### (Under construction)

### Ambiguity Logics

There are several approaches to ambiguity and logic which are interesting:

- Modal logics of multi-agent systems
- Logics where interpretation functions are not functions (or functions into power sets)

In my work, I only consider logics which extend classical logic with a binary connective || representing ambiguity. This means that ambiguity is explicitly represented in logical syntax. The underlying motivation is as follows: ambiguity, implicit in natural language, should be made explicit in the translation into a logical language. The logical language allows to define sound and complete inference systems.

### Short (Personal) History on the Topic

Van Eijck and Jaspars (1995) and van Deemter (1996) have written excellent articles on the topic of ambiguity logics in our sense, and provided important results. After this, they (and anyone else) seem to have lost interest. I conjecture the reason for this might have been twofold: firstly, my predecessors did not approach the problem algebraically (which I did by pure chance and which then lead to a paradox, see below). Secondly, my predecessors might have abandoned the propositional calculus too quickly, which I fully understand. On the contrary for me anything beyond propositional logic is messy, so I tend to stick to it.

The Ambiguity Saga started for me in 2016, together with Timm Lichte (who however soon abandoned the project). Point of departure was an apparently paradoxical result which was easy to obtain: A Boolean algebra with an ambiguity connective ||, satisfying some basic properties of ambiguity, is necessarily trivial (in a somewhat technical way, see the 2016-paper, which by now is outdated).

My next guess was to weaken the properties and use logic together with algebra. The result is presented in this 2017-paper, which is not particularly interesting: whereas using logic (instead of algebra) turned out to be crucial, I still used logics with algebraic semantics, which made the whole endeavour pointless.

The big progress came in recognizing that the problem was not some algebraic axiom, but algebra itself, in particular the fact that

1. in every class of algebras, validity of equations is preserved by uniform substitutions of atoms

2. in every algebra, identical elements can be exchanged preserving truth/validity of equations

It turned out that these two properties, together with basic properties of ambiguity, lead to triviality. Logically, this corresponds to

1′. closure under uniform substitution of atoms

2′. closure under the rule of cut

In the 2021-article, this is explained and proved; this article also generalizes the results from 2016 and 2017. The 2021-article focuses on a particular logic which is closed under uniform substitution but not cut, which I considered most apt for reasoning with ambiguity in a linguistic context.

Again I had a hard time to broaden my view: there is not one logic of ambiguity. I understood that closure under 1′. corresponds to trust, closure under 2′. corresponds to distrust. From here it was a small step to see that ambiguity needs logical pluralism. I tried to define and investigate the Family of Ambiguity Logics in the 2022-manuscript.

So here is what I think now: there is a field of ambiguity logic with some fundamental results:

- The Fundamental Theorem (proved in the 2021-article, but called such in the 2022-manuscript), which states that one basic closure property must be lacking in every ambiguity logic (AL)
- The Trust Theorem, which states that every distrustful AL can be extended to a trustful AL, but never vice versa (proved in the 2022-manuscript)
- The In-Out Lemma, which states (simplified) that every trustful AL contains a distrustful AL as inner logic
- Finally, there is the fundamental definition of Ambiguity Logics in the 2022-manuscript (section 5). This is of course arguable (as any definition), but I think it is the one which makes most sense.