Fallibilism, Verisimilitude, and the Preface Paradox

Abstract

The Preface Paradox apparently shows that it is sometimes rational to believe logically incompatible propositions. In this paper, I propose a way out of the paradox based on the ideas of fallibilism and verisimilitude (or truthlikeness). More precisely, I defend the view that a rational inquirer can fallibly believe or accept a proposition which is false, or likely false, but verisimilar; and I argue that this view makes the Preface Paradox disappear. Some possible objections to my proposal, and an alternative view of fallible belief, are briefly discussed in the final part of the paper.

Appendix

In this section, I provide formal definitions of all the notions used throughout the paper, and a proof of Theorem 1 from Sect. 4.

To keep things simple, suppose that the domain under inquiry is described by a propositional language with n atomic sentences $a_1, a_2, \ldots , a_n$.⁹ These atomic statements $a_i$ and their negations $\lnot a_i$ are called the “basic” statements of the language, and describe what we may call the “basic features” of the underlying domain. If an inquirer is only interested in the basic features of the world, his beliefs can be represented by a “basic theory”—i.e., the strongest (non-contradictory) conjunction of basic statements that he is willing to accept. The strongest possible conjunctions of this kind are the so-called state descriptions—or (propositional) constituents—of the language, which are the most informative descriptions of a possible state of affairs (a “possible world”) within the conceptual resources of the language. Each constituent $c_i$ can be written as a conjunction of n atomic sentences, negated (−) or not ($+$), as follows:

$$\begin{aligned} \pm a_1 \wedge \pm a_2 \wedge \dots \wedge \pm a_n. \end{aligned}$$

One can check that there are $2^n$ constituents, and that only one of them, call it $c_\star$, is true. Thus, $c_\star$ represents “the whole truth” about the domain, since it is the complete description of the actual world as expressed in the language. Accordingly, the verisimilitude of any statement, hypothesis or theory h can be expressed by an adequate measure of the closeness or similarity of h to $c_\star$. If h is a basic theory in the sense defined above—i.e., a conjunction of k basic statements, with $k\le n$—then there is a simple way to define such a measure.

Let $c_i$ be an arbitrary constituent and let $T(h,c_i)$ be the set of “matches” between h and $c_i$, i.e., of the conjuncts that h and $c_i$ have in common or, so to speak, of the conjuncts of h which are “true in” $c_i$. Similarly, $F(h,c_i)$ will denote the set of “mismatches” between h and $c_i$, i.e., of the conjuncts of h whose negation is true in $c_i$. Then, the similarity or closeness of h to $c_i$ can be defined as the weighted difference of the normalized number of matches and mismatches between h and $c_i$:

$$\begin{aligned} s_\phi (h,c_i) = \frac{|T(h,c_i)|}{n} -\phi \frac{|F(h,c_i)|}{n} \end{aligned}$$

(1)

where $\phi >0$. Intuitively, different values of $\phi$ reflect the relative weight assigned to truths and falsehoods: the greater $\phi$, the less similar h is to $c_i$ due to the mismatches in $F(h,c_i)$. The verisimilitude of h can then be defined as the similarity of h to the true constituent $c_\star$:

$$\begin{aligned} vs_\phi (h) = s_\phi (h,c_\star ) = \frac{|T(h,c_\star )|}{n} -\phi \frac{|F(h,c_\star )|}{n} \end{aligned}$$

(2)

Thus, $vs_\phi (h)$ is maximal (and equals 1) just in case h is the truth $c_\star$.

Since the truth is usually unknown, one cannot use Eq. 2 to calculate the actual degree of verisimilitude of h. However, if a probability distribution $p$ is defined on the set of constituents of the language—such that $p(c_i)$ expresses the rational degree of belief of the inquirer in the truth of $c_i$—, then the expected verisimilitude of h can be defined as the expected value of $vs_\phi (h)$:

$$\begin{aligned} Evs_\phi (h) = \sum _{c_i} s_\phi (h,c_i)\times p(c_i) \end{aligned}$$

(3)

In words, $Evs_\phi (h)$ expresses the inquirer’s best estimate of the actual verisimilitude of h given the available evidence.

The main formal result of this paper is Theorem 1 from Sect. 4: if the claims $b_1, b_2,\ldots ,b_m$ in the book are basic statements and b their conjunction, there is a threshold value $\sigma$ such that, if $p(b_i)>\sigma$ for all $b_i$, then $Evs_\phi (b)$ is maximal. In the following, x and y will denote arbitrary basic statements of the language, and h an arbitrary basic theory in the sense defined above.

Let us first note that the similarity measure defined in Eq. 1 is additive in the sense that:

$$\begin{aligned} s_\phi (h,c_i)=\sum _{x:h\vDash x}s_\phi (x,c_i); \end{aligned}$$

(4)

i.e., the similarity of h to $c_i$ is just the sum of the similarities of the conjuncts x of h to $c_i$ (in fact, note that $s_\phi (x,c_i)=\frac{1}{n}$ if x is true in $c_i$, and $s_\phi (x,c_i)=-\frac{\phi }{n}$ otherwise). It follows from this that the expected verisimilitude measure defined in Eq. 3 is also additive in the same sense:

Lemma 1

$Evs_\phi (h) = \sum _{x:h\vDash x} Evs_\phi (x)$

Proof

$$ \begin{aligned} Evs_{\phi } (h) & = \sum\limits_{{c_{i} }} p (c_{i} )s_{\phi } (h,c_{i} ) \\ & = \sum\limits_{{c_{i} }} p (c_{i} )\sum\limits_{{x:h{ \vDash }x}} {s_{\phi } } (x,c_{i} )\quad {\text{by}}\;{\text{eq}}{\text{.}}\;(4) \\ & = \sum\limits_{{c_{i} }} {\sum\limits_{{x:h{ \vDash }x}} p } (c_{i} )s_{\phi } (x,c_{i} ) \\ & = \sum\limits_{{x:h{ \vDash }x}} {\sum\limits_{{c_{i} }} p } (c_{i} )s_{\phi } (x,c_{i} ) \\ & = \sum\limits_{{x:h{ \vDash }x}} E vs_{\phi } (x)\quad {\text{by}}\;{\text{eq}}{\text{.}}\;(3) \\ \end{aligned} $$

The following theorem specifies under what conditions the conjunction of h with an arbitrary basic statement y (not already a conjunct of h) has greater expected verisimilitude than h itself.

Lemma 2

$Evs_\phi (h\wedge y) > Evs_\phi (h)$ iff $p(y)>\frac{\phi }{\phi +1}$.

Proof

$$ \begin{array}{*{20}l} {} \hfill & {Evs_{\phi } (h \wedge y){ \gtreqless }Evs_{\phi } (h)} \hfill & {} \hfill \\ {{\text{iff}}} \hfill & {Evs_{\phi } (h) + Evs_{\phi } (y){ \gtreqless }Evs_{\phi } (h)} \hfill & {{\text{by}}\;{\text{lemma}}\;1} \hfill \\ {{\text{iff}}} \hfill & {Evs_{\phi } (y){ \gtreqless }0} \hfill & {} \hfill \\ {{\text{iff}}} \hfill & {\sum\limits_{{c_{i} }} p (c_{i} )s_{\phi } (y,c_{i} ){ \gtreqless }0} \hfill & {{\text{by}}\;{\text{eq}}{\text{. }}\;(3)} \hfill \\ {{\text{iff}}} \hfill & {\left( {\sum\limits_{{c_{i} { \vDash }y}} p (c_{i} ) \times \frac{1}{n}} \right) + \left( {\sum\limits_{{c_{i} { \vDash }\neg y}} p (c_{i} ) \times - \frac{\phi }{n}} \right){ \gtreqless }0} \hfill & {} \hfill \\ {{\text{iff}}} \hfill & {\frac{1}{n}p(y) - \frac{\phi }{n}p(\neg y){ \gtreqless }0} \hfill & {} \hfill \\ {{\text{iff}}} \hfill & {p(y) - \phi (1 - p(y)){ \gtreqless }0} \hfill & {} \hfill \\ {{\text{iff}}} \hfill & {p(y){ \gtreqless }\frac{\phi }{{\phi + 1}}} \hfill & {} \hfill \\ \end{array} $$

Finaly, from Lemma 2, the Proof of Theorem 1 from Sect. 4 easily follows.

Proof

Let be $\sigma =\max (\frac{\phi }{\phi +1},0.5)$. For the sake of conciseness, let us say that a basic statement x is “likely” iff $p(x)>\sigma$ and “unlikely” otherwise (i.e., iff $p(x)\le \sigma$). Suppose that b is the (consistent) conjunction of all and only the likely basic statements $b_1,b_2,\ldots ,b_m$ of the language. We have to show that, provided there is one such conjunction b, $Evs_\phi (b)$ is maximal, i.e., that it is greater than $Evs_\phi (h)$ for any h different from b. Suppose first that some of the conjuncts of h are unlikely; it follows from (an iterated application of) Lemma 2 that any basic theory obtained from h by removing an unlikely conjunct has greater expected verisimilitude than h itself. So $Evs_\phi (h)$ cannot be maximal. Suppose now that h contains only likely conjuncts. If these are all the likely basic statements of the language then h is the same as b. Otherwise, if $b_i$ is a likely statement not already in h, then it follows from Lemma 2 that $h \wedge b_i$ has greater expected verisimilitude than h. In sum, $Evs_\phi (h)$ is maximal just in case h is identical with b.

A technical comment on the definition of the threshold $\sigma$ above is in order, to explain why $\sigma$ is not simply defined as $\frac{\phi }{\phi +1}$, but it is required to be the greater of 0.5 and $\frac{\phi }{\phi +1}$. As pointed out by an anonymous reviewer, if $\phi$ (the “weight of falsehood” in Eq. 2) is chosen as smaller than 1, then $\frac{\phi }{\phi +1}$ becomes smaller than 0.5. As a consequence, for some basic proposition x, both $p(x)$ and $p(\lnot x)$ may be above the threshold $\frac{\phi }{\phi +1}$. In such case, Theorem 1 implies that the conjunction b maximizing expected verisimilitude is inconsistent, since it contains both x and $\lnot x$. Assuming that a rational agent should not believe logically false propositions, one needs to exclude the case above; this can be done in at least two ways. The first is to require that $\phi \ge 1$, and hence that $\frac{\phi }{\phi +1}>0.5$; this solves the problem by restricting the application of measure $vs_\phi$ to those contexts (arguably the most common ones) in which the “loss” due to accepting a basic falsehood is greater (in absolute value) than the “gain” obtained from accepting a basic truth. The second solution—suggested by the reviewer and adopted here—is to require that $p(x)$ is greater than both $\frac{\phi }{\phi +1}$ and 0.5 or, which is the same, that x is more probable than its negation and moreover it passes the relevant threshold. This guarantees that there is at most one conjunction of all and only the basic propositions which are likely in the defined sense, and that such conjunction, if it exists, maximizes expected verisimilitude.