Part: PL1 - First Order Predicate Logic

A predicate logic of first order \(\operatorname{PL1}\) consists of all formulae, which obey a syntax defined recursively by:

The notion first order" in the name of \(\operatorname{PL1}\) means that quantifiers can be applied to variables only2.


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, e.g. the strings \("\epsilon > 0"\), \("\epsilon > 0"\), or \("|x-a|<\delta\Longrightarrow|f(x)-f(a)|<\epsilon"\) are less complex formulae.

  1. Definition: Terms in Predicate Logic
  2. Definition: Atomic Formulae in Predicate Logic
  3. Chapter: Peano Arithmetic

Chapters: 1
Definitions: 2
Lemmas: 3 4
Parts: 5 6
Proofs: 7

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


  1. Just like we can do with Boolean variables in propositional logic. Therefore, the first order predicate logic \(\operatorname{PL1}\) is an extension of a propositional logic, because Boolean variables are a simple special case of more complex formulae in \(\operatorname{PL1}\). 

  2. While \(\operatorname{PL1}\) only allows to apply quantifiers to variables, it is possible to apply them also to predicates and formulae in higher order logics