Analogy calculus
type: Item
instance of: scholarly article
title: Analogy calculus
zbMATH Open document ID: 0784.68083
DOI: 10.1016/0304-3975(93)90002-B
author: Bo Yi
author: Jiafu Xu
published in: Theoretical Computer Science
publication date: 1994-03-27T00:00:00Z
review text: The paper is devoted to the problem of analogical reasoning. The authors proposed a calculus \(LK_ A\), the system is based on Gentzen's \(LK\) calculus of sequents. First, two \(LK\)-rules are modified, then some additional initial segments, called equality axioms and analogy axioms are introduced. An analogy relation \(\sim\) is defined by the axioms (1) \(t\sim s\to s\sim t\) (2) \(t\sim s\), \(s\sim r\to t\sim r\), where \(t\), \(s\), and \(r\) are terms. Cut elimination and interpolation theorems are proven. It is shown that \(LK_ A\) is consistent and \(LK_ A\) is a conservative extension of \(LK_ e\), an equational logic calculus. Analogy correspondence is a set of sentences of the following forms: \[ \forall x_ 1\cdots\forall x_ n\forall y_ 1\cdots\forall y_ n\bigl(x_ 1\sim y_ 1\wedge\cdots\wedge x_ n\sim y_ n\supset f(x_ 1,\ldots,x_ n)\sim g(y_ 1,\ldots,y_ n)\bigr) \] \[ \forall x_ 1\cdots\forall x_ n\forall y_ 1\cdots\forall y_ n\bigl(x_ 1\sim y_ 1\wedge\cdots\wedge x_ n\sim y_ n\supset P(x_ 1,\ldots,x_ n)\equiv Q(y_ 1,\ldots,y_ n)\bigr) \] It is defined when an analogy correspondence \(A_ c\) is consistent with (suitable for, proper to) a formula \(A\). Then for any formulas \(A,B\) without analogy symbol \(\sim\) is defined when \(A\) is analogous (strongly analogous) to \(B\) based on \(A_ c\), and \(A\) is (strongly) analogous to \(B\). A model semantics of \(LK_ A\) is given, it is shown that \(LK_ A\) is sound. The final theorem characterizes the concept of analogy: Let \(A\) be analogous to \(B\) based on \(A_ c\), and \(M\) be a model of \(A_ c\), then \(M\) is a model of \(A\) iff \(M\) is a model of \(B\).
Mathematics Subject Classification ID: 68T27
Mathematics Subject Classification ID: 03F05
zbMATH DE Number: 218928
zbMATH Keywords: analogical reasoning
zbMATH Keywords: calculus of sequents
zbMATH Keywords: cut elimination
zbMATH Keywords: interpolation theorems
zbMATH Keywords: equational logic
reviewed by: Jan Šefránek
describes a project that uses: Nuprl
MaRDI profile type: Publication
cites work: Model theory
cites work: Q3723721
cites work: Computational approaches to analogical reasoning: A comparative analysis
cites work: Q3786027
cites work: Q5812175
cites work: Q5828993
cites work: Q3786003
cites work: Proof theory. 2nd ed
cites work: Q3791125
cites work: Analogy model and analogy correspondence
full work available at URL: 0304-3975(93)90002-b
OpenAlex ID: W2913641506
description: scientific article; zbMATH DE number 218928