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Logic

One area of mathematics that has its roots deep in philosophy is the study of logic. Logic is the study of formal reasoning based upon statements or propositions. (Price, Rath, Leschensky, 1992) Logic evolved out of a need to fully understand the details associated with the study of mathematics. At the most fundamental level, mathematics is a language and it is a language of choice and must be communicated with great precision. (Wheeler, 1995) The idea of logic was a major achievement of Aristotle. 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. 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. The key to his reasoning was that Aristotle used mathematical examples taken from contemporary texts of the time to illustrate his principles. 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, 1972)

Logic serves as a set of rules that govern the structure and presentation of mathematical proofs. (Fletcher, Patty, 1996) Since proofs are constructed with the English language, mathematical logic seeks to break down mathematical reasoning for a clearer understanding. By using statements or propositions that are either true or false, the English language becomes the building blocks for a mathematical language. In ordinary English, new propositions are formed from existing propositions. There are three ways that one can form new proposition, which stem directly from the basic principles of Aristotle. One can connect two propositions with the word "and." The word "or" can also serve as a connecting word. Forming negations is another way to build new propositions and involves making an assertion that a given statement is false. (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. The intuitive reasoning of these two mathematicians with regard to sets, led to paradoxes. Paradoxes are those statements that appear to be both true and false. However, logic makes the assumption that this cannot be the case. With the work of Dedekind and Cantor, a new type of scientist, the mathematical logician, came into existence. (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. It involves both inductive and deductive reasoning. Inductive reasoning is the process by which a general conclusion is arrived at by making limited observations. On the other hand, deductive reasoning is the process where one proceeds carefully from definitions and established facts to arrive at a possible conclusion. (Wheeler, 1995) Other aspects of logic include assertions, open sentences, simple statements, compound statements, conjunctions and disjunctions. (Wheeler, 1995) One way to summarize the study of logic is to use a Truth Table. A Truth table is used to summarize the possibilities of the statements one is studying to determine whether they are true or false. By using the Truth Table, one is able to replace statements or predictions with letters and symbols. For example, if you have one simple statement such as, "It is raining," then there are only two possibilities. The statement can be replaced with a simple variable to simplify the calculations. The statement can be true or it can be false. The possibilities are multiplied as a new statement or proposition is considered. Logic is about being non-contradictory, being rational, and being consistent. It is not related to personal beliefs. It is simply a means to cause us to think and apply our critical thinking skills. Logic may even force us to change our opinions and our beliefs once we have rationally thought through a particular proposition. In logic one creates formal languages for reasoning, then sets aside the reasoning and proofs to follow the rules of the new language. 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. (Christer, 1998)

In order to visualize this idea of logic; consider the following example:
 Nathan likes all red things. For all X, if X is red then Nathan likes X. The house is red. The color of the house = red.
Because of this "for all X" we can replace X with "the house" to get,
If the house is red then Nathan likes the house.
This we can write as,
If the color of the house = red, then Nathan likes the cottage.
We know that the color of the house is red, so one can conclude,
Nathan likes the house.         (Christer, 1998)
While this is just a simple illustration, one can see how logic can be applied and used to simplify the process by which a conclusion is arrived.

One important ingredient in both critical thinking and mathematical reasoning is logic. (Wheeler, 1995) While its roots are deep in philosophy, logic has valid applications in the area of mathematics. 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. (Christer, 1998) To completely discuss and fully explain logic, one would need several volumes. 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. (Fletcher, P, Patty, C.W., 1996)

Contributed by John Stockstill

References:

1. Christer's pages of logic, math and reasoning. (1998) World Wide Web. www.torget.se/users/m/mauritz/math/index.htm
2. Fletcher, P., Patty, C.W. (1996) Foundations of higher mathematics. Boston: PWS Publishing Company.
3. Kline, M. (1972) Mathematical thought from ancient to modern times. New York: Oxford University Press.
4. Price, J., Rath, J.N., Leschensky, W. (1992) Pre-algebra, a transition to algebra. Lake Forest: Macmillan / McGraw - Hill Publishing Company.
5. Wheeler, E.R., Wheeler, R.E. (1995) Modern mathematics for elementary school teachers. New York: Brooks / Cole Publishing Company.