It is often said that: “The conclusions of deductive reasoning are certain, whereas those of inductive reasoning are probable”. I think this contrast is somewhat misleading and imprecise, as the certainty of deductive conclusions just means that they necessarily follow from the premises (they are implied by the premises), but the conclusion itself might still be probabilistic.
Example: “If I have a fever, there’s a 65% probability that I have the flu. I have a fever. Therefore, there’s a 65% probability that I have the flu.”
There’s something off about this example. In deductive reasoning, if A implies B, then A and C together also imply B. But if A is “I have a fever” and C is “I have the flu” then A and C do not imply “there’s a 65% probability that I have the flu” (since actually there is a 100% chance).
I think what is going on here is that the initial statement “If I have a fever, there’s a 65% probability that I have the flu” is not actually an instance of material implication (in which case modus ponens would be applicable) but rather a ceteris paribus statement: “If I have a fever, then all else equal there’s a 65% probability that I have the flu.” And then the “deductive reasoning” part would go “I have a fever. And I don’t have any more information relevant to whether I have the flu than the fact that I have a fever. Therefore, there’s a 65% probability that I have the flu.”
There’s something off about this example. In deductive reasoning, if A implies B, then A and C together also imply B. But if A is “I have a fever” and C is “I have the flu” then A and C do not imply “there’s a 65% probability that I have the flu” (since actually there is a 100% chance).
I think what is going on here is that the initial statement “If I have a fever, there’s a 65% probability that I have the flu” is not actually an instance of material implication (in which case modus ponens would be applicable) but rather a ceteris paribus statement: “If I have a fever, then all else equal there’s a 65% probability that I have the flu.” And then the “deductive reasoning” part would go “I have a fever. And I don’t have any more information relevant to whether I have the flu than the fact that I have a fever. Therefore, there’s a 65% probability that I have the flu.”