That's this one. So that's correct.

No doubt really does capture it. There is a time when they, when they play

together and as a partner, as a partnership and they win.

Now, now you know, you can look at some sort of gent will say that sometimes this

is used exclusively to mean more than one.

I mean, I don't think that's the case. I mean you'll find people that say that

you know, language is flexible. In any case, in mathematics, we always

interpret things like some. sometimes, as at least one.

you know? That's the whole point about the way we

set things up. We, we, we, eliminate these ambiguities

by being specific. And we're specific to say that, whenever

you're asserting something exists. Sometime, some game or whatever.

You mean at least one. Okay in which case you've got existence

from, existent to quantify that means there is at least one tennis game were

they play together and they win, okay, so that one is okay.

whenever Rosario plays with Antonio, she wins the match.

Well that's really again, that's for all t wt.

So it's not that one. Rosario and Antonio win exactly one match

where they are partners. Well, that one isn't going to work,

right? Because that says exactly one match.

There's no specification here of exactly one match.

If you wanted to do that, there's a, a notation.

This notation exists a unique t. Such that wt.

You can't say it, and you can say it other way.

I mean, you can, this is just an abbreviation for, for for an expression.

You know, we've seen that in the problem sets.

Hm, yes actually one of the assignments, so you can't capture it but this doesn't

capture it, this just says is it at least one it doesn't say that's exactly one we

think when, okay. Rosario and Antonio win at least one

match when they are partners, that's it that's another one that's fine this one

when they are partners Okay. If Rosario wins the match, she must be

partnering with Antonio. Well, first of all, there's a, there's a

universal quantifier floating around here, I think.

Well it's here. because it's saying whenever she wins the

match, she must be partnered. So there's a universal quantifier here.

But it's even worse, because the universal quantifier is actually for all

of X. For all matches.

Okay? For all doubles tennis matches where she

wins something or other. So this one here is false.

So this action here what's generally known as a scope problem.

In this statement the quantification Is actually over something different.

It's over all possible double tennis matches, not just the ones where she's

partnered. So not only does this not capture it, it,

it, there's actually another issue. There's a scope issue involved.

Okay. Because here the T ranges only over games

where they've played together. In this case we're looking at games where

Rosario plays with whoever she's playing with.

Okay? So there are a couple of things that

prevent this one being the, being the right answer.

Okay? Well I, you know, as I said at the

beginning, from a mathematical perspective, this is actually very clear.

It's definitely B and it's definitely E. And the reason I can be so definitive

about that is becuase I'm familar the way that we've set up the meaning of this in

mathematics to correspond to mean at least one and we interpret in mathematics

we intepret anything that exerts an existance to mean exists one, at least

one. And so things like sometimes, some of

these, some of those [NOISE]. they're all interpreted to mean at least

one. Okay?

Let's look at the next one. Well, same setup as in question 1.

The only difference is now we're talking about for all t, w of 2.

So let's, let's run through this one. Rosario and Antonio win every match where

they are partners. So, every match where they are partners.

That would be for all t. Because that's what t captures.

T is the doubles tennis matches where they partner.

And they win. Oh, that is that.

So that one's okay. What about this one?

[LAUGH] this has really got nothing to do with that as it is.

Rosario has always got nothing to do with winning.

It's just saying she always plays together.

that would essentially say that that X and t are, are the same variable in fact.

And is just saying that this, there's no distinction between X and T.

so I think rather than cross out those, say that it's wrong.

I'll say this isn't even a candidate. I mean, it's got nothing to do with

winning. Okay.

Let's look at part c. Whenever Rosario partners with Antonio,

she wins the match. Whenever she partners, that's for all t.

She wins. She wins, they wins.

It's all the same in doubles tennis. So that's okay.

What about this one? Sometimes, Rosario wins the match.

No. I mean, first of all.

it's, it's not about t, it's about x. Sometimes she wins with whoever she's

playing with. and it's an existential one.

So it's essentially of the form, exist x, wfx.

That's really what it means. Sometimes Rosario is in the winning team.

She wins. Well, that's not that.

It's a different quantifier. And it's over x, not t.

So, well that one. I won't cross it out, because at least it

talk about winning, so it's a candidate, but not the right candidate.

OK? Rosario wins the match whenever she

partners, this is whenever she partners, that's essentially for all T, and

whenever that happens, she wins. Okay?

Bingo. That matches that.

That's correct. And finally, if Rosario wins a match, she

must be partnering with Antonio. Well essentially, you've got something

like for all x here etcetera. So as before, as in question 1.

We've got a scope issue here. this is actually about all possible

matches, not just the ones where she is, she is partnering.

And, thus, the conclusion is that she is partnering on somebody and so, so this

doesn't, I mean just talk about winning but it's seems doesn't conclude because

there is, there is a scope problem, the way I'm presuming different things and

okay so it's not that one. So in this case we've got 'a', we've got

'c' And we've got E. It's kind of unusual to have one of these

multiple choice things where three of them are correct, but there you go,

sometimes that, sometimes that happens. Okay?

Let's move on to question three. On question 3, if you look through these,

looking for something that seems to say There's no largest prime.

I think you quickly end up looking at, at this 1d.

Which says that, for any number x, there is a number y, which is prime and bigger

than x. So that certainly says there's no largest

prime. Now the question is, do any of these.

Say the same in a different way. Well let's just look at them in turn.

Let's, that says there don't exist any Xs and Ys for which X is a prime, Y is not a

prime, and X is less than one. Well there are plenty of pairs X and Y

that satisfy that. So this is actually false.

>> So, I mean, we weren't, we weren't ask to say whether things are true or

false. But this is false, and we do actually

know there was no largest prime. That's Euclid's theorem, that the primes

are infinite. and in, in, and the list of primes goes

on forever. so it can't be this, because this is

actually false, but in any case, it doesn't mean the same.

It just means something just, nonsensical.

of course, there exists pairs x and y. With x a prime, and y not a prime.

And x less than y. So, to say it's not the case is, is, is,

is clearly wrong. What does this say?

for every x, there is a y. Such that x.

Well, first of all, that would say, for all x, x is prime.

That would say every number is a prime. So that's false as well.

So that can't possibly be it, that can't be it, that can't be it.

What does this one say? For all x and for all y, x is, what that

says as well, every number is a prime. That's false.