Simpson Department of Mathematics April 30, 1999 Pennsylvania State University This article is an overview of logic and the philosophy of mathematics. It is intended for the general reader. It has appeared in the volume The Examined Life: Readings from Western Philosophy from Plato to Kant, edited by Stanley Rosen, published in 2000 by Random House. • • • • • • • • • • • Logic Logic is the science of formal principles of reasoning or correct inference.

Historically, logic originated with the ancient Greek philosopher Aristotle. Logic was further developed and systematized by the Stoics and by the medieval scholastic philosophers. In the late 19th and 20th centuries, logic saw explosive growth, which has continued up to the present. One may ask whether logic is part of philosophy or independent of it. According to Bochenski [, §10B], this issue is nowhere explicitly raised in the writings of Aristotle. However, Aristotle did go to great pains to formulate the basic concepts of logic (terms, premises, syllogisms, etc.) in a neutral way, independent of any particular philosophical orientation.

Thus Aristotle seems to have viewed logic not as part of philosophy but rather as a tool or instrument to be used by philosophers and scientists alike. This attitude about logic is in agreement with the modern view, according to which the predicate calculus (see below) is a general method or framework not only for philosophical reasoning but also for reasoning about any subject matter whatsoever. Logic is the science of correct reasoning. What then is reasoning?

According to Aristotle [, Topics, 100a25], reasoning is any argument in which certain assumptions or premises are laid down and then something other than these necessarily follows. Thus logic is the science of necessary inference. However, when logic is applied to specific subject matter, it is important to note that not all logical inference constitutes a scientifically valid demonstration. This is because a piece of formally correct reasoning is not scientifically valid unless it is based on a true and primary starting point.

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Furthermore, any decisions about what is true and primary do not pertain to logic but rather to the specific subject matter under consideration. In this way we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences. All reasoning, both scientific and non-scientific, must take place within the logical framework, but it is only a framework, nothing more.

This is what is meant by saying that logic is a formal science. For example, consider the following inference.

Some real estate will increase in value. Anything that will increase in value is a good investment. Therefore, some real estate is a good investment. This inference is logically correct, because the conclusion ``some real estate is a good investment' necessarily follows once we accept the premises ``some real estate will increase in value' and ``anything that will increase in value is a good investment'. Yet this same inference may not be a demonstration of its conclusion, because one or both of the premises may be faulty. Thus logic can help us to clarify our reasoning, but it can only go so far.

The real issue in this particular inference is ultimately one of finance and economics, not logic. We shall now briefly indicate the basics of Aristotelean logic. Aristotelean logic Aristotle's collection of logical treatises is known as the Organon.

Of these treatises, the Prior Analytics contains the most systematic discussion of formal logic. In addition to the Organon, the Metaphysicsalso contains relevant material. See Aristotle [] and Ross [].

Subjects and predicates Aristotelean logic begins with the familiar grammatical distinction between subject and predicate. Anytoiso Converter 3 4 Cracked on this page. A subject is typically an individual entity, for instance a man or a house or a city. It may also be a class of entities, for instance all men. A predicate is a property or attribute or mode of existence which a given subject may or may not possess.

For example, an individual man (the subject) may or may not be skillful (the predicate), and all men (the subject) may or may not be brothers (the predicate). The fundamental principles of predication are: • Identity.