Bayesian Argumentation: Summary of Definitions

Below is a summary of all the terms and equations defined in the essays in this series, followed by a detailed example.

• Claim: A statement that one can agree or disagree with

• Premise: A claim intended to support or oppose some conclusion

• Conclusion: The claim supported or opposed by the premise

• Argument: A premise asserted in support of or opposition to some conclusion

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.

• 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

• 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:

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:

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