<p>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</p>

<p>J A = (A -> B) -> A</p>

<p>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. </p>

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

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

<p>Or others found on our webpages: <a href="http://www.eecs.qmul.ac.uk/~pbo/">http://www.eecs.qmul.ac.uk/~pbo/</a></p>

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

<p>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. </p>

<p>Best regards, Paulo.</p>

CC BY-SA 3.0
2012-03-27T20:17:18.410Z
2012-03-27T20:17:18.410Z
Paulo Oliva
<p>Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an <em>extensional</em> definition of the modal operator:
$\Diamond A =_{def} \neg A \to A$ (which he attributes to Tarski).</p>

<p>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.</p>

<p>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 <a href="http://www.cs.st-andrews.ac.uk/~rr/pubs/lablunch-20110308.pdf">http://www.cs.st-andrews.ac.uk/~rr/pubs/lablunch-20110308.pdf</a></p>

<p>(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.)</p>

<p>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),
<a href="http://www.math.ucla.edu/~asl/bsl/0404/0404-006.ps">http://www.math.ucla.edu/~asl/bsl/0404/0404-006.ps</a> 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.)</p>

<p>Is anyone here aware of other articles or papers on this?</p>

<p>(I don't have any applications for it, but I find the properties to be interesting enough to merit a paper.)</p>

CC BY-SA 2.5
2011-04-07T13:33:26.630Z
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).
Neel Krishnaswami
CC BY-SA 3.0
2011-04-08T09:10:20.850Z
I didn't know about MathOverflow. Thanks!
Rob
CC BY-SA 3.0
2011-04-09T11:55:14.193Z
I've posted the same question to MathOverflow http://mathoverflow.net/questions/61134/looking-for-papers-and-articles-on-the-tarskian-moglichkeit
Rob
CC BY-SA 3.0
2011-04-10T17:17:29.203Z
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...
Noam Zeilberger
CC BY-SA 3.0
2011-04-12T19:36:54.650Z
@Noam It has nothing to do with interpreting formulae in MALL. Those equivalences hold in Łukasiewicz Logic, which corresponds to AMALL plus $((A\to B)\to B)\to ((B\to A)\to A)$.
Rob
CC BY-SA 3.0
2011-04-14T20:58:46.330Z
@Rob: Am I missing something?  Above you wrote $\diamond A =_{def} \neg A \to A$ as a definition, and in the slides you explain how to give rules for diamond and box in MALL, where the rule for $\diamond A$ is identical to the MALL rule for $A^\perp \multimap A \equiv A^{\perp\perp}⅋A \equiv A ⅋ A$.  So far this has everything to do with the interpretation of $\diamond A$ as the pure MALL formula $A^\perp \multimap A$, and nothing to do with additional axioms, right?
Noam Zeilberger
CC BY-SA 3.0
2011-04-15T12:56:46.790Z
@Noam, Yes, $\Diamond A$ is identical to the pure MALL rule for $A \invamp A$. There are no additional axioms, just an operator defined in terms of other operators.
Rob
CC BY-SA 3.0
2011-04-15T14:03:14.043Z
@Rob: okay, then perhaps you misunderstood my first comment/question.  All I was saying is that even if you take $\diamond A = \neg A\to A$ as your starting point, then it is not immediately obvious that $\diamond A$ should be defined in MALL as (equivalent to) $A^\perp \multimap A$: there are other ways to interpret classical/intuitionistic negation and implication (e.g., by inserting exponential modalities).
Noam Zeilberger
CC BY-SA 3.0
2011-04-15T15:16:15.930Z
@Norm I am not saying that $\Diamond A$ *should* be defined that way, or what it means. I am only noting that the modal theorems that one can prove using that definition are interesting.
Rob
CC BY-SA 3.0
2011-05-03T13:26:15.653Z
For what it's work, a preliminary paper is online at http://arxiv.org/abs/1105.0354 and will be updated.
Rob
<reference-request><lo.logic><proof-theory><linear-logic><modal-logic>
Looking for papers and articles on the Tarskian Möglichkeit
Rob
# Looking for papers and articles on the Tarskian Möglichkeit

(Asked by: username_0 on 2011-04-07)

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.)

**Tags:** reference-request, lo.logic, proof-theory, linear-logic, modal-logic

> **Comments:**
> **username_1**: 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).  
> **username_0**: I didn't know about MathOverflow. Thanks!  
> **username_0**: I've posted the same question to MathOverflow http://mathoverflow.net/questions/61134/looking-for-papers-and-articles-on-the-tarskian-moglichkeit  
> **username_2**: 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...  
> **username_0**: @Noam It has nothing to do with interpreting formulae in MALL. Those equivalences hold in Łukasiewicz Logic, which corresponds to AMALL plus $((A\to B)\to B)\to ((B\to A)\to A)$.  
> **username_2**: @username_0: Am I missing something?  Above you wrote $\diamond A =_{def} \neg A \to A$ as a definition, and in the slides you explain how to give rules for diamond and box in MALL, where the rule for $\diamond A$ is identical to the MALL rule for $A^\perp \multimap A \equiv A^{\perp\perp}⅋A \equiv A ⅋ A$.  So far this has everything to do with the interpretation of $\diamond A$ as the pure MALL formula $A^\perp \multimap A$, and nothing to do with additional axioms, right?  
> **username_0**: @Noam, Yes, $\Diamond A$ is identical to the pure MALL rule for $A \invamp A$. There are no additional axioms, just an operator defined in terms of other operators.  
> **username_2**: @username_0: okay, then perhaps you misunderstood my first comment/question.  All I was saying is that even if you take $\diamond A = \neg A\to A$ as your starting point, then it is not immediately obvious that $\diamond A$ should be defined in MALL as (equivalent to) $A^\perp \multimap A$: there are other ways to interpret classical/intuitionistic negation and implication (e.g., by inserting exponential modalities).  
> **username_0**: @Norm I am not saying that $\Diamond A$ *should* be defined that way, or what it means. I am only noting that the modal theorems that one can prove using that definition are interesting.  
> **username_0**: For what it's work, a preliminary paper is online at http://arxiv.org/abs/1105.0354 and will be updated.  

---

## 1 answers

### Answer by username_3 on 2012-03-27. Score: 8

username_0, 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 username_3. Sequential games and optimal strategies. Proceedings of the Royal Society A, 467:1519-1545, 2011.

Martín Escardó username_3, 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.


---

