We discuss subshifts of finite type (tilings) that combine virtually opposite properties, being at once very simple and very complex. On the one hand, the combinatorial structure of these subshifts is rather simple: we require that all their configurations are quasiperiodic, or even that all configurations contain exactly the same finite patterns (in the last case a subshift is transitive, i.e., irreducible as a dynamical system). On the other hand, these subshifts are complex in the sense of computability theory: all their configurations are non periodic or even non-computable, or all their finite patterns have high Kolmogorov complexity, the Turing degree spectrum is rather sophisticated, etc.
We start with the simplest example of such centaurisme with an SFT that is minimal and contains only aperiodic (and quasiperiodic) configurations. Then we discuss how far these heterogeneous properties can be strengthened without getting mutually exclusive.
This is a joint work with Bruno Durand (Univ. de Montpellier).[-]

Goedel's theorem on the completeness of propositional and first-order logic allows one to choose between a plurality of 'equivalent' methods to evaluate the validity of arguments: for example, natural deduction in a syntactic approach, and truth tables, refutation trees, model theory in a semantic approach. However, there is a practical asymmetry between the two approaches when it comes to making actual evaluations. For example, to show that an argument is invalid, it is easier to show a semantic counterexample than to provide a proof, whereas the opposite is true when it comes to showing that it is valid. When one moves to the terrain of natural language to study ordinary argumentation, the distance between syntax and semantics further increases, because of the role played by pragmatics.
There are three basic ways to add pragmatics to logic. One can keep syntax and semantics unchanged (thus maintaining symmetry as much as possible) and add pragmatics to extend formal logic in such a way that it could satisfactorily apply to natural languages. On the other hand, one can integrate pragmatics into syntactics (by extending admissible derivation rules and schemes) and into semantics (by generalizing the notion of possible world or by including context parameters in the semantic analysis of propositions). Finally, a third way grounds semantics directly on pragmatics, abandoning the idea that one can analyze the relationship between language, thought and the world regardless of human interactions.
Three open questions then come to the forefront. From a logical point of view, the question is whether one can recover a symmetry between syntax and semantics after the pragmatic turn. Or are we forced out of Goedel's completeness paradise? From an argumentation theory perspective, the question is no longer that about the relationship between formal and informal approaches, but about how syntax, semantics, and pragmatics relate to each other. From a comparative perspective that takes into account results in linguistics and philosophy of language, how should we understand the relationship between logic and argumentation theory? Does the former explain how language can speak about the world through the senses of the words, and the latter explain how language is the result of human interactions and is therefore governed by the need to coordinate and facilitate interactions? Or do they have the same objectives?[-]