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The 7 Levels of Logical Thinking — Transcript

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0:00There are few topics as important as
0:02logic and logical thinking. So today
0:04we're going to go through the seven
0:06levels of logic. Starting with its
0:08humble beginnings through what you would
0:10learn as a student and eventually seeing
0:12a little bit of what professional
0:14logicians are working on today. Before
0:16we get started, just a quick disclaimer.
0:19Some of this video is going to be
0:20intuitive ways of explaining concepts
0:22that really only have formal
0:24definitions. Because the concepts in
0:26question can't be fully described
0:28accurately without logical symbolism,
0:30these explanations will be technically
0:32wrong or technically incomplete, but
0:35they're just intended to give you a feel
0:37for the relevant formal concept. So
0:39don't take them as gospel. But with that
0:41caveat, let's get started. My name is
0:44Joe Folly and this is unsolicited
0:47advice. One, pre-logic. This is where
0:50most of us start out. Like almost anyone
0:53will have a vague sense that some
0:55arguments are good and other arguments
0:57are bad. We may have heard someone say
0:59something like, "You're a bad person,
1:02therefore you're wrong." or try to
1:04generalize from a small set of cases to
1:06infer over a massive group of people and
1:09had this kind of rough sense that
1:12something was a miss. But we don't yet
1:14know exactly what it is that separates
1:17those bad arguments from good arguments
1:19that we've heard in the past. It's
1:21important also not to confuse logic with
1:23critical thinking here. This person
1:25might still be incredibly intelligent,
1:28but the precise science of logic itself
1:31remains a mystery. If someone were to be
1:33in this stage and they were to Google
1:36words like logic or how to tell if an
1:38argument is good to try and learn more,
1:40then they would probably move to stage
1:42two. Two, the fallacy monger. I love a
1:46good logical fallacy as much as the next
1:48insufferable internet philosophy guy.
1:50And in my experience, this is also a lot
1:53of people's first encounter with logic
1:56uh in a slightly more regimented sense.
1:59and it was also how I took my first
2:01steps in this area. So it has a special
2:03place in my heart. The term logical
2:05fallacy is derived from the work of
2:07Aristotle and effectively it's a type of
2:10argument that is always or almost always
2:13bad. Aristotle wanted to identify such
2:15arguments because often times even
2:18technically bad arguments can appear
2:20rather persuasive to the untrained eye
2:23even though strictly speaking because
2:24they're bad arguments they shouldn't
2:26persuade anyone. So there is a lot of
2:29utility in identifying such uh surfacely
2:33seeming good but ultimately poor
2:35argumentation. Take the example we used
2:38earlier where someone said you're a bad
2:40person therefore you must be wrong. I'm
2:43sure that many of you will recognize
2:44this as an ad homonym fallacy. This is
2:46where we infer from an irrelevant
2:49feature of someone's character to the
2:51conclusion that their argument is
2:52incorrect. It doesn't take a genius to
2:55figure out why this inference doesn't
2:57follow. There are awful people
2:59throughout history who have still been
3:00factually correct about some things. And
3:03there are incredibly moral people who
3:05have been factually incorrect about some
3:07things. If a mass murderer reproduced
3:10Uklid's proof that there are infinitely
3:12many crimes on the spot, well, they
3:14would be just as correct as if St.
3:16Maxmillian Kobe had done it, who is
3:18undeniably a much more moral person. And
3:21if St. Max Millian Kobe had tried to
3:23reproduce the proof and got it wrong, it
3:26would still be wrong and no amount of
3:27his moral upstandingness would
3:30compensate for that. This is where most
3:32people will probably end their study of
3:34logic because it does give them a lot of
3:36what they want, which is a rough guide
3:39to telling good arguments from bad
3:41arguments. However, if we do reflect for
3:45a little bit, we can demonstrate the
3:47limits of this approach. Take the
3:49following situation. You are listening
3:51to a Nobel Prizewinning physicist and a
3:54layman discuss quantum mechanics. They
3:56have a disagreement over the details of
3:58Heisenberg's uncertainty principle. You
4:00listen to each side, but you don't know
4:02much about quantum physics yourself. So
4:04eventually you defer to the experience
4:07and authority of the physicist over the
4:09layman. Somebody well-trained in logical
4:12fallacies will spot this as an appeal to
4:14authority. It's an attempt to use
4:16someone's authority to justify a point
4:19they have made or infer from that
4:21authority that the point is true or more
4:23likely to be true. But it also doesn't
4:26seem like you've done anything
4:27particularly unreasonable here. Surely
4:30if the physicist and the layman are
4:31having a disagreement, the physicist is
4:34more likely to be correct because of
4:36their years of experience, all else
4:37being equal. They've studied the subject
4:40at a high mathematical level. So yes,
4:42they do have authority, but that
4:44authority seems like it's been earned in
4:47the relevant sense. Should we really
4:49think that it's unjustified to appeal to
4:51that authority if we can't resolve the
4:53answer to the question ourselves? Of
4:55course, if we can resolve the answer to
4:56the question ourselves, then we don't
4:58need to look to that authority. But I'm
5:00just saying it doesn't seem like this is
5:02a totally misguided way of reasoning
5:06with limited information. I think these
5:08limits are why those who are ready to
5:10label an argument a logical fallacy have
5:13acquired such a an odious reputation
5:16online. There are many arguments that
5:19look a lot like fallacies but are in
5:21fact reasonable ways of well reasoning
5:24under uncertain conditions where you
5:26cannot find out an answer directly
5:28yourself. Whether something that looks
5:31like a fallacy actually is a fallacy is
5:34often going to depend on context and the
5:36very specific proposition a speaker is
5:39trying to demonstrate. Not every appeal
5:41to authority is facious and someone's
5:43personal character is sometimes very
5:46relevant to a debate. Say if someone is
5:48running for mayor, you might use
5:50character information to infer about
5:52whether they're going to be a corrupt
5:55mayor or an incorruptible mayor. Most of
5:57the things we call logical fallacies
6:00aren't actually full-blown errors in
6:02formal logical reasoning, but are
6:04instead what's called informal
6:06fallacies. That is, you cannot tell
6:08merely from their structure that they
6:10are fellacious. They either break an
6:12argumentative rule or are unreliable
6:14reasoning within a particular context,
6:16but they're not formal errors. Many
6:18logicians would actually consider this
6:21beyond the boundaries of logic proper.
6:23But there is an area of philosophy
6:25called informal logic which does attempt
6:27to study everyday argumentation as
6:29rigorously as possible and tries to work
6:32out exactly how you can tell when a
6:35natural language argument is facious.
6:37And I made a video roughly about that
6:39and about logical fallacies in general
6:41right here where I'll go over some of
6:42the same stuff that I go through in this
6:44video, but it is a substantively new
6:46video as well. As I said, this is where
6:48most people will probably end their
6:50studies in logic. But I think that's a
6:53real shame because in my view, this is
6:55where things really start to get good
6:57because now we can begin to talk about
7:00formal logic. But ultimately, logic is a
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8:17Three, basic formal logic. In the first
8:20few months of an undergraduate
8:22philosophy degree, you will learn the
8:24very basics of formal logic. And this
8:26will almost certainly be propositional
8:28logic. This is strictly speaking a
8:30mathematical system with a set of axioms
8:33and a set of inference rules. And those
8:36axioms and rules define what kinds of
8:38statements follow from what other
8:40statements. And it'll become clear what
8:42I mean by that as this section goes on.
8:44As I said, the first formal logic that
8:46most students will encounter is called
8:48propositional logic. And that's because
8:50it is the most basic kind of formal
8:52logic. The basic objects of
8:54propositional logic are propositions as
8:57the name suggests and they tend to be
8:59represented by letters of the alphabet
9:01like P or Q or R or S and so on.
9:04Propositional logic also contains a set
9:07of logical connectives that stand in for
9:10roughly speaking certain natural
9:12language words like and or or if then or
9:16not that are meant to connect different
9:20propositions together and thus allow you
9:22to facilitate inferences. So a very
9:25basic argument in propositional logic
9:27would run as follows. If P then Q P
9:31therefore Q. This is called a modus
9:33ponins inference. The precise
9:35definitions of the logical connectives
9:37are defined by what's called truth
9:39tables. And we'll go through one now.
9:41We'll take the truth table for and since
9:43that one is just the most intuitive. A
9:46statement like P and Q is true if and
9:49only if P is true and Q is true. In just
9:53the same way that in natural language,
9:54the statement Jon is Jane's brother and
9:57her friend is only true if Jon is both
10:00Jane's brother and Jane's threat. This
10:03might sound like it's obvious and
10:04pedantic because at its most basic,
10:06formal logic is trying to create precise
10:09and mathematical conditions for everyday
10:11reasoning. Or at least it's trying to do
10:13that at this level when you're in your
10:15kind of first term or so of an
10:17undergraduate philosophy degree. As
10:19we'll see, it gets far beyond that
10:21eventually. Some connectives are a
10:23little less intuitive like the
10:24conditional connective if then. A
10:27statement if P then Q is false. If and
10:31only if P is true and Q is false. This
10:35means rather bizarrely that if P is
10:37false then if P then Q is actually true.
10:41Some people mark out this peculiar kind
10:43of truth by saying that it is trivially
10:45true. At first, this might strike you as
10:48strange because things tend to work
10:49slightly differently in ordinary
10:51language. If I said, "If cats can fly,
10:54then 1 plus 1 equals 2." It would seem
10:57really strange to call what I've just
10:58said true. It seems like here the
11:02antecedent, the first bit of the
11:04conditional, and the consequence, the
11:05second bit of the conditional, are not
11:08connected by any kind of relevant
11:10factor. And so, the conditional looks
11:12really odd. This is actually a very old
11:15problem in the philosophy of logic that
11:17dates at least as far back as the
11:19ancient Greek stoics who are actually
11:20some of the most innovative logicians of
11:22the ancient world as well as you know
11:24creating stoicism. Some philosophers
11:26argue that we should redefine the
11:28conditional to only be true if there was
11:31a further relevant connection between
11:33the antecedent and the consequent such
11:36that the antecedent would make the
11:38consequent true. The reason why formal
11:40logic mostly sticks with the definition
11:42that we've already outlined, as weird as
11:43it seems, is because translating this
11:46idea of a relevant connection between
11:49the antecedent and the consequent is
11:52very difficult to express
11:54mathematically. Right? it it what we
11:56have without it is what's called a truth
11:58functional definition which means that
12:00we can fully define the usage and
12:03inference conditions of the conditional
12:05merely by saying what is true and false
12:08about the stuff either side of the
12:09conditional. Adding in an additional
12:11constraint uh regarding relevance or the
12:14antecedent making the consequent true
12:16would indeed make it resemble natural
12:18language more closely but it would also
12:20make it much harder to work with
12:22mathematically. And as formal logic
12:24tends to on average be of more interest
12:27as a mathematical system, especially at
12:29more kind of advanced levels, that tends
12:32to be the one that people go with. There
12:33are still people that argue for a
12:35redefinition of the conditional, but the
12:37truth functional definition, the weird
12:39looking one, is still the mainstream
12:41view. We cannot go over all the rules of
12:43propositional logic now because there
12:45are quite a few of them. But to give an
12:47example of what a very basic proof in
12:50propositional logic looks like, here's
12:52one from the open- source logic textbook
12:54for all X. What this is essentially
12:56showing is that we can prove in
12:58propositional logic that if P is true,
13:01then P and D or P and not D is also
13:05true. The labels down the side and
13:07naming the particular rules used at each
13:10point in the overall proof. At this
13:13level, students will also learn basic
13:15logical concepts like validity and
13:18soundness. A valid argument is one where
13:21if all its premises are true, then its
13:23conclusion cannot be false. For
13:25instance, if all cats can fly, then cats
13:28have wings. Cats fly, therefore cats
13:31have wings. This is a valid argument
13:34because if all its premises are true,
13:36the conclusion must also be true. It
13:38can't be false. Its conclusion is false.
13:41So it's still unpersuasive, but that's
13:43because the premises themselves are not
13:46true. A sound argument is a valid
13:48argument where in addition to that
13:50validity, all of the premises are in
13:53fact true. So something like all men are
13:55mortal, Socrates is a man, therefore
13:58Socrates is mortal is a sound argument.
14:00These terms are very helpful for
14:02evaluating precisely what is wrong with
14:05an argument. Whether it is structural
14:07and therefore the argument is invalid or
14:09whether it is a problem with the
14:11premises. If you're not into philosophy
14:13or maths or computer science, I
14:15personally think this level of logic
14:17probably fulfills like 99% of your
14:19needs. It will help you formalize a lot
14:21of arguments that you encounter in the
14:23wild and it remains very closely
14:25connected to everyday life.
14:27Additionally, you can probably get there
14:29with around 5 hours of focused work. So,
14:32it's kind of the it's the the best bang
14:36for your buck part of logic for everyday
14:38life, at least in my opinion. But even
14:41philosophy students who don't choose to
14:43make logic their focus do tend to learn
14:45a little bit more logic than this. And
14:48we'll go through some of that now. Four,
14:50first order logic and friends. The next
14:53step for most students after learning
14:55propositional logic is to learn first
14:57order logic. Whereas propositional logic
15:00deals in full propositions, first order
15:02logic can deal with more fine grain
15:04statements like all men are mortal or
15:07all cakes are delicious. That second
15:09sentence would look like this in first
15:11order logic. This means for all X if X
15:14is a cake then X is delicious where CX
15:17means is a cake and DX means is
15:19delicious. On the other hand, if you
15:21wanted to say something like some cakes
15:23are delicious, you would write it like
15:25this, which in natural language means
15:28there exists an X such that X is a cake
15:30and X is delicious. The first order
15:33logic retains all of the symbols from
15:34propositional logic, but can assign
15:37properties to objects as well. In fact,
15:39it quantifies over these objects. That's
15:41what the uh little for all x symbol
15:44means. that uh little un upside down a
15:47is called a quantifier and x is a
15:50variable uh quantifying over objects.
15:53This means that you can do more with the
15:55logic essentially. We can also express
15:57relations in first order logic which are
15:59properties that involve more than one
16:01object. So again let's use a kind of
16:03natural language example. We might
16:05formalize the phrase every person has a
16:07father like this. And in natural
16:10language, this would mean for all X, if
16:12X is a person, then there exists a Y
16:15such that Y is the father of X. It
16:18sounds really clunky to say out loud,
16:19but this kind of precision becomes
16:21helpful in a formal mathematical
16:23context. To take a very basic
16:26mathematical example, we think that a
16:28prime number, well, we know that a prime
16:29number is a number that only divides by
16:32one and itself. But we could express
16:34this precisely in logic with the
16:36following sentence. In natural language,
16:38this roughly means for any Y, if X / Y
16:42is a natural number, then Y is the same
16:45number as X or Y is one. That is, X only
16:49divides by one and itself. We've drifted
16:52slightly beyond the bounds of strict
16:54first order logic to define this, but it
16:56just is an illustration of the kind of
16:58things that basic logic can do. As a
17:00student gets used to these symbols, they
17:02tend to stop translating them into
17:04natural language in their head like I've
17:06been doing and instead just read the
17:08symbols as you would any language. It's
17:10like learning French. At some point, you
17:12stop translating French sentences into
17:14English and you begin simply reading
17:16French. Alongside this, at this level,
17:19students tend to learn things like basic
17:21probability theory and basic set theory,
17:24which I'll just kind of skim over
17:25briefly. Now set theory is an abstract
17:27mathematical framework that consists of
17:29sets and objects which belong to those
17:32sets. They're called members of those
17:34sets. So we might talk about the set of
17:37all people or the set of all prime
17:39numbers and then perform various
17:41operations with these sets. At the basic
17:43level like the very basic level we can
17:45think of these sets like ven diagrams.
17:48So say we have two sets then the
17:50intersection between those sets is
17:52what's in both sets while the union of
17:54the sets are what is in either set A or
17:57set B or both and the complement of a
18:00set is everything outside of the set.
18:02This is a very handy tool both in
18:04mathematics and philosophy though
18:06obviously it tends to come in more uh uh
18:08fleshed out versions than this. You'll
18:10also notice that each of those set
18:11theoretical operations maps quite neatly
18:13onto a logical operation. So the
18:16intersection is very much like and
18:18because it's only the stuff that's in
18:19both sets. Uh the union is very much
18:22like or because it's the stuff that is
18:23in either set or both sets. And the
18:26complement is very much like not because
18:28it's everything that's not in the set.
18:29Probability theory, as I'm sure many of
18:31you will already know, is the area of
18:33maths concerned with evaluating
18:35probabilities or the likelihood that
18:37certain things hold given certain other
18:39things holding. In a philosophical
18:41context, it is often used to make sense
18:42of reasoning under uncertainty. as well
18:45as in the philosophy of inductive
18:46reasoning. That is the kind of reasoning
18:48that uses previous experience to make
18:51uncertain inferences about the future.
18:53Take the example from earlier where we
18:55appeal to the authority of the physicist
18:57over the layman. Well, why was that
18:59justified? I suspect that the answer
19:01many of you would have instinctively
19:03given is that the physicist is more
19:05likely to have true beliefs about
19:07quantum physics than the layman and thus
19:09his testimony is stronger evidence than
19:12the layman's. Even very basic
19:13probability theory can give philosophers
19:15a framework through which to make these
19:17claims precise. This is often done
19:19through B theorem, but that's kind of a
19:21topic all on its own, and I have beef
19:23with how many philosophers use B
19:24theorem, so I'm just going to park that
19:26for now. The aim of learning logic at
19:28this level is not necessarily to become
19:30like total masters of these symbolic
19:32systems. Most of the students who learn
19:34these on a philosophy course will then
19:36branch off into various nonformal
19:38interests across like epistemology and
19:41moral philosophy and political
19:42philosophy and loads of other fields.
19:44The main reasons that we teach logical
19:46systems to undergraduates at this level
19:48is that in engaging with these
19:49frameworks, it forces you to think in
19:51very precise ways. There's no room for
19:54error or ambiguity. And this can be
19:56helpful just for practicing thinking
19:57carefully about any area and not just
20:00philosophy and not just maths. So I know
20:03that was a lot of symbols and as we go
20:04on there'll be far fewer of them because
20:06we're now going to move into explaining
20:08uh various different logical systems and
20:10sometimes more advanced logic and so I'm
20:12just going to explain it in intuitive
20:14natural language terms but as a result
20:16I'm also going to give the proviso here
20:18that that will necessarily distort the
20:20concept a little bit. So uh the rest of
20:23the video is very much like going to
20:25give the feel of what these logical
20:27systems are like uh but it won't be like
20:29formal or rigorous uh because it will
20:32just get like far too complicated and
20:33this video will end up being uh loads
20:35and loads of hours long. But with that
20:37caveat uh let's move on. Five more
20:40specialized logical systems. In the
20:43later years of undergrad and potentially
20:45going into masters, students will learn
20:47a variety of different logical systems
20:49and we can't go through them all here,
20:50but we will go through a few. And
20:52potentially the most common is modal
20:54logic, which is a type of logic used to
20:56talk about possibility. Philosophers
20:58quite often talk about whether something
21:00is possible or necessary and what makes
21:03a possible or necessary statement. But
21:06as it stands, those terms are often
21:08quite vague and difficult to get a
21:10handle on. A modal audition will
21:12formalize necessary to mean true in all
21:15possible worlds accessible to the agent
21:17and they'll formalize possible to mean
21:20true in at least one possible world
21:21accessible to the agent. This in turn is
21:24cached out in terms of what's called a
21:26crypty model. We can think of a crypt
21:28model as a collection of worlds and
21:30things that are true at these worlds. So
21:33there might be a world where pigs can
21:34fly and another where Elvis is still
21:36alive and so on. In technical terms,
21:39these words are sets of propositions.
21:41The worlds are then connected by things
21:43called accessibility relations. We can
21:46think of it as like a series of balls
21:48connected by lines where the balls are
21:50possible worlds and the lines denote
21:52which worlds access which other worlds.
21:55A proposition is necessary at a given
21:57world just if it is true in all the
22:00worlds the initial world can access. So
22:02if I'm standing at world X and I can see
22:05worlds Y and Zed as well as my own world
22:08and the statement pigs can't fly is true
22:11in all three of these worlds, then
22:13within the context of that model, we
22:15would call that proposition necessary.
22:17This framework is remarkably flexible.
22:20For instance, you can use it to model
22:22belief and knowledge rather than
22:23possibility. If we think about what
22:25happens when we fully believe something,
22:27we treat it as an assumption in every
22:29potential course of action that we're
22:31considering. If I believe that pigs
22:33can't fly, then I'm never going to plan
22:35for a pig to come shooting through my
22:37bedroom window. So, we can formalize
22:39this by saying that I live at world X
22:42and I'm considering a certain set of
22:44worlds when I plan my actions. And in
22:46all of those worlds, the proposition
22:48pigs can't fly is true. With certain
22:51alterations, we can use this same
22:52framework to model what someone knows.
22:54These logics called doxastic and
22:57epistemic logics respectively have seen
22:59a surprisingly wide variety of uses
23:01across areas of computer science as well
23:03as game theory which sometimes involve
23:05modeling knowledge or belief or
23:07knowledge like or beliefike states.
23:09You'll also notice that in both of these
23:11cases it's not like the logic has just
23:14solved the philosophical issue
23:15completely. People still talk about what
23:17it means for something to be necessary
23:18or possible or what it means to know or
23:21believe things. But these provide
23:23tentative formal frameworks within which
23:25to explore these questions and so they
23:27can be useful to philosophers. On the
23:29other hand, if you are a linguist, you
23:31might end up using logic in a field
23:33called formal semantics, which is the
23:35area of linguistics that creates
23:37relatively precise formal structures
23:39with which to analyze our speech.
23:41Remember earlier in the video when we
23:42were formalizing various natural
23:44language statements as formal logical
23:46ones? Well, formal semantics takes that
23:49general principle and expands it
23:51massively to encompass more and more of
23:54our natural speech. They have ways of
23:56formalizing things like questions as
23:57well as more complex linguistic
23:59phenomena like gradable adjectives. For
24:02instance, the former semanticist Gayla
24:04Cassoon has spent much of her career
24:06trying to understand what are called
24:08multi-dimensional gradable adjectives.
24:10So, we'll just go through kind of what
24:12that means. We often say things like
24:14Adam is more athletic than Steve or
24:16compare people in this way. But the term
24:18athletic is a complex word. Athleticism
24:21is not a single property like height. It
24:24is an amalgamation of various different
24:26more simple properties like
24:28cardiovascular fitness and musculature
24:30and skill at various sports and so on.
24:33So using empirical evidence concerning
24:36how people actually do use these words,
24:39Cissoon has come up with various formal
24:41and informal ways to analyze them and
24:43she's drawn on certain logical kind of
24:46symbology in order to do so. We won't go
24:48into any more detail about that here
24:50because formal semantics is just like a
24:52huge field and I'm only really familiar
24:54with small parts of it. But I think it's
24:55worth mentioning because it is a
24:57non-fillosophical and non-mathematical
24:59use of formal logic. And so it's kind of
25:01worth bearing in mind this stretches
25:03beyond philosophy and maths. At this
25:05level, philosophy and logic students
25:07will also often learn model theory,
25:09which is a way of talking about abstract
25:12mathematical structures in the language
25:14of mathematics. This can get incredibly
25:16complicated incredibly quickly, but we
25:18can illustrate it with an example. So
25:20the model theoretic structure of a
25:22certain kind of basic arithmetic might
25:24be written as follows where zed is the
25:27set of integers that is whole numbers
25:29plus is addition. This asterisk is
25:32multiplication and the zero and one are
25:34the neutral elements for addition and
25:36multiplication. A neutral element here
25:38means like an element that does nothing.
25:40So if you add something by zero then you
25:44just get the same number again. And if
25:45you multiply something by zero then you
25:47also just get the same number again.
25:49Intuitively speaking, what we have here
25:51is basic integer arithmetic, but boiled
25:54down to its most basic and essential
25:56components. To put it another way,
25:58contained within that basic set of
26:00symbols are the sort of ingredients for
26:03the whole of basic integer arithmetic.
26:05Model theory is also used in logic to
26:08provide a formal logical system with
26:10what's called a semantics. That is, we
26:12create a model to clarify exactly what
26:14the symbols in a logic are referring to.
26:17The relationship between the syntax of a
26:18logic, that is the symbols themselves,
26:20and the semantics of a logic, that is
26:22what the symbols pick out, will become
26:24increasingly important when we move on
26:26to kind of meta mathematics and metal
26:28logic in the next section. Someone might
26:30also choose to learn things like more
26:32advanced set theory at this level, which
26:34is quite a lot of fun, but I haven't
26:36done it in a very long time. So, if I
26:37try and talk about it in any real
26:39detail, then I will get it massively
26:41wrong. Set theory gives us the tools to
26:43speak about quite a lot of seemingly
26:45mysterious concepts in surprisingly
26:48clear ways. For example, the concept of
26:50infinity has famously vexed philosophers
26:53since Aristotle. But set theory can give
26:55us a way of talking about infinities in
26:58really quite precise ways. And we can
27:00even prove things like that some
27:02infinities are bigger than others. And
27:04we can see which infinities are the same
27:07size. For example, we can actually prove
27:09that the set of rational numbers, that
27:12is the numbers that can be represented
27:13by fractions, is actually the same size
27:16as the set of integers, that is the set
27:18of whole numbers. This is a deeply
27:21unintuitive result, right? Surely there
27:23are more fractions than there are whole
27:25numbers because there are multiple
27:26fractions in between each whole number,
27:29but it turns out that we can prove
27:30they're the same size using some pretty
27:32basic tools from set theory. So yeah,
27:34it's it's pretty trippy stuff, but it is
27:36pretty cool. Obviously, there is a lot
27:38more to say about all of these fields,
27:40and I have been distorting them slightly
27:42because I've been talking about them at
27:44a kind of intuitive level, but you can
27:47see just how much logic balloons beyond
27:49merely formalizing natural language
27:51statements. Moreover, different people
27:53at this level will get vastly different
27:55experiences in logic. They will learn
27:57very different things. If you choose to
27:59focus on model theory, you'll end up
28:01learning a very different field to
28:03someone who chooses to focus on formal
28:04semantics. Another theme at this level,
28:07as we've sort of seen, is that logic
28:09becomes increasingly disconnected from
28:11our everyday reasoning. We set out at
28:14the beginning of this video simply
28:15trying to make our ordinary thinking
28:17more precise. But over time, we've
28:19become increasingly interested in logic
28:21as its own mathematical field of study.
28:24This happens with quite a lot of areas
28:25of maths. If we kind of stop and think
28:27about it, we begin by asking what a
28:29prime number is, and we end up with the
28:31whole sprawling arena of full-blown
28:33number theory. Our intuitions are thus
28:36increasingly unreliable guides to
28:38answering higher level logical
28:40questions, which is why the formal
28:41apparatus is so very important. But
28:44there are some more advanced logical
28:46results that almost every budding
28:48logician will encounter at some point,
28:51but ones that are famously difficult to
28:53make head or tails of. And I'd like to
28:55go through a few of them now as they
28:57tend to present challenges for the logic
28:59student. Six, metathematics and meta
29:03logic. If you choose to pursue logic
29:05towards the end of an undergrad degree
29:07or the beginning of a master's degree,
29:08you'll also begin learning what's called
29:10metamatics and metalogic. And this is
29:13where we use maths and logic to study
29:15mathematical and logical stuff. We
29:19already saw this a little bit in the
29:20last section when we looked at model
29:22theory very briefly. And to be honest,
29:24this field gets incredibly odd very
29:27quickly, and we're only going to be able
29:29to go over the very basics of it and a
29:30few little examples here. And yeah, the
29:33idea is just to give you a a flavor of
29:35this kind of odd area of logic. There's
29:38also an argument to be made that this
29:40level and the previous level could be
29:42swapped since the order you learn these
29:43sorts of things in really does depend on
29:45which institution you're at. And
29:47finally, and I know this video is full
29:49of disclaimers, but I just want to
29:50reiterate that I'm going to be really
29:52quite loose with my logical terminology
29:53here. So, I apologize in advance for
29:55that for anyone in the audience who is a
29:57kind of technician. This is just
29:59intended to give you a feel for these
30:01ideas rather than be exactly precisely
30:03correct, which would involve defining a
30:05lot of the terms I'm using formally. And
30:07at times, I have sacrificed precision
30:09and technical correctness for the sake
30:11of accessibility. I just want to again
30:13uh remind everyone of that. Now, you
30:15might understandably ask why anyone
30:17would want to use logical systems to
30:19study logical systems. What could
30:21possibly be the point of this beyond
30:23meaningless naval gazing? I mean, what
30:26questions would you even ask? Well,
30:28here's one basic question we might be
30:30interested in. Does our logic contradict
30:33itself? So, imagine that I had a formal
30:36system that was the same as
30:38propositional logic, but included an
30:40extra connective and an extra inference
30:42rule called Derf. My sister's favorite
30:45TV program growing up was IIC Carly. And
30:47in one episode, they invent a new number
30:49called DUR. So I think the name is
30:50appropriate. For the sake of this
30:52example, I stipulate that the statement
30:54P durf Q can be introduced whenever we
30:58have P or we have Q. And it can be
31:00eliminated to give you either P or to
31:03give you Q. So if I have P in this
31:06logic, I can infer that P durf Q. And if
31:10I have P derf Q, I can infer that Q. You
31:13might think this is a completely
31:15ridiculous rule to introduce into a
31:17logic and you would be absolutely right.
31:19In fact, derf would make our logic
31:21contradictory. So say we had proposition
31:24P. Well, we can infer from this that P
31:27durf not P. But then we can infer from P
31:30durf not P that not P. But then we've
31:34proven not P from P. And so we've
31:36contradicted ourselves. This is to put
31:38it very mildly not what we want from a
31:41logical system. Thus, a basic meta
31:44result that we want to demonstrate for a
31:46given logic is consistency. A more
31:48powerful result that is equally as
31:50important is soundness. A formal system
31:53is sound when it doesn't prove anything
31:56false. Now, false doesn't mean false in
31:59our ordinary sense in this context. It
32:01means false in the semantics of the
32:04logic. As we saw earlier, logics have a
32:06syntax which consists of their symbols
32:09as well as a proof theory which is their
32:11inference rules and a semantics which is
32:15the intended set of mathematical
32:17structures that the logic is meant to
32:18refer to. So technically logics are not
32:22sound or unound but they are sound
32:25regarding a given class of models. That
32:27is they don't prove anything that is
32:30false in those models. This is again
32:33pretty abstract but it's best
32:35illustrated with an example. Take the
32:37perfectly ordinary maths of basic
32:39arithmetic. We can create mathematical
32:41structures where basic arithmetic is
32:44definitely unound for those structures.
32:47For instance, take your ordinary
32:49everyday clock whose numbers run from 1
32:51to 12 and then go back to one again and
32:54you know so on and so forth. Ordinary
32:56arithmetic would be unsound regarding
32:58this model because in ordinary
33:01arithmetic 12 + 1 equals 13 whereas in
33:04clock arithmetic 12 + 1 equals 1. To
33:07model clock arithmetic we would need a
33:09slightly different system called modular
33:11arithmetic. Obviously I've skimmed over
33:13some details here. For one thing we
33:15would normally have to define a specific
33:17formal system and not just handwave by
33:19saying ordinary arithmetic. But this is
33:21just to give you an idea of what it
33:23means for a logic to be sound or unound
33:25relative to a given class of models. I
33:27nicked this clock example from one of my
33:29old logic teachers back in my first year
33:31of undergrad and I can't for the life of
33:33me remember his name but thank you to
33:35him. Related to soundness is
33:37completeness. A formal system is
33:39complete regarding a given semantics if
33:42it can prove everything that is true in
33:44that semantics. It's sort of the flip
33:46side of soundness. Soundness is not
33:48proving any false things and
33:49completeness is proving all true things.
33:52Let's again illustrate this with an
33:53informal sort of toy example. Imagine
33:56that there was a model that consisted
33:58simply of a traffic light and a set of
34:01rules about how cars should behave in
34:03response to that traffic light. The
34:05traffic light has two colors. It's
34:06either red or green. Only one color can
34:09show at a time and there is always one
34:12color showing. When the light is red,
34:14car should stop. And when the light is
34:16green, car should go. We can imagine a
34:18set of rules that would be complete
34:20regarding this model. It would probably
34:22consist of something like the following.
34:24If the light is green, it is not red and
34:26vice versa. If the light is green, the
34:29car should go. If the light is red, the
34:31cars should stop. We can think of this
34:34as a very simple semiformal system for
34:38the traffic lights. And it's immediately
34:40clear that this set of statements plus
34:42some other basic formal machinery that
34:43we're just kind of skimming over would
34:45allow us to say any true statement about
34:47the traffic light model. That means this
34:50set of rules is complete for this model.
34:53A complete logic of the traffic lights
34:55is thus one that intuitively speaking
34:58reaches every fact about the traffic
35:00lights. By extension, a complete formal
35:02system of say arithmetic would be
35:05something that reaches all the truths
35:07about arithmetic. However, we know that
35:11there is no formal system that is
35:13complete for arithmetic in this way
35:16because of Girdle's incompleteness
35:17theorems. These are too complex to go
35:20over in a short section like this and I
35:22will write a full video on them at some
35:23point. But essentially intuitively,
35:26Girdle proved there is no consistent
35:29effectively axiomatized system that can
35:31prove all the truths of arithmetic.
35:33Don't worry about what effectively
35:35aiomatized means here because we'll go
35:37over that in a future video.
35:39Essentially, this means there will
35:41always be an arithmetical proposition
35:43left over that is true but cannot be
35:45proven within this formal system. The
35:47reason this is such an important result
35:49is well partly for historical reasons
35:51but also because it just really seems
35:53like arithmetic is simple enough that we
35:55ought to be able to come up with a
35:57complete axiomatizable system that will
35:59prove everything in it. But we can't do
36:02this. Moreover, through a further proof,
36:04Girdle demonstrated that no consistent
36:07axiomatized formal theory expressive
36:09enough to capture basic arithmetic will
36:11be able to prove its own consistency
36:13either. So if we want to prove the
36:15consistency of arithmetic, we will need
36:17to appeal to a more powerful logical
36:19system to do so, which in turn cannot
36:21prove its own consistency and so on and
36:23so forth. However, I I want to be clear
36:26here. I will end up saying this in the
36:27whole video on girdles incompleteness
36:28theorems, but I I'm just going to
36:29reiterate it now. Any kind of claim that
36:32girdles incompleteness theorems broke
36:33maths or anything like that are very
36:36overblown. This is actually kind of
36:37quite a limited result. And while it is
36:39of intense logical and philosophical
36:41interest, it didn't like detonate the
36:43world of maths. Maths is is absolutely
36:45fine. And these meta-matical and
36:48metalological results are investigated
36:51for each logical system that you study
36:53individually along with proving certain
36:55important formal results within that
36:57particular system. So some other classic
37:00questions we might be interested in are
37:02whether certain statements can or cannot
37:05be proven within a given formal system.
37:08For instance, the most popular type of
37:10set theory is called ZFC or Zlo Frankle
37:13set theory with choice. A research
37:16program that's emerged in the past 60 or
37:18so years, I think, has been
37:20demonstrating that various results are
37:23independent of the axioms of ZFC,
37:25meaning that they cannot be proven nor
37:27disproven by this mathematical system.
37:30This helps mathematicians and logicians
37:32better understand the relationships
37:34between certain formal results and
37:36systems that may or may not be able to
37:38demonstrate these particular results. Uh
37:40the most famous example of this is the
37:42proof that the continuum hypothesis is
37:44independent of ZFC. The continuum
37:46hypothesis states that there is no
37:48infinity that is larger than the set of
37:50natural numbers but smaller than the set
37:52of real numbers. The truth or falsity of
37:54this is a very interesting mathematical
37:56question within the study of infinity.
37:58But it's been proven that ZFC can
38:00neither prove nor disprove this. And at
38:03this level, there will also be a focus
38:05on learning various different proof
38:06techniques like forcing and
38:08diagonalization as well as more basic
38:10ones like induction on the length of a
38:12formula. Like many areas of maths, the
38:14same proof techniques tend to reappear
38:16in many different proofs and many
38:17different areas. So students and
38:19researchers will become acquainted with
38:20a great number of them. In philosophy,
38:23there are many debates within
38:24philosophical logic about which is the
38:26best kind of logical system for either
38:29kind of the best in total or the best
38:32for particular things. This is a
38:34philosophical methological question
38:36rather than a mathematical one. For
38:38example, a logician named Stuart Shapiro
38:40has put forward a number of influential
38:43arguments that second order logic should
38:45be used far more often in philosophy and
38:48logic and the foundations of
38:49mathematics. We met first order logic
38:51earlier which could quantify over
38:53objects. When we were saying things like
38:55for all x, that's what the logic was
38:57doing. It was quantifying over the
38:59object variable x. Second order logic
39:01can talk both about objects and sets of
39:04objects or properties or relations which
39:06makes it much more expressively
39:08powerful. So in second order logic, we
39:10can express statements like this, which
39:12denotes that for any x, x has at least
39:15one property. We simply couldn't say
39:18this in first order logic, but it might
39:20be quite a useful sentence to have at
39:22your disposal. Even though I think in
39:24most models, this this one's basically
39:26trivial. Second order logic also helps
39:28us to define important mathematical
39:30properties that we might be interested
39:31in. The classic example of this is
39:33mathematical induction, which can be
39:35written in second order logic as
39:37follows. In rough natural language
39:39terms, this means that for any property,
39:42if zero has that property and for any
39:44number, if that number has it, then that
39:47number plus one has it, then that
39:49property holds for every number.
39:51However, there are also various
39:53downsides to second order logic, which
39:55makes some mathematicians and
39:56philosophers less likely to want to use
39:58it. For one thing, all that extra
40:00expressiveness comes with more
40:01heavyweight formal machinery, uh, for
40:04one of a better term. Within the
40:05philosophy of logic, philosophers will
40:07often discuss the various merits of
40:09different logical systems in this
40:10manner. Another philosophical debate
40:12concerning logic is between logical
40:15monists and logical plurists. Logical
40:17monism holds that there is one true
40:20logic which captures what logic
40:22intuitively actually is. While logical
40:24pluralists tend to view logic more like
40:26a set of formal tools which are better
40:28and worse at different kinds of tasks.
40:30This may sound like a non-technical
40:32debate, but it can get surprisingly
40:34mathematical, especially when
40:36philosophers are discussing which
40:37candidates are plausible for the one
40:39true logic, or in the case of logical
40:41pluralists, demonstrating the utility
40:43and philosophical value of various
40:45different types of logic. An old
40:47supervisor of mine called Owen Griffiths
40:49wrote a book with Alexander Pazo called
40:51one true logic, which argued that a
40:53particular kind of logic was the one
40:55true logic and then defended that
40:57thesis. In addition to all of this, all
40:59of the logics that we encountered back
41:01at level five along with many other
41:03kinds of formal system can be pushed far
41:06far further. There will be meta results
41:08as well as just ordinary results that
41:10are important to prove in areas like
41:12model theory and intuitionistic logic
41:15and modal logic as well as game theory
41:17and formal linguistics and topics in
41:19theoretical computer science. All of
41:21these can obviously be explored in so
41:23much more detail. We really have like
41:25just skated the surface of each of those
41:28fields in this video. If you want to get
41:30a broad overview of the kind of things
41:32that people study in logic at the mast's
41:35level, then you can check out the
41:36University of Amsterdam's logic,
41:38language, and cognition website as they
41:40run one of the most highly acclaimed
41:41logic masters in the world. And you can
41:43kind of see what topics they offer. But
41:46what comes next? Well, this is where we
41:48move far beyond my level and start to
41:50look at what active logicians are
41:53working on today. Seven, current logic
41:56research. There is a wide gulf between
41:58level six and seven here. So far, we've
42:01been looking at the kind of logic that
42:02you would learn at a university, but
42:04this level is what actual working
42:07researchers in formal logic are doing
42:09with their time and energy. Specifying
42:12beyond this is a little bit difficult
42:13because logic is a huge field. It's a
42:16little bit like asking what do
42:17researchers in neuroscience do. It
42:20heavily depends on the individual
42:21researcher and the area that they've
42:23specialized in. But to give you just an
42:25idea of what modern logic looks like,
42:28here is an abstract from a recent
42:30featured article in the NRAAM journal of
42:33formal logic. We explore several model
42:35theoretic aspects of D sets which were
42:37studied in detail by Adela and Noman. We
42:40characterize ultra homogeneity in the
42:42class of colored D sets and classify
42:44unbounded order in disccernible
42:47sequences in such structures. We use
42:49these results to provide a
42:51characterization of distal colored D
42:53sets and prove that all colored D sets
42:56are manatically NIP. It would be very
42:59difficult for anyone to pass what this
43:01means without a significant background
43:02in logic and some detailed knowledge of
43:05the particular question being tackled. I
43:08don't understand it myself. It's just
43:10like any specialized area of research.
43:12The questions become increasingly
43:13sophisticated and they also become
43:15tailored towards a particular research
43:17program. Let's take a different example
43:20of modern logic work that is more
43:22philosophical. This is a 2018 book by
43:25Tim Button and Sha Walsh called
43:27Philosophy and Model Theory. I've
43:29actually not read the whole thing, but
43:31it surveys different uses for model
43:32theory within philosophy and also
43:34philosophizes about model theory itself.
43:37It is a more accessible book than most,
43:39at least from the parts of it that I've
43:40read, and it doesn't assume an advanced
43:42mathematical background. But it is still
43:44quite tricky stuff for your average
43:46reader. It's also an example of a major
43:48part of modern philosophical logic
43:50research, which is seeing how certain
43:53formal results might be relevant to
43:54various philosophical questions and
43:56tasks. At the more technical end, logic
43:59just becomes continuous with other areas
44:01of mathematics and often theoretical
44:03computer science. So, it's not unusual
44:05to see those who focus on logic in their
44:07research hold a joint position in the
44:10philosophy and maths faculties at their
44:12respective universities. And this is
44:15kind of what I love about logic. We
44:17began the video by simply asking what a
44:19good argument was. And we've ended with
44:22a highly advanced field of study which
44:24crosses multiple disciplines precisely
44:27because it started with such a simple
44:29question. Way back at the foundation of
44:31logic, when Aristotle began to consider
44:33it its own specialized field of study,
44:36he said that logic should ideally be
44:38topic neutral, meaning that it can apply
44:40to anything. While the advanced results
44:42in logic today don't always live up to
44:44the letter of total topic neutrality, I
44:47think it still has some of the spirit of
44:49that topic neutrality by impinging on
44:51and applying to so many different fields
44:54at once. There aren't too many places
44:56where linguistics and computer science
44:58and philosophy and maths are constantly
45:00combining and communicating, but logic
45:03is one of those places. And it's
45:05remarkable that over 2,000 years of
45:07people asking what is a good argument
45:09has brought us to this place. Now,
45:12obviously, there is so much at every
45:14level that we've not been able to talk
45:15about here. What I really hope is that
45:18this video has kind of hyped you up to
45:21learn some logic for yourself. And if
45:23so, there are a few key resources that I
45:25recommend. First, there is the Open
45:28Logic Project. I mention these guys
45:30constantly because they are absolutely
45:31phenomenal. They make some of the best
45:33logic textbooks in the world. And what's
45:35more, they are completely free. So, have
45:38a look at their website and maybe try
45:40out some of their textbooks. Next, a
45:42good bridging textbook to some of the
45:44more advanced stuff is the textbook a
45:47friendly introduction to mathematical
45:48logic. It will take you up to girdles
45:50and completeness theorems and it should
45:52put you in a reasonably strong position
45:53to start venturing out for yourself. A
45:56word of warning though, if this is your
45:57first time learning mathematical or
45:59formal content, there are a couple of
46:01things to bear in mind. The first is
46:03that reading a maths book is slow work.
46:05It's not like a novel where you can zoom
46:07through or even a philosophy book which
46:09might be slowgoing but you can still
46:11read it like a normal book. Even the
46:13best written math textbooks are dense
46:15and they tend to build cumulatively,
46:18meaning that if you haven't got the hang
46:19of a concept, the rest of the book will
46:21be very hard to understand because a lot
46:23of it will rely on that concept. There
46:25is just no getting around this. Uh I
46:27suppose one word of advice here would
46:29be, you know, do the exercises cuz they
46:32really do help. Uh without doing the
46:33exercises,
46:35it is just very difficult to take in the
46:37content fully. The second is that
46:40mathematical knowledge fades really
46:42fast. It's something that I've kind of
46:44had to come to terms with over the past
46:45few years as I've been doing much less
46:47formal logic and much more of other
46:49areas of philosophy. It always takes me
46:51a while to get back into the swing of
46:54areas of logic that I used to know in
46:56quite a lot of detail. I don't say this
46:58to discourage you. It is far easier to
47:00relearn some logic than it is to learn
47:02it for the first time. But I just
47:03thought it was worth mentioning. Logic
47:05isn't the kind of thing that you can
47:06learn once and then it's with you
47:08forever. It's more like cardiovascular
47:10fitness or muscle mass where it's easier
47:12to rebuild than build, but it will still
47:14fade without proper exercise. That being
47:17said, I really do hope this video has
47:19given enough of a taste of what logic is
47:21is roughly about for you to give it a
47:23go. I can promise with pretty much
47:26complete certainty that you won't regret
47:28it. But if you do want to hear more
47:30about the flaws of simply labeling
47:32things as logical fallacies and why I
47:35think it's important to learn some logic
47:36beyond that point, you can watch my
47:38video on that very topic right here.
47:40Thank you so much for watching and have
47:42a wonderful

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