The argument to the matter address the relevant facts and the central issue under discussion, that is the substance of the controversy, the merits of the case, the heart of the matter under discussion.

Discussion of the matter is avoided, for example, when somebody accused of corruption and embezzlement of public money answers to the charge by a counter accusation of misogyny, using a classical argument substituting a private and potentially shameful ulterior motive for a public and honorable good reason.
In that sense, an argument to the matter is a quite different thing from an argument drawn from the subject matter of the law.

Like the ad judicium and ad rem arguments, arguments addressing the matter under discussion are not argument schemes, that is, forms of reasoning leading to a conclusion, such as the argument by analogy or by opposites. Any argument scheme may or may not be used to discuss the substance or form of a debate.

The three labels ad rem, ad judicium and to the matter could be taken absolutely, in the Lockian style, as referring to arguments producing knowledge based on natural objects by scientific method, see ad judicium.
On the global opposition ad judicium – to the matter – ad rem  / to the letter, S. Ad judicium.

1. Argument to the matter and validity

To say that an argument is to the matter means that it is relevant to the debate, and that it constitutes a substantial contribution to the discussion.
From the immanent evaluative perspective of the third parties, such arguments are the only ones whose strength and value are worthy of discussion and should be registered or mentioned in the argument map of the discussion.

This does not mean however that they are automatically validated. A party can invoke a precedent, for example, which is clearly a legitimate and substantial move, when dealing with the matter. The precedent can, however, be criticized and rejected, by an argument showing that the so-called precedent are not similar enough to the actual facts. This argument although on the matter is finally declared irrelevant to the issue under discussion Nonetheless, it can be registered as an interesting objection.

2. Form and substance (matter) of the discussion

Arguments dealing with the substance of the discussion are complementary to arguments about its form of the discussion,  which relate to the conditions in which the discussion is taking place. They have to do with the framework, procedure and rules according to which the issue is dealt with. For example, participants may object that they could not get in time the needed documents; or that the quorum was not reached.

3. Logos-based arguments and arguments to the matter

Misleading associations could lead one to think that the arguments related to the logos are “logical” and therefore objective, dealing with objects, and, as a consequence, with the matter and substance of things. As such, logos-derived arguments would be opposed to ethotic and pathemic arguments, the latter being  subjective and arbitrary.

In everyday argumentation, as well as logo-ic arguments, ethotic and pathemic arguments exploit the logos, understood as language and discourse. In an argumentative situation, however, it is the question which determines what the object, the matter and substance, of the debate is.
Arguments referring to persons, their values and emotions are substantial (ad rem and ad judicium) to the extent that they are relevant to the question. Recalling the previous convictions of a person is not irrelevant in all contexts. The description of the state of emotional shock in which the victim was found, for example, might be relevant in court. The problem is distinguishing between the aspects of a personality which are relevant to the discussion, and those which are not. This process is particularly complicated when the persons involved are parties in the argument process.

4. Indirect arguments, peripheral arguments,
arguments to the matter

Indirect argument such as an argument from the absurd, or from ignorance, and they can be to the matter or not.

The same is true for peripheral arguments, exploiting indices accidentally associated with action. A peripheral argument on the person, for example, is to the matter if relevant to the discussion: a witness saw him near the scene of the crime; or not really so: a witness says the suspect is a good friend of his, S. Circumstances.

[1] http://carm.org/dictionary-argumentum-ad-judicium (20-09-13).