0:27

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,

1:50

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.

2:06

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.

2:21

So you have to strike a balance.

But always, if there's going to be any

ambiguity, put the parentheses in, and mix parentheses

2:28

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.

3:17

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.

3:58

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.

4:21

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.

4:59

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.

5:35

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.

6:26

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.

7:12

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.

7:24

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.

7:42

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.

8:05

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.

8:57

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?

10:13

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.

10:44

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

11:07

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?

12:17

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.

12:37

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.

12:52

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.

13:13

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.

14:35

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.

14:55

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.

15:55

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

16:32

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.

16:58

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.

17:35

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.

18:42

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.

19:16

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.

19:37

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.

19:55

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.

21:29

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.

22:15

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.

23:20

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.

23:44

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,

24:49

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.

25:02

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.

26:08

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.

26:52

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.