Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz 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>


Anyhow, the only non-critical work that I found a reference to is a talk by A. Turquette, "A generalization of Tarski's Möglichkeit" 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.)


I don't have any applications for it, but I find the properties to be interesting enough to merit a paper (in progress) on adding this operator to various substructural logics, and comparing those logics with themselves augmented by Lewsian modal operators.


**My question:** Is anyone here aware of other articles or papers on Tarski's Möglichkeit or "extensional" modalities?


Note: This is a question from CS Theory @ Stack Overflow <https://cstheory.stackexchange.com/questions/5928/looking-for-papers-and-articles-on-the-tarskian-moglichkeit> which a commentator suggested I post in Math Overflow.



Looking for papers and articles on the Tarskian Möglichkeit
Rob, I didn't know this was called the Tarskian Möglichkeit, but Martin Escardo and I have been studying this operator (A -> B) -> A, in the more general case when falsity is an arbitrary formula B, for the past few years, mainly in connection with computational interpretations of classical theorems. If we let B be fixed, then we define


J A = (A -> B) -> A


It is easy to show that this is a strong monad. We call it the "selection monad" or the "Peirce monad", as J A -> A is Peirce's law. In fact, the seemingly absurd theorem you mentioned in your post is the cornerstone for our work on interpreting ineffective principles such as the Tychonoff theorem, for instance. Have a look at some of our papers, e.g.


Martín Escardó and Paulo Oliva. Sequential games and optimal strategies. Proceedings of the Royal Society A, 467:1519-1545, 2011.


Martín Escardó Paulo Oliva, The Pierce translation. Annals of Pure and Applied Logic, 163(6):681-692, 2012.


Or others found on our webpages: <http://www.eecs.qmul.ac.uk/~pbo/>


Any paper which mentions "selection functions" or "game" is related to the operator you are asking about.


I must warn we have been studying this operator in the setting of intuitonistic (minimal) logic. But I find it very interesting that you are looking at this in the more refined (substructural) settings of linear logic and Lukasiewicz logic.


Best regards, Paulo.


