NMR - Defeasible Logic

Nonmonotonic Reasoning

Why Defeasible Logic? (Motivation and Scope)

Classical (monotonic) logic assumes that once a conclusion is derived it can never be invalidated by adding more information. Real-world reasoning violates this assumption constantly: rules admit exceptions, evidence is incomplete, and priorities between rules (e.g., statutory vs. case law; general vs. specific medical heuristics) matter. Defeasible Logic (DL) is a family of non-monotonic formalisms engineered to reason with such retractable knowledge while keeping inference transparent, explainable, and computationally manageable.

Knowledge with exceptions. “Birds fly” is informative yet false for penguins, injured birds, heavy-laden birds, etc. We want to use the default unless there is a counter-reason.
Dynamic evidence. New information (e.g., “this bird is a penguin”) must be able to defeat earlier tentative conclusions (“it flies”).
Competing sources. A more specific or higher-priority rule should overrule a general or lower-priority one.
Explainability. Stakeholders (lawyers, clinicians, auditors) need arguments: explicit, minimal sets of premises/rules that justify a conclusion and can be compared, attacked, and defended.

Crucially, retractability is not mere choice. Having two options (“yes/no”) and committing to one is different from adopting a tentative conclusion that may later be withdrawn upon encountering defeating information. DL makes this retractability first-class.

Core Language and Knowledge Base

Let $\mathsf{At}$ be a set of ground atoms. A literal is either $p \in \mathsf{At}$ or its negation $\neg p$.

Facts. A set $\mathsf{F}$ of literals (immutable truths).
Strict rules. $\;L_0 \leftarrow L_1,\dots,L_n$
(Whenever $L_1,\dots,L_n$ hold, $L_0$ must hold.)
Defeasible rules. $\;L_0 \;-!<\; L_1,\dots,L_n$
(Typically, if $L_1,\dots,L_n$ hold and nothing defeats the rule, infer $L_0$.)
Defeaters (optional). $\;\;\sim! L_0 \;-!<\; L_1,\dots,L_n$
(Used to block conclusions without supporting the opposite.)
Priorities (optional). A binary acyclic relation $\succ$ over rules, where $r_1 \succ r_2$ means “$r_1$ overrides $r_2$” when they conflict (e.g., specificity or authority).

A knowledge base is $KB=(\mathsf{F},\mathsf{S},\mathsf{D},\mathsf{Def},\succ)$ for facts, strict rules, defeasible rules, defeaters, and a (possibly empty) priority relation.

Proof-Theoretic Intuition (Tagged Derivability)

A convenient DL presentation distinguishes definitive vs defeasible provability with tags (one common style):

$+!\Delta L$ : $L$ is definitely provable (from facts $+$ strict rules).
$-!\Delta L$ : $L$ is definitely not provable.
$+!\partial L$ : $L$ is defeasibly provable (there is a supporting chain of defeasible/strict rules and all counter-arguments are defeated).
$-!\partial L$ : $L$ is not defeasibly provable (every attempt is blocked by a strictly undefeated counter-argument).

Sketch:

Definitive layer: close under $\leftarrow$ using $\mathsf{F}\cup\mathsf{S}$ to compute what is indisputable.
Defeasible layer: tentatively apply $-!<$ rules when their premises are (definitely or defeasibly) available and no undefeated counter-derivation exists for $\neg L$ (considering $\succ$).

This two-tier view makes explicit why strictly derivable conclusions need no argument: they are not tentative.

Arguments: The Unit of Justification and Debate

An argument is a compact, defensible justification for a tentative conclusion.

Definition (Argument).
An argument is a pair $\langle A, h \rangle$ such that:

(Support) $h \in \mathsf{Cn}(A \cup \mathsf{S} \cup \mathsf{F})$ where $A \subseteq \mathsf{D}$ (and optionally $\mathsf{Def}$) and $\mathsf{Cn}$ is the closure under the proof theory.
(Non-triviality) $A \cup \mathsf{S} \cup \mathsf{F}$ is consistent.
(Minimality) For every strict subset $A’ \subsetneq A$, $h \notin \mathsf{Cn}(A’ \cup \mathsf{S} \cup \mathsf{F})$.
(Strict non-derivability) $h \notin \mathsf{Cn}(\mathsf{S} \cup \mathsf{F})$ (otherwise $h$ is definitive, not defeasible).

Intuition: an argument is the minimal set of defeasible commitments that, together with indisputable knowledge, suffices to conclude $h$.

Attacks and Defeats (How Arguments Interact)

Let $\mathcal{A}=\langle A,h\rangle$ and $\mathcal{B}=\langle B,k\rangle$ be arguments.

Rebuttal: $\mathcal{B}$ rebuts $\mathcal{A}$ if $k$ is (logically equivalent to) $\neg h$.
Undercut: $\mathcal{B}$ undercuts $\mathcal{A}$ if it attacks the applicability of some rule in $A$ (e.g., a premise/exceptions literal is shown to fail).
Undermine: $\mathcal{B}$ undermines a premise of $\mathcal{A}$ (e.g., attacks a non-strict premise literal used by $A$).

A defeat is an attack that survives priority comparison: $\mathcal{B}$ defeats $\mathcal{A}$ if it attacks $\mathcal{A}$ and there is no rule in $A$ that (by $\succ$ or specificity) overrides the attack. Formally, for rebuttals, $\text{Defeat}(\mathcal{B},\mathcal{A}) \iff \big(k \equiv \neg h\big)\;\wedge\; \neg\exists\, r_A\in A, r_B\in B\; \text{with}\; r_A \succ r_B.$ (Analogous conditions are stated for undercuts/undermines.)

Dialectical Evaluation (Which Conclusions Survive?)

Arguments form a graph whose nodes are arguments and edges are defeats. Two standard evaluation modes:

Credulous acceptance: $h$ is accepted if there exists at least one admissible (self-defending) set of arguments supporting $h$.
Skeptical acceptance: $h$ is accepted only if it appears in all acceptable (e.g., grounded/preferred) extensions.

Using Dung-style semantics, compute extensions (grounded, preferred, stable) of the defeat graph; the accepted literals are those supported by arguments in the chosen extension(s). In DeLP-style dialectics, one builds a dialectical tree: the root is $\langle A,h\rangle$; children are its defeaters; grandchildren are counter-defeaters; the warranted status of $h$ is obtained by a bottom-up marking (defeater defeated ⇒ parent justified).

Why Defeasible Logic Needs Arguments

Conflict management. When defaults collide (e.g., “birds fly” vs. “penguins do not fly”), arguments let us compare the justifications rather than raw rule sets, enabling fine-grained defeat via rebut/undercut/priority.
Explainability and accountability. Each accepted conclusion comes with a minimal, human-auditable proof object $\langle A,h\rangle$ explaining why it holds and what would defeat it.
Modularity. Arguments localize reasoning: we can refine or retract specific defeasible rules in $A$ without recomputing the entire closure.
Alignment with human reasoning. Legal, medical, and ethical deliberations naturally proceed via claims + supporting reasons + counter-reasons. DL with arguments is a faithful, formal counterpart.

Worked Example (Specificity and Defeat)

Let

$\mathsf{F}={\textit{bird}(t), \textit{penguin}(t)}$ and $\mathsf{S}={\textit{penguin}(x)\leftarrow \textit{bird}(x)}$ (or the reverse, as needed).
Defeasible rules: $r_1:\; \textit{flies}(x)\;-\!<\;\textit{bird}(x) \quad\quad r_2:\; \neg\textit{flies}(x)\;-\!<\;\textit{penguin}(x)$ Assume specificity gives $r_2 \succ r_1$.
Argument $\mathcal{A}=\langle {r_1}, \textit{flies}(t)\rangle$ is supported by $\textit{bird}(t)$.
Argument $\mathcal{B}=\langle {r_2}, \neg\textit{flies}(t)\rangle$ is supported by $\textit{penguin}(t)$.
$\mathcal{B}$ rebuts $\mathcal{A}$, and since $r_2 \succ r_1$, $\mathcal{B}$ defeats $\mathcal{A}$.
Skeptically, we accept $\neg\textit{flies}(t)$; credulously, we might still accept $\textit{flies}(t)$ only if $\mathcal{A}$ remains undefeated in some extension (it doesn’t here).

Add an undercutter: $r_3:\; \neg\textit{flies}(x)\;-\!<\;\textit{injured}(x).$ If later $\textit{injured}(t)$ becomes known, a new argument undercuts any rule concluding $\textit{flies}(t)$, even outside the penguin case.

Design Degrees of Freedom (and Why They Matter)

Priorities $\succ$ (statute > guideline; specific > general; recent > old; trusted > untrusted).
Ambiguity propagation vs. blocking. Should a tie between conflicting arguments block both conclusions, or allow both to persist credulously?
Defeaters vs. full rebuttals. Some knowledge should only prevent a conclusion without entailing its negation.
Skeptical vs. credulous acceptance. Safety-critical applications often prefer skeptical acceptance; exploratory reasoning may use credulous acceptance to surface possibilities.

These are not mere stylistic choices; they determine whether reasoning is cautious, adventurous, or balanced, and they directly impact which arguments become warranted.

Formal Summary (Ready-to-Cite Statements)

Strict vs. Defeasible Inference. $\frac{L_1,\dots,L_n \quad (L_0 \leftarrow L_1,\dots,L_n)\in \mathsf{S}}{+\!\Delta L_0} \qquad \frac{L_1,\dots,L_n \quad (L_0 -\!< L_1,\dots,L_n)\in \mathsf{D}\quad \text{No undefeated counter}}{+\!\partial L_0}$
Argument (Minimal, Consistent, Strict-Non-Derivable).
$\text{Arg}(A,h) \iff \begin{cases} h \in \mathsf{Cn}(A \cup \mathsf{S} \cup \mathsf{F}),\\ A \cup \mathsf{S} \cup \mathsf{F}\ \text{consistent},\\ \forall A' \subsetneq A:\; h \notin \mathsf{Cn}(A' \cup \mathsf{S} \cup \mathsf{F}),\\ h \notin \mathsf{Cn}(\mathsf{S} \cup \mathsf{F}). \end{cases}$
Defeat via Priority.
If $\mathcal{B}$ attacks $\mathcal{A}$ and $\nexists r_A\in A, r_B\in B$ with $r_A \succ r_B$, then $\mathcal{B}$ defeats $\mathcal{A}$.
Acceptance.
$h$ is skeptically accepted iff for all acceptable extensions $E$, there exists $\langle A,h\rangle \in E$; credulously accepted iff there exists at least one acceptable extension $E$ with $\langle A,h\rangle \in E$.

Takeaway

Defeasible Logic captures the tentative, revisable nature of practical reasoning. Arguments are the core explanatory artifact: minimal, consistent packages of defeasible commitments that justify conclusions, expose where they can be defeated, and support principled acceptance under dialectical evaluation. This pairing (DL + arguments) yields systems that are not only robust to exceptions and new evidence, but also auditable and aligned with human reasoning.

Allaway, Evaluating Defeasible Reasoning in LLMs with DEFREASING, 2025