This is another brief, but important, note within our discussion of evaluating claims that people make while trying to prove a point. The shortest way to say this is that there is more than one type of claim. You should evaluate claims based upon the category into which they fit. I could see this summary statement being a little too vague unless you can see what I mean, so let me explain.

As a mathematician, I am trained to look at mathematics with certain standards. I am taught to rely on nothing more than definitions, foundational mathematical axioms, and the basic rules of logic. Because of the goal of mathematicians – to know with logical rigor things that are true about the mathematical realm – this is an entirely appropriate way to look at mathematics. I say this is appropriate because this way of looking at mathematics is designed specifically to achieve the goal of knowing with logical rigor the correct answer. In fewer, words, the method fits the goal.

Why do I point this out? Well, not every field of academia uses the same standard. For instance, if I were to use my mathematical standards while studying history, I would learn literally nothing – I would just point out that it is at least possible that someone made everything up, and therefore on the grounds of the rules of logic I would reject all of history – even events that happened five minutes ago! If I apply mathematical standards to studies in literature, then the exact same thing will happen. Even if I apply mathematical standards to science, I will accuse the scientist of fallacy, since empirical evidence has no place within the rules of logic. In fact, but using these standards I will destroy all of human knowledge except for mathematics itself and a couple areas of philosophy.

This should not tell you that everything outside of mathematics ought to be placed in doubt. It just tells you that the standard used by mathematicians is not appropriate outside of mathematics. There are plenty of other standards outside of mathematics that are appropriate in different contexts. In a courtroom, the standard used for evidence is that it point to its conclusion beyond a reasonable doubt. This is still a very high bar, but it isn’t as high as the mathematical standard. Most scientific disciplines have the standards of empirical verification of theoretical predictions – which is significantly different than either the courtroom of mathematics. When studying history, you use various kinds of historical techniques that are entirely inappropriate anywhere outside of historical studies.

So, if we are evaluating a claim that someone else makes, we must be very careful to understand what kind of claim they are making, and what kinds of evidence are relevant to the question. It is possible that an idea will have multiple dimensions – for example, evaluating ancient history involves, geography, literary studies, anthropology, historical studies, and archaeology (at least these – perhaps even more). Each of these areas has their own methods and techniques that are designed to be helpful at answering very specific kinds of questions, and we should always be careful that we are taking all of this into account.

Of course, in order to do this we must take some time to learn about various kinds of techniques and learning about where those techniques are viewed as relevant by academic professionals. If you want to learn about whether a certain mathematical technique is valid, you ask a mathematician, not a scientist, because there are statements known by mathematicians to be false that the scientific method would end up regarding as true, because counterexamples to these statements, while we know they exist, are beyond the power of modern computers to locate. A scientific approach will get you a lot of the right ideas, but a lot of wrong ones too. If you want to know about scientific techniques, mathematicians and historical studies can be helpful, but you’re going to learn the most from a scientist. The same goes for everything you can think of that humans study today.

This is why I take so much time attempting to explain careful methods of thinking. We will often go astray if we use the scientific method when it isn’t appropriate, or if we apply a non-scientific methodology in science. We must be careful if we want to know the truth about reality, because the deeper truths about reality are not always simple.

One of the most common and comical aspects of the lives of very young children that I find quite joyful can be summarized in one word… “Why?” Well, it is more like a sequence of these “Why” questions over and over again until your head is about to explode. Even though there is a silliness to asking “why” so many times in a row, the young child doing so also shows a joyful curiosity that we all share. And, as it turns out, answering the simple question “Why?” is sometimes very hard.

A similar brand of question that receives detailed attention in philosophy is “how do you know?” More specifically, in philosophy there is an entire area of study known as epistemology that aims to analyze what it actually means to know something. While I do not plan to go super deep into epistemology here, it is important to understand such words as believe, know, and certain when we are trying to learn how to carefully have intellectual discussions with others.

My only goal here is to clarify some proper ways to use terms like these within this particular context, because many times these ideas are not understood in a properly nuanced manner.

What is Belief?

This one ought to be the simplest of all, and yet for a couple reasons this has become difficult. The reason is that the word “belief” in popular thought has obtained a new meaning in the religious context. To a great many, the term belief in the context of religions denotes at least a complete vacuity of any evidence or support for the religious position, if not (in extreme cases) an outright denial of all existing evidence. This definition is so biased. I would compare it to a fan of a particular sports team saying that their team is the best team to be a fan of. I honestly think it could even be argued that this definition is a form of either propaganda or outright intellectual discrimination against religious people. A more popular colloquial term for the thing I am trying to describe here is blind faith. Please, please do not take that definition at all seriously if you have been exposed to it. It is completely alien from EVERY religious person, and from every person who has ever lived for that matter. Nobody that I have ever heard of consciously rejects what he or she knows to be evidence without some kind of counterevidence. Let’s respect one another enough to understand the basic idea that everyone has reasons for thinking what they do.

The word belief is really quite simple. A quick Google search gave me the definition an acceptance that a statement is true or that something exists. I am perfectly happy with this definition, as is anyone who holds to a particular religious tradition. When I say that I am a Christian, what I mean by that is that I accept that the core tenets of the Christian religion are true. When a person says that they believe in this religion or that religion, that is all that they mean.

I hate to even have to define the word belief, but I have seen that in some circles (even if very small ones) the word belief has been corrupted in an overtly biased manner, and so I felt the need to clarify outright what I mean when I use this term.

What is Certainty?

Another word that gets misused sometimes is certainty. Clearing this one up is similar to the previous situation – but it is a little different because the terms certain and certainty have two genuine meanings, but these meanings sometimes get blended into one meaning sometimes.

What do I mean by this? The easiest way to distinguish them is to say that certainty has a logical definition and a psychological definition that are both legitimate, but need to be recognized as different. The logical use of the word certain is the kind of thing you get out of formal logic and mathematics – mathematicians are certain that 2 + 2 = 4 because according to fundamental laws of mathematics and logic, this statement must be true. The psychological use of the word certain is not at all related to whether something is true – it relates to what we think or feel about a particular idea. Using the psychological definition, it is entirely possible that I am certain that X is true and you are certain that X is not true. This happens all the time – just listen to any political or religious discussion in the public sphere and you will find people who are (psychologically) certain of two opposing viewpoints. But neither of them are (logically) certain about their views, because logical certainty has no role whatsoever in pretty much every area of thought – even science! We cannot even be logically certain that gravity exists. Logical certainty cannot be obtained by any amount of evidence – only logic. You can be logically certain of Y because X is logically known to be true (i.e. 2 + 2 = 4) and X implies Y (i.e. 2 + 2 = 4 implies that 2 + 3 = 5). That’s about as far as you can go with logical certainty.

The important thing to realize with this distinction is that it is very, very easy for us to treat our own psychological certainty as if it were logical certainty. We should do our best to avoid doing this. We must recognize that we can be wrong about a lot of things, even things that we feel certain about. We must be willing to listen to those who disagree with us – even if we turn out to be right and they wrong, more times than not listening to those who disagree with you will help you understand more deeply what you believe and why you believe it.

What is Knowledge?

I have moved throughout this article from the easy definitions to the difficult ones. The leap from defining belief to defining certainly was in my estimation fairly small, but defining the word knowledge is very nearly impossible. There is an entire field of academic study called epistemology that is built upon attempting to clearly define what knowledge is.

The most fundamental attempt at a definition of what knowledge is is the JTB model – which stands for justified true belief. I’ll give a sketch of the train of thought behind this attempt at defining knowledge. It is fairly obvious that if you know something, then you believe that thing. Furthermore, you can’t actually know something in a proper sense of the word unless it turns out to be true (if what you believe turns out to be false, it will be an example of psychological certainty that fails to qualify as genuine knowledge). But a true belief wouldn’t always count as knowledge. For instance, someone could gain strange but strong belief about something they know nothing about while in a psychedelic trance, but even if they turn out to be correct, none of us would say that they knew that thing was true. Upon reflection, we realize that this is because this individual didn’t have any good reason to believe what they did – their belief was basically random and just turned out to be right. This ‘good reason’ is what the term justified is meant to denote.

As it turns out, epistemologists almost universally reject JTB as a sufficient definition for what knowledge is. Roughly, one big reason for this is that something that appears to be good evidence might not actually be connected to the thing you believe. These are sometimes called Gettier-type counterexamples in honor of a philosopher Edmund Gettier who challenged the JTB model. One popular example of this kind of situation involves watching a sports game. Imagine that you know your favorite team is going to play their rival on TV today, and so you turn on the TV to watch the game. But, unknown to you, there was a channel playing a rerun of last year’s game between the two teams, and instead of watching the game going on this year, you watch last year’s game. As it turns out, this year’s game had a final score of 70-60, which is exactly the same as last year’s game. So, by watching last year’s game on TV, you come to believe that your favorite team won the game this year by a score of 70-60. This belief of yours is true, and since you watched a game on TV in which your team beat their team 70-60, you have a very good reason to believe that this is true. And yet it doesn’t seem like it can be said that you know this.

There are lots of examples like this one, each showing that good justification for believing something might not connect to the thing you believe in the way that you think it does. This is strong evidence that, while JTB is an extremely useful model for defining knowledge, it isn’t always quite right. What ought to be added or modified to fix these situations is debated and is worth thinking about, but for the sake of learning critical thinking skills, using JTB as a background of what it means to know will be good enough to avoid most mistakes.

How Do We Know a Premise?

In a philosophical argument, then, what does it mean to say that we know a premise is true? We must believe the premise, the premise must actually be true, and we must have some good reasons to back up our believing the premise. While epistemology has gone much further than this in discussions of what knowledge is, the simple JTB model will be good enough for the purposes of thinking carefully through our beliefs and through philosophical arguments.

One important aspect of philosophical arguments is whether or not there exist any “default positions.” That is, if I am trying to convince someone about X and that person is trying to convince me of the opposite of X, is either of us in some kind of default position? When is an idea innocent until proven guilty? Guilty until proven innocent? Or when does an idea start off somewhere in the middle – where all relevant viewpoints equally are required to prove their point?

The technical term used to lay out these distinctions is the burden of proof. The burden of proof is exactly what it sounds like – it is a burden or obligation to prove what you are saying. So, the questions of the previous paragraph are summarized in one question – in any disagreement, who has the burden of proof?

The most commonly discussed and well-known application of the idea of the burden of proof comes in criminal trials. In a criminal trial, the jury is instructed that the accused is to be considered innocent until proven guilty, and that the evidence must point to guilt beyond a reasonable doubt. This is a clear affirmation that between the two sides of a trial – the defense and the prosecution – only the prosecution carries a burden of proof. There is, in this case, a default assumption that the man or woman on trial is innocent. The defense is not required to prove innocence, only to show that the prosecution cannot prove guilt.

But is this always so? Are we to always use the innocent until proven guilty standard? And if so, how do we even know which side is the “innocent” side? Well, really the answer is that no, we do not always use that standard. There are other situations where it is appropriate to apply a similar standard, but most situations are not like that. In most situations, the fact of the matter is that both sides of a debate have a burden of proof. Why is this so? To explain, it would help to provide a rule of thumb for when a person involved in a debate carries a burden of proof. After making the rule of thumb clear, we can go back and add some additional details about some special situations where the rule of thumb is not considered applicable.

Generally speaking, the burden of proof falls on any person who makes an objective truth claim. A truth claim is just any claim that “such-and-such is the case,” “such-and-such is true,” “such-and-such is false,” or “such-and-such really happened.” Not everything is a truth claim. For example, if you mother tells you to clean your room, that isn’t a truth claim – rather, it is a command. Questions also are not truth claims – they are just questions. Perhaps a person asking the question is trying to get you to think of some truth claim – maybe they are asking a leading question – but in and of itself a question makes no claims. (To be grammatically precise, truth claims use the indicative mood, but don’t get bogged down in fancy language, we all have a pretty strong intuitive understanding of what truth claims are and are not when we slow down and think about it).

What then do we mean by an objective truth claim? In this context, by objective I mean external to our own consciousness. For instance, the sentence “I am happy today” is a truth claim, but it is not objective in the sense I am using it, because emotional states like ‘happiness’ reside in your consciousness. The reason that this kind of distinction is important is because it isn’t really possible for anybody to bring any evidence against your claim that you are happy – they do not have access to your thoughts and feelings, so how would they know? For this reason, a person who says “I am happy right now” has no burden of proof at all – their statement is to be accepted at face value as a statement of their perceptions of their own emotions. Of course, sometimes people are in denial about things, so people can be mistaken about certain things – but to show that this is the case would require a great deal of convincing from the outside.

Putting aside for now the abstract discussion – let’s look at examples that matter. Let’s take for instance politics. If a Democrat and Republican are debating some topic, who has the burden of proof? For very nearly every political issue I can think of, both sides have equal burdens of proof, since each side is making a claim about the way the world around us is. To decide whether it would be better to raise or lower taxes, for example, both sides must bring forward their best evidence of their own conviction, and then the evidence can be evaluated and conclusions can be drawn.

Another common example is in matters of religious debate. For the most part, in the context of a public discourse, every religious perspective has its own burden of proof. Christians, Muslims, Hindus, Jews, and atheists are all making claims that a certain deity exists or doesn’t exist, that certain things do or don’t happen in the afterlife, that certain events in history happened or didn’t happen. The exception to that rule is agnostics – because the position of an agnostic is that they don’t know what to believe, they aren’t making an truth claim about the outside world. So agnostics carry no burden of proof – everyone else does. So far as I can tell, every group seems to realize this except for some subset of the atheists. Many of the atheists that say things like they “only lack a belief in God” and that “you can’t prove negatives” are, whether they realize it or not, putting up a smokescreen to make themselves look like agnostics when they aren’t. Surely, many of these people really do believe these things and many who say this actually are agnostics. But many of those who say they merely “lack belief” will also actively try to convince you that there is no God – which is a truth claim and carries with it a burden of proof. The claim that there is no God and the burden of proof that comes with it has been part of the very definition atheism for the entire history of philosophy (and still is – some very recent popular-level thinkers have tried to redefine the word atheist to mean something more like agnostic, but academic philosophy doesn’t take that seriously as far as I know).

In summary, very nearly all actual debates that people engage in and nearly all questions that people really care about will be like the examples above – these are situations in which all sides carry a burden of proof. But there are some areas where this is not so. The epistemological term basic belief, or properly basic belief, is often used in these situations. In short, a properly basic belief is a belief that is so foundational and so deeply integrated into our ability to think at all that no burden of proof is required. Going back to the criminal trial example, a properly basic belief is like the “innocent” verdict – we are allowed to assume that our properly basic beliefs are correct unless it is shown to us beyond a reasonable doubt that they are mistaken. Broadly speaking, properly basic beliefs include things like our sensory perceptions and beliefs about our own thoughts and feelings. It may help to see some more specific examples.

One helpful example of what properly basic beliefs look like is my belief that I have a body. While technically someone could try to convince you that you’re only hallucinating your body or that you are in “the matrix” and that you don’t actually have a body (at least not like the one you think you have), the immediate experience of seeing and controlling your body every day is enough to safely conclude that you actually do have a body. We don’t have to prove to people that we have a body – we just know that we do. A similar example is the belief the people you talk to every day are not “robots” but have thoughts and feelings in the same kind of way that you do – in other words, that there are other minds that besides my own. Is it possible that I am really the only thing in the universe and am constantly hallucinating everything around me? Well, you can’t really disprove that. But that doesn’t mean we have to take such an accusation seriously either. We are being perfectly reasonable when we assume that people we talk to also have minds, because this is a properly basic belief and carries no burden of proof.

Hopefully, this discussion of the burden of proof, truth claims, and properly basic beliefs helps to shed light on the reality that, more times than not, being intellectually responsible requires bringing forth some form of evidence for that which we believe. And for my Christian readers, the New Testament explicitly tells us that our faith in Jesus is one of the things that we should be prepared to defend using evidence, for we are told:

“Always be prepared to give an answer to everyone who asks you to give the reason for the hope that you have.“ – 1 Peter 3:15 (NIV)

Having put down at least an initial discussion about having intellectual integrity in the importance of using evidence to defend what we believe, we can move on to a discussion about how we can think about evidence itself.

This is the first post I am making in what will hopefully become a long and detailed series of posts about how to think more clearly about difficult questions. Since we ought to try to think clearly in every domain of life, we must begin the discussion at the broadest level, with the most important things to understand about clear and logical thinking. The obvious place to start here is the idea of a philosophical argument.

Defining ‘Argument’ and Related Terminology

First things first, understand that this is not an argument in the popular definition of two people bickering back-and-forth about some topic. This is nothing like arguments you see on the internet, or arguing with your parents or siblings. There need not be any anger at all behind a philosophical argument, just as there doesn’t have to be (and effectively never is) any anger behind a mathematical argument. When I use the word argument in the context of a philosophical argument, all that I mean is an effort to systematically put together a collection of information to draw some sort of meaningful conclusion.

A philosophical argument can come in a variety of levels of detail, sophistication, and length. However, since at its heart all we mean by an argument is an effort to combine some information to draw a meaningful conclusion, any philosophical argument worth its salt can be boiled down into a format called a syllogism – which is more or less just a fancy way of saying a bullet-by-bullet presentation of the pieces of information that are relevant to the argument. In a syllogism, the initial pieces of information are called premises, and the culminating new information drawn from these facts is called the conclusion. Most often, a philosophical argument will consist of a syllogism along extended discussions of why each premise should be considered to be true, and these discussions are called defenses of the premises. Here is a basic example of a philosophical argument that illustrates these definitions.

1. If it is raining right now, the ground is wet.
2. It is raining right now.
3. Therefore, the ground is wet.

The collection of statements 1-3 is a syllogism. Statements 1 and 2 are premises, and statement 3 is the conclusion. To make the argument more detailed, you could make a defense of Premise 1 by convincing your listener that water makes things wet, and you could defend Premise 2 by looking out a window, walking outside, or checking a weather app.

Also important in discussion of arguments are objections to an argument. An objection to an argument is basically just anything that is meant to show that the conclusion in the argument is actually invalid. For instance, in the above argument, you could object to Premise 1 by noting that ground inside of a building or underneath a shelter won’t become wet when it is raining, or you can object to Premise 2 by showing your friend that there are no clouds in the sky. These considerations would, if they are accepted as true, make the conclusion “Therefore, the ground is wet” not necessarily true.

Objections can come in many forms, and methods of objection are a topic for later discussion. But before we can embark on that discussion, we first must lay out clearly what a philosophical argument actually has to accomplish in order to be considered as success. There are two basic criteria that arguments aim to satisfy – an argument must be valid and sound before it should be trusted as correct.

What is a Valid Argument?

Of the two conditions mentioned, validity is weaker (in the sense that being sound requires validity). An argument is valid if it is the case that, if we take as absolute truth every premise, then we cannot help but affirm the truth of the conclusion. Sometimes, the term valid is instead replaced with logically valid, because what we mean is that we can use the rules laid down by the formal theory of logic to move us from premises to conclusions. The argument about rain I presented earlier is logically valid in this sense – it has the format

1. If A, then B
2. A
3. Therefore, B

In the philosophy of logic, this form is given the title modus ponens – Latin for the mode of affirming. I’m not going to try to explain why this is accepted as logically valid – I quite frankly don’t know how to boil this down any more than it already is. If anyone reading this believes 1 and 2 but not 3, please do try to explain to me why. So long as A and B genuinely mean the same thing in all three statements, this is unavoidable.

Why does validity matter? Aren’t there other ways to know something than pure logic? Well, yes, sort of. It is true that very, very little can be known using pure logic alone. But the “impure” aspects of a philosophical argument should never, ever be found in the logical validity of an argument. Rather, uncertainty ought to arise out of the truth of a premise – which is where the idea of a sound argument comes in.

What is a Sound Argument?

In order to define a sound argument, I find it helpful to first show examples of arguments that satisfy validity but not soundness. This is an “argument” that Steve went to jail – but upon reading the argument it should be instantly clear that something is awry (which is the whole point):

1. Everyone who goes to jail commits a crime.
2. Steve was sent to jail.
3. Therefore, Steve committed a crime.

Anyone who has studied logic will affirm that this argument is logically valid – if you accept as true both Premise 1 and Premise 2, then the conclusion that Steve committed a crime cannot be avoided. And yet, we all know there is something wrong here. As much as we would like Premise 1 to be true, it is not true. Sometimes, people who have not committed any crimes go to jail. So, Premise 1 is false.

This example shows what we mean by an unsound argument – which we ought to reject. Now, informally, a philosophical argument should be called sound if it is logically valid, is based upon truth and which genuinely leads a thinking person to truth. To further clarify this, here is a definition of a sound argument that I find quite reasonable:

An argument is sound if and only if it is logically valid, all its premises are in fact true, and its premises are more likely to be true than false in light of the evidence.

Now, I will provide some examples to show why I include all the points in this definition that I do. If the argument is not logically valid, then by definition I am perfectly within my intellectual rights to accept as true every premise and to still reject the conclusion – so logical validity is required. If there is any confusion here, I will provide examples later in this series of logically invalid arguments that sometimes pass as valid (google the genetic fallacy, the ad hominem fallacy, or the fallacy of composition for some examples). I hope that I do not need to defend here how important it is that our beliefs line up with the basic laws of logic (which is really what logical validity means).

The argument about jail already shows how an argument can be logically valid but that it might be based on a false premise. Interestingly, we might here note that if an argument is logically invalid, we cannot therefore deny the conclusion. In the jail argument, it is quite possible that Steve is actually in jail. It is quite possible that Premise 2 and the conclusion are both correct. The important point here is that Premise 1 is not correct, so the argument is not valid because you cannot deduce the conclusion from the premises – since one of them is false, you can’t use it to meaningfully learn anything by this method.

So, this example shows why in my definition of a sound argument I require that all the premises be actually true. However, there is yet another way that an argument can be unsound, which is more complicated. To see what I mean, suppose that we go back in time 1000 years – to the time when everyone believed that the Sun revolved around the earth. People believed this, among other reasons, because no evidence had yet been produced that the earth actually revolves around the Sun. Now, in this situation, suppose I propose the following argument:

1. If the Earth revolves around the Sun, the Sun would appear to rise and set in the sky.
2. The Earth revolves around the Sun.
3. Therefore, the Sun appears to rise and set in the sky.

To a modern reader, this is a sound explanation of why the Sun appears to rise and set (albeit massively simplified – of course you’d want to give a visual or mathematical demonstration of how Premise 1 actually works). However, to a person living 1000 years ago, Premise 2 will seem very highly unlikely in light of the evidence that they had available to them. So, the man living 1000 years ago need not believe that the conclusion is true because of this argument – they can instead believe that the Sun appears to rotate around the Earth because it actually does – because to this man, that is where all his evidence points.

It is for this reason that in discussing whether an argument is sound – roughly speaking, whether it ought to convince intelligent and informed people of its conclusions – we have to take into account what sorts of evidence are available to us. This is why in the definition of soundness that I give, I specify that every premise must be more likely than not given the evidence. To be fair here, you could require a standard other than “more likely than not” – for instance, in a court of law very often this standard is increased to “beyond a reasonable doubt” because of the horrible consequences of a false conviction. But despite the potential for changing the “degree of likelihood” we demand of our evidence, I find the “more likely than not” approach most reasonable.

Conclusion

From this post, we have defined the minimal criteria that an argument should meet in order to be convincing. In summary, in order to convince your listener of a point, your line of reasoning – your argument – should line up with the laws of logic, it shouldn’t rely on any false premise, and all premises in your argument should be supported by evidence. In the future, I will spend more time elaborating on how to properly approach the premises of an argument and evidence for each premise carefully and even mathematically.

We hear so often how important critical thinking is for succeeding in the job market, in the world, however you want to say it. Critical thinking is put forward as a difference-maker that separates success from failure. And for good reason – by definition, critical thinking is the practice of not taking statements at face value and using careful, tried-and-true methods of sifting through information, finding the truth amidst confusion, and finding good solutions amidst a sea of bad ones.

As much as there is talk about things like critical thinking… what actually is critical thinking? And how do we go about developing our critical thinking skills?

One way to engage in critical thinking, and the one I perceive to be most common in education, is through problem-solving. By forcing yourself to think through a complicated problem, there is a natural tendency to develop stronger critical thinking skills, just as going to the gym has the tendency to make one stronger and more athletic. This is one very important aspect of critical thinking. Learning problem-solving helps you think more quickly and helps to assemble useful ideas in useful ways.

But this is not all that critical thinking is. There is more, and it is the aspect that I’ve noticed seems to be much more lacking in modern society. This is the critical thinking skill of recognizing and avoiding bad reasoning. There are a great many ‘patterns of thought’ out there – and they are not all equal. Some help you find truth, others are obviously flawed. There are some that look like they work, but really don’t, and some that don’t look like they work but actually do. There are some that apply in every situation, and some that apply in some cases but not others. And I cannot remember even once in my education learning anything about any of these critical thinking tools, at least not prior to studying mathematics in college. And even then, I still didn’t know most of the philosophical tools that I use in my writing and thinking now – and I fell victim to many of these sorts of bad ways of thinking. Hopefully, I don’t anymore, but at least by knowing about them I can do a better job trying to prevent myself from slipping into them.

I’ve previously done a series on types of mathematical proofs, and this series is meant as an extension of that into a landscape of ideas broader than mathematics. Through the weeks, months, and possibly even years that I hope to work on this series, I hope to cover a wide variety of areas and how experts utilize critical thinking in their own fields, as well as a variety of commonplace flawed patterns of thinking that we all instinctually fall prey to. I also hope to provide useful examples of commonplace fallacies in order to help people think more clearly about issues in the public square as well as in more personal matters.