# Formal logic

As formal logic in general, is logic called that deal with the relationship between the logical form of statements busy and the validity of drainage and inferring relationships between these forms. In a narrower sense, this is the name given to logics that use a formalized representation of statements and conclusions.

## General

In addition to the doctrine of judgment and the terms used in it, logic is particularly concerned with the analysis and construction of logical conclusions , whereby here formal aspects, without reference to the semantic content of the statements considered, are in the foreground, such as in the so-called modus ponens , which allows one to infer the correctness of " B " from the implication " A follows B " and the existence of the statement " A " . Such conclusions, their justification and scope are the subject of formal logic. This has its origins in antiquity and found through Aristotle in the syllogistics a form that is valid up to modern times, even if there were important additions to Aristotle's point of view in the history of logic . Since the algebraization of logic by George Boole and Gottlob Frege , formal logic has primarily been understood to be mathematical logic , which is defined as classical logic from the previous traditional logic . The basis for both was Georg Cantor's emerging set theory and the interpretation of concepts as sets of things that fall under them . Formal logic, however, soon branches out into calculus of proof , philosophical logics and non-classical logics . There is also a tradition of conceptual logic to this day .

## Formalized logic

“Formal logic” denotes a notation of inferences using a formal language that often introduces special symbols. It is usually specified exactly how well-formed expressions in this language are formed ( syntax ). The Aristotelian syllogistics is an attempt at such a formalization, which can be seen as a special case of inference in predicate logic , which in turn contains propositional logic .

A counter-term to formalized logic is informal or informal logic, which examines arguments that are actually expressed in natural language in their factual context and not prepared in formal language. This discipline can also be traced back to Aristotle, namely to the exposition in the Topik and the Sophistic refutations .

## In contrast to material logic

Immanuel Kant used the expression “formal logic” for a rule-based reasoning that “abstracts from all the content of intellectual knowledge and the diversity of its objects”, that is “has nothing to do with anything other than the mere form of thinking”. He distinguished from this a project that he called " transcendental logic " and that also deals with the content of statements.

In contrast to previous idioms, experts today mean by the word “logic” - if no further qualification is added - normally a non-material or non-transcendental logic.

## Formal and Mathematical Logic

Gottlob Frege developed formal logic in his conceptual writing (1879) up to a first almost complete axiomatization of predicate logic, which served as a model for subsequent axiomatizations by Bertrand Russell ( Principia Mathematica ) or David Hilbert ( Hilbert program ). In the 1930s, Alfred Tarski succeeded in completely abstracting the formulas created according to syntactic rules from their semantics by using the concept of a model to specify the interpretations of the formulas and clearly distinguishing them from the formulas themselves (see first-level predicate logic ). There is also a consistent separation of object language and metalanguage here . These and the work of Kurt Gödel , which ultimately led to the failure of the Hilbert program, are the cornerstones of modern mathematical logic .

## literature

• JM Bocheński : Formal Logic . 5th unchanged edition. Alber, Freiburg (Breisgau) et al. 1996, ISBN 3-495-44115-8 , ( Orbis academicus 3, 2).
• Walter Bröcker : Formal, transcendental and speculative logic . Klostermann, Frankfurt am Main 1962.
• Paul Hoyningen-Huene : Formal Logic. A philosophical introduction . Reclam, Stuttgart 1998, ISBN 3-15-009692-8 .
• Edmund Husserl : Formal and transcendental logic. Attempt a critique of logical reason . 2nd Edition. Unchanged reprint of the 1st edition in 1929. Niemeyer, Tübingen 1981, ISBN 3-484-70129-3 .
• Richard Jeffrey: Formal Logic. Its scope and limits . 2nd Edition. McGraw-Hill, New York NY 1981, ISBN 0-07-032321-6 .
• Paul Lorenzen : Formal Logic . 4th improved edition. de Gruyter, Berlin 1970, ( Göschen Collection 1176 / 1176a).
• Albert Menne : Introduction to Formal Logic. An orientation on the doctrine of consistency, its history, structures and applications . Scientific Book Society, Darmstadt 1985, ISBN 3-534-05203-X .
• Albert Menne, Niels Openenberger (Ed.): Formal and non-formal logic in Aristotle . Olms, Hildesheim-Zürich et al. 1985, ISBN 3-487-07266-1 .
• Thomas Zoglauer : Introduction to Formal Logic for Philosophers . 4th revised edition. Vandenhoeck & Ruprecht, Göttingen 2008, ISBN 978-3-525-03293-0 , ( UTB for science - university paperbacks - philosophy 1999).