Logical System

Definition

A logical system is a formal framework for reasoning that typically consists of three main components:

1. A formal language specifying the syntax of well-formed formulas
2. A proof system defined inductively over formulas
3. A semantics providing meaning through models, interpretations, or valuation systems

Components

Syntax of Logic

The syntactic component defines:

• An alphabet of symbols (propositional variables, connectives, quantifiers)
• Formation rules for constructing well-formed formulas
• Structural properties of the language (decidability of well-formedness)

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Proof Theory

The deductive component specifies:

• Axioms or axiom schemas
• Rules of inference (modus ponens, universal generalization, etc.)
• Derivation procedures for establishing theorems
• The consequence relation $⊢$ between premises and conclusions

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Semantics of Logic

The semantic component provides:

• Models or interpretations that assign meaning to formulas
• Truth conditions or satisfaction relations
• The semantic entailment relation $\models$
• Validity and satisfiability concepts

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Examples

Classical Logic

• Language: Terms, predicates and connectives $\{\neg ,\wedge ,\vee ,\to ,\leftrightarrow ,\forall ,\exists \}$
• Proof System: Natural deduction, sequent calculus, or axiomatic systems
• Semantics: Truth tables for propositional logic, Tarski semantics for First-Order Logic

Intuitionistic Logic

• Language: The same as in classical logic
• Proof System: Natural deduction without Law of the Excluded Middle
• Semantics: Kripke semantics, topological semantics, or Beth Trees

Modal Logic

• Language: Classical connectives plus modal operators $\square$ (necessity) and $◊$ (possibility)
• Proof System: Extensions of classical systems with modal rules
• Semantics: Kripke models with accessibility relations

Type Theory

• Language: Terms with types, dependent products $\Pi x:A.B(x)$, dependent sums $\Sigma x:A.B(x)$
• Proof System: Type derivation rules, term formation rules following the Curry-Howard correspondence
• Semantics: Propositions as Types where proofs are programs and types are propositions
• Connection to Intuitionistic Logic: Provides computational interpretation of intuitionistic proofs through the BHK interpretation

Key Properties

Soundness

A logical system is sound if every provable formula is semantically valid: $\text{If }\Gamma ⊢\varphi \text{ then }\Gamma \models \varphi$

Completeness

A logical system is complete if every semantically valid formula is provable (Semantic Completeness): $\text{If }\Gamma \models \varphi \text{ then }\Gamma ⊢\varphi$

Consistency

A logical system is consistent if no contradiction is derivable from the axioms.

decidability

A logical system is decidable if there exists an algorithm to determine whether any given formula is a theorem.

Types of Logical Systems

Classical Systems

• Satisfy the law of excluded middle
• Support classical reasoning principles
• Complete with respect to two-valued semantics

Non-Classical Systems

• intuitionistic logic: Constructive reasoning, rejection of excluded middle
• Relevance Logic: Requires relevant connection between premises and conclusion
• Paraconsistent: Tolerates certain contradictions without explosion
• Many-Valued: Allow truth values beyond true and false

Applied Systems

• Temporal Logic: Reasoning about time-dependent propositions
• Epistemic Logic: Reasoning about knowledge and belief
• Deontic Logic: Reasoning about obligation and permission

Relationship to Completeness Concepts

Different completeness notions apply to logical systems:

• Semantic Completeness: Captures all valid inferences
• Syntactic Completeness: Decides every formula
• Functional Completeness: Expressive adequacy of connectives

Applications

Mathematics

• Peano Arithmetic for number theory
• Zermelo-Fraenkel Set Theory for foundations
• Type Theory for constructive mathematics and intuitionistic foundations
• Martin-Löf Type Theory as foundation for constructive mathematics

Computer Science

• Program Verification: Hoare logic, separation logic
• Type Systems: Dependent types, Linear types, session types
• Proof Assistants: Coq, Agda, Lean implementing type-theoretic foundations
• Database Theory: First-order logic with equality
• Artificial Intelligence: Description logics, non-monotonic reasoning

Philosophy

• Formalization of philosophical arguments
• Analysis of logical paradoxes
• Study of meaning and truth

Metatheory

The study of logical systems themselves involves:

• Proof Theory: Structural analysis of proofs and derivations
• Model Theory: Study of interpretations and semantic structures
• Computation: Algorithmic aspects of logical systems
• Complexity Theory: Resource bounds for logical decision problems

Historical Development

Key developments in logical systems:

• Aristotelian Logic: Syllogistic reasoning
• Frege’s System: First-order predicate logic
• Hilbert’s Program: Formalization of mathematics
• Gödel’s Theorems: Limitations of formal systems
• Intuitionistic Logic: Brouwer’s constructive approach, leading to rejection of excluded middle
• Curry-Howard correspondence: Discovery that proofs in intuitionistic logic correspond to programs in typed lambda calculus
• Martin-Löf Type Theory: Unification of logic and programming through dependent types
• Modern Type Theory: Extensions with inductive types, homotopy type theory, and computer-verified mathematics

Logical systems continue to evolve with applications in computer science, artificial intelligence, and foundations of mathematics, balancing expressiveness with computational tractability.