Background

In large measure dialectic is to our factual knowledge what logic is to our formal knowledge: a mechanism of rational validation.

Nicholas Rescher, Dialectics. A Controversy-Oriented Approach to the Theory of Knowledge, p. xiii

Argumentation theory studies how conclusions can be justified through structured reasoning, particularly in the presence of conflicting views. To achieve this, it must account for:

1. How arguments can be constructed;
2. How arguments can be defeated;
3. How arguments can be defended against defeating counterarguments.

Early work produced many different systems with as many ways of representing knowledge, and this diversity reflects the limitations of formal logic in modeling real-life debate—defeasible, context-sensitive, and embedded in dialogue. This section surveys key responses to that challenge, tracing how different models have reconceived rational justification beyond the bounds of deductive form.

Models

Contextual approaches

The modern study of argumentation has been marked by a contention over the capacity of formal logic to represent real-world reasoning. The most skeptical in this regard was Stephen Toulmin, who in 1958 introduced a model of argument consisting of three core elements: the claim, the grounds (evidence supporting the claim), and the warrant (the principle explaining why the grounds support the claim). The model also included optional components such as backing (support for the warrant), rebuttal (conditions under which the claim might not hold), and qualifier (indicating the strength of the claim).

This is exemplified in Toulmin’s well-known case:

Claim: Harry is a British subject.

Grounds: Harry was born in Bermuda.

Warrant: A man born in Bermuda will generally be a British subject.

Backing: Statute and case law covering nationality.

Qualifier: Presumably.

Rebuttal: Unless his parents were foreign diplomats, or he has renounced his nationality.

Toulmin’s emphasis on warrants and backing drew directly from the structure of legal reasoning, where the strength of an argument rests not on its formal validity but on its grounding in the accepted procedures and norms of justification within a particular field. In this way, his model emphasized that reasoning varies across disciplines, depends on context, and must be judged by practical standards, laying the groundwork for a pragmatic and normative theory of argumentation.

Building on this approach, Douglas Walton extended the analysis of context-sensitive reasoning through a systematic catalog of argumentation schemes—structured templates for common types of arguments. Each scheme includes typical premises and conclusions, along with critical questions designed to test the argument’s strength and uncover hidden assumptions. For example, the appeal to expert opinion scheme can be represented as:

E is an expert in domain D.

E asserts that proposition P is true.

P is within D.

Therefore, P is true.

To evaluate this argument, one may ask: How credible is E a source? Is E truly an expert in domain D? What exactly did E assert that implies P? Is E personally reliable? Is P consistent with what other experts assert? Is E’s assertion of P based on evidence?

Walton regarded argument schemes as dialogical devices: each argument is a move in a dialogue, and the scheme it instantiates defines the set of appropriate responses (counterarguments) to that move. Schemes such as the appeal to expert opinion model source-based reasoning, and thereby offer a casuistic refinement of Toulmin’s notion of backing for warrants.

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Pollock’s attacks

In the mid-1980s, John Pollock played a central role in applying defeasible reasoning to argumentation, addressing the limitations that made classical deductive logic unsuitable for representing real-world reasoning. In practice, argumentation follows the principles of non-monotonic logic, where the addition of new information, describing a more particular case, can invalidate previously valid conclusions.

One of Pollock’s key examples illustrates this:

The object looks red.

Therefore, the object is red.

But the object is illuminated by a red light.

So, it is not the case that the object would look red only if it were red.

In this case, the object appears red and is therefore presumed to be red until it is discovered that it is illuminated by a red light. The object may still be red, but it can no longer be concluded solely from its appearance. This kind of situation cannot be represented within classical deductive logic, where valid inferences are monotonic: if a conclusion follows from a premise, it continues to follow regardless of the addition of new premises. If A implies B, and A is true, then B must be true—no further information (C) can defeat B. But in everyday reasoning, conclusions are often provisional: they hold until new, relevant information defeats them. This is the core insight motivating non-monotonic logics and their application to argumentation.

Pollock formalized defeasible reasoning by distinguishing two kinds of attacks on arguments:

• Undercut, such as the red-light case, which challenges the connection between premise and conclusion without offering an alternative conclusion (under those conditions, one cannot know what color the object really is).

• Rebuttal, or distinction, which accepts the validity of the inference A → B but introduces a new premise C that leads to conclude that the case under consideration is a special case, an exception, and therefore to reach the opposite conclusion: A ∧ C → ¬B.

A canonical example of the latter is the Tweety case, first introduced by Raymond Reiter in 1980:

Tweety is a bird.

Birds typically fly.

Therefore, Tweety flies.

But Tweety is a penguin.

Penguins don’t fly.

So Tweety doesn’t fly.

Here, the initial generalization is rebutted by a specific exception. Notice also that in this case the premise Tweety is a bird is not only conceded but implicitly reaffirmed by the added information that Tweety is a penguin. This shows how deductive and defeasible reasoning can coexist within a single discourse: while the inference Tweety is a penguin → Tweety is a bird is deductively valid, the defeasible rule Birds fly is overridden by a more specific case.

Pollock’s system did not allow attacks on premises, but later formalisations recognized the need for a third type of attack to account for plausible reasoning. This attack, now known as undermining, negates an argument’s non-axiomatic premise and removes the support for its conclusion. Unlike undercuts, it does not invalidate the inference itself, but the conclusion remains unsupported unless the premise is defended. Undermining is currently being investigated in the context of belief revision, as it reduces the informational base from which agents draw further inferences.

Dung’s abstract argumentation

In 1995, Phan Minh Dung introduced an abstract formalism for argumentation that separated the acceptability of arguments from both their internal structure and the specific nature of their conflicts. In his model, arguments are nodes in a directed graph connected by binary attack relations. A calculus of opposition is then applied to determine sets of acceptable arguments, called extensions, that represent rational stances if they are internally coherent (conflict-free) and externally defended. These extensions depend solely on attack and defense: a defense occurs when an argument in the set attacks an attacker of another member. The arguments of such an admissible extension are labelled as accepted, those attacked by an argument of the extension are defeated, and the others are undefined.

This set-theoretic approach decoupled abstract from structured argumentation, revolutionizing the field and enabling further mathematical development. Dung’s semantics now underpin most formal models of argumentation. It also helped clarify the distinction between attack—a non-evaluative conflict relation—and defeat, which is assigned after computing the extensions.

ASPIC⁺

The original ASPIC (Argumentation Service Platform with Integrated Components) was developed within a European project (2004–2007) to integrate, generalize, and extend existing approaches to structured argumentation. Its successor, ASPIC^{+ 1}, first introduced in 2010, combines Dung’s abstract semantics with structured argumentation. It allows Pollock-style attacks to be formally represented while preserving a clear link to Dung’s acceptability criteria.

ASPIC⁺ is built around the following core components:

Argumentation System

Consists of a logical language, a contradiction function (not necessarily symmetric) over formulas, and a set of inference rules—either strict (deductive) or defeasible.

Knowledge Base

A subset of the language, containing the axioms (facts, not attackable) and ordinary premises (assumptions, attackable) available for argument construction.

Argumentation Theory

The combination of an argumentation system and a knowledge base, relative to which arguments are generated and evaluated.

Argument

A structure built from a set of premises, a conclusion, and a strict or defeasible inference rule connecting them. An argument is built on the basis of an argumentation theory with a given knowledge base and argumentation system.

Attack

A relation between two arguments where the conclusion of one contradicts a premise, the conclusion, or the inference rule in the other. These are called undermining, rebutting, and undercutting attacks, respectively.

ASPIC⁺ arguments generate abstract argumentation frameworks, to which Dung-like extension calculus can be applied to evaluate acceptability. Importantly, ASPIC⁺ is not a specific argumentation system, but a general framework for specifying such systems. It supports flexible instantiation, even with partial structures (e.g. only strict or defeasible rules), while ensuring that a complying system will satisfy some key rationality postualtes.

Application in dialogical contexts

ASPIC⁺ argument schemes define a notion of consequence based on a fixed body of information, which is thus represented statically. This may seem incompatible with Walton’s claim that arguments can only be evaluated in the context of a dialogue or procedure. However, the two perspectives can be reconciled by embedding argumentation logics within argumentative dialogue systems, where arguments are modeled as interactions between agents with distinct and evolving knowledge bases, and possibly changing beliefs.

In ², ASPIC⁺ is adapted to dialogue by assigning each agent their own knowledge base within a shared argumentation system that enables communication. Dialogical agents evaluate incoming arguments relative to their internal argumentation theory and the arguments it can generate. Depending on their assertion and acceptance attitudes ³, they may accept the premises and/or conclusion, respond with a counterargument, or ask for further grounds for a premise.

The practical application of ASPIC⁺ is illustrated with useful examples in ⁴. For details on our instantiation of ASPIC⁺ in Description Logic, see Model. For a broader overview of argumentation theory, see ⁵.