Bases for Structures and Theories II

In Part I of this paper (Ketland in Logica Universalis 14:357–381, 2020), I assumed we begin with a (relational) signature P={Pi}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P = \{P_i\}$$\end{document} and the corresponding language LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}, and introduced the following notions: a definition systemdΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\Phi }$$\end{document} for a set of new predicate symbols Qi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_i$$\end{document}, given by a set Φ={ϕi}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi = \{\phi _i\}$$\end{document} of defining LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-formulas (these definitions have the form: ∀x¯(Qi(x)↔ϕi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \overline{x}(Q_i(x) \leftrightarrow \phi _i)$$\end{document}); a corresponding translation functionτΦ:LQ→LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\Phi }: L_Q \rightarrow L_P$$\end{document}; the corresponding definitional image operatorDΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\Phi }$$\end{document}, applicable to LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-structures and LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-theories; and the notion of definitional equivalence itself: for structures A+dΦ≡B+dΘ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A + d_{\Phi } \equiv B + d_{\Theta }$$\end{document}; for theories, T1+dΦ≡T2+dΘ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$$\end{document}. Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a representation basis. Suppose a set Φ={ϕi}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi = \{\phi _i\}$$\end{document} of LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-formulas is given, and Θ={θi}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta = \{\theta _i\}$$\end{document} is a set of LQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_Q$$\end{document}-formulas. Then the original set Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is called a representation basis for an LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-structure A with inverse Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} iff an inverse explicit definition ∀x¯(Pi(x¯)↔θi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$$\end{document} is true in A+dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A + d_{\Phi }$$\end{document}, for each Pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i$$\end{document}. Similarly, the set Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is called a representation basis for a LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-theory T with inverse Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} iff each explicit definition ∀x¯(Pi(x¯)↔θi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$$\end{document} is provable in T+dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T + d_{\Phi }$$\end{document}. Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document} (in LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}) is definitionally equivalent to T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2$$\end{document} (in LQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_Q$$\end{document}), with respect to Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} and Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document}, if and only if Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is a representation basis for T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document} with inverse Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} and T2≡DΦT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2 \equiv D_{\Phi }T_1$$\end{document}.


Introduction
Sometimes theories are formulated with different sets of non-logical primitives and yet are definitionally equivalent. There are many examples of theoriesoften involving formalized systems of arithmetic and set theory-formulated with rather different sets of primitives (aka signatures), which are nonetheless "equivalent". Definition 4. Let a definition system d Φ be given. Define the translation, induced by Φ τ + Φ : L P,Q → L P as follows. For symbols P i , Q j , variables x, y, x, and for L P,Q -formulas α, α 1 , α 2 : where # is any binary connective, q is a quantifier and (φ i ) is the result of ensuring that the free variables appearing φ i are relabelled, to match those of Q i (x). 3 We call τ + Φ the translation induced by Φ. It maps from the enriched language L P,Q back to the original language L P . We let τ Φ be the restriction of τ + Φ to L Q : thus, τ Φ maps from the new language L Q back to the original language L P . (τ Φ is also called the translation induced by Φ.) Definition 5. Let τ Φ : L Q → L P and τ Θ : L P → L Q be translations induced by d Φ and d Θ . Let T 1 be an L P theory. Let T 2 be an L Q theory. Then τ Θ is an right inverse of τ Φ in T 1 iff, for any α ∈ L P , We write this more suggestively as: And τ Θ is an left inverse of τ Φ in T 2 iff, for any β ∈ L Q , Likewise, we write this more suggestively as: (τ Θ τ Φ = 1) T2 Definition 6. Let A be an L P -structure. Then the L Q -structure D Φ A is defined by:

Definition 7.
The definitional image of T , with respect to Φ, is the restriction of the deductive closure of T + d Φ to the new language L Q . The definitional image of T with respect to Φ is denoted D Φ T . That is, Definition 9. Theories T 1 and T 2 are definitionally equivalent wrt d Φ and d Θ iff To express this, we write: In Part I we established a fair number of "book-keeping lemmas". The three most important results can be summarized: Log. Univers. Lemma 1. The following hold always: 4 The following are equivalent: 5 The following are equivalent: 6 In addition to those "book-keeping lemmas", we established two conditions for definitional equivalence, one for structures and one for theories: Theorem 1. The following are equivalent: Theorem 2. The following are equivalent: (2) (τ Φ τ Θ = 1) T1 and T 2 ≡ D Φ T 1 .

Definitional Equivalence: Model-Theoretic Criteria
Theorem 2 above establishes a criterion for T 1 Φ ←→ Θ T 2 in terms of translation: Next, we establish model-theoretic criteria.

Lemma 7.
The following are equivalent: Recall Lemma 3, which implies: We wish to show ( We then obtain a characterization theorem: Theorem 3. The following are equivalent Proof. Reason as follows: Together, these imply the existence of a bijection (wrt ∼ =): 7 such that, for any A ∈ Mod(T 1 ), we have: To show this, first we prove a simple lemma about definitional equivalence of structures: 7 Whenever I use the notion of bijection relating classes of models, I mean "bijection up to isomorphism". We write H : Mod(T 1 ) ↔ Mod(T 2 ) to mean the map H : Vol. 14 (2020)

Bases for Structures and Theories II 467
Proof.
The assumption (T 1 , Φ) (T 2 , Θ) asserts the existence of two functions F, G which are linked by the parameters Φ, Θ. We show that G left-inverts F and F left-inverts G.
Let A ∈ Mod(T 1 ). Then F (A) ∈ Mod(T 2 ) and So, by the above cancellation lemma, So, by the above cancellation lemma again, . So, by Lemma 9, there exists a bijection H : Thus, for any A ∈ Mod(T 1 ), we have, We will show that H and D Φ are the same up to isomorphism: i.e., Furthermore, the converse is true. Log. Univers.
be a bijection such that, for any A ∈ Mod(T 1 ), We want to show, first, that for, any The two previous lemmas give a second characterization theorem: The following are equivalent:

Mutual Definability Does Not Imply Definitional Equivalence
Andréka et al ( [1]) show that mutual definability of a pair of theories in each other does not entail their definitional equivalence. First, their notion of modeltheoretic definability is explained as follows: Let T h1 and T h2 be theories, maybe on different first-order languages. An explicit definition of T h1 over T h2 is a conjunction Δ of explicit definitions of the relation symbols of T h1 in terms of the language of T h2 such that the models of T h1 are exactly the reducts of the models of T h2 ∪ Δ (to the language of T h1). Thus, we get the models of T h1 from those of T h2 by first defining the relations of T h1 via using Δ, and then forgetting the relations not present in the language of T h2. ( [1]: 591) If we switch their T h1 to T 2 , T h2 to T 1 , and Δ to d Φ , this corresponds, in our terminology, to saying that Φ model-theoretically defines T 2 in T 1 : i.e., Mod(T 2 ) = D Φ [Mod(T 1 )] (see [4], Definition 28, Part I).
It is relatively straightforward to prove: Proof. Let us suppose that T 1 We first wish to prove (1): That is, we wish to prove that, for any L Q -structure B, we have: Vol. 14 (2020) Bases for Structures and Theories II 469 So, from Lemma 3 (but relabelling in terms of an L Q -structure B instead of an L P -structure A), we have: The proof of (2) One way to understand the problem is that the translations associated with Φ and Θ are not mutual inverses. Below (Theorem 16) we show the "gap" connected to Lemma 12 and Theorem 5 closes, by requiring not only Mod(T 2 ) = D Φ [Mod(T 1 )] but also that Φ is a representation basis for T 1 with inverse Θ: if these conditions hold, then it follows that T 1 Φ ←→ Θ T 2 . 8

Summary
We quickly summarize the results of the three previous sections. Theorem 6. The following three claims are equivalent: Theorem 7. The following are equivalent: (1) (2)

Representation Basis
We next move on to defining the notion of "representation basis" for a structure and for a theory. The underlying intuitive concept is fairly simple. Given a structure A in a signature P , we may wish to consider a special set Φ of L P -formulas, and then examine the "internal structure" defined by them in A: this is what we have called the "definitional image", D Φ A. What condition should we impose if we wish to reconstruct A from its image D Φ A?
Clearly, the condition is that the definitional expansion A + d Φ -which we used to define D Φ A prior to forgetting the P -relations-should satisfy an invertibility condition, namely that each original P i be explicitly definable from a formula, say θ i , in the new Q-language. This then means that the original set Φ of L P -formulas, in some sense, does not "omit" any structural content built into A itself. The formulas φ i merely "encode" that content differently.
That condition is then, simply, that there is a set Θ = {θ i } i∈IP of Qformulas such that holds. Or, more explicitly, for each atomic formula P i (x) of L P , we have an explicit inverse definition, The same intuition motivates an analogous account for theories. The definitional image D Φ T is a smaller theory which "lives inside" the original T (though it is not a sub-theory). It is kind of filtering or projection. But what condition would permit reconstruction of the original?
Again, it is the condition that there is a set Θ = {θ i } i∈IP of Q-formulas such that Vol. 14 (2020)

Bases for Structures and Theories II 471
holds. Similarly, we can more explicitly express this by saying that, for each atomic formula P i (x) of L P , we have an explicit inverse definition, As the reader may have noticed above, throughout earlier sections, we have examined what kinds of consequences follow from precisely this invertibility assumption. Based on these informal explanations, we then given the two main definitions: In each case, the formulas θ i in the set Θ are called inversion formulas. The following two lemmas are entirely straightforward.

Lemma 13. If T is inconsistent, any Φ is a representation basis for T .
For the case where T is the empty theory in L P (i.e., pure logic), we define the notion of a logical representation basis (for L P ): It is easy to see that being a representation basis is preserved under theory extension (or structure expansion): Lemma 15. If Φ is a representation basis for T (or structure A), then Φ is also a representation basis for every extension of T (resp. every expansion of A). In particular, if Φ is a logical representation basis (for L P ), then it is a representation basis for every theory T (in L P ).
The converse of this lemma is not true. Φ might be a representation basis for T , but not a representation basis for a sub-theory or a weaker theory. Similarly, Φ might be a representation basis for A, but not a representation basis for a reduct of A.
We note in passing that while we have defined representation basis syntactically in terms of explicit definability of the P i , a mathematically equivalent model-theoretic definition can be given, in terms of implicit definability: Definition 14. A set Φ of L P -formulas is a semantic representation basis for T just in case, each primitive L P -symbol P i is implicitly definable in T + d Φ .
It then follows, using Beth's definability theorem, that the syntactic definition of representation basis (Definition 12) and the model-theoretic one (Definition 14) are equivalent: Theorem 9. Φ is a representation basis for T iff Φ is a semantic representation basis for T .

Examples
We give three examples of this notion in operation and an interesting example involving definitional image. The first three are fairly simple. The fourth provides an example of a logical representation basis Φ for (a propositional language) L P but where the definitional image of logic in L P under D Φ is not logically true. . . . , x a(Pi) )} i∈IP , consisting of the atomic formulas of the language L P . Then Φ is a logical representation basis for L P , and indeed a representation basis for any theory T in L P . In this case, each new Q i is simply defined as P i . This means they are equivalent in the extension T +d Φ . And then, trivially, the inversion conditions hold.
The next two examples are slightly modified from examples given by David Miller in his series of papers explaining the language-dependence problem for explications of the concept of truthlikeness (Miller 1974 [5], 1975 [6], 1978 [7]).

Example 2.
Let P = {p 1 , p 2 } be a propositional signature and consider the L P -formulas Then {φ 1 , φ 2 } is a logical representation basis for L P . For consider the definitions of the new symbols, Q 1 and Q 2 : So, given d Φ , p 1 and p 2 can be explicitly defined in terms of Q 1 and Q 2 (this happens essentially because p 1 ↔ (p 1 ↔ p 2 ) is logically equivalent to p 2 ). Example 3. Let P = {P 1 , P 2 }, where P 1 , P 2 are unary predicates. Consider the L P -formulas φ 1 , φ 2 : Then the pair {φ 1 , φ 2 } is a logical representation basis for L P . For the explicit definitions, Vol. 14 (2020)

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imply (in logic alone) the inversions: for the same reason as the previous example.
Example 4. Consider, given Φ, the definitional image of T when T is the empty theory. Let Log P be the set of L P -sentences which are theorems of logic in L P . The definitional image Log P under Φ is given by: Then: Observation. There is a signature P and (logical) representation basis Φ for logic in L P such that the L Q theory D Φ Log P isn't logically true.
For example, let P = {p 1 , p 2 } be a propositional signature and consider the three L P -formulas: Then one can show that {φ 1 , φ 2 , φ 3 } is a logical representation basis for L P . Introduce the definition system d Φ for the new symbols, Q 1 , Q 2 and Q 3 : Then inversions-i.e., explicit definitions of p 1 , p 2 in terms of the Q i -can be obtained as follows: However, not every L Q -theorem of d Φ is logically true. For example, In each case, we have β ∈ L Q , with d Φ β. But, for each β, we have β. 10 And thus the theory D Φ Log P is not logically true.

Basis, Translation and Equivalence
We can now begin assemble the various pieces of this rather complicated jigsaw. In Subsection 8.1, we shall establish equivalent ways of expressing "Φ is a representation basis for T with inverse Θ". Then, in Subsection 8.2, we shall see how imposing this condition leads to strengthened properties of the D Φ operator. Finally, in Subsection 8.3, we see how to include being a representation basis as a further criterion for expressing "T 1 is definitionally equivalent to T 2 , wrt Φ and Θ".

Criteria for Being a Representation Basis
First, we establish equivalents for Φ being a representation basis for T with inverse Θ.
Theorem 10. The following conditions are equivalent: (1) Φ is a representation basis for T with inverseΘ.
For (5) ⇔ (6): notice that, by Lemma 3, Next, specializing to the case where T is pure logic, we obtain: Vol. 14 (2020) Bases for Structures and Theories II 475 Theorem 11. The following are equivalent: (1) Φ is a logical representation basis for L P with inverse Θ.

Consequences of Φ Being a Representation Basis for T
Theorem 12. Let Φ be a representation basis for T with inverse Θ. Then: (1) Proof. Let us suppose Φ is a representation basis for T with inverse Θ. In particular, by Theorem 10(4), we have: Second, recall Theorem 8. Conditions (1) and (5)  We can then prove the results quickly.