Definition:Language of Propositional Logic

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.

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:


 * An alphabet $\mathcal A$
 * A collation system with the unique readability property for $\mathcal A$
 * A formal grammar (which determines the WFFs of $\mathcal L_0$)

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.

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 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.

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

 * Definition:Language of Propositional Logic/Keisler-Robbin
 * Definition:Language of Propositional Logic/Basson-O'Connor