Ketland, Jeffrey
(2020)
Bases for Structures and Theories I.
Logica Universalis, 14.
pp. 357381.
Abstract
Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or coordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature $P = \{P_i\}_{i \in I_P}$ be given. For a set $\Phi = \{\phi_i\}_{i \in I_{\Phi}}$ of $L_P$formulas, we introduce a corresponding set $Q = \{Q_i\}_{i \in I_{\Phi}}$ of new relation symbols and a set of explicit definitions of the $Q_i$ in terms of the $\phi_i$. This is called a definition system, denoted $d_{\Phi}$. A definition system $d_{\Phi}$ determines a \emph{translation function} $\tau_{\Phi} : L_Q \to L_P$. Any $L_P$structure $A$ can be uniquely definitionally expanded to a model $A^{+} \models d_{\Phi}$, called $A + d_{\Phi}$. The reduct $A + d_{\Phi}$ to the $Q$symbols is called the \emph{definitional image} $D_{\Phi}A$ of $A$. Likewise, a theory $T$ in $L_P$ may be extended a definitional extension $T + d_{\Phi}$; the restriction of this extension $T + d_{\Phi}$ to $L_Q$ is called the \emph{definitional image} $D_{\Phi}T$ of $T$. If $T_1$ and $T_2$ are in disjoint signatures and $T_1 + d_{\Phi} \equiv T_2 + d_{\Theta}$, we say that $T_1$ and $T_2$ are \emph{definitionally equivalent} (wrt the definition systems $d_{\Phi}$ and $d_{\Theta}$). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories.
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