Abductive reasoning

About: Abductive reasoning is a research topic. Over the lifetime, 1917 publications have been published within this topic receiving 44645 citations. The topic is also known as: abduction & abductive inference.

On Practical Abduction

Abstract: In this paper practical reasoning is understood in the Aristotelian sense as reasoning leading to action or to an intention to do something. Georg Henrik von Wright and a number of other philosophers have tried to assimilate certain forms of such reasoning to deductive reasoning. Many examples of practical reasoning, including some examples given by Aristotle, do not fit a deductive or quasi-deductive model. It is argued that instances of good practical reasoning often resemble abductive rather than deductive or inductive reasoning, and that the principles governing abduction, including the Principle of Economy, are applicable to practical reasoning.

Meta-Logical Reasoning in Higher-Order Logic

Abstract: The semantics of first-order logic (FOL) can be described in the meta-language of higher-order logic (HOL). Using HOL one can prove key properties of FOL such as soundness and completeness. Furthermore, one can prove sentences in FOL valid using the formalized FOL semantics. To aid in the construction of the proof an interactive proof assistant like Isabelle can be used. The proof assistant can even automate simple proofs using the formalized FOL semantics. Introduction In textbooks the language of first-order logic (FOL) is usually presented in English. FOL is the object language, since it is the logic that is described, and English is the meta-language, since it is the language that describes the object language. However, instead of using English as meta-language we can also use a meta-logic. We will show how higher-order logic (HOL) can be used as a meta-logic to describe and reason about FOL, which is possible as HOL is much more expressive than FOL, cf. Farmer (2008). Logic itself is about formalizing which arguments are valid. Thus in FOL we have a clear definition of which theorems are valid and which are not. However, it is also interesting to prove theorems about FOL, for instance the soundness and completeness of a proof system for FOL. By using HOL as metalanguage we can ensure that there is also a clear definition of which theorems about FOL are valid, cf. Harrison (1998). Furthermore, we can show the theorems about FOL to be valid by proving them in a sound proof system for HOL. Proof systems for HOL have been implemented in interactive proof assistants which are computer programs that can help their users in proving theorems. In the following we will use the Isabelle proof assistant, cf. Nipkow (2002). In addition to helping the users to construct correct proofs, the proof assistants can in some cases even do the proofs automatically. Formalization in Isabelle We consider a formalization in Isabelle of FOL with only binary predicates: theory Semantics imports Main begin type_synonym id = string datatype tm = Var nat | Con id datatype fm = Falsity | Pre id tm tm | Imp fm fm | Uni fm primrec val :: "(nat => ’u) => (id => ’u) => tm => ’u" where "val e f (Var v) = e v" | "val e f (Con c) = f c" primrec sem :: "(nat => ’u) => (id => ’u) => (id => ’u * ’u => bool) => fm => bool" where "sem e f g Falsity = False" | "sem e f g (Pre s a b) = g s (val e f a, val e f b)" | "sem e f g (Imp p q) = (if sem e f g p then sem e f g q else True)" | "sem e f g (Uni p) = (!x. sem (%n. if n=0 then x else e (n 1)) f g p)" end The terms and formulas of the FOL language are defined as the datatypes tm and fm, respectively. Variables are indexed using de Bruijn indices. The semantics of the language is defined using the function sem where e is the environment, i.e. a mapping of variables to elements of the universe ’u, f maps constants to elements of ’u, and g gives the semantics of the predicates. Most of the cases of sem should be self-explanatory, but the Uni case is complicated. The details are not important here, but it uses the universal quantifier (!) to consider all values of the universe ’u. It also uses the lambda operator (%) to keep track of the indices of the variables. Examples of Meta-Logical Reasoning By using HOL as the meta-language for FOL it is possible to make use of proof assistants such as Isabelle to reason about certain properties of FOL. For example, let syn be a proof system for FOL implemented as a predicate in Isabelle of type fm => bool (an inductive definition). If it is sound then we can prove this in Isabelle as the theorem syn fml ==> !e f g. sem e f g fml. Likewise completeness (!e f g. sem e f g fml) ==> syn fml can be proved if the proof system is indeed complete. For example, in Berghofer (2007) a natural deduction proof system for FOL is proven sound and complete. A tool for teaching logic based on natural deduction has recently been developed and proved sound by Villadsen (2015). Meta-logical reasoning using the formalized semantics also enables formal proofs of sentences using only the semantics of the object language. For example, we wish to show that the sentence ∀x.∀y. P (y, x) → P (y, x) is valid. We can prove this by first fixing two arbitrary elements u, w of the universe and show P (y, x) → P (y, x) for an arbitrary environment updated to map variable x to u and y to w, and then use the fact that any denotation g of predicates maps (P, u, w) to either true or false. In both cases P (y, x) → P (y, x) holds. This proof sketch can then be extended to a readable proof in Isabelle. The sentence can even be proved automatically in Isabelle using the formalized semantics: theorem "!e f g. sem e f g (Uni (Uni (Imp (Pre ’’P’’ (Var 0) (Var 1)) (Pre ’’P’’ (Var 0) (Var 1)))))" by auto On the other hand the sentence ∀x.∀y. P (x, y) → P (y, x) is not valid. Therefore we can ask Isabelle to search for a counterexample, using the nitpick command, and it is able to find one. Finally, the approach we have presented can even be used with HOL as the object language, that is, the semantics of HOL and a proof system for HOL can also be formalized in HOL, most extensively by Kumar (2014), although of course self-verification is not possible due to the second incompleteness theorem. Paulson (2014) has recently formalized the incompleteness theorems in Isabelle.

Practical reasoning : International Conference on Formal and Applied Practical Reasoning, FAPR '96, Bonn, Germany, June 3-7, 1996 : proceedings

Abstract: Integrating statistical audit evidence with belief function theory.- A comparative survey of default logic variants.- Modal logics with relative accessibility relations.- Geometrical structures and modal logic.- A unified framework for hypothetical and practical reasoning (1): Theoretical foundations.- A unified framework for hypothetical and practical reasoning (2): Lessons from medical applications.- General domain circumscription and its first-order reduction.- Reasoning about rational, but not logically omniscient agents (extended abstract).- Specification of nonmonotonic reasoning.- Intelligent agents in the Situation Calculus: An application to user modelling.- Talkin'bout consistency, or: When logically possible becomes possible.- The analysis and evaluation of legal argumentation from a pragma-dialectical perspective.- Reasoning about reasoning.- A resolution-based proof method for temporal logics of knowledge and belief.- A methodology for iterated theory change.- A formal framework for causal modeling and argumentation.- Goals in argumentation.- An abductive proof procedure for conditional logic programming.- Commands in dialogue logic.- Ideal and real belief about belief.- Analogical reasoning of organic reactions based on the structurized compound-reaction diagram.- Labelling ideality and subideality.- Mind, morals, and reasons.- Aristotle, Whately, and the taxonomy of fallacies.- Nonmonotonic reasoning with multiple belief sets.- SEdit - Graphically validating technical systems.- The need for a dialectical tier in arguments.- Two kinds of non-monotonic analogical inference.- The normative reconstruction of analogy argumentation in judicial decisions: A pragma-dialectical perspective.- Formal reasoning about modules, reuse and their correctness.- A tableau calculus for first-order branching time logic.- Possible world semantics for analogous reasoning.- Using temporary integrity constraints to optimize databases.- Graded inheritance nets for knowledge representation.- Defining normative systems for qualitative argumentation.- Complex argumentation in judicial decisions. Analysing conflicting arguments.- Combining partitions and modal logic for user modeling.- Reason in a changing world.- A system for defeasible argumentation, with defeasible priorities.- Modal logic for modelling actions and agents.- Formalization of reasoning about default action (preliminary report).- An architecture for argumentative dialogue planning.- Skeptical query-answering in Constrained Default Logic.- Type theoretic semantics for SemNet.- From syllogisms to audiences: The prospects for logic in a rhetorical model of argumentation.- Human reasoning with negative defaults.- On the semantics of the unknown.- System J - Revision entailment.- Deep disagreements and public demoralization.- Practical reasoning with procedural knowledge.- Towards the assessment of logics for concurrent actions.- Default reasoning and belief revision in the CIN Project.- Mechanizing multi-agent reasoning with belief contexts.- Arguments and mental models: A position paper.- Argumentation and decision making: A position paper.- The implementation of LENA.- The implementation of CondLP.- How to reason about akratic action practically?.

Abductive inference and invalidity

Abstract: L'A. defend la these selon laquelle l'inference abductive, definie comme inference a partir de la meilleure explication, appartient au raisonnement deductif et s'inscrit au coeur de la methodologie deductiviste. Examinant la dimension negative de la refutation deductive qui consiste en la determination de l'invalidite de l'argument utilise, l'A. mesure les consequences de ces pretensions negatives au regard de la methode philosophique et de la possibilite meme de la connaissance, d'une part, et montre que le deductivisme n'aboutit pas au scepticisme mais a l'inconsistance, d'autre part