Probabilistic Moore's statements

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

1. $m$ is reasonable:
   $$\forall x \in \mathbb{I}:m(\bar{T}) = C_1$$
   
2. $$\forall x \in \mathbb{I}:m(\bar{F}) = C_0$$
3. $m$ is consistent:
   $m$ is distributive with $\land,\lor,\lnot,\oplus$
4. $m$ 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.

An extention to Moore's statements

This post is a sequel to my previous post here. If you have not read it, read that one first.

In the previous post we assumed that there was a single belief function. But peoples belief can change over time and differ between people, so one should create a belief function for each person for each point in time.

Image one changes Moore's statement to use a different time or person, for example "it is raining but you don't belief it is raining" or "it was raining but I did not belief it was raining". It is obvious from the text that this is not a contradiction of an absurdity. The math reflects this:

$$M_1(s) \land M_1(M_2(s))$$

As there are two different belief functions, the self-consiousness axiom does not apply, and this cannot be reduced further, making this statement undecided.

Now that we have a working framework, one can build more Moore's statements. For example:

$$s \oplus M(s)$$

This translates to "I do not belief this statement", which looks like the classic liars paradox, but isn't a paradox, and can be true or false. But If one checks if the person who tells the statement beliefs it, one gets the following.

$$M(s) \oplus M(M(s)) = \bar{F}$$

Which is a contradiction.

A solution to Moore's paradox

Moore's paradox describes the so called Moorian sentences, like for example "It is raining but I don't believe it is raining". These are statements that can be true, and are logically consistent, yet seem (and are) absurd. But if we mathematically formalise what "believing" means, we can see the source of the paradox appear bright as day.

Lets start with some definitions:

$U$ is the set of all states of the universe. This is a set of custom objects, and does not have any other properties. 

A statement is defined as the following.

$$U \rightarrow {T,F}$$

So, for example, 'it is raining', is a statement, so it is a function that takes the current state of the universe, and converts it to either true or false, dependant on the actual state (so if it is actually raining). Call the set of all statements $S$.

Statements have the following operations:

$$\forall u \in U, \forall s,t \in S$$

$$(s \land t)(u) := s(u) \land t(u)$$

$$(s \lor t)(u) := s(u) \lor t(u)$$

$$(\lnot s)(u) :=  \lnot s(u)$$

So the classical operations on booleans also apply on statements.

Now, consider two special statements, the Tautology and the Contradiction:

$$\forall u \in U: \bar{T}(u) := T$$

$$\forall u \in U: \bar{F}(u) := F$$

We call a statement undecided if it is not a Tautology nor a Contradiction.

A belief is a function who takes a statement, for example 'it is raining' and retuns a different statement, 'I believe it is raining'.

Formally:

$$M: S \rightarrow S$$

It has by axiom the following properties:

1. Reasonability:
   $$M(\bar{T}) = \bar{T}$$
   One always believes a tautology to be true.
2. Consistency
   $$M(s \land t) = M(s) \land M(t)$$
   $$M(\lnot s) = \lnot M(s)$$
   If I believe that "apples are red" and I believe that "bananas are yellow", I believe that "Both apples are red and bananas are yellow". In addition, if I believe apples are not red, I do not believe apples are red.
3. Self-conciousness
   $$M(M(S)) = M(S)$$
   If I believe that apples are red, I believe that I believe that apples are red.

Now, lets call a statement a triviality if the believe function of the statement is a tautology, and an absurdity if the belief of the statement is a contradiction.

Each contradiction is by the reasonability axiom an absurdity, but this property is not symmetric.

Looking back at Moore's paradox. If we call the statement, 'it is raining' $s$, then Moore's paradox states:

$$s \land \lnot M(s)$$

This statement can obviously be true. But if we apply the believe function to this statement it becomes:

$$M(s \land \lnot M(s) )= M(s) \land \lnot M(M(s)) = M(s) \land \lnot M(s) = \bar{F}$$

So, the statement, while it can be true, can never be believed, and is thus absurd. In short, the reason Moore's statement seems absurd is not because it cannot be true, but because the person stating it can never believe it.