Within Reason - #118 Joe Folley - Everything You Need to Know About Logic

Episode Date: August 24, 2025

Joe Folley runs the YouTube channel Unsolicited Advice. He graduated from Cambridge University in 202with an MPhil in Philosophy, specialising in logic. TIMESTAMPS: 0:00 - What is Logic?5:04 - Aristotelian vs Stoic Logic12:47 - How Logic Provides Clarity18:42 Ambiguities in Logical Language29:07 - Validity vs Soundness in a Logical Argument39:40 Why Anything Follows From a Contradiction47:42 - The Law of Non-Contradiction56:27 - What is Truth and Falsity in Logic?58:36 - Does Your Mum Know You’re Gay?1:05:05 What is Fuzzy Logic?01:08:14 - What is Modal Logic?01:13:40 - Informal Rules of Logic01:29:15 - Resources to Learn About Logic Learn more about your ad choices. Visit megaphone.fm/adchoices

Bankmore on course when you switch to a Scotia Bank banking package. Learn more at scotiabank.com slash banking packages. Conditions apply. Scotia Bank, you're richer than you think. Joseph Foley, welcome back to the show. Thank you very much having me. What is logic? Of course, there's no foreplay today.

Never. Never with us. I suppose, like many things in philosophy, it depends who you ask. So, as it's traditionally concerned, see, sort of going all the way back to Aristotle. Logic is supposed to be the most general principles of reasoning such that when you have a set of stuff you already know, or you're already asserting, you can through a series of indubitable steps get to a further conclusion that follows without any doubt and without any possibility of being wrong from those things

you initially know. So, classic example from Aristotelian logic is all minimal. mortal. Socrates is a man. Therefore, Socrates is mortal. And effectively, you know, Aristotle's point here is that you can't even imagine that being false. There's no situation where all of the premises are true and the conclusion is false. So you are carried in some sense from the premises to the conclusion without any kind of possibility for error entering in. That's kind of the classic Aristotelian picture. And amongst philosophers, This has remained, I would say, relatively orthodox, right up until maybe the end of the 20th century. And even now, it has really, really staunched offenders.

So this idea that there is a one true logic, and that's the thing that philosophers are interested in when they talk about logic. A recent book was put out, recently called the one true logic that was defending this idea by Owen Griffiths and Alexander Pazzo, I think his name is. you remember Owen Griffiths because he supervised my undergrad dissertation, and he wrote it with this other guy who I'm sure is very, very clever, but I've never met him, so his name doesn't stick in my mind. But another alternative conception of logic to this kind of historical orthodoxy, but in the modern day, I would say it's become less orthodox, is in some sense, logic as tools. So logic as tools for modelling or logic as ways of making sense and specifying different areas of discourse. So under that conception, you might have slightly different logics for different areas

of life. And you might have logics that are, you know, you can have different formal systems and different logics for dealing with different situations and different topics. So I don't think of a good example. So a logical pluralist might say, for instance, that say you have a certain set of logical rules, so a certain set of axioms and rules of inference, that is just as good as another set of axioms and rules for inference, they're just dealing with different stuff. So, you know, this is kind of an analogous position, which is similar enough and maybe familiar enough for audiences to kind of get intuitive,

hook onto this would be something like, you know, moral relativism, right? The idea that there are different moral systems, each of which are, in some sense, as good as one another, and that is determined by, in a more relativist case, you know, your kind of moral community, in this case, maybe by the context of what you're doing. So logic can sort of be variable as to what it is and what its goals are, depending on what it's being used as a tool to do. Yes. So it's not some like true thing that exists and can be ascertained, but rather it's like this methodology. It's a tool we use to perform particular tasks in particular areas.

Yes, so that would be a kind of classic pluralist conception of logic. And different pluralists will cash this out in different ways. But that's kind of, as far as I can tell, the sort of mainstream. pluralist view is to kind of roughly speaking logic as tool. And, you know, this is kind of an ongoing dispute in the philosophy of logic is which of these conceptions is true. Is there a one true logic? And, you know, there's a lot of very technical arguments going back and forth. I know in this conversation we're going to avoid technical stuff generally because it's just very difficult to speak out loud without writing anything down in a way that either is true from the speaker's

perspective and also makes any kind of sense to anyone listening. If you want to say a really, really complicated series of symbols and numbers and esoteric notation. I'll just get Alex to put on screen, isn't there, right, mate? But a classic kind of argument that might go back and forth between logical monists and logical pluralist is the logical monists might say something like, well, you've got to have some kind of criteria, criterion or set of criteria for what logic is strong in different areas. And in some sense, that presupposes some kind of reasoning system. So, you know, one monist argument is to say to logical pluralist, you are at least committed to some kind of pre-existing logic when you are choosing

between these different logics. And logical pluralists tends to kind of respond by appealing to the fact that seemingly different logical rules treat different domains very, very well according to different rules. So that's kind of a kind of quick gloss of some of the different conceptions of what logic is. Yeah, I think that's, yeah, I think, yeah. Yeah, logic can be thought of as like a set of rules, like rules of engagement for not just philosophy, but kind of for thinking. They're thought of as like a prerequisite for thought in order to have a thought. There needs to be like some logical rule by which that thought like functions. But it's also as well as being thought of as like methodological, like rules governing thought.

It's also thought of as like the foundation of thought as like a thing that actually sort of exists. But I guess the thing that exists are constraints. I mean, in a way, logic could be thought of as a system of, like, constraining axioms. Yes. So I think that there's, again, this kind of, I don't know, touches against the fact that we use the term logic in a kind of number of different ways. We kind of have a loose everyday sense of just something being logical as sort of a by word for sensible or good or cognitively. Oh, yeah. We should clarify.

And that, you know, that does bear a passing resemblance to what philosophers mean by logic, especially pre-19th century philosophers. Oh, yeah. Quite often, because one of the most interesting developments over the history of logic is, you know, very, very early on, you've got that Aristotelian picture and, you know, developing alongside this Aristotelian picture, or slightly later, but effectively kind of taking up alongside Aristotelian logic is Stoic logic. You know, people think about the Stoics in terms of their ethical system. That's kind of the thing that exists in public consciousness. but Stoics also were very keen logicians. And many of the kind of standard terminology that we have today around logic is inherited from the Stoics. So things like modus ponens, modus tolans, these are stoic logical rules. And those of you with a kind of background in maybe more, if you've done like an introductory course in formal logic,

you might recognize that the Aristotelian picture is very predicate-based. It's very, you know, all-A-R-B, you know, that sort of structure. whereas this modus ponens picture exists more at the level of proposition. Okay, so help us understand this. So modus ponens is a form of argument. Yes, modus ponens is if p then Q, P, therefore Q. It's like, you know, the kind of, it's such an obvious rule that we use it all the time and basically barely think about it. Premise one, if this, then that, premise to this conclusion, therefore.

Exactly, yeah. And that's called modus ponens. That's like a form of argument is known as modus ponens. Yes. And if you know, that's, that argument schema is occurring at the level of propositions. That's kind of full sentences, you know, if it's light outside, then the sun has risen. It's light outside, therefore the sun has risen. It's like, you know, regardless of whether you think that's a sound argument, it is at the very least a valid argument. And that's using modus ponens. And whereas the Aristotelian picture is more about predicates and terms, you know, in Aristotle's terminology belonging to. predicates and things like that. And that's, and the interesting kind of historical tidbit is that these two systems don't get unified until 1879, with the publication of Frege's Begriff-Shrift. Wait, so this predicate version of logic, it is more sort of like, you know, all cats are black. You're sort of predicating things of objects, you know. All cats are black, like there is a cat,

therefore there is a black cat, that kind of way of thing? That kind of thing, yeah. So Aristotle has four types of statements in his logic. He has all statements, so you know, all X or Y. He has no statements, no X or Y. He has some statements, some X or Y, and he has, you know, the kind of denial of some. So some are not one. And these, you know, have interesting logical relations and stuff like that. But again, you notice this is all... Some cats are black. No cats are black.

Yes. And some cats aren't black. Some cats are not black. And you can kind of chart out the inferences between different formulations of these sentences and stuff about that. But my broader point is that that's all occurring kind of in the sentence. This is, these are charting relationships between terms and properties of terms. Within sentence.

Yes. Whereas the stoic system. The if p, then Q. Yeah, that's kind of taking the sentences. Taking sentences or propositions already. and linking them together. And for a long time, Stoic Logic was just really neglected, you know, kind of Kant went so far as to say

that logic never advanced a step after Aristotle, which is like not only disregarding all of the kind of medieval thinkers are very interested in logic. Like William of Occam, you know, Occam's Razor was also a medieval logician, but also kind of Kant simply does not care about this whole branch of Stoic logic. But in 1879, Götlob Frege, who's a kind of German logician and philosopher and mathematician, manages to unify these all into an incredibly elegant technical system.

And it's just kind of, it's a very beautiful bit of history in the sense. You've got these two systems that emerge kind of fourth century BC. And then over 2,000 years later, they're kind of reconciled. They're kind of remarried when intuitively, you know, these feel like they should go together, you know, it feels like this should be reasonably easy to reconcile, but it takes an awful lot of mathematical machinery to make that work such that it's not until the 19th century that these two systems, both of which are obviously correct in some sense, can be reconciled. You know, you have this kind of very difficult to think of now, the idea of there being, you know, two rival true logics in a sense that clearly should be able to to melt. But, you know, They just, just nobody can quite figure out how. It would be analogous to like the modern scientific problem of general relativity and quantum mechanics. They're like, they're both true, but like they just don't. And businesses are searching for some way to put them together.

And Frater, one of the reasons why he's gone down as kind of the father of modern logic, is because he manages to stick these two things together. And the way he does that is kind of technical. I kind of don't know if it would be worth talking it through particularly. But anyone that's interested in it could go and have a crack at the Begriff shift. Homework. Yeah, it's fascinating. But one of the developments in logic is that initially people like Aristotle and the Stoics are largely conceiving of logic has something to do with language.

So, you know, codifying patterns of reasoning in language that are obviously true. And as we get more towards modern logic, if I use the term logic in a philosophy faculty today, it will immediately conjure up images of formal systems. So axioms, rules of inference, proof systems, model theory, that kind of thing is the world of modern logic. And in a lot of ways over the course of the 19th century, you get this kind of increasing mathematician of logic and then kind of into the 20th century. Logic kind of is both, is firmly philosophy, but also firmly maths, which is one of the reasons why it's so fascinating. You know, if you have a passion for technicalia, then logic is very interesting. or so if you have a passion for philosophy, it's very interesting. And the other thing that I think, you know, I realize I'm kind of pitching logic now.

And that's kind of my goal with this. I really do think that logic is an eye-opening thing to learn. And I think that what makes it so brilliant is this eye-opening to learn even if you only spend a couple of hours learning it. Really, you know, if you're not interested in kind of the mathematical side of it, even just five hours spent at, you know, cumulatively over the course of a year, say, at a desk looking at a logic textbook will fundamentally transform the way you see the world. We'll get back to the show in just a moment,

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This discounted link will also give you four months extra on a two-year plan. And with that said, back to the show. Well, what does it mean? I mean, okay, so people will listen and go, okay, logic is the process of deriving conclusions from premises. And I can do that, you know, I can look at, even if something is not true, I can derive conclusions. like, you know, all cats are black, Tom is a cat, therefore Tom is black. I'm clever enough to recognize that that conclusion follows from the premises, even if the premises aren't actually true.

The conclusion follows from them. So what do you mean, like, learn logic? What kind of thing would I be learning if I sit down and send five hours at a textbook? So you'd be learning how to put things in... Well, yeah, that too. But you'd be learning how to put things in propositional form and also spot going beyond kind of Looking at individual arguments and being like, oh, I can tell that that argument's pretty good and I can tell that argument's pretty bad. Learning the structural characteristics that make an argument, good or bad.

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But you'll find this in pretty much any philosophical discussion. People kind of talk in paragraphs, how we naturally speak. And to put something in propositional form is to take that paragraph and extract from it the argumentatively relevant parts and say, right, here is the stuff that they're assuming. And here is the conclusion that they're drawing. And here are the rules that they're invoking to draw it. And the reason I think this can be very helpful is because. Quite a lot of the time, I think, just in everyday life, we are often mistaken about, I think, what the people we're talking to are arguing. Yep.

And one of the things, I think, putting things in propositional form can do is you can then, you know, you have a working model of what your interlocutor's argument is. Yeah, it offers clarity. Yeah, I think you can agree with your interlocut. Before you agree on whether the argument works or not, like, you can first agree that this basic, clear form. with clear, like, premises and conclusions, you can at least agree, is that the argument we're having? And then you can have the argument. Yes, I think that in kind of non-mathematical practice, this is one of the best uses for logic ever. Leibniz was so keen on this, on this use of logic. He thought that one of the, you know, the, this is before we kind of had modern

symbolic logic. He thought, one of the things that we need to do if we're going to have conversations about philosophy or anything, really, is we need to have a language that, to the greatest extent possible eliminates ambiguity. And, you know, in theory, we could take a very, very complicated argument and put it in like, you know, second order formal logic or something like that. And that would be about as clear as you could ever get with formalizing an argument. That would take a very long time. Putting something in propositional form is sort of one step along that journey. So, yeah, there are still things that can remain unclified if you put something in propositional form. Because it's a series of propositions. Exactly, yeah. But the very least,

are one step closer to having clarity. And, you know, I think that, yeah, I think that if you don't have clarity on what arguments you're using and what problems you're tackling, it's just very hard to make any kind of progress whatsoever. Yeah. And notice the way that you, like, when you were just thinking about how to put that sentence together, you were like, well, you know, I think that if, if you don't have that kind of clarity, then it's difficult to put arguments together.

That's a, that's an if then statement. Yes. And I think people who are good at communicating will be good at sort of instantaneously putting things into clear propositional form. So instead of saying, like, you know, man, like, I just feel like if, like, God has to exist because if, like, the universe obviously had a beginning, right? And if the universe has a beginning, then, like, God has to exist because what caused the universe?

You could translate that into premise one. Everything which begins to exist has a cause, premise two, the universe began to exist, conclusion, therefore the universe has a cause. And that's way clearer and you can interrogate those premises versus is that word salad I said a second ago. You get what I'm saying, but you'd have to be like, well, wait, so when you said, are you saying this? Are you saying that?

And it would be a big sort of old jumble. And putting it into propositional form is the first step towards, as you say, clarity. Yes. And I think that one of the great things about putting things in propositional form is, as I say, but think basically anyone could learn how to do that with almost any argument, almost any philosophical argument in an afternoon. If that, really, I mean, you just kind of, after a while, it becomes almost second nature. And it will instantane.

I mean, it doesn't mean you have to start talking in terms of premise. But it means that instead of me saying like, oh, God exists because I think that like, you know, the universe began and like obviously God must be the, I don't need to put it into propositional form, but practicing propositional form will at least allow you to say something a bit more clearly. Like you might say like, well, I think that, you know, the universe had a beginning and I think if it had a beginning, that beginning was God. That's not a logical argument. That's not that's not fully like propositionalized. but the practice of pulling out, as you say, the argumentatively relevant, like, statements, the atomic sentences that make up the argument and just dealing with those as clearly as possible is a wonderful exorcist. Yes. And this is, this lusting after clarity is really reflected in the types of works that logicians have tended to write outside of logic.

So, again, to take the example of Frege. Frege is not only concerned with the idea of creating this perfect logical language, and he's also very, he was some one. He wants to ground arithmetic in self-evident logical axioms, which he doesn't end up doing because of Russell's paradox and then because of Gerdell's Incompleteness theorem later, which showed that he wouldn't be able to do it anyway. But alongside that, alongside these aims, he is incredibly obsessed with clarity and language. A really good example of one of the methods that Freya uses to clear up ambiguity in language is he introduces the notion of a sense. So we think about a word and its reference, right? So I have a word, Alex, it picks out you. When I say Alex, I'm picking out you in the world. Frege notes that if this is our only picture of language, then we're going to run into some logical confusions. And these are very, very rarely. For 99% of your experience, this isn't going to matter, but occasionally it really will. So I think the classic example of this is Hesperus and phosphorus are two names for Venus in the ancient world. but people didn't realize they were the same planet. So Frege basically goes, well, they weren't obviously the same to the people. They had to then go out and verify that they were both, and they both happened to be Venus.

So they are, they, Hesperus equals phosphorus. It's a statement that is true out in the world. But Frieger essentially says, well, they must have had a mode of identification for picking out Hesperus or phosphorus. And that's what he calls the sense. and that's a very abstract example. It's not immediately clear how it might be helpful. But let's take, for example, you know, you're just talking to someone actually about Descartes and Descartes' arguments about the separation of mind and body.

So, you know, to go back over it, Descartes' argument about the mind being separate from the body is that he could conceive of the mind existing and the body not existing. So he says they must at the very least not be equivalent. These are conceptually separate entities. And you could talk about, you know, you could reiterate that argument today with the nervous system. You know, the mind must be separate from the nervous system because of this conceptual freedom that I've got to imagine the mind separate from nervous system. And one way to, and, you know, that kind of, people immediately think that there's something quite fishy about that argument. Like, what's going on there?

This seems like it seems like it's in some way tricking yourself. in drawing a substantive conclusion from mere facts about language. And Frege's senses can come in real handy here because he can say, like, well, they might pick out the same objects, but your modes of identification, your senses for these two words are different. As in you're saying the same thing, but in a different sense. Yeah, exactly. And he has a very metaphysical picture of what these senses are.

He kind of thinks they're in this kind of platonic realm. But again, we don't need to get into that. So a moment ago you said it was phosphorus and hesphorus? Hesperus, which you told me earlier is, I can't remember which way around it is, but one means the morning star, one means the evening star. And so people would look in the morning and say, that's the morning star. People look in the evening and say, that's the evening star. Turns out it's actually the same star.

So they are talking about the same thing, but they're not talking about the same thing. Because one person thinks it's the evening star, one person thinks it's the morning star. And unbeknownst to them, it's the same star. So I guess the paradoxical question is, are they talking about the same thing or not? And so Frager is saying, yeah, but no. Yeah, they're talking about the same thing, but in a different sense. Essentially. And it's, as I say, in a situation like Hesperus and Phosphorus, it's kind of quite intuitive that there's something like that going on. I think that the utility in something like Frege's sense and reference distinction comes in these more philosophical arguments whereby somebody's making a point about conceptual distinctness and maybe they're right and maybe they're wrong, but it seems like they're inferring too much from the linguistic facts at play. I tell it, a really good, you know, you've talked about meta ethics a lot on the channel before. And a key argument in meta-ethics is the open question argument, right?

Of any given definition or any given naturalistic definition of goodness, somebody can ask, is that really good? And we don't ask that for questions like, for identity statements like a triangle has three sides. If you were to say, does a triangle really have three sides? What does that mean? Of course it does. That doesn't have traction. It's sort of like, you know, if you've got a definition of an object, like for that definition to be like complete and full you wouldn't be left with the question

but is that really the defined thing so if you say you know a triangle is a three-sided object with straight lines and you go yeah but there's three-sided objects with straight lines is it really a triangle yeah of course it is whereas it seems for g e more who comes up with this argument that any definition you could have given the word good well good is you know pleasure if it is pleasure really good it seems like that's still a legitimate question which means we haven't got a sufficient definition. Well, good is what God commands. Is what God commands like really good?

Are you sure? And you don't look with confusion and go like, what the hell are you talking about? It's definitional. It seems like, as Moore puts it, an open question, which more uses as an argument to say that, therefore, there can be no naturalistic definition of what good is because whenever we define good,

we're always left with this open question. But is it really good? Yes. We're not when we do have sufficient definitions like that of a triangle. And say you wanted to solve this argument, well, one of the things you might appeal to is senses. You might say, well, it's not, there is still a gap there to find a substantive definition of goodness that isn't just intuitionistic and kind of in this kind of more style irreducibility of the good. You would have to posit that the good and this other property are in fact referring to the same thing, but they're getting at that thing through different sense. And this is one way that people have attempted to solve the open question argument.

I don't kind of, I wouldn't be able to kind of kind of rattle it off the top of my head. But for example, say, you know, to construct an argument like that, a utilitarian trying to solve the open question argument might draw on this kind of phrygian backdrop to say something like, well, yeah, no, I accept right, that you can ask whether the greatest good for the greatest or the greatest net pleasure is good. I can, I understand that when you're just presented with the words, you can ask, oh, is that really good. But say I went out into the world, and I found that everything that elicited the kind of the good token also coincided extentionally, totally with the greatest net pleasure, then, or even like mostly, then a utilitarian could appeal to this Phrygian sense thing and say, well, maybe they're getting at the same thing, sorry, maybe they're getting at the same reference through different senses. So that, again, there's a couple of philosophical arguments. An example that comes up in other context is the difference between water and H2O. Like, they are the same thing, but they can kind of apply in different contexts. For example, somebody a thousand years ago, if you described to them a hydrogen molecule and an oxygen molecule, 2 hydrogen and one oxygen molecule put together, they wouldn't think of water.

And yet they'd be thinking about H2O, but they wouldn't be thinking about water, even though H2O and water are the same thing. So they are the same thing, but they're also kind of not, depending on what like sense you mean the term. Exactly. So, and I'd say, Frege has his own technical definition of sense. But I wanted to draw on this just as an example of how the kind of logicians' obsession with specificity can be very helpful for either clearing up arguments that you kind of think, oh, that's a bit spurious, but I don't quite know why it's a bit spurious. or potentially solving problems or opening up new avenues to solving philosophical problems

that might not have necessarily been spotted before. I think that another, you know, people online are often very keen on the idea of kind of logical fallacy. If you scroll through a lot of comment sections on philosophy videos, see people having arguments, you might say something like. That's a fallacy.

That's a fallacy. I think that this is, in some sense, sort of attempting to identify unsound logical structures. These are logical structures that we want to avoid. And, you know, I think that, sorry, invalid logical structures. And one of the benefits of potentially, you know, just dipping your toe into something like even just propositional logic and maybe a tiny bit of predicate calculus is that it kind of becomes immediately clear why a fallacy might be a fallacy. You know, in theory, any formal fallacy is just reducible to. non sequitur. So it just doesn't follow. So, you know, imagine affirming the

consequent. I'm pretty sure is the name for the fallacy where you say A implies B, not A, therefore not B. Okay. So let's slow down for a second. We're talking about a logical, we're talking about logical fallacies. Yes, yeah. And affirming the consequent. Is that what's called? Where you say, so if A then B, and then you say not A, therefore not B, which doesn't actually follow. Yes, that So, for example, if it's raining outside, then I'll get wet. It's not raining outside, therefore I won't get wet. Yeah, so you think that, you know, there might be some kind of... It doesn't follow.

Machiavellian person outside there with a bucket waiting to my tip. Yeah, I could just go and have a shower. Exactly, yeah. Yeah, that's a much better example. But it's, it kind of casually, it kind of feels like it should. And there are contexts where you can imagine if somebody said really quickly, like, oh, well, if it's raining, I'm going to get wet. And you'd look and go like, but it's not raining.

So, okay, cool, I won't get wet then. It's like, all right, nice. when actually like oh slow down there and in that circumstance like maybe it doesn't matter like you kind of you know what I mean it's fine but in an argument where you're like arguing for God's existence or meta ethics which is really important you could be technically that doesn't follow we need to be more careful yes it doesn't you're not going to spot that unless you when you hear somebody say gosh I'll get I'll get wet if it's raining like outside like if you are not able to translate that into if it's if it's raining then I will get wet it is raining therefore I will get wet it is not raining if you can't sort of do that translation in your head you won't spot that it's not yeah the very least it's going to be slightly harder right

as in I imagine that you know people yeah you can still spot it the white chocolate macadamia cream cold brew from Starbucks is made just the way you like it handcrafted cold foam topped with toasted cookie crumble it's a sweet summer twist on iced coffee your cold brew is ready at Starbucks

but it's just a more systematic way it's much easier to spot it yeah yeah and the The important thing here is that the conclusion isn't guaranteed by the premises. You can imagine a situation where all the premises are true, but the conclusion is false, which is the logical definition of validity. Okay, because this is really important, right? So what does logic apply to? Like, we often talk about logical arguments, right? Is it like logic is a thing which applies to arguments?

Does it apply to sentences? Does it like, what is logic? I would suggest that used on logic applies to reasoning. and quite broadly construed. If you're dealing with something like propositional logic, its subject matter is propositions. If you're dealing with something like predicate logic, its subjects are going to be predicates and objects

and the logical relations between them. In theory, you know, you can kind of say, well, any discourse whereby we want to guarantee the truth of our conclusion from our premises is going to be governed by some kind of logic. It is, or at least it claims to be the kind of normative rules of thought. Yeah. You know, what the original kind of intuition that Aristotle's having, or one of them, when he kind of founds logic in the Western philosophical tradition, is basically just how do I, how do I know that I'm thinking well?

How do I know when I'm thinking poor? And, you know, there are plenty of other non-logical aspects to that question. So you could say, well, one of the things you might want to figure out if you're thinking, if you want to, you're thinking to be precise would be something like, well, I ought to define all of my terms very precisely. Because if I don't, then I'm not, I'm not going to be quite clear on what I mean. And that's not a, that's not a matter for formal logic, but it's nonetheless a kind of general rule of general bit of cognitive hygiene, I suppose. and so yeah but logic I would say tends to deal more with the idea of reasoning so relationships between statements propositions terms

basically you know if this thing is true what do I know from that yeah like what can you derive from one thing what can you grow out of it and so there's a term that you've used a few times now which is the term valid you've also used the word sound which does not mean the same thing so let's take an argument like modus ponens

So if P, then Q, P, therefore Q. The reason we use letters there is because it doesn't matter what you plug in, it's still going to work the same, right? So you could say, if the sky is red, then my bathtub is filled, the sky is red, therefore the bathtub is filled, right? It doesn't matter what sentences you put in, if P then Q, P, therefore Q. That argument is an example of a valid argument. What is validity? What is validity in... I was saying the word valid, so I went to say, like, what is validity?

What is validity in a logical argument? So validity in a logical argument is, is there a situation whereby the premises of my argument are true, and yet the conclusion is false? So, take, what was your example there again? Oh, let's pick a better one. Like, you know, if it's raining outside, then I'll get wet. Yes, yeah. So premise two, it is raining outside conclusion, therefore I will get wet. Yes. So if you accept the first two statements, you are bound by consistency to accept the final statement.

There's not a situation where the first two statements are true, yet the conclusion is false. Crucially, I've got to separate this out because we just talked about a logical fallacy, right? This is not the same thing. So earlier we had this argument, which was, if it's raining outside, then I'll get wet. And it was then it's not raining outside, therefore I will not get wet. Yes. That's invalid. Yes. We're doing a positive modus ponies.

So it starts with the same first premise. If it's raining outside, then I'll get wet. But this time, the second premise is, it is raining outside. Therefore, I will get wet. And that whole first premise can be thought of as one. So the first premise is this if then. If it's raining, then I'll get wet. So let's say that's true.

Let's say it's just true that if it's raining, I'll get wet. Then premise two, it's raining. Because it's true that if it's raining, I'll get wet. It means that I must get wet, right? Yes, exactly. And that's what validity is. It means that essentially the conclusion is guaranteed by the premises. Yes.

And then that kind of guaranteed is then cashed out as I can't, there's no situation where the premises can be true and the conclusion. And it's important that it's defined in that way. We'll get into that in a moment. But the technical definition then of validity, an argument is valid if there is like no sense in which the premises can be true and the conclusion be false. If it's impossible for the premises to be true and the conclusion false, it means it's valid.

Importantly, this does not mean that the argument is sound. Yes, or even in any intuitive sense, good. Yeah, it doesn't mean that the, to say an argument is valid, does not mean that its premises and conclusion are true. Yes, it simply denotes. It just means that if the premises are true, the conclusion has to be true. It denotes something about its structure. We can kind of broadly, you know, logicians often think about this in terms of syntax and semantics,

but we can kind of just think about it in terms of form and content for the sake of the sake of the sake of argument today. So we can think about our previous fallacy there, affirming the consequence, is a fallacy of form. It's a formal fallacy. It is an error in the reasoning structure at play here. It's not about the content. It's not about the actual. No, exactly. And you can have an unsound argument.

Right. So just to be clear, soundness. If an argument is sound, it means it means it's. it's valid and also true. Yes, exactly. So there are kind of two definitions of soundness and rather unhelpfully, the one that I think is most useful is also the less popular one. So the most popular use of soundness is to say an argument is sound, if and only if it's valid and also all of the premises are true. And that will guarantee the truth of the conclusion whatever's the case, right? So as I say, all men are mortal, Socrates is a man, therefore

Socrates is mortal? Not only is that a valid argumentative structure, it's also sound. The other way that some people use sound is just to denote the premises are true without saying anything about validity. Unhelpfully, that's not standard usage, but I also think it's very popular. Sorry, unhelpfully, that's not standard usage, but I also think that it is, in some sense, more useful. Because then you can say an argument's sort of sound but not valid. Yeah, so we've got to slow down for a second because there's a lot of words being thrown around. We're talking about this concept of validity, and I think it's really important because it comes up a lot to distinguish validity and soundness. they're often used interchangeably, they're not the same thing.

So just to give an example, let's take an argument which is not sound, but is valid. So I'm going to produce an argument which has a valid logical structure but is not true. So, premise one, if it's raining outside, then Joe Foley is a type of bird. That's my first premise. Premise two, it is raining outside. Conclusion, Joe Foley is a type of bird. type of bird because the premise one is if it's raining outside then joe folly is a type of bird and it is raining outside therefore joe folly is a type of bird that is a valid

because there's no way for those premises to be true and the conclusion be false so it is valid but clearly those premises aren't true it's not true that if it's raining outside you're a type of bird it's also not true that it's raining outside so the argument is unsound but it's still valid so there's a separate question as to whether a logical argument is valid does the conclusion follow from the premises, it's then a separate question whether those premises are actually true or not. Yes, logicians tend to be much more interested in validity and sounds because, you know, if you're a logician attempting to craft or a final logical system, you tend to want that logical system to apply to almost anything. You can plug almost anything you want into the variables,

your arguments will still work. That's why they're kind of mainly concerned with validity rather than soundless. This is sometimes called the ideal of topic neutrality for logic. Yeah. Where if you are, if you at home happen to be designing a logical system, you ideally want the rules to be broad enough that you can stick literally anything in the structures and it will still work. And that's kind of one of the ideals, historical ideals for logical system. So just to be clear, when people are arguing on the internet, if somebody says, for example, something like that, if somebody says, you know, well, if the universe began to exist, then it must have a cause.

And the universe did begin to exist, therefore it has a cause. And somebody goes like, oh, no, that's invalid because the universe didn't have a cause. No, no, it's not invalid. It's unsound. And I do think that's... Stop using the word invalid to describe arguments that are false. Only use the word invalid to describe arguments which are structurally, like, the conclusion doesn't follow from the premises.

And this may seem kind of nitpicky, but it does tell you something very, very helpful about what it would take for the argument to be true. Because say, you have an invalid argument in front of you, then effectively you can just kind of stop that, right? It may be that the premises are true, but it doesn't matter because the conclusion doesn't follow from the premises. However, if you're saying that an argument, you're saying that an argument is, you're saying, argument is unsound, but nonetheless that it's valid, that does tell you something. It tells

you that the denial of that argument is based on your assertion that one of the premises is false. And then you can identify which premise is false. Here you might dispute that the universe had a beginning, for example. That was, you know, this was Aquinas's argument against, you know, the clam cosmological argument. Of course, it wasn't known as the clam cosmological argument. This kind of linear cosmological argument, Aquinas basically said, no, I can, I don't think that it's, it's, it's guaranteed by the, um, by natural reason that the universe had a beginning. He did think that it had a beginning because of the Bible, but he didn't think that you could know that just, just through reason. And, um, so that would be, you know, we think

about what, what's Aquinas doing there? Well, he's saying, I'm questioning one of those premises. I'm saying it's a valid argument, but I'm, I'm, I'm, unsound. Exactly. Or it's potentially at the very least, I'm not confident enough to grant you all of those. And like you say, importantly, if an argument is invalid, like you said, you can just stop there. So you might waste a lot of time because if I made an argument that said like, you know, all, all blogarts are thogarts and X is a blogart, therefore X is also a flogart, you might start asking me like, oh, well, you know, what's a blogart? What's a flogart? How do you define that? But you could actually just like look at the argumentative structure. And if I've said something invalid in the

structure, you don't even need to bother with that. Well, the very least, you know that argument's wrong, it might still be that their conclusion happens to be true, but it won't be guaranteed by an invalid argument. So it's important to separate out validity and soundness. Yeah, there's a, can we talk about the concept of ex-falso-quoddibet? We can. Yeah, I think that it's, it's ex-fossor quadilibet. Is that going to be too much? No, no, no, let's give it a go.

The definition of a valid argument is, as we said, that there's no sense in which the premises can both be true, and yet the conclusion. be false. So, like, the reason why contradictions, or one reason why contradictions aren't allowed in logic, like, why can't you have a contradiction? Why can't I say premise one, it's raining outside? Premise 2, it's not raining outside. One of the reasons you can't say that is because of this principle called X-Falso-Quadli-Bet. From the false, anything. And as far as I understand it, the reason that what that means is that from two contradictory premises, anything logically follows. So let's take this argument. Premise 1, it's raining outside. Premise 2,

it's not raining outside. Conclusion, therefore, I have a cup of tea in my hand. Is that a valid argument? You might listen to that and go like, no, of course that's not valid. That doesn't follow. But the definition of a valid argument is that there's no interpretation under which the premises can both be true and the conclusion be false. And for that argument, there is no interpretation in which the premises can be true and the conclusion be false, because there's no interpretation in which the premises can both be true

because they contradict each other. So if the definition of a logical argument is no interpretation under which the premises can be true and the conclusion false, then that argument, it's raining outside, it's not raining outside,

therefore there's a cup of tea in my hand, is valid. Yes, and if you wanted to get a, just briefly dip our toes into something slightly technical, in some logical systems, X-Fosso-quodilibet is taken as red. It's like a kind of, it's just a rule of the system. For others, they will start with something called the disjunctive syllogism, which is the idea that, say, I've got A or B, and in logic, or is inclusive.

So, A or B is true, if A is true, if both A and B are true. The disjunctive syllogism, which some logical systems will take as primitive, others will prove this through X false-o-quodilibet. but it's far more intuitive. If you start with the disjunctive syllogism for the purpose of kind of getting an intuitive grip on what ex-fossor cord a bet, how you might conceive of that. The disjunctive syllogism states if you've got A or B and I've got not A,

well then by definition I have to have B because A or B is true if A is true, B is true or both are true. I know that not A is true and so A is not true. And so I'm left with B. So just to just to be clear, in logic an or statement.

so like, you know, P or Q, that whole statement, P or Q is true, even if just only one of those is true. Yes, yeah, or both of them is true. So, so it's inclusive. So the statement, there's a car outside or it's raining. Yes, it's true. Say there is a car outside, but it's not raining. It's still true that there's a car outside or it's raining.

Yes, and if we are going to, if we have this inclusive or, that intuitively, a number of things basically follow from that. one of which is the disjunctive syllogism if I've got A or B then if I have not A then I've got B and if I've got not B then I've got A. So premise one, there's a car outside or it's raining? Yes.

Premise 2, it's not raining. Conclusion, therefore there's a car outside. Another consequence of having an inclusive or is that if I know that P is true like I just know it then I know that P or literally anything else is also true. So if I

you know, Alex is sitting on a brown chair or guerrillas are rampaging through the streets of London as we speak is true, because one of the disjunctions is true. Now, how this allows us to get an intuitive gloss of ex-volta quadilibet. And again, I want to kind of emphasize that this is different in different logical systems because some define the disjunctive syllogism through this, or we can get it if we assume the disjunctive syllogism, is that say we've got P and not peace. So we've got our contradiction. We've kind of somehow ended up in this situation. As a BMO Eclipse Visa Infinite card holder, you don't just earn points.

You earn five times the points. On the must-haves like groceries and gas and little extras like takeout and ride share. So you build your points faster. And then you can redeem your points on things like travel and more. And we could all use a vacation. Apply now and get up to 60,000 points. So many points. For more info, visit bemo.com slash eclipse. Visit us today. Terms and conditions apply. Well, I can assume P, and I can introduce literally anything else. P or literally anything else. So P or guerrillas are rampaging through the streets of London.

Then I have not P. So via the disjunctive syllogism, I can get rid of P or guerrillas are running through the streets of London. And I'm just left with a proof that guerrillas are running through the streets of London. So that's kind of an intuitive glass. So it's a it's a consequence of like this system of logic that if you allow a contradiction literally anything follows from it. Yes. And as I say, from a contradiction it follows that guerrillas are rampaging through the streets of London.

From a contradiction, it follows that Spain doesn't exist. From a contradiction, it follows that Spain does exist. Like everything follows from a contradiction. It's also known as the principle of explosion, right? Because it just explodes into proving absolutely anything, which is why, given the, the system of logic that we're talking about, you can't allow contradictions. Contradictions are not allowed, because if you allow a contradiction, the whole thing just blows up. Exactly. And I think that a nice way of, again, because you look at the rule of explosion, you think, well,

what on earth is going on there? Like, why it does not seem intuitive at all that from a contradiction, anything follows. A way of, again, just kind of intuitively getting an idea for this is it's a bit like saying, well, if that's true, then I'd give up. Anything can be true if we admit of contradictions, we've given up any kind of restrictive principles on what can be the case, and thus logic no longer gets me purchased. Now, there are logics,

formal logical systems, that will get rid of ex-Vosoccalibat and attempt to accommodate contradictions in some sense. These are called paraconsistent logics, and I'm not overly familiar with them, but I just wanted to point out that they do exist. Yeah, it's worth re-emphasizing that there are different forms.

Yes, forms. The kinds of logic, different systems, of logic that interpret different things differently, you know, because there are all kinds of problems for creating a system of logic. It's a bit like how in mathematics, like, you're not allowed to divide by zero. That's like a rule, but that seems to be a consequence of like our conception of zero. Like, it's kind of, it's true that in our system of mathematics, you can't divide by zero. But it's like, it's just this kind of like rule that exists. Because if you do, if you do divide by zero, then you end up with all these paradoxical conclusions.

So we just, it's undefined. So we just accept as a rule that you can't do it, you know. And similarly in some logical systems, we sort of have a similar approach where we say there are rules. Like you can't have a contradiction. Why? Because if you do, anything follows. And it might have been a bit too technical and it's kind of hard to follow. So if anybody's interested, just Google, you know, X falso quadli bet or from the false anything or the principle of explosion.

And just look at it written down, look at a proof. But basically, if you allow a contradiction, according to the laws of logic, anything. follows from a contradiction. And this is kind of a good reason to not get mired in contradictions. To not allow contradictions, which is also why if you want to disprove an argument if you want to show that something can't be the case,

one way to do that is to show that it entails a contradiction. Yeah, so that's kind of a reductio ad absurdum. Or I think that other people have called it a reductio per impossibly, which I think is like, I think it's just a slightly snazier term. Yeah, I think it might be. But whatever Whatever you want to call it, if somebody holds, you know, two ideas, if they say like, well, I don't know, I don't want to give an example because it might get complicated. But if somebody holds two ideas and you want to show that, like, they contradict each other, then you have shown that they can't hold both of those ideas. Or if you show that something that they believe necessarily leads to a contradiction, then since we're not allowed contradictions, you know, that's a way of showing that it's doesn't exist.

And it's really interesting how people have kind of Aristotle's justification for the law of non-contradiction is fascinating. Because I know that, you know, you're very interested in kind of global emotivism as an idea. Aristotle's justification for not allowing contradictions is eminently pragmatic. He sort of goes, well, I don't know, man, try and think a contradiction. Try and think in terms of contradictions. You absolutely cannot do it. So, you know, there is a sense in which one of the interesting things about the philosophy of logic is that because you're at this kind of incredibly abstract level where you're almost trying to get a grip on the laws of thought, you end up having to say a fair amount is just self-evidently true. The Stoics called their logical rules, the indomonstrables for this reason, because they basically sort of went, well, if you think that if P, sorry, if you think that if P then Q, P, P,

therefore Q isn't a good argument I don't know what to tell you I can't show that it's the case I can just I can just point to it and sort of say it's just there it's just there like come on it's self-evident I don't know how to think apart from using these rules

but logic also applies so we're talking about logical arguments right so like these rules of inference if P then Q P therefore Q yeah that has to be true but there are also like more atomic logical rules that we accept, not as the result of some kind of argument or inference, but they just exist as they are. These are like what oftentimes are called like the foundational laws of logic. So one of the most

important of these is the law of non-contradiction. A proposition can't be true and false at the same time. That's not an inference. That's not an argument. That is just a statement. Like, the proposition P cannot be true and false in the same way at the same time. Like, what is, what is that? What is this axiom of logic? Is that like, do we reason our way into that? Is it a prerequisite of reason? Like, how do we justify these axioms? Well, as I said, I think this is one of the exciting things about the philosophy of logic is that if you're trying to ground logic, you can't appeal to logic. You know, if you're trying to ground in something prior to logic, that's really, really hard. As I say, an awful lot of ancient logicians, and it's a prime, you know, a lot of modern

logicians will just kind of say, like, try and present to me a way of thinking. because the problem is somebody might just say okay well I'll just I'll imagine that it's yeah okay maybe P and P can be true and false at the same time cool okay right let's try that for a second so you are accepting that a proposition can be true and false at the same time right so you think it's true that a proposition can be true and false at the same time right so you think it's also false so it's false so it's false that a proposition can be true or false at the same time yeah but if it could be true or false at the same time yeah but if it could be true or false at the same time yeah but if it could be true or false at the same time yeah but if it could be true or false at the same time then that can also be false. So it's true and it's false. Another way of kind of attempting to justify this would be to say something like, okay, go out and kind of live your life and never infer from learning something

that its opposite could not be true. You're just going to run into an awful lot of difficulties. That's just not how the human brain works. So these fundamental laws of logic, like the law of non-contradictions it's called, seem to be bedrock. They seem to be foundational. And these are the laws of logic with which we use, with which we sort of construct arguments and thought and sort of have an ability to use reason.

But like, I don't know, like, what are these laws of logic? Like, where do they come from? Are they psychological concepts? Do they exist in the universe? Like, do they exist before humans existed? What do you think they are? So I think that probably one of the best ways of going about attempting to answer this question. is to begin by ruling out the kind of things that they probably can't be.

So they can't merely be descriptions of how we, in fact, do think because we think non-logically an awful lot of the time. In fact, I think one of the really interesting parts of, you know, Kahnem and Tversky's work in psychology is just to demonstrate the air. A lot of the time we're not thinking logically. We're thinking puristically. We're drawing from association and emotion. We're not, you know, man is a rational animal in the sense that he can do,

rationality, but we're not perfectly rational. That's not, that's not by any means our dominant mode of thought. So this was summed up in a sentence by the, the American logician and pragmatist philosopher C.S. Purse or Pierce, and I don't quite know which one to pronounce it, where he said that logic is a normative science, which is to say it's not simply describing how you think, it's describing how you ought to think. And cashing out that ought, philosophically is incredibly interesting. Because, you know, a lot of people are very comfortable, for example, saying something like, well, you know, moral aughts, they're a bit fishy. Like, you know, kind of, I'm happy to be a moral anti-realist. I'm happy to be a normative anti-realist in that sense. But

you go take something like logic. You say, well, can you be a normative anti-realist about logic? You know, that strikes me as something that's, at least most people are significantly less fond of doing. Because if somebody is being, like, being illogical, like literally illogical. And you tell them like, hey, that, that's not logical. And they go, I know, like, so what? You want to say, well, what do you mean so? Well, you should be acting logically. And they're like, why? It's like, that's just how thinking is supposed to be done. That's just what, and you can't say, well, that's just what thinking is, because they're not thinking logically. So, so it's not what thinking is, but you want to say that they're incorrect. Well, to say that

they're incorrect in the way that they're doing something is to say that there's a way that they should be doing it, that they're not. And, you know, you can kind of try and cash us out as a hypothetical imperative. Something like, you know, if you want to preserve truth, then you ought to try and think logically in this kind of, you know, narrower sense of what logic means. And, you know, that does leave it open for somebody to say, I'm not interested in preserving truth. Well, they could say I'm also interested in preserving truth. Yes. I'm, I'm interested in preserving truth and I'm also not interested in preserving truth because I've abandoned the laws of logic. At which point you, at the very least can say, well, you know, you clearly not

interest in preserving truth, you know, whatever you say. And you want to say to them, like, you are, your thinking is going wrong. Yes, yeah. As in your thinking has become detached from the kind of inferences that preserve truth from a premise to, a set of premises to yeah. So, so there seems to be the sort of normative undertone. Yes, yeah, yeah.

Which is, you know, just, you know, which is, I should say, normativity is significantly more commonplace than we often give it credit for. Languages. What do you mean by normativity? Normative is oughtness, is the kind of way of putting it. which I don't, by which I don't necessarily mean kind of moral aughtness. My point is that actually normativity is a lot more commonplace than you might think. And it's not just in the ethical sphere.

It's not just in the ethical sphere. Language concerns, contains within it a sort of element of normativity. Even though, you know, language tracks how people actually use language, it is also possible to use language incorrectly. So if I was to say, you know, the, what's that line from the Jabberwocky, the slidey toves, doth Geyer and gimbal in the wave to sort of mean, um, uh, I fancy a cup of tea. Then that's, I'm using language incorrectly by which, you know, we're kind of implicitly assuming a function there, which is that the purpose of language is to communicate. And I'm using language in such a way that I am so unlikely to be understood that I'm essentially going against

the assumed function for language. And I think we can. So that's normativity because you should be using language differently, but it's not ethical. It's not, no, no. My point is that it can at first be sort of very mysterious when we say, oh, you ought to be thinking like this or you're kind of using language wrong. It can be very, it can at first seem very mysterious. I think that that mysteriousness is in some sense a confusion of language. The concept of should and should not or ought and ought not can be separated from morality. Of course, yeah. And, you know, we can partly do this because we can quite naturally define a function for a lot of language. We can quite naturally define a function for logic in a way that allows you to have this hypothetical part of the imperative.

You can say, if you want to do this, then you ought to do this. If you want to be understood, then speak in such a way such that to the best of your knowledge, people will be able to understand you. If you want to preserve truth in your reasoning, then reason in such a reason according to these logical rules that we have found preserve truth. That's significantly less, to my mind, mysterious than the kind of ethical you ought to do this and it doesn't matter any kind of ifs. No if sputs or maybe, so you really ought to do this. In the context of logic, we've said that logic applies to like arguments and sentences and their relations, but it also obviously involves truth. Like logical validity requires, you know, if the premises are true, the conclusion is true, soundness requires the premises being true.

Logic doesn't seem to apply to expressions like, you know, hooray or go over there. Logic applies to the kinds of statements which have truth value, which can be. true or false. So in the context of talking about logic and learning about what logic is, what is truth and falsity? It's complicated. I suppose it depends on, again, probably depends on who you ask. One of the really fascinating findings of a guy called Alfred Tarski in the 20th century was that you cannot define, for an awful lot of logic, you cannot define truth within the scope of the logic. You have to appeal to what's called a meta-theory or a meta language to define truth within the object language, which is the logic you're talking about.

So in terms of a kind of more philosophical sense, I suppose it collapses to the question of what you think truth is in philosophy. You know, do you think that it is a matter of correspondence? Some people certainly did. You know, I think Frege has a kind of, actually, I don't know, I probably wouldn't say that. But, you know, because he also critiqued a lot of correspondence theory. So, you know, but a lot of people would say that the truth is cashed out as correspondence between a set of a proposition and a state of affairs in the world. Some people would say that it's deflationary. So when I say the snow is white is true, I'm saying nothing more than snow is white.

Yeah. So for some people to say snow is true is to say that that sentence snow is white corresponds with the thing about the world that snow is actually white. Yes, but to say that it's deflationary means it actually adds nothing. different ideas of what truth actually is, right? But also within logic, like, we need to be able to say that, okay, whatever your theory of truth is, if a premise is true and the second premise is true, and the conclusion follows from the premises, then the conclusion is true. There are some ambiguous cases as to whether to call something true or false, right? So, so the famous example is the sentence, like, the current king of France is bold.

hmm like the current king of France is bold I'm going to ask like is that true or false and obviously the problem is that France doesn't have a king summer's here and you can now get almost anything you need for your sunny days delivered with Uber eats what do we mean by almost well you can't get a well-groom lawn delivered but you can get a chicken parmesan delivered a cabana that's a no but a banana that's a yes

a nice tan sorry nope but a box fan happily yes a day of sunshine no a box of fine wines? Yes. Uber Eats can definitely get you that. Get almost, almost anything delivered with Uber Eats. Order now. Alcohol and select markets. Product availability may vary by Regency app for details. I'm like predicating something of this non-existent king that he's bold. So it seems like it's false that the current king of France is bold. I want to say that's false. But it doesn't seem right to say, no, no, no, the current king of France isn't bold. That's because I'm sort of referring to an object

that doesn't exist. So for a statement like that, is it true? Is it false? Is it undefined? So this is, statements like these, that particular one was kind of analysed quite heavily by Bertrand Russell his theory of definite descriptions. The specific statement, the present King of France is bald

as there exists a present King of France there exists only one thing the present King of France and that thing the present King of France is bold. It's kind of very, very broadly and non-technically kind of roughly what the theory of definite descriptions states the general class of statements where it's very difficult to tell

whether they're true or false because in some sense they're underspecified or they decompose into other statements are known as exponable statements and these have been a problem since basically forever in logic they're kind of one of the classic logical difficulties that they cause a lot of headache

for some medieval logicians and you know that their kind of classic example stuff like when did Socrates begin X, Y, and so. Actually, you know what? Teenage boys are like absolute masters at coming up with Exponable Statements. I think that the classic one,

when I was a teenager, was, does your mum know you're gay? And I don't know if you've seen it. There's this, because this is like, this is fantastic Instagram real for anyone that's interested about the kind of logical decomposition of Exponable Statements,

which is that there's a guy And he says to somebody across the table, when I click my fingers, you'll forget that you're ever gay. And then he clicks his fingers. And then the guy goes, oh, what are you talking about? I was never gay. He goes, see, it works. And, you know, my, my, and, you know, I often, I remember thinking about this when I was a teenager a little bit. So sort of, well, this seems like this happened to you a lot, did it?

Yeah, of course. You know, look at me. This is, this is exactly the kind of insult that is this finely tuned to my presentation and manner. And this, you know, the statement like, does your mum know you're gay, effectively decomposes into, you know, Proposition 1, you are gay, and Proposition B, your mum knows about it. So. Yeah.

Because there's no way to, you can't say, like, yes, my mum knows I'm gay. But you also can't say, no, my mum doesn't know I'm gay. Well, that is if you're not gay, if you're gay, then you have a perfect, perfectly sensible answer. You can say yes to at least one of those, yeah. The idea is that it's got a kind of nesting. claim within the broader structure. Right. And that's, yeah, an exponable statement. So, you know, if any of you are watching and people say this at school, you can say,

I'm afraid, good sir, that's an exponable statement. And as a result, I will decompose it into its separate propositions and choose to deny or affirm them individually. And I'm sure that the bullying will stop after you say something like that. But, you know, I just kind of use that as an illustrative example to show that, you know, This is, exponable statements are very, very commonplace there. They're kind of all over the shop, and Russell is dealing with just one of them. And if you want to give a kind of truth functional analysis where you maintain your sharp division between true and false,

then you tend to decompose exponable statements into something else. So how would a logician approach a question like that? Well, they would say there's an implicit claim within the present king of France. is bald, that there is a present king of France, and that he is bought. So would they say it's, would they say it's false? Well, they would say it's false that the current king of France is gay. And if, I'm sorry. Well, that would also be an exponable statement.

That also works, because the reason it's false is just because there's no present king of France. Okay, okay. But then then then if I asked you, Joe, like, does your mom know that the king of France is gay, that would that same logician say yes or no? Or would they just rejects the question? They'd say it was false. They'd say that one of the claims, well,

Well, if they were giving a kind of decompositional analysis of this, they would say that one of the implicit claims is that the present King of France is gay, and that that's false. Really, the phrase would not be like, my mum doesn't know the King of France is gay. Yeah, exactly. So, it is not the case that my mum knows the King of France is gay. Yeah, so they would say that this is a conjunction of two claims, one, the present King of France is gay or bald, literally whatever property you want to assign to the present King of France. And my mom knows about this. And of course, the conjunction of those is false because the left hand of it is. is false and that you know conjunction is only true if both sides are true so that's how that would be

one way of but suppose i said like the current king of france is non-existent yes that's true yeah but that will be analyzed as there does not exist and exist and this is where the language of logic becomes really important because you need to be able to translate a statement like that into a form that makes sense. Whereas I said, you know, the current king of France is bold, and the current king of France does not exist. That kind of sounds like I'm saying the same kind of thing twice, but I'm not. In the first case, I'm saying, there is a king of France such that he is bold. And in the other case, it's like, there is not a king of France. Yes, exactly. You know, one way of thinking about that way, you can see they're not the same kind of statement. That's to do with how you, how you

formalize the sentence. And they think that intuitively, you know, a lot of listeners may have encountered this when they're looking at the ontological argument. Famously, one of the responses to the ontological argument is that existence isn't a predicate. And this is just kind of the same sort of thing, saying that, you know, existence isn't a property of the thing. When you say the present king of France does not exist, you're not saying that there is a present king of France and he doesn't exist. That's absurd. You're saying that there is no present king of France. Those statements are equivalent.

Yeah. So that's kind of, but these are another, this does just illustrate an example. another incredibly useful tool for logical analysis, which is that you can take statements that at first seem very, very confusing, and you can break them down into parts and then get a clearer grip on what exactly they meet. Alternatives, when you, because there are logicians that hold that there are genuine vagueness in the world, or that there are statements that are genuinely of indeterminate or gradable truth value, and these are, these are a fuzzy logicians.

Fuzzy logic. Well, they're not fuzzy logicians. Well, they're logicians in fuzzy logic. Yeah, exactly. That's a better way of putting it. And, you know, many-valued logic. So what is fuzzy logic? What is that? So fuzzy logic is a form of many-valued logic. So many-valued logic, very, very broadly, is just any logic that has more than two truth values. So there's a Polish logician, and I can't pronounce his name, but it's spelled like

it's pronounced like it, and it's definitely not pronounced like that. But it's one of the ones I've only seen written down. and he came up with lots and lots of different many valued logics probably the two most important ones are a three valued logic where you've got true false and half or zero one and half because obviously he's kind of assigning it in numbers and an infinite valued logic whereby you have you know nought 0.1 and 0.01 and you've got literally the whole

you've got this continuum between 0 and 1. Yes you've got the whole 0 is false 1 is true and you've got this Yes, you've got this entire continuum in between the two. What do you mean then? So take a form of logic, a fuzzy logic that says that there are three kinds of truth value. There's 0, 1 and 0.5. So 0 means it's false, 1 means it's true. What does 0.5 mean?

So what does that actually mean? Well, the logic itself would not impose an interpretation on that. You know, it's just a series of rules. But, you know, you might think of that as, depending on what interpretation you're using, you might think of that as indeterminate. You might think of it as vague. You might, you know, I'm sure that there are some interpretation, but it's just like, I don't know.

There are, you know, the infinitely many valued ones. You know, you might think that you might use that if you were to embed that in something like a doxastic or epistemic logic, you might want to use great ability to represent confidence or something like that. There are types of fuzzy logic that mesh quite well with probability theory. So some people want to use them in that way. The interesting thing about fuzzy logic doesn't denote a single logical system or even a single definition for logical connectors. It denotes more of a kind of general vibe, I suppose.

It's any logic that has this kind of continuous truth values between zero and one. And the actual definitions of the connectives like and or implication or by implication will be different. They will cash those out in slightly different ways, and as a result, they'll end up with slightly different values for their sentences. But, yeah, it's a logic that is quite often used to deal with vagueness or uncertainty or something like that. You just mentioned epistemic logic? Yes. So epistemic logic is, well, let's start with modal logic, because epistemic logic uses a lot of the same tools as modal logic. Again, I'm keeping things very, very non-technical.

modal logic is a logic that is aimed to study necessity and possibility. So, you know, we talk about this kind of thing in philosophy a lot. You might say, well, is X necessarily true? Is it really possibly true? Is it true in this world? And modal logic allows you to talk about this kind of stuff. You know, the logic we've been talking about so far is like a statement is either true or false. Yes, that's classical logic.

But a statement can be true. Like, it's true that to. and two is five. It's also true that this chair is brown. Yes. But it seems it's not necessarily true that the chair is brown. It could have been blue, but it seems necessarily true that two and two is four. So there seems to be this different kind of truth.

There's necessary truth and there's possible truth. And so modal logic is attempting to take that into consideration. Yes. And it's the logic of possibility and necessity. Yes, exactly. And so the way that this is cashed out on a kind of model system, the model theory for modal logic uses something called Cripkey modeling, which I assume was invented by sort Cripke. I actually don't know. But it's called Cripke modeling, so you kind of hope so. And the idea is that you have a world where you intuitively live at. So you've got a world where Alex lives. And then you've got a bunch of possible worlds around you. And so. some of those and the what's what is necessary is the stuff that is true in all of the worlds that are accessible to your world. So this allows you to define different notions of possibility. So say you

wanted to restrict your notion of possibility to or your restrict your notion of accessible worlds to only those worlds that share the same physical laws as your world. Well then you've got a very, very neat, very systematic definition of physical possibility. Say you wanted to examine possibility but keep a few different propositions fixed, then you would effectively say, right, well, you know, let's only have those worlds be accessible to our current world where these few propositions are fixed. You can kind of play around with the modelling a bit like that.

There are different restrictions on how the relationship of accessibility works, which gives you slightly different modal logics. But again, I don't know. So starting simply, like we're using these ideas of other worlds. We're not talking about like a multiverse, like, you know, physical universe existing. No, no, no. It's just a way of thinking, right?

So, like, there's another possible world that could have existed where, you know, like, that table doesn't exist. So we just sort of conceptually think of it as like there is this possible world that is the same as this one except that table doesn't exist. Yes, we can think of them as, well, for the purpose of discussion here anyway, we can think of them as just sets of propositions. Yeah. So we think of them as, you know, you've got a series of propositions. A consistent set of propositions. Yes, well, depending on who you ask, there are people that do impossible world semantics where in order to mold semantics where in order to mold. model, you know, impossibilities. They will, they will allow for certain worlds to have inconsistent

sentences. They're also modal realists who believe that, actually, David Lewis, who think that, you know, these possible worlds are actually real worlds. They actually, like, truly exist. That's a little bit fringe, but it's usually just a way of thinking about the world. So, if I want to say, like, one way of me saying it's possible that, you know, I could have not existed, that was possible, I can rephrase that as there is a possible world in which I do not exist. Yes, and that would allow you to treat that sentence in a modal logic rigorously and comfortably. This same framework ends up being really, really handy for a whole bunch of other things. So, you know, classic question, what do you mean by believe? It comes up a lot at the

moment. And a doxastic or epistemic logician would look at that question, say, I know exactly what I mean by believe. By believe, I mean it's true at every possible world that is accessible to me, which is effectively saying, imagine that you're an agent living at world one, and you've got a series of worlds that you're considering, let's say, to give it a kind of psychological interpretation. Well, then everything that's true at every world that you're considering is effectively what you believe. Is the kind of reasoning that an epistemic or doxasic logician might use to justify their kind of their usage of this, um, framework of necessity to deal with things like belief and knowledge. They're not saying that

they're using the same logical apparatus, but they're giving a different interpretation to the symbols. And this is, I think, again, something that really illustrates just how handy logic can be, is that sometimes you design something that's to be used for something totally different to a new usage, and then suddenly you find that actually, this kind of broad idea of necessity, if you kind of relativize the interpretation to an agent and you don't think of possible worlds as possibilities, but instead as worlds that this agent could be considering, that actually have a very neat formal system for dealing with belief and knowledge. Take that, Jordan Peterson. Well, that's just one way of dealing with belief. You could talk about it, a whole series of

other ways. But I think that this is a, this is at the very least a very good way of formally treating belief. So we've got kind of, and this is back to what we said at the beginning, it's beginning to sound like we've got different logics or different kinds of logic. depending on what our task is. If we're interested in epistemology, we might use epistemological logic. If we're interested in possibility, we might use modal logic. If we're interested in sentence construction, we might use predicate logic. And so, oh, hi, buddy. Who's the best you are? I wish I could spend all day with you instead. Uh, Dave, you're off mute. Hey, happens to the best of us.

Enjoy some goldfish cheddar crackers. Goldfish have short memories. Be like goldfish. So when we spoke earlier and you were talking about how there might be like different kinds of logic for different kinds of circumstances, might just sound like, wait, what are you talking about? Surely there's logic and there's non-logic,

but I hope people can begin to see how the way we construct a logical system can be very different. I mean, the whole conversation we had about P's and Q's if that and that and logical policies didn't even consider or incorporate the concept of possibility versus necessity, for example. And so you need this sort of other way of thinking about logic, modal logic, in order to incorporate that. Yes. And in some sense, you can still have these different logics even if you are a logical monist and you think that ultimately there is one true way of thought. You could still say, oh, you know, these are still very

useful tools, but ultimately there is a logical theory of everything. Yes, exactly. And different people oppose different kind of logics to fill this role. I think that Owen Griffiths and Alexander Pazzo's one was in first-order infinitary logic, which is very interesting, but I can't remember how exactly it functions. But I remember enjoying it at the time. And, you know, as I say, there can be different candidates for this ultimate monist logic. But, you know, you don't have to, if you find all these tools useful, that doesn't philosophically commit you to logical pluralism. Right.

Yes. Oh, yes, to logical fallacies. Sorry, I just wanted to point out as well that there are also a huge, there's a huge camp of things that we call logical fallacies that resist formal treatment quite well. So something like a no true Scotsman fallacy. So, you know, I say something like no true philosopher would fear death. And you point to me a bunch of philosophers that did fear death. And I say, oh, well, you know, that's, that's, you know, that's, you know, they just not. They're just not really philosophers. We tend to say, okay, there's something wrong with that. And people have used the no true Scotsman fallacy to label this. And I think, you know, we all agree that's dodgy in some way. And actually, it's dodgy in a way that's quite difficult to treat formally because logic doesn't prescribe definition. No, because you could construct an argument that says, like, if somebody is a real philosopher, then they don't feel fair death. This person does fear death, therefore they're not a real philosopher.

And that's valid. Yeah, you wouldn't want logic to dictate. definitions either because it would be you would have a logic then that was not topic neutral at all it's dictating an entire language the whole point about logic is that you can have the P's and it's like algebra like you have P's and Q's and all this kind and the idea is that like you can substitute in any sentences and it will still work if it's like well if P has this definition then it works but if P has this definition it doesn't work you haven't got a universal system of logic and this is where some fallacies or kind of other bits of faulty reason

are more readily treated in a field called informal logic, which is, you know, which a lot of people would just say isn't logic. And I kind of sympathise with that, but I like it as a time. I think so, too. Informal logic is looking at arguments as they actually occur in the world and extending their scope slightly beyond formalism to sort of more broadly ask the question, what is a good argument, what is good logical reasoning even in taking into account extra formal criteria. So an informal logician might look at that, the kind of no true's got some fantasy, and say something like, right, okay, well, we're not just considering formal constraints. What is a logical argument for, that is in a logical discussion for, or reasonable discussion for? So we kind of get out of our formal heads. It is for the communication of ideas under the assumption that the ideas with the greatest support.

will be accepted, the ideas where less support would be rejected, et cetera, et cetera, et cetera. And you can, you know, informal logicians and philosophers will define certain rules that dictate the kind of things that go on in a, in a idealized reasonable discussion. And we can almost think of this as a language game if you wanted. You know, there are, you know, one of them might be that you can't, if you, you can't dismiss an opponent's point without giving a formal argument as to why it's not true. that's not a that's a rule of discussion in a reasonable discussion but it's not it's not a rule of logic it's not it's not it's not an axiom and there's no kind of logician that's going to kind of go around your house and beat you up if you don't follow it but but you're departing from the conventions of um of a reasonable discussion yeah another one might be you know you've got to ensure that your opponent um and you were using similar terminology or you might say you've got to ensure that you you and your opponent are you are making your point clear to your opponent in terms that they will understand.

And you could say, well, the no true Scotsman fallacy, if you want to accept this as a rule of conversation, a reasonable conversation, you could say, well, the no true Scotsman fallacy is in contradiction to this. You know, it's not that it's logically... Yeah, I mean, you haven't committed a logical contradiction by going, well, they're not really philosophers. There's no logical contradiction,

but it feels like we want to be able to incorporate the wrongness of doing something like that, and that's where you get these informal logical fallacies. Yes, another one is like the Mottombele. In fact, I think the majority of logical fallacies, people have heard of, are probably informal logical fallacies. Yes, because a lot of logical fantasies will depend on context. What's the Mott and Bailey?

Oh, Mottombele fantasy is where you have two versions of an argument, one of which is modest and very defensible, which is the Mott, and another which is more ambitious, but significantly harder to defend. So say I am a, you know, say I'm a particular kind of theist. I don't, you know, I, again, I don't want to kind of rag on anyone here. But like, you know, this is a kind of a type of argument you might have encountered is where someone says something like, okay, there is, you know, there must be an origin of a per se causal chain. Therefore, we have a ground level metaphysical point and we call that point got. And if one challenges them on any theistic matter, it's very tempting to retreat to that.

And then the Bailey might be something like the God of the Bible is true, significantly more ambitious, significantly harder to defend. And, you know, Aquinas has arguments as to why these, you know, I'm not accusing Aquinas of this. But it's sort of the idea is you put forward an argument like, you know, the God of the Bible exists. And someone says, well, are you sure about that? And they start interrogating them and that person starts retreating back towards. Oh, well, you know, I think there's a first cause of the universe. Exactly. And it's sort of like, okay, but you've just retreated into a much simpler art.

So you're now defending kind of a different, more modest version of the argument. That's the Mott and Bailey fallacy. But again, you haven't, at least strictly speaking, you haven't done anything like illogical there. Yes, because you've switched, if you were to analyze that in terms of propositions, nothing is, you know, you've not made any kind of invalid point. And another helpful example might be like the ad hominum fallacy, right? If I say, you know, like, Joe, like, I just think you're stupid and therefore I don't agree. agree with you. Someone would say, well, that's the ad hominem fallacy. I haven't committed a logical fallacy there. I've just said, you know, Joe is stupid, therefore he's wrong. And, you know,

you can render that as a formal fallacy, if you like. And there are, there are a number of arguments whereby it's the line between a fallacious usage of something and a non-falacious usage of something is surprisingly thick. Yeah, so you could formalize it. You could say something like, you know, premise one, Joe thinks the sky is blue, premise two, Joe is stupid, conclusion, therefore the sky is not blue. Exactly. That's an invalid. And that would be a logical fallacy.

But that would, as you say, like all genuine logical fallacies kind of just collapse into the non sequitur. The non sequitur is a fallacy. The non sequester fallacy is committed when a conclusion doesn't follow from the premises. Yes, exactly. And all fallacy is. If it's raining outside, then I'll get wet. it is raining outside

therefore my raincoat is red that's a non-sequitur the conclusion just doesn't follow from the premises and so a logical fallacy a formal logical fallacy will involve some kind of problem with the structure

whereas an informal logical fallacy will be harder to pin down in logical language and you can say and some things that people call logical fallacies are arguments where one of the premises is unsound on its face

so some ad hominem fallacies might be rendered like this you say something like this person is stupid premise to everything that stupid people say is false conclusion anything this stupid person says is false and you say well that's you know structurally fine is a valid argument soundness wise most people kind of go premise too that's obviously not true yeah so you know you can kind of you can analyze that's the ad hominem fallacy but you have to be careful to remember that that's if like I don't think we should call them logical fallacies

when they're like, as you say, informal, logical, they're just like fallacies of a different kind, you know, they're unsound, but there's nothing wrong with the logical structure. So I think people, this is actually one of the problems is when you're having an online debate and someone says, oh, that's an ad hominem fallacy, or that's a Motten Bailey, or that's a this, or that's no true Scotsman, they often throw those out as if they have the same force as proving to someone that their argument is literally logically invalid. Yes, which is not. I mean, it's still bad to do that, and that might show that.

that their argument is faulty, but it doesn't have the same, like, this is absolutely, like, demonstrably invalid in the way a formal logical fallacy does. And, of course, in either case, you're just proving that an argument is fallacious isn't sufficient to show that the conclusion is false. That's also important to specify, yeah. A huge number of very, very bad arguments for true conclusions. Yeah. So, like, the fallacy of composition is an important one.

Like, you know, you might say that, so some people say, like, everything in the universe has a cause, therefore the universe has a cause. So that commits the fallacy of composition. It's like saying every brick in a wall is small, therefore the wall is small. That doesn't follow. But sometimes you can use that fallacious argument that where the conclusion doesn't follow from the premises, but the conclusion is true. You know, all the bricks in a wall are red, therefore the wall is red. That is actually true, but it's still sort of committing this fallacy. But fallacies don't always mean the conclusion is false. It just means that the inference of that conclusion from those premises is like illegitimate.

Yes. And also, you know, for the, for the wall example, you could, you could analyze that as having an implicit premise in it, right? Which is, you know, say you wanted to add premise number three, you could say that, well, a hidden premise in that argument is that redness is the kind of property that transfers from parts to holes. And then, you know, you can cash out the fallacy of composition the other side saying, well, smallness is the kind of property that transfers from parts to holes. You can say, no, that's not true. That's false, which is why that one's invalid and this one is valid. Yes. So there is a fair amount of freedom in how you take natural language arguments and choose to treat them. Yeah. Kind of using kind of propositional form and stuff

like that. I think that oftentimes one, you know, criterion, sorry, a criteria for how you might want to go about this. Oftentimes it's kind of generally to go with like informativity. How how much does it tell you about the other person's argument is a pretty good metric for in practice how it's going to be most helpful to formalise that argument. So, you know, one of the, one of the dangers of formalizing every ad hominem fallacy, for example, as simply this person's stupid, therefore what they say is wrong, is that, although, you know, that that gets to the representation that that argument is invalid and therefore a bad argument, it might, you might discover that your interlocutor, the hidden premises

supporting your interlocutor's ad hominem fallacy might tell you something about. their wider views or the other kind of moving parts of their philosophical framework. And that can be very handy. Yeah. And it's important to note that like it kind of depends exactly what's being meant because like there are times when the so-called ad hominem fallacy might kind of be appropriate, which is to say like, you know, if somebody says, I believe this proposition and I say, well, why do you believe that?

I mean, I believe the earth is flat. Well, why do you believe that? Well, because Mark told me. I'm like, well, you know, Mark is a complete idiot and I wouldn't trust what he says. Well, there you go. You could go, that's the ad hominem fallacy. But maybe the argument could be formally cashed out as like, you know, if somebody is stupid, then what they say is less likely to be true. Yes, you know, that's a...

And so this is a statement that this person said, therefore this statement is less likely to be true. Yeah, so if we were going to formalize that. Yeah, we'd say something like, if someone is stupid, then their statements are less likely to be true, which is like plausible. In this plausible in sense that actually, I don't quite know what we would mean by stupid. if we didn't, if it didn't entail that. You know, Mark is stupid. Therefore, the things that Mark says are less likely to be true. Where, you know, we're saying less likely than the average person, I suppose it'll be something.

Kind of a way that we could extra specify that. And then you've got an argument that, at the very least, isn't obviously unsound on its face. I think the context in which this particular problem comes up the most is the appeal to authority fallacy. So there's this fallacy that says that if you say, well, I believe this is true because this authoritative. or the appeal to popularity. I believe this is because loads of people believe it. Yeah, that's kind of a fallacy, sure, but there are some context in which it is actually sensible to trust authorities, you know? Like, I, like, I don't know, you might believe in climate change because of scientific consensus, even though I'm not a scientist and I have absolutely no

idea how to interpret, you know, CO2 data, but no idea what's going on there. Like, it still might be reasonable to believe in climate change if the popular consensus of authoritative figures say that somebody could say that's an appeal to authority. And in some context, you kind of want to be like, yeah, it is, but that's kind of fine. Well, all of the idea of something being logically fallacious would mean here is that it doesn't guarantee the truth of the conclusion. So, you know, as say, if you were to chunk that down into propositional form, you kind of have the flip side of the argument we just gave about this hypothetical mark. I'm so sorry if anyone in the audience is called Mark. He'd say something like, okay, if something,

is first premise if there's expert consensus on something it is more likely to be true second premise there's expert consensus on climate change or any other proposition you want to stick in there conclusion

therefore that's more likely to be true and you know I think that's a very a nice way of breaking it down partly because it actually immediately exposes the fault lines of a debate that you would have with somebody who doesn't agree

because most of the time, if somebody is questioning expert consensus, they normally are questioning that first premise. They're normally saying, no, I think that even when there is expert consensus on something, that doesn't make it more likely to be true. So again, you can kind of see how breaking things down to propositional form in this way can do a lot to clarify the contours of a debate, what the substantive disagreements between you and your interlocutor might be. Yes. Everybody, learn logic. Where should they learn logic? What can you recommend that they read? If someone's no introduction to this. Well, so the open logic project is a website with free logic textbooks. I wish I was involved. I'm not. Then they're there, download them for free, very, very readable. To give you an idea of how readable and kind of useful they are,

the for all X free logic textbook was literally the textbook for the first undergraduate Cambridge Logic course. Yeah, when I did logic in first year of university, it was very, Volker Halbach's logic manual, which I thought was actually quite a good introduction. That's where I would point people to, but I've done a lot less reading on logic than you have. So that sounds like a really cool project. Go and check it out. And there are, there's ones on set theory. There's ones on category theory.

I don't understand category theory at all. I wish I did, but I don't. There are there ones on probability theory. It's a fantastic resource. Another one, if you're more mathematically minded, the book, I can't remember who it's by, it's called A Friendly Introduction to Mathematical Law. And that will take you right the way from, you know, formalizing statements through propositional logic, predicate logic, and all the way through to girdles and completeness

theorem, which is a tricky old proof. So by the time you've got there, you're doing pretty well. Cool. Well, there you have it, ladies and gentlemen, and good luck to you. Joe, thanks. This has been fun. I think, you know, I'd say that at the end of a lot of podcasts. Like, yeah, it's been fun. And it always is. But I think that I like podcasts that are quite a lot of the time sort of discuss all over the place, but it's nice to have a particular task, intro to logic, what's it all about?

I think we've covered most of the important bases, at least to give people a basic idea, resources in the description. Hopefully, they agree, that we've done a relatively good job here, and thank you for the time. Oh, thank you for having me.