# What is a Proposition in philosophy, logic and mathematics

## What is a proposition?

A proposition, broadly speaking, is something that is proposed. That is, it is an equivalent expression of a simple assertive sentence, a sentence in which it is stated that something is, that something exists or that it has a certain characteristic. Therefore, it can be judged as true (if it agrees with reality) or false (if it does not).

It is a term widely used in different contexts of knowledge, such as certain formal disciplines (logic, mathematics) or linguistics and philosophy. The idea is that, taking different propositions as background, certain conclusions can be obtained, and the procedure through which we have obtained them can also be carefully studied.

In any case, a proposition must be understood as a chain of signs that belong to the same language, whether they are sounds or characters (in a natural language) or signs and representations (in a formal language).

While, in colloquial language, a proposition is understood as a proposal: an invitation that we make to another or others and that can be accepted or rejected.

Finally, we should not confuse a proposition with a preposition. The latter is just a grammatical category, that is, a type of words, which have a more or less obvious grammatical meaning, and which serve to establish relationships between things. Examples of prepositions are: of, for, against, between, by, on, under, in, etc.

## Proposition in philosophy

Within the field of philosophical debate, a proposition is spoken of to refer to a mental act through which a judgment regarding reality is expressed in a specific language, allowing the establishment of a relationship of some type between a specific subject and predicate.

In this sense, the proposition should not be confused with the sentence through which it is expressed, since the same judgment can be expressed through different sentences, as in:

• Ana is a woman.
• Ana is not a man.

## Proposition in logic

Logic studies the relationships between propositions and the reasoning mechanisms that allow us to arrive at some from others. In themselves, propositions differ from judgments, since the former propose something about reality and the latter affirm or deny something about it. That is, propositions are the logical product of judgments.

Formal logic represents propositions through letters of the alphabet, in order to study the logical connections between them abstracted from their semantic content: “if p then q ”.

From this relationship, it can then be determined in which cases the expressed content is true, and in which cases it is false, through the so-called “truth tables”, which assign true (V) or false (F) values to the content. established relationship, to study its possible results.

## Simple and compound propositions

Logic classifies propositions into two types: simple and compound, depending on their formation.

• Simple propositions. They are those that are composed of a subject and a predicate directly related, without factors of negation (no), conjunction (and), disjunction (or) or implication (if… then) appearing. In sentence terms, they correspond to simple sentences without subordinate clauses. For example: “The dog is black.”
• Composite propositions. They are those of a complex type, which incorporate additional elements through factors of negation, conjunction, disjunction or implication, and which in sentence terms consist of sentences with subordinate clauses and other components. For example: “If the dog is black, the dog is not blue or red.”

## Proposition in mathematics

Since mathematics is a formal language very close to logic, its approach to propositions is not too different, with the exception that it uses numbers, variables and mathematical signs to express the relationship and connections between the terms of a proposition, or one with others. Thus, mathematical propositions also affirm or deny something, establishing a connection that can be judged as true or false.

For example, the expression 4 + 5 = 7 affirms a formal relationship between said quantities, which in this case can be considered false, since its resolution indicates that 4 + 5 = 9. However, despite being false, it can be stated , that is, it can be proposed.

Mathematical propositions can be made more complex with the incorporation of variables, such as equations, expressing relations of possibility and variation. For example, in the expression x = 3y + z the meanings of true or false will depend on the values that we assign to the variables, although their proportion and meaning will remain the same regardless.

#### References

• “Proposition” on Wikipedia.
• “Proposition” in the Dictionary of the language of the Royal Spanish Academy.
• “What is a proposition” at the Technological University of Panama.
• “Proposition” in the Cervantes Virtual Center.
• “Types of propositions” at the National Autonomous University of Mexico (UNAM).