You are now faced with a cute cat that has been checked by a veterinarian who says this cat definitely does not have toxoplasmosis\. If you decide to pet the cat, an impartial observer watching you will conclude that you are 10% more likely to have toxoplasmosis, which can be a fairly detrimental infection\. If you don't pet the cat, you'll miss out on the hedonic enjoyment of petting it\. Do you pet the cat?
This example is flawed because the analysis does not condition on all the information you have. The analysis assumes that
P(toxoplasmosis | pet cat, B) > P(toxoplasmosis | not pet cat, B)
where B is your background information. Why should this be so? If B is the background information of an outside observer who does not have access to your inner thoughts and feelings, then this is a plausible claim, because petting or not petting the cat provides evidence as to how fond you are of cats.
But you already know how fond you are of cats. Even if your introspective powers are week, once you feel (or do not feel) the urge to pet the cat, the Bayesian update for probability of toxoplasmosis has already happened. If a flawed decision analysis causes you to not pet the cat, that decision provides no further evidence about your degree of fondness for cats.