## Quantifying Argument

What makes for a *good* argument?

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. But how can this be measured?

In this series of essays, we present a Bayesian model of argumentation, where arguments are treated as information that may cause a Bayesian rational agent to modify their beliefs. Given a model of the beliefs of some Bayesian “subject”, we can **objectively** measure **subjective** aspects of an argument’s quality or strength, such as **relevance**, **persuasiveness**, and **sufficiency**.

This perspective can provide some powerful insights about argumentation for people working in artificial intelligence, law, argument mapping software, or in our case, design of social protocols.

Consider the following example.

## Introductory Example 1

Consider the argument *this is a good candidate for the job* because *he has a pulse*.

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

- The argument is not very
**persuasive**. - Nor is it
**informative**.*He has a pulse*is probably not new information to the subject. - 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.

- If the subject learned that the subject did

As everyone knows, a Bayesian rational agent updates their beliefs when they acquire new information. An argument that is not informative to the agent therefore can’t be persuasive. But a Bayesian model allows us to calculate what an agent **would** believe if they had different information (e.g. if they thought that the candidate *didn’t* have a pulse). This simple insight cracks open a number of ways of measuring argument **strength** other than just persuasiveness. In the essays on
Relevance and Corelevance and Necessity and Sufficiency we will define these measures and see how they all relate mathematically to the informativeness of the argument.

The Bayesian model of argumentation also allows us to take into account the reliability of the arguer themselves. A Bayesian rational agent will only update their beliefs if they believe the information given them. So for an argument to be informative it must not just be **new information**, it must also be believable. This perspective shows us when what looks like an *ad hominim* fallacy may somteimes be perfectly rational^{1}. These ideas are discussed in the essays on informativeness and persuasiveness.

## Introductory Example 2

Now consider another example argument: *the car won’t start* because *the car is out of gas*. If the subject previously believed the car had gas, then this new information might well be persuasive. But suppose the subject accepts this information, but also believes that *the car’s battery is dead*? With this assumption, the car being out of gas is now in a sense irrelevant.

Clearly the relevance of an argument depends on context: it depends on other beliefs the subject has about the state of world.

Theoretically, if we have a model of some subject’s beliefs about the world, we can identify the **corelevant** beliefs – the beliefs cause the argument to be relevant. We will define this more precisely in the essay on relevance and corelevance.

## Why Bayesian Argumentation?

Like all models, the Bayesian model of subjective belief is an incomplete description of the human mind. But it is clearly defined. Building clear terminology on top of a clear model helps clarify our thinking, facilitate discussion, and sharpen our intuition about what argument actually is.

There is a lot of recent academic work on Bayesian argumentation^{2}^{3}. These essay are intended not as an overview of current theory, but as a useful set of definitions and formulas for practitioners: specifically software engineers building practical applications of argumentation for AI, argument mapping systems, or in our particular case, design of social protocols. Our goal is to provide a useful and clear vocabulary, with common-sense but precise definitions for common concepts related to argument strength. This can hopefully help clarify discussion among collaborators and prove useful in documentation and code.

## Argumentation Theory and Warrants

This idea of unexpressed beliefs that justify an argument evokes the idea of the **warrant** from the field of argumentation theory. Argumentation theory views argument as a kind of flexible, informal logic. People don’t argue with logical syllogisms – instead they make simple statements, or **claims** which support other claims. For example, I might claim *people are wearing jackets* to support the claim *it’s probably cold outside*.

We use the terms **premise** and **conclusion** to differentiate between the supporting and supported claims. An **argument** is just a premise stated in support of some conclusion.

In every argument there is an unstated claim that *this premise supports this conclusion*. This doesn’t need to be stated because it’s implied by the fact that the argument was made. 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 premise that justifies the inferential leap from premise to conclusion 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.

## A Bayesian Definition of Warrant

In Bayesian terms, a rational agent is said to acquire **evidence**, which causes them to change their belief in the probability of some **hypothesis** (see this Bayesian Inference Primer).

There is clearly an analogy here: **evidence is premise as to hypothesis is to conclusion**. But argumentation theory also has warrants. What is the warrant in a Bayesian model?

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 is the inferential rule that actually justifies the inference? Is it a deductive inference? Inductive? 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 belief structure, 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**:

$$ P(A|B) ≠ P(A|\bar{B}) $$

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

## Summary

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 premise – then there must be some **warrant** justifying the inference from premise to conclusion.

## This Series

In the next essay in this series, we will formally define a measure of **relevance** from a Bayesian perspective and discuss some of its mathematical properties. In the remaining articles in this series we will define measures of **necessity**, **sufficiency**, **informativeness**, and **persuasiveness**, all of which relate back to this central concept of relevance.

## Summary of Definitions

Below is a summary of all the terms that will be defined in the above essays.

*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**$P(A \vert B) ≠ P(A \vert \bar{B})$- Otherwise, the premise is
**irrelevant**to the conclusion (or, the argument is irrelevant)- Irrelevance implies statistical independence of A and B.

- Otherwise, the premise is
**Support**: The premise**supports**the conclusion**iff**$P(A \vert B) > P(A \vert \bar{B})$**Oppose**: The premise**opposes**the conclusion**iff**$P(A \vert B) < P(A \vert \bar{B})$- If 𝐵 supports 𝐴, then 𝐵 opposes $\bar{A}$

**Relevance**: The relevance of the premise to the conclusion is $R(A,B) = P(A \vert B) - P(A \vert \bar{B})$**Conditional Relevance**:*Given some third premise 𝐶*: $R(A,B \vert C) = P(A \vert B,C) - P(A \vert \bar{B},C)$**Corelevant**: The premises 𝐵 and 𝐶 are corelevant to the conclusion 𝐴 iff: $R(A,B \vert C) ≠ R(A,B \vert \bar{C})$**Corelevance**: $CR(A;B,C) = R(A,B \vert C) - R(A,B \vert \bar{C}) = R(A,C \vert B) - R(A,C \vert \bar{B})$**Necessity**: The necessity of the premise to the conclusion is $N(A,B) = P(A) - P(A \vert \bar{B}) = P(B)R(A,B)$**Sufficiency**: The sufficiency of the premise for the conclusion is $S(A,B) = P(A \vert B) - P(A) = P(\bar{B})R(A,B)$**Testimony Event**: The event, directly observed by the subject, that the arguer asserted the premise in support of the conclusion.**Post-Argument Belief**:*Given the testimony event I*: $P_i(∙) = P(∙ \vert I)$- e.g. $P_i(B) = P(B \vert I)$ 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 informativeness of the argument is $P_i(B) - P(B)$**Persuasive**: The argument is**persuasive****iff**$P_i(A) > P(A)$- Alternatively, the argument is
**persuasive**if the argument is relevant and informative

- Alternatively, the argument is
**Persuasiveness**: The persuasiveness of the argument is $P_i(A) - P(A)$

## Key Equations

And here is a summary of key equations in this series: $\label{1}$

- Jeffrey’s Rule: $$P’(A) = P(A \vert \bar{B}) + P’(B)R(A,B)\tag{1} $$
- Relevance of Rejection of Premise/Conclusion: $R(A,B) = -R(A,\bar{B}) = -R(\bar{A},B) = R(\bar{A},\bar{B})$
- Symmetry of Corelevance: $CR(A;B,C) = CR(A;C,B)$
- Necessity = Relevance × Acceptance: $N(A,B) = P(A) - P(A \vert \bar{B}) = R(A,B)P(B)$
- Sufficiency = Relevance × Rejection: $S(A,B) = P(A \vert B) - P(A) = R(A,B)P(\bar{B})$
- Relevance = Necessity + Sufficiency: $R(A,B) = N(A,B) + S(A,B)$
- Sufficiency/Necessity of Rejection of Premise/Conclusion: $N(A,B) = S(\bar{A},\bar{B})$ and $S(A,B) = N(\bar{A},\bar{B})$
- Persuasiveness = Relevance × Informativeness: $ P_i(A) - P(A) = (P_i(B) - P(B))R(A,B) $

## Numerical Example

The following example illustrates all of the concepts introduced in this series.

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

a | b | P(a,b) |
---|---|---|

$\bar{A}$ | $\bar{B}$ | .25 |

$\bar{A}$ | 𝐵 | .10 |

𝐴 | $\bar{B}$ | .25 |

𝐴 | 𝐵 | .40 |

The marginal probabilities are:

$$ \begin{aligned} P(A) &= P(A,B) + P(A,\bar{B}) = .40 + .25 = .65 \cr P(B) &= P(A,B) + P(\bar{A},B) = .40 + .10 = .50 \end{aligned} $$

And the conditional probabilities:

$$ \begin{aligned} P(A|B) &= \frac{P(A,B)}{P(B)} = \frac{.4}{.5} = .8 \cr P(A|\bar{B}) &= \frac{P(A,\bar{B})}{P(\bar{B})} = \frac{.25}{(1 - .5)} = .5 \end{aligned} $$

**Relevance**

Which lets us calculate the relevance:

$$ R(A,B) = P(A|B) - P(A|\bar{B}) = .8 - .5 = .3 $$

**Necessity and Sufficiency**

The necessity of 𝐵 to 𝐴 is:

$$ N(A,B) = P(A) - P(A|\bar{B}) = .65 - .5 = .15 $$

And the sufficiency of 𝐵 to 𝐴 is:

$$ S(A,B) = P(A|B) - P(A) = .8 - .65 = .15 $$

Notice that relevance is the sum of necessity and sufficiency:

$$ R(A,B) = N(A,B) + S(A,B) = .15 + .15 = .3 $$

And that necessity is relevance times acceptance:

$$ N(A,B) = R(A,B)P(B) = .3 \times .5 = .15 $$

And that sufficiency is relevance times rejection:

$$ N(A,B) = R(A,B)(1 - P(B)) = .3 \times (1 - .5) = .15 $$

**Post-Argument Belief**

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

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

$$ \begin{aligned} P_i(A) &= P(A|\bar{B}) + P_i(B)R(A,B) \cr &= .5 + .9 \times .3 \cr &= .77 \end{aligned} $$

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

**Informativeness**

The **informativeness** is:

$$ P_i(B) - P(B) = 0.9 - 0.5 = 0.4 $$

**Persuasiveness**

And the persuasiveness is:

$$ P_i(A) - P(A) = 0.77 - 0.65 = 0.12 $$

Notice that persuasiveness is equal to relevance times informativeness:

$$ P_i(A) - P(A) = R(A,B)(P_i(B) - P(B)) = 0.3 × (0.9 - 0.5) = 0.12 $$

**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 $P_i(A \vert \bar{B}) = P(A \vert \bar{B}) = .5$.

The post-argument necessity is therefore:

$$ N_i(A,B) = P_i(A) - P_i(A | \bar{B}) = .77 - .5 = .27 $$

And if the subject were to learn additional information causing them to accept $B$ completely, then new posterior would be $P_j(A) = P_i(A \vert B) = P(A \vert B) = .8$.

The post-argument sufficiency is therefore:

$$ S_i(A,B) = P_i(A \vert B) - P_i(A) = .8 - .77 = .03 $$

Oaksford, M., & Hahn, U. (2013). Why are we convinced by the ad hominem argument?: Bayesian source reliability and pragma-dialectical discussion rules. In F. Zenker (Ed.), Bayesian argumentation: The practical side of probability (pp. 39–58). Springer Science + Business Media. https://doi.org/10.1007/978-94-007-5357-0_3 ↩︎

Hahn, U., & Oaksford, M. (2007). The rationality of informal argumentation: A Bayesian approach to reasoning fallacies. (https://psycnet.apa.org/record/2007-10421-007) Psychological Review, 114(3), 704–732. https://doi.org/10.1037/0033-295X.114.3.704 ↩︎

Hahn, U., Oaksford, M., & Harris, A. J. L. (2013). Testimony and argument: A Bayesian perspective. (https://psycnet.apa.org/record/2013-00206-002). In F. Zenker (Ed.), Bayesian argumentation: The practical side of probability (pp. 15–38). Springer Science + Business Media. https://doi.org/10.1007/978-94-007-5357-0_2 ↩︎