Epistemology – Colby Whitted

Aumann’s agreement theorem says that two rationally acting agents with common priors and common knowledge of each of their posterior probabilities will necessarily have equal posteriors. Meaning, they will agree. They cannot disagree to agree. Since most of our arguments do not meet these criteria this theorem while surely important is not directly super useful in the real world.

In a previous post I talked about Abstract Objects and how they relate to disagreements. Since the same word can point to differing abstract objects among multiple agents this problem contributes to disagreements. I believe that starting with the problem of differing abstract objects is a pragmatic solution to reducing disagreements.

The other main problem with disagreements is their informal nature. The actual process of deriving formal statements from informal statements could never be itself a formal process but this does not make it useless. We can rely on the two disagreeing agents working together to formalize both of their claims.

My idea here only goes so far as to create a better initial dialogue. I may revise this post. In the future I may write something about a process for an ongoing debate.

I will be using an example debate here as I go through the 4 steps I think are crucial to the process. In each step a logician would work with the agent in an iterative process. In the first three steps it would be necessary for the logician to work with both agents together.

Example

I will be using a transcript from the first 2020 presidential debate between Biden and Trump to take example disagreements and process them into formal statements. I’m only going to be analyzing the first 8 minutes and 30 seconds of the debate, found here.

Political debates aren’t really about genuine dialectics. I am using this as a convenient example.

Trump and Biden begin by responding to this question from Chris Wallace: “President Trump, you nominated Amy Coney Barrett over the weekend to succeed the late Ruth Bader Ginsburg on the Court. You say the Constitution is clear about your obligation and the Senate’s to consider a nominee to the Court. Vice President Biden, you say that this is an effort by the President and Republicans to jam through on an appointment in what you call an abuse of power. My first question to both of you tonight, why are you right in the argument you make and your opponent wrong? And where do you think a Justice Barrett would take the court?”

1: Distillation

To formalize the first 8 minutes and 30 seconds of the debate we will first distill the claims made. How I would expect this to work in practice is that the arguing agents and advisors would sit down with a logician and make their informal claims and justifications as simple and direct as possible. In real life the agents would certainly evolve their arguments during the distillation process as they saw the weaknesses in them. However we are reviewing a debate where no such process happened so I have to take the arguments the agents are making at face-value and distill them with no feedback from the agent.

The distillation of Biden’s claim is that ‘Trump’s supreme court nomination is an abuse of power’.

Trump’s distilled claim is that he has an obligation to consider a court nominee which we can consider to be logically equivalent to a negation of Biden’s claim.

2: Clear definition of terms and predicates

Now that we have distilled the main claims, we will assign some constants and a predicate to pointers. It is important that both agents are part of this process.

Let A() = Abuse of power
Let C = The US Constitution
Let O = Trump’s Obligation to consider court nominee
Let N = Nomination of Amy Coney Barrett
Let T = Trump is president
Let V = American people are losing their chance to effect court nominations
Let H = We’re in the middle of an election

3: Approximate certain claims as axioms

Because of the regress argument in epistemology we must arbitrarily end the justification regress and approximate certain claims as axioms. We can safely do this with C, N, T, and H.

4: Formalize

We will now use these pointers in a formal statement using formal logic. First showing Biden’s claim, his level 1 justification and his level 2 justification. I will then show Trump’s claim, Trump’s level 1 justification and his level 2 justification.

Biden’s claim is A(N)
Biden’s justification for his claim is V
Biden’s justification for V is H ∧ N

So in whole we have these two sentences:
V ⇒ A(N)
(H ∧ N)⇒V

Trumps claim is ¬A(N)
Trump’s justification is O
Trump’s justification for O is C ∧ T

So in whole we have these two sentences:
O ⇒ ¬A(N)
(C ∧ T) ⇒ O

In conclusion my proposed solution to formalizing debate between multiple agents is a process that would look something like this: