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

hi I posted this question on mathematics stackexhange ( https://math.stackexchange.com/questions/468855/what-are-the-rosser-turquette-axioms-of-lukasiewicz-3-valued-propositional-logic ) but did not get an helping answer (but did get two down votes) hope on this site maybe somebody can help me.

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.

I try to get the axioms out of Gottwalds book "A Treatise on Many-Valued logics" but i fear it is wrong on where he describes them (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

I do know that examples of the J functions are $J_1(p) = NCpNp$ , $J_{1/2}(p) = NCCpNpNCNpp$ and $J_0(p) = NCNpp$ but for the rest it is all greek to me.

hope somebody can help me here

• I think you mean "treatise", not "treasure". – Noah Schweber Aug 19 '13 at 16:14
• oops corrected that – Willemien Aug 19 '13 at 19:39
• Rolled back the previous edit. I strongly disagree with editing other people's colloquial idiom unless it is causing offence or obfuscation – Yemon Choi Aug 21 '13 at 5:33
• @Noah: If you read Treasure Island more carefully you'll find that the treasure was actually that of the Many-Valued Logics! Arrrrr!! :-) – Asaf Karagila Aug 21 '13 at 7:36

I found the book in downloadable form at http://www.uni-leipzig.de/~logik/gottwald/treatise.pdf , and I quickly glanced through the material up to and including the page you mention. The first observation is that the axiom you quoted as $AX_{RT}5$ appears in this version of the book as $Ax_{RT}6$, so, unless there is a typo in your question, an axiom has been added after the version you saw and before the version I saw (or deleted after my version and before yours). My version includes an axiom, called $Ax_{RT}5$, that, in effect, limits the truth values to the intended set $\{0,\frac1{m-1},\dots\frac{m-2}{m-1},1\}$. It's not expressed using a disjunction as in your question but rather using an iterated implication, saying that all the implications $J_v(A)\to B$, for all $m$ truth values $v$, together imply $B$. So, since you seemed concerned about the absence of such a limitation on truth values, I would guess that this axiom might have been missing in an early version of the book and then added later.
You were also concerned about Gottwald's use of the axioms $J_s(\bf s)$, one axiom "for each truth degree $s$ and each truth degree constant $\bf s$ denoting it" since some truth degrees $s$ might not be denoted by any constant. I suspect that Gottwald meant here exactly what he wrote, so that, in the situation where $s$ is not denoted by any constant, there would be no axiom for $s$ in this axiom schema (just as, if there were three constants denoting the same $s$, then there would be three axioms for $s$ in this schema). It is, of course, possible that, in my quick glance at the earlier parts of the book, I overlooked some convention requiring every truth degree to be named by a constant, but I don't think Gottwald needs or presupposes such a convention in $Ax_{RT}6$.
Finally, you mentioned in your question a couple of specific $J$ formulas, built using negation and implication (and rendered hard to read by Polish notation). But at this point in the book, Gottwald is not assuming any particular construction of the $J$'s. He only assumes that there exists some appropriate $J_s$ for each truth value $s$ and that there is an implication connective (with appropriate semantical behavior). How the $J$'s are to be constructed will depend on what other connectives are available, and this might be quite different in various logical systems. The point of these axioms is that they permit proofs of some important facts about various logical systems without needing to go into the details of those systems --- a classic use of the (meta-)axiomatic method.
• Axiom schema 8 just says that the connectives $\phi$ behave correctly on the truth degrees. For each connective $\phi$, say $n$-ary, and each $n$-tuple of truth degrees $s_i$, there 's an axiom saying that, when the arguments of $\phi$ have the truth degrees $s_i$, then the result of applying $\phi$ has the right truth degree $t$. For example, in classical logic, there would be an axiom $J_\top(A_1)\to(J_\bot(A_2)\to J_\bot(A_1\land A_2))$, saying that a conjunction is false when the first argument is true and the second false. (Continued in next comment) – Andreas Blass Aug 21 '13 at 16:10
• For conjunction in classical logic, there would be $3$ more such axioms to cover the other possible cases of the inputs. In general, there's an axiom for every line in the truth table of every connective. – Andreas Blass Aug 21 '13 at 16:11