 All right, so we've had a look at propositions and we atomic propositions and we've figured out how to Get every possible combination of atomic propositions on a truth table Now that we've covered atomic propositions, let's move on to our first kind of complex proposition negations So the first thing to keep in mind is that a negation is Not just merely one more kind of atomic proposition Superficially kind of seems like it, but it's not so a negation negation is a complex proposition and it says either that You know a atomic proposition is false or some of the other complex propositions are false It's because you can negate an atomic proposition and you can negate other kinds of complex propositions You can even a gate a negation, but we'll get to that a little bit so Suppose we have this proposition right so we have That sculpture right that sculpture is not blue So you might think Not blue is a predicate and that sculpture is the subject Well, well no not blue is not a predicate Blue is a predicate But not blue is it is not a predicate Because you know telling us that it's not something is not telling us what it is and a predicate's job is supposed To tell us what something is so that sculpture is not blue not blue is not the predicate That sculpture is You know I'm actually a little colorblind, so I'm gonna guess that's red that sculpture is red Right that that tells us what it is right that actually gives us a predicate not blue is not a predicate So when we're dealing with negations of an atomic proposition It's important to be able to spot the atomic proposition that's embedded within the negation Because the negation is a complex proposition it takes at least an atomic and does something else with it And tells us the atomic proposition is false So in this case I have you know when I when I say that that sculpture is not blue The atomic proposition that's embedded in there is that that sculpture is blue Well that proposition is false Right that sculpture is blue that proposition is false So when a negation does is it tells us that Proposition is false and this is probably a good time to mention the truth conditions for negation That sculpture is not blue That's a negation and it's true right then negation itself that whole thing that sculpture is not blue That whole negation is true Because the atomic proposition Within it that sculpture is blue It's false. So negations are true Just in case the proposition negated is false Negations are true just in case the proposition negated is false and it's true otherwise That's right. It's false otherwise false otherwise. So that sculpture is not blue that Again, the atomic proposition is that sculpture is blue and it's negated to say not blue Equivalent negations are it is false that that sculpture is blue. It is not the case that that sculpture is blue That sculpture is not blue Those are probably the main the main kinds right there right those are all equivalent all three of those are negations So you have to be careful. You have to you know spot that negation in there Let's try another negation That sculpture is not red Well that negation is false The atomic proposition I bet it within there right is that sculpture is red So when we negate that you know that that that that sorry that atomic proposition that atomic proposition that sculpture is red It's true. And if we negate it we say that proper that sculpture is not red that negation It's false So this is what's happening when we negate an atomic proposition It's a complex proposition that means that it's composed of something more than just you know an atomic There are other complex propositions we got conjunctions right that sculpture is Red and the plants are green and that's a that's a conjunction and can we can negate that A conjunction says both the propositions are true So actually let's try a different one that sculpture is red and the plants are pink Well that whole conjunction is false So if we negate it say it is false that that sculpture is red and the plants are pink boom now we have a true negation We got conjunctions. We got disjunctions Disjunction says at least one of these is true conjunctions says both of them are true Disjunction says at least one of these is true So either that sculpture is red or the plants are pink Right, that's a true disjunction if I say something like this either that sculpture is blue or those plants are pink That whole disjunction is false because both of them are false right that whole disjunction is so that if we negate that whole thing It is false if that sculpture is blue And the plants are pink okay now I have a true disjunction and And the other kind of complex proposition we were like as conditional So it's conditional says if the first one's true then the second one must be true right so if that sculpture is red Then that sculpture is a warm color Right, that's a true conditional if the sculpture is red and that sculpture is a warm color if If a false condition would be something like this if that sculpture is red Then that sculpture is a cool color Well that conditional is false because red things are not cool things right or at least as far as the color is concerned So if I say if it is false that that sculpture is red then that sculpture is a cool color now I have a true negation Don't I just want to briefly introduce that we don't don't worry about all those other kinds of negations right now We're only going to worry about negations of atomic propositions So atomic proposition that sculpture is blue We need get that's that the time of proposition is false if we negate it say it is false that that sculpture is blue It is not the case that that sculpture is blue that sculpture is not blue Those are true negations Those are all true and equivalent negations and all the negation says is that the proposition negated is false That's what's going on So if the proposition negated is false and the negation is true at the proposition negated is true Then the whole negation falls Okay That's negations Now let's take a look at symbolizing the negations and putting them under truth table. Well apparently it's Foggy season here in the Water room I did not anticipate this I Let's deal with truth conditions our truth tables and negations Okay, so suppose at this point you've already found your atomic propositions and your argument And following all the rules up to this point. You've assigned your truth conditions for your propositions All right, so like let's just stick with something simple. I suppose we just have paint Well, now we're gonna start looking at whether or under what conditions a negation is true or false with just p So we followed all the rules we've got our truth table. We got the rose and we've distributed p across all the you know across all possibilities, you know both of them and Now we want to figure out Whether negation or you know I figure out what conditions not p is true. All right well, let's Let's introduce a new rule and so this rule six What you're gonna do is you know you have your truth assignments for p but then across the top of our truth table we now have The symbol for negation, which is the minus sign We got that minus sign and then the atomic proposition and Each gets its own column right p gives its own column and the negation gets its own column And we're looking at The truth values for negation. All right, so we take the truth values for p that we assign found the rules up to this point And rule six says Assign those same truth values to every instance of p Now in this little truth table, we just got one right so this isn't very hard But you can have true tables where you have multiple instances of that of atomic proposition And what you do is you know you copy and paste you got the truth of science for the atomic proposition You copy and paste in every instance in the truth table That's rule six Okay, remember our truth conditions for negation and negation is true just in case The proposition negated is false And it's false otherwise so Well, look at row one row one p is true Well, since in row one p is true that means the negation is false at row one at row two p is false And since p is false that means the negation of p is true at row two So next rule we got rule seven in clothes and parentheses The truth the as the truth value for that premise for that complex proposition So this one is a negation so it's easy We just you know find the column for the negation We put our truth values in there for that and then we enclose them with parentheses You do the same thing with premises and conclusions when we get to that with the truth with the with actual arguments Okay Well That gives us that tells us you know what the truth value of negation of p is in the various ways that p is assigned true or false all right Well, let's suppose we got something a little different suppose we have p and q and Both p and q are false So we follow our rules up to this point We had since we have two variables since we got two atomic propositions We've got four rows and Following the truth assignments right the first two rows for p are true the second two rows for p are false Rows one and three are true for q rows two and four are false for q The phone the rules up to this point Now we're gonna fill in not p and not q and our truth table So we got not p and then we got not q. All right Now let's put now. Let's use rule six We'll take rule six and we put The truth assignments they had for p we copy and paste them under p We got the truth assignments for q. We copy and paste them on the q. Let me put them at the truth table well Since we got two rows for true for p rows one and two for the negation of p are now false Since p is false and rows three and four the negation of p is true and rows three and four And we put those in parentheses Moving over to q right. We got true false true false Well, that means rows one and three since those are assigned true for q. Well, that means the negation of q is false Rows three and four are assigned false. So that means the negation of q is true and rows three and four So we put all that parentheses following rule seven Now if you get more complex from here, we can have three variables We got four variables who've seen that that'll be eight rows and 16 rows and 32 rows and 64 rows But you follow these rules to fill out these truth tables in exactly the same way All right, that's the negation if that's negations and truth tables. Let's keep moving on So we've looked at negations as a complex proposition You can't have a negation as an atomic proposition Since something like not blue Is not a predicate and it's not a predicate because it doesn't tell us what it is predicus describe This just says what it isn't saying what something isn't just tells us what it is. All right so Negations When they apply to atomic propositions, I say some atomic proposition is false We'll look at the negations of other complex propositions later on Now one of the things that I mentioned right you can negate an atomic proposition You can negate conjunctions disjunctions and conditionals. You can also negate a negation in fact because of the way our system is going to work Right, you can add negations on right. There's no limit to the number of negations you can Add on to a proposition. You know if it's if it's true So, uh What would this look like right? What would this look like with the negation of the negation of p? All right. Well, let's look at our truth table. So we you know we have p It's got row one is true row two is false. That's a complex truth table And then and then across the top we have the negation of the negation of p So we following rule six we take our truth values for p and we paste them under p in our truth that Copy them and then we paste them in our truth table And then we look at the negation of p Well and the negation of p since p is true the negation of p is false I'm row one and I wrote two p is false And so the negation of p in that row is true Okay, now we don't put parentheses Around rule seven doesn't apply here because this proposition is the negation of the negation of p The parentheses go at the other negation. That's the negation of the negation of p If you put the parentheses just there and that one column without just be the negation of p All right, well, let's go to the negation of the negation of p All right, so at row one The negation of p is false Since at row one the negation of p is false the negation of the negation of p is true And we put our t in there and that's where we enclose it with the parentheses following rule seven Row two the negation of p is true Since the negation of p is true at row Two the negation the negation of p is false. All right Well, uh You're looking at this and you might think well, you just showed me something I've pretty much figured out on my own that the negation of the negation of p is the same as p Right the truth by using and that's true Now what we just proven is what's called an equivalence rule An equivalence rule will let us swap out right These instances within a formula So if I have some really long formula and right in the middle of it somewhere. I've got the negation of the negation of p Well, I could swap that out for p Or suppose somewhere in a really long formula Uh, I've got p I can swap that out for the negation of the negation of p Now that might seem a little strange at first Trust me. It comes in handy They wouldn't have figured out this equivalence rule if it didn't come in handy But that's one of the equivalence rule dots. It lets us swap out equivalent formula within a complex formula We don't have to prove anything else to do that and what justifies this is our truth table Our truth table shows us that these two formula Are equivalent in all truth assignments If we had any conjunction It would be equivalent to the negation of the negation of that conjunction If we had a disjunction It would be equivalent to the equivalent it would be equivalent to the negation of the negation of the disjunction Right, we could swap out a conjunction with the double negated conjunction We can swap out a disjunction with the double negated disjunction We can swap out a double negated disjunction with the disjunction We can swap out a double negated conjecture with the conjecture, same thing with the condition. It doesn't matter what kind of proposition is negated, or what kind of proposition you have, you can swap that out with this double negation, or if you have some proposition that's double negated, you can swap that out with that proposition. All right, so we've had, up to this point, what do we have? We've had our rules for defining. Excuse me, we've had our kinds of definitions. We've had our rules for defining. We've looked at atomic propositions, and we looked at our first complex proposition, negations, and we even got our first equivalence rule, double negation. Believe it or not, we still haven't hit arguments. We'll start taking a hard look at arguments in the next chapter, in the next videos, when we look at conditionals.