Thanks, John. This is useful info. What a shame that Lindenbaum's life, like
many others back then, was cut drastically short by the Nazis, fascists, and
totalitarian Marxists. Useful reminder for us all today, when ideologues roam
the streets.
(01)
I was reminded of Isaiah Berlin's speech on his reception of an honorary
doctorate (given by U. Toronto, 1994), recently published in the New York
Review of Books, Oct. 23, 2014 issue:
http://www.nybooks.com/articles/archives/2014/oct/23/message-21st-century/.
(02)
"Let me explain. If you are truly convinced that there is some solution to all
human problems, that one can conceive an ideal society which men can reach if
only they do what is necessary to attain it, then you and your followers must
believe that no price can be too high to pay in order to open the gates of such
a paradise. Only the stupid and malevolent will resist once certain simple
truths are put to them. Those who resist must be persuaded; if they cannot be
persuaded, laws must be passed to restrain them; if that does not work, then
coercion, if need be violence, will inevitably have to be used—if necessary,
terror, slaughter. Lenin believed this after reading Das Kapital, and
consistently taught that if a just, peaceful, happy, free, virtuous society
could be created by the means he advocated, then the end justified any methods
that needed to be used, literally any."
(03)
...
"The central values by which most men have lived, in a great many lands at a
great many times—these values, almost if not entirely universal, are not
always harmonious with each other. Some are, some are not."
(04)
Thanks,
Leo
(05)
>-----Original Message-----
>From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
>bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
>Sent: Wednesday, December 31, 2014 9:32 AM
>To: ontolog-forum@xxxxxxxxxxxxxxxx; cg@xxxxxxxxxxxxx
>Subject: [ontolog-forum] The Lindenbaum lattice and a biography of Adolf
>Lindenbaum
>
>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.
>
>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.
>
>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."
>
>The bottom of the lattice is the inconsistent theory that
>contains every statement expressible in the logic (which
>includes all their negations).
>
>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:
>
> * 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:
>
> * T1 contains the subset of all classical axioms for T
> (but T may contain zero or more nonmonotonic axioms).
>
> * The proof P of S in T can be mapped to a walk W through the
> lattice of classical theories from T1 to T2.
>
> * Each step of P that uses a classical axiom remains at the
> current classical theory in the walk W.
>
> * 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.
>
> * 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.
>
> * 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.
>
>The following note links to the December issue of Logica Universalis,
>which begins with two freely downloadable articles.
>
>John
>
>-------- Forwarded Message --------
>Subject: Adolf Lindenbaum: Notes on his Life, with Bibliography
>Date: Wed, 31 Dec 2014
>From: UNILOG 2015
>
>Adolf Lindenbaum: Notes on his Life, with Bibliography
>Logica Universalis, 8, Dec 2014 (Open Access)
>Jan Zygmunt and Robert Purdy
>
>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.
>
>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.
>
>Logica Universalis
>Volume 8, Issue 3-4, December 2014
>http://link.springer.com/journal/11787/8/3/page/1
>
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(06)
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