If you don't want to assume the existence of certain propositions, you're asking for a probability theory corresponding to a co-intutionistic variant of minimal logic. (Cointuitionistic logic is the logic of affirmatively false propositions, and is sometimes called Popperian logic.) This is a logic with false, or, and (but not truth), and an operation called co-implication, which I will write a <-- b.

Take your event space L to be a distributive lattice (with ordering <), which does not necessarily have a top element, but does have dual relative pseudo-complements. Take < to be the ordering on the lattice. (a <-- b) if for all x in the lattice L,

for all x, b < (a or x) if and only if a <-- b < x

Now, we take a probability function to be a function from elements of L to the reals, satisfying the following axioms:

1. P(false) = 0
2. if A < B then P(A) <= P(B)
3. P(A or B) + P(A and B) = P(A) + P(B)

There you go. Probability theory without certainty.

This is not terribly satisfying, though, since Bayes's theorem stops working. It fails because conditional probabilities stop working -- they arise from a forced normalization that occurs when you try to construct a lattice homomorphism between an event space and a conditionalized event space.

That is, in ordinary probability theory (where L is a Boolean algebra, and P(true) = 1), you can define a conditionalization space L|A as follows:

L|A = { X in L | X < A }
true' = A
false' = false
and' = and
or' = or
not'(X) = not(X) and A
P'(X) = P(X)/P(A)

with a lattice homomorphism X|A = X and A

Then, the probability of a conditionalized event P'(X|A) = P(X and A)/P(A), which is just what we're used to. Note that the definition of P' is forced by the fact that L|A must be a probability space. In the non-certain variant, there's no unique definition of P', so conditional probabilities are not well-defined.

To regain something like this for cointuitionistic logic, we can switch to tracking degrees of disbelief, rather than degrees of belief. Say that:

1. D(false) = 1
2. for all A, D(A) > 0
3. if A < B then D(A) >= D(B)
4. D(A or B) + D(A and B) = D(A) + D(B)

This will give you the bounds you need to let you need to nail down a conditional disbelief function. I'll leave that as an exercise for the reader.

## Comments (128)

OldIf you don't want to assume the existence of certain propositions, you're asking for a probability theory corresponding to a co-intutionistic variant of minimal logic. (Cointuitionistic logic is the logic of affirmatively false propositions, and is sometimes called Popperian logic.) This is a logic with false, or, and (but not truth), and an operation called co-implication, which I will write a <-- b.

Take your event space L to be a distributive lattice (with ordering <), which does not necessarily have a top element, but does have dual relative pseudo-complements. Take < to be the ordering on the lattice. (a <-- b) if for all x in the lattice L,

for all x, b < (a or x) if and only if a <-- b < x

Now, we take a probability function to be a function from elements of L to the reals, satisfying the following axioms:

1. P(false) = 0 2. if A < B then P(A) <= P(B) 3. P(A or B) + P(A and B) = P(A) + P(B)

There you go. Probability theory without certainty.

This is not terribly satisfying, though, since Bayes's theorem stops working. It fails because

conditional probabilitiesstop working -- they arise from a forced normalization that occurs when you try to construct a lattice homomorphism between an event space and a conditionalized event space.That is, in ordinary probability theory (where L is a Boolean algebra, and P(true) = 1), you can define a conditionalization space L|A as follows:

L|A = { X in L | X < A } true' = A false' = false and' = and or' = or not'(X) = not(X) and A P'(X) = P(X)/P(A)

with a lattice homomorphism X|A = X and A

Then, the probability of a conditionalized event P'(X|A) = P(X and A)/P(A), which is just what we're used to. Note that the definition of P' is forced by the fact that L|A must be a probability space. In the non-certain variant, there's no unique definition of P', so conditional probabilities are not well-defined.

To regain something like this for cointuitionistic logic, we can switch to tracking degrees of disbelief, rather than degrees of belief. Say that:

1. D(false) = 1 2. for all A, D(A) > 0 3. if A < B then D(A) >= D(B) 4. D(A or B) + D(A and B) = D(A) + D(B)

This will give you the bounds you need to let you need to nail down a conditional disbelief function. I'll leave that as an exercise for the reader.