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Equation 5 · What Claude Code's Permission Model Actually Allows, in Plain Terms

What does this equation mean?

decision(c)={deny,c∈Dask,c∉D and c∈Aallow,c∉D∪A and c∈Wask (default),otherwise\text{decision}(c) = \begin{cases} \text{deny}, & c \in D \\ \text{ask}, & c \notin D \text{ and } c \in A \\ \text{allow}, & c \notin D \cup A \text{ and } c \in W \\ \text{ask (default)}, & \text{otherwise} \end{cases}

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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cc

Symbol c

c is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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DD

Symbol D

D is an input to the expression that computes the quantity on the left.

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AA

Symbol A

the ask set.

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WW

Symbol W

the allow set.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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What the article says around this equation

The precedence is simple enough to state directly. For a tool call c checked against the deny set D , the ask set A , and the allow set W : decision(c)={deny,c∈Dask,c∉D and c∈Aallow,c∉D∪A and c∈Wask (default),otherwise\text{decision}(c) = \begin{cases} \text{deny}, & c \in D \\ \text{ask}, & c \notin D \text{ and } c \in A \\ \text{allow}, & c \notin D \cup A \text{ and } c \in W \\ \text{ask (default)}, & \text{otherwise} \end{cases}. An unmatched command does not fail open into silent execution. It falls through to a prompt — what the documentation calls fail-closed matching [ 2 ] .

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