Learning Objectives:

1. Understand what mathematical logic is and its main applications.
2. Understand the difference between logic and a theory of truth.
3. Understand why a formal language is used in logic and how it allows the representation and analysis of arguments and reasoning in a precise and rigorous manner.
4. Understand the difference between natural languages and formal languages.

What is Mathematical Logic?

Mathematical logic is a branch of logic that deals with the study of the fundamental principles of mathematical reasoning and argumentation. It is used to analyze and evaluate the validity of reasoning and to develop formal methods useful for executing proofs of mathematical theorems. Mathematical logic also has applications in other areas, such as computer science and the philosophy of science, and is used as a basis for the development of formal language systems and automatic deduction.

Logic is not a theory of truth

It is always important to emphasize that logic is not a theory about truth; in the sense that it does not address the discussion that allows for a definition of truth or falsehood. Instead, assuming a priori that certain expressions have truth values, it studies how they relate to each other or how some are inferred from others.

Logic requires an adequate language

Before doing logic, it is necessary to possess an adequate language for its execution. This language, which we call “formal language,” has the necessary characteristics to perform valid inferences; that is, a mechanism that allows finding or producing true expressions from the truth of previous ones.

Why does logic need a formal language?

Logic needs a formal language because it is a language specifically designed to express arguments and reasoning in a clear and precise manner. By using a formal language, it is possible to rigorously and accurately represent the content of arguments and reasoning, which allows for the analysis and evaluation of their validity and consistency.

Formal language is a language based on strict and systematic rules and conventions for the representation of concepts and the relationships between them. By using a formal language, it is possible to more precisely and rigorously represent logical concepts and arguments, which helps avoid ambiguities and errors in reasoning. One of the purposes for which these types of languages are created is to avoid the imprecisions and paradoxes that arise from everyday language: flexibility and the expressive richness of everyday language are sacrificed to obtain the precision of a formal language.

Natural languages and formal languages

Natural languages are those languages that humans use to communicate orally or in writing. Some examples of natural languages are Spanish, English, French, Chinese, Arabic, among many others.

Natural languages are complex communication systems based on a set of rules and conventions that allow ideas, thoughts, and feelings to be expressed clearly and precisely. These languages are composed of a set of symbols (such as letters, words, and phrases) that are used to convey meaning and communicate information.

Unlike formal languages, which are specially designed to express arguments and reasoning clearly and precisely, natural languages are more flexible and adaptable and are used to communicate in various situations and contexts.

In mathematical logic, the use of formal languages is preferred over natural language mainly because the flexibility and expressive richness characteristic of natural languages, while their greatest strength in the expressive field, is also their greatest weakness in terms of precision: their level of expressive richness and lack of rigor give rise to countless paradoxes that are convenient to avoid in logic. Therefore, all the expressive power of natural languages is sacrificed in exchange for the precision of a formal language.

Paradoxes of language

Language paradoxes are logical problems that arise in language and are difficult to solve due to their internal contradiction. These paradoxes are usually statements that, if accepted as true, lead to contradictory or absurd conclusions.

The natural languages we use habitually are a powerful tool that allows us to communicate ideas, thoughts, and emotions, but they can also be misleading and difficult to interpret due to the ambiguity of some words and phrases. For example, some words have several different meanings, and it can sometimes be difficult to determine which meaning the speaker is referring to. Additionally, some phrases can have contradictory interpretations depending on the context in which they are used.

Language paradoxes are avoided through formal languages

One of the advantages of formal languages over natural languages is that they avoid language paradoxes due to their precision and lack of ambiguity. By using a formal language, it is possible to specify the rules and conventions that should be followed to avoid misinterpretations or contradictions. For example, in mathematical logic, a formal language called “propositional logic language” is used to represent and express propositions and reasoning based on propositions in a clear and precise manner. This language specifies the rules and conventions that must be followed to avoid some language paradoxes and is used to conduct logical tests and demonstrations in a rigorous and systematic way.

In addition to the propositional logic language, there are other languages designed for more complex situations that pursue the same goal, such as first-order and second-order predicate logic languages.

5 examples of language paradoxes

1. The no-lying paradox: It appears when it is said that “everything that is said is a lie.” If everything that is said is a lie, then the statement that everything that is said is a lie is a lie, and therefore it is false. If the statement that everything that is said is a lie is not a lie, then something that is said is true and therefore the statement is false. Consequently, if it is true, it is false, and vice versa.
2. The liar paradox: It emerges from the statement “I am lying,” which poses a logical contradiction if it is true or false. If it is true, then the person is lying and therefore the statement is false. If it is false, then the person is not lying and therefore the statement is true. Finally, like the previous one, if it is true, it is false, and vice versa.
3. Paradox of self-referential properties: Self-referential paradoxes arise from expressions that refer to themselves, inducing a contradiction, for example, as occurs when speaking of “the smallest number that cannot be written with fewer than twenty words.” This is in itself a paradox because the expression itself has fewer than twenty words.
4. The barber paradox: This is presented as follows: “In a town, there is a barber who shaves all the men in town who do not shave themselves. Does the barber shave himself?” At first glance, this statement seems unproblematic, but what about the barber himself? It is clear that the barber is a man (otherwise we would not talk about “the barber”), and if he can shave himself, then he cannot shave himself; on the other hand, if he cannot shave himself, then he can shave himself, and so on in circles again.
5. The paradox of the existence of the empty set: It is based on the statement that the empty set (or set that has no elements) exists, but in turn, none of the elements that make it up exist (because it has none). Therefore, we have an object that exists and is made up of nonexistent objects.

Mathematical or Symbolic Logic

Mathematical logic, also known as symbolic logic, is a branch of logic that deals with the use of symbols and mathematical notations to represent and analyze arguments and expressions. This form of logic is based on the idea that thought and reasoning are processes that can be modeled, analyzed, and studied mathematically, and that symbols and mathematical notations are useful for consistently and accurately representing and manipulating these processes.

The study of mathematical logic begins by reviewing the language that will be used to represent its elements, thus distinguishing the most common ones: propositional logic and first- and second-order predicate logics. In each of these, the mathematical reasoning techniques that allow for the rigorous demonstration of countless mathematical results and theorems are developed.

The study of symbolic logic is part of one of the fundamental pillars of mathematics.

The 4 Fundamental Pillars of Mathematics

Mathematical logic is an important part of the foundations of mathematics. These foundations are composed of the following four pillars:

1. Proof Theory: It focuses on the study of how mathematical and scientific arguments can be presented and evaluated. This theory is based on the idea that proofs must be rigorous, logical, and based on formal principles. Proof theory includes research on different types of proofs, such as inductive and deductive proofs, and on how these types of proofs can be used to solve mathematical and scientific problems. This is precisely what we do when studying mathematical logic.
2. Set Theory: It is a branch of mathematics that deals with the study of sets, which are collections of elements or objects. This theory includes research on how sets can be defined and classified, and on how operations can be performed with them. Set theory is a fundamental part of modern mathematics and has been used to develop and apply many of the basic concepts and principles of mathematics.
3. Computation Theory: Of which the fundamental parts are:
   1. Complexity Theory: It is a branch of computer science that deals with the study of the complexity of problems and algorithms. This theory includes research on how the complexity of different problems and algorithms can be measured and compared, and how more efficient algorithms can be developed and used to solve these problems.
   2. Computability Theory: It is the part of computer science that deals with the study of which problems and functions can be solved or evaluated by a computer and which cannot. This theory includes research on how computable problems and functions can be defined and classified, and how they can be developed and used.
4. Model Theory: In logic and mathematics, it is the study of the relationships between formal theories (statements written in a formal language, used to establish claims about some mathematical structure) and their models (which are maintained under those structures). Such mathematical structures can be groups, fields, graphs, etc. Model theory allows for a semantic interpretation of purely formal expressions and also allows the study of completeness, consistency, and independence problems among statements.

It is extremely difficult to study each of these pillars in depth without touching on some aspects of the others. The study of these pillars is usually intertwined. When we ask ourselves what mathematical logic is, it is normal to end up answering that question with a combination of studies that move between these four pillars.