Pure First-Order Logic

Overview

Pure first-order logic, also known as predicate logic without functions or identity, is a fundamental logical system. It provides a formal language for representing statements about objects and their properties, using only predicate symbols, variables, quantifiers, and logical connectives.

Key Concepts

The core components of pure first-order logic include:

• Predicate Symbols: Represent properties or relations (e.g., IsEven(x), Likes(x, y)).
• Variables: Stand for arbitrary objects (e.g., x, y, z).
• Quantifiers: Universal (∀, ‘for all’) and Existential (∃, ‘there exists’).
• Logical Connectives: AND (∧), OR (∨), NOT (¬), Implication (→), Equivalence (↔).

Unlike richer logics, pure first-order logic does not include function symbols (like f(x)) or an explicit identity predicate (=).

Deep Dive

In pure first-order logic, statements are formed by combining predicate symbols with variables and quantifiers. For instance, ‘All even numbers are integers’ might be represented as ∀x (IsEven(x) → IsInteger(x)). The absence of function symbols means that complex terms cannot be directly constructed; instead, relationships are expressed through predicates.

The lack of an identity predicate means that statements about two distinct objects being the same or different require special handling, often through more complex logical constructions.

Formulas and Interpretations

A well-formed formula (WFF) in pure first-order logic is constructed according to specific syntax rules. The truth of these formulas is evaluated within an interpretation, which consists of a domain of discourse and an assignment of meaning to predicate symbols.

Applications

Despite its limitations, pure first-order logic is crucial for:

• Foundational studies in mathematics and computer science.
• Formalizing theories where functions or identity are not essential.
• Understanding the expressive limits of logical systems.

Challenges & Misconceptions

A common misconception is that pure first-order logic is too weak for practical use. However, its simplicity makes it an excellent tool for theoretical analysis and for building more expressive logics upon it. The absence of identity requires careful formulation of statements that implicitly rely on it.

FAQs

What distinguishes pure first-order logic?

It excludes function symbols and the identity predicate, relying solely on predicate symbols.

Can you express ‘there are at least two distinct objects’?

Yes, but it requires a more complex formula using existential quantifiers and negation, as there’s no direct identity symbol.

Is it Turing complete?

Pure first-order logic itself is not Turing complete, but when combined with axioms or used in proof systems, it can be used to express computable functions.