In Our Time - Logic

Episode Date: October 21, 2010

Melvyn Bragg and his guests discuss the history of logic. Logic, the study of reasoning and argument, first became a serious area of study in the 4th century BC through the work of Aristotle. He created a formal logical system, based on a type of argument called a syllogism, which remained in use for over two thousand years. In the nineteenth century the German philosopher and mathematician Gottlob Frege revolutionised logic, turning it into a discipline much like mathematics and capable of dealing with expressing and analysing nuanced arguments. His discoveries influenced the greatest mathematicians and philosophers of the twentieth century and considerably aided the development of the electronic computer. Today logic is a subtle system with applications in fields as diverse as mathematics, philosophy, linguistics and artificial intelligence.With:A.C. GraylingProfessor of Philosophy at Birkbeck, University of LondonPeter MillicanGilbert Ryle Fellow in Philosophy at Hertford College at the University of OxfordRosanna KeefeSenior Lecturer in Philosophy at the University of Sheffield.Producer: Thomas Morris.

This BBC podcast is supported by ads outside the UK. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio 4. I hope you enjoy the programme. Hello, in 1740, the Prussian King Frederick the Great wrote, Philosophers should be the teachers of the world and the teachers of princes. They must think logically and we must act logically.

What it is to think logically has been, one of the main concerns of thinkers since the time of the Babylonians. In the 4th century BC, Aristotle turned his attention to the subject and thereby founded the modern discipline of logic. Aristotle's system was used by scholars for over 2,000 years, but in the 19th century was succeeded by something even more powerful and subtle. Today, we understand logic as the study of argument and the forms they may take. It's widely used in fields from linguistics to cognitive science,

but above all, logic is a vital tool without which computing would never have existed. with me to discuss logic and its uses are A.C. Grayling, Professor of Philosophy at Birkbeck, University of London, Peter Milliken, Gilbert Ryle Fellow in Philosophy at Hartford College at the University of Oxford, and Rosanna Keefe, Senior Lecture in Philosophy at the University of Sheffield. Anthony Grayling, before we go into the history of the subject, can you tell us what you mean by the word logic? Logic is the science of a valid inference.

By inference, I mean drawing, conclusions from premises, from assumptions we make or information that we have, and from it we try to deduce or induce a conclusion based on those premises. And what logic does is it tries to understand the best ways of doing that, the valid forms of doing that. So you're talking about reasoning and you're talking about validity. Can you give us some examples of logical thought? Well, take a very simple example like this, all men are mortal.

Socrates is a man, therefore Socrates is mortal. That's a perfectly standard example of a deduction from the two premises that all men are mortal and Socrates is a man. You deduce that Socrates is mortal. Now you notice something rather interesting about it. That's a deduction. And if you inspect it closely, you'll see that the conclusion is actually a rearrangement of the information in the premises.

So in a way there's no informational novelty there, although it might be psychological novelty, because you might not have realized that Socrates is mortal. In the case of an inductive argument, you go beyond the information given in the premises. You say, this one is white, that one is white, the next one is white, so they're all white. And there you're taking a little bit of a risk in your reasoning, but you're going from some samples to a generalisation about the whole class. But in the science of logic traditionally, the focus of attention has been on deduction,

that is, on the question, what forms of reasoning are just in virtue of the form, not in virtue of the content of what's said, can you be sure that if you were to put true premises in place, you would be guaranteed a true conclusion. Where does the word valid, how does the word valid figure in this? This is very interesting because validity is a purely formal property of an argument. That's to say it's just about the shape, the structure of the argument and not what's actually said in it.

So, for example, supposing you got a rush of blood to the head and decided to get married, you went off to the local registry office with a vehicle license registration form hoping that it would serve for a marriage license form. The people would tell you you've got the wrong form. You've got the wrong set of blanks. You need another set of blanks to be able to fill in the right kind of information of what we're doing. So purely formal matter relates to the validity of arguments. If you put in a true premise or set of premises into a valid form,

then you have what's called a sound argument because the validity of the argument. because the validity of the form plus the truth of the premises will give you a true conclusion. That's a sound argument. Can you just unravel premises for our listeners who might be worried about this? Yeah, the premises are the sort of basis, the groundwork, the information, the data, the starting point. There might be information plus some assumptions that you make, and from them you try to draw a conclusion. Drawing a conclusion is called inferring, process of inference. And that's what logic is really interested in.

It's in this process of how you can draw inferences from premises, which providing the form, the shape of the argument is right, will be such that if you've put true statements into the premise slots, you will get a true conclusion. Peter Milliken, we've talked about deductive and inductive. There's also another distinction between formal and informal. Can you tell us how that plays? Yes. Anthony has already given an example of an argument that is formally valid.

all men are mortal, Socrates is a man, therefore Socrates is mortal. Now think of the following argument. All cows are ungulates. Buttercup is a cow, therefore buttercup is an ungulate. Now you might not know what ungulate means, but you know that if the premises are true, the first two propositions, then the conclusion, the last one, has to be true as well. That's an example of an argument that's valid in virtue of its form.

need to know the meaning of the terms, as long as they're slotted incorrectly into the form you know that you'll only get from truths to truth. Now, informal logic is concerned with other kinds of good and bad reasoning, for example, vagueness, ambiguity, appeals to authority, circular arguments, the kinds of fallacies that we can make in informal reasoning. Such as? Well, an example would be a circular reasoning where you end up taking for granted issue. So, for example, suppose I want to prove that everything has a cause and I say, well, anything that didn't have a cause would have to cause itself and that's impossible. Actually, I'm arguing in a circle because I'm taking for granted that something which isn't caused by something else has to have itself as a cause. I'm taking for granted that everything has a cause in putting forward my argument.

Is there a chasm has more than a distinction between formal and informal logic? Is it a big distinction here? Well, yes it is. I mean, in the sense that with a formal argument, the idea is that you can pin it down pretty precisely. You can say, here is a form of argument where if you put the right things in the slots... As Anthony was saying about the marriage... Marriage certificate. Absolutely.

So, you know, if you put the various predicates in the right place consistently as you're meant to, then you're guaranteed, if you've got true premises, to get to a true conclusion. But are informal and logical? They don't quite seem to rest together as two words. Do you have problems with that? Not you. I do, but I mean, not you. Are there problems with that?

Well, not really, because I think here we're saying that, formal is being used as a contrast with formal. So we've got a very clear understanding of what formal logic is. But then we're aware that reasoning in ordinary life can go wrong in all sorts of ways. And we simply take all those various things that can go wrong. It's a bit of a rag bag, but put them under the heading of informal logic. In the introduction I talked about Aristotle, they seem to be talking about logic in Babylonian times from what you know.

and I know no but Aristotle began to write about it and give it space and it carried through for hundreds of years what was the nature of his contribution was there a key to his contribution and what was that? Well Aristotle was concerned

particularly with arguments which we call syllogisms which are well start from the notion of a categorical proposition and a categorical proposition is one that concerns a subject and a predicate.

So, for example, all cows eat grass. So cows is the subject, eating grass is the predicate. And there are four different types of categorical propositions. Broadly, they take the form. Every F is G, no F is G, some F is G, some F is not G. So you put something in place of F and G, cows and eating grass or whatever it might be, and you've got four different kinds of proposition. Now, what Aristotle did was look at arguments that connect two of those kinds of proposition with a third one.

It's easiest to understand it by example. So, for example, all mammals are animals, all humans are mammals, therefore all humans are animals. And that is a valid syllogism. From the first two propositions, the third one follows. Another one, very different kind, is no cows have wings, some mammals are cows, therefore some mammals do not have wings. That's another valid form of syllogism.

And what Aristotle did essentially was look through all the different kinds of arguments that you could have of this type and categorise them and tell us which are good and which aren't. And you found reading from my notes, 256 of which only 19 were valid. Yes, the exact number that one counts as valid is potentially controversial depending on various things. But essentially you've got four different types of proposition,

you can put them in different orders, and so you end up with 256. Very brief to Peter. People listening would say all, annuals of primates, that's at Socratesism, but quite simple. It seems. Now, why was this such a hard

driver of philosophy? Why did it drive it so strongly? This idea of this illogism. Well, I think it's a very attractive idea to philosophers. I mean, philosophers naturally like system. They like to understand

why things work at a deep level and we're all concerned with reasoning and logic. So I think it's understandable that it would have captured the fascination. Rosanna Keefe, can you take that on? What was he able to do Aristotle with his invention of the syllogism? Well, what's particularly important, impressive about his system is the systematicity of it and the generality of it.

He was able to give rules and methods to show that some particular form of syllogism was valid or was not valid, and he was able to give some general results about his system. So it was more than just taxonomic exercise. It was a very full system. So, for example, his rules for showing that a particular argument form were valid involved some rules of conversion, he called. So, for example, from some A is a B, you can conclude some B's are A. if some of my pets are furry, some furry things are my pets, and some and other rules like that.

And then he could show that any valid syllogism could be reduced or explained in terms of two fundamentally one, fundamental ones that he thought were obviously valid. So one of those is one that Peter gave an example of. Socrates. All humans are mammals or mammals. animals, therefore all humans are animals. That gets called Barbara. And the other one is

Celerant, which is an example of which is no dogs are robot, all poodles are dogs, therefore no poodles are robots. So those were, he thought, obviously valid, and then he could show why other forms were valid

by using those and his other rules. Was this, these syllogers, these three liners, as it were, I know I'm treading on oceans really a bit still They do seem quite simple And yet they had such a big impact Can you tell me why this way of Aristotle bringing this together

In such a neat almost haiku way Drove philosophy for so long And still to a certain extent The recent take-up was from his basis What is it about that Which organises reasoning In a way which validates that method for hundreds of years to come?

I think it's partly the level of abstraction with which he could deal with it. So all sorts of arguments can be of these various forms. And some of them aren't so obviously valid as the ones that we've been talking about. And he had methods of showing that they were nonetheless valid.

And then his hope was that reasoning would follow these forms. So, for example, his account of scientific knowledge required demonstrations that had to start from true premises, necessarily true premises, that were better known than the conclusion, and which ideally should follow the form of a syllogism. So he felt it was the foundation of science.

That's interesting. From the very beginning, he switched the idea into science, isn't it? Because that was his primary concern if he had a primary concern knowing so much about so much. There was another group of Greek philosophers examine a different type of logical argument. I'm afraid I must use this terrible

broadcasting word briefly. Can you tell us what the Stoics were interested in? Yes. They were interested in the logic of different bits of language. So they were interested in or and and and if then and how those logical connectives

work, what rules governs, them. For instance. So, yes, they had talked about five in demonstrable rules, which, again, should seem compelling. So one of them, modus ponins, here's an example, if it's raining, it's cloudy, it's raining, therefore it's cloudy. Another one is if not, it's not both Lydia's birthday, Miranda's birthday. It is Lydia's birthday, so it's not Miranda's birthday.

That's a general form of argument that's always valid, whatever you put in. So they'd express that in general terms, say in that last case they'd express it as not both the first and the second, the first, so not the second. And whatever sentence you put in for the first or the second, you get out a valid argument. And they hoped to show that all sorts of arguments were valid using these basic ones. Thank you very much. Anthony Grayland, can you tell us how this, we've got some idea. It's already complicated, it's simple. That's the fascination, isn't it? I mean, you've already complicated it, but it seems to start for simple premises. Why, I'm asking you the same

question. Why did it persist for so long? Why was it taken up by so many inordinately intelligent people for sentences? We're going to go right from the 5th, 4th century BC to the Middle Ages, where you had great thinkers they were thinking. In the context, in the context, text of religion, let us say, but they picked up his logic, as did the Islamic scholars before them. Let's talk about the Middle Ages. What was the attraction? What was the drive? Well, first, let's just say that the point about all this is that even though the examples we've been talking about here are very simple examples, it seem to be a lot of sort of poodling and animalistic trivialities. In fact, they do tell us a great deal because they firstly constitute an

instrument which helps us to think more powerfully and to govern our reasoning much more powerfully. But also, of course, they can be generalized. There's a great deal, for example, of mileage in understanding these things. What the Stoic logicians were concerned with, and a great deal of what the Stoics wrote and devised in antiquity was lost for quite a long time, which was why the Aristotelian logic kind of survived and then got rediscovered large parts of. it in medieval times so that the schoolmen is there called of the medieval era in the absence of telly and bingo in those long winter evenings were able to devote their attention to trying to explore and spell out many more of the implications so rosanna's just been talking about the

interest that the stoics had in words like and if not and so on these are known as syncategoromatic terms that is they're ones that don't denote anything out there in the world but they have functions within sentences, functions within propositions, because they combine terms in the propositions in different ways. So an example is that if you use the word and you join two propositions, let's call them P and Q, then you know that the complex proposition P and Q can only be true if both P is true and Q is true. Whereas in the case of a disjunctive proposition, P or Q, only one of them has to be true for the whole thing to be true. And these are very consequential things to recognize, even though they seem very simple,

because once you start to build up a much larger structure of reasoning from these simple things, it really provides a handrail through the argument. Imagine some very, very complex chain of reasoning where you're thinking about some really rather fascinating subject matter. Well, you would have available to you this instrument. Indeed, Aristotle's logic had been called the organon or the instrument as a kind of help or aid, to reasoning and it would help to steer you through something which might be very complicated in its own right. So it's a powerful tool and that's why it retained the fascination of thinkers who wanted to explore it in more depth and understand much more about the way that really good reasoning works.

Why were the medieval schoolmen so attracted to it and they developed it and multiplied it? Peter, Peter Milliken. Well, one of their interests was actually in things that went beyond Aristotle. I mean, they were particularly interested in certain types of inference that might superficially seem valid, but in fact aren't. So suppose I'd give you an example. I know that she's in London or Oxford. But it doesn't actually follow that I know she's in London or that I know she's in Oxford. So here we have a case where what might seem to follow doesn't.

And that's the kind of case that many of the medievals, particularly in the 14th century, were investigating. They were looking at cases where a simple-minded application of logic didn't work, and they were looking to try to understand the logic of terms like knowledge and belief and so forth. So let's take that I know that she's in London or Oxford. How would they push that? Well, unfortunately, in the 14th century, you don't have a single system, theory. You've got a lot of people doing their thing in different places, people like William of Ockham and Thomas Bradwardine and Gregory of Rimini and John Buridan. And we don't

have a sort of organised theory there. Typically what we've got is different thinkers pointing out the occurrence of various cases like this and drawing them to the attention of other philosophers. But this, they, they, they, they. explored this and then quite soon after that Aristotle went out of favour, didn't he? Yes, he went right out of favour. That can be traced to thinkers like Francis Bacon and Descartes and John Locke. Their criticism of it was that logic was sterile. They thought that the kinds of things that Aristotle was looking at,

those arguments were pretty trivial. They didn't give any real insight because they were so obvious. and what these philosophers were concerned with was more discovery of good arguments. So if you think, for example, of a geometrical proof, you start from certain assumptions, you have to find your way towards some conclusion. And typically the way you do that is by some ingenious construction.

You draw in certain lines, put in various angles, and then you find some clever way of connecting the premises, what you took for granted, with the conclusion, which is what you want to get to. And partly based on this sort of model, the early modern thinkers looked at Aristotle and said, well, it's useless for that.

Sylligism may be good for setting out arguments once you've discovered them, but what we're really interested in is finding good arguments, finding the links between ideas that enable us to derive new knowledge. So after Locke, the understanding of logic, what logic becomes thought of as is more a study of our ideas and the study of our cognitive faculties. Anthony.

The interesting thing that I think is that the 17th century, of course, 16th to 17th century is the rise of modern science. And people like Bacon were very interested in how you do science or what kind of methodology would get you to more empirical results. And it looked as though the Aristotelian syllogistic, which is specifically about forms of deductive reasoning. were much less used to you than, for example, kinds of inductive reasoning, ways of accumulating observations, for example, in superinducing theory onto them. Bacon's idea of observational experimental science which led to the royal society. Yes, indeed.

So you get a shift, if you like, of emphasis from logic as a study of valid forms of deductive reasoning to a more general conception of logic as a science of inquiry. So it seemed, Rosani Keefe, that logic was, out for a while. Maybe he still existed among some schoolmen who refused not to be schoolmen for two centuries, but Locke and Hume and Descartes were saying it's not the way to go. In fact, they were quite snooty about it. Yet in the 19th century, I suppose unexpectedly, in this dramatic story of the development of logic, it was reborn and one of the important figures was a mathematician of British mathematician called George Boul. What did he do that made it important again?

Well, one of the things he wanted was an abstract system that, as he said, showed that the laws of thought were as rigorous as the laws of mathematics. And he, so he wanted a system, something like Aristotle's only more powerful. And one thing he did was notice the similarity between algebra and logical laws. For example, the similarity between and and multiplication and between or and addition. That's interesting, isn't there? Mm, yes. So, a simple example, A or B is the same as B or A, just as X plus Y, the same as Y plus X. And then all sorts of more complex similarities.

So what he did was come up with a system of algebraic laws that he did. took to govern logic, and he could use those to assess arguments, show arguments to be valid. One of the interpretations of his system has these algebraic laws into which you can only put values zero and one, and if you think of zero as corresponding to falsity and one to truth, then you get a propositional logic, the kind of thing that. that the Stoics were looking for. Was this the first time logic and mathematics had been conjoined in that way? Well, I think Leibniz also wanted something similar.

He recognised similarities between algebra and logic too. So not altogether, no. And around the same time, there are other mathematicians who are interested in the same sorts of relations. So supposedly de Morgan published an important book on Logic on the very same day that Ball published the mathematical analysis of logic in 1847. So it was in the air, I think. Why do you think it was a...

Oh, sorry, Peter. I was just going to say, it would give a wrong impression to suppose that in the early modern period the desire for a logic in the formal sense completely disappeared. And I think Leibniz has already. been mentioned. I think we can also mention Thomas Hobbes. Hobbes said that thinking is basically calculation and Leibniz took this further and was looking for what he called a universal characteristic or a universal logical language. And the idea was a language within which

reasoning could be, as it were, computed. So he to some extent inspired Frege, who mentions and there were various people like William Stanley Jevons took Boolean ideas and actually built a logic machine in which this idea of computing things came to reality. You can see it in the Museum of the History of Science at Oxford. Steve, Anthony, Peter's mentioned Fregeard.

He is the man who supposedly revolutionises logic completely. Maybe that's too big a claim. Can you tell us about Frega and why he matters? Yes, certainly he's a very, very significant figure in numbers. Yes, but it's relevant in talking about Frega, just to follow on from Peter's point there, about the way that logic and the sort of Aristotelian kind of canon remained of some significance. Kant, for example, in the late 18th century, the great Emmanuel Kant thought that the Aristotelian-based logic was a completed science,

and he made great use of it in constructing his theory and the critique of pure, reason. But it was this move in people like George Boul and Augustus de Morgan and Jevons and others who was starting to use algebra to try to express some of these fundamental logical ideas that gave Frege his in part gave him his clue.

I mean, for example, Augustus de Morgan recognized that what had always been thought of as laws of thought, which seemed to be a rather psychological way of thinking about logic. And these laws of thought were principle of identity, which says A is A or X is X, if you

principle of the law of excluded middle, which says you can't have both X and not X, the principle of non-contradiction, and excluded middle which says either you have X or you have not X, that in fact the last two, if you put them into an algebraic form, you see that there are just versions of the same thing. So already this way of symbolizing, of using mathematical techniques to express some of these logical ideas,

was beginning to show that there is a possible increase of power in our understanding of those logical notions. And this is what Frege had capitalised on. This is the late 19th century. In the very late 19th century, yes. Frager's work showed that the traditional logic, that Aristotle-based logic was actually just a small part of logic, really. What Frege had done was to show that you could massively generalize insights into logic,

a much, much wider field of application. And he did this in a number of rather complicated ways, but one illustration of how he did it was this. If you take a very simple relational proposition like, say, Jack loves Jill, let's say. In the Aristotelian logic, when a term occurs as a subject of a proposition, it functions, it works in a very different way, from the way that a term works when it occurs in the predicate of the proposition. So on Aristotle's analysis, Jack Loves Jill gives us Jack is the subject and Loves Jill as the predicate. But what Frager was able to do by...

really sort of rather brilliant but simple insights into how you formalise this, how you symbolise it, to show that Jack and Jill can operate on the same logical level and that you can have what it would call a two-place predicate, that is loves, which applies to both of them because they stand in that relation to one another. So you can write it as predicate expression, loves and then Jack Jill. And just by this kind of technique and by the development of what he called a concept script, a griff-shift. You could formalise, symbolise,

all these relations and show how inferences work in them. And this was like the development of a machine, which produced massively more sausages than the Aristotle machine did. Rosanna, can you, this is part of a larger grunder project in Frege's system called logicism. What did he mean by that?

The idea of logicism was that arithmetic, be reduced to logic. So we can conjure up all of arithmetic from logical foundations. And this was important because the idea was to give secure foundations to mathematics. So rather than resting on axioms that you might just think were intuitively true or trying to base maths on something empirical, we can base it on logical foundations. So for example, you can, you can, you can base it on logical foundations. So, for example, you can understand a sentence with the number one in, in purely logical terms. So if I say there's one dog, exactly one dog, that gets translated logically as there's something X, which is a dog, and anything that's a dog is that X.

So it's a logical formula that gives you a... an understanding of number. And from that, you can build up to understanding of number and more complex and rational numbers, etc. Peter Milligan, I see you wanted to get in, but when you get in, can you also tell us why a letter from the British philosopher Bertrand Russell influenced Frege so heavily,

and what was the solution to the problem that Russell identified? Right, yes. First of all, I was just going to say something that Rosanna's mentioned there is pretty crucial in Frager's thought, the introduction of variables X's and Y's. So where if you take something like all cows eat grass, for Frager this is interpreted as for all X. If X is a cow, then X eats grass. What does that do? It helps. Well, what it does, it enables the all and some propositions that Aristotle had dealt with to be incorporated within a much larger system.

And as Anthony was saying, you can actually build up propositions of arbitrary complexity by putting brackets within brackets with these quantifiers and variables in such a way that you get a much, much more general system. You can express, well, at least Frager wanted to express all the truths of mathematics. Now, what Bertrand Russell pointed out, unfortunately, is that a particular proposition that can be expressed within Frege's logic turns out to produce a contradiction. And he wrote in 1902 to Frege saying, here's this contradiction I've found. And clearly, if what you're trying to do is to build mathematics on the foundation of logic, the last thing you want to find is a contradiction. your system of logic.

Briefly, Anthony, what was the contradiction and what was Srega's response to it? Well, the contradiction is this. Russell was trying to define number in terms of classes of things. So you try to define, say, the number, naught, by saying that naught is the class of all those classes which are empty, which have no members.

And the number one is the class of all those classes that have just a single member. Number two, class of all these classes that have Doubleton members and so on. So it looks as though you're reasoning circulately here, but actually you're not using the notion of number to try and define number in that way. So crucially, you're making a use of the idea of a class. And then it turns out that you get this difficulty. Some classes are members of the class of classes, okay.

But some classes are a class of teaspoons are not themselves teaspoons. So you can ask yourself the following question. What can one say about the class of all those classes that are not members of themselves? Would it be a member of itself if it weren't? Answer, yes. And can it be a member of itself if it isn't a member of itself? Answer, yes. So you've got a contradiction.

Now, a way of trying to illustrate this is not an exact parallel of this, so do be careful not making it exact parallel. But suppose you have a barber in a village who shaves all and only the men, who shave themselves. Does he shave himself or not? If he does, then he doesn't, and if he doesn't, then he does. So it's in a way a rather similar illustration of what the difficulty was here. If this idea of classes can generate a paradox in that way, then you can't use it to try to define the basic notions of arithmetic. And remember that arithmetic can take translations from all the other branches of mathematics. So this is very, very fundamental.

Russell's solution was to say, well, somewhat arbitrary. I'm not going to allow classes of one type to take as members other classes of the same type. So you've got to have a kind of hierarchy of kinds of class and the higher levels of the hierarchy can take as members of the lower levels, but they can't take members at the same level. So you avoid the sort of problem you get when you say what I am now saying is false.

Because if it is false, then it's true, and if it's true that it's false, then etc., etc., and you get into that kind of muddle. To avoid it, you have to introduce this rather ad hoc maneuver of inventing a hierarchy. As I understand it, Rosanna Keith, Fregear introduced this as an appendix to his last work, and that was his last work on this subject. He found, he thought that the paradox, as it were. He stopped him in his tracks in some way, yet his influence on the scope of philosophy went on very powerfully.

Can you just give us an outline of how he interested, influence the field of philosophy in the 20th century? Yes. In fact, he wasn't that widely read initially. He did influence directly influence Russell and Wittgenstein and Karnap, which is obviously important influences. But certainly in the last 50 years, he is one of the most important, seen as one of the most important influential philosophers. A number of reasons for that. One, his logical system, allows us to formulate theses much more rigorously.

We can make these subtle distinctions, and that is used across all sorts of areas of philosophy. Another thing is his conception of logic, whereas, as has been mentioned, some of his predecessors, saw logic as really a part of psychology. It was interested in how we think, laws of actual thinking,

Frege was very opposed to that psychologist and wanted to take logic as autonomous. There are laws of logic that apply even if nobody, people didn't reason by them, they would apply. And that conception of logic is very widely held today.

Other aspects of his influence relate to his work in the philosophy of language which is closely tied in with his work in logic. Peter Milliken, we did a program on Imaginary Numbers a few weeks ago and we're talking there about abstract thought going on among persons

who wanted to think about imaginary numbers for centuries and then it turned out to have an extraordinary practical application which actually helped to make the world in which we live, helped in a big way. Something the same seems to happen here, doesn't it? We're into... Hobbes mentioned computation, I think, or calculation. That's right. Yes. And now the logic makes its entrance. here. Can we use Alan Turing as an example

of how that happened? Yes, yes, it's actually a beautiful example. There was a lot of thinking going on in the early 20th century about the foundations of mathematics. We've already touched on that with Frager and Russell and their logist enterprise. And

one of the big puzzles in the 1930s was whether it's possible to produce a decision procedure whereby you can find out with absolute certainty whether a particular set of axioms and rules will or will not give you a proof of a particular formula.

So once trying to find out the limits of what can be demonstrated, now Alan Turing, in order to investigate this question, came up with a very simple model of what a computer might be, called a Turing machine. It has certain very well-defined operations that limit what it can do. He showed that it can do wonderful things, all sorts of things that we intuitively think of as computable are computable by a Turing machine.

But then he showed that there are some things that cannot be computed even in principle by such a machine. Now what then happened much, much later, is that people looked at this and said, oh, here is a model for a universal computer. Here is a theory which can provide the basis for a lot of computer science. And the Turing machine is still discussed in those terms today. and Anthony, are there other, shall we stick with this, the idea of the practicalities,

because we haven't gone much time left and it is interesting. Can you tell us what other, as it were, practical developments have come from these centuries of thinking? Well, firstly, Rosanna mentioned earlier, Boole and Boone's ones and zeros in his algebra. This is a sort of digitization of thinking about how you might have a, a calculating device or a computing device early on. And Turing's machine wasn't a machine. It didn't have any cogs and wheels of smoke coming out of it. It was actually a piece of paper with some pencil marks on it.

And these ideas fed into the possibility that we now all take for granted. The mobile telephones that we've got switched off in our pockets here have got more computing power than the Apollo rocket that went to the moon back in 1969. And all this is a direct outcome of a great, deal of the work that went into interlogic and the foundations of mathematics. Very, very quickly coming to be applied. It looks very abstract when we were talking earlier about Aristotelian syllogistic, seemed very simple, seemed very abstract and remote from ordinary thinking and applications. But here is a wonderful example of how it feeds into that. Also, however, into our understanding

of even of neuropsychology, let's say, because if it's true that the brain is a neural network and functions in a sort of parallel processing kind of way, then this is something which is added on to our understanding of the world and important features of it, which use these ideas, very fundamental ideas from logic. Peter Milligan. One point that's perhaps worth making here is what a difference the computer makes. Aristotelian logic was criticised very much in the early modern period

for being so trivial. But what the computer can do is take millions and billions of trivial operations and make something interesting from. them. So, for example, everything that goes on in a chess computer, each little operation is tiny and trivial. String them all together and you get a grandmaster. So the computer has really brought logic to fruition. It enables what seemed to be trivial rules to be combined in remarkably fruitful ways in almost every field of human endeavour. And takes us into artificial intelligence? Oh, absolutely. Yes, artificial intelligence is based a lot on logic. You get developments of logic to

handle new things to represent knowledge, inferential relationships, cause effect, spatial events, and so on, probability and uncertainty. It's very much growing in that field. Well, thank you all very much. Thank you, Rosanna Keith, Anthony Grayling, Peter Milliken. And next week we'll be talking about the unicorn. Thanks for listening. If you've enjoyed this Radio 4 podcast, why not try others, such as Thinking Aloud, where Laurie Taylor discusses the latest social science research. To find out more, visit BBC.co.com.uk forward slash radio four.