Definition: Atomic Formulae in Predicate Logic 

Let \(s_1,s_2,\ldots, s_n\) be terms in predicate logic and let \(P\) be a predicate with an arity \(n\ge 1\). Then \(P(s_1,s_2,\ldots,s_n)\) is called an atomic formula in predicate logic.

Example

Take real numbers as the domain of discourse, and consider the \(\epsilon-\delta\) definition of continuous real functions:

A real function \(f:D\to\mathbb R\) is continuous at the point \(a\in D\), if for every \(\epsilon > 0\) there is a \(\delta > 0\) such that \(|f(x)-f(a)| < \epsilon\) for all \(x\in D\) with \(|x-a| < \delta.\)

This proposition can be codified using a formula like this:

\[\forall\epsilon\,(\epsilon > 0)\,\exists\delta\,(\delta > 0)\,\forall x\,(x\in D)\,(|x-a|<\delta\Longrightarrow|f(x)-f(a)|<\epsilon).\]

In this formula, the strings \(“x\in D”\), \(“\epsilon > 0”\), \(“\delta > 0”\) and \(“|x-a|<\delta”\) and \(“|f(x)-f(a)|<\epsilon”\) are atomic formulae, because they are unary and binary predicates of the terms \(“x”\), \(“\epsilon”\), \(“\delta”\), \(“|x-a|”\), and \(“|f(x)-f(a)|”\).

Other Examples of Atomic Formulae in Predicate Logic

• \(P(0)\)
• \(P(1)\)
• \(P(x)\)
• \(P(y)\)
• \(P(f(x,x),x)\)
• \(P(1,f(0,1))\)
• \(P(x,f(x,y))\)
• \(P(f(x,x),f(0,1),x,y,0,1)\)
• …

| | | | | created: 2016-10-09 21:34:21 | modified: 2018-02-11 15:35:13 | by: guest | references: [656]