The 7 Levels of Logical Thinking

There are few topics as important as logic and logical thinking. So today we're going to go through the seven levels of logic. Starting with its humble beginnings through what you would learn as a student and eventually seeing a little bit of what professional logicians are working on today.

Before we get started, just a quick disclaimer. Some of this video is going to be intuitive ways of explaining concepts that really only have formal definitions. Because the concepts in question can't be fully described accurately without logical symbolism, these explanations will be technically wrong or technically incomplete, but they're just intended to give you a feel for the relevant formal concept. So don't take them as gospel. But with that caveat, let's get started. My name is Joe Folly and this is unsolicited advice.

One, pre-logic. This is where most of us start out. Like almost anyone will have a vague sense that some arguments are good and other arguments are bad. We may have heard someone say something like, "You're a bad person, therefore you're wrong." or try to generalize from a small set of cases to infer over a massive group of people and had this kind of rough sense that something was a miss. But we don't yet know exactly what it is that separates those bad arguments from good arguments that we've heard in the past. It's important also not to confuse logic with critical thinking here. This person might still be incredibly intelligent, but the precise science of logic itself remains a mystery. If someone were to be in this stage and they were to Google words like logic or how to tell if an argument is good to try and learn more, then they would probably move to stage two.

Two, the fallacy monger. I love a good logical fallacy as much as the next insufferable internet philosophy guy. And in my experience, this is also a lot of people's first encounter with logic uh in a slightly more regimented sense. and it was also how I took my first steps in this area. So it has a special place in my heart. The term logical fallacy is derived from the work of Aristotle and effectively it's a type of argument that is always or almost always bad. Aristotle wanted to identify such arguments because often times even technically bad arguments can appear rather persuasive to the untrained eye even though strictly speaking because they're bad arguments they shouldn't persuade anyone. So there is a lot of utility in identifying such uh surfacely seeming good but ultimately poor argumentation. Take the example we used earlier where someone said you're a bad person therefore you must be wrong. I'm sure that many of you will recognize this as an ad hominem fallacy. This is where we infer from an irrelevant feature of someone's character to the conclusion that their argument is incorrect. It doesn't take a genius to figure out why this inference doesn't follow. There are awful people throughout history who have still been factually correct about some things. And there are incredibly moral people who have been factually incorrect about some things. If a mass murderer reproduced Euclid's proof that there are infinitely many crimes on the spot, well, they would be just as correct as if St. Maximilian Kolbe had done it, who is undeniably a much more moral person. And if St. Maximilian Kolbe had tried to reproduce the proof and got it wrong, it would still be wrong and no amount of his moral upstandingness would compensate for that. This is where most people will probably end their study of logic because it does give them a lot of what they want, which is a rough guide to telling good arguments from bad arguments. However, if we do reflect for a little bit, we can demonstrate the limits of this approach. Take the following situation. You are listening to a Nobel Prize-winning physicist and a layman discuss quantum mechanics. They have a disagreement over the details of Heisenberg's uncertainty principle. You listen to each side, but you don't know much about quantum physics yourself. So eventually you defer to the experience and authority of the physicist over the layman. Somebody well-trained in logical fallacies will spot this as an appeal to authority. It's an attempt to use someone's authority to justify a point they have made or infer from that authority that the point is true or more likely to be true. But it also doesn't seem like you've done anything particularly unreasonable here. Surely if the physicist and the layman are having a disagreement, the physicist is more likely to be correct because of their years of experience, all else being equal. They've studied the subject at a high mathematical level. So yes, they do have authority, but that authority seems like it's been earned in the relevant sense. Should we really think that it's unjustified to appeal to that authority if we can't resolve the answer to the question ourselves? Of course, if we can resolve the answer to the question ourselves, then we don't need to look to that authority. But I'm just saying it doesn't seem like this is a totally misguided way of reasoning with limited information. I think these limits are why those who are ready to label an argument a logical fallacy have acquired such an odious reputation online. There are many arguments that look a lot like fallacies but are in fact reasonable ways of well-reasoned under uncertain conditions where you cannot find out an answer directly yourself. Whether something that looks like a fallacy actually is a fallacy is often going to depend on context and the very specific proposition a speaker is trying to demonstrate. Not every appeal to authority is fallacious and someone's personal character is sometimes very relevant to a debate. Say if someone is running for mayor, you might use character information to infer about whether they're going to be a corrupt mayor or an incorruptible mayor. Most of the things we call logical fallacies aren't actually full-blown errors in formal logical reasoning, but are instead what's called informal fallacies. That is, you cannot tell merely from their structure that they are fallacious. They either break an argumentative rule or are unreliable reasoning within a particular context, but they're not formal errors. Many logicians would actually consider this beyond the boundaries of logic proper. But there is an area of philosophy called informal logic which does attempt to study everyday argumentation as rigorously as possible and tries to work out exactly how you can tell when a natural language argument is fallacious. And I made a video roughly about that and about logical fallacies in general right here where I'll go over some of the same stuff that I go through in this video, but it is a substantively new video as well. As I said, this is where most people will probably end their studies in logic. But I think that's a real shame because in my view, this is where things really start to get good because now we can begin to talk about formal logic. But ultimately, logic is a formal field. And if formal fields are the kind of thing that you struggle with, then you should check out today's sponsor, Brilliant. We all know that studying STEM subjects like maths and science and programming can be difficult and unintuitive. 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Three, basic formal logic. In the first few months of an undergraduate philosophy degree, you will learn the very basics of formal logic. And this will almost certainly be propositional logic. This is strictly speaking a mathematical system with a set of axioms and a set of inference rules. And those axioms and rules define what kinds of statements follow from what other statements. And it'll become clear what I mean by that as this section goes on. As I said, the first formal logic that most students will encounter is called propositional logic. And that's because it is the most basic kind of formal logic. The basic objects of propositional logic are propositions as the name suggests and they tend to be represented by letters of the alphabet like P or Q or R or S and so on. Propositional logic also contains a set of logical connectives that stand in for roughly speaking certain natural language words like and or or if then or not that are meant to connect different propositions together and thus allow you to facilitate inferences. So a very basic argument in propositional logic would run as follows. If P then Q. P. Therefore Q. This is called a modus ponens inference. The precise definitions of the logical connectives are defined by what's called truth tables. And we'll go through one now. We'll take the truth table for "and" since that one is just the most intuitive. A statement like P and Q is true if and only if P is true and Q is true. In just the same way that in natural language, the statement "Jon is Jane's brother and her friend" is only true if Jon is both Jane's brother and Jane's friend. This might sound like it's obvious and pedantic because at its most basic, formal logic is trying to create precise and mathematical conditions for everyday reasoning. Or at least it's trying to do that at this level when you're in your kind of first term or so of an undergraduate philosophy degree. As we'll see, it gets far beyond that eventually. Some connectives are a little less intuitive like the conditional connective "if then." A statement "if P then Q" is false if and only if P is true and Q is false. This means rather bizarrely that if P is false, then "if P then Q" is actually true. Some people mark out this peculiar kind of truth by saying that it is trivially true. At first, this might strike you as strange because things tend to work slightly differently in ordinary language. If I said, "If cats can fly, then 1 plus 1 equals 2." It would seem really strange to call what I've just said true. It seems like here the antecedent, the first bit of the conditional, and the consequent, the second bit of the conditional, are not connected by any kind of relevant factor. And so, the conditional looks really odd. This is actually a very old problem in the philosophy of logic that dates at least as far back as the ancient Greek Stoics who are actually some of the most innovative logicians of the ancient world as well as, you know, creating Stoicism. Some philosophers argue that we should redefine the conditional to only be true if there was a further relevant connection between the antecedent and the consequent such that the antecedent would make the consequent true. The reason why formal logic mostly sticks with the definition that we've already outlined, as weird as it seems, is because translating this idea of a relevant connection between the antecedent and the consequent is very difficult to express mathematically. Right? It it what we have without it is what's called a truth-functional definition, which means that we can fully define the usage and inference conditions of the conditional merely by saying what is true and false about the stuff either side of the conditional. Adding in an additional constraint uh regarding relevance or the antecedent making the consequent true would indeed make it resemble natural language more closely, but it would also make it much harder to work with mathematically. And as formal logic tends to on average be of more interest as a mathematical system, especially at more kind of advanced levels, that tends to be the one that people go with. There are still people that argue for a redefinition of the conditional, but the truth-functional definition, the weird-looking one, is still the mainstream view. We cannot go over all the rules of propositional logic now because there are quite a few of them. But to give an example of what a very basic proof in propositional logic looks like, here's one from the open-source logic textbook for all X. What this is essentially showing is that we can prove in propositional logic that if P is true, then P and D or P and not D is also true. The labels down the side and naming the particular rules used at each point in the overall proof. At this level, students will also learn basic logical concepts like validity and soundness. A valid argument is one where if all its premises are true, then its conclusion cannot be false. For instance, if all cats can fly, then cats have wings. Cats fly, therefore cats have wings. This is a valid argument because if all its premises are true, the conclusion must also be true. It can't be false. Its conclusion is false. So it's still unpersuasive, but that's because the premises themselves are not true. A sound argument is a valid argument where in addition to that validity, all of the premises are in fact true. So something like "all men are mortal, Socrates is a man, therefore Socrates is mortal" is a sound argument. These terms are very helpful for evaluating precisely what is wrong with an argument. Whether it is structural and therefore the argument is invalid or whether it is a problem with the premises. If you're not into philosophy or maths or computer science, I personally think this level of logic probably fulfills like 99% of your needs. It will help you formalize a lot of arguments that you encounter in the wild and it remains very closely connected to everyday life. Additionally, you can probably get there with around 5 hours of focused work. So, it's kind of the it's the the best bang for your buck part of logic for everyday life, at least in my opinion. But even philosophy students who don't choose to make logic their focus do tend to learn a little bit more logic than this. And we'll go through some of that now.

Four, first-order logic and friends. The next step for most students after learning propositional logic is to learn first-order logic. Whereas propositional logic deals in full propositions, first-order logic can deal with more fine-grained statements like "all men are mortal" or "all cakes are delicious." That second sentence would look like this in first-order logic: For all X, if X is a cake, then X is delicious (where CX means "X is a cake" and DX means "X is delicious"). On the other hand, if you wanted to say something like "some cakes are delicious," you would write it like this: There exists an X such that X is a cake and X is delicious. The first-order logic retains all of the symbols from propositional logic, but can assign properties to objects as well. In fact, it quantifies over these objects. That's what the uh little "for all X" symbol means. That uh little upside-down A is called a quantifier and X is a variable uh quantifying over objects. This means that you can do more with the logic essentially. We can also express relations in first-order logic, which are properties that involve more than one object. So again, let's use a kind of natural language example. We might formalize the phrase "every person has a father" like this: For all X, if X is a person, then there exists a Y such that Y is the father of X. It sounds really clunky to say out loud, but this kind of precision becomes helpful in a formal mathematical context. To take a very basic mathematical example, we think that a prime number, well, we know that a prime number is a number that only divides by one and itself. But we could express this precisely in logic with the following sentence: For any Y, if X / Y is a natural number, then Y is the same number as X or Y is one. That is, X only divides by one and itself. We've drifted slightly beyond the bounds of strict first-order logic to define this, but it just is an illustration of the kind of things that basic logic can do. As a student gets used to these symbols, they tend to stop translating them into natural language in their head like I've been doing and instead just read the symbols as you would any language. It's like learning French. At some point, you stop translating French sentences into English and you begin simply reading French. Alongside this, at this level, students tend to learn things like basic probability theory and basic set theory, which I'll just kind of skim over briefly. Now, set theory is an abstract mathematical framework that consists of sets and objects which belong to those sets. They're called members of those sets. So we might talk about the set of all people or the set of all prime numbers and then perform various operations with these sets. At the basic level, like the very basic level, we can think of these sets like Venn diagrams. So say we have two sets, then the intersection between those sets is what's in both sets, while the union of the sets are what is in either set A or set B or both, and the complement of a set is everything outside of the set. This is a very handy tool both in mathematics and philosophy, though obviously it tends to come in more uh uh fleshed-out versions than this. You'll also notice that each of those set-theoretical operations maps quite neatly onto a logical operation. So the intersection is very much like "and" because it's only the stuff that's in both sets. Uh the union is very much like "or" because it's the stuff that is in either set or both sets. And the complement is very much like "not" because it's everything that's not in the set. Probability theory, as I'm sure many of you will already know, is the area of maths concerned with evaluating probabilities or the likelihood that certain things hold given certain other things holding. In a philosophical context, it is often used to make sense of reasoning under uncertainty, as well as in the philosophy of inductive reasoning. That is, the kind of reasoning that uses previous experience to make uncertain inferences about the future. Take the example from earlier where we appeal to the authority of the physicist over the layman. Well, why was that justified? I suspect that the answer many of you would have instinctively given is that the physicist is more likely to have true beliefs about quantum physics than the layman and thus his testimony is stronger evidence than the layman's. Even very basic probability theory can give philosophers a framework through which to make these claims precise. This is often done through Bayes' theorem, but that's kind of a topic all on its own, and I have beef with how many philosophers use Bayes' theorem, so I'm just going to park that for now. The aim of learning logic at this level is not necessarily to become like total masters of these symbolic systems. Most of the students who learn these on a philosophy course will then branch off into various non-formal interests across like epistemology and moral philosophy and political philosophy and loads of other fields. The main reasons that we teach logical systems to undergraduates at this level is that in engaging with these frameworks, it forces you to think in very precise ways. There's no room for error or ambiguity. And this can be helpful just for practicing thinking carefully about any area and not just philosophy and not just maths. So I know that was a lot of symbols and as we go on, there'll be far fewer of them because we're now going to move into explaining uh various different logical systems and sometimes more advanced logic and so I'm just going to explain it in intuitive natural language terms, but as a result, I'm also going to give the proviso here that that will necessarily distort the concept a little bit. So uh the rest of the video is very much like going to give the feel of what these logical systems are like, but it won't be like formal or rigorous, uh because it will just get like far too complicated and this video will end up being uh loads and loads of hours long. But with that caveat, let's move on.

Five, more specialized logical systems. In the later years of undergrad and potentially going into masters, students will learn a variety of different logical systems and we can't go through them all here, but we will go through a few. And potentially the most common is modal logic, which is a type of logic used to talk about possibility. Philosophers quite often talk about whether something is possible or necessary and what makes a possible or necessary statement. But as it stands, those terms are often quite vague and difficult to get a handle on. A modal logician will formalize "necessary" to mean "true in all possible worlds accessible to the agent" and they'll formalize "possible" to mean "true in at least one possible world accessible to the agent." This in turn is cashed out in terms of what's called a Kripke model. We can think of a Kripke model as a collection of worlds and things that are true at these worlds. So there might be a world where pigs can fly and another where Elvis is still alive and so on. In technical terms, these worlds are sets of propositions. The worlds are then connected by things called accessibility relations. We can think of it as like a series of balls connected by lines where the balls are possible worlds and the lines denote which worlds access which other worlds. A proposition is necessary at a given world just if it is true in all the worlds the initial world can access. So if I'm standing at world X and I can see worlds Y and Z as well as my own world, and the statement "pigs can't fly" is true in all three of these worlds, then within the context of that model, we would call that proposition necessary. This framework is remarkably flexible. For instance, you can use it to model belief and knowledge rather than possibility. If we think about what happens when we fully believe something, we treat it as an assumption in every potential course of action that we're considering. If I believe that pigs can't fly, then I'm never going to plan for a pig to come shooting through my bedroom window. So, we can formalize this by saying that I live at world X and I'm considering a certain set of worlds when I plan my actions. And in all of those worlds, the proposition "pigs can't fly" is true. With certain alterations, we can use this same framework to model what someone knows. These logics called doxastic and epistemic logics respectively have seen a surprisingly wide variety of uses across areas of computer science as well as game theory, which sometimes involve modeling knowledge or belief or knowledge-like or belief-like states. You'll also notice that in both of these cases, it's not like the logic has just solved the philosophical issue completely. People still talk about what it means for something to be necessary or possible or what it means to know or believe things. But these provide tentative formal frameworks within which to explore these questions and so they can be useful to philosophers. On the other hand, if you are a linguist, you might end up using logic in a field called formal semantics, which is the area of linguistics that creates relatively precise formal structures with which to analyze our speech. Remember earlier in the video when we were formalizing various natural language statements as formal logical ones? Well, formal semantics takes that general principle and expands it massively to encompass more and more of our natural speech. They have ways of formalizing things like questions as well as more complex linguistic phenomena like gradable adjectives. For instance, the former semanticist Gila Casan has spent much of her career trying to understand what are called multi-dimensional gradable adjectives. So, we'll just go through kind of what that means. We often say things like "Adam is more athletic than Steve" or compare people in this way. But the term "athletic" is a complex word. Athleticism is not a single property like height. It is an amalgamation of various different more simple properties like cardiovascular fitness and musculature and skill at various sports and so on. So using empirical evidence concerning how people actually do use these words, Casan has come up with various formal and informal ways to analyze them and she's drawn on certain logical kind of symbology in order to do so. We won't go into any more detail about that here because formal semantics is just like a huge field and I'm only really familiar with small parts of it. But I think it's worth mentioning because it is a non-philosophical and non-mathematical use of formal logic. And so it's kind of worth bearing in mind this stretches beyond philosophy and maths. At this level, philosophy and logic students will also often learn model theory, which is a way of talking about abstract mathematical structures in the language of mathematics. This can get incredibly complicated incredibly quickly, but we can illustrate it with an example. So the model-theoretic structure of a certain kind of basic arithmetic might be written as follows: Z is the set of integers (that is, whole numbers), + is addition, * is multiplication, and 0 and 1 are the neutral elements for addition and multiplication. A neutral element here means like an element that does nothing. So if you add something by zero, then you just get the same number again. And if you multiply something by zero, then you also just get the same number again. Intuitively speaking, what we have here is basic integer arithmetic, but boiled down to its most basic and essential components. To put it another way, contained within that basic set of symbols are the sort of ingredients for the whole of basic integer arithmetic. Model theory is also used in logic to provide a formal logical system with what's called a semantics. That is, we create a model to clarify exactly what the symbols in a logic are referring to. The relationship between the syntax of a logic, that is the symbols themselves, and the semantics of a logic, that is what the symbols pick out, will become increasingly important when we move on to kind of meta-mathematics and meta-logic in the next section. Someone might also choose to learn things like more advanced set theory at this level, which is quite a lot of fun, but I haven't done it in a very long time. So, if I try and talk about it in any real detail, then I will get it massively wrong. Set theory gives us the tools to speak about quite a lot of seemingly mysterious concepts in surprisingly clear ways. For example, the concept of infinity has famously vexed philosophers since Aristotle. But set theory can give us a way of talking about infinities in really quite precise ways. And we can even prove things like that some infinities are bigger than others. And we can see which infinities are the same size. For example, we can actually prove that the set of rational numbers, that is the numbers that can be represented by fractions, is actually the same size as the set of integers, that is the set of whole numbers. This is a deeply unintuitive result, right? Surely there are more fractions than there are whole numbers because there are multiple fractions in between each whole number, but it turns out that we can prove they're the same size using some pretty basic tools from set theory. So yeah, it's it's pretty trippy stuff, but it is pretty cool. Obviously, there is a lot more to say about all of these fields, and I have been distorting them slightly because I've been talking about them at a kind of intuitive level, but you can see just how much logic balloons beyond merely formalizing natural language statements. Moreover, different people at this level will get vastly different experiences in logic. They will learn very different things. If you choose to focus on model theory, you'll end up learning a very different field to someone who chooses to focus on formal semantics. Another theme at this level, as we've sort of seen, is that logic becomes increasingly disconnected from our everyday reasoning. We set out at the beginning of this video simply trying to make our ordinary thinking more precise. But over time, we've become increasingly interested in logic as its own mathematical field of study. This happens with quite a lot of areas of maths. If we kind of stop and think about it, we begin by asking what a prime number is, and we end up with the whole sprawling arena of full-blown number theory. Our intuitions are thus increasingly unreliable guides to answering higher-level logical questions, which is why the formal apparatus is so very important. But there are some more advanced logical results that almost every budding logician will encounter at some point, but ones that are famously difficult to make head or tails of. And I'd like to go through a few of them now as they tend to present challenges for the logic student.

Six, metamathematics and meta-logic. If you choose to pursue logic towards the end of an undergrad degree or the beginning of a master's degree, you'll also begin learning what's called metamathematics and meta-logic. And this is where we use maths and logic to study mathematical and logical stuff. We already saw this a little bit in the last section when we looked at model theory very briefly. And to be honest, this field gets incredibly odd very quickly, and we're only going to be able to go over the very basics of it and a few little examples here. And yeah, the idea is just to give you a a flavor of this kind of odd area of logic. There's also an argument to be made that this level and the previous level could be swapped since the order you learn these sorts of things in really does depend on which institution you're at. And finally, and I know this video is full of disclaimers, but I just want to reiterate that I'm going to be really quite loose with my logical terminology here. So, I apologize in advance for that for anyone in the audience who is a kind of technician. This is just intended to give you a feel for these ideas rather than be exactly precisely correct, which would involve defining a lot of the terms I'm using formally. And at times, I have sacrificed precision and technical correctness for the sake of accessibility. I just want to again uh remind everyone of that. Now, you might understandably ask why anyone would want to use logical systems to study logical systems. What could possibly be the point of this beyond meaningless naval gazing? I mean, what questions would you even ask? Well, here's one basic question we might be interested in. Does our logic contradict itself? So, imagine that I had a formal system that was the same as propositional logic, but included an extra connective and an extra inference rule called "derf." My sister's favorite TV program growing up was iCarly. And in one episode, they invent a new number called "DUR." So I think the name is appropriate. For the sake of this example, I stipulate that the statement "P derf Q" can be introduced whenever we have P or we have Q. And it can be eliminated to give you either P or to give you Q. So if I have P in this logic, I can infer that "P derf Q." And if I have "P derf Q," I can infer that Q. You might think this is a completely ridiculous rule to introduce into a logic and you would be absolutely right. In fact, "derf" would make our logic contradictory. So say we had proposition P. Well, we can infer from this that "P derf not P." But then we can infer from "P derf not P" that not P. But then we've proven not P from P. And so we've contradicted ourselves. This is, to put it very mildly, not what we want from a logical system. Thus, a basic meta-result that we want to demonstrate for a given logic is consistency. A more powerful result that is equally as important is soundness. A formal system is sound when it doesn't prove anything false. Now, "false" doesn't mean false in our ordinary sense in this context. It means false in the semantics of the logic. As we saw earlier, logics have a syntax which consists of their symbols, as well as a proof theory, which is their inference rules, and a semantics, which is the intended set of mathematical structures that the logic is meant to refer to. So technically, logics are not sound or unsound, but they are sound regarding a given class of models. That is, they don't prove anything that is false in those models. This is again pretty abstract, but it's best illustrated with an example. Take the perfectly ordinary math of basic arithmetic. We can create mathematical structures where basic arithmetic is definitely unsound for those structures. For instance, take your ordinary everyday clock whose numbers run from 1 to 12 and then go back to one again and you know, so on and so forth. Ordinary arithmetic would be unsound regarding this model because in ordinary arithmetic, 12 + 1 equals 13, whereas in clock arithmetic, 12 + 1 equals 1. To model clock arithmetic, we would need a slightly different system called modular arithmetic. Obviously, I've skimmed over some details here. For one thing, we would normally have to define a specific formal system and not just handwave by saying "ordinary arithmetic." But this is just to give you an idea of what it means for a logic to be sound or unsound relative to a given class of models. I nicked this clock example from one of my old logic teachers back in my first year of undergrad and I can't for the life of me remember his name, but thank you to him. Related to soundness is completeness. A formal system is complete regarding a given semantics if it can prove everything that is true in that semantics. It's sort of the flip side of soundness. Soundness is not proving any false things, and completeness is proving all true things. Let's again illustrate this with an informal sort of toy example. Imagine that there was a model that consisted simply of a traffic light and a set of rules about how cars should behave in response to that traffic light. The traffic light has two colors. It's either red or green. Only one color can show at a time and there is always one color showing. When the light is red, cars should stop. And when the light is green, cars should go. We can imagine a set of rules that would be complete regarding this model. It would probably consist of something like the following: If the light is green, it is not red and vice versa. If the light is green, the car should go. If the light is red, the cars should stop. We can think of this as a very simple semi-formal system for the traffic lights. And it's immediately clear that this set of statements plus some other basic formal machinery that we're just kind of skimming over would allow us to say any true statement about the traffic light model. That means this set of rules is complete for this model. A complete logic of the traffic lights is thus one that, intuitively speaking, reaches every fact about the traffic lights. By extension, a complete formal system of say, arithmetic, would be something that reaches all the truths about arithmetic. However, we know that there is no formal system that is complete for arithmetic in this way because of Gödel's incompleteness theorems. These are too complex to go over in a short section like this and I will write a full video on them at some point. But essentially, intuitively, Gödel proved there is no consistent, effectively axiomatized system that can prove all the truths of arithmetic. Don't worry about what "effectively axiomatized" means here because we'll go over that in a future video. Essentially, this means there will always be an arithmetical proposition left over that is true but cannot be proven within this formal system. The reason this is such an important result is well, partly for historical reasons, but also because it just really seems like arithmetic is simple enough that we ought to be able to come up with a complete, axiomatizable system that will prove everything in it. But we can't do this. Moreover, through a further proof, Gödel demonstrated that no consistent, axiomatized formal theory expressive enough to capture basic arithmetic will be able to prove its own consistency either. So if we want to prove the consistency of arithmetic, we will need to appeal to a more powerful logical system to do so, which in turn cannot prove its own consistency and so on and so forth. However, I I want to be clear here. I will end up saying this in the whole video on Gödel's incompleteness theorems, but I I'm just going to reiterate it now. Any kind of claim that Gödel's incompleteness theorems broke math or anything like that are very overblown. This is actually kind of quite a limited result. And while it is of intense logical and philosophical interest, it didn't like detonate the world of math. Math is is absolutely fine. And these meta-mathematical and meta-logical results are investigated for each logical system that you study individually, along with proving certain important formal results within that particular system. So some other classic questions we might be interested in are whether certain statements can or cannot be proven within a given formal system. For instance, the most popular type of set theory is called ZFC, or Zermelo–Fraenkel set theory with choice. A research program that's emerged in the past 60 or so years, I think, has been demonstrating that various results are independent of the axioms of ZFC, meaning that they cannot be proven nor disproven by this mathematical system. This helps mathematicians and logicians better understand the relationships between certain formal results and systems that may or may not be able to demonstrate these particular results. Uh the most famous example of this is the proof that the continuum hypothesis is independent of ZFC. The continuum hypothesis states that there is no infinity that is larger than the set of natural numbers but smaller than the set of real numbers. The truth or falsity of this is a very interesting mathematical question within the study of infinity. But it's been proven that ZFC can neither prove nor disprove this. And at this level, there will also be a focus on learning various different proof techniques like forcing and diagonalization, as well as more basic ones like induction on the length of a formula. Like many areas of math, the same proof techniques tend to reappear in many different proofs and many different areas. So students and researchers will become acquainted with a great number of them. In philosophy, there are many debates within philosophical logic about which is the best kind of logical system for either kind of the best in total or the best for particular things. This is a philosophical methodological question rather than a mathematical one. For example, a logician named Stuart Shapiro has put forward a number of influential arguments that second-order logic should be used far more often in philosophy and logic and the foundations of mathematics. We met first-order logic earlier, which could quantify over objects. When we were saying things like "for all X," that's what the logic was doing. It was quantifying over the object variable X. Second-order logic can talk both about objects and sets of objects or properties or relations, which makes it much more expressively powerful. So in second-order logic, we can express statements like this: For any X, X has at least one property. We simply couldn't say this in first-order logic, but it might be quite a useful sentence to have at your disposal. Even though I think in most models, this this one's basically trivial. Second-order logic also helps us to define important mathematical properties that we might be interested in. The classic example of this is mathematical induction, which can be written in second-order logic as follows: For any property, if zero has that property and for any number, if that number has it, then that number plus one has it, then that property holds for every number. However, there are also various downsides to second-order logic, which makes some mathematicians and philosophers less likely to want to use it. For one thing, all that extra expressiveness comes with more heavyweight formal machinery, uh, for one of a better term. Within the philosophy of logic, philosophers will often discuss the various merits of different logical systems in this manner. Another philosophical debate concerning logic is between logical monists and logical pluralists. Logical monism holds that there is one true logic which captures what logic intuitively actually is. While logical pluralists tend to view logic more like a set of formal tools which are better and worse at different kinds of tasks. This may sound like a non-technical debate, but it can get surprisingly mathematical, especially when philosophers are discussing which candidates are plausible for the one true logic, or in the case of logical pluralists, demonstrating the utility and philosophical value of various different types of logic. An old supervisor of mine called Owen Griffiths wrote a book with Alexander Paseau called "One True Logic," which argued that a particular kind of logic was the one true logic and then defended that thesis. In addition to all of this, all of the logics that we encountered back at level five, along with many other kinds of formal system, can be pushed far, far further. There will be meta-results as well as just ordinary results that are important to prove in areas like model theory and intuitionistic logic and modal logic, as well as game theory and formal linguistics and topics in theoretical computer science. All of these can obviously be explored in so much more detail. We really have like just skated the surface of each of those fields in this video. If you want to get a broad overview of the kind of things that people study in logic at the master's level, then you can check out the University of Amsterdam's Logic, Language, and Cognition website, as they run one of the most highly acclaimed logic master's in the world. And you can kind of see what topics they offer. But what comes next? Well, this is where we move far beyond my level and start to look at what active logicians are working on today.

Seven, current logic research. There is a wide gulf between level six and seven here. So far, we've been looking at the kind of logic that you would learn at a university, but this level is what actual working researchers in formal logic are doing with their time and energy. Specifying beyond this is a little bit difficult because logic is a huge field. It's a little bit like asking what do researchers in neuroscience do. It heavily depends on the individual researcher and the area that they've specialized in. But to give you just an idea of what modern logic looks like, here is an abstract from a recent featured article in the Israel Journal of Formal Logic. "We explore several model-theoretic aspects of D-sets, which were studied in detail by Adler and Noman. We characterize ultra-homogeneity in the class of colored D-sets and classify unbounded order in discernible sequences in such structures. We use these results to provide a characterization of distal colored D-sets and prove that all colored D-sets are monotonically NIP." It would be very difficult for anyone to grasp what this means without a significant background in logic and some detailed knowledge of the particular question being tackled. I don't understand it myself. It's just like any specialized area of research. The questions become increasingly sophisticated and they also become tailored towards a particular research program. Let's take a different example of modern logic work that is more philosophical. This is a 2018 book by Tim Button and Sean Walsh called "Philosophy and Model Theory." I've actually not read the whole thing, but it surveys different uses for model theory within philosophy and also philosophizes about model theory itself. It is a more accessible book than most, at least from the parts of it that I've read, and it doesn't assume an advanced mathematical background. But it is still quite tricky stuff for your average reader. It's also an example of a major part of modern philosophical logic research, which is seeing how certain formal results might be relevant to various philosophical questions and tasks. At the more technical end, logic just becomes continuous with other areas of mathematics and often theoretical computer science. So, it's not unusual to see those who focus on logic in their research hold a joint position in the philosophy and math faculties at their respective universities. And this is kind of what I love about logic. We began the video by simply asking what a good argument was. And we've ended with a highly advanced field of study which crosses multiple disciplines precisely because it started with such a simple question. Way back at the foundation of logic, when Aristotle began to consider it its own specialized field of study, he said that logic should ideally be topic-neutral, meaning that it can apply to anything. While the advanced results in logic today don't always live up to the letter of total topic neutrality, I think it still has some of the spirit of that topic neutrality by impinging on and applying to so many different fields at once. There aren't too many places where linguistics and computer science and philosophy and math are constantly combining and communicating, but logic is one of those places. And it's remarkable that over 2,000 years of people asking "what is a good argument" has brought us to this place. Now, obviously, there is so much at every level that we've not been able to talk about here. What I really hope is that this video has kind of hyped you up to learn some logic for yourself. And if so, there are a few key resources that I recommend. First, there is the Open Logic Project. I mention these guys constantly because they are absolutely phenomenal. They make some of the best logic textbooks in the world. And what's more, they are completely free. So, have a look at their website and maybe try out some of their textbooks. Next, a good bridging textbook to some of the more advanced stuff is the textbook "A Friendly Introduction to Mathematical Logic." It will take you up to Gödel's and completeness theorems and it should put you in a reasonably strong position to start venturing out for yourself. A word of warning though, if this is your first time learning mathematical or formal content, there are a couple of things to bear in mind. The first is that reading a math book is slow work. It's not like a novel where you can zoom through or even a philosophy book which might be slow-going but you can still read it like a normal book. Even the best-written math textbooks are dense and they tend to build cumulatively, meaning that if you haven't got the hang of a concept, the rest of the book will be very hard to understand because a lot of it will rely on that concept. There is just no getting around this. Uh I suppose one word of advice here would be, you know, do the exercises, because they really do help. Uh without doing the exercises, it is just very difficult to take in the content fully. The second is that mathematical knowledge fades really fast. It's something that I've kind of had to come to terms with over the past few years as I've been doing much less formal logic and much more of other areas of philosophy. It always takes me a while to get back into the swing of areas of logic that I used to know in quite a lot of detail. I don't say this to discourage you. It is far easier to relearn some logic than it is to learn it for the first time. But I just thought it was worth mentioning. Logic isn't the kind of thing that you can learn once and then it's with you forever. It's more like cardiovascular fitness or muscle mass where it's easier to rebuild than build, but it will still fade without proper exercise. That being said, I really do hope this video has given enough of a taste of what logic is is roughly about for you to give it a go. I can promise with pretty much complete certainty that you won't regret it. But if you do want to hear more about the flaws of simply labeling things as logical fallacies and why I think it's important to learn some logic beyond that point, you can watch my video on that very topic right here. Thank you so much for watching and have a wonderful day.