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 translation function\(\tau _{\Phi } : L_Q \rightarrow L_P\). 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 Qsymbols is called the 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 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 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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1 Introduction
Sometimes theories are formulated with different sets of primitives and yet are definitionally equivalent. The nonlogical primitives of a formalized language L are called its signature. There are many examples of theories (often involving formalized systems of arithmetic and set theory) formulated in very different signatures, which are nonetheless “equivalent”. To take a simple example, consider the theory \(T_1\) of a reflexive relation:
expressed with a binary relation symbol P. Suppose we introduce a new binary relation symbol, Q, and give an explicit definition of it—call this definition \(d_1\)—in terms of P as follows:
Call the extended theory \(T_1 + d_1\). Then \(T_1 + d_1 \vdash \forall x \lnot Q(x,x)\).
Let \(T_2\) be the theory saying that Q is an irreflexive relation:
Consider following “inverse definition” of P in terms of Q, call it \(d_2\):
Call the extended theory \(T_2 + d_2\). Then \(T_2 + d_2 \vdash \forall x P(x,x)\). Moreover, we have:
These two theories, \(T_1\) and \(T_2\), are in fact equivalent with respect to these definitions. That is, \(T_1 + d_1\) and \(T_2 + d_2\) are logically equivalent:
We say that \(T_1\) and \(T_2\) have a “common definitional extension” and are therefore “definitionally equivalent” (see Definition 27 below).
As the reader may have guessed, this example derives from the standard mathematical method of passing between a nonstrict preorder \(\le \) and its corresponding strict preorder <: they may be defined in terms of each other. To put the above in the more standard notation:
In effect, the explicit definitions \(d_1\) and \(d_2\) are “mutual inverses”. This notion will play a major role later.
Moving on to increasingly sophisticated examples, there are equivalent reformulations of Peano arithmetic \( \mathsf {PA}\) with different primitives from the usual signature \(\{0,S,+,\times \}\). For example, one might take exponentiation as the single basic notion, along with certain axioms for exponentiation, along with explicit definitions for \(0,S,+\) and \(\times \) (along with induction). The result is equivalent to \( \mathsf {PA}\).
As is wellknown, one can interpret a modification we shall call \( \mathsf {ZF}_0\) of \( \mathsf {ZF}\) set theory, obtained by removing the axiom of infinity, adding its negation, along with an additional axiom of “transitive containment”, into \( \mathsf {PA}\). Moreover, the inverse of this interpretation interprets \( \mathsf {PA}\) into \( \mathsf {ZF}_0\).^{Footnote 1} So \( \mathsf {PA}\) is definitionally equivalent to the theory of finite sets\( \mathsf {ZF}_0\).
A similar relationship holds between the theory of formalized syntax and \( \mathsf {PA}\). Suppose \(\textsf {S}_A\) is the theory of concatenation for strings from alphabet A, with \(A \ge 2\), and with the appropriate induction principle. Then \(\textsf {S}_A\) is definitionally equivalent to \( \mathsf {PA}\).^{Footnote 2}
There are other examples—from mathematics, logic and philosophy of science.^{Footnote 3} To return to the broader point, the transformations between such equivalent formulations are rather like “basis transformations” in linear algebra and other parts of mathematics. In this paper and the followup, an analogous idea is investigated.^{Footnote 4}
2 Syntax, Structures and Theories
Throughout, everything we consider is 1sorted, relational and firstorder.
Definition 1
Let \(P = \{P_i\}_{i \in I}\) be a set and let \(a: P \rightarrow \mathbb {N}\). The pair \(S = (P, a)\) is called a onesorted signature, and a is called the arity function for S. The multiset \(t = (a(P_i) \mid P_i \in P)\) is called the similarity type of the signature S. If \(a(P_i) = 0\), then \(P_i\) is called a sentence letter (or a propositional atom). The alphabet of S is P.
Definition 2
\((P^c,a^c)\) is a copy of (P, a) iff the similarity types of (P, a) and \((P^c,a^c)\) are the same. \((P^c,a^c)\) is a disjoint copy of (P, a) when, in addition, \(P \cap P^c = \varnothing \).
Definition 3
\(L_P\) is the firstorder language over the signature \(S = (P,a)\), where each symbol \(P_i\) is a primitive relation symbol of arity \(a(P_i)\). We will sometimes call \(L_P\) “the Planguage”.
Definition 4
Let
be a set of \(L_P\)formulas, which will be called defining formulas. Given the \(\phi _i\), a corresponding set
of new relation symbols \(Q_i\) is introduced, such that the arity of each \(Q_i\) matches the arity of its corresponding \(\phi _i\). The new language \(L_Q\) will sometimes be called “the Qlanguage”. The combined language is then called \(L_{P,Q}\).
A theory T in L is a set of Lsentences. When we require deductive closure, we say so. We use a deductive system such that, if \(\Delta \vdash \alpha \), then \(\Delta \vdash \forall x \alpha \), so long as x doesn’t appear free in any formulas in \(\Delta \).^{Footnote 5}\(T \vdash \alpha \) means: there exists a derivation of \(\alpha \) from the axioms/rules of T. An Lstructure A will always interpret all variables. So, we can always write \(A \models \alpha \), even where \(\alpha \) has free variables, since A will assign a value \(x^A\) to each variable \(x \in \text {FV}(\alpha )\).^{Footnote 6} The Completeness Theorem holds in the usual form: \(\Delta \vdash \alpha \text { iff, for any } A \models \Delta , A \models \alpha \). A theory T in \(L_P\) is said to be deductively closed iff, for all \(\alpha \in \text {Sent}(L_P)\), if \(T \vdash \alpha \), then \(\alpha \in T\). The deductive closure of T (written \(\text {DedCl}(T)\)) is \(\{\alpha \in \text {Sent}(L_P) \mid T \vdash \alpha \}\).
So far as I can tell, nothing in this paper uses either proof theoretic methods or model theoretic methods beyond what is taught at intermediate logic.^{Footnote 7} We do introduce specific new terminology for the following notions:
Definition systems  \(d_{\Phi }\). 
Definitional expansions/extensions  \(A \mapsto A + d_{\Phi }\) and \(T \mapsto T + d_{\Phi }\). 
Definitional image operator  \(A \mapsto D_{\Phi }A\) and \(T \mapsto D_{\Phi }T\). 
Definitional equivalence  \(A \underset{\Theta }{\overset{\Phi }{\longleftrightarrow }} B\) and \(T_1 \underset{\Theta }{\overset{\Phi }{\longleftrightarrow }} T_2\). 
Definition 5
A structureA for the language \(L_P\) specifies a nonempty domain, \(\text {dom}(A)\); and interprets each variable x of \(L_P\) as an element \(x^{A} \in \text {dom}(A)\); and interprets each nary relation symbol \(P_i\) as a nary relation \((P_i)^{A} \subseteq (\text {dom}(A))^n\).
Definition 6
Given signature P, let \(P^c\) be a disjoint copy of P. Let A be an \(L_P\)structure. Then the disjoint copy\(A^c\) of A in \(L_{P^c}\) is defined by setting \(\text {dom}(A^c) = \text {dom}(A)\) and, for each \(P_i\), setting \((P^c_i)^{A^c} = (P_i)^A\). Let T be an \(L_P\)theory. Then the disjoint copy\(T^c\) of T in \(L_{P^c}\) is defined by replacing every occurrence of \(P_i\), in any sentence in T by the new symbol \(P^c_i\).
We may next give inductive definitions of the denotation function \(t \mapsto t^{A}\) (specifying what any term t in \(L_P\) refers to in A) and the satisfaction relation \(\models \) between A and \(L_P\)formulas.
Definition 7
If \(A_1\) and \(A_2\) are \(L_{P}\)structures, then an isomorphism
is a bijection from \(\text {dom}(A_1)\) to \(\text {dom}(A_2)\) satisfying the preservation condition that \(f[(P_i)^{A_1}] = (P_i)^{A_2}\), for each relation symbol \(P_i\) in the signature P. This is written
(or simply \(A_1 \cong A_2\) if the isomorphism is left implicit).
Definition 8
If A is an \(L_P\)structure and \(T, T_1,T_2\) are sets of \(L_P\)sentences:
(1)  \(\text {Th}_{L_P}(A)\)  :=  the set of \(L_P\)sentences true in A. 
(2)  \(A \models T\)  :=  for all \(\alpha \in T\), \(A \models \alpha \). 
(3)  \(\text {Mod}_{L_P}(T)\)  :=  the class of \(L_P\)structures \(A \models T\) 
(4)  \(T_1 \equiv T_2\)  :=  \(\text {Mod}(T_1) = \text {Mod}(T_2)\). 
(5)  \(T_1 \vdash T_2\)  :=  for all \(\alpha \in T_2\), \(T_1 \vdash \alpha \). 
\(\text {Th}_{L_P}(A)\) is called \(\text {Th}(A)\) if it’s clear from context what language is involved. Likewise, \(\text {Mod}_{L_P}(T)\) is called \(\text {Mod}(T)\) if it’s clear from context what language is involved.
If \(T_1\) and \(T_2\) are deductively closed theories, then \(T_1 \vdash T_2 \text { iff } T_2 \subseteq T_1\). The Completeness Theorem tells us that \(T_1 \equiv T_2\) iff \(T_1 \vdash T_2\) and \(T_2 \vdash T_1\).
Definition 9
Let A be an \(L_P\)structure and \(A^{+}\) be an \(L_{P,Q}\)structure. Then \(A^{+}\) is an expansion of A iff for all \(P_i\), \((P_i)^{A^{+}} = (P_i)^A\). This is equivalent to saying that A is a reduct of \(A^{+}\). If \(A^{+}\) is an \(L_{P,Q}\)structure, its reduct to P is denoted \(A^{+} \upharpoonright _{P}\) (an \(L_P\)structure) and its reduct to Q is denoted \(A^{+} \upharpoonright _{Q}\) (an \(L_Q\)structure) and we have: \((P_i)^{A^{+} \upharpoonright _{P}} = (P_i)^{A^+}\) and \((Q_i)^{A^{+} \upharpoonright _{Q}} = (Q_i)^{A^+}\).
The central property of expansions is that the truth value of a formula in the smaller language \(L_P\) remains invariant as we pass from an \(L_P\)structure to an expanded structure for \(L_{P,Q}\): if \(\alpha \in L_P\) and an \(L_{P,Q}\)structure \(A^{+}\) is an expansion of an \(L_P\)structure A, then \(A^{+} \models \alpha \text { iff } A \models \alpha \).
Definition 10
A theory \(T^{+}\) is an extension of T iff T is a subset of \(T^{+}\). Let signatures P, Q, and corresponding languages \(L_P, L_Q\) and \(L_{P,Q}\) be given. An extension \(T^{+}\) in \(L_{P,Q}\) of T in \(L_P\) is called a conservative extension of T with respect to \(L_P\)formulas iff, for any \(L_P\)formula \(\alpha \),
3 Definition
Definition 11
Given the set \(\Phi = \{\phi _i\}_{i \in I}\) of \(L_P\)formulas, we introduce a disjoint set \(Q = \{Q_i\}_{i \in I}\) of new relation symbols, with \(\text {card} \ Q = \text {card} \ \Phi \), and with the arity of \(Q_i\) matching the arity of \(\phi _i\), and let \(n_i\) be \(a(\phi _i)\). The definition system over \(\Phi \), which we write as,
is the set of explicit definitions,
where \(\{x_1,\dots x_{n_i}\} = \text {FV}(\phi _i)\). These define the new symbols \(Q_i\) in terms of the \(L_P\)formulas \(\phi _i\). We shall sometimes write \(\forall \overline{x}(Q_i( \overline{x}) \leftrightarrow \phi _{i})\) instead of \(\forall x_1 \dots x_n(Q_i(x_1, \dots , x_n)\leftrightarrow \phi _{i})\).^{Footnote 8}
Definition 12
If A is an \(L_P\)structure and \(\phi (x_1,\dots ,x_n)\) is an \(L_P\)formula, then
is the relation that \(\phi \) defines in A.
Definition 13
An \(L_{P,Q}\)structure \(A^{+}\) is a definitional expansion of an \(L_P\)structure A with respect to the definition system \(d_{\Phi }\) just if \(A^{+}\) is an expansion of A interpreting each \(Q_i\), and \(A^{+} \models d_{\Phi }\). This ensures that \((Q_i)^{A^{+}} = \phi _i(A^{+})\), for each \(Q_i\).
Given any \(L_P\)structure A, it is clear that there is a unique definitional expansion \(A^{+} \models d_{\Phi }\). We introduce the following notation for this expansion:
Definition 14
\(A + d_{\Phi }\) := the unique definitional expansion \(A^{+} \models d_{\Phi }\).
We are going to treat the definitional expansion map
as a unary operator \(+ d_{\Phi }\) (indexed by \(\Phi \)), taking us from \(L_P\)structures to \(L_{P,Q}\)structures. It is clear that it is welldefined (i.e., unique, given \(\Phi \)). It also satisfies the following useful “right cancellation” law (this amounts, in essence, to taking a reduct):
Lemma 1
If \(A_1 + d_{\Phi } \cong A_2 + d_{\Phi }\) then \(A_1 \cong A_2\).
Definition 15
A relation symbol \(P_i\) in the signature P is explicitly definable in T just if
for some formula \(\theta \) in the language of the subsignature \(P {\setminus } P_i\). We say that \(\theta \) is a defining formula for \(P_i\) in T.
Definition 16
A relation symbol \(P_i\) in the signature P is implicitly definable in T in \(L_P\) just in case, given any pair of models \(A, B \models T\), with \(\text {dom}(A)=\text {dom}(B)\) and which assign the same extension to all \(P_j\)except\(P_i\), we have \((P_i)^A = (P_i)^B\).
Beth’s Theorem states that a relation symbol \(P_i\) is implicitly definable in T iff \(P_i\) is explicitly definable in T.^{Footnote 9}
Definition 17
Given a definition system \(d_{\Phi }\), the definitional extension of T wrt \(\Phi \) is \(T + d_{\Phi }\). We say that \(T^{+}\) in \(L_{P,Q}\) is a definitional extension of T in \(L_P\) just if
for some definition system \(d_{\Phi }\), where \(\Phi \) is some set of \(L_P\)formulas.
So, each new relation symbol \(Q_i\) is explicitly defined in \(T + d_{\Phi }\) and the defining formula for \(Q_i( \overline{x})\) is simply \(\phi _i\). Analogous to what we did with structures, we are going to treat the definitional extension map
for theories as a unary operator \(+ d_{\Phi }\) (indexed by \(\Phi \)), taking us from \(L_P\)theories to \(L_{P,Q}\)theories. Again, it is welldefined (i.e., unique, given \(\Phi \)) and satisfies an analogous “right cancellation” law for \(L_P\)theories:
Lemma 2
The following are straightforward consequences of the definitions:

(1)
If \(T_1 + d_{\Phi } \equiv T_2 + d_{\Phi }\) then \(T_1 \equiv T_2\).

(2)
\(A \models T\) iff \(A + d_{\Phi } \models T + d_{\Phi }\).

(3)
If \(B \models T + d_{\Phi }\), then \(B = A + d_{\Phi }\), for some \(A \models T\).
Before moving on to translations, we give three standard lemmas about definitional and conservative extensions (the converse of Lemma 3 is far from being true):
Lemma 3
If \(T^{+}\) in \(L_{P,Q}\) is a definitional extension of T in \(L_P\), then \(T^{+}\) is a conservative extension of T wrt \(L_P\)formulas.
Lemma 4
\(T + d_{\Phi }\) is a conservative extension of T for \(L_P\)formulas.
Lemma 5
Let T in \(L_P\) and \(T^{+}\) in \(L_{P,Q}\) be such that \(T \subseteq T^{+}\). Suppose that, for any model \(A \models T\), there is an expansion \(A^{+} \models T^{+}\). Then \(T^{+}\) is a conservative extension of T for \(L_P\)formulas.
4 Translation
Definition 18
Let a definition system \(d_{\Phi }\) be given. Define the translation, induced by \(\Phi \)
as follows. For symbols \(Q_i\), variables \(x,y, \overline{x}\), and for \(L_Q\)formulas \(\alpha , \alpha _1, \alpha _2\):
(1)  \(\tau _{\Phi }(Q_i(\overline{x}))\)  \( := \)  \((\phi _i)^{\prime }\) 
(2)  \(\tau _{\Phi }(x=y)\)  \( := \)  \((x=y)\) 
(3)  \(\tau _{\Phi }(\lnot \alpha )\)  \( := \)  \(\lnot \tau _{\Phi }(\alpha )\) 
(4)  \(\tau _{\Phi }(\alpha _1 \# \alpha _2)\)  \( := \)  \(\tau _{\Phi }(\alpha _1) \# \tau _{\Phi }(\alpha _2)\) 
(5)  \(\tau _{\Phi }(\mathbf {q} x \alpha )\)  \( := \)  \(\mathbf {q} x \tau _{\Phi }(\alpha )\). 
\(\#\) is any binary connective, \(\mathbf {q}\) is a quantifier and \((\phi _i)^{\prime }\) is the result of ensuring that the free variables appearing \(\phi _i\) are relabelled, to match those of \(Q_i( \overline{x})\). We call \(\tau _{\Phi }\) the translation induced by \(\Phi \). It maps from the new language \(L_Q\) back to the original language \(L_P\).^{Footnote 10}
Lemma 6
For any \(\alpha , \beta \in L_Q\), if \(\alpha \vdash \beta \) then \(\tau _{\Phi }(\alpha ) \vdash \tau _{\Phi }(\beta )\).
Proof
We will prove this using a lemma below. Suppose \(\tau _{\Phi }(\alpha ) \nvdash \tau _{\Phi }(\beta )\). This gives us a model \(A \models \tau _{\Phi }(\alpha ) \) and \(A \not \models \tau _{\Phi }(\beta )\). By Lemma 15(1) below, \(D_{\Phi }A \models \alpha \) and \(D_{\Phi } A \not \models \beta \). So, \(\alpha \nvdash \beta \). \(\square \)
Lemma 6 is a general property of translations, but its converse is not true in general.
Corresponding to a translation \(\tau _{\Phi } : L_Q \rightarrow L_P\) is its “lift” \(\tau ^{+}_{\Phi } : L_{P,Q} \rightarrow L_P\) from the combined language \(L_{P,Q}\) down to \(L_P\):
Definition 19
Let \(\Phi \) be given, along with definition system \(d_{\Phi }\). Define the lifted translation\(\tau ^{+}_{\Phi }\) induced by \(\Phi \)
as follows. For symbols \(Q_i\), \(P_i\), variables \(x,y, \overline{x}\):
(1)  \(\tau ^{+}_{\Phi }(Q_i(\overline{x}))\)  \( := \)  \((\phi _i)^{\prime }\) 
(2)  \(\tau ^{+}_{\Phi }(x=y)\)  \( := \)  \((x=y)\) 
(3)  \(\tau ^{+}_{\Phi }(P_i(\overline{x}))\)  \( := \)  \(P_i( \overline{x})\) 
Along with the requirement that \(\tau ^{+}_{\Phi }\) commutes with the logical operators on the full language \(L_{P,Q}\).
Thus, the translation \(\tau _{\Phi }\) is the restriction to \(L_Q\) of its lift, \(\tau ^{+}_{\Phi }\).
Note that because the translations we are interested in always act as the identity on equations, it is always the case that \(\vdash \alpha \leftrightarrow \tau _{\Phi }(\alpha )\) if \(\alpha \) is an equation. Thus, in inductive proofs establishing biconditionals of the form \(\alpha \leftrightarrow \tau _{\Phi }(\alpha )\), we only need to check the condition holds for atomic formulas which are not identity formulas.
Lemma 7
We have:
(1)  \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Phi }(\alpha )\)  for any \(\alpha \in L_Q\). 
(2)  \(A + d_{\Phi } \models \alpha \leftrightarrow \tau ^{+}_{\Phi }(\alpha )\)  for any \(\alpha \in L_{P,Q}\). 
(3)  \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau _{\Phi }(\alpha )\)  for any \(\alpha \in L_Q\). 
(4)  \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau ^{+}_{\Phi }(\alpha )\)  for any \(\alpha \in L_{P,Q}\). 
Proof
For (2), we reason by induction. Let \(\alpha \) be an atomic \(L_{P,Q}\)formula. As noted above, for equations \((x=y)\), the translation \(\tau _{\Phi }(x=y)\) is trivially \((x=y)\): so \(\alpha \leftrightarrow \tau _{\Phi }(\alpha )\) always holds for equations. Suppose \(\alpha \) is an atomic formula of the form \(Q_i( \overline{x})\). Then its translation \(\tau ^{+}_{\Phi }(\alpha )\) is \(\phi _i\). Since we have \(A + d_{\Phi } \models \forall \overline{x}(Q_i( \overline{x}) \leftrightarrow \phi _i)\), we have: \(A + d_{\Phi } \models Q_i( \overline{x}) \leftrightarrow \tau _{\Phi }(Q_i( \overline{x}))\). Instead let \(\alpha \) be \(P_i( \overline{x})\). Then its translation \(\tau ^{+}_{\Phi }(\alpha )\) is simply \(\alpha \). So, \(A+ d_{\Phi } \models \alpha \leftrightarrow \tau ^{+}_{\Phi }(\alpha )\). The other cases are shown by induction on the construction of \(\alpha \).
For (1), the result follows from (2), by restricting to \(L_Q\)formulas (since \(\tau ^{+}_{\Phi }(\alpha ) = \tau _{\Phi }(\alpha )\) for \(\alpha \in L_Q\)).
For (4), reasoning by induction, let \(\alpha \) be an atomic \(L_P,Q\)formula. If \(\alpha \) is atomic, then the condition is trivial. Suppose \(\alpha \) has the form \(P_i( \overline{x})\). Again, he condition is trivial, since \(\tau ^{+}_{\Phi }(P_i( \overline{x}))\) is \(P_i( \overline{x})\). Instead, suppose \(\alpha \) has the form \(Q_i( \overline{x})\). Then its translation \(\tau ^{+}_{\Phi }(\alpha )\) is \(\phi _i\). \(Q_i( \overline{x}) \leftrightarrow \phi _i\) is a theorem of \(T + d_{\Phi }\), by construction. The other cases are shown by induction on the construction of \(\alpha \).
For (3), the result follows from (4), by restricting to \(L_Q\)formulas. \(\square \)
Definition 20
Let \(\tau _{\Phi } : L_Q \rightarrow L_P\) be the translation induced by \(d_{\Phi }\). If \(T_2\) is a theory in \(L_Q\), then the image of \(T_2\) under \(\tau _{\Phi }\) is the set of \(L_P\)sentences:
If \(T_1\) is a theory in \(L_P\), then the preimage of \(T_1\) under \(\tau _{\Phi }\) is the set of \(L_Q\)sentences:
Similarly, if \(\Theta = \{\theta _i\}_{i \in I_P}\) is a set of \(L_Q\)formulas and \(d_{\Theta }\) is the corresponding definition system over \(\Theta \) (for the primitives \(P_i\) of \(L_P\)), we can define a translation
by requiring that \(\tau _{\Theta }\) commute with the logical operators and, for atomic \(L_P\)formulas:
(i)  \(\tau _{\Theta }(P_i( \overline{x})) := \)  \((\theta _i)^{\prime }\) 
(ii)  \(\tau _{\Theta }(x=y) := \)  \((x=y)\) 
Likewise, we can also define the lifted translation \(\tau ^{+}_{\Theta }: L_{P,Q} \rightarrow L_Q\).
Definition 21
Let \(\tau : L_Q \rightarrow L_P\) be a translation. Let \(T_1\) be a theory in \(L_P\) and \(T_2\) be a theory in \(L_Q\). Then we say:

(1)
\(\tau \)interprets\(T_2\) into \(T_1\) iff \(T_1 \vdash \tau [T_2]\).

(2)
\(\tau \)faithfully interprets\(T_2\) into \(T_1\) iff \(T_1 \equiv \tau [T_2]\).
One may compare the condition \(T_1 \equiv \tau [T_2]\) with Visser’s definition of faithful interpretability:
We write \(K : U \triangleleft _{\text { faith}} V\) for: K is a faithful interpretation of U in V. This means that: for all Usentences A, we have: \(U \vdash A\) iff \( V \vdash A^{\tau _{K}}\). (Visser [19], p. 6).
Thus \(\tau _{\Phi } : T_2 \triangleleft _{\text { faith}} T_1\) holds iff, for all \(\alpha \in L_{T_2}\), we have: \(T_2 \vdash \alpha \) iff \(T_1 \vdash \tau _{\Phi }(\alpha )\). Thus, \(\tau _{\Phi } : T_2 \triangleleft _{\text { faith}} T_1\) iff \(T_1 \equiv \tau _{\Phi }[T_2]\). This establishes:
Lemma 8
\(\tau _{\Phi }\) faithfully interprets \(T_2\) into \(T_1\) iff, for all \(\alpha \in L_Q\), we have: \(T_2 \vdash \alpha \) iff \(T_1 \vdash \tau _{\Phi }(\alpha )\).
Definition 22
Let \(\tau _{\Phi }: L_Q \rightarrow L_P\) and \(\tau _{\Theta } : L_P \rightarrow L_Q\) be translations induced by \(d_{\Phi }\) and \(d_{\Theta }\). Let \(T_1\) be an \(L_{P}\) theory. Let \(T_2\) be an \(L_Q\) theory. Then \(\tau _{\Theta }\) is an right inverse of \(\tau _{\Phi }\) in \(T_1\) iff, for any \(\alpha \in L_P\),
We write this more suggestively as:
And \(\tau _{\Theta }\) is an left inverse of \(\tau _{\Phi }\) in \(T_2\) iff, for any \(\beta \in L_Q\),
Likewise, we write this more suggestively as:
The following two lemmas are easy to prove, and yet hold to the key to much that follows. Both lemmas use “invertibility conditions”, of the form:
As we see later, these conditions express a very strong constraint on the set \(\Phi \) of defining \(L_P\)formulas involved—the property of being a “representation basis” for A (or T) with inverse \(\Theta \).
Lemma 9
Suppose that \(A + d_{\Phi } \models d_{\Theta }\). Then:
(1)  \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Phi }(\alpha )\)  for \(\alpha \in L_Q\). 
(2)  \(A+ d_{\Phi } \models \alpha \leftrightarrow \tau ^{+}_{\Phi }(\alpha )\)  for \(\alpha \in L_{P,Q}\). 
(3)  \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Theta }(\alpha )\)  for \(\alpha \in L_P\). 
(4)  \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\)  for \(\alpha \in L_Q\). 
(5)  \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\)  for \(\alpha \in L_P\). 
(6)  \(A + d_{\Phi } \models \tau _{\Phi }(\theta _i) \leftrightarrow \theta _i\).  
(7)  \(A + d_{\Phi } \models \tau _{\Theta }(\phi _i) \leftrightarrow \phi _i \).  
(8)  \(A \models \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\)  for \(\alpha \in L_P\). 
(9)  \(A+ d_{\Phi } \models \alpha \leftrightarrow \tau ^{+}_{\Theta }(\alpha )\)  for \(\alpha \in L_{P,Q}\). 
Proof
Claims (1) and (2) are already established in Lemma 7(1,2) and do not need the side condition. (They are included for convenience of reference.)
For (3), the proof is analogous to the proof of Lemma 7(1), but using the fact that \(A + d_{\Phi } \models d_{\Theta }\). Reasoning by induction, let \(\alpha \) be an atomic \(L_P\)sentence, say \(P_i( \overline{x})\). Then its translation \(\tau _{\Theta }(\alpha )\) is \(\theta _i\). And \(A + d_{\Phi } \models \forall \overline{x}(P_i( \overline{x}) \leftrightarrow \theta _i)\). So \(A + d_{\Phi } \models P_i( \overline{x}) \leftrightarrow \theta _i\). That is, \(A + d_{\Phi } \models P_i( \overline{x}) \leftrightarrow \tau _{\Theta }(P_i( \overline{x}))\). The other cases are shown by induction on the construction of \(\alpha \).
For (4), we already have that \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Phi }(\alpha )\), for any \(\alpha \in L_Q\). But \(\tau _{\Phi }(\alpha )\) is an \(L_P\)formula. So, by (3), we have: \(A + d_{\Phi } \models \tau _{\Phi }(\alpha ) \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\). So, \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\), as required.
For (5), the reasoning is analogous to that for (4). And (6) and (7) are merely applications of (1) and (3).
For (8), using (5), we have \(A + d_{\Phi } \models \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\), for any \(\alpha \in L_P\). But \(A + d_{\Phi }\) is an expansion of the \(L_P\)structure A, and \(\alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\) is an \(L_P\)formula. Thus, \(A \models \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\).
For (9), the proof is analogous to the proof of (2), but is applied to the “lift” \(\tau ^{+}_{\Theta }: L_{P,Q} \rightarrow L_Q\) of \(\tau _{\Theta }\). Reasoning by induction, let \(\alpha \) be an atomic \(L_P\)sentence, say \(P_i( \overline{x})\). Then its translation \(\tau ^{+}_{\Theta }(\alpha )\) is \(\theta _i\). And \(A + d_{\Phi } \models P_i( \overline{x}) \leftrightarrow \theta _i\), since \(A + d_{\Phi } \models d_{\Theta }\). Instead, let \(\alpha \) be an atomic \(L_Q\)sentence, say \(Q_i( \overline{x})\). Then its translation \(\tau ^{+}_{\Theta }(\alpha )\) is \(Q_i( \overline{x})\). Trivially, \(A + d_{\Phi } \models Q_i( \overline{x}) \leftrightarrow Q_i( \overline{x})\). The other cases are shown by induction on the construction of \(\alpha \). \(\square \)
The following lemma, and the corresponding proofs, is a near repetition of the previous one, except that it deals with theories:
Lemma 10
Suppose that \(T + d_{\Phi } \vdash d_{\Theta }\). Then:
(1)  \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau _{\Phi }(\alpha )\)  for \(\alpha \in L_Q\). 
(2)  \(T+ d_{\Phi } \vdash \alpha \leftrightarrow \tau ^{+}_{\Phi }(\alpha )\)  for \(\alpha \in L_{P,Q}\). 
(3)  \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau _{\Theta }(\alpha )\)  for \(\alpha \in L_P\). 
(4)  \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\)  for \(\alpha \in L_Q\). 
(5)  \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\)  for \(\alpha \in L_P\). 
(6)  \(T + d_{\Phi } \vdash \tau _{\Phi }(\theta _i) \leftrightarrow \theta _i\).  
(7)  \(T + d_{\Phi } \vdash \tau _{\Theta }(\phi _i) \leftrightarrow \phi _i \).  
(8)  \(T \vdash \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\)  for \(\alpha \in L_P\). 
(9)  \(T+ d_{\Phi } \vdash \alpha \leftrightarrow \tau ^{+}_{\Theta }(\alpha )\)  for \(\alpha \in L_{P,Q}\). 
Proof
Essentially, a repetition of the proofs for Lemma 9.\(\square \)
Lemma 11
Suppose \(T_1\) is an \(L_P\)theory and \(T_2\) is an \(L_Q\)theory. Then:
(1)  If \((\tau _{\Phi } \tau _{\Theta }=1)_{T_1}\) then \(T_1 + d_{\Phi } \vdash d_{\Theta }\). 
(2)  If \((\tau _{\Theta } \tau _{\Phi } =1)_{T_2}\) then \(T_2 + d_{\Theta } \vdash d_{\Phi }\). 
Proof
For (1), we assume \(\tau _{\Theta }\) is a right inverse of \(\tau _{\Phi }\) in \(T_1\). I.e., for any \(\alpha \in L_P\), \(T_1 \vdash \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\). Thus, \(T_1 \vdash P_i( \overline{x}) \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(P_i( \overline{x})))\). Since \(\tau _{\Theta }(P_i( \overline{x}))\) is \(\theta _i\), we have \(T_1 \vdash P_i( \overline{x}) \leftrightarrow \tau _{\Phi }(\theta _i)\). Taking the definitional extension, \(T_1 + d_{\Phi } \vdash P_i( \overline{x}) \leftrightarrow \tau _{\Phi }(\theta _i)\). Now, for any \(L_Q\)formula \(\beta \), we have \(T_1 + d_{\Phi } \vdash \beta \leftrightarrow \tau _{\Phi }(\beta )\) from Lemma 7(2). So, since \(\theta _i \in L_Q\), \(T_1 + d_{\Phi } \vdash \theta _i \leftrightarrow \tau _{\Phi }(\theta _i)\). Thus, \(T_1 + d_{\Phi } \vdash P_i( \overline{x}) \leftrightarrow \theta _i\), as required.
We obtain (2) by relabelling everything (\(T_2\) is now a theory in \(L_Q\)). \(\square \)
Lemma 11 says that if \(\tau _{\Theta }\) is a rightinverse of \(\tau _{\Phi }\), relative to T, then every relation symbol \(P_i\) from the original language can be explicitly defined from the \(\theta _i\). In a sense, the original definition system, \(d_{\Phi }\) is a kind of inverse of \(d_{\Theta }\).
5 Definitional Images
Definition 23
Let A be an \(L_P\)structure. Then the \(L_Q\)structure \(D_{\Phi }A\) is defined by:
\(D_{\Phi }A\) is called the definitional image of A with respect to \(\Phi \).
Immediately, we see that the following three conditions provide an equivalent characterization of \(D_{\Phi }A\):
(1)  \(\text {dom}(D_{\Phi }A) = \text {dom}(A)\) 
(2)  \(x^{D_{\Phi }A} = x^{A}\), for any variable x 
(3)  \((Q_i)^{D_{\Phi }A} = \phi _i(A)\), for each new symbol \(Q_i\). 
Lemma 12
For any \(\alpha \in L_Q\): \(\alpha (D_{\Phi }A) = \alpha (A + d_{\Phi })\).
Although \(A + d_{\Phi }\) is by construction a definitional expansion of A, it is by no means automatically true that \(A + d_{\Phi }\) is a definitional expansion of \(D_{\Phi }A\). The requirement for this to hold is that each primitive relation \((P_i)^A\) be definable in \(D_{\Phi }A\), by some formula, say \(\theta _i\).
Turning to theories, we introduce analogous concepts:
Definition 24
The definitional image of T, with respect to \(\Phi \), is the restriction of the deductive closure of \(T + d_{\Phi }\) to the new language \(L_Q\). The definitional image of T with respect to \(\Phi \) is denoted \(D_{\Phi }T\). That is,
The definitional image \(D_{\Phi }T\) of a theory T in \(L_P\) is, essentially, the preimage\((\tau _{\Phi })^{1}[T]\) of T under the translation \(\tau _{\Phi }\) induced by \(\Phi \):
Lemma 13
\((\tau _{\Phi })^{1}[T] \equiv D_{\Phi }T\).
Proof
Suppose \(D_{\Phi }T \vdash \beta \), for \(\beta \in L_Q\). Thus, \(T + d_{\Phi } \vdash \beta \). But \(T + d_{\Phi } \vdash \beta \leftrightarrow \tau _{\Phi }(\beta )\), by Lemma 10(1). So, \(T + d_{\Phi } \vdash \tau _{\Phi }(\beta )\). Since \(T + d_{\Phi }\) conservatively extends T for \(L_P\)formulas, \(T \vdash \tau _{\Phi }(\beta )\). Hence, \(\beta \in (\tau _{\Phi })^{1}[T]\). And so \((\tau _{\Phi })^{1}[T_1] \vdash \beta \). Conversely, suppose \(\beta \in (\tau _{\Phi })^{1}[T]\). So, \(T \vdash \tau _{\Phi }(\beta )\). So, \(T + d_{\Phi } \vdash \tau _{\Phi }(\beta )\). But \(T + d_{\Phi } \vdash \beta \leftrightarrow \tau _{\Phi }(\beta )\), by Lemma 10(1) again. Thus, \(D_{\Phi }T \vdash \beta \). \(\square \)
Definition 25
Let A be an \(L_P\)structure, B an \(L_Q\)structure, \(T_1\) an \(L_P\)theory and \(T_2\) an \(L_Q\)theory. Then we say:
(1)  \(\Phi \)definesB in A iff \(B \cong D_{\Phi }A\). 
(2)  \(\Phi \)prooftheoretically defines\(T_2\) in \(T_1\) iff \(T_2 \equiv D_{\Phi }T_1\). 
(3)  \(\Phi \)modeltheoretically defines\(T_2\) in \(T_1\) iff \(\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]\). 
The second of these, (2), amounts to saying that \(T_2 \vdash D_{\Phi }T_1 \text { and } D_{\Phi }T_1 \vdash T_2\). The third is equivalent to saying that
Note that the restriction \(\upharpoonright _{L_Q}\) is taken after the models are extracted from the definitional extension \(T_1+d_{\Phi }\). If the restriction is taken first, we get the rather different set \(\text {Mod}((\text {DedCl}(T_1+d_{\Phi })) \upharpoonright _{L_Q})\) of models: i.e., \(\text {Mod}(D_{\Phi }T_1)\). Indeed, this is generally a superset of \(D_{\Phi }[\text {Mod}(T)]\). As Lemma 15(3) will show, we have: \(D_{\Phi }[\text {Mod}(T)] \subseteq \text {Mod}(D_{\Phi }T)\).
6 Some BookKeeping Lemmas
We next provide several groups of “bookkeeping” lemmas about translations and definitional images. The first, Lemma 14, primarily concerns theories. The second group, Lemma 15, concerns semantics and models. The third group (in particular, Lemma 16) establishes five calculationally useful equivalences for the “definition invertibility condition” on structures:
The fourth group establishes some analogous results for theories, including the main equivalence (Lemma 19) for the “definition invertibility condition” on theories:
Lemma 14
Each of the following is true:
(1)  \(T + d_{\Phi }\) is a conservative extension of \(D_{\Phi }T\) for \(L_Q\)formulas. 
(2)  \(D_{\Phi }T_1 \vdash T_2\) iff \(T_1 \vdash \tau _{\Phi }[T_2]\). 
(3)  \(\tau _{\Phi }\) interprets \(T_2\) into \(T_1\) iff \(D_{\Phi }T_1 \vdash T_2\). 
(4)  \(\tau _{\Phi }\) interprets \(D_{\Phi }T\) into T. 
(5)  \(D_{\Phi }T = \{\beta \in L_Q \mid T \vdash \tau _{\Phi }(\beta )\}\). 
(6)  \(D_{\Phi }T \vdash \beta \) iff \(T \vdash \tau _{\Phi }(\beta )\) (for any \(\beta \in L_Q\)). 
(7)  \(T_2 \equiv D_{\Phi }T_1\) iff, for all \(\beta \in L_Q\): \(T_2 \vdash \beta \text { iff } T_1 \vdash \tau _{\Phi }(\beta )\). 
(8)  \(T_2 \equiv D_{\Phi }T_1\) iff \(\tau _{\Phi }\) faithfully interprets \(T_2\) into \(T_1\). 
Proof
To establish (1), note that \(D_{\Phi }T\) is simply the restriction of the deductive closure of \(T + d_{\Phi }\) to \(L_Q\)sentences. So if \(T + d_{\Phi } \vdash \alpha \), where \(\alpha \in L_Q\), then \(D_{\Phi }T \vdash \alpha \), as required. Statement (2) simply expresses the relationship between images and preimages. Statement (3) is an immediate corollary of (2). Statement (4) is an immediate corollary of (3).
For (5), note that \(D_{\Phi }T = (\tau _{\Phi })^{1}[T]\), from Lemma 13 above. Thus, \(D_{\Phi }T = \{\beta \in L_Q \mid T \vdash \tau _{\Phi }(\beta )\}\). For (6), let \(\beta \in L_Q\). Then, from (5), we have \(D_{\Phi }T \vdash \beta \text { iff } T \vdash \tau _{\Phi }(\beta )\), as required.
For (7), suppose first that \(T_2 \equiv D_{\Phi }T_1\). So, for all \(\beta \in L_Q\), \(T_2 \vdash \beta \) iff \(D_{\Phi }T_1 \vdash \beta \). But from (6), \(D_{\Phi }T_1 \vdash \beta \) iff \(T_1 \vdash \tau _{\Phi }(\beta )\). So, for all \(\beta \in L_Q\), \(T_2 \vdash \beta \) iff \(T_1 \vdash \tau _{\Phi }(\beta )\), as required.
Instead suppose that for all \(\beta \in L_Q\), we have: \(T_2 \vdash \beta \text { iff } T_1 \vdash \tau _{\Phi }(\beta )\). Then reason as follows:
\(T_2\vdash \beta \)  iff \(T_1 \vdash \tau _{\Phi }(\beta )\) (from the hypothesis) 
iff \(T_1 + d_{\Phi } \vdash \beta \) (as \(T_1 + d_{\Phi } \vdash \beta \leftrightarrow \tau _{\Phi }(\beta )\); Lemma 10(1))  
iff \(D_{\Phi }T_1 \vdash \beta \) (from the definition of \(D_{\Phi }T_1\); i.e., Definition 24). 
So, \(T_2 \equiv D_{\Phi }T_1\), as required.
(8) is an immediate corollary of (7) and Lemma 8. \(\square \)
Lemma 15
Each of the following holds:
(1)  For any \(\alpha \in L_Q\): \(D_{\Phi }A \models \alpha \) iff \(A \models \tau _{\Phi }(\alpha )\). 
(2)  If \(A \models T\), then \(D_{\Phi }A \models D_{\Phi }T\). 
(3)  \(D_{\Phi }[\text {Mod}(T)] \subseteq \text {Mod}(D_{\Phi }T)\). 
(4)  Let \(T_2\) be a theory in \(L_Q\). If \(A + d_{\Phi } \models T_2\) then \(D_{\Phi }A \models T_2\). 
Proof
(1): By induction. Suppose \(\alpha = Q_i( \overline{x})\). Thus,
The case of equality, and the induction steps are routine.
(2): Let \(A \models T\) and \(D_{\Phi }T \vdash \alpha \), for some \(\alpha \in L_Q\). We want to show \(D_{\Phi }A \models \alpha \). We do this using the translation \(\tau _{\Phi }\). Since \(D_{\Phi }T \vdash \alpha \), it follows, by Lemma 14(4), that \(T \vdash \tau _{\Phi }(\alpha )\). And so, \(A \models \tau _{\Phi }(\alpha )\). From (1), \(D_{\Phi }A \models \alpha \) iff \(A \models \tau _{\Phi }(\alpha )\). So, \(D_{\Phi }A \models \alpha \), as required.
(3): Let \(B \in D_{\Phi }[\text {Mod}(T)]\). So, there is some \(A \models T\) with \(B \cong D_{\Phi }A\). Since \(A \models T\), we have \(D_{\Phi } A \models D_{\Phi }T\), by (2). So, \(B \in \text {Mod}(D_{\Phi }T)\).
(4): Suppose that for any \(\beta \) such that \(T_2 \vdash \beta \), we have \(A + d_{\Phi } \models \beta \). We want to show that for any \(\beta \) such that \(T_2 \vdash \beta \), we have \(D_{\Phi }A \models \beta \). So, let \(T_2 \vdash \beta \) and \(A + d_{\Phi } \models \beta \). By Lemma 7(1), \(A + d_{\Phi } \models \tau _{\Phi }(\beta )\). So, \(A \models \tau _{\Phi }(\beta )\). By (1), \(D_{\Phi }A \models \beta \). \(\square \)
Neither the converse of Lemma 15(2) nor the converse of the inclusion in Lemma 15(3) is true.
Lemma 16
The following are equivalent:
(1)  \(A + d_{\Phi } \models d_{\Theta }\). 
(2)  For all \(P_i\), for all \(Q_j\): \((P_i)^{A + d_{\Phi }} = \theta _i(A + d_{\Phi })\) & \((Q_j)^{A + d_{\Phi }} = \phi _j(A + d_{\Phi })\). 
(3)  \(A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }\). 
(4)  \(D_{\Theta }D_{\Phi }A = A\). 
(5)  \(A \models \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\) (with \(\alpha \in L_P\)) 
Proof
For \((1) \Leftrightarrow (2)\). This is simply unwinding the definitions.
For \((1) \Rightarrow (3)\), suppose \(A + d_{\Phi } \models d_{\Theta }\). Then, from (2), we have, for all \(P_i\), for all \(Q_j\),
\((P_i)^{A + d_{\Phi }} = \theta _i(A + d_{\Phi })\) & \((Q_j)^{A + d_{\Phi }} = \phi _j(A + d_{\Phi })\). 
So, since \(\theta _i(A + d_{\Phi }) = \theta _i(D_{\Phi }A + d_{\Theta }) = (P_i)^{D_{\Phi }A +d_{\Theta } }\), we have, for all \(P_i\), for all \(Q_j\),
\((P_i)^{A + d_{\Phi }} = (P_i)^{D_{\Phi }A +d_{\Theta } }\) & \((Q_j)^{A + d_{\Phi }} = \phi _j(A + d_{\Phi })\). 
So, since \(\phi _j(A + d_{\Phi }) = (Q_j)^{D_{\Phi }A + d_{\Theta }}\), we have, for all \(P_i\), for all \(Q_j\),
\((P_i)^{A + d_{\Phi }} = (P_i)^{D_{\Phi }A +d_{\Theta } }\) & \((Q_j)^{A + d_{\Phi }} = (Q_j)^{D_{\Phi }A + d_{\Theta }}\). 
So, \(A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }\), as required.
For \((3) \Rightarrow (1)\). This reverses the reasoning \((1) \Rightarrow (3)\). Suppose \(A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }\). That is, for all \(P_i\), for all \(Q_j\),
\((P_i)^{A + d_{\Phi }} = (P_i)^{D_{\Phi }A +d_{\Theta } }\) & \((Q_j)^{A + d_{\Phi }} = (Q_j)^{D_{\Phi }A + d_{\Theta }}\). 
Now \((Q_j)^{D_{\Phi }A + d_{\Theta }} = (Q_j)^{D_{\Phi }A} = \phi _j(A) = \phi _j(A + d_{\Phi })\). So, for all \(P_i\), for all \(Q_j\),
\((P_i)^{A + d_{\Phi }} = (P_i)^{D_{\Phi }A +d_{\Theta } }\) & \((Q_j)^{A + d_{\Phi }} = \phi _j(A + d_{\Theta })\). 
But trivially, \(D_{\Phi }A +d_{\Theta } \models d_{\Theta }\). So, \( D_{\Phi }A +d_{\Theta } \models P_i( \overline{x}) \leftrightarrow \theta _i\), for each \(P_i\). Thus,
But we also have
Therefore,
Thus, for all \(P_i\), for all \(Q_j\),
\((P_i)^{A + d_{\Phi }} = \theta _i(A +d_{\Theta })\) & \((Q_j)^{A + d_{\Phi }} = \phi _j(A + d_{\Theta })\). 
And therefore, \(A + d_{\Phi } \models d_{\Theta }\), as required.
For \((1) \Rightarrow (4)\). Suppose \(A + d_{\Phi } \models d_{\Theta }\). So, from (3) above, we have:
Therefore, \(D_{\Phi }A + d_{\Theta } \models d_{\Phi }\). A proof analogous to that of (3) above gives us that if \(B + d_{\Theta } \models d_{\Phi }\) then \( B + d_{\Theta } \cong D_{\Theta } B + d_{\Phi }\), and so, if \(D_{\Phi }A + d_{\Theta } \models d_{\Phi } \text { then } D_{\Phi }A + d_{\Theta } \cong D_{\Theta } D_{\Phi }A + d_{\Phi }\). And so, we may conclude,
And thus,
And right cancellation gives,
But since the map \(D_{\Phi }\) leaves the domain invariant, we have \(A = D_{\Theta } D_{\Phi }A\).
For \((4) \Rightarrow (5)\), suppose \(A = D_{\Theta } D_{\Phi }A\). We have, applying Lemma 15(1) twice, for any \(\alpha \in L_P\), \(\beta \in L_Q\):
So, for \(\alpha \in L_P\),
But \(A = D_{\Theta } D_{\Phi }A\), and so,
And thus,
as required.
For \((5) \Rightarrow (1)\), assume \(A \models \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\). So,
But \(\tau _{\Theta }(P_i( \overline{x})) = \theta _i\). So,
And \(\theta _i \in L_Q\), and so, from Lemma 7(1),
and thus,
as required.\(\square \)
Turning next to theories:
Lemma 17
\(D_{\Phi }T + d_{\Theta } \vdash \alpha \leftrightarrow \tau _{\Theta }(\alpha )\), for all \(\alpha \in L_P\).
Proof
We reason by induction on the construction of \(\alpha \). Let \(\alpha = P_i( \overline{x})\). Then we have: \(D_{\Phi }T + d_{\Theta } \vdash P_i( \overline{x}) \leftrightarrow \theta _i\). But \(\tau _{\Theta }(P_i( \overline{x})) = \theta _i\). And thus, \(D_{\Phi }T + d_{\Theta } \vdash P_i( \overline{x}) \leftrightarrow \tau _{\Theta }(P_i( \overline{x}))\). The equality case and compound cases proceed routinely. (Notice this is analogous to Lemma 10(1).) \(\square \)
Lemma 18
Suppose that \(T + d_{\Phi } \vdash d_{\Theta }\). Then, for any \(\alpha \in L_Q\):
(1)  \(D_{\Phi }T \vdash \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\) 
(2)  \(D_{\Phi }T + d_{\Theta } \vdash \alpha \leftrightarrow \tau _{\Phi }(\alpha )\). 
Proof
For (1), we have, for any \(\beta \in L_Q\), if \(T + d_{\Phi } \vdash \beta \), then \(D_{\Phi }T \vdash \beta \), since \(D_{\Phi }T\) is simply the restriction of the set of theorems of \(T + d_{\Phi }\) to \(L_Q\). By Lemma 10(4), we have \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\), for any \(\alpha \in L_Q\). Now \(\alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\) is also in \(L_Q\). Thus, \(D_{\Phi }T \vdash \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\).
For (2), from Lemma 17, we have, for any \(\alpha \in L_Q\): \(D_{\Phi }T + d_{\Theta } \vdash \tau _{\Phi }(\alpha ) \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\), since \(\tau _{\Phi }(\alpha ) \in L_P\). By (1), we have \(D_{\Phi }T + d_{\Theta } \vdash \alpha \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\alpha ))\). Hence, \(D_{\Phi }T + d_{\Theta } \vdash \alpha \leftrightarrow \tau _{\Phi }(\alpha )\), as required. \(\square \)
The next lemma is the most important result needed for Theorem 2 given in Sect. 8:
Lemma 19
The following are equivalent
(1)  \(T + d_{\Phi } \vdash d_{\Theta }\). 
(2)  \(T + d_{\Phi } \equiv D_{\Phi }T + d_{\Theta }\). 
Proof
\((1) \Rightarrow (2)\). Suppose \(T + d_{\Phi } \vdash d_{\Theta }\). We want to show
(a)  \(T + d_{\Phi } \vdash D_{\Phi }T + d_{\Theta }\). 
(b)  \(D_{\Phi }T + d_{\Theta } \vdash T + d_{\Phi }\). 
First, for (a), let \(A + d_{\Phi } \models T + d_{\Phi }\). So, since \(T + d_{\Phi } \vdash d_{\Theta }\), we have \(A + d_{\Phi } \models d_{\Theta }\). So, by Lemma 16(3), we have \(A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }\). Since \(A \models T\), we have \(D_{\Phi }A \models D_{\Phi }T\). So, \(D_{\Phi }A + d_{\Theta } \models D_{\Phi }T + d_{\Theta }\). Thus, \(A + d_{\Phi } \models D_{\Phi }T + d_{\Theta }\). And therefore, since A was arbitrary, \(T + d_{\Phi } \vdash D_{\Phi }T + d_{\Theta }\), as required.
For (b), we want to show that \(D_{\Phi }T + d_{\Theta } \vdash T+d_{\Phi }\). That is, for for any \(L_{P,Q}\)formula \(\alpha \),
First, note that we may relabel Lemma 9(2) in terms of some \(L_Q\)structure B and definition system \(d_{\Theta }\), rather than A and \(d_{\Phi }\), to obtain: for any \(\alpha \in L_{P,Q}\),
(i)  \(B + d_{\Theta } \models \alpha \leftrightarrow \tau ^{+}_{\Theta }(\alpha )\). 
For a contradiction, suppose we have some \(\alpha \in L_{P,Q}\) such that
(ii)  \(T + d_{\Phi } \vdash \alpha \) 
(iii)  \(D_{\Phi }T + d_{\Theta } \nvdash \alpha \). 
Since \(T + d_{\Phi } \vdash d_{\Theta }\), we have, from Lemma 10(9), that \(T + d_{\Phi } \vdash \alpha \leftrightarrow \tau ^{+}_{\Theta }(\alpha )\) and since \(T + d_{\Phi } \vdash \alpha \), we have:
(iv)  \(T + d_{\Phi } \vdash \tau ^{+}_{\Theta }(\alpha )\). 
From (iii), there exists an \(L_Q\)structure \(B \models D_{\Phi }T\) with \(B + d_{\Theta } \not \models \alpha \). So, from (i), \(B + d_{\Theta } \not \models \tau ^{+}_{\Theta }(\alpha )\). And since \(\tau ^{+}_{\Theta }(\alpha ) \in L_Q\), we have:
(v)  \(B \not \models \tau ^{+}_{\Theta }(\alpha )\). 
Recall that \(D_{\Phi }T = \{\beta \in L_Q \mid T + d_{\Phi } \vdash \beta \}\). So, since \(B \models D_{\Phi }T\), it follows that, for any \(\beta \in L_Q\), if \(T + d_{\Phi } \vdash \beta \), then \(B \models \beta \). Thus, if \(T + d_{\Phi } \vdash \tau ^{+}_{\Theta }(\alpha )\), then \(B \models \tau ^{+}_{\Theta }(\alpha )\). So, from (iv), we infer:
(vi)  \(B \models \tau ^{+}_{\Theta }(\alpha )\). 
Contradiction.
\((2) \Rightarrow (1)\). Let \(T + d_{\Phi } \equiv D_{\Phi }T + d_{\Theta }\). Then \(T + d_{\Phi } \vdash D_{\Phi }T + d_{\Theta }\). Thus, \(T + d_{\Phi } \vdash d_{\Theta }\), as required. \(\square \)
Lemma 20
\(D_{\Theta }D_{\Phi }T \vdash \alpha \text { iff } T \vdash \tau _{\Phi }(\tau _{\Theta }(\alpha ))\), for \(\alpha \in L_P\).
Proof
If we examine the definitions of \(D_{\Phi }T\) and \(D_{\Theta }T_2\) (where \(T_2\) is in \(L_Q\)), we get
Together, these imply that \(D_{\Theta }D_{\Phi }T \vdash \alpha \) iff \(T \vdash \tau _{\Phi }(\tau _{\Theta }(\alpha ))\). \(\square \)
Lemma 21
Suppose \(T + d_{\Phi } \vdash d_{\Theta }\). Then
(1)  \(D_{\Phi }T + d_{\Theta } \vdash d_{\Phi }\). 
(2)  \(D_{\Theta }D_{\Phi }T \equiv T\). 
Proof
For (1), suppose \(T + d_{\Phi } \vdash d_{\Theta }\). By Lemma 18(2), for any \(\alpha \in L_Q\),
Thus,
Thus,
as required.
For (2), suppose \(T + d_{\Phi } \vdash d_{\Theta }\). We want to show that, for any \(\alpha \in L_P\), we have: \(D_{\Theta }D_{\Phi }T \vdash \alpha \text { iff } T \vdash \alpha \). By Lemma 20, we have
Since \(T + d_{\Phi } \vdash d_{\Theta }\), by Lemma 10(8), we have
Hence,
And therefore,
as required. \(\square \)
Lemma 22
The following are equivalent:
(1)  \(T \equiv D_{\Theta } D_{\Phi } T\). 
(2)  \(T \vdash \alpha \) iff \(T \vdash \tau _{\Phi }(\tau _{\Theta }(\alpha ))\), with \(\alpha \in L_P\). 
Proof
\((1) \Rightarrow (2)\). From the previous lemma, \(D_{\Theta }D_{\Phi }T \vdash \alpha \text { iff } T \vdash \tau _{\Phi }(\tau _{\Theta }(\alpha ))\). So, if \(T \equiv D_{\Theta } D_{\Phi } T\), we infer that \(T \vdash \alpha \) iff \(T \vdash \tau _{\Phi }(\tau _{\Theta }(\alpha ))\), for \(\alpha \in L_P\).
\((2) \Rightarrow (1)\). Suppose, for \(\alpha \in L_P\),
We already have
So, for \(\alpha \in L_P\)
Thus, \(D_{\Theta }D_{\Phi }T \equiv T\). \(\square \)
Notice that, in the case of structures, we have:
However, unlike that case, for theories, we only have established:
So, we have not established the provability of the biconditional \(\alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\) inside T itself.^{Footnote 11}
7 Definitional Equivalence
We next explain what it means for structures and theories to be definitionally equivalent.^{Footnote 12} Intuitively, a pair of structures are definitionally equivalent when they have a common definitional expansion. And a pair of theories are definitionally equivalent when they have a common definitional extension.
Throughout the next two definitions, A is an \(L_P\)structure, \(\Phi = \{\phi _i\}_{i \in I_1}\) is a set of \(L_P\)formulas; Q is the new disjoint signature corresponding to \(\Phi \); and B is an \(L_Q\)structure. Similarly, \(\Theta = \{\theta _i\}_{i \in I_2}\) is a set of \(L_Q\)formulas. \(d_{\Phi }\) is the definition system of the \(Q_i\) primitives in terms of the \(\phi _i\), and \(d_{\Theta }\) is the definition system the \(P_i\) primitives in terms of the \(\theta _i\). Similarly, \(T_1\) is an \(L_P\)theory and \(T_2\) is an \(L_Q\)theory. Please note that the definitions of these notions given below assume disjoint signatures.
Definition 26
Structures A and B are definitionally equivalent wrt \(d_{\Phi }\) and \(d_{\Theta }\) iff
If this is so, we write:
Definition 27
Theories \(T_1\) and \(T_2\) are definitionally equivalent wrt \(d_{\Phi }\) and \(d_{\Theta }\) iff
To express this, we write:
These definitions require that the signatures of A and B (or \(T_1\) and \(T_2\)) be disjoint. But is not A obviously definitionally equivalent to itself? Is not a theory definitionally equivalent to itself? Well, one can always arrange for a pair of structures A and B in overlapping signatures to be reformulated as copies\(A^c\) and \(B^c\) in entirely disjoint signatures (see Definitions 2, 6 above). If \(A^c\) is a disjoint copy of A in \(L_{P^c}\), then clearly \(A^c\) is definitionally equivalent to A with respect to the trivial definition systems:
\(d^{P^c}_{P}\) :  \(\forall \overline{x}(P^c_i( \overline{x}) \leftrightarrow P_i( \overline{x}))\) 
\(d^{P}_{P^c}\) :  \(\forall \overline{x}(P_i( \overline{x}) \leftrightarrow P^c_i( \overline{x}))\). 
That is,
A similar copying procedure can be adopted for theories too. If we have a theory T in \(L_P\), and \(T^c\) is a disjoint copy of T in \(L_{P^c}\), then clearly \(T^c\) is definitionally equivalent to T with respect to the definitions:
So, one can give a more general definition of definitional equivalence by first applying this copying procedure to both structures, and then applying the definition above.^{Footnote 13}
8 Main Results
We finally give two main results, which characterize definitional equivalence.
Theorem 1
The following are equivalent:
(1)  \(A \underset{\Theta }{\overset{\Phi }{\longleftrightarrow }} B\) 
(2)  \(D_{\Phi }A \cong B\) and \(D_{\Theta }B \cong A\) 
Proof
\((1) \Rightarrow (2)\). Let \(A \underset{\Theta }{\overset{\Phi }{\longleftrightarrow }} B\). Thus, \(A + d_{\Phi } \cong B + d_{\Theta }\). So,
Thus, by Lemma 16(3) (switching labels in the second case), we have:
Thus,
And by right cancellation,
\((2) \Rightarrow (1)\). Let \(D_{\Phi }A \cong B\) and \(D_{\Theta }B \cong A\). Since we have two isomorphisms, we can arrange, without loss of generality, for \(\text {dom}(D_{\Phi }A) = \text {dom}(B)\), and \(\text {dom}(D_{\Theta }B) = \text {dom}(A)\). We want to show
(a)  \((P_i)^{A + d_{\Phi }} = (P_i)^{B+ d_{\Theta }}\) 
(b)  \((Q_i)^{A + d_{\Phi }} = (Q_i)^{B + d_{\Theta }}\) 
First, we work out the extensions of \(P_i\) and \(Q_i\) in \(A + d_{\Phi }\). For \(Q_i\), we have \((Q_i)^{A + d_{\Phi }} = \phi _i(A)\). Using the assumption \(D_{\Theta }B = A\), we have \((P_i)^{A + d_{\Phi }} = (P_i)^{D_{\Theta }B+ d_{\Phi }}\). So, \((P_i)^{A + d_{\Phi }} = \theta _i(B)\). And using the assumption \(D_{\Phi }A = B\), we have \((P_i)^{A+ d_{\Phi }} = \theta _i(D_{\Phi }A)\). Summarizing:
(i)  \((P_i)^{A+ d_{\Phi }} = \theta _i(D_{\Phi }A)\). 
(ii)  \((Q_i)^{A + d_{\Phi }} = \phi _i(A)\). 
Next working out the extensions in \(B + d_{\Theta }\), and using the assumption \(D_{\Phi }A = B\), we have \((P_i)^{B+ d_{\Theta }} = (P_i)^{D_{\Phi }A + d_{\Theta }}\) and \((Q_i)^{B + d_{\Theta }} = (Q_i)^{D_{\Phi }A + d_{\Theta }}\). I.e.,
(iii)  \((P_i)^{B+ d_{\Theta }} = \theta _i(D_{\Phi }A)\). 
(iv)  \((Q_i)^{B + d_{\Theta }} = \phi _i(A)\). 
And (i) and (iii) imply (a), while (ii) and (iv) imply (b). \(\square \)
Theorem 2
The following are equivalent:
(1)  \(T_1 \underset{\Theta }{\overset{\Phi }{\longleftrightarrow }} T_2\). 
(2)  \((\tau _{\Phi } \tau _{\Theta }=1)_{T_1}\) and \(T_2 \equiv D_{\Phi }T_1\). 
Proof
For \((1) \Rightarrow (2)\), suppose \(T_1 \underset{\Theta }{\overset{\Phi }{\longleftrightarrow }} T_2\). Thus, \(T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }\). Thus, \(T_1 + d_{\Phi } \vdash d_{\Theta }\).
Then, from Lemma 10(8), we have \(T_1 \vdash \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha ))\), for \(\alpha \in L_P\). I.e., \((\tau _{\Phi } \tau _{\Theta }=1)_{T_1}\). And secondly, since \(T_1 + d_{\Phi } \vdash d_{\Theta }\), we have \(T_1 + d_{\Phi } \equiv D_{\Phi }T_1 + d_{\Theta }\), by Lemma 19. So, \(T_2 + d_{\Theta } \equiv D_{\Phi }T_1 + d_{\Theta }\), and by right cancellation, \(T_2 \equiv D_{\Phi }T_1\).
For \((2) \Rightarrow (1)\), suppose \((\tau _{\Phi } \tau _{\Theta }=1)_{T_1}\) and \(T_2 \equiv D_{\Phi }T_1\). From Lemma 11(1), we may conclude that \(T_1 + d_{\Phi } \vdash d_{\Theta }\), and from this, we may conclude that \(T_1 + d_{\Phi } \equiv D_{\Phi }T_1 + d_{\Theta }\), via Lemma 19. But since \(T_2 \equiv D_{\Phi }T_1\), we may conclude that \(T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }\), as required.\(\square \)
Indeed, from \(T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }\), we may also conclude \((\tau _{\Phi } \tau _{\Theta }=1)_{T_1}\), \((\tau _{\Theta } \tau _{\Phi } =1)_{T_2}\), \(T_2 \equiv D_{\Phi }T_1\) and \(T_1 \equiv D_{\Theta }T_2\). For example, if \(T_2 + d_{\Theta } \vdash d_{\Phi }\), from Lemma 10(8), by relabelling, we have \(T_2 \vdash \beta \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\beta ))\), for \(\beta \in L_Q\), and thus \((\tau _{\Theta } \tau _{\Phi } =1)_{T_2}\).
Notes
A detailed analysis and proof of this “mathematical folklore” result is given in Kaye and Wong [13], who show that there exist translation functions between \( \mathsf {PA}\) and \( \mathsf {ZF}_0\) which are mutual inverses (Theorem 20 of their paper). A fairly detailed discussion of the equivalence of Peano arithmetic and finite set theory \( \mathsf {ZF}_0\) may also be found in the lecture notes Andréka and Németi [2], pp. 41–45.
A weak theory of concatenation without the induction principle is now usually called \( \mathsf {TC}\) (Grzegorczyk [8]), and the precise interpretability relationship of \( \mathsf {TC}\) and Robinson arithmetic \( \mathsf {Q}\) has recently been clarified. Although similar systems had been studied before (Quine [15]; Tarski et al. [17]), the undecidability of \( \mathsf {TC}\) is demonstrated in Grzegorczyk [8]; in Grzegorczyk and Zdanowski [9], the essential undecidability of \( \mathsf {TC}\) is demonstrated through an interpretation of \( \mathsf {TC}\) into Robinson arithmetic \( \mathsf {Q}\). Subsequently, Visser and Sterken [18], Ganea [7] and Švejdar [16] demonstrated the interpretability of \( \mathsf {Q}\) into \( \mathsf {TC}\).
As another example, there is the theory \(\textsf {CEM}\), of “classical extensional mereology”, in a language with basic binary relation symbol \(\preceq \) (\(x \preceq y\) means “x is part of y”). There is a definitionally equivalent theory I shall call \(\textsf {F}\) (for “fusions”) with a basic binary operation symbol \(\oplus \) (where \(x \oplus y\) can be read “the fusion of x and y”). The detailed formulation and proof of definitional equivalence are given in Ketland and Schindler [12].
Often, we have translations/interpretations \(T_1 \rightarrow T_2\) and \(T_2 \rightarrow T_1\), for theories which appear, on the surface, to be quite different. But it is not automatically true that \(T_1\) and \(T_2\) are definitionally equivalent: the translations involved must be mutual inverses of each other. (See our Theorem 2 below.) A valuable discussion of this point, and a criterion for it to hold, is Friedman and Visser [6], who show that when two theories are “biinterpretable via identitypreserving interpretations”, then they are definitionally equivalent (Sect. 5 of their paper). Moreover, they give an example of two finitely axiomatized sequential theories that are biinterpretable but not definitionally equivalent (Sect. 7 of their paper). An example of a pair of theories which “define each others’ models” but which are not definitionally equivalent is given in Andréka et al. [1].
For any variable x, the denotation of x in A is \(x^{A}\). We define \(A^x_a\) to be the structure just like A, except that, for the variable x, we have \(x^{A^x_a} = a\).
As explained below, \(d_{\Theta }\) will be an abbreviation for the set of definitions of the form \(\forall \overline{x}(P_i( \overline{x}) \leftrightarrow \theta _{i})\), where the \(\theta _i\) are \(L_Q\)formulas.
For a proof, based on Craig’s interpolation lemma, see Boolos and Jeffrey [3], Ch. 25.
The clauses (3)–(5) are usually read as saying “\(\tau _{\Phi }\) commutes with the logical operators”.
That said, I do not have a counterexample. It is conceivable that the example given in Andréka et al. [1] might yield such a counterexample.
See Andréka and Németi [2] for a related but slightly different procedure.
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The author acknowledges support from a grant from the National Science Centre in Kraków (NCN), Grant Number 2018/29/B/HS1/01832.
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Ketland, J. Bases for Structures and Theories I. Log. Univers. 14, 357–381 (2020). https://doi.org/10.1007/s1178702000257y
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DOI: https://doi.org/10.1007/s1178702000257y