ConjectureIn mathematics, a conjecture is a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjecturesProofFormal mathematics is based on provable truth.
TheoremIn mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. 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 the axioms and previously proved theorems.
AxiomAn axiom, postulate, or assumption, is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study.
Natural languageA natural language or ordinary language is any spoken language or signed language used organically in a human community, first emerging without conscious premeditation and subject to: replication across generations of people in the community, regional expansion or contraction, and gradual internal and structural changes. The vast majority of languages in the world are natural languages.
Inductive reasoningInductive reasoning refers to a variety of methods of reasoning in which the conclusion of an argument is supported not with deductive certainty, but at best with some degree of probability. Unlike deductive reasoning (such as mathematical induction), where the conclusion is certain, given the premises are correct, inductive reasoning produces conclusions that are at best probable, given the premises provided.
LogicStudy of correct reasoningLogic studies valid forms of inference like modus ponens.Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory.
Deductive reasoningForm of reasoningDeductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is sound if it is valid and all its premises are true.
PropositionBearer of truth valuesPropositions are the meanings of declarative sentences, objects of beliefs, and bearers of truth values. They explain how different sentences, such as the English "Snow is white" and the German "Schnee ist weiß", can have identical meaning by expressing the same proposition. Similarly, they ground the fact that different people can share a belief by being directed at the same content.Argument–deduction–proof distinctionsArgument–deduction–proof distinctions originated with logic itself. Naturally, the terminology evolved.Empirical evidenceEmpirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. The terms evidence and empirical are used variably across different fields according to their methods and paradigms. In epistemology, evidence is what justifiesbeliefs or what determines whether holding a certain belief is rational.
InferenceInferences are steps in logical reasoning, moving from premises to logical consequences. Inference is traditionally divided into deduction and induction, a distinction that dates at least to Aristotle (300s BC). A third type of inference, abduction, has been proposed, notably by Charles Sanders Peirce. Deduction is inference derivinglogical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic.
RigourAdhering absolutely to certain constraints with consistency Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.
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