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A proof is sufficient
evidence Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidenc ...
or a sufficient
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
for the
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belief ...
of a
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written
communication Communication (from la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inqui ...
such as
conversation Conversation is interactive communication between two or more people. The development of conversational skills and etiquette is an important part of socialization. The development of conversational skills in a new language is a frequent focus ...
, dialog,
rhetoric Rhetoric () is the art of persuasion, which along with grammar and logic (or dialectic), is one of the three ancient arts of discourse. Rhetoric aims to study the techniques writers or speakers utilize to inform, persuade, or motivate par ...
, etc., a proof is a
persuasive Persuasion or persuasion arts is an umbrella term for influence. Persuasion can influence a person's beliefs, attitudes, intentions, motivations, or behaviours. Persuasion is studied in many disciplines. Rhetoric studies modes of persuasio ...
perlocutionary
speech act In the philosophy of language and linguistics, speech act is something expressed by an individual that not only presents information but performs an action as well. For example, the phrase "I would like the kimchi; could you please pass it to me? ...
, which demonstrates the truth of a proposition. In any area of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
defined by its assumptions or axioms, a proof is an argument establishing a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
of that area via accepted rules of inference starting from those axioms and from other previously established theorems. The subject of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, in particular
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, formalizes and studies the notion of
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seq ...
. In some areas of
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Epi ...
and
theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing th ...
, the notion of justification plays approximately the role of proof, while in
jurisprudence Jurisprudence, or legal theory, is the theoretical study of the propriety of law. Scholars of jurisprudence seek to explain the nature of law in its most general form and they also seek to achieve a deeper understanding of legal reasoning ...
the corresponding term is
evidence Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidenc ...
, with "burden of proof" as a concept common to both
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
and law. In most disciplines, evidence is required to prove something. Evidence is drawn from the experience of the world around us, with
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
obtaining its evidence from
nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
, law obtaining its evidence from
witness In law, a witness is someone who has knowledge about a matter, whether they have sensed it or are testifying on another witnesses' behalf. In law a witness is someone who, either voluntarily or under compulsion, provides testimonial evidence, e ...
es and forensic investigation, and so on. A notable exception is mathematics, whose proofs are drawn from a mathematical world begun with axioms and further developed and enriched by theorems proved earlier. Exactly what evidence is sufficient to prove something is also strongly area-dependent, usually with no absolute threshold of sufficiency at which evidence becomes proof. In law, the same evidence that may convince one
jury A jury is a sworn body of people (jurors) convened to hear evidence and render an impartial verdict (a finding of fact on a question) officially submitted to them by a court, or to set a penalty or judgment. Juries developed in England d ...
may not persuade another.
Formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seq ...
provides the main exception, where the criteria for proofhood are ironclad and it is impermissible to defend any step in the reasoning as "obvious" (except for the necessary ability of the one proving and the one being proven to, to correctly identify any symbol used in the proof.); for a
well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...
to qualify as part of a formal proof, it must be the result of applying a rule of the deductive apparatus of some
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...
to the previous well-formed formulae in the proof sequence. Proofs have been presented since antiquity.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
used the observation that patterns of nature never display the machine-like uniformity of
determinism Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
as proof that chance is an inherent part of nature. On the other hand,
Thomas Aquinas Thomas Aquinas, Dominican Order, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino, Italy, Aquino; 1225 – 7 March 1274) was an Italian Dominican Order, Dominican friar and Catholic priest, priest who was an influential List of Catholic philo ...
used the observation of the existence of rich patterns in nature as proof that nature is ''not'' ruled by chance. Proofs need not be verbal. Before
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulat ...
, people took the apparent motion of the Sun across the sky as proof that the Sun went round the Earth. Suitably incriminating evidence left at the scene of a
crime In ordinary language, a crime is an unlawful act punishable by a state or other authority. The term ''crime'' does not, in modern criminal law, have any simple and universally accepted definition,Farmer, Lindsay: "Crime, definitions of", in C ...
may serve as proof of the identity of the perpetrator. Conversely, a verbal entity need not assert a proposition to constitute a proof of that proposition. For example, a
signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
constitutes direct proof of authorship; less directly, handwriting analysis may be submitted as proof of authorship of a document.
Privileged information Privileged may refer to: Film and television * ''Privileged'' (TV series), a 2008 US television series * ''Privileged'' (film), a 1982 Hollywood film Other uses * Immunologically privileged site, a body location where immune response to ...
in a document can serve as proof that the document's author had access to that information; such access might in turn establish the location of the author at certain time, which might then provide the author with an alibi.


Proof vs evidence

18th-century Scottish philosopher
David Hume David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" '' Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment ph ...
built on
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
's separation of
belief A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to tak ...
from
knowledge Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distin ...
, recognizing that one can be said to "know" something only if one has firsthand experience with it, in a strict sense proof, while one can infer that something is true and therefore "believe" it without knowing, via evidence or supposition. This speaks to one way of separating proof from evidence: If one cannot find their chocolate bar, and sees chocolate on their napping roommate's face, this evidence can cause one to believe their roommate ate the chocolate bar. But they do not ''know'' their roommate ate it. It may turn out that the roommate put the candy away when straightening up, but was thus inspired to go eat their own chocolate. Only if one directly experiences ''proof'' of the roommate eating it, perhaps by walking in on them doing so, does one ''know'' the roommate did it. In an absolute sense, one can be argued not to "know" anything, except for the existence of one's own thoughts, as 17th-century philosopher
John Locke John Locke (; 29 August 1632 – 28 October 1704) was an English philosopher and physician, widely regarded as one of the most influential of Enlightenment thinkers and commonly known as the "father of liberalism". Considered one of ...
pointed out.Locke: Knowledge of the External World
/ref> Even earlier, Descartes addressed when saying ''cogito, ergo sum'' (I think, therefore I am). While Descartes was attempting to "prove" logically that the world exists, his legacy in doing so is to have shown that one cannot have such proof, because all of one's perceptions could be false (such as under the evil demon or
simulated reality The simulation theory is the hypothesis that reality could be simulated—for example by quantum computer simulation—to a degree indistinguishable from "true" reality. It could contain conscious minds that may or may not know that they live i ...
hypotheses). But one at least has proof of one's own thoughts existing, and strong evidence that the world exists, enough to be considered "proof" by practical standards, though always indirect and impossible to objectively confirm.


See also

*
Mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...
*
Proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
* Proof of concept * Provability logic *
Evidence Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidenc ...
, information which tends to determine or demonstrate the truth of a proposition *
Proof procedure In logic, and in particular proof theory, a proof procedure for a given logic is a systematic method for producing proofs in some proof calculus of (provable) statements. Types of proof calculi used There are several types of proof calculi. The mo ...
*
Proof complexity In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. ...
* Standard of proof


References

{{DEFAULTSORT:Proof (Truth) Automated theorem proving Computational complexity theory Concepts in epistemology Evidence Formal languages Formal systems Legal reasoning Logical truth Metatheory Proof theory