If denial is a primitive speech act coordinate with assertion, the negation introduction rule gains a purely inferential formulation that requires no relaxation of the purity condition.

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Coordinate with(in logical analysis): Equal in status or importance; standing on the same level as something else rather than being derived from it.
Denial(in logic and language): The act of saying something is not true or rejecting a statement that was previously made.
Inferential formulation(in logic): A way of expressing something based purely on what you can conclude or deduce from other statements, without adding extra assumptions.
Negation introduction rule(in formal logic): A logical rule that explains how to derive a negative statement (something that says 'not') from other statements.
Primitive speech act(in logic and philosophy of language): A basic, fundamental way of using language that can't be broken down into simpler parts—like how 'asserting' (stating something is true) is primitive because you can't define it using other simpler language acts.
Purity condition(in formal logic and proof theory): A requirement in logic that a rule should work in a perfectly consistent way without needing exceptions or special adjustments.
assertion(Philosophy of language; speech act theory): An illocutionary act characterized by assertoric force together with the applicability of cognitive and social safeguards that distinguish it from other illocutionary acts and other forms of information transfer

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