OK. Here we are at week five. Now, this is going to be the most difficult week you'll have done.
We're going to be looking at evaluating arguments again this time deductive arguments.
And in particular, we've got to look at the notion of validity. And this is a difficult notion to wrap your mind around.
So expect to have a hard slog. But if you keep at it, you'll be fine.
Right. Just a recap, as usual, or last week, if you remember, last week, we evaluated inductive arguments.
Remind me what an inductive argument is. A percentage chance of being right now, I'm not sure that's quite the way I put it.
That's more or less what's more or less likely, given the premises, the conclusion is more or less likely.
Exactly. As opposed to a deduction where if the premises of true, the conclusion must be true.
Okay. So we looked at inductive generalisations and causal generalisations,
arguments from analogy and arguments from authority, all different types of inductive arguments.
And if you remember, we looked at these things. Now, I realised at the end of last week that I hadn't added that one on.
So if you've got you might want to go back to the handouts that you had from last
week and just note that this week's handouts will have that one on as well.
And it's it's fairly obvious what that is. And if it isn't, it will be by the end of today.
OK. So those were the tests for an inductive generalisation.
These were the tests for a causal generalisation. These for the argument from analogy and these for the arguments for authority.
I won't go over them again because as we've started a bit late, I would rather get straight on to today.
But if we have time left at the end of today for questions, if you want to go over any of the ones that we did last week and you can't remember them,
or you're just like a reminder, ask me then and we'll we'll certainly come back to those.
Okay. But this week, we're going to be looking at the distinction between validity and truth.
And that's why videl its validity is important. And we'll we're going to finish both this week and next week with evaluating deductive arguments.
We've actually already mentioned the distinction between validity and truth.
Would anyone like to have a go at telling me what it is? It was about three weeks ago, so quite reasonable, if you don't remember,
and quite reasonably, if you get it wrong, because we only mentioned it glancingly. Good.
Yeah. OK. An argument can be valid, but not true.
Why is that the case? Were right.
Okay, you do. Let's not push it.
No, you were doing very well. You're absolutely right. Statements or sentences or propositions can be true or false.
But arguments can't be true or false. Arguments can only be valid or invalid.
Do you remember that? We talked about that in the first week.
And I said then that one of the things that will give you a way every time to a philosopher or a logician
is if you talk about arguments being true or false or if you talk about sentences being valid or invalid,
because it's like my talking about the table being loud, it makes it clear that I don't understand the meaning of loud or I don't
understand the meaning of table or I'm speaking metaphorically might also help.
So and sentences statements can be true or false.
So premises can be true or false.
Conclusions can be true or false, but arguments can't be an argument can be valid or invalid, but they can't be either true or false.
Okay, so we'll be looking at why that is in a little more depth today and we'll look at why validity is important,
because one of the interesting things about valid arguments, as you'll see, is that they can have false conclusions.
And you might ask yourself, why should an argument that has a false conclusion be a useful argument?
Never mind a good argument, but we'll see why in this session.
Okay. Right. First, we're going to deal with the idea of soundness and an argument,
a good deductive argument is sound if and only if it's both valid and it has two premises.
So soundness is to do with both validity and the truth of the premises.
So. Okay, is this argument sound?
Put up your hands rather than yell out. Put up your hands if you think this argument is sound.
Do you see what I mean? This argument is valid and has false premises.
Could this argument be sound? Put up your hand rather than shout out.
Right. Anyway, you've all done it now. And you're quite right. No. That can't be sounds, can it?
Okay. What about this one? Put up your hand. Because some people think more quickly than others.
So give everyone a chance to think. So you've decided. Put up your hand.
OK. That's most of you. What is it? No, it's got to be unsound, hasn't it?
It couldn't possibly be both. What about this one? Okay.
What are you saying to this one? No. Okay. Again, this one can't be valid.
So what about this one? Yes. You can yell out this time.
Okay. Yes, that's right. That's got to be sound good. Okay.
So there's the same thing filled in.
Okay. Now. So the two things that are both necessary for a sound argument or a good argument
insofar as sounds and ESCOs truth of the premises and the validity of the argument.
Now the truth of the premises, and I've mentioned this before,
is not an interesting matter for logicians or for you, given that your interest is in critical reasoning.
And the reason for that is that when we determine the truth or falsity of a premise, it's not necessarily a logical matter.
It's not necessarily for philosophical matter at all. So if I say, is the chair blue, how are you going to determine the truth or falsehood of that?
It's dead easy. There's no trick question here. You're going to look at the chair.
That's right. Okay. If I say two plus two equals four.
Is that true or not? Okay. Yes, it is.
Okay. And how do you know? Yes.
I mean, you will you will know your basic arithmetic. OK. And you might say, OK.
All swans are white. How do you determine the truth of that answer? Inductively and so on.
So there are lots of different ways in which a premise would be determined to be true or false.
And my job in teaching critical reasoning is not to teach you how to look at a premise and decide whether it's true or false.
My job is to teach you how to look at an argument and determine whether it's valid.
OK, you with me. So. So when two characteristics of a of a sound argument, one is true premises, one is validity.
I leave it to you to ask about true premises. All I can teach you is what validity looks like, what validity is.
Okay. So. Yeah. OK, validity is of interest to logicians because validity preserves truth.
If an argument is valid, then if its premises are true, we can be certain that its conclusion is true.
So it's the F here that's important. If the premises are true, then the conclusion must be true.
And so the validity of an argument preserves the truth of the premises.
If there's truth there, it will preserve it.
If there isn't truth there. It won't say anything about the truth of the conclusion.
So we'll see more about that in a minute. Okay.
So validity is truth preserving. That's why it's useful.
So validity is of interest to anyone who's concerned with truth.
It's not just true premises that concern you if you're interested in truth.
It's also the validity of arguments. And that's because we often don't know the truth of our premises and we often test the truth
of our premises by constructing valid arguments and testing the truth of the conclusion.
So let me go through that again. Okay. If an argument is valid, then if the premises are true, the conclusion must be true.
Okay, so we might have a premise that we don't know whether the whether the premise is true or not.
But if we can use that premise as the premise of a valid argument and then test the conclusion of that argument, see whether that's true or false.
And then if we can show that the conclusion of a valid argument is false, what do we know?
If we can show that the conclusion of a valid. Put your hand up, if we can show that the conclusion of a valid argument is false.
What do we know? Let me remind you again of the argument is valid if and only if if the premises are true.
The conclusion must be true. So if we see that the conclusion of a valid argument is false.
What do we know? Put your hands up when you've got the answer.
A little bit longer, it's it's quite difficult to play with these concepts in your mind if you're not used to so.
OK. Right. What do we know? Who wants to give me the answer?
George can have a. Right.
At least one of the premises must be false. That's right.
They might all be false or it might be just one of them is false. But we know that at least one of them must be false.
So if an argument is valid and the conclusion is false, then we know that at least one of the premises must be false.
And here's a couple of examples. OK. Smoking causes cancer.
That's a hypothesis. We've base that. Okay. Why do we get that from.
What sort of arguments might that be? The conclusion of an inductive argument.
Good. So we see that this smokers got cancer, this smokers got cancer, the smokers got cancer, et cetera.
And we formed the hypothesis. Smoking causes cancer.
Okay. That generates a prediction, doesn't it? If we put that hypothesis as starkly as that smoking causes cancer, it generates a prediction.
If smoking causes cancer, then every smoker will get cancer. Doesn't it?
Oh, yes, it does. No.
No, remember I said if we put the conclusion as starkly as that, sorry, the premises starkly is that that's the prediction it generates.
OK, the test or the conclusion of this argument is each smoker gets cancer.
OK, now if we look at smokers, we see that some of them get cancer and some of them don't.
Don't we? OK. So what do we know? This is a valid argument, isn't it?
Yeah. Okay. But we've seen that the conclusion is false.
So what do we know? That one of the premises is now?
It might be this. It's not the case that smoking causes cancer or it might be the case that smoking
could cause cancer without its being the case that every smoker would cause cancer.
So what do we do if we're researchers at this point? What might be the case?
We've got a correlation between smoking and cancer, haven't we?
Otherwise, we wouldn't have that first premise. But it was not going to be an exception.
This correlation, is it? We've seen that. So we could say we could decide to jettison the idea that smoke that causation is sufficient.
So we might want to say, well, okay, it's possible for A to cause B without its B in the case that A is sufficient for B.
What I mean. So we could jettison premise, too.
So possibly A can cause B without its being the case that you always get B if you get A.
Actually we probably wouldn't want to jettison that would we. Why not?
OK. I think I'm going about this, looking at your faces, I'm getting a bit wrong.
Okay. Let let's say that we decide to jettison that. Does that mean we're going to say that's false?
What are we going to say? Instead, we could say that all we could say that smoking plus something else we could say we.
We know the hypothesis is going in the right direction. But it's it's smoking.
Plus, I mean, maybe there's some genetic characteristic of people who smoke and get cancer that's
different from those who smoke and don't get cancer or something along those lines.
Do you see what I mean? You've got so we've got something for which we've got we've got reason to believe it's true,
but we're not sure or actually I mean, given what we know, we're pretty sure it isn't entirely true.
But if we put these together in a valid argument, we get that which we know to be false.
So we know that either one of those has got to be false.
We could play around with something that's the business of philosophers to play around with, namely our concept of causation and say,
well, okay, there can be causes that don't always have their effects, that don't necessitate their effects.
Or we could stick to being empirical and say there's something in addition to smoking that that.
Causes cancer. With me. If smoking always caused cancer, you couldn't have a smoker that didn't get cancer.
So that would be true, wouldn't it? So because that's false.
We know that one or the other of these is true.
And if you're a philosopher, you might want to say, well, maybe causes don't necessitate.
Their effects could premise to be true. Sorry. False.
If you're a scientist, you're much more likely to say actually causes do necessitate their effects.
Therefore, it's not smoking on its own that causes cancer.
There must be something in addition to smoking that goes to causing cancer.
By definition. Necessitates his effect.
You mean? Well, you say by definition. But of course, part of the job of a philosopher is to say, well, is it true that that's the correct definition?
Cause. Could there be it?
No. If it doesn't cause it is not a cause it. But if it doesn't necessitate it, might it cause it?
I mean, if we look at quantum mechanics, for example, here, here are some ideas that there may be causation that doesn't necessitate.
And one of the big questions of philosophy and if you ever do an introduction
to philosophy as opposed to an introduction to critical reasoning course,
you might look at questions of could there be backwards causation, for example?
Could the effects come before a cause? And could there be non necessitating causes?
What actually is the causal relation? These are all philosophical questions that are true.
No, no, they don't have a different concept, of course. What they do is they unpack the concept, of course, that we think we have.
And I quite agree that the common concept, the cause is that causes necessitating.
Absolutely. Well, then what's the call what do you mean by random here?
Because, well, then your say you're suggesting that a cause isn't necessitating, which is what I've got your name.
Sorry, Brian is is questioning that.
Well. So you disagree on the meaning of the word, cause you can have that conversation later in the common room.
But I'll tell you now that if you want to find out what the answer to that is,
you should come to a philosophy weekend on the nature of causation, because there it's a big issue.
That one. Very big one. You had a question. Well, that that would lead you.
That's why you know that each smoker doesn't you.
If you're evaluating the inductive argument that leads to that, you might want to deal with percentages and so on.
Exactly. But what I'm pointing out here is that you form a hypothesis.
You generate a prediction which is going to be based on on your understanding of causation and so on.
And then you test that prediction. And look, you've what you've done is you've constructed a deductive argument.
And as long as it's valid and you test the conclusion, if the conclusion comes out false, then you know that at least one of your premises is false.
Either your hypothesis is false or there's something fishy about causation.
And you're almost certainly not going to go for that one in this. You're a philosopher. Okay, here's another one.
All women are passive. Mrs. Thatcher is a woman. Therefore, Mrs. Thatcher is passive.
OK. You've got a hypothesis here, OK? You've generated that on.
On what grounds? None at all.
None at all, he says again. The grounds of prejudice or something like those say inductive grounds as well here.
This is observation, isn't it? And do we have a valid argument here?
Yeah. If these premises are true, this conclusion must be true.
Okay. The the conclusion isn't true. I think you'd probably agree with me.
Therefore, one or other of these must be false.
And as we know, an awful lot of people talked about, Mrs. Thatcher has been the best man in the cabinet.
Door to door, in effect, they were relying on this argument and suggesting that instead of that one being false, it's this one that's false.
And here's another one. Tony Blair is sorry. All Labour Party members are left to all socialists.
The left wing Tony Blair is a socialist.
Therefore, Tony Blair's left wing.
And if you think Tony Blair is far too right-wing to be a socialist, you're going to say that Tony Blair is into socialist rather than not.
All socialists darted off. That came off the top of my head.
I may have got bits of that wrong, but I hope you can see the general thrust of it, as we're going to see in a minute.
You can have a valid arguments, the conclusion of which is false. So you might ask, well, then what's what's the point of valid argument?
Why should we be interested in validity if it can generate arguments, the conclusion to which is false.
And I'm saying, well, actually the discovery of a valid argument that its conclusion is false can be hugely important.
And the reason it can be important is that we often test our premises by constructing a valid arguments and testing the conclusion.
Okay, so I just. Well, I just made it clear just to make it clear that we were testing a hypothesis by constructing a valid argument
showing the conclusion was false and then seeing that we can question either or both of the premises.
Okay, so you can just substitute premise, premise, conclusion, no problem.
Exactly the same. So it's just that we use premises in different ways, don't we?
Sometimes we use premises as hypotheses. Sometimes we use them as predictions and so on.
Okay. So I hope I've convinced you that the relation of validity is important to you.
If you're concerned about truth and what makes it important is that validity preserves truth if there's truth in the premises.
There will be truth in the conclusion if there isn't truth in the premises.
There won't necessarily be truth in the conclusion. And that might tell you that the premises are not true, which might be very useful information.
In fact, as is often very useful information, because, of course,
the whole of science involves testing hypotheses and and hoping, some would say, to falsify them.
Okay, so we're. So why is this relationship of validity so important?
What exactly is it? Well, there are different theories of the nature of validity.
And there are some problems, as we'll see later on, with the notion of validity.
But what I'm going to give you now is the best theory that we can come up with.
An argument is valid.
If and only if a notice this logical phrase here, if and only if there is no possible situation in which all its premises are true.
Remember, it must be all of them. And its conclusion false. Okay, that's that's a claim about a valid argument.
Okay, let's move on. Okay. Two things to beware of.
Firstly, it's the possibility of the combination of true premises and false conclusion that's ruled out by an argument's being valid.
Okay. So it's not that. It must have true premises or it must have a true conclusion.
What it must have in order to be valid is true premises.
So what it must not have in order to be valid is the combination of true premises and false conclusion.
That's completely ruled out by an arguments being valid.
And we're going to do some exercise in a minute. Also, the second thing to note and second thing to beware of, and this is going to trip you up, boy,
are you going to leave today having felt you've really had a mental gymnastics because you're all going to get this wrong?
Probably. Maybe you won't. I mean, maybe you cleverer than most people are.
But every class I've ever taught this to starts off by getting it wrong in almost every aspect.
So don't be worried if you do. It's the possibility of the combination of true premises and false conclusions.
That route that's ruled out, not just the actuality of that combination.
OK. So it's not just the premises, as a matter of fact. All true.
And the conclusion, as a matter of fact, is false. That's ruled out.
It's the very possibility of the premises being true.
And the conclusion false that's ruled out. Okay.
Big difference between actuality and possibility. Marianne is actually wearing a skirt today, but she might have been wearing jeans.
Okay. So there is a possible world in which Marion's wearing jeans, even though in the actual world she's wearing a skirt.
Okay. So faced with an argument whose validity we're trying to determine, we've got to ask.
Just all the premises. True. And the conclusion false together in actuality.
But could the premises be true? And the conclusion false together in some situation.
Okay. Do you see the difference? You're all looking very worried.
Okay, let's let's try a few. Let's test a few, okay?
I'm going to leave that with you. You have a look.
Put your hands up when you think. Well, actually, let's do the first one.
Do you think the arguments could an argument could be valid if all its premises are false?
Put your hands up. If you think you've got the answer, don't yell out.
And don't worry if you're a bit slower. Everyone thinks different speeds.
OK. If the premises of an argument are false. Could the argument be valid?
Could the argument be valid if the premises are false?
OK. Those who said yes. All right. OK.
You could have a valid argument in which the premises are false.
The only thing that's ruled out by an argument is being valid is the possibility
of the combination of the premises being true and the conclusion false.
So the only thing that's ruled out by an argument's being valid is the possibility, not the actuality of the combination of the premises being true.
And the conclusion false. So you could have a valid argument.
The premises, which are all false. If it's Friday, Marianne's wearing jeans.
It is Friday. Therefore, Marianne is wearing jeans. Isn't that an argument that has the two false premises but is valid?
Okay. Right. What about the false the premises of the argument are true and the conclusion is true.
Okay. Remember the question. Do you think an argument of this sort could be valid?
The premises are true and the conclusion is true. OK.
Could it be valid or not? Yes. Well done, well done.
Getting there. OK. That's pretty spectacular. OK, good.
What about the last one? The premises of the arguments are true and the conclusion false.
Put your hands up. Don't yell out the answer. OK.
Some of you say yes, some of you say no. The ones who say no in this case are right.
Okay. The only thing that's ruled out by the definition of validity is an argument that's valid where there's a possibility.
And in this case, there's an actuality of the combination of premises that are true and conclusion false.
That's the only one that's ruled out. And that is this one.
So an argument of this kind couldn't be valid.
The other two could. Okay. That's why there's plenty of time to practise with its finesses.
OK. So so let's think again if the premises could be true together with.
So do you remember I said two things you need to beware of. What are they.
Two things. I said you've got to beware of. What are they?
Look back on your notes if you can't remember. Because I really meant it that you must be aware of these things.
Two things you must be aware of. It's the combination of true premises and false conclusion that's ruled out.
Not either one or the other, but both together. So if the premises could be true, together with the conclusions being false.
Okay, that's that's one of the things you're looking out for. What's the other one? It's the possibility of the combination rather than the actuality.
So if the conclusion could be false.
Together with the premises being true, if they could be the case.
Not just that. If it is the case, if it could be the case, then the argument.
Is valid, is that right? Sorry, I lost track myself.
Yes, because there's only deductive arguments that are truth preserving that whether conclusion is guaranteed by the truth of the premises.
Some people do talk about inductive validity and deductive validity.
But if they do, they're using valid in two different senses. And I think it's actually much better to stick to the idea of validity with deductive
arguments and talk about inductive strengths rather than inductive validity.
Because validity is an either all thing where whereas strength is a matter of degree,
an inductive arguments are more or less strong, whereas deductive arguments are either valid or invalid.
OK. What is it? What does an argument have to have to be sounds.
Can anyone tell me two things it needs in order to be sound true them to premises and be valid?
That's right. So that's soundness and that's fairly easy. We can put that on one side.
We're now looking at the validity, one of those characteristics.
Last night is because, well, as I am about to show you, it doesn't mean it necessarily is valid.
It could be valid. Values is an either or thing.
For every argument. But if if you have a situation where the premises have one truth,
value and the conclusion, another truth value, that could encompass different arguments.
I'll show you exactly what I mean. OK. So just keep this in mind.
If the premises could be true together with the conclusions being false.
Then the argument is invalid. Otherwise, it could be valid.
Okay. Now, this is a bit of a trick question. But and I'll explain why.
Actually, it isn't a trick question afterwards. OK, here's an argument.
Now, remember, I told you right at the very beginning that anything can be an argument depending on the context.
Okay, so you may not find this a very convincing argument, but believe me, it's an argument.
These two plus two equals five. Therefore, grass is green.
Okay, that's no argument. Could that be valid?
OK. Now, don't try and be clever here. Don't try and be clever here.
Work it out actually from the definition which you can't read there, but you can read in your notes.
So have a look at that. Is there a situation in which the premises could be true and the conclusion false?
Is there a situation, any situation in which that is true?
And that is false. Is it? Could it be that that is true and that false in any possible world?
No. Okay, so is that argument valid?
Who said yes? Somebody said yes. And if they would admit to it, I would then give them a clap.
Well done. It is. That's a valid argument. And what that shows you is that actually for an argument to be good, it means it needs more than validity.
OK. That is a perfectly valid argument because there is no possible situation in which that premise is true and that conclusion false.
And the reason for that is there is no possible situation in which that premise is true.
Is the. That premise is a contradiction. And you may have heard the saying anything follows from a contradiction.
And the reason anything follows from a contradiction is that a contradiction must be false.
So there's no possible situation in which the premise is true. Therefore, there couldn't possibly be a situation in which the premise is true.
And the conclusion false. Could there. So that satisfies the definition of validity.
And that's what I want you to go on. I did. Because what shall I tell you why you're all sitting there looking so worried.
I feel for you. I really feel for you. But I guess I went through this. I honestly, I did.
I'll tell you, when I was an undergraduate, we did logic in my first term.
And I sat there looking bit like some of you looking through the whole term thinking, oh, what have I got myself in for?
And then during the vacation, I sat there with my logic book and I worked through it step by step by step.
And I went over it again and again and again and again.
And the thing that I had been missing all along was that I should have been taking more account of definitions than I had been.
I was doing what you're almost certainly doing now. True is a good thing, isn't it?
True sort of gives you a nice warm feeling, all sort of warm and cuddly and validity gives you that same sort of nice, warm feeling, doesn't it?
You know, so therefore, anything that's true is valid and anything that's valid is true.
And all this business about false premises, you know, has nothing to do with.
Is that do you think this might be behind what. Yeah. Yes.
Okay. Does that help? So try and drop the fact that both truth and validity make you feel all warm inside and just go for the definition.
What does this argument satisfy the definition of validity or does it not?
Could this argument satisfied the definition of validity or could it not?
That's what you answer or not on your nice, warm feelings of truth and validity.
Going together because sadly they don't, except in the one case of truth, being preserved by true premises.
Okay. Does everyone see why this is a valid argument? Would you like me to go over it again?
Yeah. OK. Sorry. OK.
OK. The definition of validity is there is no possible such an argument is valid.
If and only if there is no possible situation in which the premises are true.
And the conclusion false. And what matters is that it's a possible situation, not an actual situation.
And that is the combination of true premises and false conclusion that that's ruled out.
OK. That's the definition of validity. Now, here we have an argument.
OK. Could this be valid? Well, is there a situation?
Could it be the case? Could it be the case? That's what I'm asking.
That this premise is true together with this premise being false.
Well, you look at this and you see that there actually isn't any situation ever anywhere in which this could be true.
Is that true? So how could there ever be a situation in which that's true and that's false?
Couldn't be. Could the. With me. Yeah, I can see it's getting there, isn't it?
I can see understanding dawning on about 50 percent of faces.
The other 50 percent are looking pretty sick. Okay.
OK. If I say that something's P and Q.
And then I say that, okay. Is that true?
You don't know do. But if I then say not pee and say, is this true, what do you know?
No, you don't. No, not you. But do you know the truth value of this?
If PE's falls, what's the truth? Value of P and Q?
False. It must be, mustn't it, because a necessary condition for P and Q being true is that the P and Q True.
Sorry, but it isn't that you knew that immediately in the same way.
If I say there is no possible situation where this is true and this is false.
Well, if there's no possible situation in which this is true. Then how could there be a situation which this is true.
And this is false. Another couple of nights went on, then I'll do another one.
Let's let's see if another one thing on. Could this argument be valid?
Put your hands up when you've got the answer. Don't yell it out. Don't show yourself.
Just. Nobody's got it yet.
Oh, yeah. One up to. Well done.
So you're asking yourself, could there be a possible situation in which that is true and that's false?
Is that any possible situation in which that's true and that's false together?
So any situation which. That's true. And that's false together.
Put your hand up when you've got the answer. Yes.
No, you're you're doing exactly what I'm warning against. They're yours.
Did you hear what lady said? She said she thought a valid argument was where the premise was were true and the conclusion true.
Now, that's a classic example of thinking that because validity is good and truth is good.
Therefore, a valid argument must have true premises and true conclusion.
Do you see how that works? But actually, a valid argument doesn't have true premises and true conclusions.
What is a valid argument? Give me give you give me the definition of a valid argument.
You've got it written down in front of you. Oh, no, you haven't got it written there anymore. It's on slide 19.
I'm glad you asked me that question, because you will be far from the only person in the room who's thinking, well, exactly what you're thinking.
And the complete unknown, the two things that are important is that we're looking at the combination of true premises and false conclusion.
And we're looking at the possibility of not the actuality of this is why the definition is important.
And as soon as you understand that, as soon as you stop thinking of a valid arguments as having true premises and true conclusion,
you will see what's going on here. I promise you.
It's a bit rash, isn't it? Addenda don't give me the answer yet.
OK, put your hands up if you've got the answer to that. Is that all? Or could that argument be valid?
Hand up if you got it right up so I can see you. OK.
Is it? Good.
Yes, it is valid. Those who said yes have got it right. OK.
Now, why is it valid? Okay. Would anyone like to give me a crisp and clear account of why it's valuable or should I do it?
Would you like to have a go? That's right.
This can't possibly be false. Can it? So there is no possible situation in which the conclusion is false.
So how could there be a possible situation in which the premise is true? And the conclusion false?
See what I mean? Could this argument be valid? Answer yes.
Actually, as a matter of fact, this argument must be valid. Can anyone tell me why it must be valid rather than just could be valid?
That can never be false. That's necessarily true, isn't it, given that that's necessarily true?
How could there be a possible possible situation in which that's true and that's false.
Given that that cannot be false. Cannot be false.
It's not just that it is false. It's that it can't be false. And actually, the same is true of the other one.
If you look at this, given that that is is a contradiction, it's necessarily false.
Again, there is no possible situation in which that's true.
So how could it be the case that that's true and that's false? It couldn't be.
So if we say could this argument be valid in each of those cases, the answer is no.
It could. Sorry. Yes, it is. Not only is it valid, but it must be valid.
And why is that? Because there is no possible situation in which the premises are true and the conclusion false.
So when you ask me as the answer to those two questions different, they are different questions.
OK. Let me just tell you that the two arguments I've given you here are called the paradoxes of entailment, the paradoxes of entailment.
And a paradox is something that you cannot understand.
It's just something you can't wrap your mind around.
And the reason you'd want to look at these and tell me that they're not arguments, OK, is because they couldn't convince you of anything.
Could they? I mean, as arguments, they would be pretty lousy because they wouldn't be convincing.
But they're still valid because they satisfy the definition of validity.
So in the same way, we might say, if you think, you know, whales, for example, aren't fish, are they?
Well, they jolly well should be, shouldn't they? I mean, it seems to me they jolly well should be.
They they go swimming around and, you know, if they're fish only, but they're not because they don't satisfy the definition of fish.
They satisfy the definition of mammal.
Well, you might want say, well, surely we should fiddle with our definitions so that whales come out fish because they're fish.
Vesely. But they're not. And do you see what I mean?
Sometimes things that really rather irritating are thrown up by definitions that work for every other case.
And therefore you think, oh, sod it will just put up with these anomalies.
Well, these are the paradoxes of entailment because we wish that our definition of validity didn't generate these two arguments as valid.
But they do. And actually, given that think about everything follows from a contradiction that that is that makes sense, doesn't it?
Because if I say Marion's wearing a skirt and it's not the case, Marion's wearing a skirt that leaves open every possibility, doesn't it?
Everything follows from a contradiction because a contradiction doesn't rule out any possibility at all.
Okay, so those are the paradoxes of entailment. Actually, these are a pretty good test of your understanding and validity,
because when you see that those two arguments, the paradoxes of entailment must be valid.
Then you'll see you'll have understood validity. So that's quite a good little way of testing your own understanding of validity to say yes.
Yes, yes, absolutely. That's right. It really doesn't matter.
And actually, that's true anywhere because. Let me.
This is actually a theorem. If you have an argument.
I've just seen a can of worms opening up in front of me. I haven't thought to do this, but let's see if I can do it.
OK. If you have a premise here and the premise here.
Sorry. Let's. No, let's do it this way.
And then to draw a truth table, I have to actually introduce you to truth tables.
But it doesn't really matter because they're very easy. True, true.
False, false teeth. Q Okay, I've got each of these structures is a different world.
Okay. This is the world in which Q is true and P is true.
Okay. What's the truth value of P and Q in this world.
True. Good. Well done. Okay. This is the world in which Q is true and P is false.
What's the truth. Value of P and Q in this world.
Brilliant. Okay. This is the world in which Q is false and P is true.
Okay. What's the truth. Value of. P and Q in this world, and here's the world where P and Q are both false.
What's the truth? One false again. Okay, so we had true, false, false, false.
Didn't we remember that. OK. Now this is the world where Q is true and appears true.
What's the truth. Five Q In this world. True.
What about this world. True. What about this world?
False. False. That's just Q.
So in the world where Q is false, it's got to be false, doesn't it? Okay.
What about P? True.
False. True, false. Okay. That's just copying out that bit of the truth.
Now, here's an argument. Q and P.
Q therefore P. Okay. You're with me now.
We can use this truth table to test whether this argument is valid here.
The two premises are true and the conclusion is true.
So that's okay. That that seems to be valid here. One premise is true, the other is false and the conclusions false.
So that seems to be valid as well. Is that reasonable?
That's not that's not a possible situation in which the premises are both true and the conclusion false.
Yeah. Is this a situation in which the premises are both true and the conclusion false?
No, because the premises are both false and the conclusion true. The other way round.
And what about this? The premises are both false and the conclusion false. So that's.
That could also be valid. So as you'd expect, that argument is valid.
But notice, if I put in any other premise here, it's going to stay valid.
Any other premise? What would you like me to put in here in order to make the argument invalid?
Not P. Yeah. Okay. If I've put in not P.
Okay. This is the world in which P is true. So not P will be.
No. This is the world in which P is true. So not P will be false.
Okay. This is the world where P is false. So not P will be true.
Good. This is a world where P is true. So not P will be.
And it's false here. So it will be true. So let's test for validity again.
Okay. Now this one in this world. The premises are true, true and false.
And the conclusion is true. Okay. Is that all right or is that a counterexample?
Is that still valid? So it's not the case that all the premises are true.
Okay, the conclusion is true. That's okay. It's still valid, isn't it?
What about false? True, true as the premises and the conclusion is false.
Still a case that we haven't yet found, one where we've got true, true, true and then false as a conclusion, which is what we'd be looking for.
Here we've got false, false, false, true. Yes.
And here we've got false, false, true, false. So, you see, I could add in even a negation of the conclusion.
And it hasn't changed the validity. Once you've got a valid argument, you can add anything you like and it'll still be valid.
I don't know why I went into telling you all that is this is this is information you really didn't need to know.
But it's so interesting, isn't it? I think it's interesting.
Let's move on. Okay. So let's let's test you again.
OK? Don't shout out your answer. So is the argument valid if we've got true premises and a true conclusion?
Could it be valid? It's what I should've asked. Yeah. Hands up if you think.
Yes. Yes. Okay. You're quite right. False premises. True conclusion.
Hand up. OK.
The premise is a false, the conclusions true. Could it still be valid?
Yep, good. Well done. What about false conclusion, false premises?
Could it still be valid? Yes.
Yes. And true premise is false conclusion.
Could it be valid? No. Some of you getting this.
Some of you. Well, I thought I'd got there. Have got myself lost in the meantime.
OK, let's. OK. I'm going to look at this another way of looking at it.
Just because you're already probably exhausted. I'm already exhausted. We're going to look at Venn diagrams.
And we're actually going to run out of time in this session. OK, here's an example here.
Here we have two arguments, in both cases, the premises are actually true.
And the conclusion is actually true. OK. But one of them is valid and the other one isn't valid.
And I'm going to show that to you by means Venn diagrams, if I can find my pen.
So all cat's miaow. OK, here's the class of me hours.
Right? All Katsumi hours.
So where where should I draw the class of all cuts inside, outside or overlapping inside,
if all cuts me out and then all cuts must be inside the class of things that miaow.
Is that right? Okay, Bow. Incidently, I hope you noticed my topical reference to the dog of the United States of America.
Less has died since I. Okay. Bo does not miaow.
So where do I draw the bow here.
Okay. Bo is not a cat. Well is that true.
Yeah. Okay. That's true. So if all cat's miaow and Bo does not miaow then Bo is not a cat.
There is no possibility is there, in which those are true. And that's not true.
If those are true, that's got to be true. Okay, so that's a valid argument.
Accepted. Okay. And yet the truth value of that is true.
Yes. Truthfully, that is true. Assuming you don't doesn't mean that.
And that's true as well. Okay. So you accept my claim about the truth values and you accept my claim about the validity.
Okay, let's have a look at this one. All Cat's miaow. So here's the mea hours.
Where do I draw the thing of cats? Okay.
Cats. OK. Second premise. Dogs are not cats.
They're outside the class of caps, aren't they? But but they they could be either inside or outside or overlapping.
The class of me hours. Couldn't they? So there is a possible situation in which that's true.
And that's true. But that isn't true. And that's the possible situation in which Cat's Miaow.
Now we know that cats don't to actually miaow. So glad you're all with me.
But you see, dogs could miaow. Could they as far as that.
I mean, these do not guarantee that. Do they. In the way that these guarantee that it's that.
Right. OK. And yet would you agree with me that that's true.
Yeah, that's true. Is that true?
Oh, come on. Is it true? Is it actually true? Yes, it is.
Dogs do not miaow. But but that's not a necessary truth, is it?
Dogs could miaow. So there is a possible situation.
And this does not rule out that possible situations. So here we have premises where the actual truth values the premises are all true.
The conclusion is true. But here's one of an argument exactly like that that's valid.
And his arguments exactly like that is not valid. What's the difference between them?
Answer. In this case, if the premises are true, the conclusion couldn't be false.
Okay. Whereas here, if the premises are true, the conclusion could still be false.
OK, if we didn't know I'm talking about dogs here.
But what if I was talking about. Name an animal, you wouldn't know anything about Griffin's.
Achievable watched your Harry Potter. So that's a bad example, too.
But okay, I'm talking about a mythological animal or an animal Venu just discovered.
I can tell you it's a. Unicorns are not cats, but unicorns don't miaow.
You actually wouldn't know the truth value of that, would you? And you'd have to recognise that the truth of those two leaves that one open.
Okay, that's. That's the importance of validity. Here's the next one.
OK. I've got two arguments again. The premises are actually false, I hope, and the conclusion is actually true.
OK. Do you accept that that's false? Is that false? Is that true?
It is. Is that false?
There are fish without scales. I looked it up and and Jewish people either can or can't eat them.
I can't remember which is. But there's a whole table of which fish have scales in which don't.
Okay. Whales have scales. Is that false? Good whales are not fish.
True or false. True. Okay, good. So you you agree with me that the truth value is the actual truth.
Values are like that. Yes. OK.
As I described, now let's look at the validity of these arguments, all fish have lungs.
This is a class of lungs, things. Where do I draw the class of fish?
Inside his fish. If whales have lungs, where have I got to draw the.
Which hung a little. All fish have lungs. Whales off.
Oh, yeah. Whales are fish. It's got to be there, isn't it? Okay.
So it's got to be then if it's there, what's got to be the case.
It's got to have lungs, haven't it, hasn't it? So if those two are true, that's got to be true, except that.
Now, let's have a look at the. Final one.
Sorry, not the fun one. All fish scales, this is class of scales, things, class fish in there.
Okay. Fish whales have scales. So where do I draw whales?
It's got to be within there. Has it got to be here or here? It could be an either couldn't sit and whales are not fish.
Okay. That tells us that it it is there. But it could be either, couldn't it?
Again, with the. The truth of these two, if they were true, doesn't guarantee the truth of that because you can have scales without being a fish.
Okay, so, again, if you look at the explanation underneath, in both cases, the premises are actually false.
And the conclusion is actually true. But in the first place, if the premises were true, the truth is a print of the conclusion.
I should have said there. Can you see? Scrub it out on your copy.
That should be conclusion. The truth is the conclusion would be guaranteed.
In the second case, even if the premises were true, the conclusion could still be false.
OK, it isn't false, but it could be. Are you beginning to follow?
And here, I'll actually I'll skip over these because, oh, would you like me to do these with you as I've done the others?
You would. Okay. Your gluttons for punishment, aren't you? OK, firstly, do you agree with me that the truth values are, as I say.
OK. I've said everything is false. Is that false? All fish have wings.
Yep. Wales of fish. Is that false? Whales have wings.
False. False. OK. All fish have scales.
False whales have. False whales, a fish.
Okay, said all the premises and the conclusions are false here.
But this argument is valid and this argument is invalid.
OK. Why do I say that? Well, if I find my Pentagon.
All fish have wings. Here's the class of winged things, the class of fish is inside.
Whales have wings. Oh, sorry.
Yes. Thank you. Have suddenly think, oh, my God, it's not valid.
All fish have wings, whales are fish. Okay. Therefore, they've got to have wings, haven't they?
That guarantees it. They've got to be inside the glass of winged things.
What have we got over here? We've got all we're all fish have scales.
So there's the scales things. And there's the fish.
Whales have scales. Well. They could be either or.
Couldn't they? Whales are not fish. Well, OK, that tells us that whales are here.
But is that guaranteed? Other than the fact that we're saying that that's true.
You know, actually, it could be false. Couldn't it? Even though it is true, it could be false.
And it's not guaranteed by those. Is it?
Because as far as the fact that all fish are in the class of things that have scales, that doesn't tell us where the whales are out here.
So they have scales but are not fish or whether they're in here and they're fish with scales.
We don't know that. This is the structure that you're talking about bending down to see.
No. It happens that all the arguments I'm using are straightforward cases and modus Poland.
Poland's. All piece A, Q, p, therefore Q No, hang on.
No, I'm sorry, the go. I've got past the stage or I'm thinking your homework this week is to go home and work out what the form of that argument is,
because I'm using the same one for each one. What I'm trying to point out is that what matters is the possibility, not the actuality.
It's the possible truth values, not the actual truth values that matter, possible truth values, not the actual truth values.
Okay, here's another way. We've used Venn diagrams to have a look, but let's have another look.
Another way of determining validity is to create a counter example set and then determine consistency.
Let's have a look at what we mean by that. Firstly, to determine the counterexample set, we set out the argument logic bookstall.
Okay, here are two arguments set out. Logic bookstall. You familiar with these.
You've seen them before. Okay. Now to create.
That's the argument set out logic bookstall.
Now I'm going to negate the conclusion and thereby set out the counterexample set so we negate the conclusion by tacking.
It is not the case that in front of the conclusion. So here we are.
This is the counter example set of the two arguments I gave you.
And all I've given you is exactly the same arguments, but with the words it is not the case that tact in front of them.
The next thing we do is we need to consider whether the set consisting of the premises and the negation of the conclusion.
In other words, the counter example set is consistent, i.e., could they only be true together?
If the counter example set is consistent, then the original argument is invalid.
Can anyone see why that's the case, huh? This is really tough.
Why am I doing this to. If the counter example set isn't consistent, then the original argument isn't valid.
Now, again, let me warn you that consistency is a bit like truth.
You think of it as a nice, good, warm thing. And so you tend to think that if it's valid, it must be consistent.
But what I'm suggesting here is if the counterexamples set is consistent.
In other words, if the counter example set can't all be true together, then the argument is invalid.
Let's have a look. Why? OK.
This is the counterexample set, okay, consisting of the premises, plus the negation of the conclusion to accept that you're with me on that.
Now, what I'm claiming is if this is consistent, then the original arguments must have been valid.
Okay, let's have a look at here with you'll agree with me that the original argument here is invalid, isn't it?
Do you agree with that? Okay. So if it's snowing, the mail will be late.
The mail will be late. Therefore, it's snowing. That's invalid, isn't it?
Because there is a possible situation where the premises are true and the conclusion false.
Well, what we're doing here. Is we're saying, well, let's assume that the conclusion is false.
And let's see if these can all be true together. Well, if they can, then it can't be the case that the original argument was valid.
Could it? The I think this is an argument too far, frankly.
So if you're not with me. Don't. But actually what you can do is you can because this bit is dead simple, actually.
And if you go away yourself and you work out, there's several ways in which you can test your understanding of what I've said today.
The first one is if you understand why the paradoxes of entailment, the two arguments I gave you and told you were the paradoxes of intelligence.
If you can work out why they are valid, you will have understood validity.
Second thing is if you understand why this is the case,
why you can test the validity of an argument by appeal to a counterexample sets and why you and you can understand why.
This is the case. In other words, if the counterexample set is consistent, then the original argument is invalid.
And if a counterexample set isn't consistence. Ow, ow, ow, ow, ow.
So you can still see it. My hands, can't you? That's terrible. Right. Okay.
You will all see that I put a knot in where I shouldn't to put a knot, which is the worst thing you can do when you're teaching people logic.
So if. If. Will you please all note that that not shouldn't be there.
And so the second test of whether you've understood today's work is whether you can see how this works.
Okay. So if you haven't understood where you are now and if you're feeling I'm never gonna understand another thing again.
I don't ever want to see that tool. But woman ever again quite understands.
But go home and have a look at it yourself.
And I promise you, if you work through it step by step and just think of the definitions, you will understand.
And if you don't email me and we'll go through it all again by e-mail.
Okay. I'll stop there. That there's five minutes for questions. If anyone has still got some energy.