# what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it.

Maybe if somebody can give me the axioms for 3 and 4 valued logic then I can figure out the others by myself. (polish notation is not a problem)

I think the description in Gottwalds "A Treatise on Many-Valued Logics" is wrong. page 109
$AX_{RT} 5 : J_s(s)$ for each truth degree s and each truth degree constant s denoting it,

To me it makes no sense because it is an axiomatisation and then there is no truth degree constant s for an s that is not a designated truthvalue.

As far as I remember from another text there is an axiom $\bigvee_{s \epsilon W} J_s(p)$ to fix the number of truth degrees to n

but I maybe wrong.

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I don't understand this at all at present, but scanning their text on many-valued logic, putting up all the definitions for their capital letter $\gamma$ subscripted and superscripted, and "J" subscripted I think would help make this more understandable. Gottwald, I think, uses symbol that looks like "J" where Rosser and Turquette write "J", and I think uses a circled arrow where Rosser and Turquette use capital $\gamma$. You might want other relevant definitions in your question also. –  Doug Spoonwood Aug 17 '13 at 2:16
Rosser-Turquette axioms are based on unary connectives that uniquely identify each truth-value. Assume $n$-ary Lukasiewicz logic to be defined with truth-values of the form $\frac{i}{n-1}$, where $0\leq i< n$, in a language having symbols for implication and for negation, with the usual interpretations suggested by Lukasiewicz. Then one might consider unary connective symbols $J_i$, for $0\leq i< n$, interpreted by setting $J_i(v)=0$ if $v=i$ and $J_i(v)=1$ otherwise (roughly as you may find in p.18 of the 1952 book by Rosser & Turquette). The axioms of the $n$-ary Lukasiewicz logic may then be easily written with the help of such additional connectives corresponding to the $J_i$ operators. It is worth noticing that such operators are not so much of an addition to Lukasiewicz logics, as they can be routinely defined by way of native Lukasiewicz implication and negation. Notice also that nowadays such axiomatization mechanisms are known to be extendable from Lukasiewicz logics to any other finite-valued logic (you might want to check in particular the literature on algorithmic versions of Suszko's Thesis).
@DougSpoonwood Rosser & Turquette's notation is certainly appalling, and in this case the corresponding axioms $Ax_{RT}8$ from Gottwald probably do not look so much better. (Trying to understand their role in proving the completeness result would seem to constitute a better strategy, in any case.) For all I know, the axioms are there in order to provide a modular description of the truth-tables of each connective: for each tuple of truth-values taken as input, the corresponding "implied" truth-value given as output is described with the help of the $J$ operators. –  J Marcos Aug 21 '13 at 15:08