Proof (truth)
Encyclopedia
A proof is sufficient evidence
or argument
for the truth
of a proposition
.
The concept arises in a variety of areas, 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
such as conversation
, dialog, rhetoric
, etc., a proof is a persuasive perlocutionary
speech act
, which demonstrates the truth of a proposition. In any area of mathematics
defined by its assumptions or axioms, a proof is an argument
establishing a theorem
of that area via accepted rules of inference starting from those axioms and other previously established theorems. The subject of logic
, in particular proof theory
, formalizes and studies the notion of formal proof
. In the areas of epistemology
and theology
, the notion of justification
plays approximately the role of proof, while in jurisprudence
the corresponding term is evidence
, with burden of proof as a concept common to both philosophy
and law.
In most areas, evidence is drawn from experience of the world around us, with science
obtaining its evidence from nature, law obtaining its evidence from witness
es and forensic investigation
, and so on. A notable exception is mathematics, whose evidence is drawn from a mathematical world begun with postulates and further developed and enriched by theorems proved earlier.
As with evidence itself, the criteria for sufficiency of evidence are also strongly area-dependent, usually with no absolute threshold of sufficiency at which evidence becomes proof. The same evidence that may convince one jury
may not persuade another. Formal proof
provides the main exception, where the criteria for proofhood are ironclad and it is impermissible to defend any step in the reasoning as "obvious"; for a well-formed formula
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
to the previous well-formed formulae in the proof sequence.
Proofs have been presented since antiquity. Aristotle
used the observation that patterns of nature never display the machine-like uniformity of determinism
as proof that chance is an inherent part of nature. On the other hand, Thomas Aquinas
used the observation of the existence of rich patterns in nature as proof that nature is not ruled by chance. Augustine of Hippo
provides a good case study in early uses of informal proofs in theology. He argued that given the assumption that Christ had risen
, there is resurrection of the dead and he provided further arguments to prove that the death of Jesus was for the salvation of man.
Proofs need not be verbal. Before Galileo, 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
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
constitutes direct proof of authorship; less directly, handwriting analysis may be submitted as proof of authorship of a document. Privileged information 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
.
Evidence
Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...
or argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
for the truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
of a proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
.
The concept arises in a variety of areas, 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
Communication is the activity of conveying meaningful information. Communication requires a sender, a message, and an intended recipient, although the receiver need not be present or aware of the sender's intent to communicate at the time of communication; thus communication can occur across vast...
such as conversation
Conversation
Conversation is a form of interactive, spontaneous communication between two or more people who are following rules of etiquette.Conversation analysis is a branch of sociology which studies the structure and organization of human interaction, with a more specific focus on conversational...
, dialog, rhetoric
Rhetoric
Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...
, etc., a proof is a persuasive perlocutionary
Perlocutionary act
A perlocutionary act is a speech act, as viewed at the level of its psychological consequences, such as persuading, convincing, scaring, enlightening, inspiring, or otherwise getting someone to do or realize something...
speech act
Speech act
Speech Act is a technical term in linguistics and the philosophy of language. The contemporary use of the term goes back to John L. Austin's doctrine of locutionary, illocutionary, and perlocutionary acts...
, which demonstrates the truth of a proposition. In any area of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
defined by its assumptions or axioms, a proof is an argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
establishing a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
of that area via accepted rules of inference starting from those axioms and other previously established theorems. The subject of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, in particular proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, formalizes and studies the notion of formal proof
Formal proof
A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...
. In the areas of epistemology
Theory of justification
Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs. Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification, warrant, rationality, and probability...
and theology
Existence of God
Arguments for and against the existence of God have been proposed by philosophers, theologians, scientists, and others. In philosophical terms, arguments for and against the existence of God involve primarily the sub-disciplines of epistemology and ontology , but also of the theory of value, since...
, the notion of justification
Justification
Justification may refer to:*Theory of justification, a part of epistemology that attempts to understand the justification of propositions and beliefs*Justification , defence in a prosecution for a criminal offense...
plays approximately the role of proof, while in jurisprudence
Jurisprudence
Jurisprudence is the theory and philosophy of law. Scholars of jurisprudence, or legal theorists , hope to obtain a deeper understanding of the nature of law, of legal reasoning, legal systems and of legal institutions...
the corresponding term is evidence
Evidence (law)
The law of evidence encompasses the rules and legal principles that govern the proof of facts in a legal proceeding. These rules determine what evidence can be considered by the trier of fact in reaching its decision and, sometimes, the weight that may be given to that evidence...
, with burden of proof as a concept common to both philosophy
Philosophic burden of proof
The philosophic burden of proof is the obligation on a party in an epistemic dispute to provide sufficient warrant for their position.-Holder of the burden:When debating any issue, there is an implicit burden of proof on the person asserting a claim...
and law.
In most areas, evidence is drawn from experience of the world around us, with science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
obtaining its evidence from nature, law obtaining its evidence from witness
Witness
A witness is someone who has firsthand knowledge about an event, or in the criminal justice systems usually a crime, through his or her senses and can help certify important considerations about the crime or event. A witness who has seen the event first hand is known as an eyewitness...
es and forensic investigation
Forensics
Forensic science is the application of a broad spectrum of sciences to answer questions of interest to a legal system. This may be in relation to a crime or a civil action...
, and so on. A notable exception is mathematics, whose evidence is drawn from a mathematical world begun with postulates and further developed and enriched by theorems proved earlier.
As with evidence itself, the criteria for sufficiency of evidence are also strongly area-dependent, usually with no absolute threshold of sufficiency at which evidence becomes proof. The same evidence that may convince one jury
Jury
A jury is a sworn body of people convened to render an impartial verdict officially submitted to them by a court, or to set a penalty or judgment. Modern juries tend to be found in courts to ascertain the guilt, or lack thereof, in a crime. In Anglophone jurisdictions, the verdict may be guilty,...
may not persuade another. Formal proof
Formal proof
A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system...
provides the main exception, where the criteria for proofhood are ironclad and it is impermissible to defend any step in the reasoning as "obvious"; for a well-formed formula
Well-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
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
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
to the previous well-formed formulae in the proof sequence.
Proofs have been presented since antiquity. Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
used the observation that patterns of nature never display the machine-like uniformity of determinism
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...
as proof that chance is an inherent part of nature. On the other hand, Thomas Aquinas
Thomas Aquinas
Thomas Aquinas, O.P. , also Thomas of Aquin or Aquino, was an Italian Dominican priest of the Catholic Church, and an immensely influential philosopher and theologian in the tradition of scholasticism, known as Doctor Angelicus, Doctor Communis, or Doctor Universalis...
used the observation of the existence of rich patterns in nature as proof that nature is not ruled by chance. Augustine of Hippo
Augustine of Hippo
Augustine of Hippo , also known as Augustine, St. Augustine, St. Austin, St. Augoustinos, Blessed Augustine, or St. Augustine the Blessed, was Bishop of Hippo Regius . He was a Latin-speaking philosopher and theologian who lived in the Roman Africa Province...
provides a good case study in early uses of informal proofs in theology. He argued that given the assumption that Christ had risen
Resurrection of Jesus
The Christian belief in the resurrection of Jesus states that Jesus returned to bodily life on the third day following his death by crucifixion. It is a key element of Christian faith and theology and part of the Nicene Creed: "On the third day he rose again in fulfillment of the Scriptures"...
, there is resurrection of the dead and he provided further arguments to prove that the death of Jesus was for the salvation of man.
Proofs need not be verbal. Before Galileo, people took the apparent motion of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
across the sky as proof that the Sun went round the Earth
Geocentric model
In astronomy, the geocentric model , is the superseded theory that the Earth is the center of the universe, and that all other objects orbit around it. This geocentric model served as the predominant cosmological system in many ancient civilizations such as ancient Greece...
. Suitably incriminating evidence left at the scene of a crime
Crime
Crime is the breach of rules or laws for which some governing authority can ultimately prescribe a conviction...
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
Signature
A signature is a handwritten depiction of someone's name, nickname, or even a simple "X" that a person writes on documents as a proof of identity and intent. The writer of a signature is a signatory. Similar to a handwritten signature, a signature work describes the work as readily identifying...
constitutes direct proof of authorship; less directly, handwriting analysis may be submitted as proof of authorship of a document. Privileged information 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
Alibi
Alibi is a 1929 American crime film directed by Roland West. The screenplay was written by West and C. Gardner Sullivan, who adapted the 1927 Broadway stage play, Nightstick, written by Elaine Sterne Carrington, J.C...
.
See also
- Mathematical proofMathematical proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
- Proof of conceptProof of conceptA proof of concept or a proof of principle is a realization of a certain method or idea to demonstrate its feasibility, or a demonstration in principle, whose purpose is to verify that some concept or theory that has the potential of being used...
- Provability logicProvability logicProvability logic is a modal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic....
- EvidenceEvidenceEvidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or were themselves proven via evidence, to demonstrate an assertion's truth...
, information which tends to determine or demonstrate the truth of a proposition - Proof procedureProof procedureIn 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 statements....
- Proof complexityProof complexityIn computer science, proof complexity is a measure of efficiency of automated theorem proving methods that is based on the size of the proofs they produce. The methods for proving contradiction in propositional logic are the most analyzed...