Symbolic Logic Lecture #1: Basic Concepts of Logic

you let me start out by asking you what your understanding of philosophy is this is a philosophy course after all one of you remarked earlier that you thought of philosophy as a study of knowledge and it it does that I mean philosophy examines you know what sort of things might count is knowledge it examines in fact whether there is any knowledge at all has anyone ever taken the skepticism course that was offered there's a I think a special topics course in skepticism or honors course and skepticism but you can you can study the idea about whether there is any knowledge at all you can spend an awful lot of time on that and we'll offer all of ink has been spilled on that particular topic so that's one of the things that philosophers are interested in but the there are others max I can't hear you give any definitive being with philosophy is a very different philosophy than you know it's just the self whereas so I think it's really developed finding the right questions asking questions so philosophy is asking questions that's certainly seems to characterize philosophy you know the history of laws we just filled with questions and questions questions but they're also of course is the attempt to give answers and that's kind of important to me as a philosopher is to try and figure out what the answers these questions are yes sir that's really right at home with logic anyway but the creation and analysis of arguments I guess we would be doing the analysis part more than the creation of arguments whereas then we take these tools and then we use them elsewhere so creation and analysis of arguments let me try another one one more friendship with wisdom rather than a love of wisdom because philosophy says I mean that the basic word is filo Sophia and Sophia is maybe knowledge or wisdom depending on how you translate it and filo is love of or a friendship with so I'll put it your way I like that a friendship or close acquaintance ship with with wisdom I think was will you put it right those are interesting candidates but there's also stuff like people you know we're talking about academic philosophy but the word in English goes a lot further than that the word in English embraces things like philosophy of life people have a philosophy of life how would that fit in I mean how would that fit into some of these ways of defining the field or you want to just sort of banish that that's just some sort of a popular and ignorant understanding of the of the word we shouldn't be using that when we're in a philosophy classroom a philosophy of life is a conceptual framework safe a few more words about that what conceptual framework meaning there's sort of assumptions or ideas that you interpret okay does that what is that now I think that's part of what a philosophy of life is how does that relate let me go back how does that go back to any of these things or any other version of what philosophy is I mean how do we link this stuff up I'd like to have an embracing at least enough of some sort of core sense of the word philosophy so that we get we'll be able to understand why it would be used in these many many different ways Lizzie sure yeah and I think but what's your name again hey Nathan I think what Nathan was getting at was served like it's a consider it's a way of looking at stuff I mean it's a philosophy of life is your way of looking at stuff which may change may evolve over time you may not be aware of it you may not think much about it but people must have some sort of basic way of looking about you might be able to identify it even if they haven't thought about it themselves if you talk to people for a little while you get some idea of what their philosophy of life is their framework their way of looking at things that's kind of close to and trying to figure out way max what were you going to say what just asking some of the questions even though they might not be explicitly asking why is this going on let me throw something else another curve your way what's the most advanced degree you can get an engineering it's the Doctor of Philosophy PhD Doctor of Philosophy in engineering huh why is the most advanced degree you can get in physics or in math or Orion or an engineering why why is that the Doctor of Philosophy is that strange never think about that those of you who of courses have might have thought about that why can one of those who have been like in other courses of mine where we've gone over this in the introduction can anyone offer a suggestion as to why fields that don't seem very philosophical on the face of it why would they have the most advanced degree be called a doctor philosophy all right so that there's a bunch of philosophical things that perhaps the things that philosophers deal is like like for example the study of knowledge and and and perhaps something like the scientific method methods for acquiring knowledge in reliable ways things that philosophers do think about supposedly people who have PhDs are prepared to apply these in their several fields I think that that's right but I think there's more to it than this I think you know what people are supposed to have a PhD in something and you name it people are supposed to have the PhD in any field are supposed to be at the cutting edge right of their field they're supposed to be able anyway they're prepared to go out there and do work at the cutting edge of their field whatever that field is yeah creation of new knowledge creation of new knowledge and by some where we had creation and analysis of arguments anyway as part of what philosophy is all about so maybe it's something like that and I think that's kind of what I'm what I think and this is a philosophical subject by the way what is philosophy it's a philosophical topic so this is one of those topics that I'm going to now try to embrace with this core general definition I'd say philosophy it's the it's the examination of questions that don't yet have clear-cut answers the examination of issues and questions and matters that don't yet have clear-cut answers those are the philosophical questions and so you get people like Niels Bohr and Albert Einstein arguing over the course of their entire careers about the basis of quantum theory for example but whether quantum theory is complete or not Einstein said no Bohr said it's about as complete as you can get it's therefore complete you couldn't ask for more but asking for more misunderstands what's going on down there at the subatomic level but it was a philosophical discussion they both thought of themselves of course they came out a year and they came out of older traditions but they both thought of themselves first and foremost as natural philosophers they described themselves in this way rather than as physicist physicist is an invented and scientist for that matter those are invented words that are a couple of centuries old Isaac Newton was the chair of Natural Philosophy at Cambridge University another interesting thing Adam Smith was the was the chair of moral philosophy at that where uh I think was Edinburgh I'm not sure might have been Glasgow who's Adam Smith like the founder of modern economic theory more or less to the extent that theories really do have founders but he's one of the early greats in in the establishing economics as a science but he with thought himself as a moral philosopher and Isaac Newton was a natural philosopher and and born Einstein even in the 20th century still thought of themselves as natural philosophers they're working on the cutting edge questions questions that didn't yet have clear-cut answers questions that still people who are in the know they could still argue about them so that people like Bohr and Einstein who were certainly in the know these were things that weren't settled yet they still had they weren't yet they didn't have clear-cut answers so that's that's the way I want to put this and I think that that reaches all the way down into stuff like philosophies of life things like ethics questions like philosophy of religion these are things that you know they don't have you know you can't turn over a rock and find out the answer some of these questions like which things are right which things are wrong or what establishes the basis of right and wrong or whether there's a god or not or you know how you should proceed in your life what you should be or try to be or what what your relationship to society should be as an individual I mean should be you know should be trying to further the ends of society and working very hard to sacrifice your own goals in order to do something good for society or should it be the other way around a society out there to support to support individuals needs the philosophical questions because they don't yet have clear-cut answers now I want to stress that word yet a problem for me with definitions like Max's number two lots of different questions is that that's not why I'm in philosophy I'm not in philosophy just go around raising questions and and making trouble for people as a matter of fact there's a lot of questions that aren't worth raising they're not worth to me I mean they're just my view I mean there's things that aren't interesting to me but the things that are puzzling and these are the ones usually and they'll be in particular areas that I'm interested in but they're questions that cry out for answers they don't yet have them I mean there's a lot of there's a lot of there's very little on earth that can be proven we're going to be working with a very very limited domain of things in logic we're going to be working with proofs and things that can be proven but it doesn't go very far doesn't get us out to the world very much as we'll see in a moment but there's a lot of things that seem to have good enough answers so that I don't really want to dig into them and there's other stuff that whoa that's that's interesting that's that's you know for example questions about the completeness of quantum mechanics that's an interesting question to me still I mean I think all the pretty much the evidence points dramatically toward Bohr's understanding of things rather than to Einstein's but there's still some still it's still possible for physicists to argue about that stuff that's interesting to me there's all kinds of social and political questions that are interesting to me but these are the philosophical questions the ones that don't yet have clear-cut answers and what the philosophers business I take it is to do is to try and make a contribution toward toward the end of getting clear-cut answers and if we can get clear-cut answers to questions and I guess what that would mean is general agreement among people who are studying them if we can get fairly clear-cut answers to questions then that sort of takes them out of the field of philosophy the history of philosophy is kind of as the place where all those questions got raised in the first place couple thousand years ago by people like Aristotle Plato and it was philosophers who raised those questions and it continued to be philosophical all the way through all the way through that's probably the 17th century anyway before various disciplines first for most physics I guess with Isaac Newton they started to break away and become departments of their own become faculties of their own and and disciplines of their own and that continued through the 19th century when the social sciences started to break away from philosophy and so that you get things like economics first and psychology and disciplines that were different but used to be all those guys all those people they were guys unfortunately for the most part but all those people were indeed they thought of themselves as philosophers as the questions gained clear-cut answers they ceased to be philosophical questions they see they moved away out of the philosophy department leaving the philosophers with the old ones that hadn't yet achieved any kind of you know well widely regarded answers to them and then also there's always new questions and that's where max is right there's always new questions no matter whatever question you answer you're going to raise you know a dozen more it's a geometric expansion there are many more questions now than ever there used to be only because of the successes of the science for example the successes of some of the other disciplines that be you might not call scientific but have made some progress in understanding the world so there's where we stand with philosophy and now the contrast is what's logic in particular and where I'm going to go with this is I'm going to ask you the question see this works yep I'm going to ask you the question now that's that's our my proposed definition of philosophy general definition philosophy which you need not agree with but now I'm interested in what what logic might be and where I'm going to go with this as I was about to say is that I'm going to try and figure out why we're studying logic under the rubric of philosophy what's the relationship between logic and philosophy so maybe when you understand that that's kind of where I'm trying to get with this conversation that might help to influence your answers the question with logic yeah is a way to push clear-cut answers in there so logic you start a new one of these things you're supposed to give me no one logic allows pushing clear answers and two new areas that what you said yeah so that where we've achieved something in the way of clear-cut a clarity I'm just going to stop saying clear-cut answers again where we've achieved some clarity a logic would allow us to push it into new areas how would it allow us to do that what's your name again say Greg Greg how it allows to do that great if you had premises right so that if you had some if you had if you had some knowledge you may be able to use the tools of logic to derive as yet unclarified things or as you know new things that you didn't know for sure for sure any other thoughts about the what logic might do or might be I mean what is it I mean this is this actually leaps forward toward what I wanted to be the the goal of the conversation rather than you know sort of sticking at what is logic so what is logic let's go back to that question okay rules for deciding on the legitimacy of argument right so pretty much it how about the formal structure of reasoning I mean I think that's what you're getting at and that's what you're you're thinking you had that in the back of your mind Greg when you offered your suggestion is that well we had some formal rules that allow us to move from truths to truths then the other set of our tickets you know there are inference tickets we get to conclude if we know a and B we get to include C because we have this inference inference ticket that allows us to it if we could be absolutely sure the inference ticket that would be the rules of logic that you're referring to then we would have devices that would allow us to go from things we know now are things that we are pretty clear on that we think are true to new things that we haven't thought or true the real key here is is this whole idea of formal structure what is that I mean I mean this may be absolutely as obvious as can be but when I'd say we're trying to identify the formal structure of reasoning it's got that word form in it what are we talking about somebody who hasn't spoken yet what's the formal structure of reasoning a pattern that's very good what we're talking about basically is a pattern of reasoning as a matter of fact you might think of it what we're doing is we're trying to study ideal reasoning we're trying to study good reasoning we want to know not just you know anybody's you know efforts at reasoning we want to know first of all are there any ways of arguing ways of reasoning patterns forms structures that can be relied upon so that if we are able to fit an argument to fit a line of reasoning into those structures that we can then say aha the it fits and we are we're good to go we can actually have confidence in our ability to draw this conclusion from those premises because we know that the the way we've proceeded is a certified bona fides legitimate line of reasoning it's a form okay so that's what we're going to be studying this quarter we're trying trying to identify the structuring our reasoning should have if it's to be good reasoning and you may get frustrated from time to time because it's going to get so formal that we're going to be just dealing with letters and we're going to from time to time there will be examples but the examples would be so silly you know like Alfred and Betty both are going to college therefore you know things like that and you know these things that are just obvious obvious form fillers the only thing we're interested in here are the forms of reasoning and that's why that's why we call this that's why I say the study of logic amounts to attempt to identify the formal structure of reasoning and the formal structure of good reasoning in particular now finally what a logic and philosophy have to do with one another philosophy I really couldn't emphasize this not it's not just a bunch of weird ideas I mean I think that's one reason to be attracted to it it's because oh ho you can't believe what this guy says that's a reason to be attracted to it because it sort of might open your mind to some stuff that you never have thought about before even if it seems wacky but it might be fun to just say entertain that thought and that's fun and it's interesting but frankly beliefs and opinions and these bottom lines and the and and positions on things they are a dime a dozen everybody's got their opinions opinions I mean I'm by we're supposed this is not politically correct we're supposed to appreciate everybody's opinions I don't appreciate opinions much I appreciate lines of reasoning I appreciate opinions that are well supported and when they come up with something surprising and strange and they have good reasons for them Wow that's illuminated I mean that's very interesting and I mean when I say wait a minute what's wrong with this argument if you ever see the argument who has taken my intro course somebody to eat it in there we talked about the proof that one equals two is that have you seen this proof or some version of the proof that one equals two if you're a mathematician there's no plausibility to it at all get them to play in a line four and say well you throw it away but if you're not a mathematician you haven't taken math recently you can go through this proof and it can look pretty plausible you say wait a minute and it seems like it proves using perfectly normal mathematical algebraic procedures it seems to demonstrate that one equals two which is that's ridiculous where and so the question where is this thing go wrong and that's what an interesting argument does I mean if you if you're if you're not sold by the conclusion if you think oh no that conclusion can't be right then you're driven to figure out well where does the reasoning go wrong or which one of these assumptions which seems so plausible at first which one of them is wrong because something must be wrong here and then as you examine it more and more fully the oddest thing happens if you cannot find the error now that this is not the case with one equals two but if you cannot find the error in the argument well the conclusion begins to seem a little bit more plausible than not given to it at least say well maybe I was wrong logic and lines of reasoning are the one way where you can get movement in your own beliefs where your beliefs can grow and change or where the beliefs of people you're talking to might grow and change so it's through argument and reasoning and good argument and good reasoning that we can we can learn ourselves what to believe and and which of our beliefs might need a bit of correction beliefs that seem to be supported by evidence and good reason can be not only interesting but persuasive and we need to take such beliefs seriously because they might be right and this is something of a circle and I'm a little bit concerned about that but they say why might they right well it's just the the might comes it's because they're supported by evidence and good persuasive read reasoning and that's why they might be right it doesn't make them right but that's why we have to begin to take them seriously and believing isn't something that is just a fun habit or a fun thing to do or a thing that we could avoid believing is the human way of making our in the world it's our way of anticipating what's going to happen next and that's what we have to do if we're going to act in the world we have to have some idea as to you know what could what the world is like what how people behave in it what we can expect of this person that institution this machine whatever it is you you need beliefs and they need to be as right as they can possibly be it's really it's really a foolish venture to believe stuff that you want to believe we want to be true for that reason believing stuff because you want it to be true is not going to serve the function of belief believing is a natural biological thing that we do it's what it's but it's continuous with what other animals do although ours is a good deal more sophisticated in a lot of ways we have words and sentences and idea structures and all kinds of things that that enter into our beliefs but believing is something we do in order to successfully make our way in the world it's a survival tool and so you want your beliefs to be as correct as they can possibly be to be as accurate as they can possibly be you want them constantly checked that's why argument that's why a free press is good that's the reason the free press isn't good so I can say what I want to say is so I can hear what you have to say that might help me correct my wrong beliefs see Jon Stewart know about that I mean he was in Inn on Liberty he is that's probably one of the most brilliant essays ever written in political philosophy but among other things that explains explains free speech in the proper terms so the advantages of it and the value of it it's not that it's not valuable specifically only because well each of us has our beliefs and we ought to be able to say what we think that's all very nice but we need people's contrary opinions we need them desperately so that we don't go off believing stuff for bad reasons No all right where are we all right I think I'm here so logic is important to philosophy just because reasoning is central to philosophy and logic studies the structure of reason and why is philosophy studies as wise logic study is a philosophy course well because the very business of establishing Rome of hoping to establish reliable inference tickets reasoning structures it's a controversial matter we will have occasion from time to time as we learn about the logical connectives I won't dwell on these things but we will have occasion to sometimes see that whoa that's strange to interpret this particular connected in that particular way I mean is this the right interpretation for example I mean what is to give you an example one of the logical connectives is is for the what we mean by or the disjunctive like a or B either I'm going to go to school or I'm going to stay home and sleep that or in there is a logical connective and you could either you could interpret in a couple of ways you can interpret it in such a way so that it's exclusive so it's either one or the other but not both or you can interpret to such a way so that it's inclusive that okay that it would be the sentences that either I'm going to do this or that can be true if I'm going to do both now the one I gave you couldn't be both but you can imagine situations where you say either Joe goes to school or Betty goes to school is that supposed to be true or false if they both do well an inclusive or and that's the one we're going to use says that the sentence is true the disjunctive is true either Betty goes to school or Joe goes to school that's true in the event that they both do because it means at least one or the other goes and maybe both so it leaves that open but you could have interpreted the other way you could have said wait wait no if you're saying either Betty goes or Joe goes you mean not both either one or the other and you can hear how English is ambiguous on that particular topic it will have the same difficulties with another one logical implication material implication which will be symbolizing with a horseshoe that too has got a very strange interpretation so that anytime the conclusion is you know of an argument or any time that I'm sorry I found the consequent the second thing I'd like to say if if I go to school then I'm going to get a cup of coffee if I get the cup of coffee that implication the whole implication is true anytime so even if I don't go to school if I go to school then I'm going to get a cup of coffee is true if I get a cup of coffee now that's weird why should that happen well there are logical reasons there are systemic reasons why we have to interpret it somewhere or another and that's a good way to interpret it but these are philosophical questions and you could have different logics and there are different logics that make different interpretations about those particular logical connectives so logic is itself a philosophical subject logic is itself something that embodies lots of different questions that don't yet have you know answers that everybody agrees to we're going to follow we're going to follow one convention one set of conventions for these things you'll find them in the logic cafe as you go along and John Halpin who is the author of the logic logic cafe he will from time to time offer you some reasons for accepting the way he is doing it but those of you who have taken some logic before will find out yes this is the same these are the same conventions that are used generally in teaching sentence logic and predicate calculus so that you know this not it's not unique to John Halpin or to the logic cafe these are the normal ways of doing this stuff and please just accept it then please just say alright that's the it's just a stipulation we're going to interpret or as inclusive are not exclusive we're going to interpret material implication that is the horseshoe in this particular way that's just the rules that we're going to the rules of the road for our course and then if you have disagreements with it that's fine just but this is the way we're going to do it just so we can move on to the next step so anyway logic is necessary to philosophy for this reason and and philosophy is the domain within which we study logic just because logic itself is still open to disagreement and development there are new systems of logic that come out all the time there are multi valued logics there are dual valid valued logics true and false like ours let me consider this we're going to be dealing with truth tables and we interpret well every sentence every every proposition is either true or false not both not neither it's either true or false every proposition every sentence has to be either true or false but you can imagine lots of situations where you want to say well why can't we accommodate shades of gray why can't we accommodate true/false or maybe that would be a three-valued logic why can't we accommodate something that do is its truth value is sort of measured on a probability scale and that would be an infinite valued logic that would go from 0 to 1 and all places in between there are infinite valued logics there are three valued logics we're going to deal with a simple two valued logic true and false sentences are either true or false not both and not neither ok let's see there's a those are some of the difficulties with they're not difficulty so there's some of the interesting philosophical questions that come up about logic and you can imagine that different logics might be more or less useful for different sorts of questions like in quantum mechanics that's one of the places where many valued logics first arise in application because it's hard you know that you can't simultaneously determine for example the position and momentum of a particle can't simultaneously determine it and in fact if you believe Bohr and the dominant interpretation of quantum mechanics there's no such thing as the instantaneous position and momentum of a particle but what can you do well the more detail you get about the position of the particle the less detail you get about the momentum of the particle and the two things are coupled in that particular way so you if you want to make an assertion about either one the position of the momentum the particle you might want to make a probability statement and you might you're the the logic you might want to use for that what would be a an infinite valued probability sort of logic so all these things are out there to be examined in the deeper study of logic but in our case we're going to do a two valued logic studying the formal structure of reasoning and the unit of reasoning is as we've already noted in passing tacitly is the argument and an argument in this course is not anything particularly bellicose it's not people beating each other over the heads with bats it's nothing other than a collection of statements some of which are given as reasons for another member of the collection and the statements which are given as reasons are called the premises of the argument and the state which is supposed to supposed to be supported by these premises is referred to as the conclusion of the argument so arguments just are units of reasoning now notice here that I say the premises are statements that are given as reasons and here I say the conclusion is the statement which is supposed to be supportive there are good arguments and there are bad arguments I mean argument isn't necessarily a success term doesn't mean that just because we formed an argument we're not using it in such a such a precise fashion that we're only talking about good arguments we're going to need to be evaluating arguments are they good or are they bad so keep in mind that later on we'll be talking about you'll see in the logic cafe we're helping defines a deductive argument is one is which is intended to be valid we'll get to the word valid in a second intended to be valid that leaves open the possibility that it will fail in this intent so there are invalid deductive arguments deductive is not also not a success term nor is inductive but in any case for our purposes we're going to have a particular form that we're going to present we're gonna be dealing with deductive arguments almost exclusively with a brief foray maybe this week or next I can't remember which where it is in the logic cafe a brief foray into some you know just waving a hand that an inductive reasoning which is different but deductive reasoning is what we're going to focus our attention in and our attention is we're going to do this in what we call standard form and the premises will be we have arguments the premises will be written this way I mean there could be a number of premises it could be one there going to be two but the premises will be above the line that conclusion below so if you see something that looks like that it to conclude the thing on the bottom is supposed to follow from follow logically from the things above and the things above are supposed to exactly those things the premises they're suppose they're given as reasons they're supposed to be reasons for concluding the thing on the bottom all right now what's a good argument well we've pretty much talked about that it's one where at least the premises offers some reason for drawing the conclusion some reason for asserting the conclusion for concluding the conclusion but we're going to once again get a little bit more precise we're interested in valid arguments not just arguments that I'll offer some good reason for drawing the conclusion our attentions can be focused on arguments that are defending you know determine the conclusion of the the truth of the conclusion so an argument is valid this is a technical term an argument is valid just in case it's absolutely impossible for the conclusion to be false if the premises are true and it's invalid just in case it's not valid so let me give you an example so you can because I want to illustrate another point say all Certified Public Accountants are let's say does any who knows about the group wean anybody still yeah one person the cameraman Thank You people okay neat group strange room but neat group let's just that's one premise another one that's probably one person in the room who may be able to refute this let's take that as our second premise all we noted for dennis kucinich in 2008 probably false but this that's part of my what I want to illustrate from this we can validly conclude something right I mean I'm offering a very very simple example to illustrate a point that will emerge in a moment what yeah that's a valid argument in other words the can if the premises are true the conclusion has to be true but what I wanted to do I'm trying to lead up to another definition it's not a very good reason this does not provide a very good reason for us to believe the conclusion though does it and why because the premises are false or even if one of the premises are false that would be enough damage done for us to have doubts about the conclusion so you need more than validity validity is a strictly a strictly formal consideration about arguments this argument is valid but it's not yet a good argument for to be good we would have to have true premises and that yields our definition of soundness which is another technical term an argument is sound just in case it is both valid and has only true premises now if you got an argument like that you got a good argument and it's unsound just in case it's not sound so in order to be sound it has to be valid but it also has to have true premises now look this is where we get into the differences between logic and the sciences logic has no particular way of determining the soundness of argument we're going to be working on the formal structure of reasoning we're going to be concerned with validity that follows from what and what doesn't follow but we can't be investigating I mean for example in the case of the argument the consider we I mean how are we going to find this out logicians can't have no particular skills to find that the truth that's falsity of these things I've chosen wacky premises just because we can be pretty much be sure that they're false just for illustration here but it's the sub its observation it's experience it's the sciences when you're getting technical areas it's it's it's and if they're not sciences it's the special research that's where we may be able to uncover the truth and falsity of various interesting propositions these aren't even interesting propositions but that's what we need we can't do that with logic logic alone only tells you well if such as such we're true then such and such would follow that's all logic is empowered to do so logic while key while crucial doesn't take us very far to our knowledge it's just something that provides us a structure within which we can get an inference ticket from things that we might know as greg pointed out pointed out earlier it's an inference ticket that it takes us from where we are now or where we think we are now to the other conclusions other things that we might not have thought of it allows us to unpack the take taking these two together doesn't help you very much let's say a whole bunch of premises I might not see all of the implications of those things when taken together take geometry for example it's got five axioms that it starts with it uses exactly these principles of logic as a matter of fact logic was invented and developed by Euclid who was a student of Aristotle's some 2,000 years ago more 2,500 years ago I guess 2,500 years ago nearly twenty three or four hundred years ago invented by by Euclid who was a student of Aristotle's and he couldn't have done it except that Aristotle did immediately before him developed some of these principles of logic that we're working on and all the Euclidean geometry does and that lasted for two thousand years before the development of some more modern geometries at the end of the 19th and beginning of the 20th century but for that whole period Euclid's geometry was the only game in town and Euclid's geometry took five basic axioms and some definitions that were supposedly intuitively obvious and from that derived the most crazy things from those five axioms you can find out that for example the square on the hypotenuse is sum of the squares on the other two sides something that no one would expect it but from these five simple axioms that all of them seem like they're obvious common-sense things like the shortest distance between two points is a straight line stuff like that that may be a definition but things that are about that obvious take those five axioms and you can derive a world of very complicated very unexpected things so that's that's what this inference ticket may allow us to do if we're fortunate and not just in the areas of mathematics it seems you know if we know some things about physics we can derive now for example that's more or less the way Newton set up his particular Mathematica in geometric form the few basic axioms he attempted to derive further laws of physics that then were informative about how the you know the planets moved around the Sun things that were not obvious from the from the axioms themselves so but we need sciences to sort of fill in those blanks the inference tickets are all simply structural finally before we go on to something else I want to just go into this the matter of what was that this word here impossible let's take our argument from before all or now we'll start using letters all CPAs are weaned freaks then we had all green freaks are Kucinich voters and from that we derived all see are okay I want to this is literally impossible there may be some skeptics in the town who saying nothing is impossible maybe some skeptics in the room we figured what would get is just maybe as as as impossible as can be that they want to leave some room for doubt I claim that this kind of impossibility is is special this kind of impossibilities impossible and I want to try and illustrate it with what with Venn diagrams let's take that first circle it says that all see and what I'm going to do is I'm going to take the set of things that are see and make them say that circle is it that circle is supposed to represent all the C's that is all the Certified Public Accounts but you know I guess people that came before all present ones all future ones whatever it's all certified public accountants that's the set of them that's embodied by that circle and this is going to be the circle the set of wing freaks all past present and future weaned freaks the first sentence up here number one simply says this it says the whole set of C things is included in the set of W things that's what the first premise is is that correct okay second sentence says that's set is included in the set of Kucinich voters so that all we notice is voters right anybody have any problems with this so far okay what I want to point out to you is that I don't have to do any work here all I've done is diagram using the Venn diagrams all I've done is divide I grab premises one and two and if you'll see three is already in there it's not like I have to do anything if these two things are true that's just a way of saying that the conclusion is true what you're doing when you're when you're working in deductive logic is nothing more than unpacking the information that's already there in the premises you're not making really even an inference I mean the word inference properly should be used with inductive reasoning which is when you're going from a smaller set of data usually you're drawing a conclusion that's that's bigger like if I examine a whole bunch of crows and they're all black and I keep I just I really work hard and I examine crows from you know for weeks and weeks and weeks and months and months and months and I have people send me crows from Siberia and I have people look up historical records of crows let's say I determine you know all the ones I've examined and all of us I've read about and all the ones I've gotten worried about they're black you may be inclined to draw the conclusion I guess all crows are black that's an inductive conclusion because you don't have you don't have all that data you don't have enough to sort of you know let make the leap you're making the leap from as big a collection as you've made these crows it's not all of them and you're making an inductive leap notice that that's the way science must proceed we have all kinds of very fancy laws but we can only go on the basis of human experience up till now human experience in the areas we've examined it and we're drawing conclusions about laws of physics for saying here's the way the universe works but those laws those laws are much more general than the data permits so we had to constantly that's an important part of science is constantly looking in new areas that make sure you have see if you know they're the regularities that we've established in our prior experience see if those regularities continue to hold up that's inductive reasoning and science also has to make use of deductive reasoning insofar as it makes use mathematical methods so science especially the the physical sciences are an interesting combination of both deductive and inductive methods but what all we're worried about is these deductive methods were interested in arguments wherein it's impossible for the conclusion to be false if the premises are true and this diagram I think illustrates better than anything I can say just what that's the strength of that sense of impossibility it's impossible because we're not doing anything from from the premises the conclusion we are simply unpacking the information that's already up there in the premises just like including geometry does nothing more than unpack the information that's already in the five axioms and definitions okay any questions about any of all any of that stuff because what I want to do is to move now over into some other stuff that involves what you'll be working on in the logic campaign how many have visited the logic cafe more than just a Glanton to give it a glance and you've worked in it a little bit well how does it seem to you I mean so far it should be pretty simple and then the difficulty may come in in the exercises later on I'm hoping that the that the explanations will be straightforward one of the advantages of this is it's far less textual material and you would get an effects book it gets to the nitty-gritty of the stuff and it gets you right to the exercises and it says alright I'll stop doing these things and you don't have to worry too much about long long things to read I mean there's there sometimes quite a bit to read but it's not like if you're reading a chapter in a textbook and it tries to get you to the techniques and tries to give you some you know as it as you go it tries to give you some some opportunities to work with the exercises and things my advice is that you get in there by tomorrow at the latest I also would like to repeat my request that you you know give us a little introduction that's in the discussion area you can actually there's a link to it on the end the very first message that I put up in the news notes in my courses but if you don't want to go that way you can always go hit just hit the discussions link sort of to the top of the of the screen it's not at the topic mode the quarters the way up or something hit on that discussion link and it'll take it a an area that will show you all the different discussions there's a discussion subject for every week but go to the introduction the student introductions one you just sort of say a few words about yourself maybe respond to somebody else if they if somebody else something interesting and just let's get to know one another a little bit there but for now I offer you a sample argument now these are taken from the logic cafe and what I would like you to do is to take a few moments and break this down so that in that in that formal standard form I'm going to I put that slide back up the standard form whatever now what there's several exercises here some things in the sentences maybe you're relevant sometimes the whole thing isn't an argument even tell you first of all decide whether this is an argument and if it is break this into the standard form so that you've got premises up and conclusion down it could be one or more then you can abbreviate you not to write all those words out just as long as you know what you're doing who would like to share with us what basically their suggestion about this particular argument okay say what okay I'm going to call that Sam has attended each class up until today and what is okay Sam is here today okay that's more or less it you could you know might have written a lot more many questions about this one okay try this one and somebody else what's the solution this one promises change who an implied premise huh since the implied premise wasn't on there I don't expect that all of you will have said something like that but it's true there's an implied premise but the way that this argument runs it doesn't make mention of that employing presences and so whether this is a good complete argument or not as it stands is a good question but right now it says the use of the Native American names for sports team is degrading and the conclusion of sports teams like the Atlanta Braves and Washington residents should change their name so as a single premise and a single conclusion that's offered and as has been pointed out that may not be enough to make this a complete argument there's a there's a suppressed premise here does everyone get this is this okay okay and one more look this one over Andrew Steele oh where are you Andrew Steele yeah you say you don't understand yeah tell me why anybody else having that problem it's not an argument yeah okay so those of you who are agonizing over this one you have good reason to agonize over it so this is not an argument there's no effort to offer premises for a conclusion no attempt is made to do that so this just isn't an argument and one of the things in the logic cafe among the exercises he will occasionally throw that this is one of the ones he throws at you just to make anybody always ask the question there's another thing I want to let you know about the mechanics of the logic cafe when you get to some of these exercises there's things going off all over the place I mean I'm not sure that this was wise for him to design it this way but at least on my browser and my screen when I go into this particular set of exercises starts off with I think I get whole black bottom of the screen and something on the top and he says choose choose an exercise so I choose an exercise and then I get something like this and he's got it so that there links and you can click on them he says re says first thing to do is this an argument but you can't always I the first time I ran through this program last fall I didn't see these boxes there's a little box that pops up the upper left-hand corner when I do it and it asks the question is this an argument or not and if you don't answer that question you can't go anywhere you get to anything with this thing so watch out for that when you when he has these little exercises is to choose e1 e2 e3 e4 whatever it is pick the exercise but then the often the first thing he's going to do is he's going to ask you a question up there in the box and you have to say either yes or it is an argument or cancel for it's not an argument and then you close the box and he said Rio if you say you know cancel or if you say yes if you say yes and it is an argument he's going to say all right now first pick out the conclusion then pick out the premises sometimes there will be more lines and things said something irrelevant things will be said you say no that's not one of the premises for this argument but there'll be something irrelevant in the middle of the little thing that he's asking me about so watch out for stuff like that any questions about this not being an argument you all see that okay let's do something different this would be the last thing I believe just make sure yep this is this is gonna this is this represents a thing that he does see this came second comes out all right he'll have it a drag thing he wants you to drag the right number into the right line this is a three line argument and it uses a particular technique that I wanted to sort of familiarize you mill yer eyes you with now before you get started with working with it if we're trying we're going to demonstrate something here and in order to demonstrate is that a sound argument has no false premises now that goes without saying it's not something that that should cause you any trouble substantively but the the thing that we're trying to exercise here is our ability to construct a derivation to construct an argument so what we're going to do is to do this we say all right let's say we've got a sound argument and then we're going to try and derive the that it has no false premises on line three so there's a bunch of there's some ringers in here and there's some other things that you that you can use and the order matters of course so what I'd like you to do is to now try your hand at that one by filling in a number on each of those lines that will reveal the structure of the argument the derivation part knee right once again someone who hasn't offered suggestions before about either the problems you've been working on someone else yeah two five three so suppose to by the definition of sound a is both valid and has to has only true premises premises over there - all right and then five because as a has only true premises it has no false premises and then three a is a sound argument okay any any commentary on this Lizzie two four five but it alright but I want to focus on one thing at a time what do you say was less than senator yeah but you but you use the definition of valid in that particular place okay yes sir Boston so your supposition should be a reckoning then you want to conclude right though the way to proceed with this kind of dumbest derivation is to say you're supposed to say all right let's let's demonstrate that a sound argument has no false set premises all right let's start with this suppose we've got a sound argument so the first line should actually be three how many got that how many did that way okay good first line you can erase what the backs of these things up oddly the first line should be three so let's say we've got a sound argument that's that that's the point here suppose that a and we're going to call it a suppose we've got a sound argument now we're able to do what now we can use yes yes now that we know C now by the definition of sound a is both valid and has only true premises and then what somebody else yeah because a has only through premises which is what we just said in 2a has no false premises which is what we were trying to prove q e anyone familiar with those letters what do they mean you could tell me what they mean you don't have to give me the Latin yeah its quod erat demonstrandum which proves what we were trying to prove that's that's what we're trying to do and we got it does anybody have any questions about this 3-3 line proof in this technique the way to go about these things and least this step the way to go about this one if you're trying to demonstrate something all right let's say we got a sound argument well since it's a sound argument you've got it's valid and has only true premises but it has only true premises then it doesn't have any false premises that's just what that means so that's hence that order does anyone have any to see any problems with this okay any questions at all about anything you need to store up your questions for these last minutes of the class because I'll answer it all know about logic you have any questions at all as you work if anything gets frustrating please raise questions in in the discussion areas up in my courses that way you may be able to continue working a little bit more quickly than if you had to wait until class time all right so long see you Monday good luck with the cafe

Channel: Jack Sanders

Views: 42,942

Rating: 4.9275122 out of 5

Keywords: Symbolic Logic, Philosophy

Id: ExE8ucCfmH0

Length: 69min 10sec (4150 seconds)

Published: Sun Sep 11 2016