forum 1, week of Jan 8, Dretske
For question 1, the fact that one has not excluded the possibility that the zebras are painted mules shows that the claim "I know that these animals are zebras" is not carefully backed up. By carefully backed up, I mean no existence of a track record of examining the animals more closely, checking with the zoo officials, or some kind of 'evidence' of performing tasks that would give more confidence and trust to why the claim should be regarded as true. On the other hand, whether not excluding the possibility that the zebras are painted mules is equivalent to claiming "you don't know that they are zebras" depends on how one defines what it is to 'know' something. I do not take 'know'ing something to be binary (i.e., 'yes' or 'no') - I would attach a degree (between 0 and 1) on how confident that person knows that these animals are zebras. Hence, I would answer yes to the first question as follows: the fact that you have not excluded the possibility that the zebras are painted mules shows (with degree p) that you don't know that they are zebras, where 0 ≤ p ≤ 1. The reason for including a degree of 'confidence' (or something along those lines) in 'know'ing a claim is due to my background in statistics or probability theory, as well as the existence of a recent epistemological movement towards Bayesian methods. However, it should be noted that this epistemological movement comes with numerous philosophical problems and is nowhere near consensus - the following link gives a small glimpse of demonstrating the lack of consensus on using Bayesian methods (just in case anyone is interested): http://errorstatistics.blogspot.com/2011/12/jim-berger-on-jim-berger.html#disqus_thread
If there are a number of propositions on knowing the animals are zebras, each known by degree between 0 and 1, such as the 36 number example used by Dr. Morton in lecture; does the probability of the conclusion become the impact of the progression of going from one proposition to the next, each with a probability between 0 and 1, and assuming less than 1 in probability for each proposition; in a list of 36 related propositions for example, does one arrive at a residual quantified probability degree between 0 and 1, for the conclusion to the 36 propositions.
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.