Editing Theory (section)

==Formality==
{{Main|Theory (mathematical logic)}}
Theories are [[analysis|analytical]] tools for [[understanding]], [[explanation|explaining]], and making [[prediction]]s about a given subject matter. There are theories in many and varied fields of study, including the arts and sciences. A formal theory is [[syntax (logic)|syntactic]] in nature and is only meaningful when given a [[semantics|semantic]] component by applying it to some content (e.g., [[fact]]s and relationships of the actual historical world as it is unfolding). Theories in various fields of study are often expressed in [[natural language]], but can be constructed in such a way that their general form is identical to a theory as it is expressed in the [[formal language]] of [[mathematical logic]]. Theories may be expressed mathematically, symbolically, or in common language, but are generally expected to follow principles of [[reason|rational thought]] or [[logic]].

Theory is constructed of a set of [[sentence (linguistics)|sentences]] that are thought to be true statements about the subject under consideration. However, the truth of any one of these statements is always relative to the whole theory. Therefore, the same statement may be true with respect to one theory, and not true with respect to another. This is, in ordinary language, where statements such as "He is a terrible person" cannot be judged as true or false without reference to some [[interpretation (logic)|interpretation]] of who "He" is and for that matter what a "terrible person" is under the theory.<ref name="curry">Curry, Haskell, ''Foundations of Mathematical Logic''</ref>

Sometimes two theories have exactly the same [[explanatory power]] because they make the same predictions. A pair of such theories is called indistinguishable or [[observational equivalence|observationally equivalent]], and the choice between them reduces to convenience or philosophical preference.{{cn|date=May 2023}}

The [[metatheory|form of theories]] is studied formally in mathematical logic, especially in [[model theory]]. When theories are studied in mathematics, they are usually expressed in some formal language and their statements are [[closure (mathematics)|closed]] under application of certain procedures called [[rules of inference]]. A special case of this, an axiomatic theory, consists of [[axioms]] (or axiom schemata) and rules of inference. A [[theorem]] is a statement that can be derived from those axioms by application of these rules of inference. Theories used in applications are [[abstraction]]s of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include [[arithmetic]] (abstracting concepts of number), [[geometry]] (concepts of space), and [[probability]] (concepts of randomness and likelihood).

[[Gödel's incompleteness theorem]] shows that no consistent, [[recursively enumerable]] theory (that is, one whose theorems form a recursively enumerable set) in which the concept of [[natural numbers]] can be expressed, can include all [[truth|true]] statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories. (Here, formalizing accurately and completely means that all true propositions—and only true propositions—are derivable within the mathematical system.) This limitation, however, in no way precludes the construction of mathematical theories that formalize large bodies of scientific knowledge.

===Underdetermination===
{{Main|Underdetermination}}
A theory is ''underdetermined'' (also called ''indeterminacy of data to theory'') if a rival, inconsistent theory is at least as consistent with the evidence. Underdetermination is an [[epistemology|epistemological]] issue about the relation of [[evidence]] to conclusions.{{cn|date=May 2023}}

A theory that lacks supporting evidence is generally, more properly, referred to as a [[hypothesis]].<ref>{{Cite web |title=This is the Difference Between a Hypothesis and a Theory |url=https://www.merriam-webster.com/grammar/difference-between-hypothesis-and-theory-usage |access-date=2024-04-08 |website=www.merriam-webster.com |language=en}}</ref>

===Intertheoretic reduction and elimination===
{{Main|Intertheoretic reduction}}
If a new theory better explains and predicts a phenomenon than an old theory (i.e., it has more [[explanatory power]]), we are [[Theory of justification|justified]] in believing that the newer theory describes reality more correctly. This is called an ''intertheoretic reduction'' because the terms of the old theory can be reduced to the terms of the new one. For instance, our historical understanding about ''sound'', ''light'' and ''heat'' have been reduced to ''wave compressions and rarefactions'', ''electromagnetic waves'', and ''molecular kinetic energy'', respectively. These terms, which are identified with each other, are called ''intertheoretic identities.'' When an old and new theory are parallel in this way, we can conclude that the new one describes the same reality, only more completely.

When a new theory uses new terms that do not reduce to terms of an older theory, but rather replace them because they misrepresent reality, it is called an ''intertheoretic elimination.'' For instance, the [[superseded scientific theories|obsolete scientific theory]] that put forward an understanding of heat transfer in terms of the movement of [[caloric fluid]] was eliminated when a theory of heat as energy replaced it. Also, the theory that [[phlogiston]] is a substance released from burning and rusting material was eliminated with the new understanding of the reactivity of oxygen.

===Versus theorems===
Theories are distinct from [[theorem]]s. A ''theorem'' is [[formal proof|derived]] deductively from [[axiom]]s (basic assumptions) according to a [[formal system]] of rules, sometimes as an end in itself and sometimes as a first step toward being tested or applied in a concrete situation; theorems are said to be true in the sense that the conclusions of a theorem are logical consequences of the axioms. ''Theories'' are abstract and conceptual, and are supported or challenged by observations in the world. They are '[[rigor]]ously tentative', meaning that they are proposed as true and expected to satisfy careful examination to account for the possibility of faulty inference or incorrect observation. Sometimes theories are incorrect, meaning that an explicit set of observations contradicts some fundamental objection or application of the theory, but more often theories are corrected to conform to new observations, by restricting the class of phenomena the theory applies to or changing the assertions made. An example of the former is the restriction of classical mechanics to phenomena involving macroscopic length scales and particle speeds much lower than the speed of light.