Standard translation

In modal logic, standard translation is a way of transforming formulas of modal logic into formulas of first-order logic which capture the meaning of the modal formulas. Standard translation is defined inductively on the structure of the formula. In short, atomic formulas are mapped onto unary predicates and the objects in the first-order language are the accessible worlds. The logical connectives from propositional logic remain untouched and the modal operators are transformed into first-order formulas according to their semantics.

Definition

Standard translation is defined as follows:

$ST_x(p) \equiv P(x)$ , where $p$ is an atomic formula; P(x) is true when $p$ holds in world $x$ .
$ST_x(\top) \equiv \top$
$ST_x(\bot) \equiv \bot$
$ST_x(\neg \varphi) \equiv \neg ST_x(\varphi)$
$ST_x(\varphi \wedge \psi) \equiv ST_x(\varphi) \wedge ST_x(\psi)$
$ST_x(\varphi \vee \psi) \equiv ST_x(\varphi) \vee ST_x(\psi)$
$ST_x(\varphi \rightarrow \psi) \equiv ST_x(\varphi) \rightarrow ST_x(\psi)$
$ST_x(\Diamond_m \varphi) \equiv \exists y ( R_m(x, y) \wedge ST_y(\varphi))$
$ST_x(\Box_m \varphi) \equiv \forall y ( R_m(x, y) \rightarrow ST_y(\varphi))$

In the above, $x$ is the world from which the formula is evaluated. Initially, a free variable $x$ is used and whenever a modal operator needs to be translated, a fresh variable is introduced to indicate that the remainder of the formula needs to be evaluated from that world. Here, the subscript $m$ refers to the accessibility relation that should be used: normally, $\Box$ and $\Diamond$ refer to a relation $R$ of the Kripke model but more than one accessibility relation can exist (a multimodal logic) in which case subscripts are used. For example, $\Box_a$ and $\Diamond_a$ refer to an accessibility relation $R_a$ and $\Box_b$ and $\Diamond_b$ to $R_b$ in the model. Alternatively, it can also be placed inside the modal symbol.

Example

As an example, when standard translation is applied to $\Box \Box p$ , we expand the outer box to get

\forall y ( R(x, y) \rightarrow ST_y(\Box p))

meaning that we have now moved from $x$ to an accessible world $y$ and we now evaluate the remainder of the formula, $\Box p$ , in each of those accessible worlds.

The whole standard translation of this example becomes

\forall y ( R(x, y) \rightarrow (\forall z ( R(y, z) \rightarrow P(z))))

which precisely captures the semantics of two boxes in modal logic. The formula $\Box \Box p$ holds in $x$ when for all accessible worlds $y$ from $x$ and for all accessible worlds $z$ from $y$ , the predicate $P$ is true for $z$ . Note that the formula is also true when no such accessible worlds exist. In case $x$ has no accessible worlds then $R(x,y)$ is false but the whole formula is vacuously true: an implication is also true when the antecedent is false.

Standard translation and modal depth

The modal depth of a formula also becomes apparent in the translation to first-order logic. When the modal depth of a formula is k, then the first-order logic formula contains a 'chain' of k transitions from the starting world $x$ . The worlds are 'chained' in the sense that these worlds are visited by going from accessible to accessible world. Informally, the number of transitions in the 'longest chain' of transitions in the first-order formula is the modal depth of the formula.

The modal depth of the formula used in the example above is two. The first-order formula indicates that the transitions from $x$ to $y$ and from $y$ to $z$ are needed to verify the validity of the formula. This is also the modal depth of the formula as each modal operator increases the modal depth by one.

References

Modal Logic: A Semantic Perspective, Patrick Blackburn and Johan van Benthem

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