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Showing posts from August, 2008

An example of an adaptive proof

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…

A gentle introduction to adaptive logics

Here's a brief explanation of what adaptive logics are that I wrote up as a part of another paper. It covers less ground than Batens's stuff, but it is meant to be a tad more accessible and introductory. If you need more gory details, check out the centre's writings list. If I have some time, I'll post some some examples that show how the machinery works. So, here we go (let me know if you find typos or unclarities):


An adaptive logics adapts itself to the premises it's applied to: the correctness of some rules or steps depends on the choice of premises. The basic idea is that while reasoning using an adaptive logic we "swing between" two simpler logics (called the `lower limit logic,' LLL, and the `upper limit logic,' ULL), ULL being a strengthening of LLL, so that when no problematic formula (details to follow) is derived from a set of premises, we apply ULL, and once some premises turn out to lead to difficulties, we restrict ourselves only to …