Differences between law, theory and theorem
These concepts from the legal and philosophical fields may have points in common, but they are not equivalent.
What is a law, what is a theory, and what are the theorems? These concepts are handled daily in academic environments, institutes and universities, but sometimes we are not clear about the differences and what each of them means. Are theories and laws irrefutable? What is a theorem based on to be considered as such?
In this article we explain what is the meaning of concepts such as law, theory and theorem, and what are their main differences.
What is a theorem?
A theorem consists of a proposition or statement whose validity or "truth" can be demonstrated within a logical framework. and from the inclusion of axioms or other theorems that have been previously validated or demonstrated.
Axioms or axiomatic sets are propositions or statements so evident that it is considered that they do not need any demonstration to be considered valid. For example, when we want to play a game of chess, the rules of this game constitute an axiomatic system, since both participants take its validity for granted without it being questioned at any time.
In order to consider a theorem as valid, it has to be proved by means of a procedure and rules of inference, which are used to deduce from one or more premises (statements or ideas that serve as a basis for reasoning and a subsequent deduction), a valid conclusion. However, until an assertion is demonstrated, it is defined as a hypothesis or conjecture.
In mathematics, for example, a theorem is proved true by the application of logical operations and arguments.. One of the best known, the Pythagorean theorem, states that in any right triangle (one that has an angle of 90º) its hypotenuse (the side with the longest length) can be calculated in relation to the value of its legs (the sides that form the 90º angle).
What is a theory?
A theory is a logically structured system of knowledge, established on the basis of a set of axioms, empirical data and postulates.Its objective is to state under what conditions certain assumptions are generated; that is, to try to describe, explain and understand a part of the objective reality or of a particular scientific field.
Theories can be developed from different starting points: conjectures, which are suppositions or ideas that do not have empirical support, i.e., they are not supported by observation; and hypotheses, which are supported by different observations and empirical data. However, a theory cannot be inferred solely from one or several axioms within a logical system, as is the case with theorems.
The function of a theory is to explain reality (or at least part of it), to answer basic questions (such as what, how, when or where the phenomenon to be understood and explained occurs) and to order this reality into a series of understandable and accessible concepts and ideas.
The set of rules that constitute a theory must be capable of describing and predicting the behavior of a given system.. For example, Charles Darwin's theory of evolution explains how living beings have a specific origin and slowly change and evolve, and how these changes cause different species to emerge from the same ancestor, in what he came to call natural selection.
In science, theories are built using the hypotheticodeductive system or method, which consists of the following steps:

The phenomenon to be studied is observed.

One or more hypotheses are generated to explain the phenomenon.

Taking the hypothesis/s as a starting point, the most basic consequences or statements are deduced.

These statements are tested and validated by comparing them with the empirical data emanating from observation and experience.
Law: definition and characteristics
By law we mean a rule, a standard or a set of standards, which describe the relationships that exist between the components involved in a particular phenomenon or system. or a particular system. Although in popular culture it is common to think of laws as a kind of universal and absolute truths (above theories), this is not exactly so.
Laws, in the realm of science, must be invariant rules (that cannot be modified), universal (that must be valid for all elements of the phenomenon they describe) and necessary (that must be sufficient by themselves to describe the phenomenon in question). However, a law is considered as a particular rule, present in all theories (hence its universality), not as an assumption of higher rank.
For example, in a science like physics, there are multiple theories that explain certain phenomena and realities; the theory of quantum mechanics (which explains the nature of the smallest), the theory of special relativity or the theory of general relativity (both necessary to explain the nature of the largest). All of them share a common law: the conservation of energy, as a particular and universal rule in all three theories.
All in all, the laws maintain their provisional status and can be disprovedThe laws can be disproved, since in science nothing is absolute or written in stone, and any statement, be it a theory or a law, can be dismantled with the necessary proofs and the pertinent demonstration.
Differences between theorem, theory and law
The differences between the concepts of theorem, theory and law can be somewhat blurred, but let us look at some of them.
Regarding the difference between a theorem and a theory, the following should be noted: while a theory can be defined on the basis of a pattern of natural events or phenomena that cannot be demonstrated using an axiom or a set of basic statements, a theorem is a proposition of an event or phenomenon that can be determined from a set of axioms, within a logical framework or criterion.
Another subtle difference between theory and law is that, although both are based on hypotheses and empirical data, the theory is established to explain a phenomenon, theory is established to explain an observed phenomenon, while laws seek to describe such a phenomenon.. For example, Kepler described mathematically the motion of the planets in their orbits around the sun, formulating the wellknown Kepler's Laws; however, these do not provide an explanation of planetary motions.
Finally, it is worth noting a basic difference between the concepts of theorem and law, which is that a theorem is composed of demonstrable propositions (by means of axioms, in a logical system); and a law is a series of established, constant and invariable rules, based on observations and empirical data that can be validated or refuted.
Bibliographic references:

AcevedoDíaz, J. A., VázquezAlonso, Á., ManasseroMas, M. A., & AcevedoRomero, P. (2007). Consensus on the nature of science: epistemological aspects. Eureka journal on science education and popularization, 4(2), 202225.

Chalmers, A. F., Villate, J. A. P., Máñez, P. L., & Sedeño, E. P. (2000). What is that thing called science? Madrid: siglo XXI.
(Updated at Mar 28 / 2023)