Introduction to Formal Logic

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Podcast Transcript

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 to the musings of Mister Spock. 

However, there is more to logic than just making sense and avoiding fallacies. It can also be a highly formal system using symbols and variables to represent statements.

Learn more about formal logic, its ancient roots, and its modern applications on this episode of Everything Everywhere Daily.

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 in which they occur, and the pragmatic aspects of argumentation. Informal logic includes such things as the study of fallacies, argumentation theory, and 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 is one that I will never forget. 

However, it should be noted that this doesn’t mean 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. This is also known as symbolic logic. 

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 has been a branch of philosophy, but you might find a formal or symbolic 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 used only in formal logic. 

The origins of formal logic date back to ancient Greece, and the man who is considered to be the founder of formal logic is Aristotle. His writings on logic can be found in a collection of his works known as the “Organon.”

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 is 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.

The conclusion that can be drawn from these two premises would be that, therefore, Socrates is mortal. 

That is a very simple example, but they can be more complex. You can have something called a pollysyllogism. You could add other layers.

For example:

Socrates is a Greek.
All Greeks are Humans
All Humans are Mortal

Therefore, Socrates is mortal.

Syllogisms are a pretty simple case, but after Aristotle, the development of formal logic continued. 

Newer developments in logic didn’t replace what came before it. 

Stoic philosophers continued to develop formal logic by developing propositional logic. Whereas syllogisms involve categorical logic (ie, 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 didn’t invalidate anything that came before it, only built off of it.

There will be more on propositional logic in a bit.

The development of formal logic wasn’t linear over time. In fact, from the 1000-year period going from the 3rd BC to the 8th century, very little work was done.

Islamic scholars took on the revival of formal logic during the Islamic Golden Age from the 8th to 13th centuries. Muslim philosophers such as Al-Farabi, Ibn Sinna, and Ibn Rushd, all took the works of 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 Ockham. 

In the 17th century, Gottfried Leibniz, the co-discoverer of calculus, was also actually probably the most prominent logician since Aristotle. If he hadn’t discovered calculus, he probably would still be known today for his work on formal logic, even though he never actually published anything on the subject. Everything we know of his work came from unpublished papers discovered after his death.

Leibniz believed that there were a small number of simple ideas that consisted of what he called the alphabet of human thought. These simple ideas could then be logically combined into more complex ideas. 

His goal was nothing else than to create a universal logical language that he called characteristica universalis.

Needless to say, he failed.

The big advancement in formal logic took place in the 19th century. An English mathematician by the name of George Boole, working as a professor at Queen’s College in Cork, Ireland, developed a system of algebra that involved true/false statements 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 Gottlob Frege.

Frege 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’s syllogisms, laying the groundwork for modern analytic philosophy and mathematical logic. 

His seminal works, “Begriffsschrift” 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 on a foundation of logic.

This was later picked up by Alfred North Whitehead and Bertrand Russell, who published the Principia Mathematica. The Principia Mathematica attempted to base all of mathematics on a set of logical axioms, and the book famously took 360 pages to definitively prove that 1 + 1 = 2.

However, all the attempts at creating a logical basis for all mathematic and human thought came crashing down in 1931 when Kurt Gödel discovered that in any axiomatic system, there will always be some propositions that cannot be proven or disproven. These are known as Gödel’s incompleteness theorems. 

So, what I hope you get from this is that logic as a discipline is something that has been around for a very long time, and major advances have been made in it within just 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 basics of formal logic, which really are not 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. The is usually expressed as A=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 be true. 

Another closely related principle is the Law of Excluded Middle. According to this principle, for any particular proposition, either that proposition is true or its negation is true. There’s no middle ground or third option. 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 expressed in the form of “if P, then Q.”

For example, a conditional statement could be, “If it is raining, then I use an umbrella.”

I should note that such a statement doesn’t have to be true. It could be false. A conditional statement such as “if it is raining, then the stock market goes up” is a valid statement, even if it is false.

All that is required is a dependent clause, in this case, “if it is raining,” and a main clause that expresses the result, in the above example, “then I use an umbrella.”

With a conditional statement, there are several simple things you can do to analyze it, and they involve terms you are 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.”

The inverse of the 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 guaranteed to be true, but it could be true. In this example, it could be true.

If my original statement was “If something is a bird, then it is an animal,” then the inverse of the statement, “If something is not a bird, then it is not an animal,” is clearly false because a mammal is not a bird but is an animal. 

If you flip P and Q around, you get the converse of a statement. This would be “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 is not guaranteed to be true. To use my second example, “If something is an animal, then it is a bird” is clearly false. 

What is interesting is what happens when you take both the inverse and converse of a statement together. This 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 always be true. 

For example, if the 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 be true. If even if you use an umbrella for other reasons, if you aren’t using an umbrella, then it must not be raining. 

To use my second example, the contrapositive would be “if you are not an animal, then you are not a bird” has to be true. 

These are very simple examples, and I am just scratching the surface of what you can do with formal and symbolic logic. From here, you can add conjunction statements using “and,” disjunctive statements using “or,” and operations which can get very complex. 

The importance of this field of study is pretty obvious if you have 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 based on formal logic inside of it. 

That is not too bad for a system of thinking which was developed over 2000 years ago. 

The Executive Producer of Everything Everywhere Daily is Charles Daniel. 

The associate producers are Peter Bennett and Cameron Kieffer. 

Today’s review comes from listener P1BK, over on Apple Podcasts in the United States. They write: 


As a new member of the Completionist Club, I just want to say thank you, Gary! You have made every episode interesting and fun. Keep up the great work!

Thanks, P1BK! As a member of the Completionist Club, you are now entitled to all the rights and privileges of membership including the use of the Completionist Club pool. However, please remember that a lifeguard may not always be on duty. 

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