A Bayesian Account of Argumentation

Quantifying Argument

What makes for a good argument? How can this be quantified?

From a logical point of view, a good argument is logically sound. But in the real-world people rarely argue with pure logic.

From a rhetorical point of view, a good argument is one that is convincing, whether through emotion, logic, or ethics. But how can this be measured?

In this essay, we present an account of argumentation as the exchange of information between Bayesian rational agents. The basic idea of the Bayesian view of probability is that probabilities represent subjective degrees of belief. So if we know the beliefs of some rational “subject”, we can precisely define and measure various concepts relating to the strength or quality of an argument in the mind of the subject. in other words we can objectively measure subjective beliefs.

Economists and data scientists increasingly use the Bayesian rational agent as a model of human behavior. Like all models it is imperfect, but it has the advantage of being well defined. Building clear terminology on top of a clear model helps clarify our thinking and sharpen our intuition about what argument actually is.

Introductory Example 1

Consider the argument this is a good candidate for the job because he has a pulse. This is a pretty lame argument. But what if the candidate did not have a pulse?

If our subject is a Bayesian rational agent with common sense, then probably:

• The argument is not persuasive.
• He has a pulse is not news to the subject. It is not informative.
• Yet the argument is clearly relevant, because:
  • If the subject learned that the subject did not have a pulse, this would be sufficient to reject him as a candidate.
  • Alternatively, the belief that he probably has a pulse is necessary for the belief that he might be a good candidate.

In this series of asseys, we will precisely define the concepts of relevance, informativeness, persuasiveness, sufficiency, and necessity and quantify these with some concrete numerical examples.

Introductory Example 2

Now consider another example argument: the car won’t start because the car is out of gas. And suppose that in this case, this is news to the subject. So it is probably going to be persuasive.

Except what if the the subject previously believed that the car’s battery is dead? With this assumption, the car being out of gas is now in a sense now irrelevant.

Clearly the relevance of an argument depends on context, or more specifically the various related prior beliefs of the subject. If we have a model of the subject’s prior beliefs, we can identify the corelevant beliefs – the prior beliefs cause the argument to be relevant.

This idea of unexpressed beliefs that justify an argument evokes the idea of the warrant from the field of argumentation theory. In this essay, we’ll introduce the basic concepts of premise, conclusion, and warrant from argumentation theory, and connect these to the Bayesian concepts of evidence, hypothesis, and priors.

Argumentation Terminology

Traditional logic, with its ideas of validity and soundness requires logical propositions are either true are false, whereas the beliefs of a Bayesian reasoner allow uncertainty.

Modern Argumentation Theory views argument as a kind of more flexible, informal logic. Arguments are typically generalized as involving some stated premise, which is asserted as a reason for accepting a conclusion. The inferential leap from premise to conclusion is justified by some unexpressed premise, called the warrant. The warrant can be any kind of inferential rule: deductive, inductive, intuitive – whatever justifies the inference in the mind of the arguer. Some academics use different terms for these concepts: our terminology is influenced by the influential Toulmin model, except we prefer the traditional terms premise and conclusion. More precise definitions of our terms are given in the Deliberati Argument Model.

In Bayesian terms, a rational agent is said to acquire evidence, which causes them to change their belief in the probability of some hypothesis. There is clearly an analogy here: evidence is to hypothesis as premise is to conclusion. But what is the warrant?

Warrants

The warrant clearly has to do with the subject’s prior beliefs, because a Bayesian agent’s priors are precisely what justify, in their mind, any inferential leap from premise to conclusion.

For example, if our subject is more likely to believe that (𝐴) it is going to rain today if they believe that (𝐵) the sky is cloudy than if they do not, then there clearly exists a warrant justifying, in the subject’s mind, the inferential leap from 𝐵 to 𝐴.

But why does this warrant exist in the subject’s mind? What actually justifies the inference? Is it a deductive inference? Inductive inference? Gut feeling?

We can’t necessarily answer this question, because a Bayesian agent’s beliefs are modeled by a simple probability distribution, which gives us the end result of the agent’s internal reasoning process, but not how they got there.

If the prior beliefs of our subject are represented by the probability measure $P$, then we can at least say that, in the mind of the subject, a warrant exists justifying the inference from premise 𝐵 to conclusion 𝐴 iff:

If the warrant exists, we say that 𝐵 is relevant to 𝐴. Otherwise, we say it is irrelevant.

Next in this Series

So in a Bayesian argument, an arguer asserts a premise in support/opposition to some conclusion, and if the premise is relevant – the subject is more likely to believe the conclusion if they believe the conclusion – then there must be some warrant justifying the inference from premise to conclusion.

In the next essay, we will formally define relevance and discuss some of its mathematical properties. In the remaining articles in this series we will define the concepts of necessity, sufficiency, informativeness, and persuasiveness, all of which relate back to this central concept of relevance. A summary of these definitions is below.

Summary of Definitions

Here is a summary of definitions in this series:

For an argument with premise 𝐵 and conclusion 𝐴, and a subject whose beliefs are represented by probability measure P…

• Relevant: The premise is relevant to the conclusion (or, the argument is relevant) iff $P(A|B)\ne P(A|\overline{B})$

• Irrelevant: The premise is irrelevant to the conclusion (or, the argument is irrelevant) iff $P(A|B)=P(A|\overline{B})$.
  • Irrelevance implies statistical independence of A and B.

• Support: The premise supports the conclusion iff $P(A|B)>P(A|\overline{B})$

• Oppose: The premise opposes the conclusion iff $P(A|B)<P(A|\overline{B})$
  • If 𝐵 supports 𝐴, then 𝐵 opposes 𝐴̅

• Relevance: $R(A,B)=P(A|B)-P(A|\overline{B})$

• Conditional Relevance: Given some third premise 𝐶: $R(A,B|C)=P(A|B,C)-P(A|\overline{B},C)$

• Corelevant: The premises 𝐵 and 𝐶 are corelevant to the conclusion 𝐴 iff: $R(A,B|C)\ne R(A,B|\overline{C})$

• Corelevance: $CR(A;B,C)=R(A,B|C)-R(A,B|\overline{C})=R(A,C|B)-R(A,C|\overline{B})$

• Necessity: $N(A,B)=P(A)-P(A|\overline{B})=P(B)R(A,B)$

• Sufficiency: $S(A,B)=P(A|B)-P(A)=P(\overline{B})R(A,B)$

• Argument Event: The event, directly observed by the subject, that the arguer asserted the premise in support of the conclusion.
  • This subtle concept is is discussed in more detail in Argument as Exchange of Information
• Post-Argument Belief: Given the argument event I: $P_i(\bullet )=P(\bullet |I)$
  • e.g. $P_i(B)=P(B|I)$ is the post-argument belief in 𝐵.

• Informative: The assertion of the premise is informative (the argument is informative) iff $P_i(B)>P(B)$

• Informativeness: The difference (absolute, percent, relative entropy. etc.) between $P_i(B)$ and $P(B)$

• Persuasive: The argument is persuasive iff $P_i(A)>P(A)$
  • Alternatively, the argument is persuasive if the argument is relevant and informative
• Persuasiveness: The difference (absolute, percent, relative entropy. etc.) between $P_i(A)$ and $P(A)$

Key Equations

And here is a summary of key equations in this series:

• Jeffrey’s Rule: $\begin{array}{}\text{(1)}& P^{\prime }(A)=P(A|\overline{B})+P^{\prime }(B)R(A,B)\end{array}$
• Relevance of Rejection of Premise/Conclusion: $R(A,B)=-R(A,\overline{B})=-R(\overline{A},B)=R(\overline{A},\overline{B})$
• Symmetry of Corelevance: $CR(A;B,C)=CR(A;C,B)$
• Necessity = Relevance × Acceptance: $N(A,B)=P(A)-P(A|\overline{B})=R(A,B)P(B)$
• Sufficiency = Relevance × Rejection: $S(A,B)=P(A|B)-P(A)=R(A,B)P(\overline{B})$
• Relevance = Necessity + Sufficiency: $R(A,B)=N(A,B)+S(A,B)$
• Sufficiency/Necessity of Rejection of Premise/Conclusion: $S(A,B)=N(\overline{A},\overline{B})$ and $N(\overline{A},\overline{B})=N(A,B)$
• Persuasiveness = Relevance × Informativeness: $P_i(A)-P(A)=(P_i(B)-P(B))R(A,B)$

Numerical Example

Suppose the priors of the subject are modeled by the probability measure 𝑃 given in this table:

The marginal probabilities are:

And the conditional probabilities:

Relevance

Which lets us calculate the relevance:

Necessity and Sufficiency

The necessity of 𝐵 to 𝐴 is:

And the sufficiency of 𝐵 to 𝐴 is:

Notice that relevance is the sum of necessity and sufficiency:

And that necessity is relevance times acceptance:

And that sufficiency is relevance times rejection:

Post-Argument Belief

Now suppose the assertion of 𝐵 in support of 𝐴 causes the subject to increase their belief in 𝐵 from $P(B)=50\mathrm{\%}$ to $P_i(B)=90\mathrm{\%}$.

The subject’s post-argument belief in 𝐴 will be, according to formula $\text{(1)}$:

This is slightly less than $P(A|B)=.8$ because the subject still harbors some doubt about 𝐵.

Informativeness

The informativeness is:

Persuasiveness

And the persuasiveness is:

Notice that persuasiveness is equal to relevance times informativeness:

Post-Argument Necessity and Sufficiency

If after the argument the subject were to learn additional information causing them to reject 𝐵, the new posterior would be $P_i(A|\overline{B})=P(A|\overline{B})=50\mathrm{\%}$.

The post-argument necessity is therefore:

And if the subject were to learn additional information causing them to accept $B$ 100%, then new posterior would be $P_j(A)=P_i(A|B)=P(A|B)=80\mathrm{\%}$.

The post-argument sufficiency is therefore: