Some background: ukasiewicz many-valued logics were intended as modal logics, and ukasiewicz gave an extensional definition of the modal operator:$\Diamond A =_{def} \neg A \to A$ (which he attributes to Tarski).This gives a weird modal logic, with some paradoxical, if not seemingly absurd theorems, notably $(\Diamond A\land \Diamond B) \to \Diamond (A\land B)$. Substitute $\neg A$ for $B$ to see why it's been relegated to a footnote in the history of modal logic.However, I've realised that it's less absurd when that definition of a possibility operator is applied to Linear Logic and other substructural logics.I have an informal talk about this earlier in the month. A link to the talk is at http://www.cs.st-andrews.ac.uk/~rr/pubs/lablunch-20110308.pdf(One of the reasons that I asked about substructural modal logics was to compare the expressiveness of those logics with the use of this operator.)Anyhow, the only non-critical work that I found a reference to is a talk by A. Turquette, A generalization of Tarski's Mglichkeit at the Australasian Association for Logic 1997 Annual Conference. The abstract is in the BSL 4 (4),http://www.math.ucla.edu/~asl/bsl/0404/0404-006.ps Basically Turquette suggested applications in $m$-valued logics for $m$-state systems. (I've not been able to obtain any notes, slides or other content of this talk, so I would appreciate hearing from anyone who has more information.)Is anyone here aware of other articles or papers on this?(I don't have any applications for it, but I find the properties to be interesting enough to merit a paper.)
Looking for papers and articles on the Tarskian Mglichkeit
reference request;lo.logic;proof theory;linear logic;modal logic