We explain what this logic that is applied in philosophy, mathematics and science consists of and what its function is. Plus, examples of truth tables and more.
What is formal logic?
Formal logic or mathematical logic is the logic used in philosophy and applied to mathematics and science.. Its task is to “translate” natural language into logical language. For example, if the natural language sentence “It is raining and the street is wet” is taken, formal logic has to translate the information into logical language (in this case: “p ∧ q“).
To represent natural language in logical language, formal logic uses logical symbols. There are different formal systems of representation. The most used are the propositional logic system, the first order logic system and the modal logic system. Each of these systems operates on different elements (propositions, predicates and truth values) and variables (propositional, quantifiers and individual).
Formal logic encompasses four fundamental areas that explore different aspects of mathematics and logic:
- Model theory. It is the analysis of structures. This discipline investigates the study of axiomatic theories and mathematical logic through the analysis of complex mathematical structures, such as sets, networks or dynamic systems. This makes it possible to attribute semantic content to the formal constructions of logic and understand their meaning in a broader mathematical context.
- Demonstration theory. It is the investigation of validation methods. This area seeks to develop methods and techniques for the validation of logical problems and the demonstration of theorems through the use of advanced mathematical tools. These methods are essential to guarantee the solidity and reliability of logical and mathematical reasoning.
- Set theory. It is the exploration of sets and relationships. This branch of logic focuses on the study of sets and their interrelationships, as well as the associated basic properties and operations. Rigorous set analysis provides a solid foundation for the development of mathematical theories and is essential in multiple areas of mathematics and logic.
- Computability theory. It is the connection between mathematics and computer science. This area of research examines the relationship between mathematics and computer science, especially with regard to decision problems and computability. It studies how algorithms and calculating machines can face and solve complex logical and mathematical problems, which implies the understanding of computable and non-computable sets.
These four areas of formal logic open a vast field of research and application in which various aspects of mathematics, logic and their connection with computer science are explored.
Key points
- Formal logic is used in philosophy, mathematics, and science.
- Your task is to translate natural language into logical language.
- The four areas that operate according to formal logic are model theory, proof theory, set theory and computability theory.
Propositional logic
Propositional logic works with logical propositions. Each logical proposition is symbolized by a lowercase letter and has a truth value (true or false). There are simple and compound propositions.
Examples of logical propositions:
- p = Aristotle is a man.
- q = 4 is an even number.
- r = 5 is the sum of 3 and 2.
- yes = The rain comes out of the earth.
- t = Juan is Mexican and María is Spanish.
In the examples provided, p, q, r and yes are cases of simple propositions, while t It is a compound proposition. Furthermore, the first three can be said to be true propositions, while yes It is a false proposition.
logical connectives
Compound propositions are made up of more than one simple proposition, brought together through logical connectives. In the case of tthe union of two simple propositions is shown: “Juan is Mexican” and “María is Spanish”, joined by the logical connective “and”, which is called “conjunction”.
There are other logical connectives, such as negation, disjunction, material condition and biconditional. All of them are symbolized by their own sign:
- Denial (no): ¬
- Logical conjunction (and): ∧
- Logical disjunction (o): ∨
It has two different meanings:
Inclusive. For the proposition to be true, one or all of the elements of the premise must be true.
Exclusive. On one premise, “p either q” is true, but it cannot be the case that both can be true simultaneously.
- Material conditional (If…then): →
- Biconditional (if and only if): ↔
Logical connectives are used in the formal language of propositional logic. They fulfill the function of denying a proposition or relating two or more propositions.
Below is an example of a sentence from natural language translated into the language of propositional logic, in which a conjunction occurs:
- It is raining and the street is wet.
- p ∧ q
truth tables
In addition to logical connectives, propositional logic operates according to truth tables. If each premise acquires a truth value (true or false), when two or more premises are put in relation, the truth value that each one has assigned will vary the truth value that must be assigned to the relationship established between the two.
Thus, the meaning of each logical connective can be shown through the truth tables that represent them: negation, conjunction, disjunction, conditional, biconditional and exclusive disjunction.
Examples of propositional logic
- Juan went to the river and María too.
p: Juan went to the river
q:Mary went to the river
p∧q - Juan went to the river or María went to the river.
p: Juan went to the river
q:Mary went to the river
p∨q - John went to the river only if Mary went to the river.
p: Juan went to the river
q:Mary went to the river
p↔q - If Juan went to the river, Mary went to the river.
p: Juan went to the river
q:Mary went to the river
p→q
First order logic
First-order logic or predicate logic works with first-order languages. First-order languages are formal languages that operate on predicates that operate on arguments that are only constants or variables of individuals, and quantifiers that reach to variables of individuals.
Predicates
A predicate is a linguistic expression that, connected to another expression, forms a sentence. For example, when it is stated “The water is wet”, the expression “moja” is a predicate that is connected to the expression “The water”. It may be the case that a predicate connects to two or more expressions, as in “Water is wetter than earth.”
When a predicate is connected to a single expression, it is said to express a property (in this case, the property of wet). When connected to two or more, it is said to express a relationship (in the second case, the relationship of wet more than).
In first-order logic or predicate logic, predicates are treated as functions. The functions take expressions like “water” or “earth” and transform them as follows:
- “Water is wet”
Wet (Water)
M(a) - “Water is wetter than land”
Wet (water, earth)
M(a,t)
Constants and individual variables
Individual constants are linguistic expressions that refer to an entity. For example, “water” or “earth” are constants of the individual. On the other hand, individual variables do not have a specific reference, but can be assigned to any entity. This is exemplified by saying “this is wet,” and it is written:
- M(x)
Since it is not clear what is meant by the expression “this”. Furthermore, variables, like constants, serve to formalize sentences. For example, if we say “This is wetter than that,” and we assign to “this” the x and “that” the andwe can symbolize like this:
- M(x,and)
Quantifiers
Quantifiers are operators that act on a set of individuals. It is a resource that allows constructing propositions that affirm that a condition is met for a certain number of individuals. The most used quantifiers are the universal quantifier and the existential quantifier.
- Universal quantifier. Asserts that a condition is met for all the individuals being talked about.
The expression “for everything” x” is a universal quantifier that is written as follows: ∀x.
When used in a predicate, for example, “for everything x, x is wet”, we write: ∀x M(x). Note that in this case the “M” is the variable that represents the condition of being wet. - Existential quantifier. Asserts that a condition is met for at least one of the individuals being talked about.
The expression “there exists at least one x” is an existential quantifier that is written as follows: ∃x.
When used in a predicate, for example, “there exists at least one xsuch that x It is wet”, it is written: ∃x M(x). Note that in this case the “M” is the variable that represents the condition of being wet.
Modal logic
Modal logic works on the truth value of different propositions and predicates. It is the logic that is responsible for studying the ways in which different propositions and predicates can be true or false.
This happens because, although it is true that predicates and propositions are either true or false, it is also true that they are not always true in the same way. When we talk about truth and falsehood, we talk about need, possibility, contingency and impossibility.
Each of the modalities expresses a way in which a truth or falsehood is manifested. For example, if it is stated that “Socrates died in hand-to-hand combat,” the statement is false but possible. On the other hand, by saying that “2+2=5”, the falsity of the proposition is necessary since 2 +2 can never be equal to 5.
References
- Gamut, LTF, & Durán, C. (2002). Introduction to logic. Buenos Aires, Argentina: Eudeba.
- Obiols, G. (1997). New Logic and Philosophy course. Kapelusz.
- Díez, JA (2002). Initiation to Logic. Ariel.
