Andrés Cordón-Franco

In the present paper, we address the following general question in the framework of classical first-order logic. Assume that a certain mathematical principle can be formalized in a first-order language by a set E of conditional formulas of the form $\alpha \left(v\right)\to \beta \left(v\right)$. Given a base theory T, we can use the set of conditional formulas E to extend the base theory in two natural ways. Either we add to T each formula in E as a new axiom (thus obtaining a theory denoted by $T+E$) or we extend T by using the formulas in E as instances of an inference rule (thus obtaining a theory denoted by $T+E\text{–}\mathrm{Rule}$). The theory $T+E$ will be stronger than $T+E\text{–}\mathrm{Rule}$, but how much stronger can $T+E$ be? More specifically, is $T+E$ conservative over $T+E\text{–}\mathrm{Rule}$ for theorems of some fixed syntactical complexity $\mathsf{\Gamma }$? Under very general assumptions on the set of conditional formulas E, we obtain two main conservation results in this regard. Firstly, if the formulas in E have low syntactical complexity with respect to some prescribed class of formulas $\mathsf{\Pi }$ and in the applications of $E\text{–}\mathrm{Rule}$ side formulas from the class $\mathsf{\Pi }$ and can be eliminated (in a certain precise sense), then $T+E$ is $\forall \mathcal{B}\left(\mathsf{\Pi }\right)$-conservative over $T+E\text{–}\mathrm{Rule}$. Secondly, if, in addition, E is a finite set with m conditional sentences, then nested applications of $E\text{–}\mathrm{Rule}$ of a depth at most of m suffice to obtain $\forall \mathcal{B}\left(\mathsf{\Pi }\right)$ conservativity. These conservation results between axioms and inference rules extend well-known conservation theorems for fragments of first-order arithmetics to a general, purely logical framework.

In van Benthem (2008), van Benthem proposes a dynamic consequence relation defined as ${\psi _1}, \ldots ,{\psi _n}{ \models ^d}\phi \,{\rm{iff}}{ \models ^{pa}}[{\psi _1}] \ldots [{\psi _n}]\phi ,$ where the latter denotes consequence in public announcement logic, a dynamic epistemic logic. In this paper we investigate the structural properties of a conditional dynamic consequence relation $\models _{\rm{\Gamma }}^d$ extending van Benthem’s proposal. It takes into account a set of background conditions Γ, inspired by Makinson (2003) wherein Makinson calls this reasoning ‘modulo’ a set Γ. In the presence of common knowledge, conditional dynamic consequence is definable from (unconditional) dynamic consequence. An open question is whether dynamic consequence is compact. We further investigate a dynamic consequence relation for soft instead of hard announcements. Surprisingly, it shares many properties with (hard) dynamic consequence. Dynamic consequence relations provide a novel perspective on reasoning about protocols in multi-agent systems.