Definition:Language of Propositional Logic/Keisler-Robbin

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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 $\mathcal L_0$ 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 $\mathcal P_0$.

See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.


Signs

Brackets

The brackets used are square brackets:

\(\displaystyle \bullet \ \ \) \(\displaystyle [\) \(:\) \(\displaystyle \)the left bracket sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle ]\) \(:\) \(\displaystyle \)the right bracket sign\(\)

See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.


Connectives

The following connectives are used:

\(\displaystyle \bullet \ \ \) \(\displaystyle \land\) \(:\) \(\displaystyle \)the conjunction sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \lor\) \(:\) \(\displaystyle \)the disjunction sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \implies\) \(:\) \(\displaystyle \)the conditional sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \iff\) \(:\) \(\displaystyle \)the biconditional sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \neg\) \(:\) \(\displaystyle \)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 $\mathcal P_0$ be the vocabulary of $\mathcal L_0$.

Let $Op = \left\{{\land, \lor, \implies, \iff}\right\}$.


The rules are:

$\mathbf W: \mathcal P_0$ $:$ If $p \in \mathcal P_0$, 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 $\left[{\mathbf A \circ \mathbf B}\right]$ is a WFF.

See the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.


Also see


Sources