Mathematical Logic Fundamentals

Overview

Mathematical logic is a subfield of mathematics that deals with the use of formal systems to study the principles of valid inference and reasoning.

Key Concepts

Propositional Logic

• Deals with statements that can be either true or false
• Uses logical operators:
  • ¬ (not)
  • ∧ (and)
  • ∨ (or)
  • → (implies)
  • (if and only if)
• Truth tables are used to evaluate the truth value of statements

Predicate Logic

• Deals with statements that contain variables and predicates (properties or relations)
• Uses quantifiers:
  • ∀ (for all)
  • ∃ (there exists)
• Examples:
  • ∀x (P(x) → Q(x)) (for all x, if P(x) then Q(x))
  • ∃x (P(x) ∧ Q(x)) (there exists x such that P(x) and Q(x))

Formal Systems

• A formal system consists of:
  • A language (set of symbols and rules for forming expressions)
  • A set of axioms (self-evident truths)
  • Rules of inference (used to derive theorems from axioms)
• Examples of formal systems:
  • Propositional logic
  • Predicate logic
  • Set theory
  • Peano arithmetic

Model Theory

• Deals with the relationship between formal systems and their interpretations
• A model is an interpretation of a formal system that assigns meaning to the symbols and satisfies the axioms
• Examples:
  • A model of Peano arithmetic is the set of natural numbers with the usual operations and relations
  • A model of set theory is a collection of sets with the usual operations and relations

Proof Theory

• Deals with the study of proofs and their properties
• A proof is a sequence of formulas that follows the rules of inference and leads to a theorem
• Examples:
  • A proof of a theorem in propositional logic might involve a series of applications of the rules of inference
  • A proof of a theorem in set theory might involve a series of applications of the axioms and rules of inference

Applications

• Computer science: mathematical logic is used in the design of programming languages, artificial intelligence, and software verification
• Philosophy: mathematical logic is used to study the foundations of mathematics and the nature of truth and reasoning
• Mathematics: mathematical logic is used to study the foundations of mathematics and the properties of formal systems

Overview

• Mathematical logic is a subfield of mathematics that deals with the use of formal systems to study the principles of valid inference and reasoning.

Key Concepts

Propositional Logic

• Deals with statements that can be either true or false.
• Uses logical operators:
  • ¬ (not)
  • ∧ (and)
  • ∨ (or)
  • → (implies)
  • (if and only if)
• Truth tables are used to evaluate the truth value of statements.

Predicate Logic

• Deals with statements that contain variables and predicates (properties or relations).
• Uses quantifiers:
  • ∀ (for all)
  • ∃ (there exists)
• Examples:
  • ∀x (P(x) → Q(x)) (for all x, if P(x) then Q(x))
  • ∃x (P(x) ∧ Q(x)) (there exists x such that P(x) and Q(x))

Formal Systems

• A formal system consists of:
  • A language (set of symbols and rules for forming expressions)
  • A set of axioms (self-evident truths)
  • Rules of inference (used to derive theorems from axioms)
• Examples of formal systems:
  • Propositional logic
  • Predicate logic
  • Set theory
  • Peano arithmetic

Model Theory

• Deals with the relationship between formal systems and their interpretations.
• A model is an interpretation of a formal system that assigns meaning to the symbols and satisfies the axioms.
• Examples:
  • A model of Peano arithmetic is the set of natural numbers with the usual operations and relations
  • A model of set theory is a collection of sets with the usual operations and relations

Proof Theory

• Deals with the study of proofs and their properties.
• A proof is a sequence of formulas that follows the rules of inference and leads to a theorem.
• Examples:
  • A proof of a theorem in propositional logic might involve a series of applications of the rules of inference
  • A proof of a theorem in set theory might involve a series of applications of the axioms and rules of inference

Applications

• Computer science: mathematical logic is used in the design of programming languages, artificial intelligence, and software verification.
• Philosophy: mathematical logic is used to study the foundations of mathematics and the nature of truth and reasoning.
• Mathematics: mathematical logic is used to study the foundations of mathematics and the properties of formal systems.