In this video, I'll explain how you read mathematical formulas. Just as you're probably familiar with in arithmetic and algebra, you need to know the conventions regarding the order in which logical operators apply. Well, let me begin by just quickly summarizing the precedence order for applying the logical connectives. The ones that bind the most tightly, the ones that are sort of the strongest, if you want, that, that hold something something close together, are the quantifiers for all and exists. And a quantifier applies to whatever comes adjacent to it. Okay? Now typically, what comes adjacent to it involves various other things, like and's and or's and not's, so you would put them in in parentheses, or brackets, square brackets or whatever. So very often, in fact alm, I, I almost always make a habit of putting whatever I want next to it in parentheses, because the for all then applies to everything that comes there. Okay? It binds tightly to this applies to everything, the same with exists in here. Now, if there's only something very simple coming next, for example, supposing I wanted to say, all the balls are red. I could say, for all balls Red B, if Red is a predicate that applies to balls. So I could say for all B, Red B, and that would apply to the red balls. And if there was something else here, you know, if it was, and dah dah dah, then the for all would not apply to that. If I wanted the for all to apply to that, I'd have to put parentheses around there. I didn't put parentheses here. I could do. again, especially when I'm giving introductory-level courses, I usually put parentheses around quantifiers, but if you look at some of my research work and advanced courses, you'll find I often don't do that. That's fairly consistent among among instructors. You know, the golden rule is, if there's any doubt whatsoever, and if you're beginning on this material, there certainly will be doubts, if there's any doubts, put parentheses in. You know, you can't have too many paren, well, that's not quite true. If you have too many parentheses, it gets hard to read it. So you have to strike a balance. But always, if there's going to be any ambiguity, put the parentheses in, and mix parentheses along square brackets or even spaces. I'll, I'll try to remember to give an example in a minute with a space, because you can sometimes use spaces to disambiguate. But the golden rule must be you want to avoid someone being left un, unclear as to what the meaning is. Okay? [COUGH] negation is about the same strength as, well, it is the same strength as, as the quantifiers. So the negation applies to whatever's immediately next to it. And since we usually want a whole bunch of things to be negated, then the negation is followed by parentheses, and then it applies to everything between there. Let me give you the following example. Suppose you wanted to say, not the case that 3 is greater than 0 and 3 is less than 0. Okay. Well, is 3 bigger than 0? Yes. Is 3 less than 0? No. So here I've got a conjunction of something that's true and something that's false. So this conjunction is false, so its negation is true. So this guy is true. But supposing I wrote it this way: not the case, 3 greater than 0 and 3 less than 0. Supposing the parentheses didn't include both of those; they just included one of them. In that case, what I have got? This guy is false. This guy, 3 is greater than 0, is true, so that guy's false. So here I've got a conjunction of false things, so I've got something that's false. So these clearly aren't the same, because this one's true and that one's false. Here, the negation applies to everything in between, which makes it true. Here, the negation only applies to the thing next to it. Now, I could've gone back here, and put parentheses here, and I'll mention that in a moment. Actually comes up in the, the next, the next priority. That wouldn't have changed things. That wouldn't have changed things, because the negation would apply to what was in the next parentheses. Negation applies to whatever comes next, and what comes next is the whole thing, because the parentheses includes the whole thing. So simply putting parentheses inside doesn't change anything. It makes it maybe a little bit clearer, although this is one of those cases where adding parentheses arguably makes things a little bit less clear. But in terms of the logic the issue between these two wasn't whether there were parentheses around the 3 greater than or less than 0. The issue was whether the parentheses governed everything that was next, or just the one thing that was next. Okay? So these are not the same. Okay. The next one is conjunction. Let me now just pick up that thing I mentioned before. When I did that the first time, I wrote this: I said, 3 is bigger than 0 and 3 is less than 0. Now I, in fact, left a space: if you watch what I did, I left a space. And I realized at the time I was doing it, that that's what I was doing, which is why I decided to pick it up now. So this says that 3 is bigger than 0 and 3 is less than 0. You actually don't, strictly speaking, need parentheses around here, because this in an atomic formula, as we sometimes call it. This is a basic building block out of which we're building more complex formulas. This simply states a fact: 3 greater than 0, an atomic fact, a single fact. This states another atomic fact: 3 less than 0. So when you have basic facts about arithmetic, or whatever, they are, they stand on their own. The conjunctions the kind of quantifiers, are what combine these basic facts. So you, strictly speaking, don't need to put parentheses around these. This is a case where I typically would just leave a bit of extra space in here, to sort of make it clear that this is a unit, and that's a unit. On the other hand, if you want to be safe, and it's always wise to be safe if you're at all unsure, you could put those things in. Okay. And this here: I went back and put them in just to make it clear. Okay? Then well co, some mathematicians will say that conjunction, disjunction are, are more or less the same, or conjunction, disjunction, implication. We, we're getting down to a sort of a general grouping now, where everything has roughly the same strength. There are actually some arguments that say that conjunction should be tighter than disjunction, but it's, it's, it's not particularly strong. In any case, I think in, this is a case where when one should always use parentheses to disambiguate. The point is, the, the conjunction applies to whatever's to the left of it, and whatever's to the right of it, and if you want it to apply to a whole bunch of things, you put them inside parentheses. Likewise here: you would have a whole bunch of things. And the same is true for disjunction and implication. So regardless of whether you think that's stronger than those the issue should never really arise, because you should always put things in parentheses to just say, it's this guy or this guy. And in here, it could be a whole bunch of things. And that whole bunch of things will be disjoined with this. And likewise here, if you have an implication or a conditional, this whole thing would be the antecedent, and this will be the consequent. Now, in here, there may be all sorts of conjunctions and disjunctions and stuff. There may be quantifiers in here. There may be quantifiers in here; there could be all sorts of stuff in here, negation signs inside. This whole thing would imply that whole thing. So whenever you look into, I mean, the sort of, the basic thing with all of these is, when you've got a, a, a, a, a, a quantifier or a negation symbol or a conjunction or a disjunction or an implication, or equivalence, actually. I didn't talk about equivalence, but equivalence is just the conjunction of two implications, the biconditional. So you cou, we could put that one in here as well. Strictly speaking if you did assume that that had more tight binding than this, you could write something like this. You could write, A and B or C and D. And if you put space in there that, I would think, is fairly fairly clear that it meant to be this or that. So this is, I would say, at least the way I was brought up, let's put it that way, as a mathematician, I was brought up to, to say that that actually is, is okay and it's, it's not ambiguous. But I would almost certainly now, I think I've cured myself of that, that childhood sin, I, I would always put in parentheses, and say its A and B or C and D. I mean, you just have to be very careful about making sure that things are nice, and not ambiguous. Okay? so, golden rule, put parentheses in. And then everything applies to whatever comes between the next parentheses. It's kind of optional as whether you have parentheses around the quantifiers. But the parentheses that follow, providing you always do it, is, it shouldn't be any problem. Okay? Now let me illustrate those points with examples that are all related to that quiz at the end of Lecture Five. Okay? So let's let's take these one at a time. This was actually from that particular quiz. I've got a for all, and a for all applies to everything that's adjacent to it. Now, that parenthesis there teams up with that parenthesis there. And I've actually written them not as parentheses, but as square brackets to, to make it absolutely clear. That is the thing, that the for all applies to. Okay? For all applies to all of this. In particular, what it's going to say is that, pick any licence. And then it's going to say something about the licences. Now, the licence L is going to appear on both sides of this conditional, and it'll be the same L, because the L has been picked here, and once you've picked the L, it'll apply to everything here. So that L is bound by that quantifier. Okay? So it would say, L is bound by the quantifier for all L. Okay. A quantifier binding. Okay, so let's read it now in in English. It says, for any licence L, if there is a state in which L is valid, if L is valid in some state, at least one state, then L is valid in every state. Okay? For any licence L, if L is valid in some state, the exists here simply applies to this. The exists binds what's next to it. So the exists simply binds this thing. Likewise, the for all binds what's next to it, which is this thing. So what this says is, for every licence, for any licence, or for every licence L, something happens. What happens? If that licence L is valid in some state, then that licence, that same licence, is valid in every state. So this is the one that actually says, a licence that's valid in one state is valid in every state which is true in the United States, by the way. okay. That's the first one. Let's look at the second one. What's the difference? Let's see. Well, we've got for all applies to something in the middle, so, for all applies to everything here, because I wrote the parentheses. The only difference is that instead of having a conditional or an implication, I've got a conjunction. So let's see what that, how we would read that. okay? Coming down, the binding is the same. The for all applies to everything here. This exists applies to this thing. This for all applies to this thing. And the L is the same L here as here, because once you've said the for all L, within this expression here, the L is determined by that. The L is still bound. So as, as was the case there, the L inside here is bound. okay. But what does it, how do, how would we read it? We'd say, for every licence L, there is a state in which the licence is valid and the licence is valid in all states. So let me just write that down: for any licence L, there is a state in which L is valid and L is valid in every state. Now, it doesn't mean the same as the previous one. Just think about this. For any licence L, for any licence, there is a state in which that licence is valid and L is, in fact, valid in every state. But this is actually false. This precludes the fact that there are invalid licences. For example, if you go to California, and you drive with too much alcohol in your bloodstream, you will find yourself with an invalid licence. Not every licence is valid. So this is actually a false statement. The one above was true. This is true in, in the United States. This is false. This means something different. Okay. In fact, really it, it's the, it's the first thing that was the problem. For every licence, there is a state in which it's valid. Well, that's simply not the case. Already the first conjunct makes it invalid. Didn't arise in the first one, because in the first one, the par, the part that says it's valid in a state was the antecedent. If it's valid in a state, then it's valid in all states. So that said, for every licence, if it's valid in a state. This says, for every licence, it is valid in a state. Well, that's not the case. Not all licences are valid. Okay. So there is a distinction between these two, and in fact the distinction is a meaningful one in terms of validity of licences and so forth. Let's look at this one. Well, here we don't have these brackets. So let's see what's going on. This means for all L, for every licence there is a state that the licence is valid in that state. Well, [LAUGH] is that true? Is it true? Remember, this is, this is a unit. The for all and the exists apply to whatever's next. So there's a line here. The for all and the exists don't apply to that. They apply to what's next. And there was no bracket, so it doesn't include here. So what this actually says is, for every licence, there is a state in which that licence is valid. So what this really says, is that all licences are valid somewhere. Well, okay. That's not true. And, and it's, the statement is, if that's the case, then for all S2, that would say, L, well, ha ha, this would say that L is valid in all states. Well, now we've got all sorts of things gone wrong. As a conditional, this guy would look, on the face of it, as if it was going to be true, because the antecedent is false. It's not the case that all licences are valid somewhere. There can be invalid licences. So this is a false antecedent. Now, you might be tempted to say, since it's a false antecedent, the conditional is true. But not quite, because this guy isn't even defined. This is undefined. This is meaningless. What's L? What is that L? It's not governed by that quantifier. This is just an orphan. It's just sitting there. We don't know where it comes from. We don't know what it means. It's just a letter. It has no internal meaning to this formula. So it's not the case that this is a valid conditional. It's actually undefined. This is meaningless unless you know what L is. If you know what L is, you can assign meaning. And once you know what L is, then, you know, if L referred to my licence, if, if that L there was my licence then we would have a, a, a meaningful, and, in fact, a true conditional. But as it stands, you've just got a, a completely, an unbound variable. So the L there is what we might sometimes call just an unbound or free variable. Unbound variable, free variable. Okay? And until you assign a value to it, it doesn't have any real meaning. And then let's finally look at this one: for all L1 for all S. What's the difference here? Somewhat similar to the one up here, but not quite. Okay. Let's just read it. So it says, for every licence and for all pairs of states, the licence is valid in one state and the licence is valid in two states. well, that really just means all licences are valid in all states. Again, this is not the case in United States, because you can have invalid licences so we've got something that's, that's actually false. And it's, it's over, I mean, there's redundancy here, because the second S adds nothing new. It simply says, for all licences and for all states, the licence is valid in that state, and it's valid in the other state. So we could just scrap that, and scrap that, and we'd have the meaning without any of that stuff. So there's nothing actually wrong with this. It's just I mean, it's a false statement, but it's it's got redundant clauses. The second clause says, adds nothing that the first one didn't already state. Okay. Finally, we just try to distinguish between four cases that beginners typically get find to be very confusing. They're actually really very distinct. And if you find the, there's confusion between these four, that's a sure sign that you haven't yet mastered the, the notations and the, and what they mean. Okay. Let me just write down a transcription of what it means, and then let's just ask ourselves exactly what that signifies. So in English, that would say, for every x, if P of x then Q of x. If, P. Okay? For every x, if P of x then Q of x. This is very common. Okay, for every for every number, for every real number, if that number is non nonnegative, then it has a square root etc, etc, etc. So this occurs a lot in mathematics, this kind of statement: for every x, if P of x then Q of x. Okay? Very meaningful. And it's the same x here, notice. Once you've got that for all, the x here is the x here. So, whatever x, providing the x satisfies P, then it satisfies Q. So this establishes a relationship between P and Q. Because if you've got an x that satisfies P, then that x will definitely satisfy Q. So this is a very strong and very common statement to make. This is also pretty common. This says, for every x, P of x and Q of x. It says that every x satisfies P and Q. This is kind of strong. I mean, it, it doesn't occur terribly frequently, because that's really the same. And, and, I mean, you could equally, you could just as equally say, for all x P of x and for all x Q of x. Because you're basically saying everything satisfies P and Q, and that's equivalent to everything satisfies P and everything satisfies Q. notice, by the way, that this is nonambiguous, because of the binding, the for all binds what's next to it, so the for all can only bind that. The for all binds what's next to it. And so I don't need the parentheses here, because in this case, the for all absolutely can't be confused with that, so here's a case where you don't need extra parentheses. I didn't even write the parentheses here. You don't need them. This is totally clear in this case. Okay? And it's equivalent to that. So you don't see this very often, because it really is just saying everything satisfies P and everything satisfies Q, but it's okay. If that arises, don't worry about it. It might, in a context, it might be sensible to write that down. But it doesn't have the same sort of logical force that this does. This has real logical force. It establishes that there's a relationship between P's and Q's. Okay, what about this one? This says, there is an x for which P of x and Q of x. Now, this is, again, pretty common in mathematics. This is quite a strong statement. It says you can find a single x which satisfies P and satisfies Q. Okay? So you can find an x which has the property P and which has the property Q. So this is a strong statement. That one was strong. That one's strong. I mean, this one's strong, but it's only strong because each part is strong. So the, there's a re, there's almost a redundancy in the way it's written. So this is maybe I'd better put strong in quotation marks, just sort of say, well, yes it is strong, but it's not strong because of the logical structure. It's strong simply because it's making a statement about P and Q both being satisfied by all x's. Okay. What about this last guy? This is one that people often write down, and it, this, this really means nothing, in, in any real sense. It says what: there is an x such that if P of x, then Q of x. Okay. There is an x such that if P of x, then Q of x. It's pretty rare to have a, to have a need to say that, actually. This really doesn't arise particularly frequently that you would need to say something like this. If you see yourself writing an exists with an implication the chances are very high that you've sort of got confused. it, this is, it's really let me just put it, let me just say that this is weak. Okay? It's not on the same strength as, as, as these guys, because this says for every x, if it satisfies P, then it satisfies Q. Now that's making a strong statement. For every x, there's an implication. This simply says, there's one x for which there's an implication. Well, in a, in a sense, there's, the implication is almost vacuous then. I mean, one thing to say, for example, is that if you can find an x that does not satisfy P, in other words, you can find an x for which P of x is false, if you can find an x anywhere for which P of x is false, then you have a conditional that's necessarily true. So this can be made true by finding an x that doesn't satisfy P. Okay, so that's all it would take to make this thing true. So if you're trying to make a, stronger statement, if you're trying to make an existence statement, if you're using this to say there is an x with a certain property, then you could make this guy true simply by finding an x that does not satisfy P. Because if you can make that part false, the conditional becomes true. So that's one of the reasons, really, why this is weak. it, it's you know, that I'm sure there will be circumstances where this is this could have some significance, but basically my message for you would be, forget that one. It's just if you see yourself writing an exists with an implication after it, the chances are very high that you've got confused. You know, always be prepared to override what I say. You know, all sorts of circumstances can arise. But in general, these guys are all quite significant. That's particularly significant. That one is very significant. This one is sort of less so, because it really just reduces to the two separate things. And this one is really pretty weak. So exists combined with implications, if you see that, flag it and say, do I really mean what I am writing there? Okay. Well, I hope that's helped clear up some of the basic issues about reading formulas. But like many things at this stage really, the only way to get rid of any confusions is to just do a whole bunch of examples for yourself. Okay? Good luck on sorting these things out as you work through the rest of the course.
