Deductive reasoning Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from, i.e., there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning. Simple example An example of a deductive argument: All men are mortal.Socrates is a man.Therefore, Socrates is mortal. Law of detachment P → Q.
Abductive reasoning Form of logical inference which seeks the simplest and most likely explanation Abductive reasoning (also called abduction, abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century. It starts with an observation or set of observations and then seeks the simplest and most likely conclusion from the observations. This process, unlike deductive reasoning, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of 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, computer science, and artificial intelligence research spurred renewed interest in the subject of abduction. Diagnostic expert systems frequently employ abduction. Deduction, induction, and abduction .
Logical reasoning Informally, two kinds of logical reasoning can be distinguished in addition to formal deduction: induction and abduction. Given a precondition or premise, a conclusion or logical consequence and a rule or material conditional that implies the conclusion given the precondition, one can explain that: Deductive reasoning determines whether the truth of a conclusion can be determined for that rule, based solely on the truth of the premises. Example: "When it rains, things outside get wet. The grass is outside, therefore: when it rains, the grass gets wet." Mathematical logic and philosophical logic are commonly associated with this style of reasoning.Inductive reasoning attempts to support a determination of the rule. See also References T.
Proof assistant An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. 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. Comparison of systems User interface See also Notes Jump up ^ Hunt, Warren; Matt Kaufmann; Robert Bellarmine Krug; J Moore; Eric W. References Henk Barendregt and Herman Geuvers (2001). External links Catalogues
Defeasible reasoning Defeasible reasoning is a kind of reasoning that is based on reasons that are defeasible, as opposed to the indefeasible reasons of deductive logic. Defeasible reasoning is a particular kind of non-demonstrative reasoning, where the reasoning does not produce a full, complete, or final demonstration of a claim, i.e., where fallibility and corrigibility of a conclusion are acknowledged. In other words defeasible reasoning produces a contingent statement or claim. Other kinds of non-demonstrative reasoning are probabilistic reasoning, inductive reasoning, statistical reasoning, abductive reasoning, and paraconsistent reasoning. Defeasible reasoning is also a kind of ampliative reasoning because its conclusions reach beyond the pure meanings of the premises. The differences between these kinds of reasoning correspond to differences about the conditional that each kind of reasoning uses, and on what premise (or on what authority) the conditional is adopted: History Specificity
Inference Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Statistical inference uses mathematics to draw conclusions in the presence of uncertainty. Definition 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. Examples Example for definition #1 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. Now we turn to an invalid form. ? (where ? ?
Subjective logic Subjective logic is a type of probabilistic logic that explicitly takes uncertainty and belief ownership into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and incomplete knowledge. For example, it can be used for modeling trust networks and for analysing Bayesian networks. Arguments in subjective logic are subjective opinions about propositions. A binomial opinion applies to a single proposition, and can be represented as a Beta distribution. A fundamental aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about the world is true or false. Subjective opinions Subjective opinions express subjective beliefs about the truth of propositions with degrees of uncertainty, and can indicate subjective belief ownership whenever required. where is the subject, also called the belief owner, and is the proposition to which the opinion applies. . Binomial opinions
Logical consequence 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 . Formal accounts The most widely prevailing view on how to best 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 If you know that follows logically from no information about the possible interpretations of or will affect that knowledge. is a logical consequence of cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. Proofs and models Syntactic consequence A formula of a set of formulas if there is a formal proof in of
Knowledge representation and reasoning Knowledge representation and reasoning (KR) is the field of artificial intelligence (AI) devoted to representing information about the world in a form that a computer system can utilize to solve complex tasks such as diagnosing a medical condition or having a dialog in a natural language. Knowledge representation incorporates findings from psychology about how humans solve problems and represent knowledge in order to design formalisms that will make complex systems easier to design and build. Knowledge representation and reasoning also incorporates findings from logic to automate various kinds of reasoning, such as the application of rules or the relations of sets and subsets. Examples of knowledge representation formalisms include semantic nets, Frames, Rules, and ontologies. Examples of automated reasoning engines include inference engines, theorem provers, and classifiers. Overview This hypothesis was not always taken as a given by researchers. History Characteristics