Editing Falsifiability (section)

==Basic statements and the definition of falsifiability==
===Basic statements===

In Popper's view of science, statements of observation can be analyzed within a logical structure independently of any factual observations.<ref name="fourlinesoftesting" group=upper-alpha/><ref name="poppernoapriori" group=upper-alpha /> The set of all purely logical observations that are considered constitutes the empirical basis. Popper calls them the ''basic statements'' or ''test statements''. They are the statements that can be used to show the falsifiability of a theory. Popper says that basic statements do not have to be possible in practice. It is sufficient that they are accepted by convention as belonging to the empirical language, a language that allows [[intersubjective verifiability]]: "they must be testable by intersubjective observation (the material requirement)".{{sfn|Shea|2020|loc=[https://iep.utm.edu/pop-sci/#SH2c Sec. 2.c]}}<ref name="keuthbasicstatementsdependontechnology" group=upper-alpha /> See the examples in section {{slink||Examples of demarcation and applications}}.

In more than twelve pages of ''The Logic of Scientific Discovery'',{{sfn|Popper|1959|loc=sec. 13–15, 28}} Popper discusses informally which statements among those that are considered in the logical structure are basic statements. A logical structure uses universal classes to define laws. For example, in the law "all swans are white" the concept of swans is a universal class. It corresponds to a set of properties that every swan must have. It is not restricted to the swans that exist, existed or will exist. Informally, a basic statement is simply a statement that concerns only a finite number of specific instances in universal classes. In particular, an existential statement such as "there exists a black swan" is not a basic statement, because it is not specific about the instance. On the other hand, "this swan here is black" is a basic statement. Popper says that it is a singular existential statement or simply a singular statement. So, basic statements are singular (existential) statements.

===The definition of falsifiability===

Thornton says that basic statements are statements that correspond to particular "observation-reports". He then gives Popper's definition of falsifiability:
{{Blockquote | text= "A theory is scientific if and only if it divides the class of basic statements into the following two non-empty sub-classes: (a) the class of all those basic statements with which it is inconsistent, or which it prohibits—this is the class of its potential falsifiers (i.e., those statements which, if true, falsify the whole theory), and (b) the class of those basic statements with which it is consistent, or which it permits (i.e., those statements which, if true, corroborate it, or bear it out)."| author= Thornton, Stephen | source = {{harvnb|Thornton|2016|loc= at the end of section 3}} }}

As in the case of actual falsifiers, decisions must be taken by scientists to accept a logical structure and its associated empirical basis, but these are usually part of a background knowledge that scientists have in common and, often, no discussion is even necessary.<ref name="nodiscussionneededforbasicstatements" group=upper-alpha/> The first decision described by Lakatos{{sfn|Lakatos|1978|p=22}} is implicit in this agreement, but the other decisions are not needed. This agreement, if one can speak of agreement when there is not even a discussion, exists only in principle. This is where the distinction between the logical and methodological sides of science becomes important. When an actual falsifier is proposed, the technology used is considered in detail and, as described in section {{slink||Dogmatic falsificationism}}, an actual agreement is needed. This may require using a deeper empirical basis,<ref name="nosolidgrown" group="upper-alpha"/> hidden within the current empirical basis, to make sure that the properties or values used in the falsifier were obtained correctly ({{harvnb|Andersson|2016}} gives some examples).

Popper says that despite the fact that the empirical basis can be shaky, more comparable to a swamp than to solid ground,<ref name="nosolidgrown" group="upper-alpha"/> the definition that is given above is simply the formalization of a natural requirement on scientific theories, without which the whole logical process of science<ref name="fourlinesoftesting" group=upper-alpha/> would not be possible.

===Initial condition and prediction in falsifiers of laws===

In his analysis of the scientific nature of universal laws, Popper arrived at the conclusion that laws must "allow us to deduce, roughly speaking, more ''empirical'' singular statements than we can deduce from the initial conditions alone."{{sfn|Popper|1959|pp=[{{Google book|id=LWSBAgAAQBAJ|plainurl=yes|page=64}} 64–65]}} A singular statement that has one part only cannot contradict a universal law. A falsifier of a law has always two parts: the initial condition and the singular statement that contradicts the prediction.

However, there is no need to require that falsifiers have two parts in the definition itself. This removes the requirement that a falsifiable statement must make prediction. In this way, the definition is more general and allows the basic statements themselves to be falsifiable.{{sfn|Popper|1959|pp=[{{Google book|id=LWSBAgAAQBAJ|plainurl=yes|page=64}} 64–65]}} Criteria that require that ''a law'' must be predictive, just as is required by falsifiability (when applied to laws), Popper wrote, "have been put forward as criteria of the meaningfulness of sentences (rather than as criteria of demarcation applicable to theoretical systems) again and again after the publication of my book, even by critics who pooh-poohed my criterion of falsifiability."{{sfn|Popper|1959|loc=[{{Google book|id=LWSBAgAAQBAJ|plainurl=yes|page=65}} p. 65 Footnote *1]}}

===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**-->