A Brief History of Logic

hello in 1740 the Prussian King Frederick the Great wrote philosophers should be the teachers of the world and the teachers of printers 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 our subtle 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 the logic as the study of argument and the forms they may take it's widely used in fields from linguistics to cognitive times but above all logic is a vital tool without which computing would never have existed with me to discuss logic and its uses are AC 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 Rosana Keefe senior lecturer in philosophy at the University of Sheffield and integrating before again of the history of the subject can you tell us what you mean by the word logic logic is the science of valid inference by inference I mean drawing conclusions from premise premises from assumptions we make information that we have and from it we try to deduce or induce a conclusion based on those premises and what sigh what logic does is it tries to understand the best ways of doing that that the valid forms of doing that so you're talking about reasoning in terms of validity can you give us some examples of logical sort well take a very simple example like this all men are mortal Socrates is a man therefore Socrates is mortal and to perfectly stand an example of a deduction from the two premises that all men are mortal and Socrates is a man you deduce Socrates is mortal now you measure something or the interesting about it that's a deduction and if you inspect it closely you will see that the conclusion is actually a rearrangement of the information in the premises so in a way there's no informational novelty there whether 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's one is white the next one is white so they wore white and there you're taking a little bit of a risk in your reasoning but you're going from some samples to a generalization about the whole class but in the science of logic traditionally the focus of attention has been on deduction that is only on the question what forms of reasoning are just in virtue of the form not in virtue of the content of what said can you be sure that if you were to put true premises in place you would be guaranteed a true conclusion what is the word valid how does the word valid for your release how was it this is the interesting because validity is a purely formal property of an argument that Cersei's just about it just about the shape the structure of the argument and not what's actually said in it so for example supposing you've 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 will tell you you got the wrong form you've got the wrong set of blanks that you need another set of blanks and they were filling the right kind of information for what we're doing so purely formal matter relates to the validity of arguments if you put in a true premise of set of premises into a valid form then you have what's called a sound 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 use some Ravel premises for our listeners who might be worried about this yeah the premises are of the sort of basis they 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 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 put true statements into the premise slots you will get a true conclusion Peter Milligan we've talked about deductive and inductive what there's also another district formal and informal can you tell us how that plays yes um 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 ungulate buttercup is a cow therefore buttercup is none Gillett now you might not know what ungulate means that 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 you don't need to know the meaning of the terms as long as they're slotted in correctly into the form you know that you'll only get from truths to truth now in formal 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 circular reasoning where you end up taking for granted the pointed 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 as more than a distinction between formal and informal logic it is a big distinction here well yes it is I mean in the sense that with a formal argument the idea is that you can 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 exactly they were saying about the absolute very difficult absolute just 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 through premises to get to a true conclusion but 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 other problems with that well not really because I think here we're we're saying that informal 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 rat rag bag but put them under the heading of informal logic in the introduction I talked about the that Aristotle who seemed to be talking about logic in Babylonian times from what you know and nine Anna but Irish total began to write about it and and give it space and it carried through for hundreds of years what was the nature of his contribution what was it was the key to his contribution what was that well Aristotle was concerned particularly with arguments which we call syllogisms reguar 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 um 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 form every F is G no F is G some F is G some F is not G so you put um something in place of s and G cows meeting 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 or 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 categorize them and tell us which are good and which aren't I'm inclined early over my notes 256 of return in 1900 valent yes the exact number that one counts as valid is potentially controversial depending on on various things that essentially you've got four different types of proposition you can put them in different orders and so you end up with 256 very briefly bidat people listening would say all animals are primates us as Socrates ISM with quite simple it seems now why was this such a hard driver of philosophy why did it drive it so strongly this idea of the syllogism 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 Masonic Eve can you take that on and what was he able to do as subtle with his invasion 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 is more than just a taxonomic exercise it was a very full system so for example his rules for sharing with a particular argument form were valid involved some rules for conversions they call so for example from some A to B you can conclude some B's array if some of my pets are furry some clary things for my pet 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 form that Peter gave an example of Socrates the all humans are mammals or my mother and also for all humans are animals that gets called Barbara and the other one is feller Inc which is an example of which is no dog the robot all poodles are dogs therefore no poodles the robots so those were he thought obviously valid and then he could show why other forms were valid by using those and his other rule was this these these villages these three lines as it were I know I'm treading on oceans are really a bit still they do seem quite simple yet they had such a big impact can you tell me why this way of I started bringing this together in such a neat almost haikyuu way drove philosophy or so long still to a certain extent it was recent take a push from his basis what is it about that which organizes reasoning in a way which validates that method for hundreds of years to come I think it's probably the level of abstraction with which he could deal with it so all sorts of arguments can be of the 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 non the less valid and then his hope was that reasoning would follow these forms so for example his account at fine to knowledge required required demonstrations that had to start from three premises necessarily true premises that were better known the concern the conclusion and which ideally should for follow the form of the syllogism so anywhere else it was the foundation of all of science I'd interesting from the very beginning he switched the idea into science didn't it because he was that was his primary concern that if he had a private Osen knowing so much about so much you switched it industry there's an eligible Greek philosophers examine a different type of logical argument afraid I must use his 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 an if/then and how those logical connectives work what what rules govern them for a loan so yes they had talked about v in multiple rules which again should should seem compelling so one of them modus ponens 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 so 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 for the second you get out a valid argument and they hope to show that all sorts of arguments were valid using these basic ones thank you very much underneath grow me can you tell us how this we've got some idea it's already complicated a too simple that's the fascination isn't it I mean you've already complicated it but it's such a such machine to start with simple premises why I'm asking the same question why did it persists Oh long why have you taken up by so many in order me in charge people sense ISM we're going to go right from the fourth century BC to the Middle Ages where you had great thinkers they were thinking in the context of religion that is over those who they picked up his logic as did the the Islamic scholars before the most over the Middle Ages what was a attraction what was really Drive well first let's just say that that the point about all this is that even though the examples we've been talking about here very simple examples it seems to be a lot of sort of crude lling and and animalistic trivialities in fact they do tell us a great deal because they they purposely constitute an instrument which helps us to think more powerfully in to govern our reasoning much more powerfully but also the course they can be generalized is a great deal for example of mileage in understanding these things what the stoic traditions were concerned with that 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 a large part of it in medieval times so that the the schoolmen is they're called of the medieval era in the absence of telly and bingo and those long winter evenings were able to devote their attention to trying to explore and spell out many more of the of the implications so Rosanna just been talking about the interest that the Stoics and in words like and if not and so on these are known as sync category magics terms that is they're ones that don't denotes anything out there in the world that they have functions within a sentence as functions within propositions because they combine two terms in the propositions in different ways so that an example is that if you use the word and and you join two propositions that's called them P and Q then you know that the complex proposition P and Q can only be true it's 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 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 all Ganon or the 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 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 but why was the medieval school man so attracted to it and they developed it and multiplied it Peter Peter Milligan well one of their interests was actually in things that went beyond Aristotle and they were particularly interested in certain types of inference that might superficially seem valid but in fact aren't so suppose I give you an example I know that she's in London or Oxford but he 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 dunt and that's the kind of case that many of the medieval 's 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 Jews in London or Oxford can you how would they push that well unfortunately in the 14th century you don't have a single systematic theory you've got a lot of people doing their thing in different places people like William of Ockham Thomas Bradford Island Gregory of Rimini and John burden and we don't have a sort of organized theory they're 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 explored this and then quite soon after that Aristotle and out of favor in there yes he went right out of favor and that can be traced to thinkers like Francis Bacon and Descartes and John Locke their 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 know you grow 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 to that syllogism 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 and so they're very interesting I think that I think is that the 17th century of course 16 to 17th century ously those are the rise of modern science and people like bacon were very interested in how you how you do science so what kind of methodology would get you to more empirical results and it looked as there would be arsed Italian solar gistic which is specifically about forms of deductive reasoning and were much less used to you then for example kinds of inductive reasoning ways of accumulating observations for example in super inducing theory on to the makers idea of observational experiments also Andres led to the Royal Society yes indeed so the you you get a shiftless be 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 tune grows Anarchy that logic was out for a while maybe still existed among some school Manuel refused not to be school moon for two three centuries but Locke and Hume and Descartes were saying it's not the way to go in fact they were quite sneaky 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 English mathematician called George Boole 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 we're 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 a similarity between algebra and logical laws for example the similarity between and and multiplication and between or and addition interests mmm mmm yes so a simple example a or B as same as being or a just as X plus y the same y plus X and then all sorts of more complex and molarity so what he did was come up with a system of algebraic laws that he took to govern logic and he could use those to assess arguments to arguments to feed to be valid one of the interpretations of his system has these algebraic laws in to which you can only put values 0 and 1 and if you think of zeros corresponding to falsity and 1 to truth then you get propositional logic the kind of things that and the Stoics were looking for which is the first ontological mathematic have been conjoined in our mind and well I think Leibniz also wanted something similarly recognized similarities between algebra logic to so not all together no and round 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 the ball published his the mathematical analysis of logic in 1847 so it was in the air I think why did he was sorry Peter is just going to say it would give a wrong impression to suppose that in the early modern period the desire for 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 the universal characteristic or a universal logical language and the idea was a language within which reasoning could be as it were computed so um he to some extent inspired Frager um who mentions him and the 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 who can see it in the Museum of the history of science at Oxford seemed um Anthony Peters mentioned Frager can you he is the man who supposedly revolutionizes logic completely maybe that's too big a claim can you tell us about Riga and why he matters this certainly is a very very significant thing very nice yes but it's relevant in talking about Vega just to follow on from Peters point there about the way that logic and the sort of Aristotelian kind of cannon remained of some significance Kent for example in the late 18th century the great Immanuel Kant fought that the Arcadian logic was a completed science and he made great use of it in constructing his theory in the critique of pure reason but it was this move in people our George Boole and Augustus de Morgan and James and others who were starting to use algebra to try to express some of these fundamental logical ideas that gave Frager his in part gave him his clue I mean for example Augustus de Morgan recognized that what have always been thought of as laws of thought which think could be promised a rather psychological way of thinking about the logic and the laws of thought were principle of identity which says a is a or X is X P right principle of the law of excluded middle which says you can't have both X and not X what's the principle of non-contradiction and excluded middle which says either you have X or you have logics 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 who is beginning to show that there is a possible increase of power in our understanding of those logical notions and this is what fragleft on the phrases in essay below 19th century in the bay late 1970s frege's work showed that the traditional logic that Aristotle based logic was actually just a small part of logic really what were fragrant done was to show that you could massively generalize insights into logic a much much wider field of application and you did this in in in a number of rather complicated ways but one illustration of how he did it with this if you take a very simple relational proposition like say Jack love's Jill let's say in the Aristotelian logic when a term occurs as the subject of a proposition it functions it works in a very different way from the way that the term works when it occurs in the predicate of the proposition so on Estados analysis Jack love's Jill gives us Jack as the subject and loves Jill is the predicate but what phrase I was able to do by a really sort of Robert brilliant but simple insights into how you formalize this how you symbolize it to show that Jack and Jill can operate on the same logical level and that you can have what it would go to place pretty clear that is loves which applies to both of them because they stand in attack relation to one another so you can you can write it as predicate expression loves and then Jack Jill and it just by this kind of technique and by the development of what he called a concept script a brief shift you could formalize to symbolize all these relations and show how inferences work in them this was like the development of a machine which you know produced massively more sausages than the our bottled machine did because I know can you this is part of a larger grander project and Brega system called logis ISM what did he mean by that federal auditors and was that arithmetic can be reduced to two logic so we can conjure up all of arithmetic from logical foundations and this was important because they the idea was to give secure foundation for mathematics so rather than resting on axioms that you might just think or intuitively true or trying to base maps on something empirical we can we can base it on logical foundations so for example you can understand a sentence with the number one in into your logical terms so if I say there's one dog exactly one dog that gets translated logically as there's something with X which is a dog and anything that's a dog is that X so the logical formula that gives you an understanding of a number and from that you can build up to understanding of of number and more complex and rational knowledge etc in American like see you wanted to get in but when you're getting can you also tell us why a letter from the bridgewater Bertrand Russell includes figure so hemily 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 frege'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 as long as either the hell well what it does it enables these 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 but 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 atleast 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 Frager saying here's this contradiction I 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 in your system of logic briefly and only what was the contradiction and what was frege's responses well that the connotation is this and Russell was trying to define number in terms of classes of things so and you try to define say that the number naught by saying that nor is the class of all those classes which empty which have no members and the number 1 is the class of all those classes that have just a single member number 2 class all these classes that have double tuned members and so on so it looks at that your reasoning circularly here but actually you're not using the notion of number to try and define number in that way so crucially making a use of the idea of a class many turns out that you get this difficulty some some classes are members of the class of classes ok but but some classes or class of teaspoons are not themselves piece prints so you can ask yourself the following question and what what is what what can one say about the class of all those classes that are not members of themselves um would it be a member of itself if it weren't answer yes'm 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 this so do be careful and not making it exact parallel but but supposing you have a barber in a village few shades all and only the men you 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 of arithmetic and remember that arithmetic can take translations from all the other branches of mathematics editors this is very very fundamental Russell's solution was to say well somewhat arbitrarily I'm not going to allow classes of one type to take as members other classes of the same type so you would have a kind of hierarchy of kinds of class and the higher levels of the hierarchy can take as members 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 extract receptor and you get into that kind of muddle to avoid it you have to introduce this wrong rather ad hoc maneuver of inventing a hierarchy as I understand that the resign if Prager introduced this as an appendix to his last work and that was his last work on this subject he apparently taught it but the paradox had 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 influenced appeal to philosophy in the 20th century yes in fact he wasn't that widely-read initially he did influence directly influence Russell and the inch line and Carnap the important influences but in certainly in the last 50 years he is one of the most important team is 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 it that is used across all sorts of areas of philosophy another thing is his conception of logic where as as has been mentioned some of his predecessors saw logic as really a part of psychology was interested in how we think laws of actual thinking fragrant about psychology ism and wanted to take logic as autonomous there are laws of logic that apply even if even if nobody didn't 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 his work in the philosophy of language which is closely tied in with with his working in logic EDA Milligan but we do the programming imagine your numbers of you weeks ago and we're talking about about abstract thought going on among persons we wanted to think about imagined numbers for centuries and then he 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 were into Hobbes mentioned computation nautical calculation and that's right yes and now the logic makes his entrance here can we use Island curing 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 Lodge assist 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 give me 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 if the 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 curing regime is still discussed in those trends today um and underneath are the other others shall we stick with this the idea of the practicalities because we haven't got much time left and it is interesting can you tell us what other as it were practical developments have come from those centuries of thinking well firstly a result mentioned earlier rule and rules ones and zeros in in his algebra this is digitization of of thinking about how you might have a calculating device or a computing device early on and the Turing machine what are the machines in Level II cogs and wheels of smoke coming out of it or is 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 telephone into the people switched off in our pockets here I've 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 inter logic and and the foundations of mathematics very very quickly coming to be applied looks very abstract when we were talking earlier about Aristotelian syllogistic seem very simple seem very abstract and remote from ordinary thinking on 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 never added on to our understanding of the world and important features of it which used these ideas very fundamental ideas from logic is a really good yet one point that's perhaps worth making here is is what a difference for computer makes era stop Aristotelian logic was criticized 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 endeavor 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 influential relationships cause-effect spatial event and so on probability and uncertainty it's very much growing in that field well thank you all very much thank you Rosanna key fountain in 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 Lori Taylor discusses the latest social science research to find out more visit bbc.co.uk/topgear

Channel: Philosophy Overdose

Views: 228,459

Rating: 4.9059329 out of 5

Keywords: Philosophy, Analytic Philosophy, History of Philosophy, Logic, History of Logic, Aristotle, Frege, Deductive Logic, Inductive Logic, Formal Logic, Informal Logic, Epistemology, Fallacy, Square of Opposition, Syllogism, Predicate Logic, Deduction, Fallacies, Laws of Thought, Theory of Knowledge, Validity, Leibniz, Russell's Paradox, Alan Turing, Bertrand Russell, Inference, Turing Machine, Boole

Id: IBLzwKtfF0Y

Length: 42min 3sec (2523 seconds)

Published: Sat Feb 18 2017

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