Semiotics, Language Structure, and the Foundations of Logic

Understanding Signs and Semiotics

The signs are defined as everything that represents or evokes something else in any aspect for someone. For something to be considered a sign, it is necessary to have meaning and that there is a receiving agency that can interpret the sign.

Types of Signs

• Vestiges (Indices): The relation of a trace with its meaning is natural. Example: Smoke indicates fire.
• Pictures (Icons): Maintains a relationship of similarity with the meaning. Example: A photograph or drawing.
• Symbols: These signs have an arbitrary relationship with their meaning. Example: Traffic lights or letters of the alphabet.

Language: Natural and Artificial Systems

Language is a social phenomenon based on the capacity that some animal species have to communicate through symbols. This capability is more advanced in humans, allowing them to even refer to themselves.

Natural Language

We understand natural language as the language used by a linguistic community, which is a language we learn.

• Definition: Languages are natural language elements that consist of a finite set of symbols or language signs, and a certain number of morphosyntactic rules to form sentences.
• Usage: The uses of natural language are practically limitless.

Sentences and Statements

A sentence is a linguistic expression that is grammatically correct and makes sense.

• Sentences can be true or false, but only in the case of statements.
• Statements: Statements are language segments that convey complete sense. We call declarative sentences those sentences that are susceptible to always being true or false.

Artificial Language

Artificial languages are constructed to overcome the shortcomings of natural language and to equip science with rigor and accuracy. These languages have a well-defined and efficient operating structure.

1. Syntactic Rules: These are a set of effective rules for the formation of sentences, allowing one to know at any time if an expression in the artificial language is well-formed. This set of transformation rules allows moving from some expressions to other chains to build deductive, rigorous, and accurate systems.
2. Signs: Signs used in artificial languages are well defined.

Formal Language

Formal language is an artificial language that uses a formal symbol table and syntactic rules that are operational and efficient for calculation.

• A formal symbol table is the set of variables and signs used by the formal language. It is established by agreement and contains as many symbols as are necessary to operate within that artificial language.

Constants and Variables in Formal Systems

Constant

These are signs with a fixed meaning that are used to bind to each variable symbol.

Variable

These signs have no fixed meaning and therefore can receive unlimited content.

Logic and Reasoning

Logic is defined as the formal science responsible for establishing the conditions for valid formal reasoning.

1. Reasoning: Reasoning is a mental process characterized by a step from one or more statements (called premises) to a subsequent statement (called the conclusion), which necessarily derives from the premises.