Truthlikeness and the Lottery Paradox via the Preface Paradox

ABSTRACT

In a 2017 AJP paper, Cevolani and Schurz (C&S) propose a novel solution to the Preface Paradox that appeals to the notion of expected truthlikeness. This discussion note extends and analyses their approach by applying it to the related Lottery Paradox.

KEYWORDS:

• truthlikeness
• lottery paradox
• preface paradox

Notes

¹ When an atom $p$ is made true by a state $w$, $\mathrm{T}\mathrm{r}(p,w)=\frac{1}{n}$. When calculating $ETr$, if—for all $w_i$ that add non-zero to ${\Sigma }_{w_i}\mathrm{P}\left(w_i\right)—$it is the case that $\mathrm{T}\mathrm{r}(p,w_i)=\frac{1}{n}$, then, since ${\Sigma }_{w_i}\mathrm{P}\left(w_i\right)=1$, the maximum is ${\Sigma }_{w_i}\mathrm{P}\left(w_i\right)\frac{1}{n}=\frac{1}{n}$. The converse applies for the lowest of $-\frac{1}{n}$.

² The TO truthlikeness measure of a statement $A$ against the true state $w$ is $\mathrm{T}\mathrm{r}(A,w)=1-\Delta (A,w)$, where $\Delta (A,w)$ is the average difference of atom valuations between $w$ and the models of $A$.

³ C-monotonicity is a property whereby, given two conjunctive statements $A$ and $B$, if $B$ has more true atoms and fewer false atoms than $A$, then the truthlikeness of $B$ is greater than the truthlikeness of $A$.