forum 1, week of Jan 8, Dretske
I am not sure if I understand the question you are asking (e.g., what you mean by a "residual quantified probability"). Nevertheless, here is my attempt in answering your question: When the propositions are not known with certainty, then the probability of obtaining the conclusion is not necessarily a linear combination of the probability of the propositions. In other words, the propositions are not necessarily related linearly (or in some 'straightforward' fashion if we want to use non-mathematical terms) to the conclusion when the probability of the propositions is less than 1 (i.e., when the propositions are NOT known with certainty). The reason for the nonexistence of a straightforward relation between the propositions and conclusion in a probabilistic setting is because all kinds of alternative conclusions can come up with varying probabilities attached to them IF the propositions are not known with certainty. I hope this answered your question. If not, I will see if I can come up with an example to present in class tomorrow in front of the entire class, just in case anyone else has a similar question.
The question relates to the overall impact of 36 propositions at less than 1 probability. The example is 36 propositions. If the propositions are a linear combination, and the second proposition is dependent or influenced by the first propsition, does the probability of the first proposition at less than 1 become the basis to apply the probability calculation of the second proposition. The residual probability being the calculation of the probability of the conclusion after 36 propositions, each successive calculation starting from the reduced probability of the preceding proposition. If there is no dependence or sequence in the 36 propositions, what methods may be used to select a probability from the 36 propositions, to quantify the probality of the conclusion.
In defining residual probability, you presuppose that each successive proposition has a probability less than its precursor. However, I don't think this presupposition must hold - each proposition has a probability that is not necessarily related to the proposition before it, if the 36 propositions are in a sequence. In reference to your first question, I will not be able to explain what a linear combination is to the layperson - knowledge in mathematics, particularly linear algebra, is needed to understand this. Despite my background in probability theory, I am unable to answer your last question on which methods to use in selecting a probability of the conclusion. To point you in the right direction, the notion of statistical dependence or independence between the propositions themselves, or between any one of the propositions and the conclusion, pretty much governs which methods to use. Lastly, I will have to end this 'conversation' because 1) the content is not interesting enough to everyone else in the course (PHIL 440), and 2) this topic on the probability of the conclusion being related to the probability of the propositions is not something that this course will cover, beyond what has been written here. Hence, this 'conversation' is now closed.