Verisimilitude and Belief Change for Conjunctive Theories

Abstract

Theory change is a central concern in contemporary epistemology and philosophy of science. In this paper, we investigate the relationships between two ongoing research programs providing formal treatments of theory change: the (post-Popperian) approach to verisimilitude and the AGM theory of belief change. We show that appropriately construed accounts emerging from those two lines of epistemological research do yield convergences relative to a specified kind of theories, here labeled “conjunctive”. In this domain, a set of plausible conditions are identified which demonstrably capture the verisimilitudinarian effectiveness of AGM belief change, i.e., its effectiveness in tracking truth approximation. We conclude by indicating some further developments and open issues arising from our results.

Appendix: Proofs

Proof of Theorem 1

The following remark will be useful in proof:

Remark 1

If X is set of literals and x is a literal of ${\fancyscript{L}_{n},}$ then $x\in \hbox{Cn}(X)$ iff $x\in X.$

• (i) Recall first that we are assuming that A is logically compatible with T. By the definition in (9), (T + A)^b = T ^b + A ^b = T ^b∪ A ^b. According to Definition 4, T ^b may be written as T ^b _oA ∪ T ^b _cA ∪ T ^b _xA and, likewise, A ^b = A ^b _oT ∪ A ^b _cT ∪ A ^b _xT . Since by hypothesis A is compatible with T, $T_{cA}^{b}=A_{cT}^{b}=\varnothing;$ moreover T ^b _oA  = A ^b _oT by Definition 4. Hence, T ^b = T ^b _oA ∪ T ^b _xA and A ^b = T ^b _oA ∪ A ^b _xT . It follows that T ^b∪ A ^b = T ^b∪ A ^b _xT , i.e., T + A = T∧ A _xT .

• (ii) According to the definition in (10), $T^{b} \ast A^{b}=((T^{b}-\neg(A^{b}))+A^{b}),$ where $\neg(A^{b})$ is the negation of A ^b. Note that, since A is a c-input, any $a\in A^{b}$ is a literal. We start by proving that T ^b _oA ∪ T ^b _xA is the unique remainder of T ^b by $\neg(A^{b})$ (see Definition 3), and hence the unique result of the contraction of T ^b by $\neg(A^{b}).$ By Definition 4, $T_{oA}^{b}\cup T_{xA}^{b}\subseteq T^{b}.$ Moreover, T ^b _oA ∪ T ^b _xA does not imply $\neg(A^{b}).$ In fact, by Definition 4, each element of T ^b _oA ∪ T ^b _xA is either the negation of a disjunct of $\neg(A^{b})$ or a logically independent literal. Finally, any set Y such that $T_{oA}^{b}\cup T_{xA}^{b}\subset Y\subseteq T^{b}$ will contain some element of T _oA , again by Definition 4. Any such element is a negation of an element of A ^b and then implies $\neg(A^{b}).$ Thus, T ^b _oA ∪ T ^b _xA is a remainder of T ^b by A ^b. To see that T ^b _oA ∪ T ^b _xA is the unique remainder, note that any other subset X of T ^b is such that either $X\subseteq T_{oA}^{b}\cup T_{xA}^{b}$ or X overlaps T _cA and then implies $\neg(A^{b}).$ In other words, either X is not maximal or implies $\neg(A^{b}).$ It follows that $(T^{b}-\neg(A^{b}))=T_{oA}^{b}\cup T_{xA}^{b}.$ According to definition 9, $(T^{b}-\neg(A^{b}))+A^{b} = T_{oA}^{b}\cup T_{xA}^{b}\cup A^{b}.$ Since, by Definition 4, $T_{oA}^{b}=A_{oT}^{b}\subseteq A_{oT}^{b},$ it follows that T ^b _oA ∪ T ^b _xA ∪ A ^b = T ^b _xA ∪ A ^b, i.e., T * A = T _xT ∧ A.

• iii) To prove the theorem it will suffice to prove that T ^b _cA ∪ T ^b _xA is the unique remainder of T ^b by A ^b (cf. Definition 3), and hence the unique result of the contraction of T ^b by A ^b. First, let us prove that $T_{cA}^{b}\cup T_{xA}^{b}\in T^{b}\bot A^{b}.$ By Definition 4, $T_{cA}^{b}\cup T_{xA}^{b}\subseteq T^{b}.$ Moreover, by Remark 1 and Definition 4, T ^b _cA ∪ T ^b _xA implies no element of A ^b. Finally, any set Y such that $T_{cA}^{b}\cup T_{xA}^{b}\subset Y\subseteq T^{b}$ will contain some element of T _cA  = A _oT and then will imply some element of A ^b. Thus, T ^b _cA ∪ T ^b _xA is a remainder of T ^b by A ^b. To see that T ^b _cA ∪ T ^b _xA is the unique remainder, note that any other subset X of T ^b is such that either $X\subseteq T_{cA}^{b}\cup T_{xA}^{b}$ or X overlaps T _oA  = A _oT and then implies some element of A ^b. In other words, X is not maximal or implies some element of A ^b. Thus, T ^b − A ^b = T ^b _cA ∪ T ^b _xA and then T − A = T _cA ∧ T _xA .

Proof of Theorem 2

First, note that if A is true then A _oT , A _cT and A _xT are also true, and that if A is completely false then A _oT , A _cT and A _xT are also completely false. Moreover, recall that we leave cases of vacuous change aside.

• (i) By hypothesis, A is compatible with T, i.e., $A_{cT}^{b}=T_{cA}^{b}=\varnothing.$ By Theorem, T + A = T∧ A _xT  = T _oA ∧ T _xA ∧ A _xT . Thus, $T^{b}\subset(T+A)^{b}=T^{b}\cup A_{xT}^{b};$ moreover, since A _xT is true by hypothesis, $t(T, C_{\star})\subset t(T+A, C_{\star}),$ whereas $f(T, C_{\star})=f(T+A, C_{\star}).$ It follows from condition (M _t ) of Definition 1 that Vs(T + A) > Vs(T).

• (ii) By Theorem 1, T − A = T _cA ∧ T _xA . Thus, $(T-A)^{b}\subset T^{b};$ moreover, since A _oT is true by hypothesis, $t(T-A, C_{\star})\subset t(T, C_{\star}),$ whereas $f(T, C_{\star})=f(T-A, C_{\star}).$ It follows from condition (M _t ) of Definition 1 that Vs(T) > Vs(T − A).

• (iii) By Theorem, T * A = A ∧ T _xA . Note that T can be written as $A_{oT}\wedge\widetilde{A_{cT}}\wedge T_{xA}$ and T * A as A _oT ∧ A _cT ∧ A _xT ∧ T _xA . In the limiting case $A_{cT}^{b}=\varnothing$ then T * A = A _oT ∧ A _xT ∧ T _xA  = T∧ A _xT  = T + A; thus Vs(T* A) > Vs(T) by the first clause of the present theorem (proved above). Otherwise, note first that since A _cT is true by hypothesis, $\widetilde{A_{cT}}$ is completely false by Definition 4. It follows that $t(T, C_{\star})\subset t(T+A, C_{\star})=t(T, C_{\star})\cup (A_{oT}\wedge A_{xT})^{b},$ and that $f(T \ast A, C_{\star})\subset f(T, C_{\star})=f(T \ast A, C_{\star})\cup (\widetilde{T_{oT}})^{b}.$ Then, from condition (M _f ) of Definition 1 that Vs(T* A) > Vs(T).

The proofs of the remaining three clauses of the theorem can be obtained from the ones above in a straightforward manner.

Proof of Theorem 4

Let us first state without proof the following useful lemma:

Lemma 1

Given a c-theory $T, Vs_{\phi}(T)=\sum_{x\in T^{b}} Vs_{\phi}(x).$ It follows that, given a c-input A, Vs _ϕ(T) = Vs _ϕ(T _oA ) + Vs _ϕ(T _cA ) + Vs _ϕ(T _xA ) and Vs _ϕ(A) = Vs _ϕ(A _oT ) + Vs _ϕ(A _cT ) + Vs _ϕ(A _xT ).

Then:

• 1. Vs _ϕ(T + A) > Vs _ϕ(T) iff (by Theorem 1) Vs _ϕ(T∧ A _xT ) > Vs _ϕ(T) iff (by Definition 4 and Lemma 1) Vs _ϕ(T) + Vs _ϕ(A _xT ) > Vs _ϕ(T) iff Vs _ϕ(A _xT ) > 0.

• 2. Vs _ϕ(T − A) > Vs _ϕ(T) iff (by Theorem 1) Vs _ϕ(T _cA ∧ T _xA ) > Vs _ϕ(T) iff (by Definition 4 and Lemma 1) Vs _ϕ(T _cA ) + Vs _ϕ(T _xA ) > Vs _ϕ(T _oA ) + Vs _ϕ(T _cA ) + Vs _ϕ(T _xA ) iff Vs _ϕ(A _oT ) = Vs _ϕ(T _oA ) < 0.

• 3. Vs _ϕ(T * A) > Vs _ϕ(T) iff (by Theorem 1) Vs _ϕ(T _xA ∧ A) > Vs _ϕ(T) iff (by Definition 4 and Lemma 1) Vs _ϕ(T _xA ) + Vs _ϕ(A _oT ) + Vs _ϕ(A _cT ) + Vs _ϕ(A _xT ) > Vs _ϕ(T _oA ) + Vs _ϕ(T _cA ) + Vs _ϕ(T _xA ) iff (recalling the definition of specular) $Vs_{\phi}(A_{xT})>Vs_{\phi}(\widetilde{A_{cT}})-Vs_{\phi}(A_{cT}).$

Proof of Theorem 5

The following results will be useful in proof:

Lemma 2

For any c-theory T:

• 1. T is verisimilar iff (by Definition 5) Vs _ϕ(T) > 0 iff (by Definition 6) $cont_{t}(T,C_{\star})>\phi cont_{f}(T,C_{\star}).$

• 2. $Vs_{\phi}(T)>Vs_{\phi}(\widetilde{T})$ iff (by Definition 6 and the definition of specular) $cont_{t}(T,C_{\star})-\phi cont_{f}(T,C_{\star})> cont_{f}(T,C_{\star})-\phi cont_{t}(T,C_{\star})$ iff $cont_{t}(T,C_{\star})> cont_{f}(T,C_{\star}).$

• 3. By the two previous results, it follows that: For ϕ = 1, T is verisimilar iff $Vs_{\phi}(T)>Vs_{\phi}(\widetilde{T})$ iff $\widetilde{T}$ is t-distant. For ϕ < 1, if T is t-distant then $Vs_{\phi}(T)<Vs_{\phi}(\widetilde{T})$ and $\widetilde{T}$ is verisimilar. For ϕ > 1, if T is verisimilar then $Vs_{\phi}(T)>Vs_{\phi}(\widetilde{T})$ and $\widetilde{T}$ is t-distant.

Accordingly:

• 1. If A _cT and A _xT are verisimilar, then Vs _ϕ(A _cT ) > 0 and Vs _ϕ(A _xT ) > 0 by Definition 5. Thus, if ϕ ≥ 1, then (by Lemma 2) $Vs_{\phi}(A_{cT})>Vs_{\phi}(\widetilde{A_{cT}})$ and $Vs_{\phi}(\widetilde{A_{cT}})-Vs_{\phi}(A_{cT})<0<Vs_{\phi}(A_{xT});$ it follows that Vs _ϕ(T* A) > Vs _ϕ(T) by Theorem 4.

• 2. If A _cT and A _xT are t-distant, then Vs _ϕ(A _cT ) < 0 and Vs _ϕ(A _xT ) < 0 by Definition 5. Thus, if ϕ ≤ 1, then (by Lemma 2) $Vs_{\phi}(A_{cT})<Vs_{\phi}(\widetilde{A_{cT}})$ and $Vs_{\phi}(\widetilde{A_{cT}})-Vs_{\phi}(A_{cT})>0>Vs_{\phi}(A_{xT});$ it follows that Vs _ϕ(T* A) < Vs _ϕ(T) by Theorem 4.