Definition:Formal Semantics

From ProofWiki
Jump to: navigation, search

Definition

Let $\mathcal L$ be a formal language.


A formal semantics for $\mathcal L$ comprises:

  • A collection of objects called structures;
  • A notion of validity of $\mathcal L$-WFFs in these structures.


Often, a formal semantics provides these by using a lot of auxiliary definitions.


Structure

Part of specifying a formal semantics $\mathscr M$ for $\mathcal L$ is to specify structures $\mathcal M$ for $\mathscr M$.


A structure can in principle be any object one can think of.

However, to get a useful formal semantics, the structures should support a meaningful definition of validity for the WFFs of $\mathcal L$.


It is common that structures are sets, often endowed with a number of relations or functions.


Validity

Part of specifying a formal semantics $\mathscr M$ for $\mathcal L$ is to define a notion of validity.


Concretely, a precise meaning needs to be assigned to the phrase:

"The $\mathcal L$-WFF $\phi$ is valid in the $\mathscr M$-structure $\mathcal M$."

It can be expressed symbolically as:

$\mathcal M \models_{\mathscr M} \phi$


Examples

Boolean Interpretations

Let $\mathcal L_0$ be the language of propositional logic.


The boolean interpretations for $\mathcal L_0$ can be interpreted as a formal semantics for $\mathcal L_0$, which we denote by $\mathrm{BI}$.


The structures of $\mathrm{BI}$ are the boolean interpretations.

A WFF $\phi$ is declared ($\mathrm{BI}$-)valid in a boolean interpretation $v$ iff:

$v \left({\phi}\right) = T$

Symbolically, this can be expressed as:

$v \models_{\mathrm{BI}} \phi$


Constructed Semantics

Let $\mathcal L$ be a formal language.


A constructed semantics for $\mathcal L$ is a formal semantics which is invented solely for proving a property about $\mathcal L$ or other entities related to $\mathcal L$.


Also see



  • Results about formal semantics can be found here.