# "Can Ordinary People Understand Advanced Logic?" Why? Radio Episode with Guest Otávio Bueno

## 3:27AM Oct 16, 2020

### Speakers:

### Announcer

### Jack Russell Weinstein

### Otávio Bueno

### Keywords:

### logic

### classical logic

### people

### logician

### contradiction

### inferences

### reasoning

### understand

### logical

### philosophy

### mathematics

### talking

### question

### symbols

### true

### premises

### form

### systems

### weinstein

### inconsistency

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#### Why philosophical discussions about everyday life is produced by the Institute for philosophy and public life, a division of the University of North Dakota's College of Arts and Sciences. Visit us online at why Radio show.org

#### Hi, welcome to wide philosophical discussions about everyday life. I'm your host, jack Russell Weinstein. Today we're asking a topic of whether ordinary people can understand advanced logic. I know a philosopher who enjoys telling everyone that they won't understand his research. When people ask him what he does, he groans exhales and waves the way it's really complicated. He says, No one understands it. It's a really impressive act of theater actually, very few people ask a second time. Nevertheless, I've always found his behavior frustrating, not because I have a particular interest in what he does. But because he's telling interested people, that philosophy isn't for them. By pushing them away by grunting and miming exhaustion, he is suggesting that he's smarter, better and more capable than they are of understanding the tough stuff. In contrast, when people ask me what my research is on, I asked them whether they want the short or the long version. I like talking about what I do. And I also believe that given the right explanation, and a little attention, anybody can understand anything. This is the premise of why radio itself, the show assumes that if someone doesn't understand an idea, the failure lies in the explanation, not in their intellect or lack of experience. The show is first and foremost a project of translation. My goal is to find the right language to explain very complicated ideas, so that we can all share together in the various expanse of philosophical thought. Today, however, we are going to put my belief to the test. We're going to talk about symbolic and mathematical logic, we're going to dive into the wild world of symbolic equations and existential quantifiers. And we'll do it on the radio without any visual aids at all. We are going face to face with philosophy at its most technical. every subject has a technical side, it's essential to progress. Specialists need to filter out extraneous ideas, establish boundaries for their research and cultivate precision. I tell the mechanic that my transmission went out, she responds by checking whether the dynamic or kinematic viscosity of the ATF is appropriate. I tell the baker that my bread isn't crusty enough, he increases the direct heat to improve sugar carmelization. These aren't jargon salads designed to keep out the great unwashed. They're precision tools developed over generations to establish unique research methods for every task. But technical language doesn't just isolate the experts. It's also a democratizing force. computer programmers and rural Bangladesh can work with their urban Swedish counterparts because they share the same language. And the poor daughter of a drug addict who clawed her way to the first cello uses it as a common theoretical framework to work with the privileged oboe player who grew up in a mansion. Sure, the wealthy oboist will find it easier to get access to school and musical instruments. But once they both get where they're going, they're in the same place. Regardless of what clothes they wear, how hard it is at home, or what prejudices they face. Once they enter into the world of technical expertise. They're both doing the same thing. Advanced logic does this for philosophy by connected claims across the disciplines. Its focus on form, not content, doesn't describe facts or impart new discoveries. It maps how one idea leads to another. It shows that fields as disparate as ethics, computer programming and chemistry all share something in common. The patterns of reasoning that makes some conclusions necessary, and others unwarranted. We'll get to all this. We'll also get to the big questions. Is logic created or discovered? Can there be a language free of error? Is there a category between truth and falsity? We've asked these before, albeit in different forms. Logic, like all philosophy investigates the human experience. It just looks really alien when it does. So you might fairly ask, if it's so difficult to understand, why not leave the technical stuff to the experts? Why talk about it here on the radio? My answer is the same as the mountain climbers, because it's there. I'm genuinely curious as to how deep into the philosophical rabbit hole we can go. I want to see what why radio can do.

#### It is important to celebrate the technical rather than abandoned to the world of things not spoken of. It deserves to be considered aesthetically, creatively and emotionally. Language is always human, no matter how narrow the subject, it can history with revealing culture, expectations and prejudices, and the more stripped down it becomes, the more portrays the values of those who created it. Today's episode will probably have the narrowest focus of all 100 episodes that came before it. But it's questions will also be the most all encompassing logic. The logician will tell us talks about everything. And now our guest ottavio bueno is Professor and Chair of the Department of Philosophy at the University of Miami. He's the author of two books in almost 160 papers on the philosophy of science, mathematics, logic, metaphysics and epistemology. ottavio. Thanks so much for joining us on why.

#### Thanks for having me, Jack.

#### We've pre recorded the show. So we won't be taking any questions. But if you'd like to send your comments, tweet us at at y radio show, post a comment on facebook.com slash y radio show or visit our live chat room at y radio show. org. So a tabia. Straight up let's deal with what I suspect many of our listeners are feeling at this very moment. Why is logic so scary?

#### Well, that's a very good question, jack. I think a lot of people tend to be scared about just the formalism and symbols. There are people who are feeling comfortable with mathematics and mathematical ideas. And of course, logic gets into this mode. No, I thought your introduction couldn't be better. Because in many ways, logic is indeed a democratizing tool. because it brings together everybody in the sense that it doesn't matter what you're talking about, it doesn't matter. The details, whether you're engaging with ethical issues, whether it's talking about how to properly check your bank, bank accounts, you need reason, we need to be able to reason, and that's an important trait that we all need to develop and acquire. Now, a lot of people may think, well, don't we already know how to do that? Why do we need object? And then when, if you're presented with a foreign language, a bunch of symbols, that don't mean anything to you, people get scared, right? We often get scared with things that we do not understand. So I think the first and most important aspect is you understand that the scariness comes, perhaps with not being properly presented with the subject matter, right. And in my own case, as an undergraduate student back in the university universe of San Paulo, in Brazil, I had this wonderful encounter with a person who arguably is the most important Brazilian logician who was teaching at the time there, Professor Newton Acosta. And he show up in one of his lectures, typically, those were given to more advanced students. But I was curious. And I, of course, I heard about him and I want to see what was it like, and it was this life changing experience, where he would show how logic connects with everything, right? every single aspect from science, from understanding, even the basic kind of philosophical reasoning, and then learning the tools, the technical vocabulary is what's needed for you then to properly probing through the details. Let me ask

#### you a question. You said something in passing. That really struck me, you said, many people ask, Well, why do I need to learn logic? Because I already think these things logic is, is looking at ourselves in the mirror in a certain sense, right? So to what extent is logic about trying to describe how we think and to what extent is logic trying to improve how we think and I'm thinking about, right, I listened to music, and I can appreciate in like it, but then if I have a conversation about why this song is good, or why this musician is outstanding, it helps me think more about the music is logic, the same thing? Is it? Is it just describing, or does it refine our thoughts? Perfect.

#### I think it does both things, right? In a way. Even if the, if we go all the way back to the very beginning of the conceptualization of logic as a as a field, all the way back to Aristotle and the ancient Greeks. They already had experience looking over different kinds of reasoning and reasoning patterns all over the place. Right? Some are okay, as you said in your introduction, some seem to be valid or just seem to be completely mistaken. So on the one hand, you want to capture something that people are already doing. They are reasoning, but sometimes a reason properly, sometimes you don't. Right. So we think the reason properly, as long as from true premises, you had to get a true conclusion. It will be improperly from true premises, you get a false conclusion. Right? So then the question is, is there something about the form the way in which your arguments, your premises, what you're assuming, and your conclusion, what you're trying to establish? Is there some relation between the two that would say, Yes, I'm properly inferring from these things that I already think are true. These are the things that I think is true as well. So sometimes people do make all kinds of mistakes, right. So in that sense, logic does have a kind of normative component of refining, in improving the way we reason. But also, he has a descriptive component in the sense that it is supposed to capture patterns of inference that we all do, right we all engage with, and we all invoke. So it needs to have these two components. In order to do the work,

#### we're going to focus in a minute on the more symbolic and technical end of it. And I know that you use a metaphor frequently, which I'll ask about in a second about the boiled egg. But when we talk about logic in normal, everyday life, we use all of these different terms. Right, we talk about thinking logically, we also think about talking about critical thinking. We talked about being reasonable and rational. To what extent are these names for the same thing? And to what extent when we're talking about logic, and its most technical sense? Is it different than these other sort of more familiar ways of talking about whether or not we can figure something out? Well,

#### right, okay. Well, there are, let's just make a couple of distinctions, right. So you can think about what's called mathematical logic as a entire domain of investigation about formal systems and their properties, and whatever features they have.

#### So what's a form, what's a formal system,

#### so we will be a system in which you introduce a special language, a formal language with symbols. And the idea behind it is that the meaning of each of these symbols is either stablish in advance, or it's going for the what's called the logical constants. Right? So those will be your primary logical symbols, such as, for all for some, if then, and or so those lots of words that play an important influential role.

#### So let me just let me let me just help folks understand that that what you're suggesting is that there are all these different terms that we use, right, that we don't think about it read, let's say, but you can, you can think about it logically in its relationship to other people. So there's not red or, and red. So so blue is not red, but the tie is blue, and red, and some are red, and none are red. And if you're wearing red, then you shouldn't be on the television camera. Right? So these logical constants are ways of thinking about other things so that we understand, I guess, the relationship or the amount or when it becomes relevant, right? That's right.

#### I mean, in a formal system, this special words would then be studied in terms of, well, what are the relation among them? And so you want to say, if you're reading, if you're wearing red, and you're, you're have a shoe, then you have a shoe? I didn't want to say okay, so it seems that that should follow, right? If the two premises the premises are true, the conclusion would have to be true. So you, you have a valid inference, right? In that, and then you realize, well, it doesn't matter, then if I'm talking about wearing red shoes, or if I'm talking about, if I'm going to the movie, and I'm having apartment eating popcorn, then I'm eating popcorn, right. So it doesn't matter what the specific content of these sentences is, but the relation that they have, with one another, I did relation of form. And in that gives us a sense of what follows from what think the notion of validity in the formal systems I was mentioning earlier, or systems that have been developed to study in the most general terms in the most general terms, what are those inferential relations? And what properties the systems have? Now you can do that purely mathematically, right as a just a branch of mathematics, which mathematical logic in many ways is

#### just 111. more one more clarification. So when you talk about an inferential relationship, what you're saying is, if I know something, then I know something else, right it. So if john is taller than Sally, and Sally is taller than Bob, I can infer that john is taller than Bob. And what you're saying, if I understand correctly, is that the value of this model of this formal system is that I can substitute john, Sally Bob with Fred Muhammad and Juan or I could substitute it in terms of building dog and water bottle, right that it's for logic, it's the relationships. And that's what an inference is that if I know something I can infer or I know something else, because I know that first thing,

#### right, or if something is true, and something else is also true, then there's a third thing, that it's also true. Excellent. Right? So yes, that's the NDA the plenty say, Well, why should we bother with that? Well, because you may not realize that certain things that are true, some of them, you may or may even believe, also have implications that you may not be aware of. Some of these implications may be surprising to you. Some of them may actually be in conflict with other things that you think you're at. You also know, in which case, you're running into trouble, right? Because you get inconsistent inconsistency attention between the things,

#### you think you know, and inconsistency is, is the worst thing in logic, right? I mean, that that's, that's, that's though Oh, no, we have to start again, right?

#### Right. And inconsistency. Again, in classical logic, you will be a statement of the form. I am ready wearing red, and I'm not wearing red,

#### right can be true at the same time, it cannot be true at the same time.

#### And the problem in classical logic with that is, if you have a contradiction like that, you can derive anything. So for example, I can prove to you that the moon is made of green cheese, if I have a contradictory premise. And it's a simple five step argument that we'll get there. And that is bad, because it's what we call trivial ism. Right? So we get everything being true. In from from a contradiction, and the problem is, we cannot do inquiry. At that point, the whole point of engaging in inquiry is to figure out what are the things that are true? What are the things that are not which things I should believe with things I should not need to draw a line? In? If everything follows from the things you believe, assume that there the thing among the things you believe there is some inconsistency, then you're unable to do that?

#### You know, it? This is this is one of for me, the coolest ideas in logic, right that, that the second you have this contradiction, the very idea of truth falls apart. So if I am red and not red at the same time, well, I can say anything, right? I can say the moon is made of green cheese. I can say I'm married to Jennifer Lawrence, you know, I can say anything, right? Because it's this, this, at least in in the classical logic, it's this, this exclusion that you can't have truth or falsity, because truth needs to be consistent. And I love that because it really goes to the foundation of how we assume the world to be logical, right? If the if a truck can hit me and kill me, then I know that if I get hit by the truck, I'm gonna be killed and I avoid the truck. But if, if the truck can kill me, and I'm still alive, I can do anything.

#### Right? It would be very good if you could pull that off.

#### Yeah, I'm working on it.

#### Okay, good.

#### So I had asked you before, um, so these this is the formal model, but then we use these other notions like critical thinking and reasonable and rational, how do those relate to the formal systems that are the more sort of very stripped stripped down mathematical symbolism that you're talking about? Right.

#### I think that's a crucial point. Because, in many ways, when we're engaging with critical reasoning when we're inferring things in ordinary language, In other words, when you're not using just a formal framework on the formal tools, we want to ensure that what we're reasoning and how your reasoning is valid. So the way I like to think is that these formal systems give you a pattern, e juice against which you compare your reasoning, and you decide to try to figure out whether your reasoning is acceptable or not. So in a way, the formal tools can be thought of as validating the informal ones. On the other hand, the informal ones are the ones that we all humans use.

#### And by informal, you mean, we're not using symbols, but I'm, you know, I looked through the window. And I see, Ashley is there, and I can see the top of her head, but I can't see the top of Skip's head. And so therefore, I can infer that Ashley is taller than skip, even though I'm not using any symbols, I just have regular. everyday thinking about things, right, no symbols, just this is how I think and that's what you mean by informal because, of course, most people who are listening are going to think of that more informal terms, right? I just think about stuff and I don't really map it. So if I'm going to think that way about the relationship between Ashley and skip, then I am formalizing the logical, the logician actually considers that informal because it doesn't use the symbols.

#### Right? That's right. So I think and I think you're quite right. Most people would even consider already formalize what is going on in a critical reasoning context, the one you just mentioned. Right. And, and, and that's an important issue, because we have in a certain way, there are levels of idealization going on. Right, we have the level in which you just have a good gut feeling and said, Yeah, Ashley Staller. I see the top of her head and you just say it, right. And then someone probes But why? Okay, how can you tell that? Right? Of course, people can press and said, Well, maybe skip is just on his knees. Right. And, and so No, No, he's not. And you tried to give some reasoning. And at that point, you can start to formalize the the argument you gave as to why, you know, Ashley Staller in the way that you just sat, right to compare consider topper era cannot see the ad, I cannot see if skips. And that allows you to give a little more structured argument, right? When you say, well, the, the taller person would, would show up as being tall from from this distance. Therefore, since that's how Ashley shows up, she's taller. Right? And, and this is an issue without symbols, right? or formal symbols done entirely in English. And then you may want to ask, well, but is that valid? Is it the case that the true of if these premises the assumptions you start from were true, the conclusion would also be true? How can I establish that, and that's where the formal stuff can help. Because you need to go beyond the specific content of Ashley and scape and their tone is or not, and think about all the possible relations between the statements in question and the logical form.

#### We're going to have to take a break. But when we get back, I want to follow up on this a little bit. And I want to talk about a phrase that I think will help people understand that that really encapsulates all this, that logic is truth preserving, and what does it mean for something to be truth preserving? I also promised that I would ask you about the boiled egg and I haven't yet but I consistently lie to my guests, so you should get used to it. And then we'll talk a little bit more about some of the philosophical things and then get into this question of whether or not truth and falsity are really exclusive. But for the moment, you are listening to a Tavi of widow and Jack Weinstein on why philosophical discussions about everyday life. We'll be back right after this.

#### The Institute for philosophy and public life bridges the gap between academic philosophy and the general public. Its mission is to cultivate discussion between philosophy professionals, and others who have an interest in the subject regardless of experience or credentials. visit us on the web at philosophy and public life.org. The Institute for philosophy and public life because there is no ivory tower

#### You're back with wide philosophical discussions about everyday life. I'm your host, jack Russell Weinstein, we're talking with a toffee of one oh about advanced logic and trying to figure out if, if it's for the rest of us, or if it's just for specialists. And every time I think about logic, I always think of the very first argument I ever had with my daughter. And of course, it was also the very first argument I ever lost to my daughter. She was about six months old or something, she must have been a little over two is learning to talk, but she was still in the backpack, you know, when we're to go take the dog for a walk and, and it's starting to rain. And she said, Park, Edina, go Park, right, she wanted to park and I said, Edina, no, sir, we can't go the park. And she said, Dina go Park. And I said, Edina, it's starting to rain. We can't go Park. And there's a pause, a beat, and she says, umbrella. And those those, there's nothing to be done. Right? She won the argument. And then four or five years later, we're sitting at the table, and she's looking at the newspaper and this big thing about war. And she asked what it was. And I told them, she said, and she asked, Why, why, why is there war? And I said, it's probably a little older than than six at this point. But she said, Oh, at the end, you know, it's breakfast. It's really complicated. I don't want to get into it. And she looked at me and she said, But Dad, you're a philosopher, it's your job to explain complicated thing. You know, just my listeners know that once I get started talking about a deal, I won't stop. But But the question that I have, and that leads to is, is logic discovered? Or is it created? Is logic something that once philosophers started trying to figure out what reasoning is, they saw the the structure of the universe the way that say Newton saw the structure of the universe, and they were basically journalists, they were reporting the necessary connections of things between the universe? Or is logic created in that it's a system made by people that can improve some things and other improvement, prove other things and focus on some things and focus on other things? Is logic a human invention? Or is it a natural law, so to speak?

#### That's a great question. And of course, it's one that people disagree about there. Depending on the interpretations you give to logic. There are indeed people who think that there are those who think about them as abstract logical forms, and we are uncovering them. So so we just in that sense, describing the patterns that are already there, even if no human being ever exists, or ever thought of them. Others deny that and would say no logic, something just like mathematics, just like fiction, or things we made up.

#### Hold on a second, you aren't you aren't, you're telling our audiences that we made mathematics up that some there's a lot of high school students who are now sitting going I told you, right, um, what do you mean that we made mathematics up? Isn't one plus one equal two? Isn't that isn't that like, equals mc squared? That

#### it is once you define the terms in these ways, but there are different ways in which you can define it. Right. So we are all we often are taught mathematics in a way that I don't think there's justice to what mathematics is, mathematics is actually a very live subject matter with a bunch of open questions, quite issues that sometimes lead to surprising discovery sometimes leads to nowhere. There's a long, sophisticated history of failed attempts, successes, development of new theories. None of that is presented when mathematics is taught. It started, okay, here's what you have to understand. And you just given a bunch of rules, that you've no idea where they came from, why they are set up in these way. And you're told, okay, those are truths. And they are necessarily so and it was always be that way. In many ways, logic is often presented along the same mode being related to mathematics. Well,

#### and and just again, to, to stop there. There is a real argument about the connection between logic and math very famously, right. Bertrand Russell and Alfred North Whitehead argued that, that if I understand it, right, that that mathematics was at route logic, and once they sort of proved that one plus one equals two, the rest came, is that right?

#### Yeah. So they are part of a certain movement that emerge in the late 19th century, early 20th century, which was a large assist movement, right, the idea that you could reduce mathematics, ultimately to logic plus some definitions. And of course, if the logic says program can be good can be carried over entirely. The there wouldn't be any sharp lines between logic and mathematics. So

#### is that why logic now looks the way that it does it has variables, it has symbols, it has equations and deductions look like mathematical problems, because the form of mathematics is a useful form of discussing logic, or is that just sort of the accidents of history?

#### No, I think, at that point, right, in late 19th century, early 20th century, people were concerned with the foundations of mathematics. Because if I could just mention briefly mentioned, so, there was a big trouble in the 17th century, when the calculus was introduced. And it's a beautiful theory. But he had infinitesimals, right, he had magnitudes that are positive. So they are different from zero, but they are larger, smaller than any positive number. In that sense. They, but because they're so small, that could be considered to be zero.

#### So So when when calculus was invented, this is this is this is brand new to me, when calculus was invented, there was this problem, because the numbers they were dealing with were so small, that they might as well be zero. And you could treat them as zero,

#### exactly. And you're finding that they're evasions at the time, for example, that the revisions of the the tape of certain polynomials, you'll find at certain points in the lines of the derivations and look, because the infinitesimals are different from zero, we can divide by that. And then a little download the derivation, you said, you have some infinitesimals left. And it's a but because they're so small, you can ignore them. So in the same derivation, you have a number being treated as different from zero and identical to zero.

#### That's, that's amazing. And that's amazing. And in a couple minutes, I'm going to ask you whether that means that things can be true and false at the same time, but so that they so in the same equation in the same process, they treat a number as more than zero, and as the same as zero, and they use it to divide it, and then they ignore it when they don't want it correct. Sounds like it sounds like a bad boyfriend.

#### It looks like though, the what's amazing is both live Nate's and Newton, correct correctly, got all the proper theory about differential and equate differential calculus on that basis. Now, Barkley at the time, complained bitterly about this. He says, Look, this is a disease, we can't we cannot have a mathematics done that way. And it was an ongoing problem until all through the 18th century, and the 19th century, and mathematicians decide to clean up their act. And then you have people such as Khushi, via straws, and, and others who've shown Okay, we can recast all the proper inferences that we need in the calculus. Without infinitesimal,

#### I just have to say that I've never enjoyed hearing anything more than the phrase mathematicians have to clean up their act. I think that in itself is it was worth this entire conversation.

#### But that's Look, that's, that's how mathematics has been. So when students are presented the calculus with all the clean up version, and they think, Oh, these came about that way, already fully formed, it's entirely inaccurate, you have to have a sense and look, there are lots of false starts. And what happened was, so when these mathematicians cushy and others clean up the the calculus, basically they showed, okay, as long as arithmetic is properly understood, then analysis, so the real numbers and all the stuff you need for the calculus will be on good footing. Now the question was on what does arithmetic, right, your theory about the natural numbers 0123 and so on, addition and subtraction, on what does that theory rest? Now frager Russell Whitehead, people you just mentioned thought, while we should rest on logic, so the and that's where the idea of using losing logic to Grand mathematic scheme came in, and that was the project of the Lord Jesus.

#### So this is this is this is actually something very familiar in the history of philosophy and I think it's really important to spend a second and think About philosophers like to build things on foundations, right? This comes from Descartes. And so if if if the, if the third floor wants to be strong, the second floor has to be strong, the second floor has to be strong, the first floor, etc. And so what you're suggesting is that, in order for calculus to work, it has to have it has to be based on arithmetic, and arithmetic has to be strong. And so the question the philosopher, and in this case, the logicians are asking, Well, okay, how do we know that arithmetic is strong? This is an interesting question that I think most people don't ask, how do we know that one plus one equals two or three times three equals nine? What makes arithmetic strong? And so Russell and Whitehead in particular frager is talking about ideal language I'm gonna ask him about in a little bit, but but Russell and Whitehead, they basically say, arithmetic is strong, because it's logic and logic is strong.

#### That's right. So the trick was to show that you had a proper conception of numbers, which is what arithmetic is about, that could be formulated out of logic and definitions alone. Now frager was the one the first one to have that idea, right? 10. And if you think about it, I bet that most people don't know why one plus one equals two,

#### I want to say because there are very few people I'll talk to who might know this off the top of their head, you may be one of them. In in Franco's little book, the foundations of arithmetic, which is an awesome book, he actually defines what a number is, right? So our audience, right? Think about? What would it mean to define the number one, how would you do that? And And do you remember what his definition is? Because I don't remember exactly. But I remember being Hmm, I'm not sure I expected that.

#### Yeah, no, it's beautiful. Yes, of course. So he, remember he has to start only off out of logic plus definitions, right? So he says, Okay, so let's consider, and if, if you have a theory, you have concepts, right? So when, of course, we, we do have some concepts. And he said, Look, let's consider them the number of things. And so we're trying to define zero, right? So you're gonna consider identity first. Things are identical to themselves. So that's a logical notion.

#### 00 equals 00 is zero.

#### Well, but before we don't have zero yet, right? So we have identity,

#### okay. So something something is the same as itself

#### itself, right? So now consider the negation of that when negation is also a logical notion, but of those logical constants. So consider how many things are different from themselves? Right? So consider not the number of things that are different from themselves?

#### No, well, if if Okay, so I'm gonna, I'm gonna do logic in the place of my audience, because I have to advocate for my audience. Let me figure this out. You said that everything is identical to itself, right? Therefore, I would assume that nothing is not identical to exactly Is that how it works?

#### Perfect. So the number of things that are not identical to themselves? Zero. And that's exactly the definition of zero that frager gave, write notes that the only thing you have is you need identity and engagement. Both are logical notions. And you define zero out of that, because there's nothing who is not identical to itself. Right. Now you can see there. Fair enough. Now we have 00, is the number of things that are not identical to themselves. Now, consider the following. How about the number of things that are identical to zero? How many things are identical to their 00? Just one, that's the new one.

#### So I just, I just want everyone to see the magic that we just did. We just created something out of nothing, right? We created one out of zero, right? The number of things that are identical to this is this is this is beautiful, right? When people talk about logic being beautiful, I think this is what they mean, this incredibly simple idea of these two definite of this one definition, leading to two definitions leading now to the numbers zero and one, right. I mean, that was that was, that was that was ballet. It was so it was so easy and obvious, once it was said,

#### exactly. That's logic for you. And that's how he should be. And, and, and he does have and in fact, some of the most beautiful results in logic, have that feel. You need to get to them. But once you get to them, they have that the same kind of elegance that you find in a good artwork, right? There's an internal dynamics, that's a motivation behind it, and suddenly you can see what's going on. And it's just as rewarding as discovering something that was already there. Right? Now frigga thought that those things were already there. Right? I happen to disagree. I think he was making them up, right in the sense that he was offering a definition. And one that actually worked very well, given where he was starting from. And in that way, you can actually ground arithmetic in logic. And as a result, there is this very close connection between logic and mathematics. Right. And so that's, that's the part of the historical motivation as to why these two fields mathematical logic, and mathematics are often so closely intertwined.

#### Right? So that the argument was, we want calculus to be founded on a strong arithmetic, we want arithmetic to be founded on logic, because its strongest. Well, I will ask you, is logic strong? x is true, always true, is false, always false? Is is does logic serve the the solid and certain grounding that these folks wanted?

#### That's good. I think these folks, right, that we have just mentioned in fraggin, Russell, they thought of logic in what I would call a more nice way, there is just one true logic. And that's it classical logic. As it turns out, I think the proper way of thinking about these is in terms of a logical pluralism that there is more than one logic, more than one answer to the question, what follows from what? And I think the most important developments of logic in the sort of second half of the 20th century was the realization that there is actually a plurality of logics, there is more than one answer to what follows from what now here, and that's where the regimentation with the formal symbols was important. Because once you lay down those symbols, right, we mentioned a minute ago, that from a contradiction, everything follows according to classical logic, right? So if you have a and not a contradiction, I'm red. And I'm not Rand, it follows that anything, the moon is made of green cheese. Now you look at an inference like that it is a valid inference, according to classical logic, but it seems irrelevant. So look, wait a minute, what does blue cheese have to do with being read right? Or I'm being read, there's no connection between the premise and the conclusion. And at that point, you may say, Well, that's right. This is an artifact of classical logic. Maybe that's an inference that we should not apart. And by probing that further, logician start to break apart classical logic and challenge some of the principles that have been up to that point accepted as necessarily true.

#### I want to I want to stop again, because you said something, again, a term that is incredibly interesting. And I and if I understand correctly, would really help connect the conversation we're having to debates that we all have all the time, which is this notion of relevance, right? relevance is a huge issue in today's life, what is prejudice, prejudice is taking an irrelevant characteristic, right? So I want to, I want to hire a bus driver, and a blind person comes and says, I'd like to drive the bus and you say, Well, I'm afraid you can't, because you have to see in order to drive the bus being blind is relevant. But if an African American person comes and says, I'd like to drive the bus, and they say you can't, because you're African American, that's an irrelevant characteristic. It has nothing to do with driving a bus. And so is relevance in logic, this same basic thing is it just basically, this has nothing to do with that. And this, this debate about this discussion or example I used about prejudice is really at root, this debate that you are talking about now, which is read has nothing to do with blue cheese, and therefore, classical logic doesn't really allow us to understand the concept of relevance. I mean, is it is it connected in that way? Well,

#### it is connected, of course, there are different ways of them fleshing it out. What that relevance is, from a logical point of view, but it is certainly connected. It's basically the idea well, there's nothing to do A with B. Why should you know we want to infer one from the other if they are not connected in any, any way. Then you will was an attempt to resist what was perceived to be some irrelevance of classical logic that some non classical logics were created. My two said, Okay, what would what would the takes? Because it was also, it's a deep held assumption throughout the history of philosophy that we should avoid contradictions at any costs, right? And most philosophers, perhaps with the exception of Hegel, and would would tend by that, that claim by contradiction. If you get a content inconsistent, that's the end of it, right?

#### And we see, and we see that in everyday life, right? If you're trying if you're, if you're if you're in a monogamous relationship, and someone has an affair, well, then you've contradicted it's no longer a monogamous relationship. And so so contradictions in our day to day lives have the same effect, it destroys the things that we want to preserve. That's right.

#### So but here's the flip side, right? Have you consider whether your own beliefs may not harbor an inconsistency, right? We believe all kinds of things. And the more you think about the kinds of things you believe, the more likely is it that you end up with, unbeknownst to you having contradictory beliefs? Right? And, and even if that were not the case, if you just think about in contemporary technology, right, you're creating a huge database, right? for medical diagnosis, people getting the information into these databases, more likely than not, that those databases are going to be inconsistent, they're going to be inflammation of the form. Well, if the patient has such and such features you treat these way, if the patient has such and such exactly the same features don't do nothing. Right? And then what do you do in cases like that?

#### So with enough information, there is inevitably going to be some form of contradiction. And and, and that's just a fact of reality in in, in learning stuff in gaining stuff, right?

#### So and then what do we doing according to classical logic, you have to, you're done, right? Because now the database is helpless? Everything follows from it, you're done. What What can I do? You have to go back and find some bitten and have inflammation and get rid of it? How can you reason if it's the whole thing is inconsistent? Right? And that's where, where paraconsistent? Large what what it's called paraconsistent. logics, logics that say, look from a contradiction, it should not be the case that everything falls. That's what motivated to say, well, that's a feature of classical logic that we need you we think, right? And not because we believe that some contradictions are indeed true, right? Although there are some people who have those beliefs, but most people don't and who say, look, having a contradiction is indeed a sign that something got mistaken. The problem is, we need to be able to reason through those inconsistency to find where the mistake is, in classical logic doesn't help us because you can derive anything, right? So we need to then to be able to introduce a distinction between inconsistency in triviality, right being between having a contradiction and everything following in to do that, you need to block the inference that from a contradiction, everything follows. And that's exactly what paraconsistent logics do.

#### Okay? So so so we have this notion, right? of consistency that we started out from very beginning and then contradiction. And then we had this notion of relevance, which we talked a little bit about. Now, we have this thing called triviality. Now, am I wrong in thinking that it has something to do with the infinitesimals? In the zeros? And well, we can treat this thing as zero what what makes something trivial in logic? That's great.

#### Well, yeah, we'll make it trivial. We've from it, you could logically derive anything whatsoever. Right? So in effect, classical logic identifies contradiction and triviality. That's a distinction without a difference from a classical logician point of view. Because of that inference, that from a contradiction, you can derive anything. So what do you want to say is, well, maybe we can tolerate contradictions as long as they do not lead to triviality as long. So

#### can you give me an example of a triviality that would help us understand this?

#### Yeah. So let's see. So the exactly the same example I mentioned about the diagnosis or the treatment, right? So if, if in your database, you say, Well, someone who has such and such symptoms do the following are in elsewhere in the database? Is it? Well, if someone has those symptoms do nothing. Right. So now you have a contradictory bit of information, do something, do nothing, right? And now, what working? Can you do in that case? Right? So if your logic is classical, then you're there's nothing else that you can reasonably reasonably do. Because you can, you can do something can do nothing, you can be ignored, nothing is now logically warranted, or in the sense everything would be because everything would follow from that contradiction, right?

#### But you don't have to assume that that's the proper way of reasoning in the presence of a contradiction. What would be the proper way not to derive everything, right. So what a paragon system logician would do is you realize that some inferences in classical logic are not do not hold in general, they actually hold very well, as long as what you're dealing with is consistent. And of course, that's the primary source of motivation for classical logicians. However, if what you're dealing with involves some kind of inconsistency, then we need to reason in a little more restricted way. Because what basically the Perkins the the non classical logics, do, they restrict inferences that are considered to be classically valid? And they say No, they are not. They do not produce new inferences, right? They don't add, they subtract inferences.

#### So let me let me let me see if I understand this. Because of course, this is really abstract. So let's say we have this database as your as you're describing. And it there's this you know, a person has the symptoms, do this person has the same symptoms do nothing. And there's a contradiction. Now, if I were a doctor who subscribed to classical logic, I look at the information that came out of the computer, and I raised my hand, if I go, I can't do anything. Medicine is dumb, and I'd walk away. Right? But But if I'm, if I'm a paraconsistent, Doctor, then what I do is I say, look, the database is still there. Let me use all of the reasoning. I know, to figure out all the information around it, I'll sort of put aside I'll ignore the contradiction. And I'll reason my way through it to see first of all, maybe what I should do. And second, why I made the error in the first place. And so the para consistent doctor is going to use all of the reasoning except that little bit it's going to he or she is going to subtract that little bit. And classical logic wouldn't have let us do that. And the new logics would is that is, am I getting this? Right?

#### Yes, that's exactly the idea.

#### I'm trying to formulate a question here. And I'm not I'm not sure I'm gonna get it right the first time. But it seems to me that for how long did it take the logicians to figure out that everything didn't have to be perfect. That's not the right question. But, but it seems like it seems like what's happening with the database, and, and the doctor is that I know the question. Okay, so so so it seems like there's a different form of reasoning going on, right? The, the classical logician is doing what we would call deduction, right? principal, to principal to principal to principal, this is the answer these The only things you consider, is the doctor in the more modern doctor doing what scientists and logicians would call abduction. Are they Is it a different kind of reasoning, that doesn't go line to line to line to line to line but rather, makes the best guess and the best information given a plurality of evidence that isn't necessarily in a perfect line? I mean, is that is Do you understand?

#### No, that's a great question. Right? But as it turns out, the non classical logics I'm considering I should did that Piff. right because what they do is you have all the valid inferences according to classical logic and they take out a few of them right. So is whatever they are Say is valid is still valid according to classical logic, what they say is that classical logic consider certain things as valid that should not be considered such as inferring everything out of out of a contradiction. So, these the non classical logicians here still reasoning in perfectly classical ways except for those bits of inference that they deemed not to be relevant, right that such that there is not a proper connection between the premises and the conclusion. Now, there is indeed, as you noted, different forms of reasoning, inductive or abductive forms of reason. And they are of a different kind in the sense that they need not be truth preserving the prep the truth of the premises need not guarantee the truth of the conclusion it just made, okay, likely.

#### And I told people, we were going to come back to this, and I'm glad we're doing it. Truth preserving means that if you have a true premise, and you follow the right principles, then or the right logical form, the form is valid. The conclusion has to be true, right? So it preserves truth throughout the argument. And the problem here is that logic may that that do how do we keep it truth preserving or we don't necessarily need to keep truth, truth preserving this is this is one of the other key concepts right? So so so please continue.

#### Right. Yes. So. So the the route I was mentioning with Paragon system logic, was to keep the truth preservation business, right. So you're preserving show classical logic. And because you want, you're not necessarily in the business of creating new truths, but you're in the business of figuring out, given the truths you already have. What are other truths that you should already endorse? Right? Of course, if you know something, if you know something, you automatically know something else. And so you're trying to figure that out? What is the other thing that you know, it's right. And we many kids may not be aware of that. Right? And in fact, mathematics is done exactly. In those terms. It's based on deductive reasoning, out of promises and principles, mathematical principles. And it comes in often came as big surprises when mathematicians realize, Oh, I didn't realize that that house Oh, hold, I intend. This is a matter of what we call logical opacity, right? It's opaque to us. What exactly are the logical consequences of things that we already have? and logic is crucial to help with that? Because he helped us figuring out what are the other things that follow that we may not know yet or may not realize that they follow?

#### Now, in contrast,

#### if let me just again, use another example. So right, this, people decided that all human beings had certain rights, let's say the right to vote, right. In a democracy, all human beings have the right to vote. And they also knew that women were human beings. But yet, they didn't follow the logic through and say, Well, if women are also human beings, then they also have the right to vote, they had to go through all these processes to see the contradiction, or to throw away the trivial things or the things that that interfered with the process, right. And so that's what you're talking about, you're talking about that when you know that human beings deserve to have the right to vote. And you know, that women are human beings, if you can't end up with the conclusion that women have the right to vote as well. Something is going on, right. And the contradiction isn't going to throw out the voting stuff. It's just going to throw out the little bit, that leads to the contradiction. Exactly.

#### No, that's exactly right. And, and, of course, that's precisely what happening in the case of women completely say, hey, why we don't have these. Right. Right. We're just as humans as you guys. So we need to change this. Right. And, and, and it's a simple matter of logic in the way you just pointed out, right? So if it's a human right, well, there's no justification as to why women are being excluded. And I think that's exactly right.

#### And so, so let me ask them and we're sort of coming to the end, but But let me ask a question, which is, you understand my reluctance, asked when I ask it is the goal of logic to make everything simple and obvious, the way that the women voting example is now simple and obvious. Is, is the goal of logic to make everything transparent? And ultimately, logically error free. But in such a way that you look at it, you say, of course. Right? How could we have not seen that? Is that the goal of logic?

#### Now? That's a good question, right? It's hard to say, whether that's a goal, or that's something that will be good to have a sort of a regulative ideal, because you're not, it's not always that you reach that point. right in. And again, I would draw a distinction between pure logic and applied logic, right. Pure logic is just this study of those mathematical formal systems that we mentioned early in the beginning of the conversation, and applied logic, it's actually everything else, when you use the patterns of reasoning, to understand what is acceptable, what kinds of inferences are acceptable, what kinds of inferences are not. And in that context of applied logic, of course, when you can boil down things to a simple, powerful argument that makes the case for something, well, you make it right, and then I think that's a really important tool that logic can provide, and help us identify what are this crucial features, right? that need to be in place, of course, some people may contest the truth of the premises, right? So in the example you gave, in, every human has the right to vote. Women are human, therefore, they have the right to vote. Of course, you could go back and change the first premise or the second, right, of course, the second be totally implausible. The first one come and say no, this is a right only for men. Right. And in fact, throughout the history, that's the way in which was understood. Right. So maybe the case is should be made is that that's not only a man's right, it is a human, right?

#### Although there are traditions in which women are not regarded as fully human. And that would be a challenge to the second premise. We just don't give that much attention anymore.

#### That's right. Yes. Right, then, thank goodness. Yes.

#### So so so logic, then, I guess like physics, like other areas of philosophy, there's the purely theoretical approach. And then there's the applied approach. And the purely theoretical, right is technical in the way that I was first talking about in that it's focused and has boundaries, and limits very specifically what we can talk about, because it's looking to identify the very characteristic of what logic it is. But the applied logic, isn't it is exists in the world, and is subject to our prejudices, the information that we have our motives. And in some ways, pure logic is harder, because it's a system, you have to learn these things, you have to talk as an expert, you have to follow these very specific rules. But in other ways, applied logic is harder, because you have all of these other things in influencing all these other variables. And it's hard to see the logic underneath all of that other extraneous stuff?

#### I think that's exactly right. But also, what's interesting is, when you consider the applied side, it's often puts pressure on the peer side, right? So I think, for example, the creation of those no non classical logics, they emerge out of particular applications, and people realize, Oh, I cannot reason out of a contradiction. I need a logic that would allow me to do that. What logic would that be? Right? So that's an applied matter that then generate a pure one, but he is now he developed a whole system of logics that would justify how to reason in the presence of inconsistencies. Right. So there is an interesting dynamics between the pier and the applied, right. So in that the applied may put pressure on the pier, and the pier in terms would help identify patterns for the applied, right.

#### So, so, alright, so I want to ask one last question, and then we'll, we'll call it a day. Why are there still problems that are unresolved in logic? And if so, fairly briefly, what is the one that you think is the most important or possibly the most interesting to you? I guess what's the most important are? Are all the problems in logic solved. What's next? on the agenda,

#### right? Well, the lots of problems that there are not solved. And in any, and I think one very important issue is understanding exactly the nature of the logic, right? And that's a philosophical question about, about logic. And the fact that nowadays, there are so many different logical systems, you have all this plurality, of different logics, each of which has its own domain of application, each of which help us elucidate some aspect of the world. And, and what is needed is a proper way of understanding that plurality, accommodating it, and then connecting it with a long standing tradition, within logic, of combining the formal resources with the less formal ones, right, and the bits and bits and bits and pieces that involve and engage with actual reasoning in bits and in reasoning that people engage with.

#### So is this is this a search for an umbrella logic that will encompass all of the other logics? Or is it just more variations of how logic interacts with actual human thought processes and different methodologies and different increase? Right,

#### great again, if the path diverged, there is a strong movement towards what's called Universal logics, which is exactly the idea of identifying these large scale logical principles that would hold across a variety of, of logical systems. On the other hand, you have a more pluralist take that would say, well, maybe while you get in these overarching systems, something so thin, so generic and abstract, that it's not going to be very helpful. We want something with more content in a way, right, it would still connect in interesting ways, with the reasoning that we all do. So in a way, as you can see, it's a sort of an open issue, right there. People were people are working within different traditions to to address that. But it's not something that people should feel intimidated, right. So I think, grab back to the beginning, when you raise the question about why people are afraid of logic, I think often they just lump it together with mathematics, they didn't and some of them feel uncomfortable with symbols, and they want to put these away, I would suggest keeping logic as another chance. And realize the logic is actually very tightly connected with everyday matters, right with fishes that we all care about. And so if I can give you an example, and that's the example about the boy dag that she also you mentioned in the beginning,

#### so then I'm then I'm not a liar, we can finish up and I won't be a liar this month.

#### So because it's a little story, right, as you know, my wife Patricia is a wonderful cook. She does all the cooking at home and I wash the dishes in one point is they look I cannot believe you don't know how to cook, you know, you need to learn how to do that. Sure. So she said I'm going to teach Alicia them how to boil an axe. Well, that seems good enough. So she, you know, being organized that she she wrote down all the instructions, right. So the first one, boil the water. Okay. So I asked her after reading the first one, how do I know that the water is boiled? And then she was livid to what you don't know that. And so why wait, not so fast. Would you say that the water is boiling? If there is one bubble? Two bubble,

#### I, I see divorce on me Just let me just say,

#### well, 20 years and it's still going strong.

#### Yeah. And, and of course, the question there was, there's a question about vagueness, right. What is the boiling point? And of course, if the word is not boiling, yet, we would just say that by adding one bubble we will count as boiling. If adding one book one bubble doesn't. How can any water ever boil? Of course, that's an issue about the vagueness of of the the notion of boiling and, and behind that, what do you have is that ready? A clear connection of how the way we reason about things is tightly, tightly connected with our experience of them. Right then, what counted boiling, how to deal with a vase. sickness involved, how to conceptualize our own experience in ways that we can learn from the things we already know and learn from the things that we don't. And I think logic is crucial in all these, these aspects of, of our experiences as he reasonings reasoners. And people who inquiry, sometimes more technically, sometimes, just as in ordinary context.

#### I tell you, I think we will leave it here. Because, as you well know, we could spend the next 25 years talking about this, this is an entire career. And frankly, and I can speak for my audience, I think we did pretty well. I think that we, we managed to talk about the stuff in a way that that made it both attractive and interesting. And I'm sure everyone got lost at one point or another, but that they came back. So thank you. People might have figured out from the conversation that you and I are old friends and I, I had asked you specifically do this because I wouldn't have trusted anyone else with the experiment and you have come through with flying colors and was exactly what I had hoped for. So Tommy, Oh, I'm so happy to talk to you. And thank you so much for being on why.

#### My pleasure. It's such a I love your show, and I was thrilled to be able to be part of it. Thank you, jack.

#### You've been listening to a toffee of one oh and jack Russell Weinstein on life philosophical discussions about everyday life, and I will be back right after this.

#### Visit IPP ELLs blog p QED philosophical questions every day. For more philosophical discussions of everyday life. Comment on the entries and share your points of view with an ever growing community of professional and amateur philosophers. You can access the blog and view more information on our schedule our broadcasts and the y radio store at www dot philosophy and public life.org.

#### You're back with wide philosophical discussions about everyday life. I'm your host, jack Russell Weinstein, we were talking with a toffee of one Oh, and ask him the question, can ordinary people understand advanced logic? Now? I can't answer that question for you. You have to decide for yourself whether not only this was a compelling discussion, but something that you really understood. I think Tommy did very well, I think as I said, we all did very well. And he said something that really intrigued me that if we had had more time, I would have spent more time talking about, which is that there are different logics that apply to different parts of the world. And I think what he means by that is that we use a certain kind of logic in science and biology and chemistry. And we use a different kind of logic, maybe in relationships, and we use a different kind of logic in art. And we use a different kind of logic in politics. And this is really intriguing, because we have a tendency to reduce all of this to one in his terms, universal logic, we have a tendency to say, Oh, you are not very logical, oh, your political opinion is this. What does that say about your relationships? Or your relationship is this you could never be a scientist not we don't say that explicitly. But we assume it because we assume that there's something about if you're logical, then you're always logical. What this opens up for us, this notion of different logics in different parts of the world, is that people learn how to think differently in different contexts. If I am going to evaluate that relationship, I have to think differently about it. If I'm going to evaluate that painting, then I'm going to have to think differently about it. What it means to understand a context is what it means to have a logic for that context. And that leads to a very basic and important philosophical question. What does it mean to isolate what we do? How do you identify how you think about relationships, as opposed to the individual relationships? If you're fighting with your partner, if you are afraid of of the relationship? How do you isolate the logic of that fear, the logic of that fighting with the fact of that fighting? Same with art? How do you isolate what it means to think about art to figure out if it's beautiful, to figure out how to create something with your experience of the art with your experience of the painting, whether you like it or not, whether you think it's beautiful or not. This is the task of logic. logic is trying to isolate our thought processes and more importantly, our thought patterns, so that we can get rid of all the other stuff and get to the root of perhaps the problem or get to The root of what we get right? That makes logic hard. Because it's disconnected. It's abstract. It doesn't hold on to the things that we feel familiar if I am talking about the logic of my relationship, it's hard for me not to think about my wife and me, right? What does that logic look like? That was this experiment. This experiment on today's show, was to try to figure out how do we talk about isolating, not specific technical aspects of any subject, but of the way that we think about subjects. When we do that we get a sense not only of what underlies philosophy, but what underlies the whole world, what underlies all human inquiry and what underlies human thought, which is why many, many of the philosophers in history have thought that the thing that makes humans humans and not animals is that we are the logical that we can reason. I don't know if this is true, but it's certainly a good guess. You've been listening to Jack Russel Weinstein on why philosophical discussions about everyday life. Thank you for listening. As always, it's an honor to be with you.

#### Why is funded by the Institute for philosophy and public life, Prairie Public Broadcasting in the University of North Dakota is College of Arts and Sciences and division of Research and Economic Development. skipwith as our studio engineer, the music is written and performed by Mark Weinstein and can be found on his album Louis soul. For more of his music, visit jazz flute weinstein.com or myspace.com slash Mark Weinstein. Philosophy is everywhere you make it and we hope we've inspired you with our discussion today. Remember, as we say at the Institute, there is no ivory tower.