Here’s an example that might help to see how the machinery described in the previous post works. Let’s take LLL to be CLuN (the positive part of classical logic plus the excluded middle). We can construct an adaptive logic where the constituents of Ω are all substitution instances of (∃)(ϕ ∧¬ ϕ), and we quite naturally obtain CL (classical logic) as the upper limit logic.

Take the adaptive logic thus obtained and consider the example where ourpremise set is:

Before we get to our adaptive proof, observe three facts:

In the proof we first write down the premises:1.

p ∨ q

Prem

∅

2.¬p Prem∅

3.r Prem∅

4.¬r Prem∅

5. r ∨ sPrem∅

Since p∧¬p is an abnormality and q ∨ (p∧¬p) LLL-follows from lines 1 and 2, we can apply Rc to lines 1 and 2 and conclude q, relying on the normal behavior of p ∧¬p:

6. q Rc: 1, 2

{p ∧¬p}

Similarly, s ∨ (r ∧¬r) is LLL-derivable from lin…

Take the adaptive logic thus obtained and consider the example where ourpremise set is:

Before we get to our adaptive proof, observe three facts:

In the proof we first write down the premises:1.

p ∨ q

Prem

∅

2.¬p Prem∅

3.r Prem∅

4.¬r Prem∅

5. r ∨ sPrem∅

Since p∧¬p is an abnormality and q ∨ (p∧¬p) LLL-follows from lines 1 and 2, we can apply Rc to lines 1 and 2 and conclude q, relying on the normal behavior of p ∧¬p:

6. q Rc: 1, 2

{p ∧¬p}

Similarly, s ∨ (r ∧¬r) is LLL-derivable from lin…