Convergence is a basic mode of organization of complex discourse supporting a conclusion, S. Convergent, Linked, Serial.
Two or more arguments are convergent when they independently support the same conclusion. The arguments are said to be convergent or co-oriented, and the argumentation is called convergent or multiple.
“Two reasons are better than one”: in a convergent argumentation, a claim is defended on the basis of several arguments which, considered separately, can be relatively inconclusive, but, considered as a whole, combine to make a stronger case: “My computer is beginning to age, there are discounts on the price of my favorite brand, I’ve just got a bonus, I will buy one! ”.

In the above diagram, each argument is represented as a whole. The following diagram spells out the transition laws according to Toulmin’s proposal, S. Layout; compare with linked argumentation:

As well as pro-arguments, counter-arguments can converge to refute a claim. S. Script.

This open structure defines the argumentative net, as opposed to the demonstrative chain. In the demonstrative chain, each step is necessary and sufficient; if one step is invalid, the constituent parts, and, in turn, the whole construction collapses. In the case of the argumentative net, if one link in the mesh breaks, the net can still be used to catch fish, at least the biggest ones.

In a convergent argumentation, the organization of the sequence of arguments is relevant. If the arguments are of a very different strength, a weak argument alongside a strong argument risks damaging the whole argumentation, especially if this argument ends the enumeration:

He’s a great hunter, he killed two deer, three wild boars and a rabbit.

In classical rhetoric, the theory of discourse general organization (Lat. dispositio) discussed the supposed different persuasive effects of the various possible textual arrangements of converging arguments of different strength, S. Rhetoric.

Convergent arguments can be merely listed (paratactic disposition):

Arg, Arg and Arg, so Concl

The argument can be connected by any listing or additive connective:

first, Arg1; second, Arg2; third, Arg3; so Concl.
Additionally, also, in addition, let alone, moreover, not only, 
besides

Connectives such as besides, not only, in addition, let alone, not to mention… not only add argument upon argument(s), they present them as if each one was actually sufficient for the conclusion, and are adduced just “for good measure” (Ducrot & al. 1980, pp. 193-232):

No, Peter will not come on Sunday, he has work, as usual, besides his car broke down.

The additive approach considers that each argument brings in a part of truth, and that these parts can be arithmetically added to create one big decisive discourse. Speech activity theory considers that by nature, an argument is presented as sufficient, and that their addition actually obeys the logic of commercial display for consumers (the audience), that is to say the speaker offers the audience a range of equally satisfying and self-sufficient arguments.

Case-by-case argument  — To refute the conclusion of a convergent argument, each of the arguments supporting this conclusion must be discarded. Thus, a convergent argument is countered by a case-by-case refutation, limited to cases that have been advanced by the proponent.