A partial consequence account of truthlikeness

Abstract

Popper’s original definition of truthlikeness relied on a central insight: that truthlikeness combines truth and information, in the sense that a proposition is closer to the truth the more true consequences and the less false consequences it entails. As intuitively compelling as this definition may be, it is untenable, as proved long ago; still, one can arguably rely on Popper’s intuition to provide an adequate account of truthlikeness. To this aim, we mobilize some classical work on partial entailment in defining a new measure of truthlikeness which satisfies a number of desiderata. The resulting account has some interesting and surprising connections with other accounts on the market, thus shedding new light on current attempts of systematizing different approaches to verisimilitude.

A Proofs

In the following, we prove all the main claims in the paper.

• Measure$vs _{}$meets P5. Let be h and g true, conjunctive propositions, i.e., conjunctions of true literals. It is then clear that $inf _{F}(h)= inf _{F}(g)=0$, since both h and g partially entail no basic falsehoods. Moreover, $inf _{T}(h)=| B _{T}(h)|/n$ and $inf _{T}(g)=| B _{T}(g)|/n$, since both h and g “fully” entail each of their own conjuncts. Suppose now that h entails g. This means that $B _{T}(h)\supset { B _{T}(g)}$, i.e., that h adds to g some true conjunct, and hence $inf _{T}(h)>{ inf _{T}(g)}$. It follows that $vs _{}(h)= inf _{T}(h)> inf _{T}(g)= vs _{}(g)$, i.e., that P5 is satisfied.

• Measure$vs _{av}$meets P5. If h is a conjunctive proposition, all constituents in the range of h are “completions” of h, in the sense that they all agree with h on each of its conjuncts. Now suppose that h and g are true, conjunctive propositions such that h is logically stronger than g. This means that h entails all (true) conjuncts of g and also some other true conjunct. In turn, this means that, by the closeness measure $\lambda _{}$, each constituent in the range of h is closer to the truth than each constituent in that of g. It follows that the average truthlikeness is greater for h than for g, i.e., that P5 is satisfied.

• Theorem 1: Measures$vs _{}$and$vs _{av}$are ordinally equivalent. In the following, we shall use b to denote arbitrary an basic proposition (literal) of $\mathcal {L}_{n}$, w to denote an arbitrary constituent, and t for the true constituent. Moreover, in order to simplify notation, we shall write $r (h)$ for $| R (h)|$, i.e., for the number of constituents entailing h.

  We start by noting that, for any h and b, $m _{}(\lnot b|h) = 1- m _{}(b|h)$; from this and the definition of $inf _{}$ it follows that $inf _{}(h,\lnot b)=- inf _{}(h,b)$. Accordingly, $vs _{}$ can be rewritten as follows:

  $$\begin{aligned} \begin{array}{lcl} vs _{}(h) &{} = &{} inf _{T}(h) - inf _{F}(h)\\ &{} = &{} \frac{1}{n}\times \sum \limits _{b\in PB _{T}(h)} inf _{}(h,b) - \frac{1}{n}\times \sum \limits _{b\in PB _{F}(h)} inf _{}(h,b) \\ &{} = &{} \frac{1}{n}\times \sum \limits _{b\in PB _{}(h): t\vDash b} inf _{}(h,b) - \frac{1}{n}\times \sum \limits _{b\in PB _{}(h): t\vDash \lnot b} inf _{}(h,b) \\ &{} = &{} \frac{1}{n}\times \sum \limits _{b\in PB _{}(h): t\vDash b} inf _{}(h,b) - \frac{1}{n}\times \sum \limits _{b\not \in PB _{}(h): t\vDash b}- inf _{}(h, b) \\ &{} = &{} \frac{1}{n}\times \sum \limits _{b\in PB _{}(h): t\vDash b} inf _{}(h,b) + \frac{1}{n}\times \sum \limits _{b\not \in PB _{}(h): t\vDash b} inf _{}(h, b) \\ &{} = &{} \frac{1}{n}\times \sum \limits _{b: t\vDash b} inf _{}(h,b)\\ \end{array} \end{aligned}$$

  Moreover, by definition of $inf _{}$:

  $$\begin{aligned} \begin{array}{lcl} vs _{}(h) &{} = &{} \frac{1}{n}\times \sum \limits _{b: t\vDash b} inf _{}(h,b)\\ &{} = &{} \frac{1}{n}\times \sum \limits _{b: t\vDash b} \left( 2( m _{}(b|h)-\frac{1}{2})\right) \\ &{} = &{} \frac{2}{n}\times \sum \limits _{b: t\vDash b} \left( m _{}(b|h)-\frac{1}{2}\right) \\ &{} = &{} \frac{2}{n}\left( \sum \limits _{b: t\vDash b} m _{}(b|h)-\frac{n}{2}\right) \\ &{} = &{} \frac{2}{n}\sum \limits _{b: t\vDash b} m _{}(b|h) - 1\\ \end{array} \end{aligned}$$

  (1)

  As for the average measure, note that, by definition of $vs _{av}{h}$, of $\lambda _{}(w,t)$, and of $B _{T}(w)$:

  $$\begin{aligned} \begin{array}{lcl} vs _{av}(h) &{} = &{} \frac{1}{ r (h)} \sum \limits _{w:w\vDash h} \lambda _{}(w,t)\\ &{} = &{} \frac{1}{ r (h)} \sum \limits _{w:w\vDash h} \frac{| B _{T}(w)|}{n}\\ \end{array} \end{aligned}$$

  Moreover, since for any constituent w, $r (b\wedge w)$ is either 1 (if $w\vDash b$) or 0 (if $w\nvDash b$), $| B _{T}(w)|$ can be written as $\sum _{b:t\vDash b} r (b\wedge w)$. Accordingly,

  $$\begin{aligned} \begin{array}{lcl} vs _{av}(h) &{} = &{} \frac{1}{ r (h)} \sum \limits _{w:w\vDash h} \frac{1}{n}\sum _{b:t\vDash b} r (b\wedge w)\\ &{} = &{} \frac{1}{n} \sum _{w:w\vDash h} \sum \limits _{b:t\vDash b} \frac{ r (b\wedge w)}{ r (h)}\\ \end{array} \end{aligned}$$

  Now note that $m _{}(b|h)=\frac{ m _{}(b\wedge h)}{ m _{}(h)}=\frac{ r (b\wedge h)}{ r (h)}$ by definition. In turn, $r (b\wedge h)$ can be written as $\sum _{w:w\vDash h} r (b\wedge w)$. It follows that $m _{}(b|h)=\sum _{w:w\vDash h} \frac{ r (b\wedge w)}{ r (h)}$ and hence that:

  $$\begin{aligned} \begin{array}{lcl} vs _{av}(h) &{} = &{} \frac{1}{n} \sum \limits _{b:t\vDash b} m _{}(b|h)\\ \end{array} \end{aligned}$$

  (2)

  From the comparison of Eqs. 1 and 2 above, it follows that:

  $$\begin{aligned} vs _{}(h) = 2\times vs _{av}(h) - 1 \end{aligned}$$

  and hence that measures $vs _{}$ and $vs _{av}$ are ordinally equivalent, which completes the proof of Theorem 1.

• Measure$vs _{}$meets P7. We start by stating the following results, which will be useful in the following. First, if h is a disjunction of k literals, the amount of information provided by h on each of its disjunct b is:

  $$\begin{aligned} \begin{array}{lcl} inf _{}(h,b) &{} = &{} 2\times \left( m _{}(b|h) - \frac{1}{2}\right) = 2\times \left( \frac{ m _{}(b\wedge h)}{ m _{}(h)} - \frac{1}{2}\right) \\ &{} = &{} 2\times \left( \frac{2^{n-1}}{2^n-2^{n-k}} - \frac{1}{2}\right) = \frac{1}{2^k - 1} \end{array} \end{aligned}$$

  (3)

  In fact, note that if b is a disjunct of h, then $b\vDash h$ and $m _{}(b\wedge h)= m _{}(b)=2^{n-1}$. Moreover, h is false only in those worlds where all k literals of h are false; since there are $2^{n-k}$ such worlds, $m _{}(h)=2^n-2^{n-k}$.

  Second, if h is either completely true or false, its truthlikeness only depends on the total number k of its disjunct and on the amount of information provided by h on each of them (as just calculated above). In fact:

  $$\begin{aligned} \begin{array}{l} \text {if }h\text { is completely true, then } vs _{}(h) = inf _{T}(h) = \frac{1}{n}\times \frac{k}{2^k - 1}\\ \text {if }h\text { is false, then } vs _{}(h) = - inf _{F}(h) = -\frac{1}{n}\times \frac{k}{2^k - 1}\\ \end{array} \end{aligned}$$

  (4)

  This is because if h is completely true (resp. false), then it entails no basic falsehoods (resp. truths), and hence $inf _{F}(h)$ (resp. $inf _{T}(h)$) is zero.

  Finally, one should note that the factor $\frac{k}{2^k - 1}$ appearing in the two expressions in 4 decreases for increasing k. To show this, we prove that increasing k by 1 (i.e., weakening h by the addition of one false disjunct) makes the above factor decrease:

  $$\begin{aligned} \begin{array}{rcll} \frac{k+1}{2^{k+1} - 1} &{}< &{} \frac{k}{2^k - 1} &{} \text {iff} \\ \left( {k+1}\right) \left( {2^k - 1}\right) &{}< &{} {k}\left( {2^{k+1} - 1}\right) &{} \text {iff} \\ 2^{k}-1 &{}< &{} k2^k &{} \text {iff} \\ 1-\frac{1}{2^{k}} &{} < &{} k &{} \end{array} \end{aligned}$$

  (5)

  Recalling that k is a positive integer ($k\ge 1$), the above inequality is always satisfied, which proves that $\frac{k}{2^k - 1}$ decreases for increasing k.

  Now, coming back to P7, let be h and g two completely true disjunctions of literals such that $h\vDash g$. This means that g weakens h by adding to it some true basic disjunct, in the sense that g has the form $h\vee b$ for at least one b different from the disjuncts of h. Since the number k of disjuncts increases by moving from h to g, truthlikeness decreases by Eqs. 4 and 5 above, and then h is more verisimilar than g, which proves that $vs _{}$ meets P7.

• Measure$vs _{}$meets P8. Let be h and g completely false conjunctive propositions, i.e., conjunctions of false literals. It is then clear that $inf _{T}(h)= inf _{T}(g)=0$, since both h and g partially entail no basic truths. Moreover, $inf _{F}(h)=| B _{F}(h)|/n$ and $inf _{F}(g)=| B _{F}(g)|/n$, since both h and g “fully” entail each of their own conjuncts. Suppose now that h entails g. This means that $B _{F}(h)\supset { B _{F}(g)}$, i.e., that h adds to g some false conjunct, and hence that $inf _{F}(h)>{ inf _{F}(g)}$. It follows in turn that $vs _{}(h)=- inf _{F}(h)<- inf _{F}(g)= vs _{}(g)$, i.e., that P8 is satisfied.

• Measure$vs _{}$meets P9. Let be h and g two false disjunctions of literals such that $h\vDash g$. This means that g has a greater number of disjuncts than h (compare the proof that $vs _{}$ meets P7 above). From Eqs. 4 and 5 above, it then follows that $inf _{F}(g)< inf _{F}(h)$ and hence that $vs _{}(g)=- inf _{F}(g)>- inf _{F}(h)= vs _{}(h)$, i.e., that g is more verisimilar than h, which proves that $vs _{}$ meets P9.