Qualitative Inductive Generalization and Confirmation

Abstract

Inductive generalization is a defeasible type of inference which we use to reason from the particular to the universal. First, a number of systems are presented that provide different ways of implementing this inference pattern within first-order logic. These systems are defined within the adaptive logics framework for modeling defeasible reasoning. Next, the logics are re-interpreted as criteria of confirmation. It is argued that they withstand the comparison with two qualitative theories of confirmation, Hempel’s satisfaction criterion and hypothetico-deductive confirmation.

Appendix: Blocking the Raven Paradox?

If a formalism defined in terms of CL behaves overly permissive, a good strategy to remedy this problem is to add further criteria of validity or relevance. For instance, in order to avoid problems of irrelevant conjunctions and disjunctions, hypothetico-deductivists may impose further demands on HD confirmation [11.32, 11.33, 11.34, 11.35].

A similar strategy could be adopted with respect to I-confirmation and the raven paradox. In this appendix, an alternative adaptive logic of induction, IC, is defined, as is a corresponding criterion of confirmation which is slightly less permissive than the criteria from Sect. 11.4. IC makes use of a non-classical conditional resembling a number of conditionals originally defined in order to avoid the so-called paradoxes of material implication. First, an extension of CL is introduced, including this new conditional connective. Next, the adaptive logic IC is defined.

The new conditional, $\rightarrow$, is fully characterized by the following rules and axiom schema’s

$$\begin{aligned}\displaystyle\frac{A,(A\rightarrow B)}{B}\;,&\displaystyle&\displaystyle(\mathrm{MP})\\ \displaystyle\frac{A\equiv B}{(A\rightarrow C)\equiv(B\rightarrow C)}\;,&\displaystyle&\displaystyle(\mathrm{RCEA})\\ \displaystyle\frac{A\equiv B}{(C\rightarrow A)\equiv(C\rightarrow B)}\;,&\displaystyle&\displaystyle(\mathrm{RCEC})\\ \displaystyle(A\rightarrow(B\wedge C))\equiv((A\rightarrow B)\wedge(A\rightarrow C))\;,&\displaystyle&\displaystyle(\mathrm{D}\wedge)\\ \displaystyle((A\vee B)\rightarrow C)\equiv((A\rightarrow C)\wedge(B\rightarrow C))\;,&\displaystyle&\displaystyle(\mathrm{D}\vee)\end{aligned}$$

((RCEA), (RCEC), and (D$\wedge$) fully characterize the conditional of Chellas’s logic CR from [11.36]. The latter was also used for capturing explanatory conditionals in [11.37]. See also [11.38, Chap. 5] for some closely related conditional logics, including an extension of Chellas’s systems that validates (MP).)

Let $\mathbf{CL^{\rightarrow}}$ be the logic resulting from adding $\rightarrow$ to the language of CL, and from adding (MP)-(D$\vee$) to the list of rules and axioms of CL. Note that the conditional $\rightarrow$ is strictly stronger than $\supset$

$$(A\rightarrow B)\supset(A\supset B)\;.$$

(11.B57)

(By (MP), $A,(A\rightarrow B)\vdash_{\mathbf{CL^{\rightarrow}}}B$. By the deduction theorem for $\supset$, $A\rightarrow B\vdash_{\mathbf{CL^{\rightarrow}}}A\supset B$. By the deduction theorem again, $\vdash_{\mathbf{CL^{\rightarrow}}}(A\rightarrow B)\supset(A\supset B)$.)

In view of this bridging principle between both conditionals it is easily seen that counter-instances to a formula of the form $\forall{x}(A(x)\supset B(x))$ form counter-instances to $\forall{x}(A(x)\rightarrow B(x))$, and falsify the latter formula as well. For instance, if $Pa\wedge\neg Qa$, then, by CL, $\neg\forall{x}(Px\supset Qx)$, and, by (11.B57), $\neg\forall{x}(Px\rightarrow Qx)$.

The adaptive logic IC is fully characterized by the lower limit logic $\mathbf{CL^{\rightarrow}}$, the set of abnormalities

$$\begin{aligned}\displaystyle\Omega_{\mathbf{IC}}&\displaystyle=_{\text{df}}\{\exists(A_{1}\wedge\ldots\wedge A_{n}\wedge A_{0})\\ \displaystyle&\displaystyle\quad\wedge\neg\forall((A_{1}\wedge\ldots\wedge A_{n})\rightarrow A_{0})\mid\\ \displaystyle&\displaystyle\quad A_{0},A_{1},\ldots,A_{n}\in\mathcal{A}^{f1};n\geq 0\},\end{aligned}$$

(11.B58)

and the adaptive strategy reliability (IC^r) or minimal abnormality (IC^m). IC is defined within the SF. All rules and definitions for its proof theory are as for the other logics defined in this chapter, except that in the definition of RU and RC, CL is replaced with $\mathbf{CL^{\rightarrow}}$.

The following proof illustrates how formulas are derived conditionally in IC

$$\begin{array}[]{lll}1&\neg Ra&\text{Prem}\\ &\emptyset&\\ 2&\neg Ba&\text{Prem}\\ &\emptyset&\\ 3&\forall{x}(\neg Bx\rightarrow\neg Rx)&1,2;\text{RC}\\ &\{\exists{x}(\neg Bx\wedge\neg Rx)\wedge\neg\forall{x}(\neg Bx\rightarrow\neg Rx)\}&\\ \end{array}$$

Given only the premises $\neg Ra$ and $\neg Ba$, there is no possible extension of this proof in which line 3 gets marked. Hence

$$\neg Ra,\neg Ba\vdash_{\mathbf{IC}}\forall{x}(\neg Bx\rightarrow\neg Rx)\;.$$

(11.B59)

However, contraposition is invalid for the new conditional $\rightarrow$, hence we cannot derive the raven hypothesis from the formula derived at line 3. Note also that, in view of (11.B60), we cannot use the conditional rule RC to derive $\forall{x}(Rx\rightarrow Bx)$ on the condition $\{\exists{x}(Rx\wedge Bx)\wedge\neg\forall{x}(Rx\rightarrow Bx)\}$ in an IC-proof, since

$$\begin{aligned}\displaystyle&\displaystyle\neg Ra,\neg Ba\not\vdash_{\mathbf{CL^{\rightarrow}}}\forall{x}(Rx\rightarrow Bx)\\ \displaystyle&\displaystyle\quad\vee(\exists{x}(Rx\wedge Bx)\wedge\neg\forall{x}(Rx\rightarrow Bx))\;.\end{aligned}$$

(11.B60)

Therefore

$$\neg Ra,\neg Ba\not\vdash_{\mathbf{IC}}\forall{x}(Rx\rightarrow Bx)\;.$$

(11.B61)

Thus, if conditional statements of the form for all x, if A(x) then B(x) are taken to be IC-confirmed only if the conditional in question is an arrow ($\rightarrow$) instead of a material implication, then the raven paradox, in its original formulation, is blocked.

An additional property of IC is that strengthening the antecedent fails for $\rightarrow$. In Sect. 11.3, for instance, we saw that

$$Pa\vdash_{\mathbf{G}}\forall{x}(Qx\supset Px)\;.$$

(11.B62)

In IC, (11.B62) still holds for the material implication, but not for the new conditional. In an IC-proof from Pa we can still derive $\forall{x}Px$ on the condition $\{\exists{x}Px\wedge\exists{x}\neg Px\}$, and since IC extends CL it still follows that $\forall{x}(Px\supset Qx)$

$$Pa \vdash_{\mathbf{IC}}\forall{x}Px\;,$$

(11.B63)

$$Pa \vdash_{\mathbf{IC}}\forall{x}(Qx\supset Px)\;.$$

(11.B64)

However, since $\forall{x}Px\not\vdash_{\mathbf{CL^{\rightarrow}}}\forall{x}(Qx\rightarrow Px)$, and since we do not have any further means to conditionally derive the formula $\forall{x}(Qx\rightarrow Px)$ in an IC-proof

$$Pa\not\vdash_{\mathbf{IC}}\forall{x}(Qx\rightarrow Px)\;.$$

(11.B65)

Originally, the logics in the G-family were constructed as logics requiring a positive instance before we are allowed to apply RC. This is reflected in the definition of the set of G-abnormalities. In order to derive a formula like $\forall{x}(Px\supset Qx)$ on its corresponding condition, a positive instance, e. g., $Pa\wedge Qa$, is needed. Examples like (11.36) and (11.B62) show, however, that such a positive instance is not always required in order to G-derive a generalization. The logic IC, it seems, does much better in this respect. However, it still does not fully live up to the requirement for a positive instance before generalizing, as the following IC-proof from $\Gamma_{9}=\{\neg Ra\wedge\neg Ba,Rb,Bc\}$ illustrates (where $A_{0},A_{1},\ldots,A_{n}\in\mathcal{A}^{f1}$, ${\dagger}((A_{1}\wedge\ldots\wedge A_{n})\rightarrow A_{0})$ abbreviates $\exists(A_{1}\wedge\ldots\wedge A_{n}\wedge A_{0})\wedge\neg\forall((A_{1}\wedge\ldots\wedge A_{n})\rightarrow A_{0})$).

$$\begin{array}[]{ll@{\kern 30mm}l}1&\neg Ra\wedge\neg Ba\kern 85.358268pt&\text{Prem}\\ &\hskip 17.071654pt\emptyset\kern 85.358268pt&\\ 2&Rb\kern 85.358268pt&\text{Prem}\\ &\hskip 17.071654pt\emptyset\kern 85.358268pt&\\ 3&Bc\kern 85.358268pt&\text{Prem}\\ &\hskip 17.071654pt\emptyset\kern 85.358268pt&\\ \end{array}$$

$$\begin{array}[]{lll}4&\forall{x}(\neg Bx\rightarrow\neg Rx)&1;\text{RC}\\ &\hskip 17.071654pt\{{\dagger}(\neg Bx\rightarrow\neg Rx)\}&\\ 5&Bb&2,4;\text{RU}\\ &\hskip 17.071654pt\{{\dagger}(\neg Bx\rightarrow\neg Rx)\}&\\ 6&\forall{x}(Rx\rightarrow Bx)&2,5;\text{RC}\\ &\hskip 17.071654pt\{{\dagger}(Rx\rightarrow Bx),{\dagger}(\neg Bx\rightarrow\neg Rx)\}&\\ \end{array}$$

The key step in this proof is the derivation of Bb at line 5, which together with Rb provides us with a positive instance of the raven hypothesis. Bb is derivable from lines 2 and 4 in view of CL and (11.B57). Except for the formulas $\exists{x}Rx\wedge\exists{x}\neg Rx$ and $\exists{x}Bx\wedge\exists{x}\neg Bx$, no minimal Dab-formulas are $\mathbf{CL^{\rightarrow}}$-derivable from Γ₉. Therefore

$$\Gamma_{9}\vdash_{\mathbf{IC}}\forall{x}(Rx\rightarrow Bx)\;.$$

(11.B66)

As (11.B61) illustrates the logic IC avoids the raven paradox in its original formulation. A possible drawback of IC is that it does not fully meet the demand for a positive instance when confirming a hypothesis (Sect. 11.4.3). It is left open whether it is possible and desirable to further extend IC so as to fully meet this demand.