1.1 Foundations of mathematical logic

Mathematical logic provides the foundation for formal reasoning in mathematics. It establishes principles like consistency and completeness, which are crucial for building reliable systems of thought. These concepts form the bedrock of mathematical reasoning.

Syntax and semantics are two key aspects of logic. Syntax deals with the structure and rules of logical systems, while semantics focuses on meaning and interpretation. Understanding both is essential for grasping how mathematical statements are constructed and evaluated.

Foundations of Mathematical Logic

Principles of mathematical logic

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• Mathematical logic studies formal systems of reasoning
  • Focuses on structure and properties of mathematical statements and proofs
• Key principles of mathematical logic:
  • Consistency: Logical system should not contain contradictions (no statement and its negation can both be proven)
  • Completeness: Every logically valid statement can be proven within the system (all true statements are provable)
  • Soundness: Every statement provable in the system is logically valid (only true statements can be proven)
• Mathematical logic provides framework for:
  • Formalizing mathematical statements and arguments (using symbols and rules)
  • Analyzing properties of mathematical theories (consistency, completeness, decidability)
  • Investigating limits of mathematical reasoning (incompleteness, undecidability)

Syntax vs semantics in logic

• Syntax refers to formal structure and rules of logical system
  • Specifies allowed symbols and formulas (variables, constants, connectives, quantifiers)
  • Defines rules for constructing well-formed formulas (wffs) (atomic formulas, complex formulas)
  • Determines valid inference rules for deriving new formulas (modus ponens, universal instantiation)
• Semantics refers to meaning and interpretation of symbols and formulas
  • Assigns truth values (true or false) to formulas based on meaning
  • Defines conditions under which formula is considered true or false (truth tables, interpretations)
  • Establishes relationship between syntax and intended interpretation (models, satisfiability)

Components of propositional and first-order logic

• Propositional logic deals with propositions and their relationships
  • Propositions are statements that can be either true or false (it is raining, 2+2=4)
  • Logical connectives combine propositions:
    • $\neg$ (negation): not
    • $\wedge$ (conjunction): and
    • $\vee$ (disjunction): or
    • $\rightarrow$ (implication): if...then
    • $\leftrightarrow$ (biconditional): if and only if
  • Allows construction of compound statements and analysis of logical properties (tautologies, contradictions)
• First-order logic extends propositional logic with predicates, variables, and quantifiers
  • Predicates represent properties or relations that can be true or false of objects ($P(x)$: x is prime, $R(x,y)$: x is greater than y)
  • Variables stand for arbitrary objects in domain of discourse ($x$, $y$, $z$)
  • Quantifiers express statements about all or some objects in domain:
    • $\forall$ (universal quantifier): for all
    • $\exists$ (existential quantifier): there exists
  • Allows representation of complex mathematical statements and reasoning about objects and properties ($\forall x (x>0 \rightarrow \exists y (y^2=x))$: for every positive real number, there exists a square root)

Construction of well-formed formulas

• Logical connectives combine propositions or predicates into complex formulas
  • Negation ($\neg$): $\neg P$ is true iff $P$ is false
  • Conjunction ($\wedge$): $P \wedge Q$ is true iff both $P$ and $Q$ are true
  • Disjunction ($\vee$): $P \vee Q$ is true iff at least one of $P$ or $Q$ is true
  • Implication ($\rightarrow$): $P \rightarrow Q$ is true iff either $P$ is false or $Q$ is true
  • Biconditional ($\leftrightarrow$): $P \leftrightarrow Q$ is true iff $P$ and $Q$ have the same truth value
• Quantifiers express statements about objects in domain
  • Universal quantifier ($\forall$): $\forall x P(x)$ is true iff $P(x)$ is true for all objects $x$ in domain
  • Existential quantifier ($\exists$): $\exists x P(x)$ is true iff there exists at least one object $x$ in domain for which $P(x)$ is true
• Well-formed formulas (wffs) constructed by following syntactic rules of logical system
  1. Atomic formulas (propositions or predicates) are basic building blocks ($P$, $Q(x)$)
  2. Complex formulas formed by applying logical connectives and quantifiers to atomic formulas or other complex formulas ($P \wedge Q$, $\forall x (P(x) \rightarrow Q(x))$)
  3. Parentheses used to specify order of operations and scope of quantifiers ($\exists x (P(x) \wedge \forall y Q(y))$)