## [Theory](/theory-1) Definition A collection of sentences in a fixed logical language, typically closed under logical consequence, whose members serve as axioms or consequences that characterize a class of models. ## Principle Principle A theory organizes syntactic commitments: it specifies which sentences are accepted as true and determines the class of structures that satisfy those sentences. ## Demonstration Demonstration The first-order theory of groups comprises the group axioms; the complete theory of the real closed fields consists of all first-order sentences true in the real numbers under the field language. ## Misapplication Misapplication Equating a theory with a single finite presentation without acknowledging closure under consequence, or treating a set of informal assumptions as a formal theory, is a misuse. ## Consequence Consequence Working with a theory lets one deduce sentences by logical consequence, compare expressiveness, and speak of completeness, decidability, and model classes determined by the theory. ## Reversal Reversal Instead of starting from sentences and asking for models, reverse the perspective by starting from a class of structures and asking for the theory of all sentences true in them (the theory of the class). ## Boundary Boundary Typically refers to a theory in a specified formal language and logic (e.g., first-order); it excludes informal axiom systems not closed under logical consequence and frameworks requiring higher-order or infinitary languages unless stated. ## Semantic Tension Semantic Tension Tension exists between 'axiomatization' (a generating set of sentences) and the whole theory closed under consequence; one emphasizes generators, the other the full closure. ## Synthesis Synthesis A theory is the syntactic locus of a mathematical subject: a set of sentences closed under consequence that both encodes assumptions and determines the semantic class of models those sentences describe.