Formal logic is the study and analysis of good reasoning. In essence, formal logic is a method by which a standardized approach can be used to decide whether or not a given statement is logical or not given a set of basic facts to work from. Formal logic is considered to be a subfield of philosophy, though it has applications in mathematics, computer science, and statistics, among other fields. In providing an overview of this topic, I am going to try to avoid the army of symbols that are often used in formal logic in order to avoid some of the nuances of the English language; instead, I will merely try to keep my words as sharp and concise as possible.
The most fundamental underlying idea behind formal logic is propositional logic. Propositional logic is a method of applying inference rules for transforming expressions containing simple statements joined together by connectives. In essence, this means that propositional logic seeks to ensure that combining together sentences makes for a resulting idea that is still true and correct. Here are a couple of examples of propositional logic.
Consider the following two statements:
I ate the pizza or Bill ate the pizza.
From this, we can easily deduce the following:
I did not eat the pizza.
Bill ate the pizza.
Now, forget about Bill and his pizza. Consider these two statements:
I ate the pizza or Sarah ate the pizza or John ate the pizza.
From this, we cannot deduce who ate the pizza. We are left with only the fact that Sarah or John ate the pizza.
I did not eat the pizza.
Obviously, purely propositional logic is very limited. We have to build upon this with another level of logic, this entitled first order predicate logic. First order predicate logic adds the notions of functions, predicates, variables, and quantifiers to the toolbox of a logician. In essence, predicate logic allows us to treat the different parts of a proposition separately. We are no longer forced to merely draw complete pieces from a statement, such as the items linked in the above example by OR statements. Instead, we can divide the statements up and manipulate them a bit. Here are two examples of first order predicate logic.
Consider the following statements:
All men are mortal.
From this, we can devise a function that states if a man is given, then what is given is mortal. I am given to the function, and thus since I am a man, the function states that I am indeed mortal. Thus,
I am a man.
I am mortal.
Now, forget about me and my mortality. Consider these two statements:
All balls are round.
From this, we can devise a function that states if a ball is given, then what is given is round. I am given to the function. Since I am not a ball (the statement says that I am round, not that I am a ball), the function can state nothing about me.
I am round.
Formal logic builds upon itself as such with many, many layers. Many actual statements of formal logic use a variety of symbols similar to those used in set theory and other mathematical and statistical realms. It also deals with the development of logical calculus, which can be used to derive new truths, as well as development of purely logical languages.
From a cultural perspective, formal logic has much in common with the underlying ideas behind Buddhism. One has a set of absolute truths, all truths beyond them are built up from the underlying absolute truths. Thus, in a way, formal logic has had a great deal of impact on worldwide cultural and religious growth as well.
Formal logic is an undeniably broad and complex field. What is discussed here only scratches the surface to provide a broad overview of the topic. Additional subtopics of interest would include first-order languages, logical calculus, tree method for logical decomposition, and formal proofs.