Examples of propositional logics with conditionals based on conditional probability?

Question

Are there any propositional logics with conditionals whose semantics are based on conditional probability?

I can see two design challenges that come with defining a conditional-probability-flavored conditional:

1. What is $P(B|A)$ when $A$ is impossible?
2. $P(B|A)$ really only makes intuitive sense when $B$ and $A$ are crisp, i.e. they're either fully true or fully false at every possible situation. Is there a way to allow $B$ and $A$ to themselves have intermediate truth values at possible worlds?

I'm interested in references, answers that directly contain semantics for such logics, and meta-logical results (if there are any) for logics whose conditionals are sufficiently $P(B|A)$-like.

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Here are two possible semantics to illustrate what I mean by (1) and (2) being design challenges.

Both resolve $1$ the same way, by having a flat hierarchy of worlds and having each primitive proposition take on a classical truth value at these worlds.

Connexive Variant

Let my truth values be $[0, 1]$ and let $1$ be the sole designated truth value.

Let $V$ be my valuation map, taking worlds $w$ and mapping them to $0$ or $1$.

For my notion of truth in a model, I'll use the syntax $V, w \models \alpha$, but I will allow $\alpha$ to take on any value in $[0, 1]$.

• $V, w \models A$ is equal to $V(w, A)$.
• $V, w \models A \land B$ is equal to the minimum of $V, w \models A$ and $V, w \models B$.
• $V, w \models A \lor B$ is equal to the maximum of $V, w \models A$ and $V, w \models B$.
• $V, w \models \lnot A$ is equal to $1 - (V, w \models A)$.
• $V, w \models A \to B$ is equal to $V, w \models A \land B$. This rule is different from usual rules.

Now, let $Q$ be a distribution over worlds with a countable domain and let $Q(w)$ be the probability of a given world.

• $V, Q \models \alpha \to \beta$ is zero if the expected value of $\alpha$ is zero.
• $V, Q \models \alpha \to \beta$ is zero if the expected value of $\beta$ is zero.
• $V, Q \models \alpha \to \beta$ is equal to $(\sum_{w \in W \; \text{where $\alpha$ holds}} Q(w))^{-1} * \sum_{w \in W \; \text{where $\alpha$ holds}} Q(w)(V, w \models \beta)$

All other connectives are defined in a straightforward way.

• $V, Q \models \lnot \alpha$ is $-1 + (V, Q \models \alpha)$.
• $V, Q \models \alpha \land \beta$ is the minimum of $V, Q \models \alpha$ and $V, Q \models \beta$
• $V, Q \models \alpha \lor \beta$ is the maximum of $V, Q \models \alpha$ and $V, Q \models \beta$

This is a really weird logic.

In it, $A \to \lnot A$ is a contradiction. This is Aristotle's Thesis and not that unusual.

However, its algebraic semantics validates the identity $(A \to B) \to C \approx (A \land B) \to C$, which is counterintuitive.

Classical Variant

The classical variant is like the connexive variant above, except that $\alpha \to \beta$ is interpreted as $\lnot \alpha \lor \beta$ in a single world and:

$$ M, Q \models \alpha \to \beta \;\;\text{is equal to}\;\; \frac{\sum_{w \in W \;\text{where $\alpha$ holds at $w$}} Q(w)(V, w \models \beta) }{\sum_{w \in W \; \text{where $\alpha$ holds at $w$}} Q(w)} $$

and $M, Q \models \alpha \to \beta$ is defined to be $1$ when the expected value of $\alpha$ is zero.

Answer 1

Do you know Ernest Adams's work? He gives a logic whose conditionals are expressly designed to be, as you put it, (|) like.

A very good way in to that body of work and its descendants, placing it in context, is via the terrific article on indicative conditionals at the Stanford Encyclopedia, here: https://plato.stanford.edu/entries/conditionals

(But if, as could well be the case, you are already familiar with all that stuff, then it isn't too clear exactly what new direction you are wanting to explore.)