Definition:Language of Propositional Logic/Keisler-Robbin
Definition
There are many formal languages expressing propositional logic.
The formal language used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is defined on Definition:Language of Propositional Logic.
This page defines the formal language $\LL_0$ used in:
Explanations are omitted as this is intended for reference use only.
Alphabet
Letters
The letters used are a non-empty set of symbols $\PP_0$.
See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Signs
Brackets
The brackets used are square brackets:
\(\ds \bullet \ \ \) | \(\ds [\) | \(:\) | \(\ds \)the left bracket sign\(\) | |||||||||||
\(\ds \bullet \ \ \) | \(\ds ]\) | \(:\) | \(\ds \)the right bracket sign\(\) |
See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Connectives
The following connectives are used:
\(\ds \bullet \ \ \) | \(\ds \land\) | \(:\) | \(\ds \)the conjunction sign\(\) | |||||||||||
\(\ds \bullet \ \ \) | \(\ds \lor\) | \(:\) | \(\ds \)the disjunction sign\(\) | |||||||||||
\(\ds \bullet \ \ \) | \(\ds \implies\) | \(:\) | \(\ds \)the conditional sign\(\) | |||||||||||
\(\ds \bullet \ \ \) | \(\ds \iff\) | \(:\) | \(\ds \)the biconditional sign\(\) | |||||||||||
\(\ds \bullet \ \ \) | \(\ds \neg\) | \(:\) | \(\ds \)the negation sign\(\) |
See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Collation System
The collation system used is that of words and concatenation.
See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Formal Grammar
The following bottom-up formal grammar is used.
Let $\PP_0$ be the vocabulary of $\LL_0$.
Let $Op = \set {\land, \lor, \implies, \iff}$.
The rules are:
$\mathbf W: \PP_0$ | $:$ | If $p \in \PP_0$, then $p$ is a WFF. | |
$\mathbf W: \neg$ | $:$ | If $\mathbf A$ is a WFF, then $\neg \mathbf A$ is a WFF. | |
$\mathbf W: Op$ | $:$ | If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\sqbrk {\mathbf A \circ \mathbf B}$ is a WFF. |
See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Also see
Sources
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.2$: Syntax of Propositional Logic