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 retreat 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 . , where . . . entails

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 ? ?

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

Incorporeality State or quality of being bodiless Incorporeality is "the state or quality of being incorporeal or bodiless; immateriality; incorporealism. In the problem of universals, universals are separable from any particular embodiment in one sense, while in another, they seem inherent nonetheless. The notion that a causally effective incorporeal body is even coherent requires the belief that something can affect what's material, without physically existing at the point of effect. Philosophy[edit] Plato depicted by Raphael. Pre-Socratic[edit] "The Love and Strife of Empedokles are no incorporeal forces. "Zeller holds, indeed, that Anaxagoras meant to speak of something incorporeal ; but he admits that he did not succeed in doing so, and that is historically the important point. On the whole of ancient philosophy and incorporeal, Zeller writes: Aristotelian[edit] Flannery in A Companion to Philosophy of Religion writes: Platonic[edit] Renehan (1980) writes: Theology[edit] See also[edit] References[edit]

Logical consequence Relationship between statements that hold true when one logically 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 . .

Infinity (philosophy) Philosophical concept ... It is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. This is often called potential infinity; however, there are two ideas mixed up with this. , which reads, "for any integer n, there exists an integer m > n such that P(m)". Sed omne continuum est actualiter existens. The parts are actually there, in some sense. Among the scholastics, Aquinas also argued against the idea that infinity could be in any sense complete or a totality. The Jain upanga āgama Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Enumerable: lowest, intermediate and highestInnumerable: nearly innumerable, truly innumerable and innumerably innumerableInfinite: nearly infinite, truly infinite, infinitely infinite Jain theory of numbers (See IIIrd section for various infinities) The Jains were the first to discard the idea that all infinities were the same or equal. ...

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 impetus 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

Information The ASCII codes for the word "Wikipedia" represented in binary, the numeral system most commonly used for encoding textual computer information In Thermodynamics, information is any kind of event that affects the state of a dynamic system that can interpret the information. Etymology[edit] The English word was apparently derived from the Latin stem (information-) of the nominative (informatio): this noun is derived from the verb informare (to inform) in the sense of "to give form to the mind", "to discipline", "instruct", "teach". The ancient Greek word for form was μορφή (morphe; cf. morph) and also εἶδος (eidos) "kind, idea, shape, set", the latter word was famously used in a technical philosophical sense by Plato (and later Aristotle) to denote the ideal identity or essence of something (see Theory of Forms). Information theory approach[edit] As sensory input[edit] Often information can be viewed as a type of input to an organism or system. As representation and complexity[edit]

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.