I have often mentioned and recommended the Lindenbaum lattice
of theories. For any classical logic L, it's possible to organize
all theories expressible in L in a Lindenbaum lattice. (01)
Since there are infinitely many theories in L, each of which has
infinitely many statements, it's not possible to store them. But
it is possible to use the lattice as a framework for relating
the axioms for a collection of theories and microtheories of any
ontology or collection of ontologies expressible in L. (02)
The top of the lattice is the theory that contains no axioms
and all possible theorems provable from zero assumptions.
These include all the Boolean tautologies, and theorems such as
"If x likes y, then x likes y, and every unicorn is a unicorn." (03)
The bottom of the lattice is the inconsistent theory that
contains every statement expressible in the logic (which
includes all their negations). (04)
An important application of the Lindenbaum lattice is its use for
relating proofs in a nonmonotonic logic N to proofs in the classical
logic L on which N is based: (05)
* For a nonmonotonic theory T in N, and a proof P of a statement S
in T, there exist two theories T1 and T2 in L: (06)
* T1 contains the subset of all classical axioms for T
(but T may contain zero or more nonmonotonic axioms). (07)
* The proof P of S in T can be mapped to a walk W through the
lattice of classical theories from T1 to T2. (08)
* Each step of P that uses a classical axiom remains at the
current classical theory in the walk W. (09)
* But each step of P that uses a nonmonotonic axiom in T,
takes one step on the path W to a theory that modifies the
previous classical theory by one of the AGM operations for
theory revision. (010)
* The final step of the proof ends at the classical theory T2,
which is a modification of T1 by one AGM operation for each
nonmonotonic step of P. Then a purely classical proof can
derive S from the axioms of the revised theory T2. (011)
* Comment: For negation as failure and for Reiter's default logic,
the walk W always adds axioms. The theory T2 is therefore a
specialization (more axioms) of T1. But defeasible logics may
use the AGM operation of *contraction*, which moves to a theory
that is higher in the lattice (fewer implications) than the
previous theory. This makes the walk more complicated, since
you have to make sure that you don't use an axiom at one step
that is deleted at a later step. (012)
The following note links to the December issue of Logica Universalis,
which begins with two freely downloadable articles. (013)
John (014)
-------- Forwarded Message --------
Subject: Adolf Lindenbaum: Notes on his Life, with Bibliography
Date: Wed, 31 Dec 2014
From: UNILOG 2015 (015)
Adolf Lindenbaum: Notes on his Life, with Bibliography
Logica Universalis, 8, Dec 2014 (Open Access)
Jan Zygmunt and Robert Purdy (016)
This paper is dedicated to Adolf Lindenbaum (1904–1941) — Polish Jewish
mathematician and logician; a member of the Warsaw school of mathematics
under Waclaw Sierpinski and Stefan Mazurkiewicz and school of
mathematical logic under Jan Lukasiewicz and Stanislaw Lesniewski;
and Alfred Tarski’s closest collaborator of the inter-war period. (017)
Our paper is divided into three main parts. The first part is
biographical and narrative in character. It gathers together what
little is known of Lindenbaum’s short life. The second part is a
bibliography of Lindenbaum’s published output, including his public
lectures. Our aim there is to be complete and definitive. The third
part is a list of selected references in the literature attesting
to his unpublished results and delineating their extent. (018)
Logica Universalis
Volume 8, Issue 3-4, December 2014
http://link.springer.com/journal/11787/8/3/page/1 (019)
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