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Propositions can be formulated in natural languages, e.g. the English sentence "It is raining" is a proposition. However, natural languages have complex grammars and are too ambiguous to strictly define a logical calculus of a propositional logic. Therefore, we will concentrate on the main elements needed to construct a propositional logic and leave everything else as a superfluous ballast out of our definition.
Definition: Signature of Propositional Logic - PL0
The signature $(V,F,P)$ of a propositional logic consists of
- A set of variables $V$. We will denote variables by (indexed) Latin letters, e.g. $a,b,\ldots$, $x_1,y_1,\ldots.$
- A set of n-ary functions:
- constants $1,0$ as nullary functions
- and functions taking $n\ge 1$ arguments as input and assigning the input either the constant $1$ or $0$ as output.
- An empty set of predicates $P=\emptyset$.
Because the set of predicates $P$ is empty, the propositional logic is also called zeroth-order predicate logic and denoted by $PL0$.
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- Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000