By McGinnis M.

This quantity of the author's mathematical discoveries makes its visual appeal on the request of many capable mathematicians, lecturers, and students through the usa, between some of whom are: J. M. Greenwood, Superintendent of Kansas urban colleges and President of the nationwide organization of schooling; N. B. Newsom, Professor of arithmetic and Languages on the Kansas country college, Lawrence; R. P. Baker, Professor of arithmetic and Languages, Lamar, and J. C. Shelton, President of Scarritt collage, Neosho, Missouri; F. С Colburn, relevant of highschool, Texarkana, Texas; and L. D. Harvey, country Superintendent of Wisconsin.

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For the infinite valued system Loo a very simple axiomatisation is available. e. sound and complete axiomatisation already in the 1930th. The proof of this fact, however, was given only in [92] in 1958. t. H1 ) -+L (Hl -+L H2), ((Hl -+L H2) -+L H2) -+L ((H2 -+L H 1 ) -+L Hl). A correspondingly simple axiomatisation exists for the other extreme case, the three-valued system L3 and was given by WAJSBERG [111]. t. H1 ) -+L H1 ) -+L H 1 . -+L H 3 )), 33 Many-valued logic Because for each set of logically valid sentences of Lm one has the inclusion taut~ ~ taut~ it is quite natural to expect to get an adequate axiomatisation for Lm by extending a suitable axiom system for Loc.

JPm = {V, "'p}. They have as their characteristic matrices 24 the structures (72) This choice of the truth degree functions is the reason that non2 sometimes is called POST negation function. 2 the POST systems are functionally complete. This is, as it seems, the most interesting property they have. The set v P of designated truth degrees is, however, not completely fixed for these systems. The present version v P = {l} is mainly chosen, but alredy POST [87] discussed also other possibilities. This (more or less open situation) is actually not an essential difficulty.

There was always a kind of uniformity in the definition of the systems of one class, there were, however, important differences in the particular approaches. e. from the choice of truth degree sets and truth degree functions constitutive for the particular systems, this is not a problem. , adequate axiomatisations for semantically determined systems. B. R. TURQUETTE in [94] for a wide class of systems S of many-valued logic with finite truth degree sets of the form W S = Wm,m ~ 2. :Js of connectives of S shall contain a binary connective --+, denoting a kind of implication connective, and unary connectives J s for each SEWs, or at least that such connectives are definable from the primitive connectives of the system S.

### Universal solution for algebraic equations of all degrees in terms of their coefficients by McGinnis M.

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