Is $P(tautology) = 1$? What are the connections between logic and probability?

Question

It's well-known that sets are "isomorphic" to logic: if we treat $\varphi(A_1, A_2)$ as a shorthand for $\forall x: \varphi(x \in A_1, x \in A_2)$ then $A \land B \equiv A \cap B$ and $A \rightarrow B \equiv A \subseteq B$ and so on.

I've noticed that a large number of true logical statements become events with probability 1 when interpreted probabilistically. For example, if $A \subseteq B$ ($\equiv A \rightarrow B$) then $\mathbb{P}(B|A) = 1$. If you squint hard enough you should see modus ponens there.

To connect a Boolean algebra with a Boolean ring you set $x \lor y := x + y - xy$, and wouldn't you know it, $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)$. That connection can't just (ahem) be a random event, can it? ;-)

If we combine some propositional calculus and/or a Boolean algebra with measure/probability theory, can we get some theorems for free? Is it e.g. the case that if $\varphi$ is some tautology then the set-theoretic interpretation of $\varphi$ always has probability 1? Is there something stronger that's also true?

I also notice that $\mathbf{0}$ and $\mathbf{1}$, by which I mean the empty set and the set of all outcomes, are independent from all other events, and that I run into problems with Huntington's equation when I set $\lnot x := 1 - x$ and try to make a Boolean algebra over $[0, 1] \subseteq \mathbb{R}$, to do particularly with higher-order terms.

What are the theorems I'm grasping at but not quite seeing?

Answer 1

Every $\sigma$-algebra over some $\Omega$ is a Boolean algebra with $\Omega = 1$. The 1-element of every Boolean algebra is unique. Thus, if $φ$ provable using e.g. the sound and complete Russel-Bernays axioms with the associated deduction rule and we uniformly substitute in members of our $\sigma$-algebra for the variables in φ and replace disjunction/negation with union/complement, the result must be the unique 1-element in our $\sigma$-algebra, i.e. $\Omega$. But $P(\Omega) = 1$ for every probability measure $P$ by definition, so every tautology has probability 1 (for this value of tautology).

It's easy to show that every $\sigma$-algebra is a Boolean algebra by using the Huntington axiomatization (the "fourth set" on page 7 in Huntington's PDF article).

I assume for now that my failure to make a (Boolean) ring homomorphism out of a probability measure is because it doesn't work in general. I'm not sure what to make of the similarity between the union rule for probability and disjunction in Boolean rings. Maybe "$\mathbb{P}$ preserves some of the structure, but at the expense of other parts".

Answer 2

"To connect a Boolean algebra with a Boolean ring you set x∨y:=x+y−xy, and wouldn't you know it, P(A∪B)=P(A)+P(B)−P(A∩B). That connection can't just (ahem) be a random event, can it?"

One can maintain that it kind of is. Setting (x$\lor$ y) := max(x, y) comes as simpler. But, P(AUB) does not equal max(A, B). This also kind of dovetails with a discussion about possibility theory, which uses possibility measures instead of probability measures.

"If we combine some propositional calculus and/or a Boolean algebra with measure/probability theory, can we get some theorems for free?"

Plenty of propositional calculi can't get combined with a Boolean algebra and consistency get maintained. The propositional calculus has to come as a classical one for that to work, and there exist plenty of non-classical propositional calculi. So, it can't be an arbitrary propositional calculus that you select for this sort of thing.

"Is it e.g. the case that if φ is some tautology then the set-theoretic interpretation of φ always has probability 1?"

Are you asking for Cantorian/classical set theory? Because there exist non-Cantorian/non-classical set theories also.

Answer 3

I see that you found out about Cox's theorem. Have you also discovered Jaynes's "Probability Theory: The Logic of Science"? Inspired by Cox, Keynes and others, Jaynes goes straight from logic to probability without going through set theory ala Kolmogorov, but he gets to the same abstract theory. For Jaynes, probability is the rational plausibility of a proposition based on all available information.

Just as expectation $E(\cdot)$ is linear, you can consider probability $P(\cdot)$ as linear and go from a logical identity about propositions $A$ and $B$ to a property of probability.

For example, start with this logical identity $$\bar{A} = 1 - A$$ and get to this property of probability $$P(\bar{A}) = 1 - P(A)$$

Start with $$A \cup B = A + B - AB$$ and get $$P(A \cup B) = P(A) + P(B) - P(AB)$$

Start with \begin{align*} A \cup B \cup C &= 1 - \bar{A}\bar{B}\bar{C}\\ &= 1 - (1-A)(1-B)(1-C)\\ \end{align*} and get Inclusion/Exclusion

In order to conform to standard textbook notation for these properties of probability, I use $\cup$ instead of $\lor$. As you already pointed out, they mean the same thing.