Formal system

A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics. The entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory. A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements.

Each formal system has a formal language, which is composed by primitive symbols. These symbols act on certain rules of formation and are developed by inference from a set of axioms. The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.

Formal systems in mathematics consist of the following elements:

Logic system

Logic system may refer to:

Mill's Methods

Mill's Methods are five methods of induction described by philosopher John Stuart Mill in his 1843 book A System of Logic. They are intended to illuminate issues of causation.

The methods

Direct method of agreement

For a property to be a necessary condition it must always be present if the effect is present. Since this is so, then we are interested in looking at cases where the effect is present and taking note of which properties, among those considered to be 'possible necessary conditions' are present and which are absent. Obviously, any properties which are absent when the effect is present cannot be necessary conditions for the effect. This method is also referred to more generally within comparative politics as the most different systems design. Symbolically, the method of agreement can be represented as:

To further illustrate this concept, consider two structurally different countries. Country A is a former colony, has a centre-left government, and has a federal system with two levels of government. Country B has never been a colony, has a centre-left government and is a unitary state. One factor that both countries have in common, the dependent variable in this case, is that they have a system of universal health care. Comparing the factors known about the countries above, a comparative political scientist would conclude that the government sitting on the centre-left of the spectrum would be the independent variable which causes a system of universal health care, since it is the only one of the factors examined which holds constant between the two countries, and the theoretical backing for that relationship is sound; social democratic (centre-left) policies often include universal health care.