Rökfræði

Rökfræði is the study of valid reasoning and the principles by which conclusions follow from premises. In everyday use, it concerns the assessment and construction of good arguments; in formal settings, it employs precise symbolic languages and rules of inference to determine when statements are logically entailed.

Historically, rökfræði has roots in Aristotle's syllogistic and was transformed by modern logicians such as Frege, Russell, Hilbert, and Gödel. The field is usually divided into subfields including formal logic (propositional, first-order, and higher-order logics), proof theory, model theory, and set theory, as well as non-classical logics (modal, temporal, intuitionistic).

Key concepts include validity (a conclusion that follows from the premises in all cases), soundness (valid and with true premises), satisfiability, provability, completeness, and decidability. Methods consist of constructing formal languages, axiomatic systems, and proof calculi, as well as semantic analyses using interpretations or models.

Applications span mathematics, computer science (program verification, automated reasoning), philosophy, linguistics, and cognitive science. The field informs both theoretical questions about the nature of truth and practical tasks such as verification of software and hardware. Gödel's incompleteness theorems illustrate fundamental limits on formal systems, showing that no sufficiently powerful system can prove all true statements and that consistency cannot be proven within the same system.

Rökfræði continues to evolve with advances in automated reasoning, complexity theory, and the study of alternative logics used to model real-world reasoning.