PROOF and the ARTS OF PROOF
The words to prove, a proof, probation, probatory come from the Latin probare and its cognates; probare means “to make good; to esteem, represent as good; to make credible, show, demonstrate; to test, inspect; to judge by trial” (OED, Prove). All of these meanings, suggesting 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 some articulations of the ordinary lexicon of proof.
Agent Names – Some names are related to their root verbs with the meaning “person who (verb)”; so are arguer; reasoner; prover. Demonstrator derives from to demonstrate_2 “to show other people how something is used or done”.
This can be interpreted as a sign of subjective involvement in the mechanism of proving, arguing, reasoning.
Verb Complementation – In “Peter reasons about P”, P is the subject, the substance of the reasoning or of the argument. In “Peter has demonstrated, or proved that P”, the P clause is true and expresses the conclusion of the demonstration. To argue allows 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 relation 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 than to look for something is a weakened form of to find something. Proof is the “terminator” of 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 presents conclusive evidence and convincing arguments. Such discursive practices place argument/evidence in continuity with proof, with proof being the end and finality of the argument: it is “a knock-down argument” (Hamblin 1970, p 249). Arguments are proof-oriented.
Positional Markers – These terms which may be considered as quasi-synonyms in some contexts, can clearly appear as markers of argumentative positions in the context of a debate. In the legal arena, the judge hears the statements and arguments of the parties; each party presents (what it considers to be) evidence and rejects the evidence presented by the opponent as quibbling and sophistry. We are no longer dealing with synonyms, but with strongly opposed antonyms. The difference between evidence/proof, argument, and quibble becomes a simple matter of perspective. Proof and evidence are now nothing more than the positive value I assign to my argument and deny to my opponent’s.
A polite but decisive rebuttal is 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.
Dialogical status — The distinction between demonstration / proof / argument seems to be 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. Proof Between Fact and Discourse
Proofs are expressed in a language, natural or formal, and presented in a discourse. Formal proof brought about by a hypothetical-deductive demonstration is often considered the archetypal proof. Its counterpart in ordinary language would be the argument based on essentialist definition used in philosophy and theology. In other fields of activity, the discourse of proof requires a reference to the world, in which case, proof is now seen as evidence and fact. The proof-evidence is constructed through a series of experiences and calculations, as suggested by the concrete metaphors used to talk about evidence: to produce, to provide evidence, to bring a proof, to make a demonstration. This connection to reality is what distinguishes proof and argumentation on the one hand from formal demonstration on the other.
The concept of proof as fact invokes non-discursive evidence of material realities, that can be seen and touched. The proof that I did not murder Peter is that he is alive and standing in front of you. Such situations would seem to render language superfluous. But 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 can be silently presented to the appropriate judges. If some facts “speak for themselves”, at other times they are not so “eloquent”, or even “silent” to many. They need to be spoken for, and discourse is needed 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 predict their acceptance, see proof, facts and belief.
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 that are not self-evident, by their integrating them into a coherent discourse in the correct language, be it a demonstration, or a story that explains what happened.
— Cognitive and even aesthetic: The proof must be relatively clear, and, if possible, “elegant ».
— Psychological: It removes doubt and inspires confidence.
— Rhetorical: It is persuasive.
— Dialectical: It removes the challenge, and closes the discussion.
— Social: It builds consensus, reassures the community affected by the problem, and strengthens their confidence in their technical ability to produce evidence especially, but not only, in the social and legal domains.
— Conversely, evidence excludes: Those who accept proof believe that those who resist it proof must be crazy, weak-minded, or 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 a discourse: arguing, demonstrating, proving, all require a semiotic medium, a language developed in a discourse. The same can be said of reasoning, although the term focuses on the cognitive aspects of the process.
— An intention: Like any discourse, the flow of demonstrative, argumentative, probative, and inferential discourse is organized by an objective, that is, an intention.
— A question: These processes begin with a problem, an uncertainty, a doubt.
— A process of illation (derivation) or inference: The notion of inference is primitive. In logic an inference is defined as the logical derivation of a statement, the conclusion, from a set of premises. The intellectual process of inference contrasts with the intuitive approach, in 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, through data or assumptions expressed by statements and supported by underlying principles, the nature of which depends on the field concerned, see self-evidence.
— Argument, proof and demonstration refer to something external; the development of the discourse is more or less determined by the external world. Anything and everything can be said, but reality sets 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 proofs differ according to the domain, the kind of technical language used and the kind of experimental method employed in that domain. The establishment of large classes of scientific proofs is the task of epistemologists. Natural language argumentation is characterized by its unique ability to combine a large variety of heterogeneous proofs, corresponding to the different argument schemes with technical proofs coming from technical domains.
5. Argumentation Among the Arts of Proof
Perelman & Olbrechts-Tyteca’s New Rhetoric contrasts “argumentation” with “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 proof, see demonstration, without considering the possibility of bridging the gap between the two discursive regimes, or of positioning them on the same truth-oriented scale. Increasingly, however, contemporary discourses, 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 to deal with such mixed data. This is true of all the varieties that the Treatise considers to be typically argumentative: “speeches [of politicians] … pleadings [of lawyers] … decisions [of judges] … treatises [of the philosophers]” (Perelman & Olbrechts-Tyteca [1958], p. 10).
Approaches to 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 framework of the situation. The study of argumentation is thus clearly distinguished from the study of scientific methodology, and from the epistemological study of proof, demonstration, explanation or justification in mathematics, science, or philosophy, see demonstration.