I am searching for all the possible features of reasoning (all of them can be expressed in logic), so far I have found the following features:

- non-monotonicity, defeasible reasoning (expressed by special features of consequence relation)
- probabilistic reasoning (expressed by assignment of probabilities to the predicates and by the operations on the probabilities to the connectives and modal operators)
- higher order logic (expressed by allowing the predicated to take other predicates and formulas as arguments)
- modal operators (expressed by the unary operators acting on the predicates and formules)
- special connectives (expressed by special connectives that can arise in linear/substructural setting)

**My question is this - are there any other features beside those 5, features that can improve the expresiveness of the logic.**

I am trying to combine all those features in one logic and initially I would be happy to know that the set of those 5 features is exhaustive and so - when I come up with the language that can express all those 5 features then there is no more general language than that. Of course, I am thinking about templates, i.e., I will leave open the final set of modal operators and connectives (and the interactions among them), because different concrete logics can arise in each concrete choice of those. My aim is to create reasoner (forward chaining engine) that could be used for such templates and that works modulo concrete set of modal operators and connectives.

Of course, I will have to find common proofs for each of the logic but I plan to automate this task by formalizing each concrete logic in Coq or Isabelle/HOL as it has already be done by linear logics. Then (semi)automatic proof search can lead to the proofs of rejections of soundness and completeness theorems and other theorems for each logic. I am even thinking about combination of genetic search (for the operators/connectives/their sequence rules) with automatic theorem proving (for the theorems about concrete logic) *(possibly - with deep learning inspired) for (semi)automatic discovery and development of logics. But to guide all this process, to predict the most general grammar for the possible logics, I should be sure that there is nothing beyond those 5 features. (Neural methods have stuck in deadlock, as can be seen from delayed introduction of autonomous vehicles, that is why strong boost of symbolic methods is necessary and automation of the discovery and research of the logics is the key process for the adoption of symbolic methods in industrial setting)*.

After that I will have to find semantics, but I am sure that set semantics (with probability distributions and set operations and relations (for modalities and substructural connectives)) is sufficient for all those 5 features, because everything in math can be expressed by (ZFC) set theory and that is why any other possible (sophisticated) semantics can be expressed via sets anyway.

Of course, I am aware of the efforts by Logica Universalis community, but the Florian Rabe, but the community of categorical logics and institution theory. But I am having hard time to find the logic that already encompass those 5 features and also I can not find definite answer that those 5 features are exhaustive or am I missing something?

I would like to add that it is very important that everyone at this time come up with some advice or suggestion, idea. Now the economic crisis is starting again and it is very important to finalize the achievement of artificial intelligence and streamline them into industry exactly now. Only the increase of the supply side productivity (by the artificial intelligence) can save the us from the coming crisis.

]]>example of nominal sets with separated tensor added, see Chapter 3.4 of Pitts monograph Nominal Sets

Alexander Kurz

]]>At the old entry *cohomotopy* used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

]]>I am resuming my old unfinished (and unublished) work on universal noncommutative flag varieties and noncommutative Grassmannians. One of the motivations has some avatars in operator theoretic setting and in relation to integrable systems. Thus I started revising pages and (re)collecting references on infinite-dimensional Grassmann varieties and creating some new pages like this one for Sato Grassmannian.

]]>I fixed a link to a pdf file that was giving a general page, and not the file!

]]>Cross-linked.

]]>removed link to old philosophy paper

steveawodey

]]>starting something. Not done yet but need to save

]]>added some details

]]>Page created, more to come.

]]>Hello. There is possible confusion between the notions of saturated set and saturated subset, which have different pages and mean different things. Any advice on how to handle this?

Thanks!

]]>created page, more to come

]]>I wrote out a proof which uses very little machinery at fundamental theorem of algebra. It is just about at the point where it is not only short and rigorous, but could be understood by an eighteenth-century mathematician. (Nothing important, just fun!)

]]>Defined a complete flag so I can link here from another page in good conscience.

Mike Pierce

]]>Page created, but author did not leave any comments.

]]>Created page, to be renamed to “stably compact space”, more to come.

]]>I added few words about the derived background behind the Duflo map (all comes from a deep insight of Kontsevich, later detailed by many authors).

]]>I split some material specific to Kashiwara-Vergne conjecture from Hausdorff series and added more references and a quoted idea from a seminar page. Related to Drinfeld associator (with which it has a serious overlap, especially in references) and few related pages I work on these days.

]]>added pointer to

- Dev Sinha, Section 1 in:
*Koszul duality in algebraic topology - an historical perspective*, J. Homotopy Relat. Struct. (2013) 8: 1 (arXiv:1001.2032)

am finally splitting this off from *Hopf degree theorem*, to make the material easier to navigate. Still much room to improve this entry further (add an actual Idea-statement to the Idea-section, add more examples, etc.)

Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a *finite* group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the **permutation representation** in the standard representation theory books like in

- Claudio Procesi,
*Lie groups, an approach through invariants and representations*, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

]]>added pointer to Bredon 72. Will add this pointer also to various related entries on equivariant homotopy theory

]]>tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

]]>added pointer to today’s

- Andrea Fontanella, Tomas Ortin,
*On the supersymmetric solutions of the Heterotic Superstring effective action*(arxiv:1910.08496)

started a bare minimum at *Poisson-Lie T-duality*, for the moment just so as to have a place to record the two original references