Inductive reasoning

Abductive reasoning Inference seeking the simplest and most likely explanation Abductive reasoning (also called abduction,[1] abductive inference,[1] or retroduction[2]) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by American philosopher and logician Charles Sanders Peirce beginning in the latter half of the 19th century. Abductive reasoning, unlike deductive reasoning, yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in terms such as "best available" or "most likely". In the 1990s, as computing power grew, the fields of law,[3] computer science, and artificial intelligence research[4] spurred renewed interest in the subject of abduction.[5] Diagnostic expert systems frequently employ abduction.[6] Deduction, induction, and abduction [edit] Deductive reasoning allows deriving from only where . from a body of knowledge . .

Proof assistant From Wikipedia, the free encyclopedia Software tool to assist with the development of formal proofs by human–machine collaboration In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.[1] A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh. Formalization extent [edit] Freek Wiedijk has been keeping a ranking of proof assistants by the amount of formalized theorems out of a list of 100 well-known theorems. Notable formalized proofs Catalogues

Inference Steps in reasoning Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. The process by which a conclusion is inferred from multiple observations is called inductive reasoning. This definition is disputable (due to its lack of clarity. Two possible definitions of "inference" are: A conclusion reached on the basis of evidence and reasoning.The process of reaching such a conclusion. Example for definition #1 [edit] Ancient Greek philosophers defined a number of syllogisms, correct three part inferences, that can be used as building blocks for more complex reasoning. All humans are mortal.All Greeks are humans.All Greeks are mortal. The validity of an inference depends on the form of the inference. Now we turn to an invalid form. ? (where ? ?

Logical consequence Relationship where one statement follows from another Logicians make precise accounts of logical consequence regarding a given language , either by constructing a deductive system for or by formal intended semantics for language . The most widely prevailing view on how best to account for logical consequence is to appeal to formality. All X are Y All Y are Z Therefore, all X are Z. This is in contrast to an argument like "Fred is Mike's brother's son. A priori property of logical consequence [edit] If it is known that follows logically from , then no information about the possible interpretations of or will affect that knowledge. is a logical consequence of cannot be influenced by empirical knowledge.[1] Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori.[1] However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. Syntactic consequence A formula of a set of from the set . . are true and

Automated theorem proving Subfield of automated reasoning and mathematical logic Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major motivating factor for the development of computer science. Logical foundations [edit] However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems (1931), showing that in any sufficiently strong axiomatic system, there are true statements that cannot be proved in the system. First implementations Decidability of the problem Depending on the underlying logic, the problem of deciding the validity of a formula varies from trivial to impossible. The above applies to first-order theories, such as Peano arithmetic. Proof assistants require a human user to give hints to the system. First-order theorem proving

Premise Statement supporting an argument An argument is meaningful for its conclusion only when all of its premises are true. If one or more premises are false, the argument says nothing about whether the conclusion is true or false. For instance, a false premise on its own does not justify rejecting an argument's conclusion; to assume otherwise is a logical fallacy called denying the antecedent. One way to prove that a proposition is false is to formulate a sound argument with a conclusion that negates that proposition. Key to evaluating the quality of an argument is determining if it is valid and sound. Aristotle held that any logical argument could be reduced to two premises and a conclusion.[2] Premises are sometimes left unstated, in which case, they are called missing premises, for example: Socrates is mortal because all men are mortal. It is evident that a tacitly understood claim is that Socrates is a man. Because all men are mortal and Socrates is a man, Socrates is mortal.

Argument In a typical deductive argument, the premises are meant to provide a guarantee of the truth of the conclusion, while in an inductive argument, they are thought to provide reasons supporting the conclusion's probable truth.[6] The standards for evaluating non-deductive arguments may rest on different or additional criteria than truth, for example, the persuasiveness of so-called "indispensability claims" in transcendental arguments,[7] the quality of hypotheses in retroduction, or even the disclosure of new possibilities for thinking and acting.[8] Formal and informal arguments[edit] Informal arguments as studied in informal logic, are presented in ordinary language and are intended for everyday discourse. Conversely, formal arguments are studied in formal logic (historically called symbolic logic, more commonly referred to as mathematical logic today) and are expressed in a formal language. Standard argument types[edit] Deductive arguments[edit] Validity[edit] For example: Soundness[edit]

Theoretical computer science Subfield of computer science and mathematics Theoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. TCS covers a wide variety of topics including algorithms, data structures, computational complexity, parallel and distributed computation, probabilistic computation, quantum computation, automata theory, information theory, cryptography, program semantics and verification, algorithmic game theory, machine learning, computational biology, computational economics, computational geometry, and computational number theory and algebra. While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. An algorithm is a step-by-step procedure for calculations. [edit] Computational geometry

Artificial intelligence Intelligence of machines Artificial intelligence (AI) is the capability of computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of research in engineering, mathematics and computer science that develops and studies methods and software that enable machines to perceive their environment and use learning and intelligence to take actions that maximize their chances of achieving defined goals. The traditional goals of AI research include learning, reasoning, knowledge representation, planning, natural language processing, and perception, as well as support for robotics. Goals The general problem of simulating (or creating) intelligence has been broken into subproblems. Reasoning and problem-solving Knowledge representation Knowledge representation and knowledge engineering[15] allow AI programs to answer questions intelligently and make deductions about real-world facts.