 Hello my friends, welcome to the 47th episode of Patterson in Pursuit. I'm coming to you today from Sydney, Australia. We're doing another interview breakdown today of a fantastic conversation I had back in Auckland with Dr. Patrick Girard, who works with Para Consistent Logic. That is a logic which is a bit more tolerant of contradiction. If you guys have been following the show, you know that my own logic is about as intolerant as contradiction as conceivably possible, but we still had a great conversation and I'm going to break it down for you. Before we start, I've got a YouTube clip recommendation for you. The CEO and founder of Praxis, which is the company sponsoring this podcast, was just invited to go on Tucker Carlson to talk about his company because as I've said before many times on the show, Praxis is exploding in popularity and for good reason. So go to YouTube and punch in Tucker Carlson, Praxis, P-R-A-X-I-S and listen to their awesome interview. 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So with this conversation with Patrick Girard, we talked for maybe about 20 minutes or so about truth and it was really interesting. I could probably dedicate a whole breakdown just to that conversation. But I want to do this breakdown starting about the 25 minute mark where we start talking about logic, para consistent logic, contradictions and classical logic. So I begin by asking him a question from the classical logical perspective, which is my own perspective, which when I think of contradictions, I immediately associate a logical contradiction with an error in thought that I have exactly 0% tolerance for any kind of logical contradiction. He disagrees and in the course of this conversation, he gives a great explanation for people that are maybe unfamiliar with this position, why he thinks maybe contradictions aren't as big a deal as somebody like me would make them out to be. So in this way of thinking, there's a very visceral reaction to contradictions, to contradictory claims. So could you give some explanation? I know in some of your work, you have maybe you're more tolerant of contradictions. Could you explain why contradictions aren't as big a deal maybe as the classical logicians make them out to be? Yeah, so contradictions are obviously a big deal and it's obviously important to care about them. So that is true. It has occupied philosophers and logicians since for a long time, because I guess for a lot of people, once you're committed to a contradiction, basically you're committed to the moon is made of blue cheese, like if it's sort of a measure of incoherence that once you have a belief that something is both true and false at the same time, that basically all bets are off. If you can commit yourself to something as bad as a contradiction, then basically you've just trivialized. Okay, so that's a great place to start and it's an excellent summation of the classical view and my view, that if anywhere in your world view you're saying, well, here is something that's true and false, you've essentially just exploded the foundations for your entire world where you've essentially, from my perspective, said true and false is now meaningless, because as it comes up later, we can only have a coherent and sensible understanding of what truth is if we distinguish it from not truth. And if you blur those lines, you've kind of blurred the only tool we have for discerning true from not true. He continues. So inconsistency and contradiction has become the measure of triviality, of incoherence, because no logician wants triviality, no one wants everything to be true. If everything is true, then what are we doing? Nothing is true, everything is true, nothing is true. It's all trivial, like I'm losing my time, I might as well just go play video games and I'll be fine with that, like I don't need to. So no logician want that. Okay, so that's a bit of technical jargon that's actually really good to keep in your mind. There is a logical theory called trivialism, and trivialism can be summarized as such. All propositions are true, that includes all contradictory propositions, so A and not A at the same time in the same way. Every single proposition is true. And as you can imagine, that's for most people not a very desirous theory, because it essentially throws out this whole idea of truth and falsehood, and it's aptly named, I think, trivialism. But that's what he's saying, is no logician wants to trivialize, wants to say, okay, now because there's a contradiction, that means everything is true. The question is, how do we prevent ourselves from getting into a trivial system? A common reaction is that once is what is called the law of explosion, that everything follows from contradiction. It's this idea that once you committed yourself to contradiction, then all bets are off, we don't know anymore. So I guess that's the intuition that has been driven a lot of research in the 20th century, and overall in history as well in mathematics, just making sure that we don't end up in contradiction. Whether or not it is, so that, I guess, that rule of explosion has become the measure of triviality. Okay, one more bit of jargon that's very good to understand if you want to grasp this, it's kind of the history of logical reasoning. It's this, the principle or the law of explosion, which says that everything follows from a contradiction or from a falsehood. So if I say, if squares are circular, then I am 35 feet tall. If squares are circular, then the moon is made of cheese. If squares are circular, if something is false as true, then anything goes. And that's the traditional way of putting it, and I agree with this. The way that I would put it is, if it's the case that there is any case of something that is true that is actually not true, then true doesn't make any sense, neither does not true. We can't know anything about anything, and any attempt at internal coherence of a thought is impossible, because without logical consistency, we're only one step away, I would say, from madness. If you want to know more in detail of my own case for why there are, certainly, I would say no logical contradictions, that is exactly the subject of my first book on philosophy, Square One, The Foundations of Knowledge, which you can pick up on Amazon. But there is sort of the dream, think of it as a dream, that you're going to, that we're going to, truth is that kind of thing that's going to carve the universe between, you know, there's the true things and the false things. And I'm just going to, like, throw a lasso in the universe, and I'm going to pull it, and what I'm going to pull back is only, all and only the true things, right? So that's, you can think of it as Tarski's dream, if you want. That would be a true spread of it. So something will be in the collection I've just had, or in my bag, only if it's true and all the true things will be in there. Okay, now this, now, it may seem esoteric, and it may not seem very important, but what he just said and what he's about to say is absolutely central. If you want to understand the history of modern mathematical and modern logical thinking. I've been on the pursuit of what I call irrationalism. This idea that there are logical contradictions, you can't actually know anything about anything, the world is gray and blurry. I have pursued what the fundamental beliefs are, which lead people to these conclusions. And my pursuits have wound up in a few areas. One area, people appeal to quantum physics. Another one they appeal to is religious mysticism. Another one that they appeal to, interestingly enough, is various arguments in mathematics. This is one example, kind of historically, of the origins of what I consider to be irrationalism coming from the Liar's Paradox. And specifically for the reason that many philosophers have claimed the Liar's Paradox, which is this sentence is false, is a true contradiction. It is true and false, and therefore, this naive idea that we could have in a particular theory or a particular way of talking about the universe that is devoid of contradictions is a little naive. You could put it another way. You could say that according to this way of thinking, and historically, this is the case, various philosophers and mathematicians and logisons have tried to claim that inconsistency is inescapable, that if you want to have a powerful enough language to say almost anything meaningful about the world, you're necessarily going to include some logical contradictions. It's something that's just the nature of language. Some people say it's the nature of truth. The nature of reality is that these contradictions are inescapable. So a great deal of what we're talking about in this interview really comes back to the Liar's Paradox, or as he puts it, the Tarski sentence. And what Tarski showed us is that you can't do that because your bag will also have false things in there, because as soon as the lasso itself will also return false things. That's the Liar's sentence. Because if you have a predicate that says a truth predicate, so a truth predicate would be a predicate that applies to propositions and it applies to the proposition when the proposition is true. It's a very simple and it just says this proposition is true, as we're talking about it. But it's isolating this is-truth things as a predicate that could be used in propositions. Once you can do that, you can create the Liar's sentence. The Liar's sentence is a sentence that says of itself that it is false. So is the Liar's sentence true? Well, if it's true, then what it says is true. What it says is that it's false, and so is false. So it's false. Well, if it's false, then what it says is false. And what it says is of itself that is false, so it's true. So you end up in that contradiction. So holding that dream that you can return only the true propositions leads you to an inconsistency. Okay, so again, the way that I would put it is the claim is that contradictions seem to be unavoidable. And therefore, the, let's say, mature response isn't to try to eliminate contradictions because they're unavoidable. The mature response is to say, let's deal with them. Let's grapple and accept that contradictions are kind of inescapable. The truth and falsehood, at the same time, that's just the way it is. Maybe it's not as big a deal as the classical logicians make it out to be. Now, one way that philosophers have tried to say, well, maybe contradictions aren't as big a deal is by attacking or challenging this idea of the principle of explosion. They say, okay, well, yes, in some circumstances, you get true contradictions, but that doesn't trivialize the system. The law of explosion doesn't hold. Yes, they might agree that trivialization is a bad thing, but they would disagree that from a contradiction, anything follows. We talked just ever so briefly about this in the conversation I had with Dr. Timothy Williamson of Oxford, where we were talking about the philosopher Grand Priest is known for doing this. He is a dialectist, which means he thinks that there are some true contradictions, but he also thinks they're kind of limited and not a very big deal. Okay, so before we move on, are there any other sentences like the liar sentence, which the lasso brings back, or is it just the liar sentence? The true sentence will bring back that one that may have some companion ones and that are contradictory, probably. Yeah, I don't know right off the top of my head. Okay, well maybe we can revisit that. Yeah. Now here's why I ask in that question, because I have a resolution for the liar's paradox, which you can read about in square one. I also have a while ago, I wrote a little article on the topic. I also have a YouTube video. My most popular YouTube video is one resolution to the liar's paradox. It essentially says it's a linguistic error, that it's kind of a trick, that this sentence is false. If you actually break it down, this sentence doesn't really refer to anything. If it refers to itself, namely the words of this sentence, then it's not true and it's not false. It's not a proposition, it's just two words, this sentence. But if this sentence refers to this sentence as false, then it generates an infinite regress, because every time you ask what is this sentence, you're left with this sentence as false. So then the liar's paradox turns into this sentence is false, is false. And you say, okay, well, what exactly, which sentence is false? Well, this sentence is false, is false, is false. Add infinitum. It's easier to see with parentheses and maybe written down. So I asked him that question specifically because it's a, if you're just in logic and truth, and we're talking about some of the most important ideas in the world, like, oh, maybe contradictions are inescapable. That's true. I mean, that's a pretty big deal. So, if the only case of this inescapable contradiction is the liar's paradox, and I think there's a satisfactory resolution to the liar's paradox, well, maybe we don't have to abandon classical logic at all. Maybe we don't have to incorporate any contradictions into our worldview. And in my research, I have not found any other types of argument for the inescapability of logical contradictions that don't ultimately come down to self-reference. There's this other paradox, which is called the barber's paradox, which is that imagine a barber who shaves everybody that does not shave himself. Does the barber shave himself? Well, he must, which means, well, he wouldn't. Which means, he must, which means he wouldn't, and so on. Oh, it's true and false at the same time. There's a few self-referential things like that, but I would say in every circumstance, this is a problem of language. It's not a problem of truth. I would say it's certainly not a problem of logic. We've got the lasso. The idea is we want to only get the true things and have this system. All and only. All and only, exactly. But in that particular system, we get this anomaly, the liar's sentence. This sentence is false. Well, it's true, but if it's true, it's false. If it's false, it's true. It's a contradiction. Now what? Well, now the 20th century, the 20th century tradition says, contradiction, warning, triviality. Now I have a trivial theory, and therefore I back away. Go away from contradictions, the lasso does not exist. Right? So that's, that has been the reaction. Okay, so there's no truth predicate. Okay, that seems like an extreme reaction. It won't set in that way. So maybe I've given the story so as to get as an extreme reaction, but that's sort of what has happened. That we said, okay, so that was a great dream, but that dream can't happen because it leads to inconsistency. So you can think of 20th century logicians as going on to shoot. So okay, we'll just make sure that we're not gonna get all truth, but we're only gonna get truth. Okay, fair enough. So we're gonna restrict our analysis such that we're gonna take a smaller lasso kind of thing. It's just gonna, what it brings back, we're just gonna make sure that they're true and only true kind of things. Okay. So they're on the shooting, right? Okay. To preserve consistency. To preserve consistency, because consistency would be as important, but to a certain extent, it kind of changes what we thought was truth, right? Okay, so I love this lasso analogy, right? The idea is that you've got this, out there you've got truth and you have this lasso, and the ideal world, you'd be able to sling the lasso out, get all the true propositions in the world, all and only the true propositions, and you wouldn't get any false propositions there, bring the lasso back and boom, now you have this perfect theory. Well, what he's saying is, well what the liar's paradox is doing is it's saying you can't have this system which includes all truths. If your lasso's that big, then it's also gonna include these contradictions. So if you want to get all the truths, then you have to accept a little bit of inconsistency in your system. Now what he says is, so the response of 20th century logicians is to undershoot. Maybe shrink the size of a lasso. So we don't get any inconsistent conclusions or any consistent truths. We throw the lasso out and what we return is only truth, but we don't get all of it. There's truth out there that we're not gonna be able to capture because our system isn't powerful enough or our lasso isn't big enough. It's a beautiful analogy. An alternative reaction is you say, okay, I'm gonna do the opposite. I don't wanna miss out any truths, but I'm gonna accept that some of them will also be false. So you overshoot. Okay. Right? So that would be, so that's where you say, okay, so you're gonna say the explosion is no longer valid. That inconsistency is no longer a measure of triviality. So you're gonna say, okay, I wanna keep my lasso, but my lasso has this funny feature that for some sentences, it returns some that are both true and false. So I need to make sure that I don't trivialize because of that and how do I keep all truths and not trivialize? So all truths and then some? And then some, some of them won't be false. And so that's a way of thinking about dialectic logic. So that's a logic that accepts that they are contradictions. So you have always a but kind of clause. Altatology is a truth, some are also false. The truth predicates only return true formulas, some are also false. There's always a but clause kind of thing. Beautiful. You won't hear it more clearly articulated anywhere. I think that dialecticist position that the goal is to have this lasso that can pull back all the truths but some of the truths are false. Now this came up a little bit when I had a conversation with Dr. Steven Hicks who I will have back on the show who said, well, the interesting thing about contradictions is how philosophers react to the contradictions. And we didn't get to dive into that though. We certainly will. My reaction is to say, okay, if some of the truths that you're pulling back are false, you've made a catastrophic error that you've put dynamite under the fundamentals of the entire idea of pulling back truth if you're saying, oh, and some truth is false. But he's saying, it's not that big a deal. Yeah, we wanna get all the truth and if that means some of the truth is false, that's just something we have to deal with. A crude way of putting this that I have heard some people say that Dr. Girard didn't say. But I've heard from irrationalists to say, well, I am large. I contain multitudes. If you catch them at a contradiction, that's what they say, well, I'm large. I contain multitudes. I contradict myself, but that's okay. Contradictions maybe aren't that big a deal. But doesn't that kind of deflate the notion of truth, though, when we say we're gonna say some things are true? Or we're gonna say in this lasso got all truths, but some of them are false and true. Doesn't that defeat the idea of what we mean by true is that they're not false? Well, we got all the truth. We wanted all the truth. Yeah, but it, okay. It doesn't make the game easy. It makes the game a lot harder. You're saying? It makes the game a lot harder because we don't wanna trivialize either, right? It's not like all of a sudden we're saying, oh, contradictions are fun. We can't be saying that all contradictions are true. If all contradictions are true, then everything is true, right? But if all contradictions are true, then everything is true. What I would say, you could say that, but I would kind of go one step below that. I would say then truth is meaningless because there seems to be this polarity between what we mean by true and false. I think I can go with you on that. So I don't want my lasso to return everything because like truth has just gone out. Now it would be genuinely meaningless. So that would be unusable because we'd have some kind of trivia, would be trivial, let's forget about truth. No, but I don't, yes, so we agree with that. And most people that earn these kind of projects to like overshoot will then try to find different kind of measures just to make sure that we don't trivialize. So we need some kind of new measures that will tell us which contradictions are acceptable and which aren't. But we can't only rely on contradictions anymore because inconsistency is no longer the measure. So I was trying to get that there as my claim isn't simply, oh well if you contradict yourself, there's the law of explosion and therefore you can't contradict yourself. My point is that if you allow any true contradictions, you have deflated the notion of truth with or without the principle of explosion. If there's any circumstance whatsoever where something is true is false. That means truth is not something that's universal, it's not absolute, it's not even a clear coherent concept. And I don't quite understand the project of saying, oh well, we're not gonna trivialize because what's the big deal about trivialization? If it's the case that you allow contradictions into your theory, I don't see what the hesitation is about trivialization because if some things are true or false, why wouldn't everything be true-false? And of course, again, but with it this way, there is no coherent way to make sense of the claim that proposition X is both true and not true at the same time. You can say something like that, but I don't think you can coherently make sense of it. Okay, so that's that I got several really important questions, but let's go on that, I'll just on that thread, by what are those other measures? So if we say contradiction is no longer the standard, what are the other ones? Okay, well, for logicians, there's been attempts at non-trudiality proofs is something that also, of course, has occupied 20th century logicians from Hilbert to Gerdoll, for instance. They wanted to make sure that they could prove that they had some kind of consistency, right? Gerdoll showed us, well, forget about it, right? We can only get relative consistency proof as in if you have a stronger theory that is consistent, then you can show that your smaller theory is also consistent, but then you're at the top, right? One of the other areas I didn't mention earlier that comes up in the rationalist worldview all the time is Gerdoll. Gerdoll and his famous incompleteness theorems. Now, I haven't written anything about the incompleteness theorems yet because this is something I've been working on for some time now, and just like with the long piece I wrote on Cantor, had to do a lot of research. I had to get my tucks sorted out really precisely before I criticized the theory, but I would claim Gerdoll's incompleteness theorems at best are profoundly abused to prove things that Gerdoll didn't prove. Gerdoll's incompleteness theorems at worst is a bunch of nonsense that almost nobody had actually worked through the proofs because they're incredibly convoluted. And my analysis is definitely somewhere near the ladder than the former. I think it's another case, just like Cantor's supposed proof of the different sizes of infinities, I think there are some fundamental conceptual errors that happen with Gerdoll's incompleteness theorems that people then compound and abuse to prove all kinds of ridiculous things that they actually can't prove. Now unlike my claim about the non-existence of the infinity of infinities, if you're familiar with Cantor's diagonal arguments, I haven't written a piece to back that up yet, so feel free to flame me in the comment section. Oh my gosh, I think Gerdoll's incompleteness theorems are actually probably misguided and I don't have anything to back it up yet. But don't worry, at some point I guarantee I will have probably multiple pieces, both explaining the Gerdoll's incompleteness theorems which I intend to do further in this series as I'm talking to other mathematicians, and you might say debunking some of the radical claims that people make by appealing to Gerdoll's incompleteness theorems. Why do I go on about all this is that it was, okay so I guess like what Gerdoll showed is that Hilbert's idea of showing that mathematics was consistent is probably unachievable, right? So 20th century logicians are in no better position to tell us that their theory is non-trivial, if it's rich enough. Okay, again if you haven't heard these ideas before I don't think it will immediately strike you just how serious and grave and profound these claims are that supposedly according to Gerdoll and others, the project of mathematics, of building this kind of universally consistent and provable system failed, it's over. This happened in the early part of the 20th century with Gerdoll's incompleteness theorems, supposedly. But if the theory of mathematics is powerful enough meaning you have enough explanatory power in your system of language for example, if you express terms like all mathematically then you inescapably run into either inconsistencies or unprovable propositions. This is one of the areas in mathematics you guys know I criticize quite a lot where there is a feverish level of dogmatism not unlike Cantor's supposed diagonal proof that if you so much as suggest that maybe it's not set in stone that Gerdoll proved that this project of founding mathematics with a consistent and universal and provable language then you're immediately seen as a heretic who is not of course grasped the highest levels of abstraction that you'll find in mathematics. As you work your way up the level of mathematical abstraction and profundity eventually at the top you discover your eyes cross and suddenly everything becomes contradictory and paradoxical or maybe Gerdoll was wrong. One of the interesting things that I challenge everybody listening to do is that they know anybody who has stakes of position if somebody mentions Gerdoll's incompleteness theorems I practically guarantee that they'll hold the position that Dr. Gerard holds that they believe that Gerdoll proved what he supposedly proved and I can practically guarantee you in fact I believe everybody that I've spoke to ever on this subject this applies to and I actively seek these people out. Nobody works through the proofs themselves. It's all second hand. It's all I wrote, I read a book on the proofs or I read a secondary thing. They actually don't dive into the actual Gerdoll incompleteness theorems themselves. It's all the second hand stuff. I think that's a problem especially because if you look at the proofs you might not quite find it as compelling as you think it should be given the importance of what is being claimed. But anyway this is not a breakdown about Gerdoll's incompleteness theorems yet. We continue. So let me ask you before we go back to contradiction about mathematics. You said 20th century, the project of putting 20th century mathematics as being this perfectly consistent thing. You think that project is toast? People are still trying to do that. It's not entirely toast but what counts as a proof of consistency had to be revisited and from what again you need to go talk to some proof theorists and probably travel to somewhere in America or Germany or you'll find proof theorists talking about it. It's just that what we take to be a consistency proof has to be adapted in ways that will not fall for Gerdoll's. So the mathematics that they use to prove consistency itself becomes exceedingly complex. All right so now I ask him a question that I should have asked earlier about this notion of the inescapability of contradiction with the relative power of the logical theory that you're dealing with. Would you say that this is a fair analogy or a fair analysis of the two areas in logic and in mathematics? That in the system of logic and classical logic our systems even if they're complex and intricate and detailed and beautiful and powerful are always going to contain in the system of logic itself an inescapable contradiction, at least one. Not all of them, no. Not all of them. Not all of them, it depends how complex it gets. Right. Okay, what about just? So you take for instance, I don't know, propositional logic. Yeah. Yeah, that's fine. Okay. You know, 20th century propositional logic. You know, we have consistency proof of that. Okay. Okay. And it's not expressive enough to be able to, it's not self-referential. Okay. Right, so you need to throw in it. You need to throw in more specificity. So quant fires and then you start adding it enough rule so that it can start like doing some mathematics. You need to be able to do enough. So this is an area that a lot of people are unaware of and this is where girdles and completeness theorems often get abused. In math and logic, there are different orders of logic. Of course, I'm speaking in a jargonistic academic perspective. This is the standard way that mathematicians and philosophers and logicians think about things. It's not necessarily the truth, but there is so-called zero-width order logic or as it's sometimes called propositional logic that is a system which doesn't allow for a great deal of expression. So it doesn't have what's called quantifiers. Then above zero-width order logic, propositional logic, you have first order logic, which does allow for things like quantifiers. Now, all that means is you can essentially say more things within your logical system as you add variables and quantifiers into your system. You're able to say more. You're able to make more statements. Propositional logic is a internally consistent complete system that this is something that everybody, at least to understand the topic, agrees to. Girdles and completeness theorems don't apply to propositional logic. It's once you get into the higher levels of logic where you start having quantifiers, where you can say things like, there exists an X such that Y and Z and so on. Then you supposedly run into Girdles and completeness theorems because you're able to generate within that system a sentence like this statement is unprovable, which is similar to the liar's sentence. So that's where he's saying, it's not that this notion that Girdles and completeness theorems or what you might say is the inescapable inconsistency or unprovability of a logical system is universal. You do have propositional logic, which is a more restricted system or to use the earlier analogy, it's a really small lasso. Okay, so then the question would be, why would it be necessary to expand the logic outside of propositional logic outside of this beautiful system that contains no contradictions? Why would we even... Because you're missing out on validities. So if you only have propositional logic, you can't even get good old syllogisms like from antiquity, all humans are mortal, Sophie is human, therefore Sophie is mortal. So if you only have propositional logic, then you can't get at the validity of that. Because all humans are mortal, has a quantifier in there. If you don't have the quantifier, you can only translate it as P. So you get P and then you get Sophie is mortal. Well, you don't have names to talk about Sophie. So that's just Q. And the conclusion is R. And you can't get R from P and Q. So that comes out as invalid. Okay, so... So we're missing out on validities. So that's why we need to throw in some quantifiers. Okay, so don't be overwhelmed by the jargon here. This is actually a really interesting point. And again, it comes up, it's throughout mathematics, but it's this desire to put our conceptual reasoning into the language of mathematics, which some people might think is synonymous with into a language of mathematical logic. And in fact, I would say there's a fundamental problem that plagues what seems to be every area of thought. And it's this desire to mathematicize everything. But unfortunately, not all of our concepts can be neatly packed into mathematical structures and formulas. So the desire to do this, I'm not exaggerating here, by the way, the desire to be able to break down our conceptual claims about the world into some kind of a mathematical structure. Has led to building these theories of propositional logic, first order logic, higher order logics. And in those systems, people conclude that inconsistency is inescapable and contradictions are inescapable. And they literally will turn around and make claims about the world like the world is fundamentally paradoxical. Or we can't know anything about anything because in their attempt to formalize, mathematicize everything, they've discovered some fundamental inconsistency in their language. This is again something I talk about in square one, The Foundations of Knowledge, that all of these claims about inconsistency are about language. They're not about the world, they're not about truth, they're not about logic, they're about language. You sort of see a similar phenomenon when people are talking about quantum physics and they're trying to justify their irrational world view by saying no, no, there are true contradictions because superposition in quantum physics, which of course usually reveals they don't know what they're talking about in quantum physics. But again, if it's the case like in quantum physics that it appears that reality is blurry, you're presented with a question, do you have a clear perception of a blurry reality? Or do you have a blurry perception of a clear reality? Is reality the way that it is and you have got some smudge on your glasses or are the glasses you're wearing perfectly clean but you're looking at a smudged reality? If you understand the analogy, that's kind of what's going, what really does go on quite a lot in this world of mathematics and mathematical logic, is people think, you know, mathematics is the language of the universe and then they find some kind of inconsistency or trouble with their mathematical language and include, therefore the world is contradictory. And the only way that they can get away with this is because of the allure of math, that it seems really intelligent, it seems kind of mystical in mathematical language. Imagine I were to say, guys, I have this concept of being a square and I have this concept of being a circle and you know what, I'm gonna put them together. There is such a thing as a square circle. Look, look at the sentence, there it is, there is such a thing as a square circle. Wow, it must be that reality is contradictory because I put those words together. People would say, Steve, what are you smoking? That's totally ridiculous and yet it happens all the time with mathematics. People will make, I think, linguistic errors in math and draw profound epistemological claims from their linguistic confusions. If you don't believe me and if you're unaware of this area of thought, don't take my word for it. Just punch in Google or punch in YouTube, you know, girdle in the limits of knowledge. I don't know, that's just made up that title but there's a billion different lectures of people talking about how magical these mathematical conclusions are that, oh man, we've developed our mathematical and logical system and it tells us this amazing thing that if we want a logical system to be powerful enough, it's gonna inevitably result in creating inconsistencies. I don't buy it folks. This is why I say a massive amount of revision has to be done in the world of modern mathematics. There's something very interesting at what you're saying there though because obviously for logicians anyway and logicians that like to devise these languages, these languages that are only truthy, right? Back to the languages that we were talking about earlier, there's a choice as to the control you can have in terms of like controlling inconsistency and triviality and how much expressivity it will give you and to sort of trying to understand valid inferences and truthy kind of things that you're kind of after, right? So as we said, when we say a propositional logic, we're cool, the problem is that we're missing out on valid arguments that we'd like to have, right? So we start putting, there's a trade-off, right? So we add the quantifiers and then we're still cool. So long as we only have the quantifiers, now we got this argument that we wanted but then we're missing out on some valid argument in mathematics and now we start throwing in some widgets to get at some mathematics and then Gerdl shows up. So I think this is another kind of central idea in this balance between the expressivity of the logical system and the internal consistency of it. There's this idea that he brought up that oh well in natural language, we can make sense sort of of the liar's paradox so we can say this sentence is false and oh well we know what that means, therefore our logical systems have to account for that and that is most definitely not my claim. My claim is that actually the liar's sentence, this sentence is false, is a trick of language. It's an illusion. When you actually examine it really, really precisely, it falls apart and the way that people think they make sense of it isn't actually correct. They don't make sense of something that they think they're making sense of. So yeah, if you try to codify a linguistic error, then you're going to have errors in that codified system. If in your mathematical system you can, you think that you can say this sentence is false, references this sentence is false without a problem, well that's a fundamental bug in the system. And again, we have these two options. Is it that the bug is in the system and that's just the way it is because all logical systems, if they're powerful enough, are going to run into this error or is it the case that yeah, it's a bug in the system because we've made a mistake, because we programmed it wrong. But doesn't it seem like that would be a statement about some kind of flaw about expressivity, that it's like the more you, if you want to say more and more and more, eventually you can say so much that it includes contradictions. Is that a flaw or is that just what is all of that? You know, if anything, English, the kind of language that you and I are using at the moment has all these things that we can express, the liar's sentence, and we can express all these kind of things, right? Once we start developing language and using them for reasoning and for trying to get at truth and validity and things like that, well, the tools that we're using can hurt us too, right? Okay. All right, so. So it's always trying to find the right balance between how you're gonna express, how you're getting at truth and the tools that you're using and how much control you have on the tools that you're using. Okay, so then we have a bit of a discussion about the specifics of the liar's paradox. I'll tell you a little bit more detail about this later. When this sentence is false, when we say this is a true contradiction and it's true and false at the same time, this is something, would you say that that's a violation of the law of identity? The old Aristotelian A is a, and maybe the law of non-contradiction, right? Obviously. I don't think so. You don't think so? I don't think so. So would you think that the law of non-contradiction can be violated and you could still preserve the law of identity? Yes. Okay, can you explain that? Because so for intuitively, when we say something like, when we use the term not A versus A, it seems like the whole meaning of what not is is a negation of A. That's the whole reason we come up with the concept is like not means a big X over it. Like it's absolutely not, which seems incompatible with A, the whole point of A, right? Okay, so this again is the center part of square one, the foundations of knowledge. That's what practically the whole book about is explaining this, that it must be in order to have the law of identity that things are what they are. It must be the case that you have the law of non-contradiction, that things cannot be the way that they are not. Because if things can be the way that they are not, then things being things, being what they are, is not some kind of universal law. If you can be what you aren't, then I'd say there's no being in the first place. I think the intuition you're getting at is that what is true and what is false are mutually exclusive, right? Sure. Right. And the liar just goes as a foot in both. So there you go. I mean, sometimes it kills me. I get flacked from people sometimes. And they say, oh Steve, nobody actually argues for logical contradictions. Nobody actually argues that true and false aren't mutually exclusive. No, you just heard it. In fact, I think there's a great number of these people that think what is true and what is false is not mutually exclusive because the liar's paradox or sometimes because quantum physics. This is a huge deal, folks. If you care about the world of ideas and you're wondering why it appears that the modern world is steeped in a bit of irrationalism, I'm suggesting this is one reason why. It is the liar's paradox and the abuse of the liar's paradox where people are thinking that sometimes what is true is false and that's okay. Now, of all the people on earth, I am probably the person who is more revolted in the exact polar opposite of this position than anybody, which is interesting because Dr. Gerard and I got along so well. I mean, he's super pleasant. He invited me up to this beautiful property. I like the guy a lot, but we are on absolute opposite ends of the spectrum. What I've also noticed, and I've also felt prey to this before, is that people who have strong opinions about the internal coherence of mathematics tend not to be mathematicians. That mathematics gets defended for being this consistent and coherent and rational discipline from people who aren't mathematicians because if you go through the training, you learn about girls and completeness theorems. You learn about the infinity of infinities. I think it immediately breeds out of you this pre-20th century notion that mathematics is kind of the perfect logical discipline. In the modern world, it isn't. If anybody would like a book recommendation that goes into a bit more detail about this state of modern mathematics, there's a book by Morris Klein called Mathematics, The Loss of Certainty. I highly recommend it, even though some of it is fairly technical. It goes through some of the history of why there is indeed a loss of certainty in modern mathematics. His position is not my position. His position is, well, it's not a damning of the profession of modern mathematics. That's just the truth of the matter. My position is, oh, well, that dams the mathematics profession. Okay, so I guess the question that I have for you is how are we to make sense of true what that means? If we're saying in at least one circumstance what is true is false. Because in the way that I'm conceiving of truth and falsehood, I would say by definition is mutually exclusive. That's the meaning of the word. Right, but that was the dream of the lasso. Right. Yeah, and that dream won't, if you want to have the lasso, then that lasso is an inconsistent thing. Or the lasso does not exist. Right, but that is the dilemma, right? But how do you make sense of the concept of true when it encompasses not true? In some cases, it's just that truth and falsity are not mutually exclusive. That's how I'm, I mean. But that's the sentence, but how do you make, so I'm saying, okay, I'm there. I'm like, I'm at the doorstep. Yep, okay, what does that mean? How can I make sense of that? I understand it perspective, we have to accept it. But I would say, okay, let's accept it. Let's act at least like we accept it. But how do I make sense of it now? Or can it be made sense of? I don't want to answer the question directly. So do you know, you know about David Lewis and his idea of modal realism? So he's committed to the existence of other possible worlds. So he does modal logic and he talks about kind of factuals. And he basically, in his picture, in his metaphysical picture, there are infinitely many possible worlds and all these worlds are sort of causely independent entities and the full stories and everything. So we are in the possible worlds and there are other possible worlds, the way this world could have been. So, you know, there could be three people in this room, there could be four people in this room, there could be nobody in this room. So each of those is a different. It's like a multiverse theory. Sort of, it's the same kind of story. Anyway, the point is that he's defended the view that these worlds exist just as much as others. And people have said exactly what you're saying. I have heard this argument, but I didn't know the name. Yeah, okay, so it's called modal realism. I'm not getting into this to rehearse the argument. Sure. I think the point is that people, to him, had the same kind of reaction, just this kind of incredulous look. What do you mean? I can't make sense of other possible worlds existing like ours. Okay. I would say, sure, me neither. That doesn't mean it's not the best explanation of what I'm after. This is a really interesting example. And when he gave it, as you'll hear in a second, I wanted to give it the benefit of the doubt and say, oh, I can make sense of that. I don't know why he's saying it. The person who proposed it is saying both, I can't make sense of this and it's the best theory, which is hard for me to understand. So I naively gave him the benefit of the doubt and said this. Well, I can make sense of that to say, I mean, we would incorporate something like the multiverse theory, that I don't think this is necessarily the case, but I can at least imagine a consistent way to say, there is this universe, there is another universe, these two universes are not the same. What we mean by a possible universe is simply a kind of a descriptor of that other universe. Yeah, so I think that's not exactly the right kind of analogy, because if we live in a multiverse, then that's one possible world, because we could live in a different multiverse. So it kind of depends on where you carve the boundaries. Yeah, yeah, yeah. Okay, so he's saying it's outside the boundaries. Yeah, yeah, yeah. Okay, now that can make sense of it. See what I mean? So, but that's the same kind of thing, with at some point, once you start exploring these kind of ideas, yeah, it's incredulous. And I sort of, yeah, I'm still a bit there with you as when I contemplate the liar sentence, yeah, it is, it's hard to make sense of it. Aside from just like repeating it, yeah, it's true. Yeah, so it's false, yeah. So it's false, yeah, and it's true. And then you just keep on going, and then eventually you just stop worrying too much about the fact that this sentence is incredulous, because you have other motivation around it for dealing with these kinds of systems. Okay. Right? So the idea is, I mean... So could you say something like this then, that it is not the case that truth is absolute in the sense that we have to accept in our conception of what truth means that you bump up into the incredulous sentences, and that's just the nature of the game? Yeah, if you dig far enough with any kind of concept, if you dig far enough, you might reach banners in which you see incredulous things. And when these things are, you know, and what to do with them. Right. Of course, that's when it becomes all fascinating, right? It's not that they were wrong in the 20th century to say, oh my God, go away. There's no truth predicate. That's perfectly fine. And like against motorists, some people will say, well, there are no possible words. That makes no sense. Go away. That's fine. I mean, what do you do with the incredulous things when you meet them? That's when all the fun begins, isn't it? And with all due respect, this, from my perspective, is a self-refutation, both of this modal realist position and of Dr. Girard's position, the pair's consistent logistician's position, to say, okay, I admit we can't make sense of it, but that's the profundity of the theory, is in my worldview to cross one's eyes, open one's mouth, and drool. You can't. The purpose of philosophizing is to best explain the phenomena that we experience. We must be able to make sense of the theories that we posit. And if it's the case that our theory is so presented, as to be incomprehensible even to us, that theory has refuted itself. And how does a theory go about doing this? Well, it contradicts itself because you can't make sense of a logical contradiction. So I tried to say with the modal realism example, like, oh, I can make sense of that. It's like the multiverse. You say, no, no, no, well, it's a possible universe. So the multiverse would be one universe, and there are these other universes that exist, but they're outside the realm of existence. They're possible universes, but they're actual. And then people said, what the hell does that mean? And his response was, well, I can't make sense of it, but that doesn't mean it's wrong. This is the mystical perspective. This is the idea that, yes, there are an infinity of infinities, and I can't really make sense of it, but that doesn't mean it's not true. Or quantum physics shows that the world is true and false at the same time, or it is and it isn't, and it is both. And I can't make sense of it, but wow, that's the profound nature of the universe. And we see it here. The liar's paradox is true and false at the same time, and he says himself that, no, I can't make sense of it, it is incredulous, and the proper response is to just repeat it and say, yeah, okay, it's true, and yeah, okay, it's false, and you move on because you have other motivation. If anybody is wondering why I spend so much time talking about mathematics and logic, this is an excellent example. If you cut at the foundations of all philosophic reasoning by admitting logical contradictions into your theory, you have thrown out rationality in its entirety. If you have created a theory that is incomprehensible, admittedly, and you still believe that theory, it is as if you are a doctor that has used a scalpel to cut off your patient's head. It is the greatest violation of the project rationality and intellectualism. You can't contradict yourself, folks, because you can't make sense of it, and it is not respectable or correct to believe an incomprehensible theory. And that being said, I don't want to imply that Dr. Girard or Grand Priest or Justin Clark Stone from Columbia when I interviewed him are all stupid as if their brains aren't working correctly. From interacting with them, I don't think they are stupid. I think they are fundamentally, profoundly, profoundly deeply misguided. As if a runner, a very talented and muscular runner whose legs are on backwards, and when the track gun fires, not only does he not move forward, he moves backwards. It's how I view these people that. I have no doubt that there is mental processing power there, they're nice people, but they're so far away from the truth that it has to be running in the opposite direction, and then amputating their own legs when they embrace the idea that it's okay to be inconsistent and incomprehensible when developing a theory. But that's where I'm gonna have to leave it for today. I hope you guys enjoyed this interview breakdown. We did talk for about another solid hour that wasn't recorded about the Liar's Paradox, which is why if you guys were listening closely to this interview, there's kind of an abrupt transition where it goes from, we talk about the Liar's Paradox, he says, hey, I think you should talk to somebody who's like an expert on this topic. And that's because there's a whole bunch of stuff in the middle that wasn't recorded. There was a magnificent conversation, honestly, one of the highlights of our entire New Zealand trip was talking to Patrick Gerard about this. I loved it. But let's just say my proposed resolution to the Liar's Paradox was not satisfactorily refuted. And so he had a good suggestion, which I will take him up on, which is finding somebody that specializes in the Liar's Paradox and can address my resolution of the Liar's Paradox and tell me why it actually is this inescapable true contradiction. So if you enjoyed this interview breakdown, you are a gigantic nerd and that's great, so am I. There's a lot of us, our movement is growing. You may also have a rational head on your shoulders. If you think that maybe my analysis is onto something, maybe it's the case that professional mathematicians and logicians might have made the largest of intellectual errors ever that had been repeated maybe for the last century, then you're not alone. Our community is growing by the week. And if you'd like to support this project of me going around talking to these people and trying to create a rational world you myself by writing books and doing YouTube videos, check out patreon.com slash Steve Patterson. You can become a patron of the show. You can become a patron of I work. So whenever I release an article or a video or one of these podcasts, you contribute a dollar or two to help make this show possible. If it's the case that my suspicions are correct about the causes of irrationalism in the past century, we have a gigantic amount of work in front of us. I'm excited to do it. I know many of you are too. So I hope you'll join me in the mutual pursuit of truth and the attempt to create a rational world view and make sure to join me next week where I'm talking with a professor about a very similar topic that I pretty much went to Australia to talk to this guy. I'm not gonna give you more details away. Just make sure to tune in next week. Have a good one.