Definition:Language of Propositional Logic

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Definition

There are a lot of different formal systems expressing propositional logic.

Although they vary wildly in complexity and even disagree (to some extent) on what expressions are valid, generally all of these use a compatible formal language.

This page defines the formal language of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$.


We will use $\mathcal L_0$ to represent the formal language of propositional logic in what follows.

In order to define $\mathcal L_0$, it is necessary to specify:


Alphabet

Letters

The letters of $\mathcal L_0$, called propositional symbols, can be any infinite collection $\mathcal P_0$ of arbitrary symbols.

It is usual to specify them as a limited subset of the English alphabet with appropriate subscripts.


A typical set of propositional symbols would be, for example:

$\mathcal P_0 = \left\{{p_1, p_2, p_3, \ldots, p_n, \ldots}\right\}$


Signs

The signs of the language of propositional logic come in two categories:


Brackets

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


Connectives

\(\displaystyle \bullet \ \ \) \(\displaystyle \land\) \(:\) \(\displaystyle \)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\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \top\) \(:\) \(\displaystyle \)the tautology sign\(\)
\(\displaystyle \bullet \ \ \) \(\displaystyle \bot\) \(:\) \(\displaystyle \)the contradiction sign\(\)

These comprise:


Collation System

The collation system for the language of propositional logic is that of words and concatenation.

To ensure the unique readability property, it is necessary that the vocabulary $\mathcal P_0$ is chosen appropriately.

In particular, such that it does not conflict with the signs.


Formal Grammar

The formal grammar of the language of propositional logic (and hence its WFFs) can be defined in the following ways.


Backus-Naur Form

In Backus-Naur form, the formal grammar of the language of propositional logic takes the following form:


\(\displaystyle \)<formula>\(\) \(\)::=\(\) \(\displaystyle p \ \mid \ \top \ \mid \ \bot\) where $p \in \mathcal P_0$ is a letter
\(\displaystyle \)<formula>\(\) \(\)::=\(\) \(\displaystyle \neg\) <formula>\(\)
\(\displaystyle \)<formula>\(\) \(\)::=\(\) \(\displaystyle (\) <formula> <op> <formula> \()\)
\(\displaystyle \)<op>\(\) \(\)::=\(\) \(\displaystyle \land \ \mid \ \lor \ \mid \implies \mid \iff\)


Note that this is a top-down grammar: we start with a metasymbol <formula> and progressively replace it with constructs containing other metasymbols and/or primitive symbols, until finally we are left with a well-formed formula of $\mathcal L_0$ consisting of nothing but primitive symbols.


Bottom-Up Specification

The following rules of formation constitute a bottom-up grammar for the language of propositional logic $\mathcal L_0$.


  • Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$.
  • Let $Op = \left\{{\land, \lor, \implies, \iff}\right\}$.


The rules are:

$\mathbf W: TF$ $:$ $\top$ is a WFF, and $\bot$ is a WFF.
$\mathbf W: \mathcal P_0$ $:$ 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 $\mathbf A$ is a WFF and $\mathbf B$ is a WFF and $\circ \in Op$, then $\left({\mathbf A \circ \mathbf B}\right)$ is a WFF.


Any string which can not be created by means of the above rules is not a WFF.


Also defined as

Since most authors concern themselves only with one formal system for propositional logic, they tend to refer to the whole formal system as propositional logic or propositional calculus.

In correspondence, a particular author may decide to use only a subset of the signs.

Generally, the other signs then are considered definitional abbreviations.


At $\mathsf{Pr} \infty \mathsf{fWiki}$ we aim to incorporate all these different approaches, and thus we have come to separately define the formal language.

For the sakes of modularity and universality, we have settled for the formal language on this page as the language of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The page Definition:Translation Scheme for Propositional Logic documents how various other approaches from the literature can be translated into ours.


If so desired, a generic such formal system may be addressed as a propositional calculus, but this has to be used with reluctance and caution.


Also see


  • Results about the language of propositional logic can be found here.


Sources