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

31:30If 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 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 claims to be the kind of normative rules of thought. 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 know that I'm thinking well? How do I know I'm thinking poorly.

32:12And 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 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 going to be quite clear on what I mean. And that's not a matter for formal logic, but it's nonetheless a kind of general rule of a 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?

32:55Yeah, like what can you derive from one thing? What can you draw 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.

33:35that argument is an example of a valid argument what is validity what is what is validity in uh 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 um take what was your what was your example there again sorry let's pick a better one like you know um if if it's 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 like 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 ponens 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 like get wet right yes exactly and that's what validity is it means that essentially the the conclusion is guaranteed by the premises yes and yeah and then that kind of guaranteed is then cashed out as 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 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.

35:52It just means that if the premises are true, the conclusion has to be true. Yes, 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 argument today. So we can think about our previous fallacy there, affirming the consequent, 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. It's about the structure. You can have an unsound argument.

36:29Right. So just to be clear, soundness, if an argument is sound, it means 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 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 um the the other way that some people use sound is just to denote the premises are true without saying anything about validity unhelpfully um that's not standard usage, but I also think it's very popular.

37:15Sorry, 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 um if it's raining outside then joe folly is a type of bird that's my first premise premise two it is raining outside conclusion joe folly is a 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 argument because there's no way for those premises to be true and the conclusion be false.

38:14So 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 than soundness. Because if you're a logician attempting to craft or refine a logical system, you tend to want that logical system to apply to almost anything.

38:45You 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 soundness. This is sometimes called the ideal of topic neutrality for logic. Where 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. That's kind of one of the ideals, historical ideals for a 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.

39:21And 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. 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 there.

39:52It may be that the premises are true, but it doesn't matter because the conclusion doesn't follow from the premises. yeah um however if you're saying that an argument is unsound but nonetheless that it's valid that does tell you something it tells you that the 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 yeah um that was you know this was aquinas's argument against you know the clam cosmological argument though of course it wasn't known as the clam cosmological argument then this kind of linear cosmological argument aquinas basically said, no, I don't think that it's guaranteed by natural reason that the universe had a beginning.

40:32He did think that it had a beginning because of the Bible, but he didn't think that you could know that just through reason. And so that would be, you know, we think about what's Aquinas doing there where he's saying, I'm questioning one of those premises. I'm saying it's a valid argument. Yes, it's a valid argument, but I'm - It's unsound. Exactly. Or it's potentially, at the very least, I'm not confident enough to grant you all of those premises. I have to 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,

41:04all Bloggarts are Fogarts and x is a Bloggart, therefore x is also a Flogart, you might start asking me, like, oh, well, you know, what's a Bloggart? 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, at 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. Can we talk about the concept of ex-falso quadlibet?

41:40We can. Yeah, I think that it's ex-falso quadlibet. 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've said, that there's no sense in which the premises can both be true and yet the conclusion be false. So, 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 two, it's not raining outside. One of the reasons you can't say that is because of this principle called ex falso quadlibet, from the false, anything.

42:16And 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 one, it's raining outside. Premise two, it's not raining outside. Therefore, I have a cup of tea in my hand. Is that a valid argument? You might listen to that and go, no, of course that's not valid, that doesn't follow. 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.

42:59So 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 wherever it's not raining outside, therefore there's a cup of tea in my hand, is valid. Yes. And if you wanted to just briefly dip our toes into something slightly technical, in some logical systems, ex forso quadlibet is taken as read. 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.

43:38If A is true, if B is true, or if both A and B are true. The disjunctive syllogism, which some logical systems take as primitive, others will prove this through ex-falso quadlibet, 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-falso quadlibet, how you might conceive of that. The disjunctive syllogism states that 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 that whole statement p or q is true if even if just only one of those is true yes yeah or both of them is true so so it's it's it's inclusive and the statement there's a car outside or it's raining.

44:33Yes. 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 have this inclusive or, 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 two, it's not raining conclusion therefore there must be a car outside um another uh 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 um gorillas are rampaging through the streets of London as we speak is true because one of the disjunctions is true.

45:30Now, how this allows us to get an intuitive gloss of ex falso quadlibet, 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 P. So we've got our contradiction. We've kind of somehow ended up in this situation through the streets of London. Then I have not P. So via the disjunctive syllogism, I can get rid of P or gorillas are running through the streets of London. And I'm just left with a proof that gorillas are running through the streets of London.

46:04So that's kind of an intuitive gloss. So it's a consequence of like the 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 system of logic that we're talking about, you can't allow contradictions.

46:39Contradictions 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? 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 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.

47:14Now, there are logics, former logical systems, that will get rid of ex falso quadlibet 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 of logic, different systems of logic that interpret different things differently, 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.

47:50Like 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 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 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 quadlibet 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 contradiction according to the laws of logic anything follows from a contradiction.

48:37And this is kind of a good reason to not get mired and contradict. 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 impossibile yeah which i think is like i don't think it's just a slightly snazzier term yeah i think it might be but whatever whatever you want to call it if if somebody if somebody holds you know two ideas if they say like um well i don't know i 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, that's a way of showing that it doesn't exist.

49:30And 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'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 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.

50:16The Stoics called their logical rules the indemonstrables for this reason, because they basically sort of went, well, if you think that if P, then Q, 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 point to it and sort of say, it's just there. It's just there. 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.

51:01These 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 something like p the 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 is it a prerequisite of reason like how do we justify these axioms well as i said i think this is one of the the exciting things about the philosophy of logic is that if you're trying to ground logic it's very you can't appeal to logic you know if trying to ground in something prior to logic, that's really, really hard.

51:44As I say, an awful lot of ancient logicians and a lot of modern logicians will just kind of say, try and present to me a way of thinking that allows me to have comprehension. Somebody might just say, okay, well, I'll imagine that it's, yeah, okay, maybe 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. I 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, which is what I was arguing all along.

52:20And you say, no, no, no, it can 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-contradiction, as it's called, seem to be bedrock. They seem to be foundational.

52:53And 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 what are these laws of logic? Where do they come from? Are they psychological concepts? Do they exist in the universe? Did 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.

53:36In fact, I think one of the really interesting parts of Karnam and Tversky's work in psychology is just to demonstrate that a lot of the time we're not thinking logically. We're thinking heuristically. We're drawing from association and emotion. Man is a rational animal in the sense that he can do rationality, but we're not perfectly rational. That's not by any means our dominant mode of thought. So this was summed up in a sentence by the American logician and pragmatist philosopher C.S. Peirce, I don't quite know which way 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.

54:14And cashing out that ought philosophically is incredibly interesting because a lot of people are very comfortable, for example, saying something like, well, you know, moral oughts, 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 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 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 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 can kind of try and cash this out as a hypothetical imperative.

55:22Something like, if you want to preserve truth, then you ought to try and think logically in this kind of narrower sense of what logic means. And that does leave it open for somebody to say, I'm not interested in preserving truth. Or they could say, I'm also interested in preserving truth. Yes. 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're clearly not interested in preserving truth, whatever you say. And you want to say to them, like, your thinking is going wrong.

55:53Yes, as in your thinking has become detached from the kind of inferences that preserve truth from a set of premises to a question. So there seems to be this sort of normative undertone. Yes, yeah, yeah. Which is, I should say, normativity is significantly more commonplace than we often give it credit for. What do you mean by normativity? Normativity is oughtness is the kind of way of putting it, which I don't necessarily mean kind of moral oughtness. 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.

56:27Language contains within it a sort of element of normativity. Even though language tracks how people actually use language, it is also possible to use language incorrectly. So if I was to say, you know, what's that line from the Jabawaki? The slidey toes doth gyre and gimble in the wave to sort of mean 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.

57:05So that's normativity because you should be using language differently, but it's not ethical. 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 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 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.

57:47You 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 according to these logical rules that we have found preserved 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 ifs, buts, or maybes. You really ought to do this. In the context of logic, we've said that logic applies to arguments and sentences and their relations, but it also obviously involves truth.

58:26Logical validity requires Whereas 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? And it's complicated. And I suppose it depends on, again, it 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 logics, you cannot define truth within the scope of the logic.

59:10You 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 truth is cashed out as correspondence between a set of proposition and a state of affairs in the world.

59:50Some people would say that it's deflationary. So when I say, you know, snow is white is true, I'm saying nothing more than snow is white. Yeah. So for some people to say snow is white 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. So there are 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.

1:00:23There are some ambiguous cases as to whether to call something true or false, right? So the famous example is the sentence like the current king of France is bold. like the current king of france is bold i'm gonna ask like is that true or false and obviously the problem is that france doesn't have a abercrombie is an official fashion partner of

1:00:50Joe Folley:the nfl and i'm cd lamb wide receiver for the dallas cowboys you know i'm here for abercrombie's cowboys gear that's not a question but i need a whole wardrobe to go with it no shade to the guys but I'm used to having the best tunnel fits. This season, Abercrombie has me covered. Shop NFL by Abercrombie in the app, online, and in store.

1:01:17I'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 bald 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 is it undefined so this is the statements like these um that particular one was kind of analyzed 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 there oh um there exists only one thing, the present king of France, and that thing, the present king of France, is bold.

1:02:01It'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. You know, they're kind of one of the classic logical difficulties. They cause a lot of headache for some medieval logicians. And, you know, their kind of classic example is stuff like, when did Socrates begin X, Y, and Z?

1:02:38Or like, actually, you know, 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 mom know you're gay? and and you know i don't know if you've seen it there's this um because this is like this is this is um a fantastic instagram reel for anyone that's interested about um the kind of logical decomposition of exponable statements um which is that there's a guy and he he says to somebody across the table um 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 what And, you know, I remember thinking about this when I was a teenager a little bit.

1:03:22Sort of, well, this seems like this happened to you a lot, did it? Yeah, of course. Look at me. This is exactly the kind of insult that is finely tuned to my presentation and manner. And this, you know, the statement like, does your mom know you're gay? effectively decomposes into, you know, proposition one, 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.

1:04:00You can say yes to at least one of those, yeah. The idea is that it's got a kind of nested 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, And expornable statements are very, very commonplace.

1:04:35They'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 expornable statements into something else. So how would a logician approach a question like that? you know well they would say there's there's an implicit claim within the present king of francis ball that there is a present king of france and he is bought so would they say it's would they say it's false well they would yeah they would say it's false it's false the current king of france is gay and if i'm sorry well that would also be an exponable statement that would be that also works because the the reason it's false is just because there's no present king of france okay okay but then then if i asked you joe like does your mom know that the king of france is gay, would that same logician say yes or no?

1:05:27Or would they just reject the question? They'd say it was false. They'd say that one of the claims, 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. Because really the phrase would not be like, my mum doesn't know the King of France is gay. It would be like, 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 or literally whatever property you want to assign to the present king of France.

1:05:53And my mum knows about this. And of course, the conjunction of those is false because the left hand of it is false. And that conjunction is only true if both sides are true. So that would be one way of - But suppose I said the current king of France is non-existent. Yes. That's true. Yeah. But that will be analysed as there does not exist an X such that. 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. Or as I said, the current King of France is bold, and the current King of France does not exist.

1:06:34That 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. One way of thinking about this is - 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 I 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.

1:07:01And 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 that those statements are equivalent um yeah so that's kind of but these these are another um this does just illustrate 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 mean alternatives um when you because there are logicians that hold that there's genuine vagueness in the world or that um there are statements that are genuinely of indeterminate or gradable truth value.

1:07:48And these are fuzzy logicians. Well, they're not fuzzy logicians. Well, they're logicians in fuzzy logic. Yeah, exactly. That's a better way of putting it. And 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 spelt like it's pronounced Lukasivix, 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.

1:08:26Probably 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 0.1 and 0.01 and you've got literally the whole... You've got this continuum between 0 and 1. So 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.

1:09:04What does 0.5 mean? What does that actually mean? Well, the logic itself would not impose an interpretation on that. It's just a series of rules. But 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. I'm sure that there are some interpretations, but it's just like, I don't know. There are the infinitely many valued ones. You might think that you might use that to, 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.

1:09:41There are types of fuzzy logic that mesh quite well with probability theory. So some people want to use them in that way. But 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.

1:10:23But 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.

1:11:03So, 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 two 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 five, 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 um there's the logic of possibility and necessity yes exactly and so the way that this is cashed out on a kind of model system um the kind of the yeah the model theory for um modal logic is uses something called kripke modeling um which i assume was invented by sort of kripke and i actually don't know um but it's called kripke modeling so you I kind of hope so.

1:11:57And 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 some of those and 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 So say you wanted to restrict your notion of possibility to, or 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.

1:12:35Say 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 modeling 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, we're using these ideas of other worlds. We're not talking about a multiverse, physical universe existing. It's just a way of thinking, right?

1:13:08So there's another possible world that could have existed where that table doesn't exist. So we just sort of conceptually think of it as 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. So we think of them as 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 model impossibilities, they will allow for certain worlds to have inconsistent sentences.

1:13:43There are also modal realists who believe that actually David Lewis, who think that these possible worlds are actually real worlds. They actually 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 one way of me saying it's possible that 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.

1:14:22So, 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 and 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 in every world that you're considering is effectively what you believe is the kind of reasoning that an epistemic or doxastic logician might use to justify their usage of this framework of necessity to deal with things like belief and knowledge.

1:15:10They'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 as and well as and that's just one way of dealing with belief you know you could you could talk about it you know a whole series of other ways but i i think that this is a this is at the very least a very good way of formally treating belief um 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 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 some 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 and if that and that and logical palaces didn't even consider or incorporate the concept of possibility versus necessity for example.

1:16:34And 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 have posed 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.

1:17:11And, 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. Yes. Oh, yes, to logical fallacies. Sorry, I just wanted to point out as well that 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 I say something like, no true philosopher would fear death. And you point to me at a bunch of philosophers that did fear death. And I say, oh, well, that's not really philosophers.

1:17:53We 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 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 fear death. This person does fear death, therefore they're not a real philosopher. Exactly. 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 a 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 if p's and q's and all this kind of 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.

1:18:46And this is where some fallacies or kind of other bits of faulty reasoning are more readily treated in a field called informal logic, which is, you know, which a lot of people just say isn't logic. And I kind of sympathize with that, but I like it as a topic. 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 Scotsman fallacy and say something like, right, okay, well, we're not just considering formal constraints.

1:19:32What 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 with less support will be rejected, etc., etc., etc. And you can, you know, informal logicians and philosophers will define certain rules that dictate the kind of things that go on in an idealized, reasonable discussion. We can almost think of this as a language game, if you wanted. One of them might be that you can't dismiss an opponent's point without giving a formal argument as to why it's not true.

1:20:20That's a rule of discussion and a reasonable discussion, but it's not a rule of logic. It's not an axiom. There's no kind of logician that's going to go around your house and beat you up if you don't follow it. But you're departing from the conventions of a reasonable discussion. Another one might be, you've got to ensure that your opponent and you were using similar terminology. Or you might say you've got to ensure that you and your opponent are, you are making your points 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.

1:21:03You 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 Motten-Bailey fallacy. So, in fact, I think the majority of logical fallacies people have heard of are probably informal logical fallacies. Yes, because a lot of logical fallacies will depend on context. What's the Motten-Bailey? Oh, Mott and Bailey fallacy 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.

1:21:47So say I'm a particular kind of theist. Again, I don't want to kind of rag on anyone here, but this is a kind of type of argument you might have encountered is where someone says something like, okay, there must be an origin of a per se causal chain. therefore we have a ground level metaphysical point and we call that point God. 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 Aquinas has arguments as to why these are these, you know, I'm not accusing Aquinas of this.

1:22:32So 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, 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, 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 illogical there. Yes, because if you were to analyze that in terms of propositions, you've not made any kind of invalid point.

1:23:04And another helpful example might be the ad hominem fallacy. If I say, you know like joe like i just think you're stupid and therefore i i don't agree with you someone say well that's the ad hominem fallacy so i haven't committed a logical fallacy there i've just said you know joe is stupid uh therefore he's wrong which like and you know you could you can render that as a formal fallacy if you like and there are there are a number of arguments whereby um it's the line between a fallacious usage of something and a non-fallacious usage of something is surprisingly thick so you could formalize it you could say something like you know So premise one, Joe thinks the sky is blue.

1:23:42Premise two, Joe is stupid. Conclusion, therefore, the sky is not blue. Yeah, exactly. And that's an invalid argument. 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 sequitur fallacy is committed when a conclusion doesn't follow from the premises. Yes, exactly. And all formal fallacies. 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 and yes 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 know you say something like this person is stupid premise two 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 soundness wise most people kind of go premise two that's obviously not true yeah so you know you can kind of you can say that's the ad hominem fallacy but you have to be careful to remember that that's it like i don't think we should call them logical fallacies uh when they're like as you say informal logical they're just like fallacies of a different kind you know they're they're unsound but there's nothing wrong with the logical structure so i think people i this is actually one of the problems is when you're having an online debate and someone says oh that 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.

1:25:29Yes. Which is not. I mean, it's still bad to do that. And that might show that their argument is faulty, but it doesn't have the same, like this is absolutely demonstrably invalid in the way that 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.

1:26:04So 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 illegitimate. And also, for the wall example, you could analyze that as having an implicit premise in it, which is, 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.

1:26:51And then 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. And 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 using propositional form and stuff like that. I think that oftentimes a criteria for how you might want to go about this, oftentimes it's kind of generally to go with informativity. 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 formalize that argument.

1:27:32So 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 that gets to the representation that that argument is invalid and therefore a bad argument, 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 it kind of depends exactly what's being meant because there are times when the so-called ad hominem fallacy might kind of be appropriate, which is to say like you know uh if somebody says i believe this proposition and i say well why do you believe that yeah 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 uh if somebody is stupid then what they say is less likely to be true yes you know And so this is a statement that this person said, therefore this statement is less likely to be true.

1:28:44Yeah. So if we were going to formalize that, we'd say something like, if someone is stupid, then their statements are less likely to be true, which is like plausible. It's plausible in the sense that actually I don't quite know what we would mean by stupid if it didn't entail that. you know mark is stupid uh therefore um 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 to be something um kind of a way that we could we could extra specify that and then you've got a you've got an argument that at the very least isn't obviously unsound on its face i think the 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 figure says so.

1:29:29Or 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 contexts in which it is actually sensible to trust authorities. 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 CO2 data, but no idea what's going on there. 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 contexts you kind of want to be like yeah it is but you know that's that's kind of fine well i all of the um all the the idea of something being logically fallacious would mean um here is that um it doesn't guarantee the truth of the conclusion so so um it doesn't you know say if you were to 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 You'd say something like, okay, if something is first premise, if there's expert consensus on something, it is more likely to be true.

1:30:33Second 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 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.

1:31:09So again, you can kind of see how breaking things down into 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 got no introduction to this? Well, so the OpenLogic project is a website with free logic textbooks. I wish I was involved. I'm not. They're there. Download them for free. Very, very readable. To give you an idea of how readable and useful they are, the For All X free logic textbook was literally the textbook for the first undergraduate Cambridge logic course.

1:31:52that was yeah when i did logic in first year university it was 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 and there are there's there's ones on set theory there's ones on category theory i don't understand category theory at all um i wish i did but i don't um 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 Logic.

1:32:27And that will take you right the way from formalizing statements through propositional logic, predicate logic, and all the way through to Gödel's incompleteness 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. You know, I say that at the end of a lot of podcasts, like it's been fun and it always is. But I think that I like podcasts that are quite a lot of time sort of discuss all over the place, but it's nice to have a particular task. Intro to logic.

1:33:02What'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, thanks for having me.