argument let's take win the lottery that'll be statement p I'll take my family on a vacation to Hawaii that'll be q and so then when you look at this argument if p then q that's the first statement the next one is I did not take my family to Hawaii that's not q therefore I did not win the lottery not p this is another example of the law of contra position this is a valid argument because it has an established valid argument so the thing is we can prove an argument's valid in one of two ways if it's a novel argument like an argument we've never seen before we can actually go and look at the truth table to then ascertain the truthfulness of this argument is it is the do the premises imply the truth of the conclusion but typically what happens is we're going to use arguments that we already know the form we can we already know the law of contra position once you've proven it to be true you don't have to prove it again and so oftentimes we determine an argument is valid not by a truth table but by recognizing it has a logical form that we already know to be valid