Source-linked AI summary
A General Non-Probabilistic Theory of Inductive Reasoning
Wolfgang Spohn
TL;DR
The paper asks how epistemic states represented by NCFs should change through information or experience. It defines NCF conditionalization and argues that NCFs preserve important probabilistic structures while being computationally simpler.
Problem
The paper asks how NCF-represented epistemic states should change through information or experience, seeking a complete answer from two plausible assumptions.
Method
It defines NCF conditionalization, including a generalized field-based form parameterized by a subfield NCF.
Results
NCFs preserve key probabilistic structures, including additive and multiplicative relations and conditional independence, while being computationally simpler than probability theory.
Takeaways & Limitations
The theory supports computationally manageable implementations of conditional independence and related techniques, while permitting coarser elicitation of subjective judgments.
Takeaways & Limitations
The paper states that statistical data are difficult to handle within an NCF framework because relative frequencies are closely tied to probabilities.
Abstract
from arXiv · showhide
Probability theory, epistemically interpreted, provides an excellent, if not the best available account of inductive reasoning. This is so because there are general and definite rules for the change of subjective probabilities through information or experience; induction and belief change are one and same topic, after all. The most basic of these rules is simply to conditionalize with respect to the information received; and there are similar and more general rules. 1 Hence, a fundamental reason for the epistemological success of probability theory is that there at all exists a well-behaved concept of conditional probability. Still, people have, and have reasons for, various concerns over probability theory. One of these is my starting point: Intuitively, we have the notion of plain belief; we believe propositions2 to be true (or to be false or neither). Probability theory, however, offers no formal counterpart to this notion. Believing A is not the same as having probability 1 for A, because probability 1 is incorrigible3; but plain belief is clearly corrigible. And believing A is not the same as giving A a probability larger than some 1 - c, because believing A and believing B is usually taken to be equivalent to believing A & B.4 Thus, it seems that the formal representation of plain belief has to take a non-probabilistic route. Indeed, representing plain belief seems easy enough: simply represent an epistemic state by the set of all propositions believed true in it or, since I make the common assumption that plain belief is deductively closed, by the conjunction of all propositions believed true in it. But this does not yet provide a theory of induction, i.e. an answer to the question how epistemic states so represented are changed tbrough information or experience. There is a convincing partial answer: if the new information is compatible with the old epistemic state, then the new epistemic state is simply represented by the conjunction of the new information and the old beliefs. This answer is partial because it does not cover the quite common case where the new information is incompatible with the old beliefs. It is, however, important to complete the answer and to cover this case, too; otherwise, we would not represent plain belief as conigible. The crucial problem is that there is no good completion. When epistemic states are represented simply by the conjunction of all propositions believed true in it, the answer cannot be completed; and though there is a lot of fruitful work, no other representation of epistemic states has been proposed, as far as I know, which provides a complete solution to this problem. In this paper, I want to suggest such a solution. In [4], I have more fully argued that this is the only solution, if certain plausible desiderata are to be satisfied. Here, in section 2, I will be content with formally defining and intuitively explaining my proposal. I will compare my proposal with probability theory in section 3. It will turn out that the theory I am proposing is structurally homomorphic to probability theory in important respects and that it is thus equally easily implementable, but moreover computationally simpler. Section 4 contains a very brief comparison with various kinds of logics, in particular conditional logic, with Shackle's functions of potential surprise and related theories, and with the Dempster - Shafer theory of belief functions.
1. Introduction
The paper argues that probability theory handles inductive belief change well but lacks a formal, corrigible counterpart to plain belief. It proposes a non-probabilistic solution for updating epistemic states, claiming structural similarity to probability theory with simpler computation.
- Motivation: Probability theory supports induction through general belief-change rules, especially conditionalization on received information.Its epistemological success is attributed to the existence of a well-behaved concept of conditional probability.
- Motivation: Plain belief cannot be identified with probability 1 or with exceeding a fixed probability threshold, because it is corrigible and typically closed under conjunction.Probability 1 is described as incorrigible, whereas plain belief can be revised.
- Problem: Representing an epistemic state as the conjunction of all propositions believed true does not determine how to revise incompatible beliefs.The introduction states that the answer cannot be completed under this representation and that no other proposed representation had solved the problem.
- Proposal: The paper proposes a solution intended to complete the theory of inductive belief change while formally representing plain belief.The author says the proposal is formally defined and intuitively explained in section 2, with a fuller uniqueness argument given elsewhere.
- Paper roadmap: The proposed theory is structurally homomorphic to probability theory in important respects, equally implementable, and computationally simpler.Section 3 compares the proposal with probability theory, while section 4 briefly compares it with several logical and belief-function approaches.
2. Theory
The theory represents epistemic states with natural conditional functions that grade disbelief, yielding deductively closed and consistent beliefs. It defines belief revision through conditionalization and shows structural parallels with probability theory, while remaining computationally simpler but less suited to statistical data.
- Formal framework: An NCF maps possibilities to natural-number disbelief grades, with value 0 assigned to at least one possibility and equal values across propositionally indistinguishable atoms.For any non-empty proposition A, its NCF value is the minimum grade among possibilities in A.
- Belief interpretation: An NCF believes A true exactly when the negation of A has positive disbelief, producing a deductively closed and consistent set of beliefs.A proposition can be neither believed true nor believed false when both it and its negation have NCF value 0.
- NCF laws: NCFs obey fundamental laws in which not both a proposition and its negation are disbelieved, while disjunction takes the minimum of the disjuncts’ disbelief values.The disjunction law is x:(A u B) = min {x:(A),x:(B)} for non-empty propositions.
- Conditionalization: Belief change is defined by conditionalization: information shifts the relative disbelief of A and -A without altering the grading within either part, and generalizes from one proposition to a whole subfield.A conditionalization parameter can be a single firmness value or an entire subfield-valued NCF.
- Computational scope: NCFs preserve computational advantages associated with conditional independence and are computationally simpler than probability theory, but they do not provide a reasonable treatment of statistical data.The framework may also make subjective expert judgments easier to elicit because it uses coarser terms.
- Relation to probability: NCF operations are structurally homomorphic to probabilistic operations: probability addition corresponds to minimum disbelief, multiplication to addition, and conditionalization to subtraction.For finite fields, every non-standard NCF can be represented by a non-standard probability measure through orders of an infinitesimal.
4. Other comparisons
The paper compares NCF-theory with conditional approaches, Shackle’s functions, Shafer’s belief functions, and fuzzy logic. It emphasizes differences in belief-change dynamics and the need for general, precise induction rules.
- General comparison: Alternative epistemic-state representations are criticized for often lacking general and precise rules governing belief change through information.The paper treats such rules as tantamount to a theory of induction.
- Conditional logic: Conditional approaches model belief change by retaining sentences whose conditionals from the new information belong to the prior epistemic state.This proposal depends crucially on the properties of the conditional, which cannot be interpreted as material or strict implication.
- Shackle’s functions: Shackle’s functions of potential surprise and the paper’s functions share core axioms but differ in their ranges, a difference with substantive mathematical consequences.The paper argues that countable-union generalization requires well-ordered ranges and avoids a maximal degree of disbelief.
- Dempster–Shafer theory: Shafer’s theory encompasses Shackle’s theory as a special case and can complete it through Dempster’s rule, but its dynamics conflict with NCF-theory.NCF dynamics remain within all NCFs, whereas consonant belief functions are not generally closed under Shafer’s dynamics.
- Fuzzy logic: NCF-theory is presented as intuitively and formally incomparable with fuzzy logic, which addresses vague expressions and approximate reasoning rather than induction as such.The comparison also notes that Zadeh’s preferred base logic is Łukasiewicz logic with infinitely many truth-values.