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Math Logic

Logic

Logic is a field of mathematics that has its origins in philosophy. It can be defined as the science of formal principles of reasoning or correct inference. Logic had its birth with the Greek philosopher Aristotle whose system of logic is referred to as Aristotelean logic. His form of logic is based upon a collection of treatises known as the Organon. One treatise in this collection, the Prior Analytics, has the most organized discussion of formal logic.

Aristotelean logic begins with the distinction between subject and predicate which is a grammatical concept as well. A subject is defined as an individual entity or a class of entities. A predicate is a property or characteristic of a mode of existence which a given subject may or may not possess. For example, an individual (the subject) can be skillful or not (predicate) and all men (subject) may or may not be brothers (predicate). The basic principles of predication include identity, non-contradiction, and either/or.

Identity simply means that everything is what it is and acts accordingly to its designation. For example, an acorn will only grow into an oak tree. Non-contradiction refers to an object only being able to be one thing and one existence. A predicate cannot belong and not belong to a subject in a given respect at a given time. For example, an honest man cannot also be a thief. Either/or refers to everything being in existence or not being in existence. It may sound complicated but it means that a predicate either belongs or does not belong to a subject in a given respect at a given time. For example, a society can either have its freedom or not have its freedom.

Aristotle believed that the basic unit of reasoning was a syllogism. In basic terms a syllogism denotes a relationship within a group of variables. For example:

Some A are B

All B are C

Therefore, some A are C

Every syllogism consists of two premises and one conclusion. Each premise and its conclusion can be subcategorized into four types. These four types include universal affirmative, or all members of a group are members of another group; universal negative, or no members of a group are members of another group; particular affirmative, or some members of a group are members of another group; and particular negative, in which some members of a group are not members of another group.

Aristotle may be known as the first to devise the idea of logic but the work of Richard Dedekind and Georg Cantor also contributed to what is known as mathematical logic. These two mathematicians reasoned that there are paradoxes within logic. Paradoxes are statements that can be both true and false. However, logic dictates that this assumption cannot be correct. The work of Dedekind and Cantor led to the profession of mathematical logician.

Logic involves both inductive and deductive reasoning. Inductive reasoning is the process by which a conclusion is derived by making limited observations. Deductive reasoning is the process where definitions and facts are used to come up with a particular conclusion. Arguments within logic can be summarized through truth tables that summarize the possibilities of the statements a person is studying, to determine whether they are true or false. When using a truth table, statements and predictions can be replaced with letters and symbols.

Sources

  1. http://www.personal.psu.edu/t20/papers/philmath/
  2. http://www.math.wichita.edu/history/topics/logic.html

 

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