🧠Model Theory Unit 1 – Model Theory: First-Order Logic Intro

Key Concepts and Definitions

• First-order logic (FOL) is a formal system used to represent and reason about mathematical structures and their properties
• FOL consists of a formal language with well-defined syntax and semantics, allowing for precise and unambiguous statements
• Signature of a first-order language includes constant symbols, function symbols, and relation symbols
  • Constant symbols represent specific elements in the domain of discourse
  • Function symbols represent mappings between elements in the domain
  • Relation symbols represent relationships between elements in the domain
• Terms are expressions built from variables, constants, and function symbols, representing elements in the domain
• Formulas are expressions built from terms, relation symbols, and logical connectives, representing statements about the elements and their relationships
• Structures (models) are mathematical objects that provide an interpretation for the symbols in a first-order language, giving meaning to the formulas
• Validity of a formula means it is true in all possible structures (models) that interpret the language
• Satisfiability of a formula means there exists at least one structure (model) in which the formula is true

Syntax of First-Order Logic

• Alphabet of a first-order language includes variables, constant symbols, function symbols, relation symbols, and logical connectives
• Variables (usually denoted by lowercase letters like $x$, $y$, $z$) represent arbitrary elements in the domain of discourse
• Terms are recursively defined as variables, constants, or function symbols applied to terms
  • Example: If $f$ is a unary function symbol and $c$ is a constant symbol, then $f(c)$ and $f(f(x))$ are terms
• Atomic formulas are formed by applying relation symbols to terms
  • Example: If $R$ is a binary relation symbol, $x$ is a variable, and $c$ is a constant symbol, then $R(x, c)$ is an atomic formula
• Complex formulas are built from atomic formulas using logical connectives and quantifiers
  • Logical connectives include negation ($\neg$), conjunction ($\wedge$), disjunction ($\vee$), implication ($\rightarrow$), and equivalence ($\leftrightarrow$)
  • Quantifiers include universal quantifier ($\forall$) and existential quantifier ($\exists$)
• Well-formed formulas (wffs) are formulas constructed according to the syntax rules of the first-order language
• Free variables are variables not bound by any quantifier, while bound variables are variables within the scope of a quantifier

Semantics and Interpretations

• Semantics of FOL specify how to interpret the symbols in a first-order language within a given structure (model)
• Interpretation (model) $\mathcal{M}$ consists of a non-empty domain $D$ and an assignment of meanings to the symbols in the language
  • Constant symbols are assigned specific elements from the domain $D$
  • Function symbols are assigned functions on the domain $D$
  • Relation symbols are assigned relations on the domain $D$
• Variable assignment $s$ is a function that maps variables to elements in the domain $D$
• Term interpretation $t^\mathcal{M}[s]$ is the element in $D$ obtained by evaluating the term $t$ under the interpretation $\mathcal{M}$ and variable assignment $s$
• Formula satisfaction $\mathcal{M} \models \varphi[s]$ means the formula $\varphi$ is true in the interpretation $\mathcal{M}$ under the variable assignment $s$
  • Atomic formulas are true if the relation holds for the interpreted terms
  • Complex formulas are evaluated based on the truth values of their subformulas and the semantics of the logical connectives and quantifiers
• Validity $\models \varphi$ means the formula $\varphi$ is true in all interpretations (models) for all variable assignments
• Satisfiability means there exists an interpretation (model) and a variable assignment under which the formula is true

Logical Connectives and Quantifiers

• Logical connectives combine formulas to create more complex formulas
• Negation ($\neg \varphi$) is true if and only if $\varphi$ is false
• Conjunction ($\varphi \wedge \psi$) is true if and only if both $\varphi$ and $\psi$ are true
• Disjunction ($\varphi \vee \psi$) is true if and only if at least one of $\varphi$ or $\psi$ is true
• Implication ($\varphi \rightarrow \psi$) is true if and only if $\varphi$ is false or $\psi$ is true
• Equivalence ($\varphi \leftrightarrow \psi$) is true if and only if $\varphi$ and $\psi$ have the same truth value
• Quantifiers express properties of elements in the domain of discourse
• Universal quantifier ($\forall x \varphi(x)$) is true if and only if $\varphi(x)$ is true for all elements in the domain
  • Example: $\forall x (x > 0)$ states that all elements in the domain are greater than zero
• Existential quantifier ($\exists x \varphi(x)$) is true if and only if there exists at least one element in the domain for which $\varphi(x)$ is true
  • Example: $\exists x (x^2 = 2)$ states that there exists an element in the domain whose square is equal to two
• Quantifier scope extends to the formula immediately following the quantifier, unless parentheses are used to specify a different scope
• Nested quantifiers allow for expressing more complex properties involving multiple elements in the domain

Formal Proofs and Inference Rules

• Formal proofs are sequences of formulas, each of which is either an axiom or derived from previous formulas using inference rules
• Axioms are formulas assumed to be true without proof, serving as the starting points for formal proofs
• Inference rules specify how to derive new formulas from existing ones, preserving truth
• Modus Ponens (MP) is an inference rule that allows deriving $\psi$ from $\varphi$ and $\varphi \rightarrow \psi$
• Universal Instantiation (UI) allows deriving $\varphi(t)$ from $\forall x \varphi(x)$, where $t$ is a term substituted for the variable $x$
• Existential Generalization (EG) allows deriving $\exists x \varphi(x)$ from $\varphi(t)$, where $t$ is a term substituted for the variable $x$
• Other inference rules include Modus Tollens, Hypothetical Syllogism, and Resolution
• Soundness of an inference rule means that if the premises are true, the conclusion derived using the rule is also true
• Completeness of a proof system means that all valid formulas can be derived using the axioms and inference rules of the system

Models and Structures

• Models (structures) are mathematical objects that provide an interpretation for the symbols in a first-order language
• A model $\mathcal{M}$ consists of a non-empty domain $D$ and an interpretation function $I$ that assigns meanings to the symbols in the language
• Isomorphic models have the same structure, differing only in the naming of elements in the domain
• Elementary equivalence means that two models satisfy the same first-order formulas
• Finite models have a finite domain, while infinite models have an infinite domain
• Decidability of a theory means there exists an algorithm to determine whether a given formula is true in all models of the theory
• Completeness of a theory means that for every formula, either the formula or its negation is provable from the axioms of the theory
• Categoricity of a theory means that all models of the theory are isomorphic
• Compactness theorem states that if every finite subset of a set of formulas has a model, then the entire set of formulas has a model

Applications and Examples

• First-order logic is widely used in mathematics, computer science, and philosophy to formalize and reason about various concepts
• Peano arithmetic is a first-order theory that axiomatizes the natural numbers and their properties
  • Example: $\forall x (x + 0 = x)$ is an axiom stating that adding zero to any number results in the same number
• Set theory (ZFC) is a first-order theory that axiomatizes the principles of sets and their operations
  • Example: $\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \rightarrow x = y)$ is the axiom of extensionality, stating that sets with the same elements are equal
• Group theory is a first-order theory that axiomatizes the properties of algebraic structures called groups
  • Example: $\forall x \forall y \forall z ((x * y) * z = x * (y * z))$ is the axiom of associativity, stating that the order of applying the group operation does not matter
• First-order logic is used in database query languages (SQL) to express constraints and retrieve information from relational databases
• Automated theorem provers use first-order logic to prove mathematical theorems and verify the correctness of software and hardware systems

Common Challenges and Tips

• Translating natural language statements into first-order logic formulas can be challenging due to ambiguity and implicit assumptions
  • Tip: Break down complex statements into simpler components and identify the key elements (objects, properties, and relationships)
• Choosing the appropriate signature (constant, function, and relation symbols) is crucial for accurately representing the problem domain
  • Tip: Consider the essential elements and their relationships, and select symbols that capture these aspects concisely
• Quantifier ordering and scope can significantly impact the meaning of formulas
  • Tip: Pay attention to the order of quantifiers and use parentheses to clarify the scope when necessary
• Proving validity or unsatisfiability of formulas can be difficult, especially for complex formulas with multiple quantifiers
  • Tip: Break down the proof into smaller steps, use inference rules strategically, and consider proof by contradiction or contrapositive
• Constructing models to show the satisfiability of formulas requires creativity and understanding of the problem domain
  • Tip: Start with simple models and gradually add complexity, ensuring that the model satisfies the given formulas
• Identifying the limitations of first-order logic, such as its inability to express certain concepts like finiteness or transitive closure
  • Tip: Be aware of the expressive power of first-order logic and consider alternative logics or extensions when necessary
• Recognizing the decidability and complexity of first-order theories is important for understanding the feasibility of automated reasoning tasks
  • Tip: Familiarize yourself with known decidable and undecidable theories, and consider the computational complexity of decision procedures
• Applying first-order logic to real-world problems requires careful modeling and interpretation of results
  • Tip: Collaborate with domain experts to ensure accurate representation and validate the conclusions drawn from logical reasoning