 Hello, I'm JJ Joaquin and welcome to Philosophy and What Matters, where we discuss things that matter from a philosophical point of view. In today's episode, we're concerned about paradoxes and hypodoxies. This very sentence I'm uttering now is false. Is it true? If it is, then what it says holds. Since what it says is that it is false, it follows that it is true. If it is true, it is false. But if it is false, then what it says does not hold. Since it says that it is false, it follows that if it is true, then it is false. So the sentence that I've uttered is true, just in case it's false, false, just in case it's true. Now this is a classic liar paradox, a semantic paradox in philosophy that involves notions of truth and falsity and the inconsistency that they may bring. One way to diffuse this paradox and avoid logical explosion, says our guest, is to convert it to a hyper-dox, which makes conundrum consistent, but its truth is still under its remit. Now to discuss paradoxes and hypodoxies and why they matter, we have Peter Eldridge-Mitt, a visiting fellow at the Australian National University, and along with his daughter Vera Unique, the co-inventor of the Pinocchio Paradox. So hello Peter, welcome to philosophy and what matters. Thank you JJ, it's a pleasure to be here, welcome the opportunity to speak with you about these things. Okay, so before getting into our main topic, let's first discuss your philosophical background. How did you get started in philosophy? I started off doing a double major in psychology and philosophy, and then I had a couple of severe bouts of glandular fever and I only just finished my major in philosophy and then went out to the workforce. But I was always passionate about developing an approach to paradoxes, particularly the lie paradox, so every now and again I would keep coming back and doing more philosophy part-time, mostly part-time. So it became a bit of an on-again, off-again lie. I mainly worked for private companies, I did some work in government. Being from a philosophical background did have some advantages. Some of the executives would recognise that I could take on original projects. So I developed a Qantas in User Systems Development Guide for Qantas. It had me as the author and steering committee of 14 people, each of which had to sign off on it. That was no mean feat. I had a bit of a mid-life career crisis and I came back and started a part-time PhD in 2000, which I completed in 2008. And I did get, I've had a fair amount of work. Early on I had some tutoring at Sydney and Wollongong and then later on some lecturing at ANU and CSU and a private institute in Sydney. So what's it like to be in a Qantas, the airline company and you are a philosopher trained in philosophy, so what is it like to be in that business commercial atmosphere? Overall it was good. At times people didn't know how to take me. I think at times I was known to ask too many questions. As I say, at other times they saw the opportunity, they saw me as someone they could give an original piece of work to such as that end user systems development guide. I got to talk to engineers and business people about what they were doing, which was very interesting in the development, in developing that guide, because I wanted to have an understanding of who I was writing it for and what they were actually doing. Yeah, so it worked both ways. I think it was an experience from which I overall I benefited. Okay, so but who will influence you to pursue a career in academic philosophy? Well, my idols I guess are Buridan, Arthur Pryor, David Lewis, you know, me, you know, the goal is to write something like chapter eight of Buridan's Suffered Vata on insolubles. But, you know, in real time I was influenced by Michael McDermott and Peter Roper. Michael McDermott was always encouraging me and saying, you know, what was my strategy, which I think was something I needed. The combination, probably the pinnacle of my philosophy career was an article in mind in January 2015. I've had particular projects, you know, the taxonomy of paradoxes and paradoxes, as well as writing about particular paradoxes. My daughters, you know, care paradox in particular, and pursuing, you know, my own approach to the lie paradox. Okay, so let's go there. Let's go to our main topic. What's, what are paradoxes? Yes. So in itself, people attribute, describe various things as paradoxes. So if you're trying to think about a particular thing, type of thing that paradoxes are, you find that people will talk about opinions against common sense, particularly those that have an argument for them. And in line with that, Quine would talk about conclusions that were sustained by absurd conclusions that were sustained by an argument. You get a very similar definition from Sainsbury, but he wants it to be a good argument, so a seemingly sound argument to an absurd conclusion would be a paradox. And there's some debate there as to whether it needs to be, you know, the premises need to be factual or whether they can be merely possible. There are, but you also find that people will describe issues or images. If you're familiar with issues, staircases where, where people appear to be able to walk continuously upward and yet ended up, end up where they started from. Some philosophers will describe them as paradoxical. Other definitions from philosophers like Nicholas Russia that paradox is a collection of statements, each of which is plausible on its own. But they can't be collectively true. So they're collectively inconsistent. Sorensen has a very accessible book on the history of paradox back from around 2001 or three, 2003, I think. And he says that paradoxes are a species of riddle with more than one good answer. And I think he means more than one good incompatible answer. So you can see if all these things are paradoxes that philosophers actually have too many good answers to what paradox is and there's work to do here. I think on the concept of paradox, which is part of my project for a taxonomy. You know, if you want to see Frank Jackson about generally about what conceptual analysis is, but you know, I think it's in. It behoves philosophers to analyze the concept of paradox and try to achieve some sort of reduction that accounts for all this divergent, the divergence of, you know, cases. And so I'm working towards that in my own work. There are a number of heuristics that are very useful. You know, saying basically his definition is that paradox is a seemingly sound argument to insert conclusion or an unacceptable conclusion, he says. So he then goes on to say that, well, that means you can, you've got three options. You can accept the conclusion after all, you can reject one of the premises. You can find full with one of the inference. Another one of the set of options. It sounds pretty close. But you know, then you find people like have like a barris and Dave Ripley who will deny transitivity of inferences paradox might in its argument might have a number of steps. Each of which represents an individual entrant inference. Dave Ripley might say that each of those inferences on their own is okay. But, you know, going from the first inference to perhaps the second or third, the trans sequence of those inferences is not acceptable. So what he transitivity of entailment or transitivity of those inferences fails. He says, this is not something in in saying covered by Sainsbury's heuristics. So, you know, philosophers have this habit of jumping out of the square. It's which hence I refer to that sort of trilemma of Sainsbury is a heuristic rather than being a complete and exhaustive classification of ways of ways you can handle paradoxes. In the in William James' lecture on pragmatism, his second letter he starts it off with an example of resolving a contradiction by making a distinction. And it's not clear, which is something you can do for paradoxes. And it's not clear how that relates to Sainsbury's trilemma, whether it's covered or slightly, as in some cases outside of it. Another useful heuristic is Kwan's trichotomy. The way he classified paradoxes was to classify them as veredical, falsitical and antinomous. Now, if Julius Caesar had been Kwan, you would have said all paradoxes are divided into three classes. The veredical, the falsitical and the antinomous. Caesar wasn't Kwan and Kwan didn't do that. He didn't say that his classification was exhaustive. And there's good reason to think that it isn't. But it's pretty good. So if you look at heuristic, it's very useful. So if you remember, Kwan's definition of paradoxes was that they're conclusions that are supported by an argument. Absurd conclusions rather that are supported by an argument. So for Kwan, a veredical paradox is their cases where, in fact, the absurd conclusion is true. And he gives examples like the birthday paradox, Gilbert's birthday paradox out of Gilbert and Sullivan's Pirates of Pansance. It's an operator from the 1800s, late 1800s, in which one of the lead characters, a chap called Frederick, who's in a comic scenario, indentured to a pirate king, as it were, as an apprentice or something. I don't know quite why, but he's indentured to a pirate king until his 21st birthday. And the catch is he was born on the 29th of February. There's this lovely little song about a paradox, a paradox, a most ingenious paradox where Frederick is debating when he actually turns 21 or more to the point when his 21st birthday is. Because although he's about to turn 21 in terms of his age, it's the pirate king is very much of the opinion that he's not 21 until his past 21 birth, as his 21st birthday, which Kwan would argue comes but once every four years. Because February 29 is a deep year. Yeah, yeah. And Kwan's argument for response to that is to say, well, that's true. In terms of his age, he's going to turn 21. But in terms of his 21st birthday, which he's indentured to, his birthday has to be, Kwan says, has to be on the exact date. And so it'll be many years hence. And the paradox is another example. There's a barber who's a villager who shaves all and only those villagers do not shave themselves. And then philosophers like to ask who shaves the barber. Well, you know, you can do and throw with this after a while, but ultimately it turns out that they can't be such a barber it's impossible. And the reason this is a radical paradox, I think, is because there's nothing to back up the existence of such a barber. There's no principle you can appeal to that says there should be such a barber. It's actually in, if you look at the formal logic of it, it actually instantiates a contradiction in the sense of being the negation of the theorem in logic. But basically so basically it's self contradictory. The Frederick paradox, Kwan says the Frederick paradox is a vertical one. If we take its proposition, not about something about Frederick, this is the birthday paradox. But as an abstract truth that a man can be for in four times the number of years he's passed old on his birthday. Similarly, the barber says Kwan paradox is a vertical one if we take his proposition about as being that no village contains such a barber. So in order for it to be vertical you've actually got to use the argument one says as a reductio. That is, you assume rather than have premises saying there is such a barber you you assume that there's such a barber because nothing guaranteed you there was such a barber. And then you reduce it to you was derive a contradiction and because you've derived a contradiction in certain logics that most logics that that will result in negating your assumption on one of your assumptions. And in this case, there's no such barber. There's an interesting example in Chinese. The word for paradox or self contradiction is an idiom, which which actually gives an example, which in some ways is better than the barber. And, you know, sort of, I'll move on at this point but I just want to mention that this is not a particularly western philosophy. I think it's worth mentioning, you know, these these concepts are not just Western, not necessarily just Western concepts. And, you know, the ancient Chinese had this idiom for it that works very, very well with the idea of a vertical, what Kwan just said about a vertical paradox. Now, moving on, Kwan's idea of a full cynical paradox is, as you might expect, one that commits a fallacy. So he gives the example of there are proofs that Bill gave of glaciers proofs of zero equal being equal to one where there's a thinly disguised use of division by zero. You might spot it right away, but it's there. And so these these are proofs in which a fallacy has been committed and Kwan regards them as political lead to a false conclusion. Why do we have anything else? We've got the antinomies left over. How could there be anything more than just true conclusions and false conclusions? And here's where, you know, you have to do a little bit of interpretation of what Kwan's saying. He says for an antinomy, that an antinomy produces a self contradiction by accepted ways of reasoning. It establishes that some tacit and trusted pattern of reasoning must be made explicit and hence forward be avoided or revised. So he has in mind his, you know, paradigm cases are cases like the lion is also Russell's paradox berries paradox. So in the case of the lion, which we'll come back to he would suggest that our we use reasoning that for trusted patterns of reasoning based on our concept of truth that have to be avoided and revised. Which for some of us is pretty drastic. But but you know this this is actually an orthodox approach. You know, I would say a large percentage of philosophers and mathematicians who think who's, if they think about these things or would would follow Kwan in this response to the live paradox. And we'll come back to that Russell's paradox. Well, it has to do with membership. So some classes or sets if you prefer members of themselves, some are not. So the class of all classes that are more fire than five members clearly has more than five members and is self member of itself, but the class of all men it's not a man so it's not a member of itself. So all classes that are not members of themselves coin says, since its members are not members, self members, it qualifies as a member itself, even if it doesn't. And the thing that's going on here coin points out is for any condition you can formulate such as not being a member of itself or being a person or being, you know, a horse or so on. This principle seems to naturally form a set of objects that would meet that condition. And it's not and he says this principle is not easily given up the almost invariable way of specifying the classes by stating unnecessary and sufficient condition for belonging to it. And I think that's right. It's not easily given up. But his attitude is yep, it's got to go. And, you know, sort of or it's or it's got to be revised. So, so, you know, most solutions to Russell's paradox follow that pattern they're either revising that principle or restricting it in some way if you like. And they're replacing it with other principles. And by principle, I mean what quite is saying here about an X, you know, are usually accepted or trusted pattern of reasoning. Okay, so, so far we have Sainsbury's dilemma. A paradox results from some kind of reasoning from innocent premises leading to an absurd conclusion. So either accept the conclusion, absurd conclusion, reject one of the premises or reject one of one of the steps leading to the conclusion. You also discuss coins, trichotomy of vertical paradoxes like the birthday paradox. And that's a fun, funny paradox, the barber paradox. You have also for cynical. You have false conclusions right away, facing from poor reasoning, and you have this kind of paradox, the antinomies, the liar paradox and, of course, Russell's paradox. Now, let's go to the liar paradox, which is the main target of our discussion. Yeah, just a little bit more of an introduction if I may. Okay. I think, and we'll get there directly. I want to take a step back from what Klein says about antinomies because I think he focuses on a narrow set of paradoxes. He says about antinomies. The term comes to us from Kant, who adopted it from legal terminology about inconsistencies arising in law or laws. So I guess what we do is we, and what Kant did was to generalize laws to be, you know, what we might call philosophically in the way Russell used to, in terms of the principles. And so, you know, you get paradoxes like or antinomies like the ship of the theses. There's an ancient story about the ship of theses having been maintained by the Athenians. And someone actually wrote up that, you know, sort of, well, they maintained the ship replacing a plank by plank. And then this person said, so what happens if they get gather all the original parts and they put them back together. So you have this, these two print competing principles, you have the continuity of something over time that's well maintained ship and something that's made up of all the original parts arranged in the original structure. And the paradoxical question is which one is the ship of theses. And it seems that, you know, these these two principles which normally would not conflict conflict in this scenario. Just flagging ahead something we will come to, I would say that the concept of antinomy in this, this sort of sense has it is inconsistently over determined. And when we these by accepted principles. So these principles about the ship, which is the ship of theses over determined which is inconsistently over determined which is super theses where it's the one that's well maintained, or whether it's the one from reassembled from original principles. And I've come up with a concept that I call hyper docs and it's similar to, to some other concepts around but the concept I have in mind when I talk about a height hyper docs is the dual. So to speak, of that conception of antinomy, hyper docs is under determined lack of an accepted principle that would determine the matter. And so perhaps we should just quickly do some examples in our introduction, and then we'll move on to the lie. So, probably, many people are probably familiar anecdotally with the grandfather paradox. It's a paradox about time travel. Tim, you know, for some misguided reason wants to go back and assassinate his grandfather. Tim's a time traveling assassin. He's got a perfect hit record. You know, you can build it all up as much as you want. But if there's just one timeline and grandfather and grandmother are not time travelers. Then, Tim, it can't go back and kill grandfather succeed in killing grandfather. It seems you can't because then you would have the inconsistent events, the event of grandfather dying back in 1923 versus grandfather being alive up until being alive in 1944. So David Lewis, a solution to this is a laissez-faire. Theory of time travel, you just can't do it. Even if time travel is possible. And he does contrast to this with other scenarios that, you know, so you could call them science fantasy, I think, those sort of scenarios. But he does contrast this with science fiction where people thought more carefully, authors have thought more carefully about it. So I've got my own sort of cut down example is the grandmother hyperdocs. So Tess has always wondered who saved her five year old grand from being run over by a tram. Time travels back to witness the event. It becomes clear to her that she's the only person in a position in time to save grand. She does so. And so part of her existence becomes forms a causal loop in a time. And this this this I say is a hyperdocs. Lewis would say they're possible. These sort of scenarios. So you begin to see that you can have paradoxes and hyperdocs as in pairs. So just as the concept was hyperdocs as dual to a certain restricted conception of paradox. For many paradoxes that fit that prescription description, you can have a paired hyperdocs. So let's read it just quickly the birthday Gilbert's birthday paradox. Right. So Quine says age is reckoned in elapsed time, whereas a birthday has to match the date. And February 29 comes less frequently than once a year. Consequently, you know, Frederick is about to turn 21, but he's 23rd birthday as many years away. Do you accept this? Well, yes, given the definition of. Given the definition. I maintain that it is an argument from authority. Fallacy. Mine is an authority on logic. He's not an authority on a birthday. I think it's been a very successful example of an argument from authority. It's one that's persisted for over 50 years. But if you go to the. So let's imagine that Frederick has a twin sister Frederick. And she's in a quandary about when a dilemma to about when to celebrate her 21st birthday. She thinks her brother's been hoodwinked by by pirates and Quine and other kill joys and she goes to the Oxford English dictionary. She looks up birthday. She finds out that birthday is an anniversary. No way to be celebrated on the exact date. It says it's an anniversary. Right. Once a year. To be sure she looks up anniversary. And it says exactly that. You know, something to celebrate it on a yearly basis. It says nothing about the exact date. You know, think of Easter. So that is an anniversary. Birthday is the same. They're an anniversary. It says nothing about the exact date. It does say the only exceptions it mentions are things like two week anniversary. Where it where it's specified. There's nothing in the as far as in those two entries at any rate to support Quine and in fact quite the opposite the implication. You know, you know, conversation or implication or what have you. The implication is that, you know, birthdays are anniversaries that are celebrated once a year and then Quine is wrong. Come back to Frederick's dilemma. Now what does she do? Well, she can celebrate on a 21st. There's no 29th of February. She can celebrate apparently on the 28th of February, last day of February or on the 1st of March being the date after the 28th of February. So there are principles that seem to help one way or the other, but there's no principle that really determines which there's a lack of a principle that determines which so this is another example of a hyperdocs. You know, so it. And, you know, I just wanted to snake that sort of introduction. What you're saying here is that the paradox over determines the two conditions so to speak of whatever the proposition is involved in a hyperdocs under determines that because you don't have a principle that would give her when is your birthday. If your birthday is on February 29 so you can celebrate on February 28 or March 1 depends on you. Is that the thing going on here. Yes, that's right. So paradox will determine in different ways. You can get cases where you've got more than one reason for the same result but in case of paradox you've got pretty good reasons for different results. So there's a kind of over determination of cases in the paradoxes and under determination of cases in hyperdocs. Yes, that's right. And you're saying that you could have for is it a general claim where in for every paradox you have you could have a hyperdocs as a jewel. I'm working on a piece where I claim for almost all antinomies. So it's a restricted conception of paradox. You do have paradoxes that don't necessarily relate and conclude with inconsistency. Despite what coin might have suggested. So here's an example of a lion like paradox 100% you know he's he's a pseudo paradox 80% of statistics lie. You can interpret that as 80% of statistics. Now, you know, most people have these sort of inkling sometimes. But if that's right, other things being equal as they say, you know, no further information relevant information then that'd be an 80% chance that that statistic, the one that says 80% of statistics. It's a kind of a self unless you've got some other basis for it's kind of a self undermining claim. Now if you take it to extremes 100 and you claim say someone does and claims 100% of statistics are false. Then that claim being itself a statistic has to be false. And when you think about it, if it has to be false that means not 100% of statistics falls which means that some statistic is true. Now this is a sort of paradox that church point Alonzo church pointed out about 1943. It's a strange inference. It's strange that you can get this sort of existential claim from, you know, the, you know, a self undermining claim. So, if all you knew is, you know, the original claim, you know, how, how, how did it, it seems to be deduction is, you know, non amplitude in the sense that it doesn't take you beyond what's contained in the premises. Here it seems to be an example of a deduction which does the way you learn something that you didn't wasn't obviously you didn't obviously know in the premise or the assumption in this case. So, actually you can extend this to the lottery paradox to the previous paradox and so on. Because. Yeah, you can. So say we say something I'm saying in this in this blog. It's overstated. Now, if nothing else is overstated I've just overstated. Made an update. So if it wasn't, I'd have a it seems as though it must be true. Now, when you put these sort of things together when you put something together which must be false and something together which must must be true, you know, cases like in cases like this involving truth. You seem to have a lot of paradox. So, you know, the usual examples of things like, you know, the original Greek example was something like I'm lying. Man just says I'm lying that's all he says, referring to what he's actually saying. And they took this to be about whether or not it was true. And there seemed to be quite a quite a debate about it. So this is a classic Cretan paradox, right? All Cretans are liars. Oh, well, that's even earlier. Epimenides Cretan around, you know, on about 600 BC. You know, in an epic poem, he says Cretans always lie. And this gets quoted, you know, rather humorously by Plymachus, who's, you know, one of the first one of the librarians from Alexandria before the library. So he had access to that poem, you know, and he made reference to it. And then it even ends up in the Bible, you know, one of them a prophet of their own and says, you know, apparently some poor would be to bad travel experience in Crete. The chariot taxi drivers with him off or something, you know, sort of, and he was very down on. So he said, you know, one of them even a prophet of their own says Cretans always lie, you know, he goes on with a few other. And then he says, the statement is true. Now we have a paradox, you know, so first, you know, so it was like, you know, like Glimicus, you know, sort of making a bit of a joke of the fact that this can't be right. But then he says it is right. And, you know, if you think that, you know, with respect, if you think that, you know, every statement in the Bible is true, then, then you have a paradox. But the usual examples here are to, you know, you have to do things like assume that every other Cretan statement is false in order to get this into a paradox, or you do what church did in 1943. You say, well, look, Cretans always lie, that can't be right. So there must be some true Cretan utterance, how strange. And isn't it amazing, you've got this, you know, potential paradox, this paradox, if you like, sitting there in ancient times, around 600 BCE, and then all the way through to 1943. And then someone draws out an extra variation, you know, and a significant variation. And, you know, this variation of churches is quite significant. Prior picks up on it and talks about it, Link. But the ones we're dealing with mostly today are ones that involve contradictions. So, you know, sort of going back to Quine's side here, and me being something that produces a contradiction, I don't know why I did that because it's not a technical term. And I guess it might be because you don't have to stop at the contradiction. So, if we look at our example, a more formal example, you know, we've got, as a premise, in some way shape or form, we've got something like the Liar Sentences, Liar Sentences, not true. And some people like to stipulate this, you know, I generally prefer example, you can have examples that aren't just a matter of stipulation, you know, like my favorite sentence is my favorite is my favorite sentence is not true. Then it just happens to be empirically true. That's my favorite sentence. I prefer those sort of examples, but it's more succinct hit on the screen if we do this slightly. So, you know, there's a principle of truth which we'll talk about a bit more strongly. You know, that if this Liar Sentences true, or rather, this Liar Sentences, in quotes, is true, if and only if Liar Sentences is not true. So I shouldn't spell that out, sorry. The Liar Sentences is not true, is true, if and only if the Liar Sentences is not true. And then you can see what's going to happen very soon. We just substitute what the Liar Sentences refers to, which was given in the first premise, for that quoted expression in the second premise. And we end up with the Liar Sentences true if and only if the Liar Sentences is not true. Now, most people think this is false, except for many logicians who work with three-valued logics. True. But personally, I think the natural interpretation is that it's false. But if you use classical logic, there's a sequence of ways in which you can infer that any sentence you like. So what sentence do you like? My favorite ice cream is vanilla. Right, so you can prove that. But it's not the case as well. You know, sort of improve any sentence you like there, JJ. By use of this inference called, you know, called explosion. Its Latin name was ex-contradict to the only quarterly bed. From a contradiction anything follows. Yep, exactly. And, you know, so it gets talked about by many logicians quite a lot. And, you know, nowadays still talked about quite a lot. And so it should be because you think about what we were saying about deduction being non-ampliative. You don't want to be able to infer anything in the conclusion that wasn't something you already really knew in the premises. So explosion really looks like a non-ampliative inference. And hence, you know, you'd really have to question whether it really is a deductive inference. Valid deductive, you know, a good deductive inference. And the classical logicians will say, oh yeah, but contradictions never obtain. So you're never going to get this conclusion. It's because the argument's not never sound. But still, you know, I think the dialecticists who will come to who are people who accept that there are some kind of tree contradictions. There are many, but you know, some, they will find fault with explosion. And you can see why they would want to do that. It just does look like it doesn't look like a very good deductive inference. So I have projected the liar argument here again. Yes. So how do we solve it? So if we want our logic to be preserved. So how would we say that there's something wrong here in this argument? There are a number of ways of doing this, right? So that principle of truth has been a focus for a lot of the solutions that most people following Quine and earlier than Quine, based on work, particularly by Tarski, but even going back to Russell, they would reject that principle. They would say it needs to be revised, avoided and replaced. And we might come to talk about what that principle is. I think at this point it's probably worth saying it's a principle about the truth. We tend to think that if something is the case, then it's true and that those two things go together. And we tend to think that if something's not the case, then it's false. It's not true. And those two things go together. So we tend to think that, you know, if some sentence is true, then it is the case. Some sentence is true just when it is the case, in fact. And on the next slide, we can see that the T-schema there, that this is something that, based on work, this is from work of Tarski, he didn't actually use the label until, you know, the label comes from his 1944 work, but I've just inserted it there in square brackets. He, you know, says, well, look, you know, there are these by conditional sentences, the if and only if sentences that seem to be sentences of special kind could serve as partial definitions of truth. So if you had all of them collectively, you might have a definition of truth. And he gives this general scheme of the T-schema. So, which is what I was saying, you know, if a sentence, a sentence X is true, if and only if it is the case and here you replace X with the name of the sentence usually for our purposes today, we're talking in natural language, mostly use a quote name. Then you replace the P with that sentence, and you get one of these T by conditionals, these an instance of the T-schema, the truth schema, if you like. And he says if all of those were true, collectively you would have a definition of truth, but they can't all be true, guess why, because of the liar paradox. So if you just flip back one slide. You can see there that our second premise was based on that principle of truth. Right, so what most logicians and mathematicians have done is reject that principle and revise it. Yeah, this is also known as the transparent truth idea. Yeah, so there are other ways of representing this sort of principle. So you might say that, and people do these days they talk about transparent truth where they're saying that for any given sentence you can always substitute that sentence with an expression saying that it's true. Right. A name of the sentence followed by is true. You'll be able to substitute that freely anywhere. And vice versa, you can swap out the claim claiming the sentence is true with the sentence itself. That's, as I understand it, transparency, which is obviously closely related to this T-schema and instances of this T-schema. Also, you can have rules of inference. So you can have a rule of inference for introducing truth. So from given the sentence, then you can always infer that that sentence is true, or given a claim that that sentence is true, you can infer the sentence. Those are the three typical ways of cashing out these sort of principles of truth. Other things that some logicians have tried to do is weaken the principle. So you might, those rules of inference I talked to, you might put restrictions on them and go from there and see if you can avoid the liar paradox. Good luck. So when you go back to Quine, what Quine wants to do is following Tarski and Russell, he wants to stratify the truth pretty good. So he wants to take out these principles and replace them with something else that works on the basis of subscripting a truth pretty good so that you have multiple of them in a hierarchy where one's further up the hierarchy can refer back to the other expressions containing the lower level truth pretty good. But the lower level truth pretty good can't refer to themselves, and they can't refer to truth pretty gets further up the hierarchy, roughly speaking. Um, you know. So he blames it basically the liar paradox basically on expression, the truth pretty good. Now, what we might say naive, our natural language use of the truth pretty good has to be replaced and regimented with a more formal use. And this is the way Tarski goes Tarski basically says natural language is hopeless I'm going off to do for the logic. Thank you very much. He does has excellent career doing exactly that. And he uses this stratification of languages. So each of these truth pretty good. The truth pretty good for the base level language is in the meta language the meta language the next level language the truth pretty good for the meta language is in the next level language the meta meta language. And you get what what's called the hierarchy of languages. And it's infinite. And formally they can work with this. But I don't think that's a terribly good idea. In terms of, well actually I do think it's a very clever idea and I do think it's a, in that respect informally it's very good idea but um, I'll, what I, what it does is is leave us in a quandary about what to do with natural language. Um, and I'll come back to that, which is, you know, the particular paradox is counter is in a sense a counter example to that sort of solution. Um, but there's also more of a motivation for, for not wanting to use that solution because I want to use truth in natural language. I want to have a truth pretty good natural language without going into an infinite having an infinite having to form lies everything in an infinite hierarchy. And I think there are very good reasons for doing that that hentikar and priests have brought out among others and I'll come back to that. There are other approaches. Um, so, you know, crypti says well, you know, um, taking the whole every instance of the truth pretty good out of the language is a bit sphere. You know, most of them are innocuous in the sense that they don't produce paradoxes. They're harmless and they're not paradoxical is what I should say. Um, so for those that are germane to, to, to, um, in the base level language you can talk about them being true or not true. Um, and you allocate those sentences to in your semantics to sets extension of truth and the anti extension of truth being those that are not true. And then you have those ones in the middle, like the lyre sentence that fall into a gap. Um, and don't get a truth value in the base level language. But because they don't get a true value in like base level language in the meta language in crypti's account, you can now say that they're not true in the meta language. Um, so he's still got this hierarchy. It ends up I haven't given you a full account of why but you know that I've given you the start of that hierarchy. And, uh, yeah, so, uh, you can't quite avoid the hierarchy on the crypti approach, even though you've got value, you know, truth value gaps. Um, truth value gaps in, in a simple multiple multi-valued logic they're associated with, um, a, you know, a third value. Yeah, it's neither true nor false. Neither true nor false. Yeah. Um, when we talk about it, we have to say it. And, uh, you know, um, so they tend, those three valued logics going to avoid, um, what's called the law of excluded middle, which, which says basically for, for every sentence, it's either the case or it's not the case. And because the liar sentence falls into this truth value gap, you know, for those three valued logic, the law of excluded middle doesn't appear to hold in the base in the base language. Um, so there's, there's a logic that, um, says we can have something like the T schema. Um, and we can conclude with that third, um, by conditional, which I, you know, intuitively, although intuitively it leaves false and a three-valued logic, you can say it's true. And then you can block, um, conclusion concluding that, uh, the, you know, sort of anything fall in. You can block the conclusion of the liar paradox using a three-valued logic with gaps. Um, yeah, because the argument would not be valid now. Yeah. And so, um, but in contrast to that, you can, the dialetheists are going to say, well, we can have this similar sort of three-valued logic, but we can consider our third value to be both true and false. And we can say that the liar sentence is both because that's accepting the conclusion that it is, um, both true and false. And we can conclude, um, that it's both true and false, but that argument to further argument to, um, using an explosion doesn't follow. Because explosion is invalid in the dialethec logic. Yeah. Yeah. So, um, you have those other responses, uh, to the liar. Um, and, uh, now I just want to return to the point about why, why do we want to solve this in natural language? Oh, yeah. Um, and someone like Hintika says, well, Hintika exactly says, uh, Tarski's theorem, which is his basically using a lie-like proof to show that you can't have, on, on his account, you can't have a truth predicate in natural language that for which all those instances of the T-scheme are true, in particular those for the liar. Tarski's theorem is formulated so as to deal only with formal but interpreted languages satisfying certain conditions. But assuming the conditions of Tarski's results are satisfied by ordinary language, then we cannot define truth for this language. The main characteristic of our own natural language, duly emphasized by Tarski, is its universality. There is therefore no stronger metal language beyond or over it in which the notion of truth for this universal language could be defined. So what, what Hintika is saying is we need natural language or natural language extended in certain ways as our ultimate metal language. That's for practical reasons that extensions of natural language are more or less formalized, you know, are the ultimate metal language in practice. And priest says something, you know, to the same net effect, his argument is somewhat different. But yeah, yeah, it's, so what, what happens is Tarski, Tarski's analysis of the liar is there are two things required. It's sort of principle of truth, and you need the usual laws of logic like this law of excluded middle we talked about. And he says the problem comes from this, this, this principle of truth together with being able to name everything in natural language and, and, and, and have all these instances of using the true predicate truth predicate like as with license. And that he calls semantic closure and he says natural language is medically closed. That that's the real problem because he wants to accept the ordinary laws of logic. Priest, being a dialectist wants to say no, there's problem with the ordinary. There's no problem with the semantic closure there's problem with the ordinary laws of logic. You know, you can see when you look, as I said, when you look at what explosion does for you and compare that what the idea of deductive logic is you can see something to that. Okay. And then there are these other other accounts where, you know, using truth value gaps, you know, which also make modifications to the law of excluded middle or other principles of logic and or other principles of logic. It seems like you need to let go either of the principle of truth, like Tarski did, or let go of your basic principles of logic, the standard logic, classical logic. So either turn Gabby or turn Glutty to use the terminologies. Yeah, when you go when you go down to this intentional logic, you've got a choice between Glutty and Gabby. You know, okay, but your strategy in solving the lawyer paradox is different from these current options. Yes, I think it's problem with predicate logic. Okay, so potential logic. So how do you solve this, your suggestion is to convert the lawyer paradox to a hyperdocs. That's right. How does it work. So, you know, what I want the the aim is to, when you think about, you know, the sort of metaphor of explosion, the aim is to diffuse the the the liar paradox to a hyperdocs because compared to the liar, the truth teller which says this, which is a sentence self referential sentence that says this self referential sentence is true. Well, it might be true, or it might be false. And you could, you know, it could consistently be either but there doesn't seem to be a principle that determines which it's a hyperdocs interpreted that way. And compared to the lie, the truth teller is a squib. You know, which is, it gets a lot of attention but nothing like a lie. And so my idea is to diffuse paradoxes like the lie to hyperdocs is like the truth teller it won't be the same as truth teller. But but it'll be one for which, you know, while you might be left with a residual issue about determining what its truth value is. You won't have a proof of a contradiction. Yes, I've got to restrict some logic. But I argue for some minimal restrictions. If you remember we use that what I on in the proof of the lie what I called substitution of identicals. And in first order logic with identity, you know, one of the one of the ways it can be referred to as this practice identity elimination summary, you know, which is abbreviated by the equal sign with the capital identity. First, there is a slight restriction on the truth teller that I the T scheme that I argue for the way Tarski defines it which I quoted there. Allows you to use non canonical names like we did and substitute them freely. Like we've got the liar sentence. That's a non canonical name for the license is true. So I do I do restrict in the in stating the principle I restrict the principle to using like quote names or canonical names like quote names. I also restrict substitution of identicals or, you know, equals equals a, you know, in a more formal system. And I spell this out in some recent papers in 2019 and 2020. So for your solution you're rejecting premise one and premise three here. No, no, no. My not premise one. So I'm saying premise one is fine. I'm saying premise two is as stated here is fine because it'll my restriction on the T scheme is not going to prevent premise to you've got the canonical name for the sentence used here. That's fine. It's the inference from, you know, as you went on to say yeah three I've got a problem with. And it's the inference by substitutions and identical so I'm going to restrict that so that you cannot in these circumstances substitute non canonical names like the license freely. And into expressionists like to. Now you can do it. When you've got a sentence standing on its own that that says you know what David said. Sorry when you when you when you know that what David said was that you know, honey, honey is manufactured in be hives. And you and someone else is saying, you know, and you've got a sentence that says, honey is manufactured in be hives is true. You can substitute and according to my rules into that in that context because those sentences aren't under an assumption and they aren't in a complex expression. And I know they they fall within the scope of the truth predicate. So, I'm allowing quite a lot of substitution of identicals and all the normal ones you should think I'm restricting it. And I have to motivate this right so what I what I've not emphasized emphasized in all these and what you know you didn't get when I talked about sainsbury she heuristic is that for each of those options he refers to you have to justify. Now, the thing that's really hard with most antonymies is most people just like a candor example to to an inference if it's going to be flashers or, you know. And, you know, it's kind of these things are kind of like, you know, division by zero, they're kind of more in the case of where it's a systematic issue. And any kind of sample you give is likely to involve division by zero. So, um, you, I do give two separate lots of arguments for both these restrictions on one one one a lot of arguments in my 2019 and another lot in my 2020. Some of these arguments are ones that have been sort of peddling since 1980s never presented papers but never managed to publish before. I'm very glad to have 2020 article finally published. So, um, okay, the arguments are, you know, sort of more than we're going to go into today. I'm just saying that I do I the onus is on me to justify. I do believe I justify it in two different ways in the 2019 article and in 2020 article. Okay, so in 2010 you and your daughter very publish an article and analysis entitled the Pinocchio paradox. I'm not sure if you know what your paradox became an internet meme. It's famous and the internet if you search. There's a wiki pija entry on this one. But you can tell us something about this paradox and has anyone saw that yeah. Yeah. I'm the proud dad over this one. You know, I started my thesis back in 2000 and I explained what I was working on as best I could to my children. And for some reason I thought it was a reasonable, you know, question to ask them to come up with their own variations we've talked about variations like this. The one about statistics, you know, church's variation on the epi amenities, you know, these things keep coming up. And anyway, for some reason, I thought it was reasonable with it. My son came back with one where he says, you know, policeman asks a suspect whether he's lying and suspect simply says yes. And that one's like one in the literature from Jonathan Kahn. So I fairly came back with Pinocchio says my nose will grow will be growing. And we talked about it a bit, you know, about the tense of it because, you know, there's some ambiguity in the story about when exactly Pinocchio's nose grows relative to his things. Because of course Pinocchio is the hero of popular Italian children mobile bike a lotty. He's a puppet that grew up into a little boy at the end of the novel when he learned to pull his own strings. It's a wonderful model. And Pinocchio's nose grows just when what when what Pinocchio is saying is untrue, you know, which, you know, verani picked up on and she's quite right. So Pinocchio, she says, my nose is growing. Then if his nose is growing, well, then by the Pinocchio principle, what he said is untrue. And then by the principle of truth, which, you know, she didn't know that, but, you know, everybody intuitively really uses. What he says is true because she knows is growing. So it's both true and false. So if Pinocchio's nose is not growing, then what he said by the Pinocchio principle is true. But by this principle of truth, it's not true. So it's both. And so you've got this paradox, which is a version of, you know, a variation of the lie. And I sort of was very impressed. And so in February 2000, when I got it to draw it up for me, as you can see there. And then I made a foil of it back in those ancient days. We used to use foils or I did, which you put on a display thing that shines light through and displays the foil up. Before I was going to give a talk, you know, just to keep people, you know, entertained a little bit before things started off for those who came on time or a little bit early. And so this is the picture. Yeah, yeah. So I had to draw that and then I put it on a foil and I used to use it at the start of my talks. You know, for example, in the 2004 Australasian Association of Philosophy in 2005, that was Australasian Association of Philosophy talks that I gave. I had this up there, you know, the start for people who were turning up, you know, you know, give them something to keep amused before I started talking. And then this, this, this is off the front page of my thesis, which was published online in 2008. And, you know, people quite, you know, kindly took an interest in this. You know, sort of, I appreciate it when people, you know, you know, refer to Veronique because she did devise it. And yeah, so, you know, I'm sort of happy that what was attention on on the, on the internet that that image is very nice and we've permission to use it today. So it's, you know, very proud dad. But the significance of the Pinocchio paradox has a variation, you know, like we said with churches variation you have an example of a paradox that doesn't result in directly in that seems paradoxical before it ever gets to a contradiction. And then we have a in the Pinocchio paradox, we have an example of a liar paradox using an empirical predicate or not a semantical predicate and not a synonym. So when people, when the original example I am lying, it was used as a synonym or at least to entail that, you know, something that involved the concept, a semantic predicate, you know, about lying or not telling the truth. I'm not an empirical predicate, whereas my nose is growing, Pinocchio's nose is growing as this empirical predicate is growing. And it's in no way a synonym for truth. You've got to add this Pinocchio principle, which makes it, you know, sort of somewhat as people point out somewhat fanciful. But nevertheless, you know, sort of doesn't make it self contradictory. So the difference between the Pinocchio and the, and the, the barber paradox I've argued is, is that the barber scenario on its own is self contradictory and this, you know, so consequently it's, there just is no, it follows as a matter of, you know, pure logic that there is no such cannot be. It doesn't follow as matter of pure logic or doesn't seem to that there is no such Pinocchio and cannot be. It's not obvious how you prove that, except by using the Pinocchio paradox, but then you've got to involve truth. And it looks very much like a variation of the lie that uses an empirical predicate, which is a counter example to Tarski's idea analysis that the problem came from the semantic predicates being in the same level of language. And in fact, in an article in 2018, I try and prove that, you know, sort of, if you're with those, if you use the Pinocchio paradox, you can give a counter example to you can actually prove the counter and this is a counter example to Tarski's account of the lie. You can prove a version of the lie. Granted that I mean Tarski wouldn't allow you to form the sentence in a formal language, but we're talking about natural language here. Even if natural language is stratified according to the rules that he, he stipulated, because it's an empirical predicate and the very sort of predicate you want in the base language that Pinocchio can utter his statements in the base language, then you can have truth for the base language in the metal language. And you can give a stratified version of the T schema as per Tarski's account, and you can prove a contradiction. So I claim. And I set it all out in 2018 so so it's in there. But has anyone solved it yet? There have been some very good attempts. There are there are lots of things out there on the internet that are reminiscent of other approaches to lie. You know, so there's there's a lot of debate about, you know, you know, given the form, you know, given if the Pinocchio principle is about lying then then what's lying really about, you know, this truth and can can we get around that. And that proves to be quite interesting. But, you know, I'm mostly interested in the version of that that deals with truth because I want to have truth in the middle in the in the baseline, you know, I want to be able to use truth in natural language in a way that natural language can be the middle language for whatever. You know, whether when I'm comparing systems of logic and talking about these paradoxes and so on. And I don't think I don't think it's. I haven't seen a complete solution, but I've seen some very interesting work. No, you know, I find it interesting to look at some of these these comments. So I would try and solve it the same way I've solved the line. So I think it's going to have an identity crisis. No case had a number of logical adventures. And so, you know, my original write up with very nice in 2010 was to use it. As I said to a counter example as a counter example to to task ease approach but I but I also tried to, you know, write short articles where I go goes on a number of logical adventures such as Pinocchio against the dialy theists, you know, we got this. You know, brave little. You know, and, and he has some success, I think, but the dialy theist think otherwise and JC bill was kind enough to take this take this on and give a response and we had a bit of a debate in the analysis journal. And the. The Franka Dugastini and Elena, who also responded on behalf of the dialy theists and find one that's quite challenging actually so I've got to get in there and defend my daughters. And, you know, and respond to this. I think, I think, you know, part of the response has been how implausible this has been and when we go some of those definitions of paradox I mentioned referred to plausibility. And so they've challenged. I feel challenged on this. You know, I mean, think about the grandfather paradox how plausible isn't Tim is going to be able to time travel anyway, I mean that never was never an objection to the time travel. And, and, and you know, this is a bit of a name hominin, but you know, it's sort of I find it ironic but the dialy theists to accept true contradictions. About the plausibility. But they do, and they do it very well. So, you know, so, so I'm pleased to have that debate. You know, it's an employee, they've got other things to pay attention to and I'm pleased that they've paid attention to this. Sorry, I know Luna has also used the paradox in some of his work to argue against physicalism for truth. And, and I think he does a good job. There are some some other comments out there that I need to take account of in various articles so you know, you know how this is far more important. You know, it gets more notorious than your own. So, it's a, it's a pleasing conundrum. Yeah, I don't know more personal though. So what's your advice for those who want to get into professional philosophy. Look, um, you know, given my experience and, you know, my midlife career change sort of thing. I would encourage people if you have an opportunity to work, do some work outside of academia to take it and then come back. I think it broadens your life experience. Other than that, I think there are people better qualified to to advice. Um, you know, based on my own experience, that's, that's, that's my advice. I think getting, getting broad experience in life being, you know, if you're going to do philosophy, you want to philosophize about things that you've got. Not only knowledge, but experience. Would you say that your career in philosophy is worth it? Yeah, look, here I am. I'm lucky enough to enjoy being, you know, an honorary visitor at ANU. I've got things that interest me that I'm pursuing with the support of philosophy at ANU. What more can you ask for? You know, it's, so in that sense, I think it's worth it. If you mean in terms of money. That's another question. So, you know, here we, here we, here we make a distinction, you know, it's not just about paradoxes. We want to make distinctions, but, you know, in terms of life projects, you know, I, yeah, I think you can find interests and pursue them with philosophy that are, you know, meaningful. Okay, so thanks again, Peter, for sharing your time with us. For you guys, join me again for another episode of philosophy and what matters where we discuss things that matter from a philosophical point of view. Thank you very much, JJ. Thanks. Bye.