First-Order Variables in Logic

Overview

In first-order logic (also known as first-order predicate calculus), variables serve as placeholders for individuals within a defined domain of discourse. Unlike propositional logic, which deals with whole propositions, first-order logic allows us to reason about objects, their properties, and relationships between them.

Key Concepts

First-order variables are crucial for:

• Representing arbitrary individuals.
• Binding by quantifiers (universal $\forall$ and existential $\exists$).
• Forming terms and predicates.

Deep Dive

A first-order variable, often denoted by letters like $x, y, z$, stands for an element from the set of objects that the logic system is concerned with. For example, in the statement ‘All humans are mortal’, we can represent this using variables:

$orall x ( ext{Human}(x) 
ightarrow  ext{Mortal}(x))$

Here, $x$ is a first-order variable that can represent any individual in the domain. The universal quantifier $\forall$ binds the variable $x$, making the statement a generalization about all elements.

Applications

First-order variables are foundational in:

• Formalizing mathematics (e.g., set theory, number theory).
• Computer science (e.g., database queries, program verification).
• Artificial intelligence (e.g., knowledge representation, automated reasoning).

Challenges & Misconceptions

A common point of confusion is the distinction between free and bound variables. A variable is free if it is not bound by a quantifier. Statements with free variables are not propositions; they represent properties or relations. For instance, $\text{Human}(x)$ has $x$ as a free variable.

FAQs

What is a domain of discourse?

It’s the set of all individuals that the variables in a first-order logic system can refer to. This could be numbers, people, geometric shapes, etc.

How do variables differ from constants?

Constants represent specific, fixed individuals (e.g., ‘Socrates’), while variables represent arbitrary individuals within the domain.