# [FOM] Lukasiewicz theorem

James T. Smith smith at math.sfsu.edu
Tue Feb 1 14:21:44 EST 2011

```In his first research paper, “A contribution to the axiomatics of well-
ordered sets,” published in 1921 as a Warsaw student, Tarski cited a
result of Łukasiewicz about the property "relatively weakest" of axiom
systems.  The result is stated below;  the sentence beginning with
"Suppose" defines that property.  Does anyone know (a) what this property
is called now, and/or (b) where Łukasiewicz's result is found, per se, in
the literature?

Łukasiewicz’s theorem.  Let  A1,...,An  be sentences;  for  m = 1,...,n
let  Tm  be the sentence  ((A1 & ... & Am-1) -> Am).  Then the sentences
(A1 & ... & An)  and  (T1 & ... & Tn)  are equivalent.  Suppose  1 ≤ m ≤
n  and  Wm  is a sentence strictly weaker than  Tm;  then the sentence  (T1
& ... & Tm-1 & Wm & Tm+1 & ... & Tn)  is strictly weaker than  (T1 & ... &
Tn).  (Empty conjunctions are regarded as true.)

I have seen an announcement of a 26 March 1920 Warsaw talk by Łukasiewicz
entitled "From the theory of axiomatization." Could that be the source?

By the way, Tarski commented that application of Łukasiewicz’s theorem in
the context of his paper yielded nothing "aesthetic."

Any help with this little question would be greatly appreciated.

--------------------------------------
James T. Smith
Professor Emeritus of Mathematics
San Francisco State University
mailto:smith at sfsu.edu
http://math.sfsu.edu/smith

```