Need a propositional formula that does satisfy the following

Question

There exists a $\epsilon$ > 0, formulas $\varphi$, $\psi$ that does satisfy

P[ψ] + ε ≥ P[φ] But not P[φ→ψ] ≥ 1−ε

So P[ψ] + ε ≥ P[φ] But P[φ→ψ] < 1−ε

Anyone an idea?

Answer 1

Consider the possible outcomes of rolling a six-sided die. Let $\psi$ denote "we rolled an even number", and let $\varphi$ denote "we rolled an odd number". Clearly we have $P[\psi] = \frac{1}{2}$ and $P[\varphi] = \frac{1}{2}$. The event $\varphi \rightarrow \psi$ happens if "we either did not roll an odd number or we rolled an even number", i.e. precisely if we rolled an even number.

Now set $\varepsilon = \frac{1}{6}$. We have $P[\psi] + \varepsilon = \frac{1}{2} + \frac{1}{6} \geq \frac{1}{2} = P[\varphi]$. However, $P[\varphi \rightarrow \psi] = \frac{1}{2} \not\geq 1 - \frac{1}{6} = \frac{5}{6}$.