The opposition, or progression, from to persuade to to convince, along with the development of audiences from particular to universal, is a major focus of the Treatise on Argumentation (Perelman & Olbrechts-Tyteca [1958]), S. Persuasion

1. To persuade a particular audience,
to convince the universal audience

Perelman & Olbrechts-Tyteca significantly restructure the concept of audience. First, the notion is broadened to encompass written communication, “every speech is addressed to an audience, and it is frequently forgotten that this applies to everything written as well” ([1958] p. 6-7). The focus put on this en- larged concept of audience explains the fact that the Treatise does not engage in the analysis of delivery (pronunciatio), the oral, face-to-faces, dimension of classi- cal rhetoric, S. Rhetoric.

The Treatise goes beyond actual audiences to consider the particular audiences and the universal audience. The former is the sole object of classical rhetoric; the latter is a philosophical projection of the essential characters of the former. The concept of audience is then extended to cover self-deliberation (exploiting the resource of polyphony:

Thus, the nature of the audience to which arguments can be successfully pre- sented will determine to a great extent the direction the arguments will take and the character, the significance that will be attributed to them. What formulation can we make of audiences, which have come to play a normative role, enabling us to judge on the convincing character of an argument? Three kinds of audiences are apparently regarded as enjoying special prerogatives as regards this function, both in current practice and in the view of philosophers. The first such audience consists of the whole of mankind, or at least, of all normal adult person; we shall refer to it as the universal audience. The second consists of the single interlocutor whom a speaker addresses in a dialogue. The third is the subject himself, when he deliberates or gives himself reasons for his actions. (Id., p. 30)

2. A normative opposition

While the translators of classical rhetorical texts use the verbs to persuade or to convince interchangeably, Perelman & Olbrechts-Tyteca differentiate between these two verbs on the basis of the quality of the audiences:

We are going to apply the term persuasive to argumentation that only claims validity for a particular audience, and the term convincing to argumentation that presumes to gain the adherence of every rational being. ([1958], p. 28)

This is a stipulative definition, based on a normative perspective. For the New Rhetoric, the norm of argumentation is constituted by the hierarchy of audiences who accept it. This position strongly distinguishes the new rhetoric from the standard theories of fallacies, for which the norm is given by logical laws, or by a system of rules defining rationality. S. Norms; Rules; Evaluation.

3. To persuade, to convince: The words

3.1 History

The Greek word used to refer to rhetorical evidence is pistis. Unlike the scientific and logical word proof, pistis belongs to a family of terms expressing the idea of “trust in others; which can be relied upon” and “proof” (Bailly, [Pistis]). The family of Greek terms translated as “persuasion” refers to “obeying”, as well as to “persuading, seducing, deceiving” (id., [Peitho]). The name of the goddess Peitho, the companion of Aphrodite, sometimes Aphrodite herself, goddess of beauty, seduction and persuasion, also belongs to this family. From this perspective, the word pistis is syncretic; it covers what is for us the field of influ- ence, proof, seduction, submission and persuasion. By definition, “rhetorical evidence is persuasive”.

The Latin verb suadere means “to advise”; the corresponding adjective, suadus, means “inviting, insinuating, persuasive” (Gaffiot [1934], Suadeo; Suadus). Persuadere is composed of suadere and the aspectual prefix per-, which indicates the completion of the process, meaning “I. Decide to do something […] II. Persuade, convince” (id., Persuadeo).

Convincere is composed of con- (cum-) “totally” + vincere “conquer”: “utterly conquer” (id., Convinco); its primary meaning is “to confound an adversary” (ibid.). Just like per- in persuadere, the prefix cum- refers to a completed action. The same meaning is expressed in to convict, coming from the Latin convictus, past participle of convincere meaning “to refute, convict” (MW , Convict, Etymology):

1: to find or prove to be guilty. The jury convicted them of fraud.
2: to convince of error or sinfulness

Both persuadere and convincere mark the completion of the action. According to the grammatical normative tradition, to convince is should be used in  situations in which beliefs are changed without action, whilst to persuade is should be used for situations in which action is undertaken; the rule is based on the etymology of the words. In practice, both terms are used as synonyms. The traditional rule may be based on the principle of superfluity, whereby there cannot be two words with the same meaning, as there cannot be two laws to the same effect. Yet two words can have the same meaning until everyday usage differentiates them.

3.2 Lexical opposition persuasion vs. conviction

The verbs to persuade and to convince belong to a lexical-semantic field including:

This lexical basis is a rich source of semantic orientations and oppositions whose exploitation could contribute to the reflection on the diversity of the expected effects of discourse.

To persuade and to convince are equivalent in many contexts.

A tries to persuade / convince B of something
A addresses a persuasive / convincing argument to B
=> then B adopts new persuasions / convictions

Nonetheless, in other contexts, they are non-equivalent:

A letter of persuasion — not *conviction
A considers that B is persuadable (-ible) — not *convince-able

The pair persuader / persuadee is not marched by a pair *convincer / *convincee. Convictive and convict are, at least etymologically, linked to convince. To persuade has not produced corresponding words.

The present participle convincing can be used as an autonomous adjective, meaning “cogent”; a conviction is “a strong belief”. “Very convincing” seems more common than “very persuasive”; nonetheless, both can be used to qualify not only an argumentative discourse but also many other kinds of discourse:

very convincing accounts, reports…
— — novels, tales, narratives…
— — portraits