Mathesis Universalis - No.4, Autumn 1997 http://www.pip.com.pl/MathUniversalis/4/
When using any part of this text - by Witold Marciszewski - refer, please, to the above original URL.
The University in Bialystok, Poland, in collaboration with the Bialystok Technical University, organizes the Workshop in the Centenary of Emil Post's Birth: Mutual Influences between Informatics and Logic, December 13/14, 1997. Post was born in 1897 in Augustow, a place in the vicinity of Bialystok, hence not only the time but also the place circumstances motivate commemorating him in this year and this city.The Editors would enjoy publishing preliminary discussions which might assist working on papers to be read at the Symposium. The text below is to encourage such a discussion.
Do Post's Logics Belong
to Alternative Logics?The following comment is made by a naive reader, being no expert in Post's logics. Anyway, even ignorance may prove useful in brain-storming to prepare a more thorough inquiry. The reader in question peeped into two reference books to compare their approach to what is called Post's logics. In the Dictionary of Logic as Applied in the Study of Language: Concepts, Methods, Theories (1981), he found an extensive article by J.Kabzinski on Many-Valued Logic in which the belonging of Post's logics to those being alternative to classical logic is simply presupposed (In the Polish counterpart of this Dictionary, viz. Logika Formalna: Zarys Encyklopedyczny, also edited by W.Marciszewski, PWN 1987, Kabzinski's contribution is found in the section on alternative approaches to logic.)
On the other hand, W.Kneale and M.Kneale in their thorough and seminal book The Development of Logic, 1962, raise some thought-provoking questions. In a passage on Post (Ch.9, Sec.5) they ask: Should Posts' m-valued system be called an alternative logic? First, they express a doubt whether Post's multi-valued system is a system of logic at all - in the sense of being a theory of deductive reasoning.
The authors remark that Post's system need have no connexion with reasoning - as a relation between propositions. If, however, one tries to defend such a connexion, one may say that Post's m-valued system is logical in a large sense. But then it constitutes no alternative to any two-valued system, like Frege's, since the interpretation of this system, as concerned with propositions, has to presuppose the Fregean (or Principia) system.
What seems especially worth considering in The Development of Logic comments, it is the following passage.
"Post himself suggests at the end of his article [`A General Theory of Elementary Propositions', Amer.J.of Math. 43, 1921, 163-85] that the entire argument might be translated into the language of some many-valued system, and that if this were done the many valued system would then seem fundamental while the common two-valued system took on the appearance of an artefact. But he has not explained how his argument could be translated into the language of a many-valued system, and it seems difficult, if not impossible, to attach any meaning to the suggestion."Now, Post experts can be addressed by a naive reader with the question, whether in the years which passed after The Development of Logic (1962) was there an attempt at answering those doubts. May there be in Post's text referred to a real insight regarding the nature of logic, even if too roughly or inadequately expressed? Should we seriously consider the claim that his many-valued system is more fundamental than classical logic, and thus offers - so to speak - a better alternative? Though the history did not confirm that claim, this Centenary is an opportunity and encouragement to re-examine historical verdicts in a thought-experiment way.
Witold Marciszewski
