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nomadreid

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- #1

nomadreid

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In modal logic, the axiom schema that is typically used to allow repeatability of the unary ##\square## operator is, labelled as:

$$\mathbf{4}:\ \ \square p\to\square\square p$$

with the schema consisting of one axiom for every well-formed formula (wff) ##p##.

Using the standard labelling convention that is set out here, we get the logic S4, which is the version I have seen used most often, that adopts that axiom schema together with others (K, N, T).

The schema 4 allows unlimited numbers of ##\square## preceding a sub-wff in a wff.

If we wanted to limit the number of instances of ##\square## to say ##n##, all we'd need to do is replace

$$\mathbf{4}^*:s\ p\to\square s\ p$$

with the schema only including cases where ##p## is a formula that does not commence with ##\square## and ##s## is a sequence of between 1 and ##n-1## squares.

Such an axiom schema would allow us to increase the number of squares prefixing a non-squared wff from 1 up to any number up to and including ##n##, but no further.

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nomadreid

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