Generalizing empirical adequacy I: multiplicity and approximation

Abstract

I provide an explicit formulation of empirical adequacy, the central concept of constructive empiricism, and point out a number of problems. Based on one of the inspirations for empirical adequacy, I generalize the notion of a theory to avoid implausible presumptions about the relation of theoretical concepts and observations, and generalize empirical adequacy with the help of approximation sets to allow for lack of knowledge, approximations, and successive gain of knowledge and precision. As a test case, I provide an application of these generalizations to a simple interference phenomenon.

Appendix: Proofs

Claim 1

For some appearances $\mathbf {A}$, some theories are idiosyncratically empirically adequate but not empirically adequate.

Proof

Let the appearances be given by the set of the two structures $\{\langle \{1, 2\},\{1,2\}\rangle ,\langle \{3,4\},\{3\}\rangle \}$. Let the theory be given by the family with members $\mathfrak {T} _1=\langle \{0,1,2, \},\{0,1,2\}\rangle$ and $\mathfrak {T} _2=\langle \{3,4,5\},\{3\}\rangle$ as well as the singleton sets of empirical substructures $\mathbf {E} _1=\{\langle \{1, 2\},\{1,2\}\rangle \}$ and $\mathbf {E} _2=\{\langle \{3,4\},\{3\}\rangle \}$. Let all other models of the theory be isomorphic to $\mathfrak {T} _1$ or $\mathfrak {T} _2$ and have the corresponding empirical substructures. Then the theory is idiosyncratically empirically adequate by virtue of the identity mapping on each of the appearances’ domains, but it is not empirically adequate. $\square$

Claim 2

For some appearances $\mathbf {A}$ and theory $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$, $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate given $\mathbf {A}$, but $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ has no model such that all appearances in $\mathbf {A}$ are empirical substructures of that model.

Proof

Let the appearances be given by the set of the two structures $\{\langle \{a, b\},\{a,b\}\rangle ,\langle \{c,d\},\{c\}\rangle \}$, where $a,b,c$, and $d$ are distinct objects. Let the theory be given by the family with the member $\mathfrak {T} _1=\langle \{1,2,3\},\{1,2\}\rangle$ and the set of empirical substructures $\mathbf {E} _1=\{\langle \{1, 2\},\{1,2\}\rangle , \langle \{2,3\},\{2\}\rangle \}$. Let all other models of the theory be isomorphic to $\mathfrak {T} _1$ and have the corresponding empirical substructures. Then the theory is empirically adequate, but every bijection from $\{1,2,3\}$—and thus every isomorphism for $\mathfrak {T} _1$—maps $2$, the object shared by the empirical substructures, to a single object. Since the domains of the appearances do not share an element, the appearances therefore can never be empirical substructures of the same model of the theory. $\square$

Claim 3

A theory $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for epistemic appearances $\mathbf {A}$ if and only if there are epistemically possible appearances $\mathbf {A} '$ such that $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for $\mathbf {A} '$.

Proof

‘$\Rightarrow$’: $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for $\mathbf {A}$ if and only if there is some $n\in N$ such that for every $\mathbf {Q} \in \mathbf {A}$, there are $\mathfrak {A} \in \mathbf {Q}$ and $\mathfrak {E} \in \mathbf {E} _n$ with $\mathfrak {E} \cong \mathfrak {A}$. For each $\mathbf {Q}$, choose $e(\mathbf {Q})=\mathfrak {A}$. Then there is some $n\in N$ such that for every $\mathfrak {A} \in \mathbf {A} '$, there is an $\mathfrak {E} \in \mathbf {E} _n$ with $\mathfrak {E} \cong \mathfrak {A}$, so that $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for $\mathbf {A} '$.

‘$\Leftarrow$’: Similar. $\square$

Claim 4

Let $\mathbf {A}$ be appearances, and $\mathbf {A} '=\{\{\mathfrak {A} \}\mathop {:}\mathfrak {A} \in \mathbf {A} \}$ be epistemic appearances. Then $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for $\mathbf {A}$ if and only if $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for $\mathbf {A} '$.

Proof

Given the epistemic appearances $\mathbf {A} '=\{\{\mathfrak {A} \}\mathop {:}\mathfrak {A} \in \mathbf {A} \}$, the only epistemically possible appearances are given by $\mathbf {A}$. Claim 4 now follows from Claim 3. $\square$

Claim 5

Let $\langle \{\mathbf {A} _l\}_{l\in L},C\rangle$ be a hierarchy of epistemic appearances. If theory $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for $\mathbf {A} _l, l\in L$, then $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for any $\mathbf {A} _k, k\in L, k\le l$. If theory $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is not epistemically empirically adequate for $\mathbf {A} _l, l\in L$, then $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is not epistemically empirically adequate for any $\mathbf {A} _m, m\in L, m\ge l$.

Proof

By Definition 14, for any $k\le l$ and any $\mathbf {Q} \in \mathbf {A} _k$, there is a $b$ such that $b(\mathbf {Q})\in \mathbf {A} _l$ and $b(\mathbf {Q})\subseteq \mathbf {Q}$. Thus, if $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for $\mathbf {A} _l$, there is some $n\in N$ such that for every $\mathbf {Q} \in \mathbf {A} _k$, there are an $\mathfrak {A} \in b(\mathbf {Q})\subseteq \mathbf {Q}$ and an $\mathfrak {E} \in \mathbf {E} _n$ with $\mathfrak {E} \cong \mathfrak {A}$. The proof of the claim’s second conjunct is similar. $\square$

Claim 6

$\langle \{\mathbf {A} _l\}_{l\in L},C\rangle$ is a (restricted) hierarchy of approximate appearances if and only if the following holds: For any $l\le m$ with $l,m\in L$, there is an injection (bijection) $b:\mathbf {A} _l\longrightarrow \mathbf {A} _m$ in $C$ such that for all $\mathbf {Q} \in \mathbf {A} _l$ with $\{R_i^+,R_i^-,F_j^{+\circ },c_k^{+\circ }\}_{i\in I,j\in J,k\in K}$ and for $b(\mathbf {Q})\in \mathbf {A} _m$ with $\{\tilde{R}_i^+,\tilde{R}_i^-,\tilde{F}_j^{+\circ },\tilde{c}_k^{+\circ }\}_{i\in I,j\in J,k\in K}$ it holds that $R_i^+\subseteq \tilde{R}_i^+,\,R_i^-\subseteq \tilde{R}_i^-,\,\tilde{F}_j^{+\circ }\subseteq F_j^{+\circ }$, and $\tilde{c}_j^{+\circ }\subseteq c_j^{+\circ }$ for all $i\in I, j\in J, k\in K$.

Proof

‘$\Rightarrow$’: By Definition 14, there is an injection (bijection) $b:\mathbf {A} _l\longrightarrow \mathbf {A} _m$ in $C$ such that for all $\mathbf {Q},\,b(\mathbf {Q})\subseteq \mathbf {Q}$. The claim follows from (14)–(17).

‘$\Leftarrow$’: Immediate. $\square$

Claim 7

$\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically/approximately empirically adequate at all points of all restricted hierarchies of epistemic/approximate appearances with the initial sequence $\langle \mathbf {A} _l\rangle _{l\le m}$ if and only if $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for all appearances that are epistemically possible given $\mathbf {A} _m$.

Proof

‘$\Rightarrow$’: Assume $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is restrictedly epistemically/approximately adequate at all points of all hierarchies of epistemic appearances with the initial sequence $\langle \mathbf {A} _l\rangle _{l\le m}$. For all appearances $\mathbf {A}$ that are epistemically possible given $\mathbf {A} _l$, the sequence $\langle \mathbf {A} _m,\mathbf {A} '\rangle _{l\le m}$ with $\mathbf {A} '=\{\{\mathfrak {A} \}\mathop {:}\mathfrak {A} \in \mathbf {A} \}$ as its last element is a hierarchy of epistemic/approximate appearances. Therefore $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically empirically adequate for $\mathbf {A} '$, and thus, by Claim 4, $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for $\mathbf {A}$.

‘$\Leftarrow$’: For any point $\mathbf {A} _r$ of any hierarchy with initial sequence $\langle \mathbf {A} _l\rangle _{l\le m}$, there is a bijection $b:\mathbf {A} _m\longrightarrow \mathbf {A} _r$ in $C$ with $b(\mathbf {Q})\subseteq \mathbf {Q}$. By assumption, there is therefore a function $e$ from $\mathbf {A} _r$ to $\bigcup \mathbf {A} _r$ with $e(\mathbf {Q})\in \mathbf {Q} \subseteq b^{-1}(\mathbf {Q})$ such that $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for $\{e(\mathbf {Q})\mathop {:}\mathbf {Q} \in \mathbf {A} _r\}$. Since $\{e(\mathbf {Q})\mathop {:}\mathbf {Q} \in \mathbf {A} _r\}$ is epistemically possible given $\mathbf {A} _r$, by Claim 3, $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is epistemically/approximately empirically adequate for $\mathbf {A} _r$. $\square$

Claim 8

$\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is at least as (restrictedly) epistemically empirically adequate as $\langle \mathfrak {T} _s,\mathbf {E} _s\rangle$ for any (restricted) hierarchy of epistemic appearances if and only if $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for all appearances $\mathbf {A}$ for which $\langle \mathfrak {T} _s,\mathbf {E} _s\rangle$ is empirically adequate.

Proof

‘$\Rightarrow$’: Choose the trivial (restricted) hierarchy of epistemic appearances $\langle \mathbf {A} \rangle$ with $\mathbf {A}$ containing all appearances for which $\langle \mathfrak {T} _s,\mathbf {E} _s\rangle$ is empirically adequate. Then all these appearances are epistemically possible appearances given $\mathbf {A}$, and by Claim 3, the claim follows.

‘$\Leftarrow$’: Immediate from the definitions and Claim 3. $\square$

Claim 9

$\langle \mathfrak {T} _s,\mathbf {E} _s\rangle$ is empirically adequate for all appearances $\mathbf {A}$ for which $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate if and only if for every $n\in N$, there is an $s\in S$ such that all empirical substructures of $\mathfrak {T} _n$ are isomorphic to empirical substructures of $\mathfrak {T} _s$.

Proof

‘$\Rightarrow$’: For each $\mathbf {E} _n$, choose $\mathbf {A} =\mathbf {E} _n$. Then $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate for $\mathbf {A}$, and thus $\langle \mathfrak {T} _s,\mathbf {E} _s\rangle$ is empirically adequate for $\mathbf {A}$. Therefore for every $\mathfrak {E} \in \mathbf {E} _n=\mathbf {A}$ there is an $s\in S$ and an $\mathfrak {E} '\in \mathbf {E} _s$ such that $\mathfrak {E} \cong \mathfrak {E} '$.

‘$\Leftarrow$’: Assume that for some $\mathbf {A}$, $\langle \mathfrak {T} _s,\mathbf {E} _s\rangle$ but not $\bigl \langle \mathfrak {T} _n,\mathbf {E} _n\bigr \rangle$ is empirically adequate. Then there is an $s\in S$ such that for all $\mathfrak {A} \in \mathbf {A}$, there is an $\mathfrak {E} '\in \mathbf {E} _s$ with $\mathfrak {A} \cong \mathfrak {E} '$. Since there is no such $n\in N$ and the isomorphism relation is transitive, there is no $n$ such that for all $\mathfrak {E} '\in \mathbf {E} _s$, there is an $\mathfrak {E} \in \mathbf {E} _n$ with $\mathfrak {E} '\cong \mathfrak {E}$. $\square$