Foundations of Language: Ordinary, Formal, and Logical Systems

Introduction: The Need for Language

Humans require language to gather information necessary for survival and communication. Communication enhances knowledge of life science and human culture. Language arises to avoid gaps in communication.

Ordinary Language

Ordinary language refers to the languages commonly used by humans to communicate with each other.

Functions of Ordinary Language

• Representative: To affirm or deny a predicate of a subject.
• Expressive: To express one's own attitudes, desires, and emotions.
• Appellative: To provoke actions in the receivers.
• Performative: To perform a linguistic act and an extra-linguistic one simultaneously.
• Metalinguistic: To speak about the language itself.

Problems with Ordinary Language

• Equivocal terms: Terms that can be attributed to different things.
• Analogous terms: Terms that designate things that are partly the same and partly different.

Formal Language

Definition and Calculation

A formal language is a set of artificial signs, empty of content, that interrelate through a defined set of rules and regulations. Calculation within this context refers to operations performed on uninterpreted formal symbols.

Propositional Elements

Elementary signs include lowercase letters that represent simple declarative sentences and operations.

Types of Signs

• Variables: Combine to form expressions within the calculation system.
• Constants: Indicate how the variables are related.

Formation Rules

These rules determine whether an expression is meaningful (well-formed) or meaningless (ill-formed) within that specific language.

Transformation Rules

These rules allow for the correct derivation of well-formed expressions from other well-formed expressions.

Axioms and Theorems

• Axiom: An expression initially admitted from which other expressions are derived.
• Theorem: An expression obtained by applying transformation rules to axioms or previously derived theorems.

Logical Language

Definition of Logic

Logic is the science that studies the formal validity of arguments.

Types of Logic

• Classical Logic: Associated with figures like Aristotle (dominant until the 19th century).
• Formal Logic (Modern Logic): Developed significantly in the 19th-20th centuries by authors such as Russell.

Categorical Syllogism

Definition

A reasoning structure consisting of two premises and one conclusion.

Terms

• Major Term: The predicate of the conclusion.
• Minor Term: The subject of the conclusion.
• Middle Term: Appears in both premises but not in the conclusion.

Premises

• Major Premise: Contains the major term and is typically stated first.
• Minor Premise: Contains the minor term and is typically stated second.

Conclusion

The proposition that is derived from the premises.

Figure

The structure resulting from the arrangement of the middle term in the premises.

Mood

The type of propositions (classified by quantity and quality - e.g., universal affirmative, particular negative) that make up the syllogism.

Natural Deduction

Definition

A method to derive a conclusion from a set of premises by applying established logical laws and inference rules.

Propositional Variables

Lowercase letters representing simple, affirmative propositions.

Propositional Constants (Connectives)

Signs used to relate two or more simple propositions, forming composite propositions (e.g., and, or, if...then).

Logical Laws

Inference schemes that are always valid, regardless of the content of the propositions involved.