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This study was designed to determine the effectiveness of teaching symbolic logic in the high school. Three hundred eighty-seven students enrolled on the University Programme in grades nine to thirteen at Como Lake High School, in School District No. 43 (Coquitlam), took part in the investigation. The students were grouped according to the mathematics course they were studying. Answers were sought to two specific questions. Do significant differences exist between the means of the final test scores of the students in each of the groups? At which grade levels can this material be effectively mastered? As a criterion for determining this, 75 per cent of the students at a particular level were required to obtain a score of 50 per cent or better on the final test. In order to answer the first question, the results were studied by analysis of covariance with scholastic aptitude being the variable controlled. The answer to the second question was obtained by comparing the performance of each group with the standard outlined. On the basis of this information, decisions were made regarding the suitability of the material for the various grade levels. All of the differences between the means were found to be significant at the one per cent level. The highest mean score was obtained by the students in Mathematics 101, followed in order by those of Mathematics 91, 30, 20, and 10. The students of Mathematics 101, 91 and 30 satisfied the requirement that 75 per cent should obtain a score of 50 per cent or better on the final test. The students of Mathematics 20 and 10 failed to satisfy this requirement. / Education, Faculty of / Graduate

According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion. / Arts, Faculty of / Philosophy, Department of / Graduate

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

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