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One can connect two propositions with the word "and." One of the successful results of this program is the ability to study mathematical language and reasoning By using statements or propositions that are either true or false, the English language becomes the building blocks for a … (1995) Modern mathematics for elementary school teachers. (Fletcher, Patty, 1996) While Aristotle was the first to focus on the idea of logic, the efforts of Richard Dedekind and Georg Cantor also contributed greatly to this study. New York: Brooks / Cole Publishing Company. a medium for communicating mathematics in a precise and clear way. (Wheeler, 1995) One way to summarize the study of logic is to use a Truth Table. h��V�O��3�B�3gϐ3��/M4E2�9��Y��gbc�,�0vP�����TS���̄:���2�D�V�FX�JS'��!�H�fB��P%�^��P�lS�����?a��s������y�W�{�� �@�=�#/�b$!B$�,\� ��X@x}Ԃ ڷſ���G\�_ �4^8c���q�g�\\. In this course we develop mathematical logic using elementary set theory as given, just as one would do with other branches of mathematics, like group theory or probability theory. (Wheeler, 1995) While its roots are deep in philosophy, logic has valid applications in the area of mathematics. However, through the efforts of Aristotle and others, mathematicians today have a way to produce correct laws of mathematical reasoning and establish rules that provide structure and govern the presentation of mathematical proofs. The basic assumptions of mathematics. (Kline, 1972), Logic serves as a set of rules that govern the structure and presentation of mathematical proofs. h�bbd``b� N@�� H0K�Xb@B�H������$R��������@� Œ� A Truth table is used to summarize the possibilities of the statements one is studying to determine whether they are true or false. 3. (Fletcher, Patty, 1996) Since proofs are constructed with the English language, mathematical logic seeks to break down mathematical reasoning for a clearer understanding. Logic is about being non-contradictory, being rational, and being consistent. It is not related to personal beliefs. The intuitive reasoning of these two mathematicians with regard to sets, led to paradoxes. There are three ways that one can form new proposition, which stem directly from the basic principles of Aristotle. Forming negations is another way to build new propositions and involves making an assertion that a given statement is false. New derivations of purposed formulas, and new conclusions may occur, or the rules of logic can lead to derive the truth-value of a formula in some kind of logical arithmetic. It involves both inductive and deductive reasoning. The word "or" can also serve as a connecting word. On the other hand, deductive reasoning is the process where one proceeds carefully from definitions and established facts to arrive at a possible conclusion. The possibilities are multiplied as a new statement or proposition is considered. [n the belief that beginners should be exposed to the easiest and most natural proofs, I have used free-swinging set-theoretic methods. At the most fundamental level, mathematics is a language and it is a language of choice and must be communicated with great precision. For all X, if X is red then Nathan likes X. Christer's pages of logic, math and reasoning. Paradoxes are those statements that appear to be both true and false. The basic principles of logic center on the law of contradiction, which states that a statement cannot be both true and false, and the law of the excluded middle, which stresses that a statement must be either true or false. Boston: PWS Publishing Company. By using statements or propositions that The statement can be true or it can be false. (1996) Foundations of higher mathematics. 2. It is simply a means to cause us to think and apply our critical thinking skills. One area of mathematics that has its roots deep in philosophy is the study of logic. The language of mathematics. (Wheeler, 1995) The idea of logic was a major achievement of Aristotle. (Price, Rath, Leschensky, 1992) Logic evolved out of a need to fully understand the details associated with the study of mathematics. (Fletcher, P, Patty, C.W., 1996), In order to visualize this idea of logic; consider the following example: One important ingredient in both critical thinking and mathematical reasoning is logic. are either true or false, the English language becomes the building blocks for a mathematical language. There are several areas of logic, which include everyday logic, formal logic, Boolean algebra, and many other areas that are considered as some type of logic. {y��O�$���܎��ێ�g>�[vv���p焲ޚ��W��ѻ���a�x݅N��X$��~Q~p�ǿܥ\���y��E�o+I��D/&�M�����MoH�Y��=�]����-^�**�s�uSi��%:-����+'�9�n9/}X����M{N㟿�9��?ʻk���_|�P���p�c���%|g��>J��9����2n"c�����z��z��щ)۟���Ovx���~������[�7�e�ɓm��]�w�6�[��_��I�꓏� In logic one creates formal languages for reasoning, then sets aside the reasoning and proofs to follow the rules of the new language. (Christer, 1998). |����]����O��3Qv��Q��G�rͩ����F�%�����ˍ�(��O,v}�ao,Pn��7. Logic may even force us to change our opinions and our beliefs once we have rationally thought through a particular proposition. In his effort to produce correct laws of mathematical reasoning, Aristotle was able to codify and systemize these laws into a separate field of study. Inductive reasoning is the process by which a general conclusion is arrived at by making limited observations. Even though the science of logic was derived from mathematics, logic eventually came to be considered as a study independent of mathematics yet applicable to all reasoning. Kline, M. (1972) Mathematical thought from ancient to modern times. 649 0 obj <> endobj (1998) World Wide Web. The statement can be replaced with a simple variable to simplify the calculations. %PDF-1.5 %���� %%EOF mathematical logic. Logic is the study of formal reasoning based upon statements or propositions. (Fletcher, Patty, 1996) Since proofs are constructed with the English language, mathematical logic seeks to break down mathematical reasoning for a clearer understanding. Mathematical logic originated as an attempt to codify and formalize the following: 1. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic … The permissible rules of proof. 656 0 obj <>/Filter/FlateDecode/ID[<647E5CE4DF1AF5409806B37C842C0F37>]/Index[649 16]/Info 648 0 R/Length 56/Prev 920981/Root 650 0 R/Size 665/Type/XRef/W[1 2 1]>>stream Price, J., Rath, J.N., Leschensky, W. (1992) Pre-algebra, a transition to algebra. 0 However, logic makes the assumption that this cannot be the case. (Fletcher, Patty, 1996), On the whole, logic is a way to improve one's critical thinking skills by not just looking at a problem, but studying the problem and implementing strategies to find a solution. Fletcher, P., Patty, C.W. open sentences, simple statements, compound statements, conjunctions and disjunctions. For example, if you have one simple statement such as, "It is raining," then there are only two possibilities. (Christer, 1998) To completely discuss and fully explain logic, one would need several volumes. (Wheeler, 1995) Other aspects of logic include assertions, The key to his reasoning was that Aristotle used mathematical examples taken from contemporary texts of the time to illustrate his principles. In ordinary English, new propositions are formed from existing propositions. Lake Forest: Macmillan / McGraw - Hill Publishing Company. New York: Oxford University Press. By using the Truth Table, one is able to replace statements or predictions with letters and symbols. endstream endobj startxref 664 0 obj <>stream Wheeler, E.R., Wheeler, R.E. With the work of Dedekind and Cantor, a new type of scientist, the mathematical logician, came into existence.

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