The Counterpart Principle of Analogical Support by Structural Similarity

Abstract

We propose and investigate an Analogy Principle in the context of Unary Inductive Logic based on a notion of support by structural similarity which is often employed to motivate scientific conjectures.

An examination of the SEP candidate analogical inference rule within pure inductive logic

“…Doing so would still give CAP (and DP) as a consequence but would, for q ≥ 2, restrict GAP, EAP and PAP down to the single probability function c L q 0 of Carnap's Continuum. Combined with previous results in [11,12] a pattern seems to be emerging with so called 'Analogy Principles', namely that they either hold, almost by chance, for some small family of otherwise (apparently) undistinguished probability functions, or they are actually consequences of some already established and acceptable principles such as ULi + SN or Ax. In other words, to our knowledge we do not currently have any analogy principles which genuinely introduce new concepts without also reducing the field of 'rational probability functions' down to almost a triviality (and leading to the conclusion that such a version of 'reasoning by analogy' is both very powerful and very dangerous).…”

“…We remark that although Theorems 1 and 2 are proved here for a unary language L it can be shown that when the additional constants a are allowed they hold too, with the standard extension of c 0 (as u 0,1,0,0,... ,L , see [20, Chapter 29]), for not purely unary languages. 12 …”

“…12 In more detail suppose that L is a polyadic language and w = c 0 is a probability function on L satisfying Px + SN. Then there must be some relation symbol P of L and a, b such that w(P ( a) ∧ ¬P ( b)) > 0.…”

“…, a j r ) from L by R i (a j 1 ). 13 Since negative analogies do not figure. 14 It is also possible to consider a formalization in which it applies to sentences as carriers, see the Counterpart Principle of [12], [20,Chapter 22].…”

The Twin Continua of Inductive Methods

“…Namely the information R (a ) ∧ R (a ) suggests that the constants look alike, since both a and a have been found to satisfy R (x), and this supports belief in a , a also being alike and so both satisfying R (x). [Actually it turns out that we will always have ≥ in (1) for any probability function that is a member of a Language Invariant family of functions satisfying Ax, see [9], so from this point of view the w …”