Editing Falsifiability (section)

===Falsifiability in model theory===

Scientists such as the [[List of Nobel laureates|Nobel laureate]] [[Herbert A. Simon]] have studied the semantic aspects of the logical side of falsifiability.{{sfn|Simon|Groen|1973}}{{sfn|Simon|1985}} Here it is proposed that there are two formal requirements for a formally defined and stringent falsifiability that a scientific theory must satisfy to qualify as scientific: that they be ''finitely'' and ''irrevocably'' testable.{{sfn|Rynasiewicz|1983|pages=225-6}} These studies were done in the perspective that a logic is a relation between formal sentences in languages and a collection of mathematical structures, each of which is considered a model within model theory.{{sfn|Rynasiewicz|1983|pages=225-6}} The relation, usually denoted <math>{\mathfrak A} \models \phi</math>, says the formal sentence <math>\phi</math> is true when interpreted in the structure <math>{\mathfrak A}</math>—it provides the semantic of the languages.<ref name=modeltheoryperspective group=upper-alpha>This perspective can be found in any text on model theory. For example, see {{harvnb|Ebbinghaus|2017}}.</ref> According to [[Robert Rynasiewicz|Rynasiewicz]], in this semantic perspective, falsifiability as defined by Popper means that in some observation structure (in the collection) there exists a set of observations which refutes the theory.{{sfn|Rynasiewicz|1983|loc=Sec. 2}} An even stronger notion of falsifiability was considered, which requires, not only that there exists one structure with a contradicting set of observations, but also that all structures in the collection that cannot be expanded to a structure that satisfies <math>\phi</math> contain such a contradicting set of observations.{{cn|date=May 2025}}<!--<ref name="Rynasiewicz"/> **INVALID, NOT DEFINED**-->