Quine’s conjecture on many-sorted logic

Abstract

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic.

Appendix

The objective of this appendix is to prove Theorem 2. We prove a special case of the result for convenience. We will assume that $\Sigma$ has only three sort symbols $\sigma _1, \sigma _2,\sigma _3$ and that $\Sigma$ does not contain function or constant symbols. A perfectly analogous (though more tedious) proof goes through in the general case.

We prove the result by explicitly constructing a “common Morita extension” $T_{4}\cong {\widehat{T}}_{4}$ of T and ${\widehat{T}}$ to the following signature.

$$\begin{aligned} \Sigma ^+=\Sigma \cup {\widehat{\Sigma }}&\cup \{\sigma _{12}\} \cup \{\rho _1, \rho _2, \rho _{12}, \rho _3\} \cup \{ i_1,i_2, i_3\} \end{aligned}$$

The symbol $\sigma _{12}\in \Sigma ^+$ is a sort symbol. The symbols denoted by subscripted $\rho$ are function symbols. Their arities are expressed in the following figure.

The symbols $i_1$, $i_2$, and $i_3$ are function symbols with arity $\sigma _1\rightarrow \sigma$, $\sigma _2\rightarrow \sigma$, and $\sigma _3\rightarrow \sigma$, respectively.

We now turn to the proof.

Proof of Theorem 2

The following figure illustrates how our proof will be organized.

Steps 1–3 define the theories ${\widehat{T}}_1,\ldots , {\widehat{T}}_{4}$, steps 4–6 define $T_1,\ldots , T_{4}$, and step 7 shows that $T_{4}$ and ${\widehat{T}}_{4}$ are logically equivalent.

• Step 1 We begin by defining the theory ${\widehat{T}}_1$. For each sort $\sigma _j\in \Sigma$ we consider the following sentence.

  

  The sentence $\theta _{\sigma _j}$ defines the symbols $\sigma _j$ and $i_j$ as the subsort of “things that are $q_{\sigma _j}$.” The auxiliary axioms $\phi _{\sigma _j}$ of ${\widehat{T}}$ guarantee that the admissibility conditions for these definitions are satisfied. The theory ${\widehat{T}}_1={\widehat{T}}\cup \{\theta _{\sigma _1}, \theta _{\sigma _2},\theta _{\sigma _3}\}$ is therefore a Morita extension of ${\widehat{T}}$ to the signature ${\widehat{\Sigma }}\cup \{\sigma _1,\sigma _2,\sigma _3, i_1,i_2,i_3\}$.

• Step 2 We now define the theories ${\widehat{T}}_2$ and ${\widehat{T}}_3$. Let $\theta _{\sigma _{12}}$ be a sentence that defines the symbols $\sigma _{12}, \rho _1, \rho _2$ as a coproduct sort. The theory ${\widehat{T}}_2={\widehat{T}}_1\cup \{\theta _{\sigma _{12}}\}$ is clearly a Morita extension of ${\widehat{T}}_1$.

  We have yet to define the function symbols $\rho _{12}$ and $\rho _3$. The following two sentences define these symbols.

  

  The sentence $\theta _{\rho _3}$ simply defines $\rho _3$ to be equal to the function $i_3$. For the sentence $\theta _{\rho _{12}}$, we define the formula $\psi (x,y)$ to be

  $$\begin{aligned} \exists _{\sigma _{1}} z_1 \big ( \rho _{1}(z_1)=x\wedge i_{1}(z_1)=y\big ) \vee \exists _{\sigma _{2}} z_2\big ( \rho _{2}(z_2)=x\wedge i_{2}(z_2)=y\big ) \end{aligned}$$

  We should take a moment here to understand the definition $\theta _{\rho _{12}}$. We want to define what the function $\rho _{12}$ does to an element a of sort $\sigma _{12}$. Since the sort $\sigma _{12}$ is the coproduct of the sorts $\sigma _1$ and $\sigma _2$, the element a must “actually be” of one of the sorts $\sigma _1$ or $\sigma _2$. (The disjuncts in the formula $\psi (x,y)$ correspond to these possibilities.) The definition $\theta _{\rho _{12}}$ stipulates that if a is “actually” of sort $\sigma _j$, then the value of $\rho _{12}$ at a is the same as the value of $i_j$ at a. One can verify that ${\widehat{T}}_{2}$ satisfies the admissibility conditions for $\theta _{\rho _3}$ and $\theta _{\rho _{12}}$, so the theory ${\widehat{T}}_3={\widehat{T}}_{2}\cup \{\theta _{\rho _3}, \theta _{\rho _{12}}\}$ is a Morita extension of ${\widehat{T}}_{2}$ to the signature

  $$\begin{aligned} {\widehat{\Sigma }}\cup \{\sigma _1,\sigma _2, \sigma _3, \sigma _{12}, i_1,i_2, i_3, \rho _1, \rho _2,\rho _3, \rho _{12}\} \end{aligned}$$

• Step 3 We now describe the $\Sigma ^+$-theory ${\widehat{T}}_{4}$. This theory defines the predicates in the signature $\Sigma$. Let $p\in \Sigma$ be a predicate symbol of arity $\sigma _{j_1}\times \cdots \times \sigma _{j_m}$. We consider the following sentence.

  

  The theory ${\widehat{T}}_{4}={\widehat{T}}_3\cup \{\theta _{p}:p\in \Sigma \}$ is therefore a Morita extension of ${\widehat{T}}_3$ to the signature $\Sigma ^+$.

• Step 4 We turn to the left-hand side of our organizational figure and define the theories $T_1$ and $T_{2}$. We proceed in an analogous manner to the first part of Step 2. The theory $T_1=T\cup \{\theta _{\sigma _{12}}\}$ is a Morita extension of T to the signature $\Sigma \cup \{\sigma _{12}, \rho _1,\rho _2\}$. Now let $\theta _\sigma$ be the sentence that defines the symbols $\sigma , \rho _{12}, \rho _3$ as a coproduct sort. The theory $T_{2}=T_{1}\cup \{\theta _{\sigma }\}$ is a Morita extension of $T_{1}$ to the signature $\Sigma \cup \{\sigma _{12}, \sigma , \rho _1, \rho _2, \rho _3, \rho _{12}\}$.

• Step 5 This step defines the function symbols $i_1$, $i_2$, and $i_3$. We consider the following sentences.

  

  The sentence $\theta _{i_3}$ defines the function symbol $i_3$ to be equal to $\rho _3$. The sentence $\theta _{i_{2}}$ defines the function symbol $i_{2}$ to be equal to the composition “$\rho _{12}\circ \rho _{2}$.” Likewise, the sentence $\theta _{i_{1}}$ defines the function symbol $i_{1}$ to be “$\rho _{12}\circ \rho _{1}$.” The theory $T_3=T_{2}\cup \{\theta _{i_1},\theta _{i_2}, \theta _{i_3}\}$ is a Morita extension of $T_{2}$ to the signature $\Sigma \cup \{\sigma _{12}, \sigma , \rho _1, \rho _2, \rho _3, \rho _{12}, i_1, i_2, i_3\}$.

• Step 6 We still need to define the predicate symbols in ${\widehat{\Sigma }}$. Let $\sigma _j\in \Sigma$ be a sort symbol and $p\in \Sigma$ a predicate symbol of arity $\sigma _{j_1}\times \cdots \times \sigma _{j_m}$. We consider the following sentences.

  

  These sentences define the predicates $q_{\sigma _j}\in {\widehat{\Sigma }}$ and $q_p\in {\widehat{\Sigma }}$. One can verify that $T_3$ satisfies the admissibility conditions for the definitions $\theta _{q_{\sigma _j}}$. And therefore the theory $T_{4}=T_3\cup \{\theta _{q_{\sigma _1}}, \theta _{q_{\sigma _2}}, \theta _{q_{\sigma _3}}\}\cup \{\theta _{q_{p}}: p\in \Sigma \}$ is a Morita extension of $T_3$ to the signature $\Sigma ^+$.

• Step 7 It only remains to show that the $\Sigma ^+$-theories $T_{4}$ and ${\widehat{T}}_{4}$ are logically equivalent. One can verify by induction on the complexity of $\psi$ that

  $$\begin{aligned} T_{4}\vDash \psi \leftrightarrow {\widehat{\psi }}\text { and } {\widehat{T}}_{4}\vDash \psi \leftrightarrow {\widehat{\psi }}. \end{aligned}$$

  (5)

  for every $\Sigma$-sentence $\psi$. One then uses (5) to show that $\text {Mod}(T_{4})=\text {Mod}({\widehat{T}}_{4})$. The argument involves a number of cases, but since each case is straightforward we leave them to the reader to verify. The theories $T_{4}$ and ${\widehat{T}}_{4}$ are logically equivalent, which implies that T and ${\widehat{T}}$ are Morita equivalent. $\square$