Model of an argument

Question

I have the thought that an informal argument is fundamentally about building a justification graph: a directed acyclic graph from premise propositions to intermediate and conclusion propositions, where each proposition is justified by its parents in the graph according to some informal, malleable notion of defeasible justification.

The justification graph may be good or bad. It is bad if the premises or inferences are insufficiently justified. It is bad if it contains a contradiction, or if a contradiction can be derived from it using the same informal notion of justification. It is bad if some of the propositions in the graph can be defeated by the introduction of more evidence. It is also bad if the justification rules used in the graph are themselves insufficiently justified, or yield contradictions when applied to other topics.

When people argue, they are trying to show that their conclusion can be supported by a good justification graph, or that the justification graph the other person is building is bad. The point of making a good justification graph is in order to persuade the other person of the conclusion, so each of the premises and justifications in the graph should be previously accepted by the other person, otherwise the graph lacks the power to persuade.

If we are concerned with making a persuasive argument, then we can reduce the question of whether a premise is justified, to the question of whether the other person accepts that premise.

Let's use some notation. Say that A is a set of premise propositions and b is a conclusion proposition. J(A, b) can be a true or false value saying whether a person considers it justified to conclude b from A in the absence of any other evidence. J is defeasible; if a is an additional premise, J(A, b) may be true while J(A ∪ {a}, b) could be false.

An instance of "J(A, b) = true" can be called a "judgment" or "inference." It may be read as, "b is justified by A," or "we judge b, based on A." b is the conclusion of the judgment, and A is the premises of the judgment.

These judgments are not necessarily valid or reasonable, and may be self-contradictory; they appeal to a particular person at the time they were made, and that's all.

One person's J might not be the same as another person's J, or one person's J might vary as he learns more things. To reflect this variation we want to parameterize J. So now instead of J(A, b) we may say J(A, b ; θ) for some parameter vector θ. We could think of J like a neural network, θ being the weight vector. We may also think of θ as a person's "background beliefs" that they bring with them into the argument.

A proposition p that is accepted as a premise would be reflected in J as J({}, p ; θ) = true, i.e. it is justified to conclude p from the empty set.

We need a notion of contradiction between propositions. We might be able to derive this from J, but for now it's easier to use a second symbol C. C(a, b ; θ) = true if propositions a and b contradict, given the parameters θ.

A justification graph that would persuade someone with background beliefs θ, is then an ordering of a set of propositions P, together with a set of inferences. Each inference is a pair (A, b), where A ⊂ P, b ∈ P, b occurs later in the ordering than every member of A, and J(A, b ; θ) is true. Each proposition b ∈ P must appear at least once as the conclusion of each judgment (the graph contains no unjustified propositions).

A justification graph, combined with parameters θ, is bad if it leads to any contradiction. That is, if it is possible to extend the graph with additional judgments in such a way that the extended graph contains two propositions a, b where C(a, b ; θ) = true.

Here we must be careful about defeasibility as distinct from contradiction. J, remember, may include defeasible judgments where J(A ∪ {a}, b ; θ) = false while J(A, b ; θ) = true. We don't want to throw out a justification graph because we added judgments that can be defeated that weren't in the original graph. Let's more specifically say that evidence A2 defeats evidence A1 with respect to b1 and b2, if J(A2, b1 ; θ) = false, J(A1, b2 ; θ) = true, and A1 ⊂ A2, and C(b1, b2 ; θ) = true. We may say that if we extend a justification graph to form a contradiction, it doesn't count unless none of the propositions we used to extend the graph, beyond the initial propositions in the graph, can be defeated in this way.

The full θ is generally unknown in practice, but parts of it are revealed through the justification graph. If the justification graph contains a judgment that b follows from A (given θ), we might use this information about θ to extend the graph into a contradiction, thus refuting it. For example, if someone reasons fallaciously, then we know their θ allows the fallacious inference. We can then apply the fallacy to a different context to produce a contradiction, thereby refuting the θ that allowed the fallacious inference.

When a justification graph persuasive to someone with beliefs θ is contradictory, then we understand the person needs to revise their θ so that the contradictory justification graph cannot be formed. I don't describe how this revision might happen.

Is this a good model of how rational argument works, or ought to work? Are there any elements of rational argument that cannot be fit into this structure?

Answer 1

Some thoughts.

Contradictions are not the only bad consequence of an inference. Propositions that are implausible or weird are also bad. So if an argument can be shown to have an implausible consequence, this counts against it. If we were using probability theory, a bad consequence might be an assignment of a low probability to a proposition considered highly plausible, or the assignment of a high probability to a proposition considered implausible.

Speaking of probabilities, are you sure you want all propositions to evaluate simply as true or false? And all justification relations to be true or false? A lot of reasoning is concerned with degrees of credence. Your analogy of neural network weightings suggests something more nuanced.

Propositions are defeasible, as well as inferences. We may rely on some premise that we later learn to be false.

The fact that your graph is acyclic suggests a foundationalist approach to epistemology. Someone might wish to argue that some propositions just hang together well and are mutually supportive without having a definite direction of flow of justification in mind.

You are making θ do a lot of work. Without some more content and structure to it, θ looks rather like a catch-all condition. θ would have to include more than just background beliefs, but also methods of reasoning, criteria of acceptability of a proposition, even the underlying logic and what qualifies as a contradiction.

The distinction between an unjustified proposition and a proposition whose justification proceeds from the empty set seems a rather fine one.

Answer 2

Is this a good model of how rational argument works, or ought to work? Are there any elements of rational argument that cannot be fit into this structure?

Yes, and what you have here, outside of the formalization of predication is the essence of the Toulmin model of argumentation that comes from his work Uses of Argument. From WP we see the model of argumentation:

What you described attempting to extend the formal notions to describe informal notions is essentially the same, though Toulmin's language is somewhat different than yours. We can note:

• Judgement is the process by which the mind applies normative criteria to embody the model.
• Normative criteria according to the Toulmin model are domain-specific; epistemic modality is intuitive.
• The rebuttal portion of the argument is an attempt to prevent the introduction of defeaters.
• The application of informal logic is broadly non-demonstrative.
• Real-world argumentation, which Toulmin calls "working logic" in contrast to "idealized logic", is full of bias, fallacy, and non-classical logics play a prominent role and thus is fallible.
• Toulmin accepts the analyticity of statements in the sphere of deductive enterprises such as pure mathematics.
• The model itself incorporates multiple forms of logical consequence since premises are guided by the general form "D since W on account of B therefore qC unless R".

While you may invoke functional notation, and seem to constrain your model with the principle of bivalence, you are certainly on the road to making an argument that applied iteratively leads the Toulmin model generating a graph.