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.

  • $\begingroup$ I have no references for you. But another context in which the seemingly absurd theorem would be reasonable is if $\square A$ were intended to mean "$A$ is provable intuitionistically." So perhaps there are connections to be had with intuitionistic provability logic (of which I know very little)? $\endgroup$
    – Ed Dean
    Apr 13, 2011 at 17:46
  • $\begingroup$ @Ed $\Box$ doesn't seem to coincide with intuitionistic provability, though I am not familiar w/intuitionistic provability logic. Do you have references? One result I have is that for modal theorems proved in MALL+K, the corresponding Tarskian-modal theorems are proved in MALL. (Though MALL can prove some modal theorems not provable in corresponding MALL+K.) A preliminary paper is online at arxiv.org/abs/1105.0354 and will be updated. $\endgroup$
    – Rob
    May 3, 2011 at 13:25

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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .