This is the final article in my series on Bayesian Argumentation. To understand this essay, read the introductory article and the article on Relevance and Acceptance.

Corelevance and Conditional Relevance

Relevance exists in the context of the subject’s other prior beliefs. For example, if the subject believes that ($\bar{𝐶}$) the car is out of gas, and also ($\bar{B}$) the battery is dead, then both of these are good reasons to believe ($\bar{A}$) the car won’t start.

Yet neither is relevant on its own by our definition of relevance! A premise is only relevant if changing acceptance of the premise changes acceptance of the conclusion. But given the car is out of gas, it makes no difference whether the battery is dead or not: the car won’t start anyway. In other words, ($\bar{B}$) the battery is dead is irrelevant to ($\bar{A}$) the car won’t start given ($\bar{𝐶}$) the car is out of gas.

But if the subject believes (𝐶) the car has gas, then ($\bar{B}$) the battery is dead will probably become relevant. When accepting one premise causes another premise to become relevant, we say that the premises are corelevant.

Definition of Conditional Relevance

To define corelevance mathematically, we need to first define the conditional relevance of 𝐵 to 𝐴 given 𝐶, $R(A,B \vert C)$:

$$ R(A,B|C) = P(A|B,C) - P(A|\bar{B},C) $$

Definition of Corelevant

Then 𝐵 and 𝐶 are corelevant to 𝐴 if:

$$ R(A,B|C) ≠ R(A,B|\bar{C}) $$

Quantifying Corelevance

We can measure the magnitude of the correlevance as the difference:

$$ CR(A;B,C) = R(A,B|C) - R(A,B|\bar{C}) $$

It’s easy show that co-relevance is symmetrical (proof).

$$ CR(A;B,C) = CR(A;C,B) $$

Warrants

This idea of a third belief that justifies the relevance of a premise to a conclusion evokes the idea of the warrant from the field of argumentation theory.

In every argument there is an unstated claim that the premise supports the conclusion. This doesn’t need to be stated because it’s implied by the fact that the argument was made at all. After asserting people are wearing jackets in support of the conclusion it’s probably cold outside, I don’t need to add, pedantically, “and you see, if people are wearing jackets it must be cold outside”. This unexpressed third premise is called the warrant.

The warrant doesn’t have to be a logical formula such as “if people are wearing jackets it must be cold outside”. It can be based on any kind of inferential rule (deductive, inductive, intuitive) or argumentation scheme (authority, analogy, example) – 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 over grounds and claim. More precise definitions of our terms are given in the Deliberati Argument Model.

Bayesian Warrants

In a Bayesian model, a warrant can be any belief that increases the relevance of the premise to the conclusion.

So (𝐶) A Swede can generally be taken not to be a Roman Catholic is a warrant for the argument that (𝐴) almost certainly, Petersen is not a Roman Catholic because (𝐵) Petersen is a Swede if

Mathematical, a warrant is any claim with positive corelevance with the premise.

Definition of (Bayesian) Warrant

𝐶 is a warrant for the argument with premise 𝐵 and conclusion 𝐴 if:

$$ CR(A;B,C) > 0 $$

Conecessity = Corelevance × Acceptance

Now presumably the arguer assumes not only that the warrant increases the relevance of premise to conclusion in the mind of the subject. They also assume that the subject accepts the warrant to some degree.

This evokes a parallel with the definition of necessity. We can define conecessity as:

$$ \begin{aligned} CN(A;B,C) &= CR(A;B,C)P(C) \end{aligned} $$

Conecessity of the warrant is a measure of how much the argument depends on the warrant.

There could be multiple warrants for an argument. Every belief that is partially accepted by the subject and positively corelevant with the premise is a warrant that helps, to paraphrase Toulmin, to “justify the inferential leap from premise to conclusion”.

Summary

So we’ve rounded out our reconciliation of Bayesianism and argumentation theory by defining Toulmin’s concept of warrant from a Bayesian perspective, mathematically defining not only what a warrant is, but also defining conecessity a measure of how strongly the argument leans on the warrant.

In this series, we’ve only linked some of the basic entities in argumentation theory. Concepts such as claim, grounds, testimony, premise, conclusion, and warrant map to Bayesian concepts of belief, evidence, hypothesis, and now Bayesian warrant. And we can now quantify the strength of beliefs (acceptance), the strength of arguments (relevance, sufficiency, necessity, informativeness, persuasiveness, and even the strength of warrants (corelevance and conecessity).

If you’ve read this far, I am impressed! I hope this has been useful.

In the next article I summarize all the definitions and mathematical formulas introduced in this series.

Proofs

Proof 1

Symmetry of Corelevance

$$ CR(A;B,C) = CR(A;C,B) $$

Proof:

$$ \begin{aligned} CR(A;B,C) &= R(A,B \vert C) - R(A,B \vert \bar{C}) \cr &= ( P(A \vert B,C) - P(A \vert \bar{B},C) ) \cr &\space\space\space\space- ( P(A \vert B,\bar{C}) - P(A \vert \bar{B},\bar{C}) ) \cr &= ( P(A \vert B,C) - P(A \vert B,\bar{C}) ) \cr &\space\space\space\space- ( P(A \vert \bar{B},C) - P(A \vert \bar{B},\bar{C}) ) \cr &= ( P(A \vert C,B) - P(A \vert \bar{C},B) \cr &\space\space\space\space- ( P(A \vert C,\bar{B}) - P(A \vert \bar{C},\bar{B}) ) \cr &= R(A,C \vert B) - R(A,C \vert \bar{B}) \cr &= CR(A;C,B) \cr \end{aligned} $$