 Can a proposition be true and false at the same time? Is there any way to make sense of a true contradiction? If our logic is inconsistent, does that mean it's incorrect? These are the questions we're addressing in the 42nd episode of Patterson in Pursuit. Hello my friends, this is another episode where I get to talk about my favorite topic in the world, logic. And I got a pretty awesome story to tell you too. So usually when I do interviews with people it'll be in their office, usually some small place at a university. Occasionally I've met in some professors homes, but this episode I got an awesome change of pace. It just so happened to work out that because I'm in New Zealand where they're celebrating summer, the professor I'm emailed happened to be out of his office and house sitting for one of his family members. And that family member happened to live in a gorgeous part of New Zealand on a mansion on a farm. So my guest Dr. Patrick Chirard kindly invited me and my wife out to this beautiful mansion on a farm. It was definitely by far the coolest environmental setting that I've been in for any one of these interviews. And in fact you'll hear in this interview there's birds chirping in the background and that's because where we conducted this interview was kind of this gazebo-type covered area that was letting some bird sounds in. So a special shout out to Dr. Chirard for his hospitality. My wife and I had a blast and it was a fantastic conversation. So if you guys aren't familiar with the relationship between mathematics and logic there might be some parts of this interview that seem a bit advanced. But don't be intimidated. You can follow along, you can Google. If you're interested in these kind of topics it just takes a little bit of research to see maybe a little bit of the history and context for this connection between math and logic. And also if you're interested if you haven't picked up a copy yet of my book on logic you can pick one up on Amazon. You can get a paper bag version for $9.99. It's called Square One, The Foundations of Knowledge and it is exactly about the topics that we're going to be talking about in this episode. And if you're new to the podcast you might be thinking who is this Steve Patterson guy who's written a book on logic and he doesn't even have a PhD in philosophy. Who does he think he is? Well I'm happy to say that the world of ideas is being revolutionized by the internet just like virtually every other industry. And if you want to be a part of that the sponsor for the show is a company called Praxis which I've spoken about before. I'm a huge fan of this company and they recognize that the college degree nowadays doesn't mean what it used to mean. Doesn't mean you're guaranteed a good job. Doesn't mean that you're even necessarily educated in some circumstances. It might mean that you're a bit anti-educated depending on what you studied in college and where you went. So Praxis is a company that takes young people that are dissatisfied with their college experience or want to avoid college altogether and they place them into the real world. They give them a six month paid apprenticeship and three months of a professional boot camp. And once you complete their program they contractually guarantee you a job offer. Now you don't get that in college but you can get that with the Praxis program. They like myself are on the cutting edge of the post academia workplace. So if that sounds like something that you're interested in go to discoverpraxis.com. Check out the program schedule a call with them and see if it would work for you. So my guest today Dr. Patrick Gerard teaches philosophy at the University of Auckland. He specializes in various areas of logic and he's written on para consistent logic which is an idea that in some limited circumstances you can have true and false propositions but maybe that's not as big a deal as somebody like myself would claim that it is. Our discussion begins with fundamental questions about the nature of truth and then it doesn't take long before we transition to talking about logic. Enjoy. So first of all thank you very much for inviting me to this beautiful location and we're going to have a great conversation on philosophy and it's so scenic. It's almost surreal. It is a pretty beautiful spot thanks for having me for the interview. So what I want to start with you is basic questions about truth and we have kind of intuitive notions about truth and about logic and about holding our beliefs in a consistent manner but before we dive into any detail I want to ask a very basic question. What is truth? Let me say that word. I want to know what the truth is. What is that word mean truth? That's a big question. Start with it. I mean yes we are interested in truth and as seeking beliefs we want to acquire true beliefs and false beliefs and we have to know a little about what truth is but if we get too much in the details of truth then we'll never get out of it because of course philosophers have been asking about what truth is for centuries and it is its own topic and it's a ginormous subject and so yes you need to know a little about truth but you don't want to get stuck too much probably in what truth is if you're in a true seeking activity. Just like a gold digger you don't need to know all the details of the chemistry about gold to go and find gold in the world I suppose. I think one thing that is important about truth is at least to realize what kind of things can be true or false. There is an intuitive notion I suppose that we can express it by saying something like everything is either true or false but what's everything? It can't be everything everything right if I say New Zealand is true right that makes no sense right? So something like New Zealand is not the kind of thing that can be true or false. So this has some kind of connection with propositions? Propositions yes so propositions would be what philosophers take to be the kind of things that are true or false and when we say everything is true or false we'll eventually identify everything means every proposition is the kind of things that are true or false so a proposition could be something like the only native mammals in New Zealand are bats so that is a sentence that I've expressed in words and that's the kind of thing that can be true or false right? So what a proposition is again is a big subject in philosophy okay what perhaps one good way to start thinking about it is to think about language and sentences right? So if I say like New Zealand the only native mammals in New Zealand are bats I'm using a sentence here and I could say of that sentence that is true or false because it says something that corresponds to sort of the reality about New Zealand okay right so that's that second part the correspondence to the reality okay so I'm not an expert on the various kind of ways but what matters here is more that it is the truth is ascribed to the sentence I think that's what I would like to focus on just now right yeah and okay so then does that mean that every sentence is true or false you know if you think about questions for instance like if I ask you would you like a coffee and you answer it false that doesn't seem to match right? What time is it? What time is it? Good morning shut the door right so there's various bits of language that we use that are not meant to be truthy okay right so if you're in a language you want to isolate that the sentences that are the kind of things that are true or false okay and logicians for instance have tried to devise very artificial kind of languages you can you know they develop some kind of mathematical languages that are meant to only talk about preposition so everything that every sentence that you can construct grammatically in those kind of languages are the kind of things that are true or false okay right so then you can start analyzing the logic and then you've isolated what counts as being true or false okay so when you say something like you know shut the door we say that's not really true or false it communicates some meaning those words have meaning but it's not really a statement about reality it's not really claiming something is or isn't the case okay so when we say something is the case let's say you know it is true that there are you know it is partly cloudy today we're making a statement about reality we're saying it as a true statement does that mean that truth has this necessary always connection with reality that you cannot have truth if it's somehow detached from reality that's wrapped into the definition of truth is statements about what about mathematical truth well what about mathematical truth well I suppose it depends on your commitment as to whether mathematical is part of reality or not right and that again is a big question of course but what's the reality of numbers are numbers and the operation of additions and subtraction and is zero part of reality that's a big controversial question okay so you couldn't dependant of that question there's still I think two plus two equals four will come out true regardless right would we use that term so could we say two plus two equals four is something like valid or I like the idea of calling it true I think it's true to say mathematics is part of reality I think we can I think we can justify that but could we explore could it you're saying that it could be the case that some some kind of claim could be true and not state anything about reality prison yeah I'm happy with that okay at least mathematical claims okay there claims that we can probably come to know to be true without knowing all that much about reality okay logical truth for instance you know everything is true or false every proposition is true or false okay so maybe and we won't go down this rabbit hole but maybe it has to do with what we mean by reality which is obviously I don't know I've never been there all right so so let's let's analyze the statement about the the it's partly cloudy that is a claim about reality and I think it's a true claim about reality so what if somebody would come along and say it is not the case that it is partly cloudy in where we are in New Zealand right now we would say that's a false claim because it doesn't doesn't correspond to reality correctly there are clouds look at them right now does that imply now we're gonna get into logic here does that imply that there is a kind of necessary relationship between something being the case and something not being the case so it in no possible way could it be cloudy and not cloudy is there any way to make sense of something like that like a strict logical contradiction in that sense about reality no sorry I'm not following you okay so we say it is true that it is partly cloudy right now if somebody were to say it is false that it is partly cloudy right now that would be an incorrect claim about reality but what I want to know is can somebody say a true statement that is contradictory so they say it is the case that it is partly cloudy and it is not the case that it is partly cloudy is there any way for that type of proposition to be true well I guess being being being cloudy is as far as what we call a vague predicate right that there are there are cases where it's definitely the case that it's cloudy and the cases where it's definitely not the case that it's cloudy like many kind of predicates like being read there are clear cases of red there are clear cases of not read there are clear clear cases of bold people and there are clear cases of non-bold people right when when we're in the partly cloudy is when we get in the fuzzy line in between right and this is for some people this is where you actually get some kind of you can actually find in people commitments to there being contradictory beliefs about that reality is okay so two questions about that so now we're in the problem of vagueness yeah okay so two questions about that the one let's start with the case where it's a clear cut the person is totally bald or it's you know totally cloudy in those kind of circumstances would you say that it's there is no contradiction present would be impossible for there to be any contradiction present that's I mean that'd be hard yeah okay I don't know how okay yeah and so then let's go straight to the vague case let's say it is kind of partly cloudy right now so can you give some examples of how we could make sense or try to make sense of a contradictory statement so intuitively I could think you know it is cloudy and it's not cloudy well I can kind of make sense or he's bald and he's not bald but I can kind of make sense that but they're both explicitly vague claims it seems like if we clear up what we mean then any kind of contradiction goes away okay so I think the way they got people to hold contradictory beliefs about vague cases is the kind of stories where so imagine you have like buckets of paints in front of you so on the left hand side they all white then from each bucket you just put a drop of red paint right so as you go along you travel to the right hand side then eventually if you go far away enough then it's all red right so when you're very far right it's very red and it's clear red and then on the left is white right and the story is set up such that every time the difference between every successive step is just one drop of paint right so if this one is white if you choose a bucket and you call it white and add one drop of paint you know and my drop of paint will be small enough that you won't even be able to perceive the difference between the two then you'll be committed to believing that the next one is also white right sure so it seems to be the case that for every successive buckets if the left one is right then the right one is also white and the first one is white so if you follow that reasoning then you end up that old buckets are white okay but obviously at some point there is a last white bucket and there is a first red bucket right there ought to be a transition between the two which we just don't have access to that transition okay okay so that brings up a couple more questions is it the case that in reality you have this vagueness that reality is somehow stuck in a mutually exclusive state or is it just a statement about our knowledge of reality that is it seems to be a bit of an interaction between the two because we have these predicates that turn to try to identify right there's got to be a limitation to how much we can discriminate we can't just choose a color for each single bucket because then I'll make a continuum full of buckets right and there's more there'll be more buckets than we can generate words for them okay so we'll run out eventually okay so it can't just be you can't just resolve it by putting more buckets but I guess I guess the point is that there has to be a last one but we don't know which one it is this is sort of it that's so it is a feature of the thing we just don't have access to which one it is okay and some kind of ideas is when you ask people on this I think that so when you put them in labs and you show them patches of color you start from the left and eventually they'll choose one to at some point they'll just commit themselves to saying yeah now it's red and then you start from the right and eventually they'll just commit themselves to saying yeah now that's why but there's an overlap between the two and a model that captures this idea of overlap is that there ought to be a last bucket which is white and that means the next one has to be red but because it's a bit fuzzy in between there's a few of those buckets at the same time so there's a few last white buckets is it the case that they are fuzzy themselves or is it the case that people have different labels so this color is an excellent example because my wife and I she claims that I'm a little bit colorblind it might be true because sometimes we disagree about both people if you are sure we do that too but could it be that what's going on here is a demonstration that what people label as white or red changes based on the person so we might you and I might look at you know a flower I say okay that's pink I think yeah that's pink and you might say no I think that's red so it's just a statement about how we use language rather than like an ambiguity in logic or in reality well it's embedded in the language the same way that when we talk about truth it's embedded in propositions right I mean this I mean to say that reality is vague is maybe a miscategoring the sense in the same sense to say that New Zealand is true right that's not where it happens okay so we're trying to gather information about reality so you would say that we need to split it in some kind of ways so you would say that reality itself reality by itself is not contradictory but claims about reality are it resolved contradiction is when you have a proposition that history and fault it's true and false so it's the contradictions is is about things that are truthy things about everything in reality is truthy right okay so and whether or not propositions of parts of reality will change how you answer that question so is it a necessary feature of reality that it isn't contradictory reality not think it's a maybe a miscategorization okay so we can't even say that it reality is something that could be contradictory or non-contradictory because we're just talking about language well okay perhaps it perhaps it adds up if you're committed to a propositions truly existing sin and being part of reality and combining it with the kind of story which I've just given where some propositions have to be true and false at the same time so perhaps that's a way you can get at some kind of inconsistent reality okay if maybe we take truth maybe truth is in the right word maybe we just talk about something being the case and something not being the case could is it something about the nature of reality that it can't be the way that it is and not the way that it is doesn't compute for me it doesn't compute because it doesn't literally doesn't make sense or it doesn't compute because that the question was not precise so if I were to say the New Zealand you know the New Zealand I mean it's lying here saying could something be and not be at the same time yeah kind of thing right yeah that that seems nonsensical to me now why existence obtains it doesn't obtain and some kind of exclusive kind of way and I agree with you yes but I want to I want to push it because I've encountered taking me into some kind of uncomfortable terrain here that's part of the part of the fun what would you say to somebody who is claiming that no no you're thinking of reality in a limited in a limited way you're you're putting constraints on reality that aren't there you're putting linguistic constraints on reality reality can just doesn't really care about you calling it contradictory or being enough being it's somehow unable to be made sense of but in some cases it is the case that somebody is bald and not bald themselves and that's a that's just a contradiction in the world what would you say to that go jump off a bridge or yes I don't know I mean you're talking about reality as I don't know what reality is and and what we're discussing about in a sense like things that are things that are yeah well this is this all makes me things about the kind of story where you know when we reason about truth and knowledge and beliefs and things we're reasoning about information we have right so with sort of categorizing if you can't categorize things then you're not getting any information right if everything is discrete as in non-categorized then you just don't have any information about anything there okay it's when you start agglomerating things together that you can say these things are not like those things and then that that's sort of by merging things together by combining things together by carving reality if you want now you start getting some kind of information that can be done in the perceptual way to a certain extent which I suppose that's how we're sort of getting some kind of information that we have in our perceptual experience of living and everything but then when we get to talking about knowledge and truth and these kind of things then we have some kind of device of categorization that treats information we have so are you is that a claim about that taxonomy how we categorize things or is that a claim about the world so can the world be meaningfully and accurately carved up or is that just something that well that it just is suppose I mean I and I'm having to say I don't know it isn't it's the sort of problem that can't have that you can't get at it okay right well then if you can get at it we need to carve it somehow we can't it's sort of beyond us like as soon as we start being in this reality we impose some kind of categorization if only by our kind of beings that we are the kind of perceptual system the kind of thinking imposes a categorization on you know it's just it's it I don't know what it is nothing but then if that's true if we can't get at it then why would we say it has some constraint in being the way that it is yeah now when the Kent Encounter project of which again I'm really far from me okay well this is that was an interesting that was interesting so let's take it from the reality and let's talk more about language let's talk about truthy things truthy things yeah let's keep it to truthy things it's it's sort of you know a truth a truth seeker will try and acquire true things and it's sort of what I was saying about the gold digger like at some point if you various people will make various choices as to how deep they want to understand what truth is and you're taking these questions far beyond where we're uncomfortable and to go to public talking about it in a kind of way it's just that you know it's not that they're not important questions they're just exceedingly hard questions to address right you're going super I was looking for a simple answer well I mean I don't have those we have simple answers to these kind of things philosophy wouldn't have existed for like this many centuries in millennia I mean okay so so if we're taking it back to truthy things when I say something like you know I wanted the statement there's a cup on the table seems to be a claim that is true and it is true by virtue of the fact that it is the case that in reality there seems to be a cup on the table so there is one system which I'm very partial to and I think intuitively most people hold which is you could call it the classical logic kind of an orthodox way of thinking the logic of the 20th century logic of the 20th century and may perhaps before a logic up until the 20th through 21st century well that has changed so much right there's so many stages in a way that it's not clear that there's even continuity from antiquity to right 20th century right but I think what these days with people call classical logic I mean I'll agree with you if we mean like the kind of logics that have been developed in the 20th century okay yeah so I'm more comfortable calling it 20th century logic than classical logic okay fair enough so in this way of thinking there's a very visceral reaction to contradictions to contradictory claims yes 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 and they've always 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 there's no 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 it's it's sort of like you become I guess I guess you kind of become an unreliable source of information kind of refute yourself in a sense if refutation is there but I mean it's more that you know I guess I guess in the 20th century it 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 you know what are we doing and nothing is true everything is true nothing's true you know 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 the question is how do we know that how do we prevent yourself from getting into a trivial system and exactly a common reaction is that once is what is called the 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 right 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 right and that comes in conflict with other kinds of notions that people have been interested about so truth is one of them for instance so coming back to truth let me just forget about reality let's just talk about truth as if we sort of understood how it matches up to reality or anything but there is sort of the dream think of it as a dream that you're gonna eat that we're gonna eat truth is that kind of thing that's gonna carve the universe between you know there's the true things and the false things and I'm just gonna like throw a lasso in the universe and I'm gonna pull it and when I get pulled back is only all and only the true things okay right so that's you can think of it as Tarski's dream if you want that would be a truth predicate so something will be in my in Michael 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 and what Tarski showed us is that you can't do that because your bag will also have false things in there okay right because as soon as you try the lasso itself isn't it will also return false things that's the liar 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 you know it applies to the proposition when the proposition is true very simple and it just says this proposition is true as we're talking about it but it's isolating isolating this is truth things as a predicate that could be used in propositions okay once you can do that you can create the liar sentence the liar sentence is a sentence that says of itself that it is false right so is the liar sentence true well if it's true then what it says it's true what it says is that it's false and so it's false right so it's false well if it's false then what it says it is is false and what it says is of itself that is false so it's true right right so you end up in that contradiction right right so if you so holding that dream that you can return all in only the true propositions leads you to an inconsistency okay so before we move on yeah are there any other sentences like the liar sentence which the lasso brings back or is it just the liar sentence the truth sentence will bring back that one that may have some companion ones and that are parking that a contradictory probably yeah I don't know right off the top of my head okay but we can maybe we can revisit that yeah so we've got the lasso we the idea is we want to only get the true things and all in only exactly but in that particular system we get this anomaly the liar 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 set this 20th century edition 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 okay so that's that has been the reaction okay so there's no truth predicate okay right that seems like an extreme reaction it's well said in that way so I'm maybe I've given the story so as to get as an extreme reaction but that's sort of what has happened okay that we said okay so that was a that was a great dream but that dream can't happen because it needs to inconsistency so you can think of 20th century logicians as going on to shoot so okay well just make sure that we're not gonna get all truth but we're only gonna get truth okay so we're gonna 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 what 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 well to preserve consistency to preserve consistency because consistency would be it is important right but to a certain extent it kind of changes what we thought was truth right well change changing here but you know we thought we had that notion of truth capture you know truth being these kind of things now we have to revise what we mean by truth okay that's okay and that has been done and a lot of people have done so and there's various stories more or less plausible and you know it's all good it's fine it's it's glorious philosophy and it's beautiful like it's fine but an alternative reaction is you say okay I'm gonna do the opposite I don't want to miss out any truth 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 the locks that so you're gonna say the locks explosion is no longer valid okay that inconsistencies is no longer a measure of triviality right so you're gonna say okay I want to keep my lasso but my lasso has this 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 how do I keep all truths right and not trivialize and all truths and then some and then some some some of them won't be false and and so that's a way of thinking about that elitic logic so that's a logic that accepts that they are contradictions and so you have always a but kind of clause you know all tautology is a truth so we're also false you know the truth predicates only return true from you last summer also falls there's always a bad clause kind of thing okay but doesn't 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 is right doesn't that kind of just doesn't that defeat the the idea of what we mean by true is that they're not we got all the truth we wanted all the truth yeah but it okay so but it's okay it what what what what is 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 want to trivialize either right it's not like all of a sudden we're saying oh contradictions are fun I can't we can't be saying that all contradictions are true okay right 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 what's there seems to be this polarity sure sure sure I think I can go with you on that right so I don't want my last to return everything because like truth that's just gonna right it's now it would be genuinely meaningless meaningless so that would be unusable because we have some kind of triviality I don't think it's gonna be meaningless meaningless so that would be unusable because we have some kind of trivia you know would be trivial forget about truth right no so but I don't yes so we agree with that and most people that are in these kind of projects to like overshoot well then try to find met 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 acceptable in which aren't okay but we can't only rely on contradictions anymore because inconsistency is no longer the measure okay so that's that I got several really important questions but let's go on that I just on that thread by what 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 proven non-trudality proofs is something that also of course as occupied 20th century logicians from Hilbert to girdle for instance like they wanted to make sure that they could prove that they had some kind of consistency right girdle 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 now with girdle is that correct me if I'm wrong but there's very strong correlate between the girdle sentence and the liar's paradox right and they're in the same ballpark yeah it's the same kind of feature and the problem with girdle sentence is more that it's it's not as it depends on your react to it I think I think like that the ties key sentence gives you a contradiction immediately which is sentence is more is a sentence that says if it's true I'm not provable right and it's just a it's a self-reference idea that it comes yeah so the self-references is there in both right but it's used in different kind of ways we said the Tarski sentence what is that sentence other liar sentence I'm okay why do I go on about all this is that it was okay so I guess like what girdle showed is that Hilbert's idea of showing that mathematics was consistent is probably unachievable right so 20th century logicians are no better position to tell us that their theory is is non-trivial if it's rich enough but there is this idea of relative consistency right so you can all I want to say is that there have there there are ways for instance script key showed that about the liar sentence that you can construct models in which you have these inconsistencies and so you can come you get relative non-triviality proofs okay so people that are interested in having the big lasso have found measures and systems for which they claim they have proof that you don't return everything okay right so there is a sense in which everybody is in the same boat here and in terms of wanting to avoid universal contradiction yes or triviality triviality yeah okay so I don't think anybody is in a better is has a privileged position yet okay yeah yeah so let me ask you before we go back to contradiction about mathematics he said 20th century the project of putting 20th century mathematics as being this perfectly consistent thing you think that's that project is 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 I again you need to go talk to some proof theorists and I probably you know 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 girdles and good old okay things so you so did the mathematics that they use to prove consistency itself becomes exceedingly complex would you say that this is a fair analogy or fail a fair analysis of the two areas and in logic and in mathematics that in the system of logic in the 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 there are some it depends how complex it gets okay okay what about what about just so you take for instance I don't know a propositional logic yeah yeah that's fine okay you know 20th century propositional logic you know we have consistency proof for that okay because it and it's not expressive enough to be able to it's not self-referential okay right so you need to you need to throw in it you need to throw in more expressivity so quant fires and then you start adding it enough rules so that it can start like doing some mathematics you need to be able to do enough so it's when you get so when you add the when you add more horsepower your logic yes that gives you the ability to make meaningful self-reference yes that's when you that's when we say okay now that system has at least one okay so that that it's a thing you can have enough horsepower that you have self-reference without having inconsistencies that can probably be built up okay it depends what horsepower you throw in okay so for instance yeah I see if you throw in the truth predicate then you then you're in trouble okay right because of Tarski's argument but that doesn't mean it's the only way to can you explain that more of the Tarski's argument that was the liar sentence that I said so if you have if you throw in a truth predicate then you can say and you have enough self-reference then then you can generate a sentence that says of itself that it's true if and only if it's false basically so then you get that contradiction okay but there's various pieces there's you need enough horsepower to generate that sentence okay but you don't need to you know so if you're missing a bit then you won't be able to generate that sentence sometimes you know there's various ways of like it's when you produce the argument when you run it all like in all its glory details there's various widgets and bits and pieces that you put there and there's various bits and pieces and which is that you can remove from the engine okay that will keep you safe okay so that has been a lot of that has been various kinds of strategies that people have adopted 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 no contradictions why would we because you're missing out on validity's so if you only have propositional logic you can't even get good old syllogisms like from antiquity you know all humans are mortal Sophie is human therefore Sophie is mortal okay right so if you only have propositional logic then you can't you can't get at the validity the validity of that okay because because all humans are mortal has a quantifier in there if you don't have the quantifier only you can only translate it as P okay so you get P and then you get Sophie is mortal well you can't you don't have names to talk about Sophie that's just Q right and the conclusion you know it's R okay and you can't get R from P and Q so that comes out as invalid okay so so we're missing out on validity's so that's why we need to throw in some quantifiers we're missing out on validity's for those propositions which include quantifiers yeah so you can still have validity's in the propositional logic but not those that when you talk about all yeah what would you think about this idea that what we've demonstrated is if we're talking about sentences where you include quantifiers like the word all that when you discover a contradiction or do you discover that that system in itself is inconsistent because it has at least one contradiction wouldn't we say oh well that's a problem with quantifiers so wouldn't wouldn't we want to throw out the quantifiers and say well we can't actually use language like all bachelors are married or whatever right yeah but we do right I mean that's the thing some people have said okay truth is truth is not a predicate but we use it as a predicate and it works just one doesn't it well but it's truth is a predicate with quantifiers I don't know 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 child 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 into sort of trying to understand valid inferences truthy kind of things that you're kind of after right so if you so as we said like when we when we say at propositional propositional logic we're cool the problem is that we're missing out on valid arguments that would like to have right so we start putting there's a trade-off right so and so we add the quantifiers and then we're still cool so long have only had the quantifiers now we got this argument that we wanted right 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 girdle shows up so let me ask you this right because this is this is an excellent way to frame it as we start going higher and higher is the reason that we're going higher and higher not because of everyday language we want to say things about the world but because of math essentially you could you could math is one of the motivations that eventually you'd want to get at but it you know like the first argument we had the syllogism about so if you are being mortal is not math it's just pure reasoning it's the kind of reasoning that you and I use another way of thinking about climbing this ladder and think about in a naive kind of way so I but just as an analogy think think about artificial intelligence right if all a little robot had was propositional logic it would definitely fail to be as intelligent and seeing so so much validity as we do right so we throw it a bit more we'd give it quantifiers right and then so how would the robot with that with standard logic when we're talking about you know all X or Y that we can say that what would it react how would it react to that gets already problematic because that kind of logic is undecidable so there's various problems the more you give it expressive power the less control you have on on on helping it make an inference is in some kind of regulated way 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 say you can say so much that it includes contradictions is that a flaw is that just what is all of that you know if anything English the kind of language it language that you and I are using at the moment has all these things that we can express the the liar 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 it's always trying to find the right balance between how you can 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 right and I know I like to think about it and sometimes you go in different departments in economics for instance they want to make predictions about I don't know markets and things like that right and there's no cons on on on the mathematics that they're gonna use all mathematics goes so long as I can get the right kind of predictions for my market and that's fine but so they will help themselves to any bits of language and mathematics that will give them the proper kind of analysis it's I don't mean by then it's arbitrary but they're not in the business of trying to isolate a language that can control like you know so they want to be as expressive as they can because what they care about is the predictions that they're making right so they don't need to like take all that super powerful baggage that they're using and put it in a little machine that can control it right logicians and well some logicians more like the computer science kind of you know there is that project of trying to find some kind of language that will isolate valid inferences the last two that isolates truth like these kind of ideas can we formalize these things that get at truth okay and and now we get into the problems of how expressive these languages can be okay right so was that make sense yeah yeah let's revisit if we can the liar's sentence because I find this a really interesting and apparently they have very high stakes right it says something if we can't grapple with the liar's sentence in if we can't just dismiss the liar's sentence essentially we're forced to revise logic in a sense is that fair we're forced to find logics that can cope right with the consequences of having the liar's sentence in the logic basically okay I don't know if it's a revision but yeah all right maybe well it it forces us to make some decision as to what kind of logic we can use basically okay so let's let's go and analyze the liar's sentence so 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 all the risk detilion 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 with the concept is like not means a big X over it absolutely not which seems in incompatible with a the whole point of a right I think I liked it is your question something like I think I think the intuition you're getting at is that true what is true and what is false and mutually exclusive yeah right right and the liar just like goes has us as a foot in both okay right so is the 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 can be 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 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 that is the dilemma right but how do you make sense so how do you make sense of the concept of true when it encompasses not true in some cases it's just that truth and falsehood you're not mutually exclusive that's how 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 say okay let's accept it let's act at least like we accept it but how do I make sense of it now 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 okay so he does modal logic and talks about kind of factual's and he basically in his in his picture of in his in his metaphysical picture there are infinitely many possible worlds in all these worlds are sort of causally independent entities and the full stories are everything so we are in the possible worlds and there are other possible worlds the way this world could have been so you know that could be three people in this room it could be four people in this room there could be nobody in this room and you know so each of those is a different so it's like a multiverse theory so it's sort of it's the same kind of story anyway the point is that he's defended the view that that these worlds exist just as much as others and people that said exactly what I have heard this argument but I didn't know the name yeah okay so it's going all over again I'm not 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 to which you would say sure me neither that doesn't mean it's not the best explanation of what I'm after 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 could I could 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 no so I think that's not exactly the right kind of analogy because a multi 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 okay so he's saying it's outside the boundaries yeah it's okay now that makes sense of it so you so but that's the 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 so it's false yeah so it's false yeah it's true and then you just keep on going and eventually you just stop worrying too much about the fact that this sentence is incredulous because you have other motivation okay around it for dealing with these kinds of systems okay right so the idea is I mean so could would 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 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 in like like against monorealists some people will say well there are no possible world that makes no sense go away that's fine I mean what do you do with the incredulous things when you made them so when all the fun begins isn't it like my suspicion though is to think if we run into something like that we've made an error there's a there's some it's a it's a if we hit the the contradiction I guess is a good marker of it okay we've made a mistake where is the mistake rather than throwing out this idea of the consistency of truth the absoluteness of truth who say okay there's got to be some funny business going on with the liar sentence yep if if once you started traveling towards the liar sentence you had a you had some kind of conception of truth and logic that once they've reached the liar sentence trivialize right well not necessarily I mean what I'm saying is you hit that contradiction and then if your logic is explosive and you have a contradiction then there you go your trivial well right so it's an excellent way of putting it you have a theory of truth and of logic which says in some circumstances I can accept a contradiction things don't explode yes I have a theory of truth it says when you bump into a contradiction it's a demonstration that there's some funny business going on now when we're the benefit that I have I think going for in my conception of truth and logic and this is what I want to hear your input on is I can say when you analyze truth when you get down to it when you when you put that when you push it as far as it can go truth still has an absolute meaning so I can say the way that truth has meaning to me because in all circumstances it is mutually exclusive with falsehood but it sounds like you're saying your conception of truth and logic you aren't able to do that and for my I'm sure a lot of people especially if they're unfamiliar with these ideas would say oh well if we're going to choose one of the two ways of thinking about truth and logic let's do the one that does that allows you to retain this mutual exclusivity of true and false how would you respond to that that is fine but that isn't that there's some that is basically one of the first question you've asked me what's truth right right I mean both of us and any one of us that reaches these when we reach these things like we have to make choices like sure so you want to preserve some kind of conception of truth that doesn't lead us to that contradiction okay and people have devised systems about it but these systems oftentimes okay oftentimes what happened is that to preserve consistency they I don't mean it in a diminishing way about about the philosophy what seems to happen is that people start throwing up a cycles at their theories basically that's a great way of putting it so so you end up with a whole lot of epicycles so yeah so you but the thing is so you need any bicycles to account for that and then what about this and then you need another epicycles so people I want to make sure that people understand what you're saying is that epicycle is the idea and the Tomeiic model of the solar system where you instead of the theory that says you know the earth is revolving the earth is not stationary with the stars are evolving around it the way that they accounted for the motion of the stars is this it's funny looking at it now but they have these you know they're in beautiful circles and then in one part of the circle there's this little squiggle it just kind of reverses course and then it goes back and they do that to try to preserve the explanatory power but at the end they had a really really good prediction about the movement of the stars in the sky and they had a Ptolemy actually had fewer epicycles than Copernicus okay you know that I didn't know that Copernicus put the center the Sun in the center but he still had the orbits being circles and it's only when afterwards that we had the what's the name elix yeah the ellipses the ellipses that then we removed all the epicycles but anyway so it's like unnecessary theoretical clutter that's probably a demonstration of the shallowness of the theory and so there is a sense it's a great analogy to to work with anyway because there's a sense in which both the Ptolemy and the Copernicus system got at the same predictions but one had well take with the ellipses so the one that has no epicycles and the other one that has and they sort of get at the same kind of predictions you probably don't agree on everything but mostly they agree together which is best right so there is these kinds of questions that so by analogy I find that reading some of the 20th century philosophy that tries to preserve consistency that they're just throwing a whole lot of epicycles and they built what I call classical monsters of truth it's just that it almost feels like you end up with some kind of ad hoc theory or patchwork it works but there's a lot of duct tape on it now could you could you go into a little bit more detail because I'm ignorant on this this is an interesting interesting claim what are the epicycles what are the claims so let's let's take for grant let's just say let's assume the liars paradoxes resolve let's say we have a satisfactory resolution for that what are the other what are the other parts of the monster so I guess for someone who meets the contradiction and wants to keep a logic that has the law of explosion so that in which the measure of triviality is consistency then one has to a fiddle or it wants to one needs a correspondent notion of truth that will not generate a contradiction because then the combination will make them trivial so what there are several strategies common kind of strategies is to start stratifying levels so you talk about truth at level zero and then and then you can have a truth predicate that is at level one and the truth predicate of level one only talks about truth at level zero then what about truth of level one well those you can capture with a predicate that is at level two okay and then you keep climbing climbing climbing and then you take the infinite union so that was a kind of strategy that quickly used for instance but this idea that there's levels of truth it preserves the consistency but now I would be the one to ask like but that doesn't make sense to me how's that truth now this I have a vague memory of Bertrand Russell trying to do something similar exactly the theory for to that's the same kind of strategy that Bertrand Russell and the type theory to deal with Russell and the Russell set right so the Russell set from Mary is the set that contains all the set that are not members of themselves you know if you repeat that every night after a couple of years like it that's the Russell set you can drop it in an interview some kind of problem so what they started doing is okay and so there's sets of levels well there's just items in the bottom and then there's you know sets of those items and then there's sets of those sets and then sets of those sets and then you just keep climbing up and so that's one way but then this you know this is a way to sort of stratify your concept so as to preserve consistency but you know you get problems because then you can start asking questions about the collection of all those together what about that right so that's one way another way is so recently I was reading your paper and the strategy was to divide the concept of truth into it was called an ascending truth and a descending truth so we need to come back to what to what's that called from Tarski so from Tarski there's this by conditional that P is true no P if and only if P is true I've heard that in your interview for instance with Williamson you've talked about that by conditional right love it so that means if P then we can get that P is true right and if we have that P is true then we can get that P right so that's about by conditional that goes in in two ways now split that you know call from P to T of P call that ascending truth and call from T of P to P descending truth okay so you've sort of split your notion of truth into ascending and descending truth and then you can modularize or control each of these levels independently right okay so and once you do that then so then then then you can show that you're not generating the contradiction that then trivializes on the explosion okay but now what is ascending and descending truth no stories have been given I mean these are very interesting stories but the point is you know because because the combination of truth and logic leads to the triviality changing one or the other will will have consequences and commitments that are not palatable for people of each cap okay so everyone has to do something about it so what about instead of the ascending or descending truth what about if I were to say this is again part of my unbelief system what about saying that the error in the liar's paradox is more about language it's more about a function of language that that if we're talking about this sentence is false this sentence doesn't actually refer to anything so it is either the case that I put it this way this is the way that I like to put it is either the case that this sentence refers to this sentence is false the whole thing or this sentence is the thing being analyzed as true or false so it's like this sentence in parentheses is true or this sentence is false right those are the one of the only two options if it's the latter option then it's neither true or false because this sentence is just two words there's nothing you're not and they're not making a proposition you're just saying true or false and the first one this sentence if what this sentence is is this sentence is false then you run into an infinite regress because what the liars paradox would actually be is this sentence is false is false and then you say okay let's analyze what this sentence is it's this sentence is false which means it's this sentence is false is false it's false so it seems to me that you're challenging the self reference the self rent you're challenging one notion we can make sense of that self-reference in some circumstances yet you could have self-reference in this thing there I can I can talk about myself but when language is talking about itself in the process of the sentence being spoken it seems like there's an error yes so then okay so I guess so in that particular attempt to resolve a liars paradox to try to work through the logic without contradiction do you do you find that uncompelling well if you want to get into the study of the lion's sentence in great detail that is a fascinating subject and it has a long history it goes back all the way to antiquity but in the modern form comes from a paper by Tarski I think in the 40s and there's several attempts and responses and confusion and there's a huge debate and I think that would make like an excellent topic for you to follow up on this like just to take on the lion's sentence itself and you know go deep with it right with some expert on thing I think that would that would be really nice podcast for you yeah I think I think I'll take you up on that I know I've spoken a little bit to a few people about it but it's never I never have spent an entire episode just focusing on the liars paradox it is a big one and it will be a really interesting one well I think that's I think that's a great way to to wrap up it kind of gives me my next the next step in the journey where I need to go I really appreciate the conversation talking about contradictions and logic this has been awesome I've had a really good time and I really appreciate you coming all the way here in New Zealand to talk about this so I wish you the best with your follow-up interviews and everything you're doing in this project it seems really cool cheers thank you all right that was my conversation with Dr. Patrick Girard I hope you guys enjoyed it I probably could have spoken with him for another two or three hours just on this topic and of course as always there's lots more to say I really can't wait to find somebody who specializes exactly in the liars paradox because whether or not we can resolve that particular apparent contradiction I think has huge implications if we can I think it has huge implications if we can if you'd like to know more about my position on the topic of logic and contradictions that is the subject of my book square one which you can pick up on Amazon immediately that's all for today thanks for listening and enjoy the rest of your day