**N** does not validate uniform substitution. Many simple failures of uniform substitution can be found: *p*→*p* is valid in **N**, but the instance (*p*∧*p*)→(*p*∧*p*) isn't. (Verify this at our dialogical logic sandbox!) There are many other failures of uniform substitution (infinitely many, in fact).

This is a somewhat awkward feature of **N**. Earlier, I asked whether uniform substitution *consistent* with **N**: is the closure of **N** under uniform substitution consistent? Another way to put this is to ask whether uniform substitution is an admissible rule of inference for **N**.

The answer is “yes”; here is a simple proof, exploiting the fact that **N** is sub-classical. Let **N**′ be the closure of **N** under uniform substitution; we need to show that **N**′ is consistent. Suppose that there were a formula φ such that both φ∈**N**′ and ¬φ∈**N**′. There are formulas α and β in **N** such that φ is obtained by an application of uniform substitution from α and likewise ¬φ is obtained by an application of uniform substitution from β. Since α∈**N**, α is a tautology; and since classical logic **CL** is closed under uniform substitution, φ is a tautology, too. Likewise, ¬φ is a tautology. But that's impossible!

(Thanks to Benedikt Löwe for this nice solution.)

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