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 only^{2}.
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.
Chapters: 1
Definitions: 2
Lemmas: 3 4
Parts: 5 6
Proofs: 7
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 \(\operatorname{PL1}\). ↩
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. ↩