# Model of an argument

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?

• What's new from your recent similar question about (defeasible) logic of reflective equilibrium besides more formalism in appearance?... Commented Sep 18, 2023 at 20:13
• @DoubleKnot What's similar? The other post doesn't even mention justification graphs. Commented Sep 18, 2023 at 20:23
• I think there's an important element that isn't being mentioned. Casual Arguments are more than following a chain of premises to conclusions, they are about how such a chain paints a picture/story that is compelling. So, what makes a justification graph good or bad isnt whether it's premises are weak or if it has contradictions- but rather if it paints an aesthetic picture and tells a compelling story. The most important part of the justification graph is, is it a sexy graph? Commented Sep 18, 2023 at 20:31
• Any logic flow could be said to be painted to be as DAG or better yet a lattice or even a simple 1-d sequence if needed though as Michael Carey mentioned it may not be impressive enough to be mentioned... Commented Sep 18, 2023 at 20:36
• Kristian Berry had several posts here on epistemic justification graphs, and one of the features was that they are not linear from premises to conclusions because different claims "support" each other mutually. Another feature of informal arguments is that directed edges do not quite fit because inferences have more complex structure (due to warrants and defeasability) captured, for example, by Toulmin's model. Commented Sep 18, 2023 at 23:08

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.

• These are some good points. (1) Degrees of credence can be modeled as propositions expressing the degree, e.g. P = "I am highly confident that the sky is cloudy." Then we just assert P. So there's no need for explicit probability in the argument model, just inside the propositions. (2) "X is likely" can be judged as just black and white contradiction with "X is unlikely." (3) Ultimately we either state a proposition in the argument or we don't state it. That's binary even if the beliefs are a matter of degree. Commented Sep 19, 2023 at 15:44
• (4) Despite (1-3), it might be a good idea to have explicit probability in the argument model (as well as fuzzy logic), and partial contradiction, but how? Most importantly, when can the person judge that the argument is "flawed enough" that their θ must be revised? That's ultimately what it's all about. (5) Yes, propositions are defeasible, I mentioned how that fits in. A proposition is defeated if its contradiction can be obtained via a judgment based on a superset of the premise set for the proposition. (6) θ does do a lot of work, it's basically "your whole mind." Seems okay to me. Commented Sep 19, 2023 at 15:51
• (7) "The distinction between an unjustified proposition and a proposition whose justification proceeds from the empty set seems a rather fine one" - I mentioned, "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." The idea is here we don't worry about ultimate epistemic justification, but only whether our argument persuades the other person. So if they accept the premise (via J({}, b, θ) ), that's justified enough for the argument to work. Commented Sep 19, 2023 at 15:53
• My point about making θ do a lot of work is that without some content, it is not useful. It would be like saying, I have an equation that describes everything that happens in the universe, it's Φ = 0. The value of a successful model is that it exhibits the component parts and the relations between them in enough detail to be highly informative. Commented Sep 19, 2023 at 16:47
• Well, I'm thinking of θ like the weights of a neural network. In ML it's fine to just represent that with one symbol. It's a black box, internally not understandable, whose meaning is revealed only by plugging in inputs and seeing what comes out (in this case, J() and C() is what comes out). I might imagine some neural network could actually be used to produce judgments and evaluate arguments, and revise its parameters, based on something like this model. To do that, the θ-revision process would need to be fleshed out (probably involving some sort of error function and gradient descent). Commented Sep 19, 2023 at 16:53

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.

• I just read the conversation with Conifold, and I see you seem reluctant to accept that informal necessarily maintains some loose degree of circularity to maintain coherence. Perhaps the motivations might make it more palatable. Knowledge is often constructed in dependent domains independently. Think about how analytical geometry relies on both arithmetic and geometric reasoning which started off as separate domains. Eventually, as the two disciplines were integrated by the introduction of the Cartesian plane, you have a situation where geometric arguments support arithmetic ones...
– J D
Commented Sep 19, 2023 at 15:44
• but that the arithmetic arguments can be seen as supporting geometric conclusions, and neither domain is foundational to the other. The Greek construction to prove the Pythagorean theorem can be used as a foundation to demonstrate the arithmetic principle of the measures of legs and the hypotenuse, or you can introduce proof in the other directional. It is the coherence (the justificational circularity) between the domains that fully characterizes and endorses the legitimacy of the Pythagorean theorem. Why? Because it shows that what we see with our eyes as sensible experience...
– J D
Commented Sep 19, 2023 at 15:47
• corresponds to the truths our PA-based language show to be true also. Making math rigorous is about showing the circular justification between multiple domains. That's why calculus was intuitive up until analysis was fully in place. Vicious circularity is fallacious NOT because it is false (circular arguments are tautological), but because IT abstracts too much from the argument and relies the triviality of the Principle of Identity. In more complex forms of justification, it is desirable because it is demonstrative of a lack of contradiction. Quine's web of belief and Popper's falsifibility.
– J D
Commented Sep 19, 2023 at 15:49
• In mathematics everything is done according to a formal system, and formal systems do not have circularity. They begin from axioms and a proof proceeds with the structure of a DAG. It may be that theorem T1 can be used to prove theorem T2 and vice versa, but this alone does not support T1 or T2. For example, in computer science we may say, "If 3SAT is not in P then subset-sum is not in P. If subset-sum is not in P then 3SAT is not in P." But neither of these statements shows 3SAT to be outside P, nor does it show subset-sum to be outside P; that remains the open question, "P=NP?" Commented Sep 19, 2023 at 19:33
• To resolve that question at least one of the two propositions (e.g. "3SAT is not in P") must be proved on its own merits starting from the axioms, which has not happened yet. The circular justification, ungrounded in the axioms, is not enough to establish either proposition. Commented Sep 19, 2023 at 19:33