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

(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 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.)

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.)

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    $\begingroup$ I've never seen anything about this modality, but I did like your slides. If nothing turns up here, you might also try MathOverflow (or even the FOM mailing list). $\endgroup$ Commented Apr 7, 2011 at 13:33
  • $\begingroup$ I didn't know about MathOverflow. Thanks! $\endgroup$
    – Rob
    Commented Apr 8, 2011 at 9:10
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    $\begingroup$ I've posted the same question to MathOverflow mathoverflow.net/questions/61134/… $\endgroup$
    – Rob
    Commented Apr 9, 2011 at 11:55
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    $\begingroup$ I've never heard about Tarski's Möglichkeit before, but I'm curious whether you are sure the interpretations $\Diamond A = A⅋A$ and $\Box A = A\otimes A$ are faithful? You know there are other possible translations of the (classical/intuitionistic?) proposition ¬A→A even into classical MALL... $\endgroup$ Commented Apr 10, 2011 at 17:17
  • $\begingroup$ @Noam It has nothing to do with interpreting formulae in MALL. Those equivalences hold in Łukasiewicz Logic, which corresponds to AMALL plus $((A\to B)\to B)\to ((B\to A)\to A)$. $\endgroup$
    – Rob
    Commented Apr 12, 2011 at 19:36

1 Answer 1


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.


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