Everything Everywhere Daily: History, Science, Geography & More - Formal Logic

Starting point is 00:00:00 Located in the area between philosophy and mathematics is the realm of logic. Logic permeates everything we do, from the work of Aristotle to modern computer programming languages to the musings of Mr. Spock. However, there's more to logic than just making sense and avoiding fallacies. It can also be a highly formal system using systems and variables to represent statements. Learn more about formal logic, its ancient roots and its modern applications on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong?

Starting point is 00:00:46 ThruLine is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night. And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. Before I get into a discussion of formal logic, I should explain the difference between formal and informal logic. Informal logic deals with the analysis of natural language arguments, where the focus is more on the content of the arguments, the context which they occur, and the pragmatic aspects of argumentation. Informal logic includes things such as the study of fallacies, argumentation theory, and

Starting point is 00:01:31 rhetoric. I did a previous episode on logical fallacies, and that would be an example of informal logic. I took a course on argumentation in college from one of the best argumentation instructors in the country, and it's one that I will never forget. However, it should be noted that it doesn't mean that there are different types of logic or that something is logical in one system and not the other. Formal logic is simply a more formalized system where you break down statements into premises and often represent statements with variables, and this is also known as symbolic logic.

Starting point is 00:02:06 Formal logic can analyze the structure of an argument independent of the content of the argument. Formal logic sits at the intersection between mathematics and philosophy. Historically, it's been a branch of philosophy, but you might find a formal or symbolic logic course in either math or philosophy departments. If you look at a page of symbolic logic statements, it would probably make absolutely no sense at first glance because there are a host of symbols that are only used in formal logic. The origins of formal logic date back to ancient Greece, and the man who's considered to be the founder of formal logic, Aristotle. His writings on logic can be found in a collection of his works known as the Organon.

Starting point is 00:02:47 Aristotle developed the first known symbolic logic. He was also the first person to use a powerful technique known as a syllogism. A syllogism is a simple technique that consists of at least two premises and a conclusion that can be logically drawn from the premises. The example that's often used to demonstrate a syllogism is the following classic one. The major premise would be that all humans are mortal. The minor premise would be that Socrates is a human, and the conclusion that can be drawn from these two premises would be that therefore, Socrates is mortal. That's a very simple example, but they can be more complex. You can have something called a polysyllogism, which could add other layers.

Starting point is 00:03:29 For example, Socrates is a Greek. All Greeks are humans, all humans are mortal, therefore Socrates is mortal. Sylligisms are a pretty simple case, but after Aristotle, the development of formal logic continued. And I should note, newer developments in logic didn't replace what came before it. It only added to it. Stoic philosophers continued to develop formal logic by developing propositional logic. Whereas syllogisms involved categorical logic, i.e. all humans are mortal. the Stoics developed propositions that involved if-then statements, as well as propositions that involved and and or. The forms of logic developed by the Stoics were more complicated, but they didn't invalidate

Starting point is 00:04:14 anything that came before it, they only built off of it. And there will be more on propositional logic in a bit. The development of form of logic was not linear over time. In fact, from the 1,000-year period going from the 3rd century BC to the 8th century, very little work was done. It was Islamic scholars who took upon themselves the revival of formal logic during the Islamic Golden Age from the 8th to 13th centuries. Muslim philosophers such as El Fibari, Abin Sina, and Ebn Rusjid, all took the works of

Starting point is 00:04:44 earlier Greek philosophers and made their own advancements. Europeans returned to the study of logic in the Middle Ages by the likes of Peter Abelard and William of Ackham. In the 17th century, Gottfried Leibnance, the co-discoverer of calculus, was also actually probably the most prominent logician since Aristotle. If he hadn't discovered calculus, he would probably still be known today for his work on formal logic, even though he never actually published anything on the subject during his lifetime. Everything we know of his work came from unpublished papers discovered after his death. Leibnance believed that there were a small number of

Starting point is 00:05:20 simple ideas that consisted of what he called the alphabet of human thought. These simple ideas he thought could then be logically combined into more complex ideas. His goal was nothing else than to create a universal logical language that he called the Characteristica Universalis. Needless to say, he failed. The next big advancement in formal logic took place in the 19th century. An English mathematician by the name of George Buell, working as a professor at Queens College in Cork, Ireland, developed a system of algebra that involved true false statements

Starting point is 00:05:54 instead of numbers. The system he developed, Boolean algebra, bears his name today. However, the towering figure in formal logic in the late 19th and early 20th centuries was the German logician Gottlieb Frege. Fraga. Frega was a pioneering figure in the field of formal logic, mathematics, and language at the turn of the 20th century. He introduced a new logical system that significantly expanded the scope of logic beyond Aristotle syllogisms, laying the groundwork for modern analytic philosophy and mathematical logic. His seminal works Begriff-Trift and the foundations of arithmetic introduced the concept of predicate logic, a formal system that allowed for the expression of statements and relations in a precise symbolic language. He attempted to base all mathematics

Starting point is 00:06:41 on a foundation of logic. This was later picked up by Alfred North Whitehead and Bertram Russell, who published the Principia Mathematica. The Principia attempted to base all of mathematics on a set of logical axioms. And the book famously took 360 pages to definitively prove that one plus one equals two. However, all the attempts at creating a logical basis for all mathematics and human thought came crashing down in 1931 when Kurt Godel discovered that in any axiomatic system, there will always be some propositions that cannot be proven or disproven. And these are known as Godel's incompleteness theorems. So what I hope you get from all this is that that logic as a discipline is something that's been around for a very long time and major

Starting point is 00:07:27 advances have been made in it within the last 100 years. As I'm guessing most of you have never studied formal logic before, I want to go over some of the very basics of formal logic, which are really not that difficult to understand. One of the first principles is known as the law of identity. The law of identity states that any concept or object is identical to itself. This is usually expressed as A equals A. Again, in this case, A doesn't represent a number, but can represent anything. Other similar principles include the law of non-contradiction, which states that a statement and its negation cannot both be true. If I say that a ball is round, then both that statement and the statement that a ball is not round cannot both be true.

Starting point is 00:08:15 Another closely related principle is the law of the excluded middle. According to this principle, for any particular proposition, either that proposition is true or its negation is true. There is no middle ground or third option. And I should note that these are for logical statements, not normal language statements that can be used in everyday language. Once we've established rules regarding statements that we can represent with a letter, we can then make conditional statements. Conditional statements are expressed in the form of an if-then statement and are usually in the form of if p then q for example a conditional statement could be if it is raining then i use an umbrella and i should note that such a statement doesn't have to be true

Starting point is 00:09:00 it could be false a conditional statement such as if it is raining then the stock market goes up is a valid statement even though it is not true all that's required is a dependent clause in this case the statement if it is raining, and a main clause that expresses the result. In this case, then I use an umbrella. With a conditional statement, there are several simple things you can do to analyze it, and they involve terms that you're probably familiar with, but didn't know the exact definition of. The first is to take the inverse of the statement. The inverse is simply the negation of each part of the conditional statement. In symbolic logic, this would be expressed, if not P, then not Q.

Starting point is 00:09:45 The inverse of my example I gave would be, if it is not raining, then I will not use an umbrella. If a conditional statement is true, then the inverse of the statement is not necessarily true. In the example I gave, the inverse could be true. But if my original statement was a little bit different, perhaps if something is a bird, then it is an animal. Then the inverse of that statement would be,

Starting point is 00:10:10 if something is not a bird, then it is not an animal. And that is clearly false because a mammal is not a bird, but it is an animal. Next would be if you flip P and Q around, you get the converse of a statement. This would be said, if Q, then P. In my original example, this would be, if I use an umbrella, then it is raining. The converse of a statement, just like the inverse, is not guaranteed to be true. To use my second example, I could say, if something is an animal, then it is a bird, and that statement is clearly false.

Starting point is 00:10:47 But what is interesting is what happens when you take both the inverse and the converse of a statement together, and that is known as the contrapositive. In symbolic logic, the contrapositive would read, if not Q, then not P. It turns out that if a conditional statement is true, then the contrapositive must all also always be true. For example, if the original statement, if it is raining, then I use an umbrella is true. Then the contrapositive of the statement, if I don't use an umbrella, then it is not raining, must also be true. Even if you use an umbrella for other reasons, if you're not using an umbrella, then you can be sure it's not raining. And to use my second example, the contrapositive would be,

Starting point is 00:11:34 if you're not an animal, then you're not a bird. And that is true. These are very simple examples, and I'm just scratching the surface of what you can do with formal and symbolic logic. From here, you can add conjunctive statements using an and, disjunctive statements using or, and operations which can get very complex. The importance of this field of study is pretty obvious if you've ever done any computer programming, which is really just an exercise in formal logic. Every computing device you use, from a smartphone to a washing machine, has some program in it based on formal logic. And that's not too bad for a system of thinking which was developed over 2,000 years ago. The executive producer of Everything Everywhere Daily is Charles Daniel.

Starting point is 00:12:25 The associate producers are Peter Bennett and Cameron Kiefer. Today's review comes from listener P1BK over on Apple Podcasts in the United States. They write, Incredible. As a new member of the Completionist Club, I just want to say thank you, Gary. You've made every episode interesting and fun. Keep up the great work. Thanks, P1BK. As a member of the Completionist Club,

Starting point is 00:12:45 you are now entitled to all the rights and privileges of membership, including use of the Completionist Club pool. However, please remember that a lifeguard may not always be on duty. Remember, if you leave a review or send me a boostagram, you two can have it read on the show.

Everything Everywhere Daily: History, Science, Geography & More - Formal Logic

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