Proposition

This is the dedicated crate for proposition building logic. It provides the mathematical foundations for constructing logical languages through formal grammars, rule systems, and semantic interpretation mechanisms.

Purpose

The proposition crate is specifically designed to handle the mathematical construction of logical systems by implementing:

• Formal Language Theory: Building blocks for defining logical languages through alphabets, strings, and grammars
• Grammar Hierarchies: Implementation of Chomsky hierarchy concepts (regular, context-free, context-sensitive, and recursively enumerable grammars)
• Rule Systems: Mechanisms to stack and compose logical rules to build increasingly complex language structures
• Mathematical Lexers: Tokenization of logical expressions according to formal syntax rules
• Semantic Interpretation: Binding meaning to syntactic structures in logical systems
• Proof Theory: Foundations for establishing validity and deriving conclusions within logical frameworks

This crate enables the construction of logical languages in a systematic way, similar to how numerical systems are built upon foundational axioms and rules, allowing for the stacking of increasingly complex rule systems to create sophisticated logical frameworks.

This crate works in conjunction with:

• logic_tracer (main crate with macros)
• digital (digital logic operations)

Mathematical Foundations

To build the logical systems in this crate, a solid understanding of the following mathematical and logical concepts is required. This list provides a roadmap for studying the necessary topics:

1. Foundational Logic

• Propositional Logic and Boolean Algebra: The groundwork for all digital logic. This is where you'll learn about basic logical operators (AND, OR, NOT), truth tables, and how to manipulate logical expressions.
• Predicate Logic and Quantification: An extension of propositional logic that introduces quantifiers (for all, there exists). This is essential for expressing more complex logical statements.

2. Formal Language and Grammar

• Formal Grammar Theory (Chomsky Hierarchy): This is the study of how to formally describe a language. You'll learn about different types of grammars (regular, context-free, etc.) and how they relate to the complexity of the language.
• Automata Theory and Recognizers: This is the study of abstract machines that can recognize formal languages. You'll learn about finite automata, pushdown automata, and Turing machines, which are the theoretical models for computers.

3. Advanced Topics

• Lambda Calculus and Combinatory Logic: These are formal systems for expressing computation based on function abstraction and application. They are the theoretical foundation for functional programming languages.
• Proof Theory and Natural Deduction: This is the study of how to construct formal proofs. You'll learn about different proof systems and how to use them to derive conclusions from a set of premises.
• Model Theory and Semantic Interpretation: This is the study of the relationship between formal languages and their interpretations. You'll learn how to assign meaning to the symbols in a formal language.
• Type Theory and Logical Frameworks: This is the study of type systems, which are used to prevent errors in programs and to express mathematical concepts in a formal way.
• Category Theory Applications: This is a branch of mathematics that studies abstract structures and their relationships. It has applications in many areas of computer science, including functional programming and type theory.
• Modal and Temporal Logics: These are extensions of classical logic that allow you to reason about possibility and necessity (modal logic) and time (temporal logic).
• Combinatory Logic and Permutation Principles: This is the study of how to combine and rearrange objects. It has applications in many areas of computer science, including cryptography and coding theory.
• Recursive Function Theory: This is the study of computable functions. You'll learn about different models of computation and how to prove that certain functions are computable.
• Decidability and Complexity Theory: This is the study of which problems can be solved by a computer and how much time and memory are required to solve them.