M.Sc. / An overview of the process of mathematical logic's growth is depicted in this dissertation. Man began at the very beginning, distinguishing only between truth and falsity (a huge leap in those days, and definitely one in the right direction). Like a sound "abc" , ropositional and predicate logic were developed to be the basis for other mathematical logics. One needs to crawl first, before one walks. Given this frame of reference, humans could let their imaginations roam free. The thought of being limited by using only truth and falsity, was not a foreign concept during the beginning of mathematical developments in logic. It did not, therefore, take very long for the first expansions of propositional and predicate logic. As time progressed, so did thoughts, visions and ideas. Soon mathematicians were developing more enhanced logics, such as modal, many-valued and nonmonotonic logics. In fact, modal logic (or the idea behind it) was considered by Aristotle himself. New developments encouraged mathematicians (and people in other fields — for example computer scientists) to broaden their thinking and produce new ideas. Relatively speaking, fuzzy logic is one of the most recent developments. A very powerful logic, given current computer strengths. Fuzzy logic is a system not without drawbacks, even with powerful computers driving fuzzy logic systems. For example, as the number of inputs in a certain system increase, so does the level of complexity, rendering even enormous computers incapable of coping. Currently the success of any fuzzy logic system depends on the model driving it. These models are built by humans, based on a variety of information gathered over time. If, for instance, in medical diagnoses, the reference base says the symptom of a sore throat is associated with an appendix, the diagnoses will certainly be wrong and you might lose your appendix while only suffering a cold. In this day and age we are standing on the verge of a totally computerized environment. The fridge will soon tell you that you are out of milk and that your brand of milk is currently on a special at a specific shop. It will be able to order and pay for new milk. Keeping an open mind, you might be able to envisage a little robot collecting the milk from your front gate, programmed with the ability to judge whether the milk is fresh (by referring to the sellby- date and the smell and colour of the milk). The robot might even be able to tell you to increase your intake of fresh fruit and vegetables as your pale skin color, dark rings under your eyes, your level of fatigue and current intake of these produce indicates a lack thereof. In the case that you did not sleep at all during the night before (dur for instance to a deadline that needed to be met) the robot can decide that this is more than enough reason for your physical appearance and fatigue

About a decade ago, Adam Elga introduced philosophers to an intriguing puzzle. In it, Sleeping Beauty, a perfectly rational agent, undergoes an experiment in which she becomes ignorant of what time it is. This situation is puzzling for two reasons: First, because there are two equally plausible views about how she will change her degree of belief given her situation and, second, because the traditional rules for updating degrees of belief don’t seem to apply to this case. In this dissertation, my goals are to settle the debate concerning this puzzle and to offer a new rule for updating some types of degrees of belief. Regarding the puzzle, I will defend a view called “the Lesser view,” a view largely favorable to the Thirders’ position in the traditional debate on the puzzle. Regarding the general rule for updating, I will present and defend a rule called “Shifted Jeffrey Conditionalization.” My discussions of the above view and rule will complement each other: On the one hand, I defend the Lesser view by making use of Shifted Jeffrey Conditionalization. On the other hand, I test Shifted Jeffrey Conditionalization by applying it to various credal transitions in the Sleeping Beauty problem and revise that rule in accordance with the results of the test application. In the end, I will present and defend an updating rule called “General Shifted Jeffrey Conditionalization,” which I suspect is the general rule for updating one’s degrees of belief in so-called tensed propositions.

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