0:27

The ones that bind the most tightly, the ones that are sort of the strongest, if

Â you want, that, that hold something something close

Â together, are the quantifiers for all and exists.

Â And a quantifier applies to whatever comes adjacent to it.

Â Okay? Now typically, what comes adjacent

Â to it involves various other things, like and's and or's and not's, so

Â you would put them in in parentheses, or brackets, square brackets or whatever.

Â So very often, in fact alm, I, I almost always make a habit of putting whatever I

Â want next to it in parentheses, because the

Â for all then applies to everything that comes there.

Â Okay?

Â It binds tightly to this applies to everything, the same

Â with exists in here.

Â Now, if there's only something very simple coming next, for

Â example, supposing I wanted to say, all the balls are red.

Â I could say, for all balls Red B, if Red is a predicate that applies to balls.

Â So I could say for all B, Red B, and that would apply to the red balls.

Â And if there was something else here,

Â 1:50

again, especially when I'm giving introductory-level courses, I

Â usually put parentheses around quantifiers, but if you

Â look at some of my research work and

Â advanced courses, you'll find I often don't do that.

Â That's fairly consistent among among instructors.

Â 2:06

You know, the golden rule is, if

Â there's any doubt whatsoever, and if you're beginning

Â on this material, there certainly will be

Â doubts, if there's any doubts, put parentheses in.

Â 2:21

So you have to strike a balance.

Â But always, if there's going to be any

Â ambiguity, put the parentheses in, and mix parentheses

Â 2:28

along square brackets or even spaces.

Â I'll, I'll try to remember to give an example in a

Â minute with a space, because you can sometimes use spaces to disambiguate.

Â But the golden rule must be you want to avoid

Â someone being left un, unclear as to what the meaning is.

Â Okay?

Â [COUGH] negation is about the same strength as, well,

Â it is the same strength as, as the quantifiers.

Â So the negation applies to whatever's immediately next to it.

Â And since we usually want a whole bunch of things to be negated, then

Â the negation is followed by parentheses, and

Â then it applies to everything between there.

Â Let me give you the following example.

Â Suppose you wanted to say, not the case that 3

Â is greater than 0 and 3 is less than 0.

Â 3:17

Okay. Well, is 3

Â bigger than 0? Yes.

Â Is 3 less than 0? No.

Â So here I've got a conjunction of

Â something that's true and something that's false.

Â So this conjunction is false, so its negation is true.

Â So this guy is true. But supposing I wrote it this way:

Â not the case, 3 greater than 0 and 3 less than 0.

Â 3:58

This guy, 3 is greater than 0, is true, so that guy's false.

Â So here I've got a conjunction of

Â false things, so I've got something that's false.

Â So these clearly aren't the same,

Â because this one's true and that one's false.

Â Here, the negation applies to everything in between, which makes it true.

Â Here, the negation only applies to the thing next to it.

Â 4:21

Now, I could've gone back here, and put

Â parentheses here, and I'll mention that in a moment.

Â Actually comes up in the, the next, the next priority.

Â That wouldn't have changed things.

Â That wouldn't

Â have changed things, because the negation would

Â apply to what was in the next parentheses.

Â Negation applies to whatever comes next, and what comes next

Â is the whole thing, because the parentheses includes the whole thing.

Â So simply putting parentheses inside doesn't change anything.

Â It makes it maybe a little bit clearer, although this is one of

Â those cases where adding parentheses arguably

Â makes things a little bit less clear.

Â 4:59

But in terms of the logic

Â the issue between these two wasn't whether there were

Â parentheses around the 3 greater than or less than 0.

Â The issue was whether the parentheses governed everything that

Â was next, or just the one thing that was next.

Â Okay? So these are not the same.

Â Okay. The next one is conjunction.

Â Let me now just pick up that thing I mentioned before.

Â When I did that the first time, I wrote this: I said, 3 is bigger than

Â 0 and 3 is less than 0.

Â Now I, in fact, left a space: if you watch what I did, I left a space.

Â 5:35

And I realized at the time I was doing it, that that's what

Â I was doing, which is why I decided to pick it up now.

Â So this says that 3 is bigger than 0 and 3 is less than 0.

Â You actually don't, strictly speaking, need parentheses around here, because

Â this in an atomic formula, as we sometimes call it.

Â This is a basic building block out

Â of which we're building more complex formulas.

Â This simply states a fact: 3 greater than 0, an atomic fact, a single fact.

Â This states another

Â atomic fact: 3 less than 0.

Â So when you have basic facts about arithmetic,

Â or whatever, they are, they stand on their own.

Â The conjunctions the kind of quantifiers, are what combine these basic facts.

Â So you, strictly speaking, don't need to put parentheses around these.

Â This is a case where I typically would just leave a bit of extra space in

Â here, to sort of make it clear that this is a unit, and that's a unit.

Â 6:26

On the other hand, if you want to be safe, and it's always wise

Â to be safe if you're at all unsure, you could put those things in.

Â Okay.

Â And this here: I went back and put them in just to make it clear.

Â Okay?

Â Then well co, some mathematicians will say that conjunction, disjunction

Â are, are more or less the same, or conjunction, disjunction, implication.

Â We, we're getting down to a sort

Â of a general grouping now, where everything has roughly the same strength.

Â There are actually some arguments that say that conjunction should

Â be tighter than disjunction, but it's, it's, it's not particularly strong.

Â 7:12

The point is, the, the conjunction applies to whatever's

Â to the left of it, and whatever's to the right

Â of it, and if you want it to apply to

Â a whole bunch of things, you put them inside parentheses.

Â 7:24

Likewise here: you would have a whole bunch of things.

Â And the same is true for disjunction and implication.

Â So regardless of whether you think that's stronger

Â than those the issue should never really arise,

Â because you should always put things in parentheses

Â to just say, it's this guy or this guy.

Â And in here, it could be a whole bunch of things.

Â 7:42

And that whole bunch of things will be disjoined with this.

Â And likewise here, if you have an implication or a conditional, this whole

Â thing would be the antecedent, and this will be the consequent.

Â Now, in here, there may be all sorts of conjunctions and disjunctions and stuff.

Â There may be quantifiers in here.

Â There may be quantifiers in here; there could be

Â all sorts of stuff in here, negation signs inside.

Â 8:05

This whole thing would imply that whole thing.

Â So whenever you look into, I mean, the sort of, the basic thing with all

Â of these is, when you've got a, a, a, a, a, a quantifier or a negation

Â symbol or a conjunction or a

Â disjunction or an implication, or equivalence, actually.

Â I didn't talk about equivalence, but equivalence is

Â just the conjunction of two implications, the biconditional.

Â So you cou, we could put that one in here as well.

Â 8:57

So this is, I would say, at least the way

Â I was brought up, let's put it that way, as

Â a mathematician, I was brought up to, to say that

Â that actually is, is okay and it's, it's not ambiguous.

Â But I would almost certainly now, I think I've cured myself of

Â that, that childhood sin, I, I would always put in parentheses, and

Â say its A and B or C and D.

Â I mean, you just have to be very careful

Â about making sure that things are nice, and not ambiguous.

Â Okay?

Â 10:13

I've got a for all, and a for all applies to everything that's adjacent to it.

Â Now, that parenthesis there teams up with that parenthesis there.

Â And I've actually written

Â them not as parentheses, but as square brackets to, to make it absolutely clear.

Â 10:44

And then it's going to say something about the licences.

Â Now, the licence L is going to appear on

Â both sides of this conditional, and it'll be the same

Â L, because the L has been picked here, and

Â once you've picked the L, it'll apply to everything here.

Â So that L is bound by that quantifier. Okay?

Â So it would say, L is bound

Â 11:07

by the quantifier for all L. Okay.

Â A quantifier binding.

Â Okay, so let's read it now in in English. It says, for any

Â licence L, if there is a state

Â in which L is valid, if L is valid

Â in some state, at least

Â one state, then L is valid

Â in every state. Okay?

Â 12:17

What happens?

Â If that licence L is valid in some state, then

Â that licence, that same licence, is valid in every state.

Â So this is the one that actually says, a licence that's valid in one state

Â is valid in every state which is true in the United States, by the way.

Â 12:37

okay. That's the first one.

Â Let's look at the second one. What's the difference?

Â Let's see.

Â Well, we've got for all applies to something in the middle,

Â so, for all applies to everything here, because I wrote the parentheses.

Â 12:52

The only difference is that instead of having

Â a conditional or an implication, I've got a conjunction.

Â So let's see what that, how we would read that.

Â okay? Coming down, the binding is the same.

Â The for all applies to everything here. This exists applies to this thing.

Â This for all applies to this thing.

Â 13:13

And the L is the same L here as here, because once you've said

Â the for all L, within this expression here, the L is determined by that.

Â The L is still bound.

Â So as, as was the case there, the L inside here

Â is bound. okay.

Â But what does it, how do, how would we read it?

Â We'd say, for every licence L, there is a state in

Â which the licence is valid and the licence is valid in all

Â states. So let me just write

Â that down: for any

Â licence L, there is a state

Â in which L is valid and L is

Â valid in every state.

Â 14:35

For example, if you go to California, and

Â you drive with too much alcohol in your bloodstream,

Â you will find yourself with an invalid licence.

Â Not every licence is valid.

Â 14:55

Okay.

Â In fact, really it, it's the, it's the first thing that was the problem.

Â For every licence, there is a state in which it's valid.

Â Well, that's simply not the case.

Â Already the first conjunct makes it invalid.

Â Didn't arise in the first one, because in the first one, the

Â par, the part that says it's valid in a state was the antecedent.

Â If it's valid in a state, then it's valid in all states.

Â So that said, for every licence, if it's valid in a state.

Â This says, for every licence, it is valid in a state.

Â Well, that's not the case.

Â Not all licences are valid. Okay.

Â So there is a distinction between these two, and in fact the distinction

Â is a meaningful one in terms of validity of licences and so forth.

Â 15:55

Well, [LAUGH] is that true? Is it true?

Â Remember, this is, this is a unit.

Â The for all and the exists apply to whatever's next.

Â So there's a line here.

Â The for all and the exists don't apply to that.

Â They apply to what's next.

Â And there was no bracket, so it doesn't include here.

Â So what this actually says is, for every licence,

Â there is a state in which that licence is valid.

Â So what this really says, is that

Â 16:32

Well, okay. That's not true.

Â And, and it's, the statement is, if that's the case, then

Â for all S2, that would say, L, well, ha ha,

Â this would say that L is valid in all states.

Â 16:58

As a conditional, this guy would look, on the face of it,

Â as if it was going to be true, because the antecedent is false.

Â It's not the case that all licences are valid somewhere.

Â There can be invalid licences. So this is a false antecedent.

Â Now, you might be tempted to say, since

Â it's a false antecedent, the conditional is true.

Â 17:35

It's not governed by that quantifier. This is just an orphan.

Â It's just sitting there.

Â We don't know where it comes from. We don't know what it means.

Â It's just a letter.

Â It has no internal meaning to this formula.

Â So it's not the case that this is a valid conditional.

Â It's actually undefined.

Â This is meaningless unless you know what L is.

Â If you know what L is, you can assign meaning.

Â And once you know what L is, then, you

Â know, if L referred to my licence, if, if that

Â L there was my licence then we would have

Â a, a, a meaningful, and, in fact, a true conditional.

Â 18:42

Somewhat similar to the one up here, but not quite.

Â Okay.

Â Let's just read it.

Â So it says, for every licence and for all

Â pairs of states, the licence is valid in one

Â state and the licence is valid in two states.

Â well, that really just means all

Â licences are valid in all states.

Â 19:16

And it's, it's over, I mean, there's redundancy

Â here, because the second S adds nothing new.

Â It simply says, for all licences and for all states, the licence

Â is valid in that state, and it's valid in the other state.

Â So we could just scrap that, and scrap that,

Â and we'd have the meaning without any of that stuff.

Â 19:37

So there's nothing actually wrong with this.

Â It's just I mean, it's a false

Â statement, but it's it's got redundant clauses.

Â The second clause says, adds nothing that the first one didn't already state.

Â 19:55

Finally, we just try to distinguish between four cases

Â that beginners typically get find to be very confusing.

Â They're actually really very distinct.

Â And if you find the, there's confusion between these four, that's a sure sign

Â that you haven't yet mastered the, the notations and the, and what they mean.

Â Okay.

Â Let me just write down a transcription of what

Â it means, and then let's just ask ourselves exactly

Â what that signifies. So in English, that would

Â say, for every x, if P of x then Q of x.

Â If, P. Okay?

Â For every x, if P of x then Q of x. This is very common.

Â Okay, for every for

Â every number, for every real number, if that number is

Â non nonnegative, then it has a square root etc, etc, etc.

Â So this occurs a lot in mathematics, this kind of statement:

Â for every x, if P of x then Q of x.

Â Okay? Very meaningful.

Â And it's the same x here, notice.

Â Once you've got that for all, the x here is the x here.

Â So,

Â whatever x, providing the x satisfies P, then it satisfies Q.

Â So this establishes a relationship between P and Q.

Â Because if you've got an x that satisfies P, then that x will definitely satisfy Q.

Â So this is a very strong and very common statement to make.

Â 21:29

This is also pretty common. This says, for every

Â x, P of x and Q of x. It says that

Â every x satisfies P and Q. This is kind of strong.

Â I mean, it, it doesn't occur

Â terribly frequently, because that's really the same.

Â And, and, I mean, you could equally, you could just as equally

Â say, for all x P of x and for all x Q of x.

Â 22:15

notice, by the way, that this is

Â nonambiguous, because of the binding, the for

Â all binds what's next to it, so the for all can only bind that.

Â The for all binds what's next to it. And so I don't need the parentheses here,

Â because in this case, the for all absolutely can't be confused

Â with that, so here's a case where you don't need extra parentheses.

Â I didn't even write the parentheses here.

Â You don't need them.

Â This is totally clear in this case. Okay?

Â And it's equivalent to that.

Â So you don't see this very often, because it really is

Â just saying everything satisfies P and everything satisfies Q, but it's okay.

Â If that arises, don't worry about it. It might, in a context,

Â it might be sensible to write that down.

Â 23:20

Now, this is, again, pretty common in mathematics.

Â This is quite a strong statement.

Â It says you can find a single x which satisfies P and satisfies Q.

Â 23:44

I mean, this one's strong, but it's only strong because each part is strong.

Â So the, there's a re, there's almost a redundancy in the way it's written.

Â So this is maybe I'd better put strong in quotation marks, just sort of

Â say, well, yes it is strong, but it's not strong because of the logical structure.

Â It's strong simply because it's making a statement about

Â P and Q both being satisfied by all x's.

Â Okay.

Â What about this last guy?

Â This is one that people often write down, and it,

Â this, this really means nothing, in, in any real sense.

Â It says what: there is an

Â x such that if P of x,

Â 24:49

This really doesn't arise particularly frequently that

Â you would need to say something like this.

Â If you see yourself writing an exists with an implication

Â the chances are very high that you've sort of got confused.

Â 25:02

it, this is, it's really let me just put it, let me just say that this is weak.

Â Okay?

Â It's not on the same strength as, as, as these guys, because this says for every x,

Â if it satisfies P, then it satisfies Q. Now that's making a strong statement.

Â For every x, there's an implication.

Â This simply says, there's one x for which there's an implication.

Â Well, in a, in a sense, there's, the implication is

Â almost vacuous then. I mean, one thing to say, for

Â example, is that if you can find an x

Â that does not satisfy P, in other words,

Â you can find an x for which P of x is false,

Â if you can find an x anywhere for which P of x

Â is false, then you have a conditional that's necessarily true.

Â So this can be made true by finding an x that doesn't satisfy P.

Â 26:08

Okay, so that's all it would take to make this thing true.

Â So if you're trying to make a, stronger statement, if you're trying to make an

Â existence statement, if you're using this to say

Â there is an x with a certain property,

Â then you could make this guy true simply by finding an x that does not satisfy P.

Â Because if you can make that part false, the conditional becomes true.

Â So that's one of the reasons, really, why this is weak.

Â it, it's you know, that I'm sure there will be circumstances where this is this

Â could have some significance, but basically my

Â message for you would be, forget that one.

Â It's just if you see

Â yourself writing an exists with an implication after it,

Â the chances are very high that you've got confused.

Â 26:52

You know, always be prepared to override what I say.

Â You know, all sorts of circumstances can arise.

Â But in general, these guys are all quite significant.

Â That's particularly significant.

Â That one is very significant.

Â This one is sort of less so, because

Â it really just reduces to the two separate things.

Â And this one is really pretty weak.

Â So exists combined with implications, if you see that,

Â flag it and say, do I really mean what I am writing there?

Â Okay.

Â Well, I hope that's helped clear up

Â some of the basic issues about reading formulas.

Â But like many things at this stage really, the only way to get rid

Â of any confusions is to just do a whole bunch of examples for yourself.

Â