Natural Conditional Functions (NCFs)

• Natural Conditional Functions are ordinal-valued functions defining plain belief and guiding qualitative belief revision.
• They replace standard arithmetic operations with (min, +, -) to mimic probabilistic updates in a non-numerical framework.
• NCFs support corrigible belief sets and deductive closure, enabling effective qualitative reasoning and counterfactual updates.

A Natural Conditional Function (NCF) is an ordinal-valued function on the set of possible worlds that represents the epistemic state of an agent with respect to “plain belief” and guides qualitative, non-probabilistic belief change under evidence or experience. NCFs are structurally analogous to probability functions, replacing arithmetic operations $(+, \times, /)$ with $(\min, +, -)$. Originating in the work of Spohn, these functions provide a rigorous model of corrigible belief and inductive revision, forming the foundation of a systematic non-probabilistic theory of inductive reasoning. NCFs generalize to Ordinal Conditional Functions (OCFs), and play a central role in formal epistemology, knowledge representation, and qualitative belief revision.

1. Formal Definition and Semantic Interpretation

Let $W$ be a non-empty set of atomic states (worlds), and $\mathcal{A}$ a field of subsets of $W$. An $\mathcal{A}$-measurable NCF is a function $k : W \to \mathbb{N} \cup \{\infty\}$ such that

• $\exists w \in W$ with $k(w) = 0$ (at least one maximally plausible world exists),
• $k$ is constant on each atom of $(\min, +, -)$0.

The function is extended to $(\min, +, -)$1 by $(\min, +, -)$2, $(\min, +, -)$3.

Epistemic semantics:

• $(\min, +, -)$4: $(\min, +, -)$5 is not disbelieved (fully plausible).
• $(\min, +, -)$6: $(\min, +, -)$7 is disbelieved to degree $(\min, +, -)$8.
• $(\min, +, -)$9: $W$0 is impossible.

Plain belief in $W$1 is defined as $W$2; that is, belief in $W$3 corresponds to all maximally plausible worlds lying inside $W$4 (Spohn, 2013).

2. Algebraic Structure and Relationship to Probability

NCFs are structurally homomorphic to probability theory via the replacement of arithmetic operations:

• Union: $W$5
• Intersection: $W$6
• Conditionalization: $W$7

This “$W$8” algebra enables the translation of classical probabilistic theorems (e.g., law of total probability, Bayes’ theorem) into the NCF framework.

In probabilistic embedding, $W$9 for infinitesimal $\mathcal{A}$0, so $\mathcal{A}$1 corresponds to the exponent in the conditional probability, formalizing NCFs as a qualitative, non-numerical counterpart to Bayesian updating (Spohn, 2013).

3. Inductive Reasoning and Belief Revision

Single-proposition updates: Conditioning on $\mathcal{A}$2 at firmness $\mathcal{A}$3 is defined by

$\mathcal{A}$4

This operation preserves the internal grading within $\mathcal{A}$5 and $\mathcal{A}$6, while ensuring $\mathcal{A}$7 is believed to degree $\mathcal{A}$8.

Generalized updates: For a partition $\mathcal{A}$9 and arbitrary assignment $W$0 of non-negative integers to blocks, the Jeffrey-type update is

$W$1

Key theorems:

• Closure: The set $W$2 is deductively closed under finite intersection.
• Independence: Order of updates on independent $W$3 commutes.
• Total probability/Bayes: $W$4 for $W$5 partitioning $W$6 (Spohn, 2013).

4. Extension to Infinite Ranks and OCFs

NCFs can be generalized to OCFs $W$7 (where $W$8 is finite or countable), and further to ordinal-valued functions $W$9 to model “nearly counterfactual” revision.

OCFs and Belief Change (Hunter, 2016):

• In standard (finite-valued) OCFs, updating by a hard report $\mathcal{A}$0 with strength $\mathcal{A}$1 is via $\mathcal{A}$2, with normalization to ensure at least one world has rank 0.
• In the ordinal $\mathcal{A}$3 setting, revision supports hypotheses that require infinitely more evidence to shift from impossible to possible (“nearly counterfactuals”).
• The basic update operation generalizes to ordinal addition, with normalization (“zeroing”) applied at all infinite levels.
• Conditional revision at a nearly counterfactual antecedent stores the conditional only at hypothetical levels where the antecedent could eventually become possible. This extends Spohn’s NCF machinery to more expressive epistemic dynamics.

5. NCFs and Qualitative Belief Models

NCFs formalize plain belief as corrigible, conjunctively closed, and structurally similar to probability, but support qualitative updating and infinite gradations of disbelief.

Comparison with other belief models:

• Unlike probability theory (where $\mathcal{A}$4 is incorrigible), beliefs in NCFs are always corrigible.
• Dempster-Shafer theory offers set-valued beliefs but lacks the algebraic revision structure of NCFs.
• Conditional logics lack the procedural belief change primitives present in NCF arithmetic (Spohn, 2013).

Recent developments (Liberatore, 2023):

• “Natural revision” on OCFs is extended from propositional inputs $\mathcal{A}$5 to conditionals $\mathcal{A}$6 by identifying “current conditions” (the set of worlds at least as plausible as the most plausible $\mathcal{A}$7-worlds).
• The operator revises only those $\mathcal{A}$8-worlds at the critical plausibility level, shifting their rank minimally, preserving the principles of minimal change and naivety.

6. Illustrative Examples and Computational Properties

Given $\mathcal{A}$9 and $k : W \to \mathbb{N} \cup \{\infty\}$0:

• $k : W \to \mathbb{N} \cup \{\infty\}$1
• Condition on $k : W \to \mathbb{N} \cup \{\infty\}$2: for $k : W \to \mathbb{N} \cup \{\infty\}$3, $k : W \to \mathbb{N} \cup \{\infty\}$4; $k : W \to \mathbb{N} \cup \{\infty\}$5, $k : W \to \mathbb{N} \cup \{\infty\}$6; $k : W \to \mathbb{N} \cup \{\infty\}$7, $k : W \to \mathbb{N} \cup \{\infty\}$8.

The update rules involve only minima and addition/subtraction, ensuring computational efficiency compared to probabilistic updating, especially in large or structured state spaces (Spohn, 2013).

7. Applications and Significance

NCFs provide a full, non-probabilistic framework for belief modeling and qualitative inductive reasoning, particularly suitable for settings where probability assignments are unavailable or inappropriate. They underpin formal approaches to belief revision, non-monotonic reasoning, and qualitative epistemic planning. The extension to transfinite ranks via OCFs enables nuanced conditional reasoning about counterfactuals and iterated belief update, with practical implications for AI, automated reasoning, and knowledge representation (Hunter, 2016, Spohn, 2013, Liberatore, 2023).

Structural Feature	Probability Theory	NCF/OCF Framework
Value Range	$k : W \to \mathbb{N} \cup \{\infty\}$9	$\exists w \in W$0, $\exists w \in W$1
Belief Update	Bayes’ rule	$\exists w \in W$2 algebra
Incorrigibility	$\exists w \in W$3 is permanent	Belief always corrigible
Computational Need	Sums, products	Minima, addition
Counterfactuals	Requires extensions	Supported via ordinal OCFs

The algebraic and operational properties of NCFs guarantee that belief sets are deductively closed, updates for both certain and uncertain evidence are well-defined, and conditional independence has a rigorous qualitative analog. This positions NCFs and their generalizations as a foundational framework in qualitative epistemology and AI.