First-order theories in logic are like building blocks for understanding complex systems. They use a special language with symbols, rules, and starting points called axioms to create a framework for reasoning about different subjects like math or science.

These theories help us model real-world stuff using logic. By setting up basic rules (axioms) and following strict reasoning steps, we can prove new things (theorems) about the topic we're studying, whether it's numbers, shapes, or abstract ideas.

First-order theories: Definition and Components

Definition and components of first-order theories

• A first-order theory is a set of sentences in first-order logic that includes the formal language, axioms, and rules of inference used to derive theorems within the theory
• The formal language of a first-order theory consists of:
  • An alphabet of symbols
  • A grammar for constructing well-formed formulas
  • A semantics for interpreting the meaning of the formulas
• Axioms are the foundational statements of a first-order theory that are assumed to be true without proof and serve as the starting points for deriving other theorems within the theory
• Rules of inference are the valid forms of reasoning used in a first-order theory to derive new theorems from the axioms and previously proven theorems

Applications of first-order theories

• First-order theories are used to model and reason about various domains by capturing the essential properties and relationships of the objects in the domain
• Examples of domains modeled by first-order theories include:
  • Arithmetic
  • Geometry
  • Set theory
• First-order theories provide a formal framework for analyzing and proving statements within these domains

Well-formed formulas in first-order logic

Syntax and alphabet of first-order logic

• The syntax of first-order logic specifies the rules for constructing well-formed formulas (wffs) from the symbols of the formal language
• The alphabet of a first-order language includes:
  • Variables (x, y, z)
  • Constants (a, b, c)
  • Function symbols (f, g, h)
  • Predicate symbols (P, Q, R)
  • Connectives (∧, ∨, →, ¬)
  • Quantifiers (∀, ∃)

Constructing terms and formulas

• Terms are the basic expressions in first-order logic and can be:
  • Variables (x)
  • Constants (a)
  • Function symbols applied to other terms (f(x), g(a, b))
• Atomic formulas are the simplest well-formed formulas and consist of a predicate symbol applied to one or more terms (P(x), Q(a, f(y)))
• Complex formulas are constructed from atomic formulas using connectives and quantifiers, following the rules of the grammar
  • Examples: $∀x(P(x) → Q(x))$, $∃y(R(y) ∧ S(y))$
• The scope of a quantifier is the part of the formula that the quantifier applies to, and the rules for determining the scope are based on:
  • The precedence of the connectives
  • The placement of parentheses

Axioms in first-order theories

Types of axioms

• Axioms are the fundamental assumptions of a first-order theory that are taken to be true without proof and serve as the starting points for deriving other theorems
• Logical axioms are the axioms that hold in all first-order theories and capture the basic properties of the connectives and quantifiers
  • Examples: $∀x(P(x) ∨ ¬P(x))$, $(P → Q) → (¬Q → ¬P)$
• Non-logical axioms, also called proper axioms or theory-specific axioms, are the axioms that capture the essential properties and relationships of the objects in a particular domain
  • Examples: $∀x(x + 0 = x)$ in arithmetic, $∀x∀y(d(x, y) = d(y, x))$ in geometry

Properties of axioms

• Axioms should be consistent, meaning that no contradiction can be derived from them
• Axioms should be independent, meaning that no axiom can be derived from the others
• The choice of axioms for a first-order theory depends on:
  • The intended interpretation of the theory
  • The properties and relationships that are considered essential for the domain being modeled

Axioms for deduction in first-order logic

Role of axioms in the deductive process

• Axioms serve as the starting points for the deductive process in first-order logic, which involves deriving new theorems from the axioms and previously proven theorems using the rules of inference
• The deductive process in first-order logic is sound, meaning that if the axioms are true and the rules of inference are applied correctly, then the derived theorems will also be true
• The deductive process is also complete for first-order logic, meaning that if a formula is a logical consequence of the axioms, then it can be derived as a theorem using the rules of inference

Implications of axiom selection

• The choice of axioms determines the scope and limitations of the theorems that can be derived in a first-order theory, as only those formulas that logically follow from the axioms can be proven as theorems
• Axioms can be used to define new concepts and relations in a first-order theory by specifying their properties and relationships to other concepts and relations in the theory
  • Example: Defining the concept of a group in abstract algebra using axioms for associativity, identity, and inverses
• The consistency and independence of the axioms are crucial for the coherence and non-redundancy of the resulting theory
  • Inconsistent axioms lead to contradictions
  • Dependent axioms lead to unnecessary complexity in the deductive process