Bases for Structures and Theories II

Abstract

In Part I of this paper (Ketland in Logica Universalis 14:357–381, 2020), I assumed we begin with a (relational) signature $P = \{P_i\}$ and the corresponding language $L_P$, and introduced the following notions: a definition system $d_{\Phi }$ for a set of new predicate symbols $Q_i$, given by a set $\Phi = \{\phi _i\}$ of defining $L_P$-formulas (these definitions have the form: $\forall \overline{x}(Q_i(x) \leftrightarrow \phi _i)$); a corresponding translation function $\tau _{\Phi }: L_Q \rightarrow L_P$; the corresponding definitional image operator $D_{\Phi }$, applicable to $L_P$-structures and $L_P$-theories; and the notion of definitional equivalence itself: for structures $A + d_{\Phi } \equiv B + d_{\Theta }$; for theories, $T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$. 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 $\Phi = \{\phi _i\}$ of $L_P$-formulas is given, and $\Theta = \{\theta _i\}$ is a set of $L_Q$-formulas. Then the original set $\Phi$ is called a representation basis for an $L_P$-structure A with inverse $\Theta$ iff an inverse explicit definition $\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$ is true in $A + d_{\Phi }$, for each $P_i$. Similarly, the set $\Phi$ is called a representation basis for a $L_P$-theory T with inverse $\Theta$ iff each explicit definition $\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$ is provable in $T + d_{\Phi }$. 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 $T_1$ (in $L_P$) is definitionally equivalent to $T_2$ (in $L_Q$), with respect to $\Phi$ and $\Theta$, if and only if $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$ and $T_2 \equiv D_{\Phi }T_1$.

1 Introduction

Sometimes theories are formulated with different sets of non-logical primitives and yet are definitionally equivalent. There are many examples of theories—often involving formalized systems of arithmetic and set theory—formulated with rather different sets of primitives (aka signatures), which are nonetheless “equivalent”.

2 Summary of Part I

In Part I ([4]), we considered a starting language $L_P$ over a relational signature $P = \{P_i\}_{i \in I_P}$, and a set $\Phi = \{\phi _i\}_{i \in I}$ of $L_P$-formulas. Given $\Phi$, 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 $\phi _i$. The extended language is denoted $L_{P,Q}$ and the language built from the new signature Q (with the implicitly induced arities) is denoted $L_Q$.¹

Definition 1

Given $\Phi = \{\phi _i\}$, the definition system over $\Phi$, which we write as,

$$\begin{aligned} d_{\Phi } \end{aligned}$$

is the set of explicit definitions,

$$\begin{aligned} \forall x_1 \dots x_{n_i}(Q_i(x_1, \dots , x_{n_i}) \leftrightarrow \phi _{i}) \end{aligned}$$

where $\{x_1,\dots x_{n_i}\} = \text {FV}(\phi _i)$, and $n_i = \text {card} \ \text {FV}(\phi _i)$ (the “arity” of $\phi _i$). These define the new symbols $Q_i$ in terms of the defining $L_P$-formulas $\phi _i$. We shall sometimes write $\forall \overline{x}(Q_i( \overline{x}) \leftrightarrow \phi _{i})$ instead of $\forall x_1 \dots x_n(Q_i(x_1, \dots , x_n)\leftrightarrow \phi _{i})$.²

Definition 2

Let A be an $L_P$-structure. Then $A + d_{\Phi }$ is the unique definitional expansion $A^{+} \models d_{\Phi }$ of A. ($A + d_{\Phi }$ is an $L_{P,Q}$-structure.)

Definition 3

Let T be an $L_P$-theory. The 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

$$\begin{aligned} T^{+} \equiv T + d_{\Phi }, \end{aligned}$$

for some definition system $d_{\Phi }$, where $\Phi$ is some set of $L_P$-formulas.

Definition 4

Let a definition system $d_{\Phi }$ be given. Define the translation, induced by $\Phi$

$$\begin{aligned} \tau ^{+}_{\Phi }: L_{P,Q} \rightarrow L_P \end{aligned}$$

as follows. For symbols $P_i, Q_j$, variables $x,y, \overline{x}$, and for $L_{P,Q}$-formulas $\alpha , \alpha _1, \alpha _2$:

$$\begin{aligned} (1) \ \ \tau ^{+}_{\Phi }(P_i(\overline{x})):= & {} P_i(\overline{x})\\ (2) \ \ \tau ^{+}_{\Phi }(Q_j(\overline{x})):= & {} (\phi _j)^{\prime }\\ (3) \ \ \tau ^{+}_{\Phi }(x=y):= & {} (x=y)\\ (4) \ \ \tau ^{+}_{\Phi }(\lnot \alpha ):= & {} \lnot \ \tau ^{+}_{\Phi }(\alpha )\\ (5) \ \ \tau ^{+}_{\Phi }(\alpha _1 \# \alpha _2):= & {} \tau ^{+}_{\Phi }(\alpha _1) \ \# \ \tau ^{+}_{\Phi }(\alpha _2)\\ (6) \ \ \tau ^{+}_{\Phi }(\mathbf {q} x \alpha ):= & {} \mathbf {q} x \ \tau ^{+}_{\Phi }(\alpha ). \end{aligned}$$

where $\#$ 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 $Q_i( \overline{x})$.³ We call $\tau ^{+}_{\Phi }$ the translation induced by $\Phi$. It maps from the enriched language $L_{P,Q}$ back to the original language $L_P$. We let $\tau _{\Phi }$ be the restriction of $\tau ^{+}_{\Phi }$ to $L_Q$: thus, $\tau _{\Phi }$ maps from the new language $L_{Q}$ back to the original language $L_P$. ($\tau _{\Phi }$ is also called the translation induced by $\Phi$.)

Definition 5

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 $\alpha \in L_P$,

$$\begin{aligned} T_1 \vdash \alpha \leftrightarrow \tau _{\Phi }(\tau _{\Theta }(\alpha )) \end{aligned}$$

We write this more suggestively as:

$$\begin{aligned} (\tau _{\Phi } \tau _{\Theta }=1)_{T_1} \end{aligned}$$

And $\tau _{\Theta }$ is an left inverse of $\tau _{\Phi }$ in $T_2$ iff, for any $\beta \in L_Q$,

$$\begin{aligned} T_2 \vdash \beta \leftrightarrow \tau _{\Theta }(\tau _{\Phi }(\beta )) \end{aligned}$$

Likewise, we write this more suggestively as:

$$\begin{aligned} (\tau _{\Theta } \tau _{\Phi } =1)_{T_2} \end{aligned}$$

Definition 6

Let A be an $L_P$-structure. Then the $L_Q$-structure $D_{\Phi }A$ is defined by:

$$\begin{aligned} D_{\Phi }A := (A + d_{\Phi }) {\upharpoonright _{L_Q}} \end{aligned}$$

$D_{\Phi }A$ is called the definitional image of A with respect to $\Phi$.

Definition 7

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,

$$\begin{aligned} D_{\Phi }T : = \text {DedCl}(T + d_{\Phi })\upharpoonright _{L_Q} = \{\beta \in L_Q \mid T + d_{\Phi } \vdash \beta \} \end{aligned}$$

Definition 8

Structures A and B are definitionally equivalent wrt $d_{\Phi }$ and $d_{\Theta }$ iff

$$\begin{aligned} A + d_{\Phi } \cong B + d_{\Theta }. \end{aligned}$$

If this is so, we write:

Definition 9

Theories $T_1$ and $T_2$ are definitionally equivalent wrt $d_{\Phi }$ and $d_{\Theta }$ iff

$$\begin{aligned} T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }. \end{aligned}$$

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:

Lemma 1

The following hold always:⁴

$$\begin{aligned} (1)&\text {If } A \models T \text { then } D_{\Phi }A \models D_{\Phi }T\\ (2)&D_{\Phi }[\text {Mod}(T)] \subseteq \text {Mod}(D_{\Phi }T) \end{aligned}$$

Lemma 2

The following are equivalent:⁵

Lemma 3

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:

3 Definitional Equivalence: Model-Theoretic Criteria

Theorem 2 above establishes a criterion for in terms of translation: $(\tau _{\Phi }\tau _{\Theta } = 1)_{T_1}$ and $T_2 \equiv D_{\Phi }T_1$. Next, we establish model-theoretic criteria.

Definition 10

We write

to mean:

$$\begin{aligned} \text {for any } A \models T_1, \text { there is a } B \models T_2 \text { st } A + d_{\Phi } \cong B + d_{\Theta }. \end{aligned}$$

We write:

to mean: and .

Lemma 4

If , then $D_{\Phi }: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ and, for any $A \in \text {Mod}(T_1)$, we have .

Proof

Let . So, for any $A \models T_1$, there is a $B \models T_2$ such that $A + d_{\Phi } \cong B + d_{\Theta }$.

Consider the operator $D_{\Phi }$. Let $A \models T_1$. So, there is a $B \models T_2$ such that $A + d_{\Phi } \cong B + d_{\Theta }$. So, $B \cong D_{\Phi }A$. So, $D_{\Phi }A \models T_2$. So, $D_{\Phi } : \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$. And since $A + d_{\Phi } \cong B + d_{\Theta }$, we have $A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }$. So, , as required. $\square$

Lemma 5

If $D_{\Phi }: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ and, for any $A \in \text {Mod}(T_1)$, we have , then .

Proof

Suppose that $D_{\Phi }: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ and, for any $A \in \text {Mod}(T_1)$, we have .

Now suppose $A \models T_1$. We claim there is a $B \models T_2 \text { st } A + d_{\Phi } \cong B + d_{\Theta }$.

Since $A \models T_1$, we have $D_{\Phi }A \models T_2$ and

And so,

$$\begin{aligned} A + d_{\Phi } \cong D_{\Phi } A + d_{\Theta } \end{aligned}$$

So there is a $B \models T_2$ such that $A + d_{\Phi } \cong B + d_{\Theta }$. So, , as claimed. $\square$

Lemma 6

The following are equivalent:

Proof

Lemma 4 and Lemma 5. $\square$

Lemma 7

The following are equivalent:

Proof

$(1) \Rightarrow (2)$. Assume $T_1 + d_{\Phi } \vdash T_2 + d_{\Theta }$. Recall Lemma 3, which implies:

$$\begin{aligned} (*) \qquad \text {If } A + d_{\Phi } \models d_{\Theta }, \text { then } A + d_{\Phi } \cong D_{\Phi }A+ d_{\Theta }. \end{aligned}$$

We wish to show . Let $A \models T_1$. Thus, $A + d_{\Phi } \models T_1 + d_{\Phi }$. Thus, $A + d_{\Phi } \models T_2 + d_{\Theta }$. And so, $A + d_{\Phi } \models d_{\Theta }$. From $(*)$, it follows that $A + d_{\Phi } \cong D_{\Phi }A+ d_{\Theta }$. Since, $A + d_{\Phi } \models T_2$, we have $D_{\Phi }A + d_{\Theta } \models T_2 + d_{\Theta }$. Thus, $D_{\Phi }A \models T_2$. And $A + d_{\Phi } \cong D_{\Phi } A+ d_{\Theta }$. I.e., . Since this holds in general, .

$(2) \Rightarrow (1)$. We suppose, for any $A \models T_1$, there is some $B \models T_2$ such that $A + d_{\Phi } \cong B + d_{\Theta }$. For a contradiction, suppose $T_2 + d_{\Theta } \vdash \alpha$ and $T_1 + d_{\Phi } \nvdash \alpha$, for some formula $\alpha$. This gives us $A + d_{\Phi } \models T_1 + d_{\Phi }$, and $A + d_{\Phi } \not \models \alpha$. Thus, $A \models T_1$. Hence, there is a model $B \models T_2$ such that $A + d_{\Phi } \cong B + d_{\Theta }$. Thus, $B + d_{\Theta } \models T_2 + d_{\Theta }$. Since $T_2 + d_{\Theta } \vdash \alpha$, we have that $B + d_{\Theta } \models \alpha$. Since, $A + d_{\Phi } \cong B + d_{\Theta }$, we get $A + d_{\Phi } \models \alpha$. Contradiction. $\square$

We then obtain a characterization theorem:

Theorem 3

The following are equivalent

Proof

Reason as follows:

The relationship asserts the existence of two functions $F: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ and $G: \text {Mod}(T_2) \rightarrow \text {Mod}(T_1)$ such that, for any $A \in \text {Mod}(T_1), B \in \text {Mod}(T_2)$, we have

Together, these imply the existence of a bijection (wrt $\cong$):⁷

$$\begin{aligned} H: \text {Mod}(T_1) \leftrightarrow \text {Mod}(T_2) \end{aligned}$$

such that, for any $A \in \text {Mod}(T_1)$, we have:

To show this, first we prove a simple lemma about definitional equivalence of structures:

Lemma 8

If and , then $A \cong A^{\prime }$.

Proof

Let $A + d_{\Phi } \cong B + d_{\Theta }$ and $A^{\prime } + d_{\Phi } \cong B + d_{\Theta }$. Thus, $A + d_{\Phi } \cong A^{\prime } + d_{\Phi }$. Thus, by right cancellation, $A \cong A^{\prime }$. $\square$

Lemma 9

If , there exists a bijection $H: \text {Mod}(T_1) \leftrightarrow \text {Mod}(T_2)$ wrt $\cong$ such that, for any $A \in \text {Mod}(T_1)$, .

Proof

The assumption asserts the existence of two functions F, G which are linked by the parameters $\Phi , \Theta$. We show that G left-inverts F and F left-inverts G.

Let $A \in \text {Mod}(T_1)$. Then $F(A) \in \text {Mod}(T_2)$ and

Now consider G(F(A)). We have $G(F(A)) \in \text {Mod}(T_1)$ and

So, by the above cancellation lemma, $G(F(A)) \cong A$. Thus G is a left-inverse of F.

We may also show that F is a right inverse of G. For suppose $B \in \text {Mod}(T_2)$. Then $G(B) \in \text {Mod}(T_1)$ and

Likewise, consider F(G(B)). We have $F(G(B)) \in \text {Mod}(T_2)$ and

So, by the above cancellation lemma again, $F(G(B)) \cong B$. Thus F is a left-inverse of G.

Now if we have $F: X \rightarrow Y$ and $G: Y \rightarrow X$, and G is a left-inverse of F and F is a left-inverse of G, then $F: X \rightarrow Y$ is a bijection. Therefore, $F : \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ satisfies the conditions stated. $\square$

Lemma 10

If , then $D_{\Phi }: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ is a bijection wrt $\cong$ such that, for any $A \in \text {Mod}(T_1)$, .

Proof

Suppose . So, by Lemma 9, there exists a bijection $H: \text {Mod}(T_1) \leftrightarrow \text {Mod}(T_2)$ wrt $\cong$ such that, for any $A \in \text {Mod}(T_1)$, . Thus, for any $A \in \text {Mod}(T_1)$, we have,

$$\begin{aligned} \text {(i)} \qquad A + d_{\Phi } \cong H(A) + d_{\Theta } \end{aligned}$$

We will show that H and $D_{\Phi }$ are the same up to isomorphism: i.e.,

$$\begin{aligned} H(A) \cong D_{\Phi }A \end{aligned}$$

Since , by Lemma 4, we know that $D_{\Phi }: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ and, for any $A \in \text {Mod}(T_1)$, we have . So, for any $A \in \text {Mod}(T_1)$, we have

$$\begin{aligned} \text {(ii)} \qquad A + d_{\Phi } \cong D_{\Phi } A + d_{\Theta } \end{aligned}$$

Combining (i) and (ii), we infer,

$$\begin{aligned} \text {(iii)} \qquad H(A) + d_{\Theta } \cong D_{\Phi } A + d_{\Theta } \end{aligned}$$

And by cancellation, $H(A) \cong D_{\Phi }A$, as required. $\square$

Furthermore, the converse is true.

Lemma 11

If $D_{\Phi }: \text {Mod}(T_1) \leftrightarrow \text {Mod}(T_2)$ such that, for any $A \in \text {Mod}(T_1)$, , then .

Proof

Let $D_{\Phi }: \text {Mod}(T_1) \rightarrow \text {Mod}(T_2)$ be a bijection such that, for any $A \in \text {Mod}(T_1)$,

$$\begin{aligned} A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }. \end{aligned}$$

We want to show, first, that for, any $A \models T_1$, there is a $B \models T_2$ st $A + d_{\Phi } \cong B + d_{\Theta }$. If $A \models T_1$, then $D_{\Phi }A$ is such a model. So, . We want to show, second, that for, any $B \models T_2$, there is an $A \models T_1$ st $A + d_{\Phi } \cong B + d_{\Theta }$. If $B \models T_2$, then $D_{\Phi }^{-1}B$ is such a model. So, . And therefore we have . $\square$

The two previous lemmas give a second characterization theorem:

Theorem 4

The following are equivalent:

4 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 model-theoretic definability is explained as follows:

Let Th1 and Th2 be theories, maybe on different first-order languages. An explicit definition of Th1 over Th2 is a conjunction $\Delta$ of explicit definitions of the relation symbols of Th1 in terms of the language of Th2 such that the models of Th1 are exactly the reducts of the models of $Th2 \cup \Delta$ (to the language of Th1). Thus, we get the models of Th1 from those of Th2 by first defining the relations of Th1 via using $\Delta$, and then forgetting the relations not present in the language of Th2. ([1]: 591)

If we switch their Th1 to $T_2$, Th2 to $T_1$, and $\Delta$ to $d_{\Phi }$, this corresponds, in our terminology, to saying that $\Phi$ model-theoretically defines $T_2$ in $T_1$: i.e., $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$ (see [4], Definition 28, Part I).

It is relatively straightforward to prove:

Lemma 12

Let . Then

$$\begin{aligned} (1)&\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]\\ (2)&\text {Mod}(T_1) = D_{\Theta }[\text {Mod}(T_2)]. \end{aligned}$$

Proof

Let us suppose that . That is, $T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$.

We first wish to prove (1): $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$. That is, we wish to prove that, for any $L_Q$-structure B, we have:

$$\begin{aligned} B \models T_2 \text { if and only if } B \cong D_{\Phi }A, \text { for some } A \models T_1. \end{aligned}$$

Suppose $B \models T_2$. Thus, $B + d_{\Theta } \models T_2 + d_{\Theta }$. So, $B + d_{\Theta } \models T_1 + d_{\Phi }$. Let $A = D_{\Theta }B$. Now $B + d_{\Theta } \models d_{\Phi }$. So, from Lemma 3 (but relabelling in terms of an $L_Q$-structure B instead of an $L_P$-structure A), we have: $B + d_{\Theta } \cong D_{\Theta }B + d_{\Phi }$. I.e., $B + d_{\Theta } \cong A + d_{\Phi }$. So, $A + d_{\Phi } \models T_1 + d_{\Phi }$. So, $A \models T_1$, as required.

Instead, suppose $B \cong D_{\Phi }A$, where $A \models T_1$. Thus, $A + d_{\Phi } \models T_1 + d_{\Phi }$. And thus, $A + d_{\Phi } \models T_2 + d_{\Theta }$. Since $A + d_{\Phi } \models d_{\Theta }$, we have, by Lemma 3 again, $A + d_{\Phi } \cong D_{\Theta }A + d_{\Theta }$. Thus, $A + d_{\Phi } \cong B + d_{\Theta }$. And so, $B + d_{\Theta } \models T_2 + d_{\Theta }$, which implies that $B \models T_2$, as required.

The proof of (2), $\text {Mod}(T_1) = D_{\Theta }[\text {Mod}(T_2)]$, is entirely analogous, just switching labels. $\square$

On the other hand, the converse of Lemma 12 is not true. Andréka et al 2005 [1] provide a counter-example:

Theorem 5

(Andréka et al 2005 [1]) There are theories $T_1, T_2$ and defining sets $\Phi , \Theta$ such that the following all hold:

One way to understand the problem is that the translations associated with $\Phi$ and $\Theta$ are not mutual inverses. Below (Theorem 16) we show the “gap” connected to Lemma 12 and Theorem 5 closes, by requiring not only $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$ but also that $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$: if these conditions hold, then it follows that .⁸

5 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:

Theorem 8

The following are equivalent:⁹

6 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 $\Phi$ of $L_P$-formulas, and then examine the “internal structure” defined by them in A: this is what we have called the “definitional image”, $D_{\Phi }A$. What condition should we impose if we wish to reconstruct A from its image $D_{\Phi }A$?

Clearly, the condition is that the definitional expansion $A + d_{\Phi }$—which we used to define $D_{\Phi }A$ prior to forgetting the P-relations—should satisfy an invertibility condition, namely that each original $P_i$ be explicitly definable from a formula, say $\theta _i$, in the new Q-language. This then means that the original set $\Phi$ of $L_P$-formulas, in some sense, does not “omit” any structural content built into A itself. The formulas $\phi _i$ merely “encode” that content differently.

That condition is then, simply, that there is a set $\Theta = \{\theta _i\}_{i \in I_P}$ of Q-formulas such that

$$\begin{aligned} \boxed {A + d_{\Phi } \models d_{\Theta }} \end{aligned}$$

holds. Or, more explicitly, for each atomic formula $P_i( \overline{x})$ of $L_P$, we have an explicit inverse definition,

$$\begin{aligned} A + d_{\Phi } \models \forall \overline{x}(P_i( \overline{x}) \leftrightarrow \theta _i) \end{aligned}$$

for some $\theta _i \in L_Q$.

The same intuition motivates an analogous account for theories. The definitional image $D_{\Phi }T$ 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 $\Theta = \{\theta _i\}_{i \in I_P}$ of Q-formulas such that

$$\begin{aligned} \boxed {T + d_{\Phi } \vdash d_{\Theta }} \end{aligned}$$

holds. Similarly, we can more explicitly express this by saying that, for each atomic formula $P_i( \overline{x})$ of $L_P$, we have an explicit inverse definition,

$$\begin{aligned} T + d_{\Phi } \vdash \forall \overline{x}(P_i( \overline{x}) \leftrightarrow \theta _i) \end{aligned}$$

for some $\theta _i \in L_Q$.

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:

Definition 11

$\Phi$ is a representation basis for A with inverse $\Theta$ iff $A + d_{\Phi } \models d_{\Theta }$.

Definition 12

$\Phi$ is a representation basis for T with inverse $\Theta$ iff $T + d_{\Phi } \vdash d_{\Theta }$.

In each case, the formulas $\theta _i$ in the set $\Theta$ are called inversion formulas.

The following two lemmas are entirely straightforward.

Lemma 13

If T is inconsistent, any $\Phi$ is a representation basis for T.

Lemma 14

Suppose $T + d_{\Phi } \vdash d_{\Theta }$ and $T + d_{\Phi } \vdash d_{\Psi }$, for inverses $\Theta = \{\theta _i\}$ and $\Psi = \{\psi _i\}$. Then, for all i, we have: $T + d_{\Phi } \vdash \theta _i \leftrightarrow \psi _i$.

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 $L_P$):

Definition 13

$\Phi$ is a logical representation basis for $L_P$ with inverse $\Theta$ iff $d_{\Phi } \vdash d_{\Theta }$.

It is easy to see that being a representation basis is preserved under theory extension (or structure expansion):

Lemma 15

If $\Phi$ is a representation basis for T (or structure A), then $\Phi$ is also a representation basis for every extension of T (resp. every expansion of A). In particular, if $\Phi$ 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. $\Phi$ might be a representation basis for T, but not a representation basis for a sub-theory or a weaker theory. Similarly, $\Phi$ 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 $\Phi$ 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 $T + d_{\Phi }$.

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

$\Phi$ is a representation basis for T iff $\Phi$ is a semantic representation basis for T.

7 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 $\Phi$ for (a propositional language) $L_P$ but where the definitional image of logic in $L_P$ under $D_{\Phi }$ is not logically true.

Example 1

Given a signature $P = \{P_i\}_{i \in I_P}$, let $\Phi = \{P_i(x_1, \dots , x_{a(P_i)})\}_{i\in I_P}$, consisting of the atomic formulas of the language $L_P$. Then $\Phi$ 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 $T + d_{\Phi }$. 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

$$\begin{aligned} \phi _1&: =&p_1\\ \phi _2&: =&p_1 \leftrightarrow p_2 \end{aligned}$$

Then $\{\phi _1, \phi _2\}$ is a logical representation basis for $L_P$. For consider the definitions of the new symbols, $Q_1$ and $Q_2$:

$$\begin{aligned} Q_1&\leftrightarrow&p_1\\ Q_2&\leftrightarrow&(p_1 \leftrightarrow p_2) \end{aligned}$$

Let $d_{\Phi }$ be $\{Q_1 \leftrightarrow p_1, Q_2 \leftrightarrow (p_1 \leftrightarrow p_2)\}$. Then $d_{\Phi }$ implies:

$$\begin{aligned} p_1&\leftrightarrow&Q_1\\ p_2&\leftrightarrow&(Q_1 \leftrightarrow Q_2) \end{aligned}$$

So, given $d_{\Phi }$, $p_1$ and $p_2$ can be explicitly defined in terms of $Q_1$ and $Q_2$ (this happens essentially because $p_1 \leftrightarrow (p_1 \leftrightarrow p_2)$ 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 $\phi _1, \phi _2$:

$$\begin{aligned} \phi _1&: =&P_1(x)\\ \phi _2&: =&P_1(x) \leftrightarrow P_2(x) \end{aligned}$$

Then the pair $\{\phi _1, \phi _2\}$ is a logical representation basis for $L_P$. For the explicit definitions,

$$\begin{aligned} Q_1(x)&\leftrightarrow&P_1(x)\\ Q_2(x)&\leftrightarrow&(P_1(x) \leftrightarrow P_2(x)) \end{aligned}$$

imply (in logic alone) the inversions:

$$\begin{aligned} P_1(x)&\leftrightarrow&Q_1(x)\\ P_2(x)&\leftrightarrow&(Q_1(x) \leftrightarrow Q_2(x)) \end{aligned}$$

for the same reason as the previous example.

Example 4

Consider, given $\Phi$, the definitional image of T when T is the empty theory. Let $\mathsf {Log}_P$ be the set of $L_P$-sentences which are theorems of logic in $L_P$. The definitional image $\mathsf {Log}_P$ under $\Phi$ is given by:

$$\begin{aligned} D_{\Phi }\mathsf {Log}_P := \{\beta \in \text {Sent}(L_Q) \mid d_{\Phi } \vdash \beta \} \end{aligned}$$

Then:

Observation

There is a signature P and (logical) representation basis $\Phi$ for logic in $L_P$ such that the $L_Q$ theory $D_{\Phi }\mathsf {Log}_P$ isn’t logically true.

For example, let $P = \{p_1,p_2\}$ be a propositional signature and consider the three $L_P$-formulas:

$$\begin{aligned} \phi _1&: =&p_1 \wedge p_2\\ \phi _2&: =&p_1 \wedge \lnot p_2\\ \phi _3&: =&\lnot p_1 \wedge p_2 \end{aligned}$$

Then one can show that $\{\phi _1, \phi _2, \phi _3\}$ is a logical representation basis for $L_P$. Introduce the definition system $d_{\Phi }$ for the new symbols, $Q_1, Q_2$ and $Q_3$:

$$\begin{aligned} Q_1&\leftrightarrow&p_1 \wedge p_2\\ Q_2&\leftrightarrow&p_1 \wedge \lnot p_2\\ Q_3&\leftrightarrow&\lnot p_1 \wedge p_2 \end{aligned}$$

Then inversions—i.e., explicit definitions of $p_1, p_2$ in terms of the $Q_i$—can be obtained as follows:

$$\begin{aligned} d_{\Phi } \vdash p_1 \leftrightarrow (Q_1 \vee Q_2)\\ d_{\Phi } \vdash p_2 \leftrightarrow (Q_1 \vee Q_3) \end{aligned}$$

However, not every $L_Q$-theorem of $d_{\Phi }$ is logically true. For example,

$$\begin{aligned} d_{\Phi } \vdash Q_1 \rightarrow \lnot Q_2\\ d_{\Phi } \vdash Q_1 \rightarrow \lnot Q_3\\ d_{\Phi } \vdash Q_2 \rightarrow \lnot Q_3 \end{aligned}$$

In each case, we have $\beta \in L_Q$, with $d_{\Phi } \vdash \beta$. But, for each $\beta$, we have $\nvdash \beta$.¹⁰ And thus the theory $D_{\Phi }\mathsf {Log}_P$ is not logically true. $\triangle$

8 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 “$\Phi$ is a representation basis for T with inverse $\Theta$”. Then, in Subsection 8.2, we shall see how imposing this condition leads to strengthened properties of the $D_{\Phi }$ 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 $\Phi$ and $\Theta$”.

8.1 Criteria for Being a Representation Basis

First, we establish equivalents for $\Phi$ being a representation basis for T with inverse $\Theta$.

Theorem 10

The following conditions are equivalent:

Proof

(1) $\Leftrightarrow$ (2) is simply Definition 12. (2) $\Leftrightarrow$ (3) follows immediately from Lemma 2. Similarly, (2) $\Leftrightarrow$ (4) follows immediately from Lemma 2.

For (2) $\Leftrightarrow$ (5): first, suppose $T + d_{\Phi } \vdash d_{\Theta }$. So, by Lemma 2: $T + d_{\Phi } \equiv D_{\Phi }T + d_{\Theta }$. Next, suppose $A \models T$. So, $A + d_{\Phi } \models d_{\Theta }$. So, by Lemma 3: $A + d_{\Phi } \cong D_{\Phi }A + d_{\Theta }$. So, , as claimed.

For the converse, suppose , for any $A \models T$. By Lemma 3, we infer that $A + d_{\Phi } \models d_{\Theta }$, for any $A \models T$. Now suppose $B \models T + d_{\Phi }$. Let $B = A' + d_{\Phi }$. So, $A' \models T$. And therefore, $A' + d_{\Phi } \models d_{\Theta }$. So, $B \models d_{\Theta }$. Therefore, $T + d_{\Phi } \vdash d_{\Theta }$, as claimed.

For $(5) \Leftrightarrow (6)$: notice that, by Lemma 3,

$\square$

Next, specializing to the case where T is pure logic, we obtain:

Theorem 11

The following are equivalent:

Proof

The equivalences of (1), (2) and (3) follow immediately from Theorem 10 by setting T as the empty theory. The equivalences of (4), (5) and (6) follow immediately from Lemma 3. And the equivalence of (2) and (5) follows from the equivalence of conditions (2) and (5) in Theorem 10, by setting T as the empty theory. $\square$

8.2 Consequences of $\Phi$ Being a Representation Basis for T

Theorem 12

Let $\Phi$ be a representation basis for T with inverse $\Theta$. Then:

$$\begin{aligned} (1)&D_{\Phi } : \text {Mod}(T) \leftrightarrow \text {Mod}(D_{\Phi }T).\\ (2)&\text {If } A,B \in \text {Mod}(T) \text { and } D_{\Phi }A \cong D_{\Phi }B, \text { then } A \cong B.\\ (3)&\text {For any } B \models D_{\Phi }T, \text { there is } A \models T \text { st } B \cong D_{\Phi }A.\\ (4)&A \models T \text{ iff } D_{\Phi }A \models D_{\Phi }T.\\ (5)&D_{\Phi }[\text {Mod}(T)] = \text {Mod}(D_{\Phi }T). \end{aligned}$$

Proof

Let us suppose $\Phi$ is a representation basis for T with inverse $\Theta$. In particular, by Theorem 10(4), we have:

Second, recall Theorem 8. Conditions (1) and (5) tell us that iff $D_{\Phi }: \text {Mod}(T_1) \leftrightarrow \text {Mod}(T_2)$ st, for any $A \in \text {Mod}(T_1)$, . So, we have, using $(*)$:

We can then prove the results quickly.

(1) is a consequence of $(**)$. And (2) and (3) are obvious consequences of (1).

For (4). We already know that if $A \models T$, then $D_{\Phi }A \models D_{\Phi }T$ (Lemma 1(1)). So, instead, suppose that $D_{\Phi }A \models D_{\Phi }T$. Let $B = D_{\Phi }A$. It follows from (**) that there is a model $A' \models T$ such that $B \cong D_{\Phi }A'$. So $B \cong D_{\Phi }A'$ and $B \cong D_{\Phi }A$, which implies: $D_{\Phi }A' \cong D_{\Phi }A$. And therefore, $A \cong A'$, since $D_{\Phi }$ is injective. So, $A \models T$.

For (5). We already know that $D_{\Phi }[\text {Mod}(T)] \subseteq \text {Mod}(D_{\Phi }T)$ (Lemma 1(2)). We wish to prove the converse: $\text {Mod}(D_{\Phi }T) \subseteq D_{\Phi }[\text {Mod}(T)]$. So let $B \in \text {Mod}(D_{\Phi }T)$. We claim $B \in D_{\Phi }[\text {Mod}(T)]$. That is, we claim there is some $A \models T$ such that $B \cong D_{\Phi }A$. But this follows immediately from (3). $\square$

The next two results reveal the sense in which moving between different bases, say $\Phi = \{\phi _i\}$ and $\Phi ^{*} = \{\phi ^{*}_i\}$, is somewhat analogous to moving between bases for a vector space or between co-ordinate system on a manifold. That is, so long as both are representation bases, then they are interdefinable:

Theorem 13

Let an $L_P$-structure A be given. Let $\Phi = \{\phi _i\}$ be a representation basis for A defining the $Q_i$ as $\phi _i$, and with inverse $\Theta$. Let $\Phi ^{*} = \{\phi ^{*}_i\}$ be a representation basis for A, defining the $Q^{*}_j$ as $\phi ^{*}_j$, and with inverse $\Theta ^{*}$. Then:

$$\begin{aligned} (1)&\text {For each } Q^{*}_j, \text { there is } \alpha \in L_Q \text { st } A + d_{\Phi } + d_{\Phi ^{*}} \models \forall \overline{x}\ (Q^{*}_j( \overline{x}) \leftrightarrow \alpha ).\\ (2)&\text {For each } Q_i, \text { there is } \beta \in L_{Q^{*}} \text { st } A + d_{\Phi } + d_{\Phi ^{*}} \models \forall \overline{x}\ (Q_i( \overline{x}) \leftrightarrow \beta ). \end{aligned}$$

Proof

One has definitions, in the definitional expansion $A + d_{\Phi } + d_{\Phi ^{*}}$, of the symbols $Q_i$ and the $Q^{*}_i$ in terms of the $P_i$ symbols, as well as “inverse definitions” of the $P_i$ in terms of the $Q_i$ and the $Q^{*}_i$. One can then verify that there are definitions of the $Q_i$ in terms of the $Q^{*}_i$ and vice versa. $\square$

Theorem 14

Let an $L_P$-theory T be given. Let $\Phi = \{\phi _i\}$ be a representation basis for T defining the $Q_i$ as $\phi _i$, and with inverse $\Theta$. Let $\Phi ^{*} = \{\phi ^{*}_i\}$ be a representation basis for T, defining the $Q^{*}_j$ as $\phi ^{*}_j$, and with inverse $\Theta ^{*}$. Then:

$$\begin{aligned} (1)&\text {For each } Q^{*}_j, \text { there is } \alpha \in L_Q \text { st } T + d_{\Phi } + d_{\Phi ^{*}} \vdash \forall \overline{x}\ (Q^{*}_j( \overline{x}) \leftrightarrow \alpha ).\\ (2)&\text {For each } Q_i, \text { there is } \beta \in L_{Q^{*}} \text { st } T+ d_{\Phi } + d_{\Phi ^{*}} \vdash \forall \overline{x}\ (Q_i( \overline{x}) \leftrightarrow \beta ). \end{aligned}$$

Proof

Analogously to the previous result, one has definitions, in the definitional extension $T + d_{\Phi } + d_{\Phi ^{*}}$, of the symbols $Q_i$ and the $Q^{*}_i$ in terms of the $P_i$ symbols, as well as “inverse definitions” of the $P_i$ in terms of the $Q_i$ and the $Q^{*}_i$. One can then verify that there are definitions of the $Q_i$ in terms of the $Q^{*}_i$ and vice versa. $\square$

8.3 Criteria for Definitional Equivalence

We extend the criteria for definitional equivalence (Theorem 8) with a sixth condition, now formulated in terms of “representation basis”:

Theorem 15

The following are equivalent:

Proof

The equivalence of the first five criteria is stated in Theorem 8. To establish $(3) \Leftrightarrow (6)$, note that, using Theorem 10(1, 3), $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$ if and only if $(\tau _{\Theta } \tau _{\Phi } = 1)_{T_1}$. $\square$

A (final) corollary is:

Theorem 16

Suppose $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$. Then

are equivalent.

Proof

For $(1) \Rightarrow (2)$, let us suppose first that . Then this already implies that $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$ and, furthermore, $T_2 \equiv D_{\Phi }T_1$. So, by Theorem 12(5), $D_{\Phi }[\text {Mod}(T_1)] = \text {Mod}(D_{\Phi }T_1)$. And since $T_2 \equiv D_{\Phi }T_1$, we conclude that $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$.

For $(2) \Rightarrow (1)$, let us suppose instead that $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$ and $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$. By Theorem 12(5), $D_{\Phi }[\text {Mod}(T_1)] = \text {Mod}(D_{\Phi }T_1)$. Hence, $\text {Mod}(T_2) = \text {Mod}(D_{\Phi }T_1)$. And so, $T_2 \equiv D_{\Phi }T_1$. And by condition (6) of Theorem 15, we conclude that . $\square$

Thus, if $\Phi$ is a representation basis for $T_1$ with inverse $\Theta$ and $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$, it follows that . However, the counterexample from [1] shows that $\text {Mod}(T_2) = D_{\Phi }[\text {Mod}(T_1)]$ is too weak for this conclusion: the missing ingredient is that the defining set $\Phi$ be a representation basis for $T_1$ (with inverse $\Theta$).

9 Theories and Basis Dependence

There are several theories which have been carefully studied in mathematical logic—usually involving arithmetic and set theory—known to be definitionally equivalent.

But this is not a minor topic of narrow interest only to mathematical logicians. First, philosophers of science have long been interested in what constitutes either the empirical equivalence, or the full equivalence, of scientific theories. In the mid 70s, David Miller introduced the “language dependence” problem for theories of truthlikeness: the core of his argument being that changing which predicates are taken as primitive can affect comparisons of truthlikeness for false theories ([5,6,7]). In metaphysics and epistemology, there is the famous example of “grue” and “bleen” predicates introduced by Nelson Goodman ([3]).

Second, there is an intuitive idea within mathematics and in physics of trying to eliminate dependence on “arbitrary choices”, and that one prefers “basis independent” descriptions of mathematical objects. In a recent paper, Albert Visser notes:

The study of interpretability is, in part, about the escape from the tyranny of signature. Specific choices for the language are implementation artifacts introduced because, after all is said an done, we have to do things one way or another. Good mathematical properties of theories should be independent of these arbitrary choices. (Visser 2015 [8], p. 2 of preprint)

An example is the basis of a vector space V. Every vector space has a basis $\{\mathbf {e}_a\}$ and the space itself is reconstructed as the linear span of the basis. But any such basis itself is an “arbitrary parametrization” or “implementation” of V and vectors in V don’t “care” what basis they are expanded relative to: given two bases $\{\mathbf {e}_a\}$ and $\{\mathbf {e}^{\prime }_a\}$, and some $\mathbf {v} \in V$, there are expansions:

$$\begin{aligned} X^a \mathbf {e}_a = \mathbf {v} = Y^a \mathbf {e}^{\prime }_a \end{aligned}$$

Sometimes one basis is much more convenient to work with than others (e.g., certain matrices get diagonalized; certain operators take simpler forms). But these efficiency features have nothing to do with vector space itself.

Similarly, any particular chart $(U,\varphi )$ in a topological manifold is, in some sense, arbitrary. For the points in M shouldn’t really “care” what co-ordinates they are given in overlapping charts; say, charts $(U,\varphi )$ and $(V,\psi )$ where $U \cap V \ne \varnothing$. And similarly, in the overlap region, there is an invertible mapping (a local homeomorphism) between the charts. If M is a 3-manifold and I am told that $\varphi (x) = (0,0,0)$, then I have been told nothing intrinsic to M itself; since any point $x \in M$ can be given those co-ordinates by some chart $\varphi$.

One wonders if structures and theories could be treated similarly. Here we have explained (Sect. 6) what it is for a set $\Phi = \{\phi _i\}_{i \in I}$ of formulas of a language L to form a “representation basis” for an L-structure A or for an L-theory T. The atomic formulas automatically do. But complex logical compounds may also form a representation basis for A or T, so long as the corresponding system of definitions is “invertible”. If the set $\Phi = \{\phi _i\}_{i \in I}$ is a representation basis, each primitive $P_i$ of the language L can be given an explicit definition of the form $\forall \overline{x}(P_i( \overline{x}) \leftrightarrow \theta _i)$, where $\theta _i$ contains only new predicates $Q_i$ introduced via explicit definition to represent the $\phi _i \in \Phi$. These intertranslatable representation bases then provide “definitionally equivalent reparametrizations” of a structure A or a theory T (relative to L).¹¹