Deliberati Argument Model

In this article we introduce a few key terms and concepts used to develop the mathematics of Distributed Bayesian Reasoning.

In the field of argumentation theory, there are many different theories and models for analyzing arguments and naming their parts. But many modern theories still incorporate the basic ideas from the influential Toulmin model first introduced in 1948. We will adopt the core parts of the Toulmin model here, but use more modern terminology.

Anatomy of an Argument

Claims

A claim is a declarative sentence that people can accept or reject (agree with or disagree with). This definition is broad enough to include not only descriptive claims about reality, such as the universe is expanding, but also normative claims such as we should go to the beach.

An argument involves at least three claims:

• A conclusion: the claim in dispute.
• A premise: the reason given to accept or reject the conclusion.
• A warrant: an unstated claim that the premise, if accepted, is a good reason to accept or reject the conclusion.

The diagram below shows a sample argument from a hypothetical jury trial, with labels for the three parts of the argument.

The Premise

Since any logical combination of premises can be treated as a single premise, we will speak of an argument as having one explicit premise.

The premise itself can be the conclusion of another argument.

An argument may be worded in such a way that the premise is unclear (sarcasm, etc.), but there is general agreement among argumentation theorists that there is always a premise hiding in an argument. Some authors use the “grounds”, “evidence”, or “data” instead of premise.

The Warrant

The warrant can be thought of as a second unexpressed premise, comprising whatever might link the premise to the conclusion in the minds of the jurors.

Anything that is actually stated in the argument is part of the explicit premise, but there is always always something that is left unexpressed, some belief that justifies the inferential leap from premise to conclusion. In his case, the unexpressed premise might be a general rule like “if people confess, they are usually guilty”, or something more specific, such as “this defendant wouldn’t confess unless they are guilty.” In any case, the warrant can be thought of as the belief, for whatever reason, that the premise is a good reason to accept the conclusion.

So in our model, for simplicity sake, we assume every argument has an implicit warrant, which is the belief that the premise is a good reason to accept the conclusion.

The idea of an argument with an unstated premise is related to the Aristotles concept of enthymeme. This means that in our model, we in a sense treat all arguments as enthymemes.

So in the diagram above, the argument in support of conclusion (𝐴) the defendant is guilty, has two halves. On the left is the premise (𝐵) the defendant signed a confession. This is the actual claim that has been made. On the right is the warrant: the belief that 𝐵 is a good reason to accept 𝐴.

Types of Arguments

Opposing and Supporting Arguments

The warrant of a supporting argument is the claim that the premise is a good reason to accept the conclusion, and the warrant of an opposing argument is the claim that the premise is a good reason to reject the conclusion.

Note that an opposing argument is just a supporting argument for the negation of the conclusion. (𝐵) the defendant signed a confession supports conclusion 𝐴, but opposes conclusion (not 𝐴) the defendant is innocent.

Premise Arguments and Warrant Arguments

Making the distinction between premise and warrant allows us to cleanly distinguish between premise arguments and warrant arguments.

In the chart below, we have added two arguments that oppose 𝐵. The argument with premise (𝐺) the signature was forged, opposes the premise (𝐵) the defendant signed a confession. It is a reason asserted for believing that 𝐵 is not true. We call this a premise argument. The argument with premise (𝐶) the defendant retracted her confession, opposes the warrant of the argument with premise 𝐵 – the belief that 𝐵 is a good reason to accept 𝐴. We call this a warrant argument.

Notes on Terminology

It is easy to confuse the terms “claim” and “premise”. A claim is any declarative statement that can be agreed with or disagreed with. A claim can also take the role of premise or conclusion in some argument.

This is a very simplified model. There are many concepts from the field of argumentation theory literature that we don’t need to address here (rebuttals, backing, etc.), and our definitions may lack nuance. But these definitions are meant to provide not as a comprehensive theory of argumentation, but a vocabulary that helps us develop the math.

Argument Notation

The same claim can be used as the premise of many different arguments. To identify arguments unambiguously, we will need a notation that represent the argument itself, and not just the premise.

Identifiers for Premise Arguments

We represent an argument that supports conclusion 𝑋 with premise 𝑌 using the notation:

Note we place the premise after the conclusion.

For example, referring to our jury trial argument graph, the argument $\text{𝐴◂-𝐵}$ could be expressed in plain English as the fact that the defendant signed a confession is a good reason to believe that she is guilty.

We represent an argument that opposes conclusion 𝑋 with some premise 𝑌 using the notation:

For example, the argument $$\text{𝐵◃-𝐺} might be expressed in plain English as the fact that the signature was forged is a good reason NOT to believe that the defendant signed a confession.

If the claim 𝐵 is also used as the premise of some other argument, that would be a separate argument. For example, an argument that opposes conclusion 𝐻 with premise 𝐵 would be $\text{𝐻◃-𝐵},whichisnotthesameargumentas$\text{𝐴◂-𝐵}, even though it uses the same premise

Identifiers for Warrants

We represent the warrant of the argument $\text{𝐴◂-𝐵}$ using the notation

It’s easy to confuse the warrant with $\text{𝐴◂𝐵}$ with the argument $\text{𝐴◂-𝐵}$. The warrant $\text{𝐴◂𝐵}$ is a claim that 𝐵, given it is accepted, supports 𝐴, whereas the argument $\text{𝐴◂-𝐵}$ is the claim that 𝐵 should be accepted, and that it supports 𝐴. The former says “if the defendant signed a confession, that would be a good reason to believe she is guilty,” whereas the latter says “the fact that the defendant signed a confession is a good reason to believe she is guilty.”

Identifiers for Warrant Arguments

Premises such as (𝐶) the defendant retracted her confession oppose the warrant of the argument $\text{𝐴◂-𝐵}$. We notate warrant arguments like this:

This identifier has the same form as the identifier for a premise argument, except that the conclusion (the part on the left of the $\text{◃-}$ or $\text{◂-}$) is a warrant.

In plain English, this argument might be read as the fact that the defendant retracted her confession is a good reason to believe that she is not guilty even if she confessed.

Argument Graphs

Given a set of arguments that have been made in some situation, we can create a graph that shows relationships between the arguments. The graph below represents our sample argument, now using our new argument notation.

Each argument in this graph has one outgoing blue arrow, pointing to the argument or claim it supports or opposes. Using the same convention we adopted for argument identifiers, solid arrow heads represent supporting arguments, and hollow arrows heads indicate opposing arguments.

Note that the blue arrows in this graph are redundant, because the relationships between arguments are revealed in the identifiers themselves. But note also that the two arguments that oppose $\text{𝐴◂-𝐵}$ oppose it in different ways: $\text{𝐵◃-𝐺}$ opposes the premise ($𝐵$), and $\text{𝐴◂𝐵◃-𝐶}$ opposes the warrant ($\text{𝐴◂𝐵}$), but they both oppose the argument overall.

Because cycles are hard to deal with, we will assume all argument graphs are acyclic, thus our argument graphs will always be DAGs.

Review of Sample Argument

Here is a brief review of the sample argument shown in the argument graph above using the terminology and notation we have introduced so far:

• The claim (𝐺) the signature was forged opposes the conclusion (𝐵) the defendant signed a confession. Since 𝐵 is the premise of the argument $\text{𝐴◂-𝐵}$, $𝐺$ opposes the premise of $\text{𝐴◂-𝐵}$. So $\text{𝐵◃-𝐺}$ is a premise argument.

• The claim (𝐶) the defendant retracted her confession, opposes the warrant of $\text{𝐴◂-𝐵}$. It says that even if the premise 𝐵 were true, it is not a good or sufficient reason to support $𝐴$. So $\text{𝐴◂𝐵◃-𝐶}$ is a warrant argument.

• Both $\text{𝐵◃-𝐺}$ and $\text{𝐴◂𝐵◃-𝐶}$ oppose their conclusions: the premise and the warrant of $\text{𝐴◂-𝐵}$, respectively. Thus they both oppose the argument $\text{𝐴◂-𝐵}$.

• In the example above, we consider $\text{𝐴◂-𝐵}$ itself to be a premise argument, because it is supporting the claim $𝐴$. While claim $𝐴$ does not play the role of premise in any argument in this graph, it is pragmatic to call $\text{𝐴◂-𝐵}$ a premise argument because we always assume that $𝐴$ could be playing the role of premise in some larger argument graph.

Argument Threads

Definition of an Argument Thread

An argument can support or oppose the warrant of another warrant argument.

In response to $\text{𝐴◂𝐵◃-𝐶}$, someone might argue (𝐷) guilty people always say they are innocent. This argument would be written as $\text{𝐴◂𝐵◃𝐶◃-𝐷}$.

And of course there could be a response to this argument, and response to the response. The result could be a long argument thread.

We define an argument thread is a premise argument followed by a chain of zero or more warrant arguments. The chart below illustrates an single argument thread.

Argument Threads are Dialogs

Argument threads proceed along the lines of “𝐴 because 𝐵, yes but not 𝐶, okay but 𝐷,” and so on. Each argument in the thread is made in the context of all the previous arguments in the thread. The thread may be long, but jurors that are adding to the thread can be assumed to have have followed the whole thread of the argument – even if they have not participated in the sub-jury about each premise.

The claims in the thread thus represents a shared context. Each argument in the thread is made in the context of all the previous claims in the thread, and presumes acceptance of all of them. For example, when someone argues that ($\text{𝐴◂𝐵◃-𝐶}$) the defendant retracted her confession it is clear from context that they accept (concede) that (𝐵) the defendant signed a confession but still don’t accept that (𝐴) the defendant is guilty.

Now, the arguer may not actually accept 𝐵 to be true, but by responding with a warrant argument instead of a premise argument (e.g. (𝐵◃-𝐺) the signature was forged), they are conceding 𝐵 for the sake of argument, providing a reason to reject 𝐴 even if 𝐵 is accepted.

Considering a longer argument thread, the argument $\text{𝐴◂𝐵◃𝐶◃-𝐷}$ presumes acceptance of 𝐵 and 𝐶: a person who makes this argument is giving 𝐷 as a reason to reject 𝐴 given (even if) 𝐵 and 𝐶 are accepted.

So the warrant of the last argument in the thread can be interpreted as the claim that the premise, given acceptance of all preceding premises in the thread, is a good reason to accept or reject the root conclusion.

Next

Understanding the concepts in this argument model, in particular the difference between premise and warrant arguments, and the idea of argument threads as dialogs with shared context, is critical to understanding the Causal Assumptions in the Distributed Bayesian Reasoning Math.