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Definition: Terms in Predicate Logic

Let \(\Sigma = (V_\Sigma,F_\Sigma,P_\Sigma)\) be a signature of a predicate logic. A set of predicate logic terms is defined recursively:

• Every variable \(x\in V_\Sigma\) is a term.
• Every constant (i.e. a function with arity \(0\)) \(f\in F_\Sigma\) is a term.
• If \(s_1,s_2,\ldots, s_n\) are terms \(f\in F_\Sigma\) is a function with arity \(n\ge 1\), then \(f(s_1,s_2,\ldots,s_n)\) is a term.

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 \("x"\), \("\delta"\), \("\epsilon"\) and \("a"\) are variables, thus they are terms. Also the constant \("0"\) as well as the functions \("f(x)"\), \("f(a)"\), \("x-a"\), \("f(x)-f(a)"\) and \("|x-a|"\), and \("|f(x)-f(a)|"\) are terms, because they are nullary, unary and binary functions.

are terms.

Other Examples of Terms in Predicate Logic

• \(0\)
• \(1\)
• \(x\)
• \(y\)
• \(f(x,x)\)
• \(f(0,1)\)
• \(f(x,y)\)
• \(f(f(x,x),f(0,1))\)
• ...

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Definitions: 1

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References

Bibliography

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