Definition:Language of Propositional Logic/Basson-O'Connor
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 round 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:
| \(\ds \bullet \ \ \) | \(\ds .\) | \(:\) | \(\ds \)the conjunction sign\(\) | |||||||||||
| \(\ds \bullet \ \ \) | \(\ds \lor\) | \(:\) | \(\ds \)the disjunction sign\(\) | |||||||||||
| \(\ds \bullet \ \ \) | \(\ds \supset\) | \(:\) | \(\ds \)the conditional sign\(\) | |||||||||||
| \(\ds \bullet \ \ \) | \(\ds \equiv\) | \(:\) | \(\ds \)the biconditional sign\(\) | |||||||||||
| \(\ds \bullet \ \ \) | \(\ds \sim\) | \(:\) | \(\ds \)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 $Op = \left\{{., \lor, \supset, \equiv}\right\}$.
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 $\sim \mathbf A$ is a WFF. | |
| $\mathbf W: Op$ | $:$ | If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF. |
See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Also see
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.2$: The Construction of an Axiom System