Logical System

A logical system is a triple $\mathcal{S}:=⟨\mathcal{L},\mathcal{K},\models ⟩$ where

• $\mathcal{L}$ is a formal language
• $\mathcal{K}$ is a set
• and $\models \subseteq \mathcal{K}\times \mathcal{L}$

Members of $\mathcal{L}$ are called formulas of $\mathcal{S}$. Members of $\mathcal{K}$ are called models for $\mathcal{S}$ And $\models$ is the satisfaction relation.

One example of such a logical system is the Propositional Logic: $⟨wff\left({\Sigma }_0,{\mathcal{V}}_0\right),\mathcal{K},\models ⟩$ where we define $\mathcal{K}$ as the set of variable assignments: $\mathcal{K}:={\mathcal{V}}_0\to {\mathcal{D}}_0$ and we say an assignment entails $A$ if its meaning interpretation is equal to true.

Now let there be a logical system with $\mathcal{M}\in \mathcal{K}$ a model and $A\in \mathcal{L}$ a formula, the we say that $A$ is:

• satisfied by $\mathcal{M}$ iff $\mathcal{M}$ entails $A$
• satisfiable if there exists a $\mathcal{M}\in \mathcal{K}$ which satisfies $A$
• unsatisfiable if there doesnt exist a $\mathcal{M}\in \mathcal{K}$ which satisfies $A$
• valid if every $\mathcal{M}$ satisfies $A$
• falsified by $\mathcal{M}$ iff $\mathcal{M}$ does not entail $A$
• falsifiable if there exists a $\mathcal{M}\in \mathcal{K}$ which falsifies $A$

Calculus

Logical System + Derivation Relation = Formal System