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 $$\epsilon &gt; 0$$ there is a $$\delta &gt; 0$$ such that $$|f(x)-f(a)| &lt; \epsilon$$ for all $$x\in D$$ with $$|x-a| &lt; \delta.$$

This proposition can be codified using a formula like this:

$\forall\epsilon\,(\epsilon &gt; 0)\,\exists\delta\,(\delta &gt; 0)\,\forall x\,(x\in D)\,(|x-a|&lt;\delta\Longrightarrow|f(x)-f(a)|&lt;\epsilon).$

In this formula, the strings $$"x\in D"$$, $$"\epsilon &gt; 0"$$, $$"\delta &gt; 0"$$ and $$"|x-a|&lt;\delta"$$ and $$"|f(x)-f(a)|&lt;\epsilon"$$ 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

• $$P(0)$$
• $$P(1)$$
• $$P(x)$$
• $$P(y)$$
• $$P(f(x,x),x)$$
• $$P(1,f(0,1))$$
• $$P(x,f(x,y))$$
• $$P(f(x,x),f(0,1),x,y,0,1)$$
• ...

Axioms: 1
Corollaries: 2
Definitions: 3
Parts: 4

Thank you to the contributors under CC BY-SA 4.0!

References

Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011