This blog is a followup on my previous blog here, if you have not read that, go read it first.
The previous axioms introduce a new paradox, the Preface Paradox, and to fix this paradox we have to forgo the assumption that all beliefs are true or false, and go to a probabilistic model.
Using fuzzy logic the following operators are defined:
$$a \land b = ab$$
$$a \lor b = a+b-ab$$
$$a \oplus b = a+b-2ab$$
$$\lnot b = 1-b$$
All these operators are closed in the unit interval $\mathbb{I}$ and consistent with De Morgan's laws.
A universe is defined the same way as before, but probabilistic statements are defined as follows:
$$P: U \rightarrow \mathbb{I}$$
Not that normal statements still exist, and are defined as follows:
$$S: U \rightarrow {T,F}$$
And let's also define
$$\Omega: S \rightarrow P$$
The space of all statement transformations, which turns a statement into how sure we are of the statement.
Now define:
$$\forall x \in \mathbb{I}, \forall p in S: F_x(s)(u) :=
\begin{cases}
1,& \text{if } p(u) = x\\
0,& \text{otherwise}
\end{cases}
$$
This function turns a probabilistic statement, for example 'it will rain tomorrow', into a normal statement, "I am 90% sure it will rain tomorrow". If there was indeed a 90% chance, the new statement will have a 100% chance to be true, otherwise it will be zero. Call this function the fuzzy function.
Note that our beliefs only apply to deterministic states, and not to probabilistic states, as it is nonsensical to clarify a belief in a probabilistic statement. For example, I am 60% sure I am 90% sure is, while mathematically valid, not something I would constitute as a belief.
Now that we have these to sets we can read our new axioms. Call the subset of $\Omega$ that matches these axioms $M$
Now is $m \in M$ if
- $m$ is reasonable:
$$\forall x \in \mathbb{I}:m(\bar{T}) = C_1$$ - $$\forall x \in \mathbb{I}:m(\bar{F}) = C_0$$
- $m$ is consistent:
$m$ is distributive with $\land,\lor,\lnot,\oplus$ - $m$ is self-conscious:
$$m(F_x(m(s))) = F_x(m(s))$$
The function $F_1 \circ m$ is equivalent to the belief function of previous blog, and by stating things like, I am quite sure, $F_{0.95} \circ m$, also works, and has no problem with the Preface Paradox.