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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.

J Marcos
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Willemien
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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 02:16

2 Answers2

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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).

J Marcos
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  • I know the Ji connectives (for 3 valued logic they are J1 = NCNpp, J(1/2) = NCCNppNCpNp, j0 = NCpNp ) but I am confused about the RT axioms themselves – Willemien Aug 19 '13 at 21:09
  • @Willemien Maybe you should try clarifying what you find so "confuse" (or even "wrong") about the axioms? – J Marcos Aug 20 '13 at 06:41
  • @JMarcos I find the notation difficult enough that I don't quite know how I could expand the notation in a particular example. I think I learn notations best when I can see how you can expand them with particular examples. I believe I could expand axioms 1-6 in the 3-valued or 4-valued case myself if I reread more closely and got out my pencil. But, I'm NOT confident I could expand their axiom schema 7 for axioms (I'm looking at Rosser and Turquette, not Gottwald). I ask you kindly to show us a few particular examples of how to write out the axioms, so we can understand the notation better. – Doug Spoonwood Aug 21 '13 at 03:06
  • @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
  • @JMarcos I did not intend my comment to come as some sort of complaint about their notation. Unless we know of a better one that does the same thing, I don't see any point in harping on them for their notation. It sounds to me like you don't have the ability to actually transform their axiom schemas into a finite set of axioms. I hope I'm wrong in that inference. If you can present an example, please do so. If you know someone else who can do so, please do so. I'll look over Gottwald, and Rosser and Turquette again and see if I can do this myself. – Doug Spoonwood Aug 21 '13 at 16:30
  • @DougSpoonwood I cannot say I am really interested in discussing the structure of the original axioms by Rosser & Turquette, which are anyway pretty useless in practice in proving anything. My responses so far all intended to bring some insight to what their "meaning" or their "intention" might have been. If you're interested in seeing useful proof systems for the same logics, based on the same intuition that I explained in my previous comment, you might want to check this (following Examples 2.2, 2.7, 2.15 and 2.17). – J Marcos Aug 22 '13 at 06:38
  • @JMarcos There's a simple way to axiomizatize any finite-valued logic which has modus ponens (we'll also want to have |=Cpp in the system). I outlined how to do so below. Note that once you prove some tautology in any such system you can then start using just ordinary substitution and detachment or condensed detachment (or tell if you need to prove more tautologies as theorems first if you can tell that an "occurs" check or a "symbol clash" would happen under a condensed detachment proof). – Doug Spoonwood Oct 21 '14 at 06:31
  • @DougSpoonwood I guess it all depends on what is your intention in axiomatizing a logic. In my earlier response I have pointed out that this can now be done for any finite-valued logic, using a uniform classic-like setting not unlike the one used by Rosser and Turquette, and giving rise to tableau systems with associated proof strategies and corresponding decision procedures. These systems would seem almost incomparable, in terms of usefulness, if you place them side by side traditional axiomatic systems. – J Marcos Oct 23 '14 at 01:21
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For any finite valued logic, where we have some connective "C" where detachment is a rule of inference for that connective, and we have truth tables, we can basically turn what the truth tables tell us into an axiomization of the logic using functioral variables (we'll also want |=Cpp to hold, otherwise we'll run into a problem with the way the transformation works).

With three-valued conditional-negation Lukasiewicz logic we have the following tables:

C  0  1  2  N
0  2  2  2  2 
1  1  2  2  1
2* 0  1  2  0

The table tells us that from an algebraic perspective, we have the following equations: C00=2, C01=2, C02=2, C10=1, C11=2, C12=2, C20=0, C21=1, C22=2, N0=2, N1=1, N2=0. It turns out that if x=y, then C$\delta$x$\delta$y, where $\delta$ is a variable functor of one argument. So, to make an axiom set for a finite-valued logic first suppose we find each of these C$\delta$x$\delta$y for the finite-valued logic under study. Make each of those C$\delta$x$\delta$y into an axiom. Next make the designated value into an axiom. Lastly, for a finite-valued logic where it's values belong to the set {a, ..., o} make the following into an axiom:

C$\delta$aC$\delta$b...C$\delta$o$\delta$p. When you've done those three steps you have a complete and sound axiomization for the logical system.

So for the four-valued C-N Lukasiewicz logic with the following table:

C  1  2  3  0  N
1* 1  2  3  0  0 
2  1  1  3  3  3
3  1  2  1  2  2
0  1  1  1  1  1

the following is a complete and sound axiomatization under the rules of substitution for propositional variables, substitution for the unary functorial variable, and detachment:

  1. 1
  2. C$\delta$1$\delta$C11.
  3. C$\delta$2$\delta$C12.
  4. C$\delta$3$\delta$C13.
  5. C$\delta$0$\delta$C10.
  6. C$\delta$1$\delta$C21.
  7. C$\delta$1$\delta$C22.
  8. C$\delta$3$\delta$C23.
  9. C$\delta$3$\delta$C20.
  10. C$\delta$1$\delta$C31.
  11. C$\delta$2$\delta$C32.
  12. C$\delta$1$\delta$C33.
  13. C$\delta$2$\delta$C30.
  14. C$\delta$1$\delta$C01.
  15. C$\delta$1$\delta$C02.
  16. C$\delta$1$\delta$C03.
  17. C$\delta$1$\delta$C00.
  18. C$\delta$0$\delta$N1.
  19. C$\delta$3$\delta$N2.
  20. C$\delta$2$\delta$N3.
  21. C$\delta$1$\delta$N0.
  22. C$\delta$1C$\delta$2C$\delta$3C$\delta$0$\delta$p.
Doug Spoonwood
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