Proof and the Arts of Proof

The words to prove, a proof, probation, probatory come from the Latin probare and its cognate words; probare means “to make good; esteem, represent as good; make credible, show, demonstrate; test, inspect; judge by trial” (OED, Prove). All these meanings evoking a practical activity are still present in the use of proof in rhetorical theory.

1. Vocabulary of the arts of proof

The following words belong to the elementary lexicon of the arts of proof.

to argue; an arguer, an argument, an argumentation; argumentative
to demonstrate; a demonstration; demonstrative
to prove; a proof, a prover; probatory
to reason; a reason, a reasoner; reasoning; reasonable; rational
evidence; evidential

The following remarks deal with the articulations of the ordinary lexicon of the arts of proof.

Agent names — Some names are related to their root verbs with the meaning “person who (Verb)”; so are arguer; reasoner; prover. Demonstrator a derivative from to demonstrate_2 “show other people how something is used or done”.

This can be interpreted as a mark of a subjective involvement in the mechanism of proving, arguing, reasoning.

Verb complementation — In “Peter reasons about P”, P is the issue, the substance of the reasoning or of the argument. “Peter has demonstrated, or proved that P”, the P clause is true and expresses the conclusion of the demonstration. To argue admits both constructions:

Peter argues about P: P is the issue,
Peter argues that P: P is the claim, but to argue does not imply that its P clause is true.

Aspectual distinction — The relationship of argument to proof is grammatically an aspectual distinction, that of unaccomplished / accomplished. To argue is no more a semantically weakened form of to prove that to look for something is a weakened form to find something. The proof is the “terminator” of the argument.

Semantic orientation —Evidence, proof, argument and demonstration, however, can function in co-orientation, as quasi-synonyms in many contexts. The lawyer is engaged in a brilliant demonstration in which he brings conclusive evidence and convincing arguments. Such discursive practices put in continuity argument/evidence and proof, the proof being the end and finality of the argument: it is “a knock-down argument” (Hamblin 1970, p 249.). Arguments are oriented towards proof.

Position markers — These terms which may be regarded as quasi-synonyms in some contexts, may clearly appear as markers of argumentative positions in the context of a debate. In the judicial field, the judge hears the statements and arguments of the parties; each party brings (what they consider to be) proofs and rejects those brought by the opponent as quibbling. We are no longer dealing with synonyms, but anti-oriented antonyms. The difference between evidence/proof, argument, and quibble becomes a simple matter of perspective. The probative value is now no more than the positive assessment I give to my argument and I refuse to grant to that of my opponent.

A polite although decisive rebuttal will be proposed as a mere objection and a simple argument. Argument is then a “lexical softener” for proof, its use implies a distance, a lesser commitment of the speaker to the claim.

Dialogic Status — The distinction demonstration / proof / argument seems primarily sensitive to the presence or absence of counter-discourse. This is why the word argument is used to describe reasoned discourse at both ends of scientific activity, in learning activities, as well as in the sharpest controversies over open questions, where two discourses both perfectly equipped theoretically and technically, revert to the status of argument, simply because there is disagreement.

2. The proof between fact and discourse

Proofs are expressed in a language, natural or formal, and put forward in a discourse. Formal evidence brought by a hypothetical-deductive demonstration is often seen as the archetypical proof. Its counterpart in ordinary language would be the argument based on essentialist definition used in philosophy and theology. In other areas of activity, probationary speech requires a reference to the world, in which case, evidence is now seen as a fact. The proof is built by a series of experiences and calculus, as suggested by the concrete metaphors used to talk about evidence — to produce proof, to provide evidence, to bring a proof, to make a demonstration. This connection with reality makes the difference between proof and argumentation on one side and formal demonstration on the other.

The concept of proof as fact invokes non-discursive evidence of material realities, perceptible to sight and touch. The proof that I did not murder Peter is that he is alive, standing before you. Such situations seem to make language superfluous. Nonetheless, facts can become evidence through discourse alone. Evidence is relative to a problem, and discourse frames the situation in which the evidence solves the problem. Evidence may be silently brought before the relevant judges. If some facts “speak for themselves”, some other times they are not so “eloquent”, or even remain “silent” for many. One must speak for them, and discourse is required to make the material evidence visible. The cruel experience of Semmelweiss has certainly shown us that the de facto existence of seemingly indisputable facts does not foretell their acceptance (Plantin, 1995, chap. 7).

 “The Wolf and the Lamb” — The La Fontaine fable “The Wolf and the Lamb” (Fables, I, X) shows how innocent people can trust material evidence, and that material evidence does not carry the day.

The reason of the strongest is always the best,
As we’ll show just now.


A lamb was quenching its thirst
In a pure water stream.
A fasting wolf came by, looking for adventure;
Attracted to this place by hunger.

The wolf starts with a violent reproach, as men do with their future victims

—What makes you so bold as to cloud my drinking?
Said this animal, full of rage,
You will be punished for your audacity.

The offense is assumed (you cloud my drinking). The request for explanation of motives (what makes you so bold […]?) appears to give the lamb the opportunity of explaining itself. Yet, the accusation is immediately followed by the sentence (you will be punished for your temerity). This incriminating speech is deeply mysterious, why does the wolf speak? It could simply take advantage of the food it was yearning for, and finally met, devouring the lamb like the lamb drinks the water. With exquisite courtesy, the lamb denies the presupposed fact and its denial is backed up by undisputable proof, S. Self-Evidence:

—Sir, answered the lamb, let Your Majesty
Not get angry.
But rather, let Her consider,
That I am quenching my thirst
In the stream
More than twenty steps below Her;
And that, consequently, in no way,
Am I clouding his beverage.

The lamb’s argument is conclusive, physical laws are such that the brook never flows upstream. But conclusive does not mean impossible to contradict:

—You do cloud it, said the cruel beast.
And I know you said bad things about me last year.

This second accusation is also rebutted in the same decisive way:

— How could I have done that, when I wasn’t born,
Answered the lamb; I am still suckling my mother

Idem for the third:

— If it wasn’t you, then it was your brother.
— I have none.

But the last accusation is irrefutable; the lamb is given no chance to refute it:

— Then it was a relative of yours;
For you have no sympathy for me,
You, your shepherds and your dogs.
I am told of that. I must avenge.

The conclusion is that good reasons do not change the course of history:

Thereupon in the dark of the forest
The wolf carries the lamb, and then eats it,
Without further ado.

3. Functional heterogeneity of the discourse of the proof

Whatever the field, the discourse of proof is functionally heterogeneous. Proof fulfills a number of functions:

— Alethic: it establishes the truth of a fact.

— Epistemic: it justifies a belief; it helps to stabilize and increase knowledge.

— Explanatory: it accounts for facts which are not self-evident, via their integration into a coherent discourse in the correct language, be it a demonstration, or a story accounting for what took place.

Cognitive and even aesthetic: proof must be relatively clear, and, if possible, “elegant”.

— Psychological: it eliminates doubt and inspires confidence.

Rhetorical: it is convincing.

Dialectical: it eliminates the challenge, and closes the discussion.

— Social: it builds consensus, assuages the community affected by the problem, and strengthens its confidence in its technical capacities to produce evidence particularly, but not only, in the social and judiciary domain.

— Conversely, evidence excludes: those who accept proof consider that those who resist the proof must be mad, feeble minded, carried away by their passions.

4. Unity of the arts of proof

The arts of proof — reasoning, arguing, demonstrating, proving — share the following characteristics.

— A language and discourse: arguing, demonstrating, proving, all require a semiotic medium, a language developing in a discourse. The same can be said for reasoning, although the term focuses on the cognitive aspects of the process.

— An intention: Like every discourse, the flow of demonstrative, argumentative, probative, reasoned discourse is organized by an objective, i.e., an intention.

— A question: These processes start with a problem, an uncertainty, a doubt.

— An illation (derivation) process or inference: The notion of inference is primitive. In logic an inference is defined as the logical derivation of one statement from a set of premises. The intellectual process of inference contrasts with the intuitive approach, for which a truth is asserted directly (without mediation) on the basis of its direct physical or intellectual perception. In the case of inference, the truth is asserted mediately, that is indirectly, via data or assumptions expressed by statements and supported by underlying principles, the nature of which depends on the area concerned. S. Self-evidence.

— Argumentation, proof and demonstration are referring to something external; the development of discourse is more or less governed by the external world. Anything and everything can be said, but reality creates limits. The practice of proof and argument is not pure linguistic virtuosity, it must confront objects and events.

— Domain dependence. As argumentation, demonstration and proof are domain dependent. The modes of production of evidence differ according to the field, the kind of technical language used and the kind of experimental method used in the considered area. The establishment of large classes of scientific proofs is the task of epistemologists. Argumentation in natural language is characterized by its capacity to combine a large variety of heterogeneous proofs, corresponding to the various argument schemes.

5. Argumentation among the arts of proof

Perelman & Olbrechts-Tyteca’s New Rhetoric opposes “argumentation” to “calculation”:

The very nature of deliberation and argumentation is opposed to necessity and self-evidence […] The domain of argumentation is that of the credible, the plausible, the probable, to the degree that the latter eludes the certainty of calculation. ([1958], p. 1).

This position leads us back to the Aristotelian opposition between rhetorical “means of pressure” and scientific proofs, S. Demonstration, without considering the possibility of bridging the gap between the two discursive regimes, or of positioning them upon the same truth oriented scale. An increasing range of contemporary discourses, however, are mixed; they seek to articulate some scientific reasoning and data, along with social values and material interests. A contemporary challenge for argumentation studies is to find a way of dealing with such mixed data. This is true of all the varieties considered to be typically argumentative in the Treatise: “speeches [of politicians] … pleadings [of lawyers] … decisions [of judges] … treaties [of the philosophers]” (Perelman & Olbrechts-Tyteca [1958], p. 10).

The approaches of argumentation as a set of “discursive techniques” (Perelman, Olbrechts-Tyteca), as discourse orientation (Ducrot) or discursive microstructure, as dialogue or interaction, anchor the study of argumentation in ordinary linguistic practices, structured by rules and norms depending on the genre of discourse and on the frame of the situation. Argumentation studies are thus clearly distinguished from research in scientific methodology, and from the epistemological study of proof, demonstration, explanation or justification in mathematics, science, or philosophy, S. Demonstration.