Informal logic

Informal logic or non-formal logic is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial, formal, or technical language (see formal logic). Johnson and Blair (1987) define informal logic as "a branch of logic whose task is to develop non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation in everyday discourse."

Opinion pieces of newspapers provide illustrative textbook examples of informal logic (Walton 1989), usually because these pieces are short and often fallacious. However, informal logic is also used to reason about events in the human and social sciences. In fact, most reasoning from known facts to unknown facts that uses natural language, even if combined with mathematical or statistical reasoning, can be regarded as an application of informal logic so long as it does not rely on additional empirical evidence.

Mathematics and the natural sciences
In mathematics the reasoning that occurs in proofs, though informal, is often regarded as a close approximation to a formal proof, that is, one which is carried out in a formal system of logic. Note that in practice, however, the separation between an informal mathematical proof and its formal idealization is so large that hardly anyone attempts to bridge that gap. This gap arises because most steps in informal proofs accumulate an enormous number of simple logical inferences, or other proof steps which are straightforward to most readers with enough mathematical experience. Moreover, many mathematical researchers regard proof as something other than a sequence of inference steps. Nevertheless, one of the goals of the Mizar project is to formalize the entire body of informal proofs of mathematics.

In theoretical physics, arguments are used to derive new formulas or physical principles. These arguments often use mathematics, although in many cases the relations between assertions in a derivation contain mathematically serious gaps. Examples of these mathematical gaps are failure to prove convergence of an infinite series or an integral (or worse, rely on an expression whose value is known to be divergent) or ignoring quantities which are small in a limiting sense. Despite mathematical gaps, arguments used in physical derivations are generally considered to be valid arguments.

Social sciences
In the social sciences many arguments are based on applications of statistics to demonstrate correlation or lack thereof between sets of variables, such as levels of income and education, ethnicity and wealth and so on. Such arguments are based on theories of statistical hypothesis testing together with empirical data accumulated by polling, collection of historical records, long term studies etc. Econometrics is the branch of economics that applies statistics to economics. Besides statistics, economists use a wide variety of analytical tools, for example, calculus, qualitative reasoning about systems of equations, asymptotic analysis (theories of growth), and so on.

Law and politics
An extremely intricate form of reasoning is legal reasoning since it involves such considerations as legal precedent and existing law. The nature of the propositions used in legal reasoning is one of the concerns of legal theory.