Mathematical Logic: Origin, What It Studies, Types

The mathematical logic or symbolic logic is a mathematical language that covers the tools through which one can affirm or deny a mathematical reasoning.

It is well known that there are no ambiguities in mathematics. Given a mathematical argument, it is either valid or it simply is not. It cannot be false and true at the same time.

A particular aspect of mathematics is that it has a formal and rigorous language by which the validity of an argument can be determined. What is it that makes a certain reasoning or any mathematical proof irrefutable? That’s what mathematical logic is all about.

Thus, logic is the discipline of mathematics that is responsible for studying mathematical reasoning and proofs, and providing the tools to be able to infer a correct conclusion from previous statements or propositions.

To do this, use is made of axioms and other mathematical aspects that will be developed later.

Origin and history

The exact dates with respect to many aspects of mathematical logic are uncertain. However, most of the bibliographies on the subject trace its origin to ancient Greece.

Aristotle

The beginning of the rigorous treatment of logic is attributed, in part, to Aristotle, who wrote a set of works of logic, which were later collected and developed by different philosophers and scientists, until the Middle Ages . This could be considered “the old logic”.

Later, in what is known as the Contemporary Age, Leibniz, moved by a deep desire to establish a universal language to reason mathematically, and other mathematicians such as Gottlob Frege and Giuseppe Peano, notably influenced the development of mathematical logic with great contributions , among them, the Peano Axioms, which formulate indispensable properties of natural numbers.

Mathematicians George Boole and Georg Cantor were also of great influence at this time, with important contributions in set theory and truth tables, highlighting, among other aspects, Boolean Algebra (by George Boole) and the Axiom of Choice (by George Cantor).

There is also Augustus De Morgan with the well-known Morgan laws, which contemplate negations, conjunctions, disjunctions and conditionals between propositions, keys to the development of Symbolic Logic, and Jhon Venn with the famous Venn diagrams.

In the 20th century, approximately between 1910 and 1913, Bertrand Russell and Alfred North Whitehead stand out with their publication of Principia mathematica , a set of books that collects, develops and postulates a series of axioms and results of logic.

What does mathematical logic study?

Propositions

Mathematical logic begins with the study of propositions. A proposition is a statement that can be said without any ambiguity if it is true or not. The following are examples of propositions:

  • 2 + 4 = 6.
  • 5 2 = 35.
  • In 1930 there was an earthquake in Europe.

The first is a true statement and the second is a false statement. The third, even though the person reading it may not know if it is true or immediately, is a statement that can be tested and determined whether or not it really happened.

The following are examples of expressions that are not propositions:

  • She is blonde.
  • 2x = 6.
  • Let’s play!
  • Do you like movies

In the first proposition, it is not specified who “she” is, therefore nothing can be affirmed. In the second proposition, what “x” represents has not been specified. If instead it were said that 2x = 6 for some natural number x, in this case it would correspond to a proposition, in fact true, since for x = 3 it is fulfilled.

The last two statements do not correspond to a proposition, since there is no way to deny or affirm them.

Two or more propositions can be combined (or connected) using well-known logical connectives (or connectors). These are:

  • Denial: “It is not raining.”
  • Disjunction: “Luisa bought a white or gray bag.”
  • Conjunction: “4 2 = 16 and 2 × 5 = 10″.
  • Conditional: “If it rains, then I’m not going to the gym this afternoon.”
  • Biconditional: “I go to the gym this afternoon if, and only if, it doesn’t rain.”

A proposition that does not have any of the previous connectives is called a simple (or atomic) proposition. For example, “2 is less than 4” is a simple proposition. The propositions that have some connective are called compound propositions, such as “1 + 3 = 4 and 4 is an even number.”

Statements made by means of propositions are usually long, so it is tedious to always write them as seen so far. For this reason, a symbolic language is used. Propositions are usually represented by capital letters such as P, Q, R, S , etc. And the symbolic connectives as follows:

So that

The converse of a conditional proposition

Original text


is the proposition

And the counter-reciprocal (or contrapositive) of a proposition

is the proposition

Truth tables

Another important concept in logic is that of truth tables. The truth values ​​of a proposition are the two possibilities for a proposition: true (which will be denoted by V and it will be said that its truth value is V) or false (which will be denoted by F and it will be said that its value really is F).

The truth value of a compound proposition depends exclusively on the truth values ​​of the simple propositions that appear in it.

To work more generally, we will not consider specific propositions, but propositional variables p, q, r, s , etc., which will represent any propositions.

With these variables and the logical connectives, the well-known propositional formulas are formed, just as compound propositions are built.

If each of the variables that appear in a propositional formula is replaced by a proposition, a compound proposition is obtained.

Below are the truth tables for logical connectives:

There are propositional formulas that receive only the value V in their truth table, that is, the last column of their truth table only has the value V. These types of formulas are known as tautologies. For example:

The following is the truth table of the formula

A formula α is said to logically imply another formula β, if α is true each time β is true. That is to say, in the truth table of α and β, the rows where α has a V, β also has a V. We are only interested in the rows in which α has the value V. The notation for the logical implication is the following :

The following table summarizes the properties of logical implication:

Two propositional formulas are said to be logically equivalent if their truth tables are identical. The following notation is used to express logical equivalence:

The following tables summarize the properties of logical equivalence:

Types of mathematical logic

There are different types of logic, especially if one takes into account the pragmatic or informal logic that points to philosophy, among other areas.

As far as mathematics is concerned, the types of logic could be summarized as:

  • Formal or Aristotelian logic (ancient logic).
  • Propositional logic: it is responsible for the study of everything related to the validity of arguments and propositions using formal and symbolic language.
  • Symbolic logic: focused on the study of sets and their properties, also with a formal and symbolic language, and is deeply linked to propositional logic.
  • Combinatorial logic: one of the most recently developed, involves results that can be developed using algorithms.
  • Logical programming: used in the various packages and programming languages.

Areas

Among the areas that make use of mathematical logic in an indispensable way in the development of their reasoning and arguments, stand out philosophy, set theory, number theory, constructive algebraic mathematics and programming languages.

References

  1. Aylwin, CU (2011). Logic, Sets and Numbers. Mérida – Venezuela: Publications Council, Universidad de Los Andes.
  2. Barrantes, H., Díaz, P., Murillo, M., & Soto, A. (1998). Introduction to Number Theory. EUNED.
  3. Castañeda, S. (2016). Basic course of number theory. Northern University.
  4. Cofré, A., & Tapia, L. (1995). How to Develop Mathematical Logical Reasoning. University Editorial.
  5. Zaragoza, AC (sf). Number theory Editorial Vision Libros.

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