## A formal language for reasoning about indiscernibility

B. Konikowska. A formal language for reasoning about indiscernibility. Bulletin of the Polish Academy of Sciences, 35:239–249, 1987.

#### Abstract:

Fix $m$ and $n$, and consider structures consisting of a set of $n$ entities and a set of $m$ properties, and for each set $P$ of properties an equivalence relation $\romanind(P)$ on the set $E$ of entities (thought of as indiscernibility with respect to properties in $P$; thus $\romanind(P\cup Q)= \romanind(P)\cap\romanind(Q)$, and $\romanind(\varnothing)$ is the universal relation on $E$). Now consider a language having expressions denoting each set of properties (using the Boolean operations) and whose formulas denote subsets of $E$. The formulas include constants denoting the singleton subsets, are closed under the Boolean operations, and if $F$ is a formula denoting the set $S$ of entities and $A$ is an expression denoting the set $P$ of properties then $\romanind\sb *(A)F$ denotes the set $\e\in E\:(\forall f)$ (if $(e,f)\in\romanind(P)$ then $f\in S)\$, and $\romanind\sp *(A)F$ denotes $\e\in E\:$ $(\exists f)((e,f)\in\romanind(P)$ and $f\in S)\$. A formula is valid if it always denotes the whole set $E$. The author obtains an axiomatization of the valid formulas.

@ARTICLE{konikowska87,
author = {B. Konikowska},
title = {A formal language for reasoning about indiscernibility},
journal = {Bulletin of the Polish Academy of Sciences},
year = {1987},
volume = {35},
pages = {239--249},
abstract = {
Fix $m$ and $n$, and consider structures consisting of a set of $n$
entities and a set of $m$ properties, and for each set $P$ of properties
an equivalence relation $\roman{ind}(P)$ on the set $E$ of entities
(thought of as indiscernibility with respect to properties in $P$;
thus $\roman{ind}(P\cup Q)= \roman{ind}(P)\cap\roman{ind}(Q)$, and
$\roman{ind}(\varnothing)$ is the universal relation on $E$). Now
consider a language having expressions denoting each set of properties
(using the Boolean operations) and whose formulas denote subsets
of $E$. The formulas include constants denoting the singleton subsets,
are closed under the Boolean operations, and if $F$ is a formula
denoting the set $S$ of entities and $A$ is an expression denoting
the set $P$ of properties then $\roman{ind}\sb *(A)F$ denotes the
set $\{e\in E\:(\forall f)$ (if $(e,f)\in\roman{ind}(P)$ then $f\in S)\}$, and $\roman{ind}\sp *(A)F$ denotes $\{e\in E\:$ $(\exists f)((e,f)\in\roman{ind}(P)$ and $f\in S)\}$. A formula is valid if
it always denotes the whole set $E$. The author obtains an axiomatization
of the valid formulas. }
}


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