All the components of written or spoken discourse in a given situation can be used or manipulated by the opponent in order to present this discourse as untenable, S. Destruction.
The word refutation is used to designate a reactive speech act covering the explicit forms of discursive rejection of positions, opinions, charges or projects. The possible use of  rejection or denial as a synonym of refutation rdoes not imply the absence of argument. As non-preferred second pairs, denials and rejections are also characterized by the presence of accounts. In fully argumentative interactions, refutation is in particular characterized by its explicitness and careful elaboration.

From a scientific point of view, a proposition is refuted if it is proved to be false; the calculation from which it derives contains an error; it affirms something that is contradictory to the observed facts.
From the point of view of ordinary interaction, an argumentative line is contextually refuted if, after being discussed, it is given up by the adversary, either explicitly or implicitly. Accordingly, the question itself disappears, and the interaction progresses to another structuring topic.

As a reactive speech act, refutation can be dealt with in only a verbal (face to face) or written (text to text) dialogue. Monological discourse knows only the concession, there are no refutative subordinate clauses, and concessive clauses reduce the refutation to an objection.

1. Structural refutation

Each component of the propositional argumentative model may be targeted by the act of refutation, S. Argumentation (III)); Layout.

1.1 Turning down the argument

An argument supporting a conclusion may be rejected in different ways.

(i) The argument is declared false:

S1 — Peter will surely arrive on Tuesday; he has been invited to Paul’s birthday.
S2 — But Paul’s birthday is on Monday.

(ii) The argument is rejected as irrelevant to the conclusion, S. Relevance:

S1 — He is very intelligent, he read all of Proust’s work within three days.
S2 — Intelligence has nothing to do with reading speed.

(iii) The argument can be accepted as such, recognized as somehow relevant to the conclusion but may be dismissed as too weak, or of poor quality:

S1 — The President has spoken, the stock market will go up.
S2 — Yes, and what he says goes! (said sarcastically).

The rejection of the argument may lead to a new argumentative question (sub-debate), about the truth, strength or relevance of the former argument.

Turning down the argument does not mean renouncing the conclusion. This is often the case in factual argumentation:

S1 — Peter will surely arrive on Tuesday, he wants to be there for Paul’s birthday.
S2 — Paul’s birthday is on Monday, but sure, Peter will arrive on Tuesday, I bought him his flight tickets.

Nonetheless, in ideological debates, only the most ascetic arguers will refute questionable or bad arguments made in favor of conclusions which they consider to be good or virtuous.

1.2 Turning down the backing

The backing invoked, implicitly or not, is declared false:

S1 — Pedro was born in the Malvinas Islands, so he is an Argentine citizen
S2 — The Falkland Islands are British territory.

The adverbs exaclty, precisely (not) can substitute one backing to another (Ducrot & al., 1982), S. Orientation:

S1 — Noodles for dinner!
S2 — Again! We had noodles for lunch!
S1 — Exactly, we need to finish the leftovers, we don’t want to waste food.

The resulting stasis is produced by the conflict of two topoi:

Dietary, or gustatory pleasure principle: « you have to vary one’s diet« .
Economy principle, against the waste « food should not be thrown« .

1.3 Turning down the conclusion

Conclusions may be dismissed even though some validity is granted to the argument:

S1 — Cannabis should be legalized; the taxes will pay off the National Health Service deficit
S2 — It will certainly increase tax revenues, but it will further increase the number of drug addicts. The prohibition must be maintained.

The counter-argumentation establishes a counter-conclusion leaving the argument it opposes intact, S. Counter-argumentation.

2. Weak refutation protecting the claim

By generalizing of the law of weakness, a weak refutation confirms the attacked position, S. Scale. This principle applies to various interpretative schemes, whose analysis must take into account the whole corpus produced by the argumentative question.

(i) Weak refutation of a poorly re-constructed attacked position

The wise man concludes that the refutation is not worth much, to say nothing of its author, and the problem remains intact.

(ii) Weak refutation of an outstanding exposition of the attacked position

The conclusion is that the attacked position is reinforced by this attempt at refutation. The interpretive calculation is based on the fact that the arguer is qualified.

— The poor refutation is standard, while the quality of the exposition, clearly indicates a good arguer. Since the given refutation is taken to be the best possible (according to Grice’s maxims), and since it is weak or even ridiculous, the conclusion will be that, “since even such an arguer finds nothing else to say, then, the criticized position must actually be correct”, even if this derivation is ad ignorantiam, S. Counter-argumentation.

— The poor refutation is bizarre. It contains obvious errors alerting the careful reader; there is a contrast between the quality and care of the exposition and the scanty character of the refutation. Moreover, this refutation is not put forward in the usual argumentative style of the author. For example, a fine theologian develops in a dialectical and detailed manner, a position condemned by the official authorities of his religion, and refutes it only by arguments drawn from various authorities (which the reader may be aware are considered questionable), so the careful reader is led to think that this oddity is strategic. The speech is apparently refuted, only to be better asserted in reality, the negation serving then to cover the author. This case of indirectness has been theorized by Strauss (1953). If, under special historical, social, or religious circumstances, a discourse is banned, it is nevertheless possible to give it a voice under the cover of its refutation, the negation then serving to protect the speaker from tyrannical authorities.

This strategy of confirmation, or argumentation by weak refutation, is dangerous to maintain. The authorities are not necessarily naive nor uninformed, and they may be well aware of the intended purpose of the pseudo refutation, which will be rightly interpreted as a denial of a belief which is actually held by the speaker: “How can you so be such an expert about heterodox positions and such a fool when dealing with orthodoxy?”.

Such a strategy, based on the opacity of the writer’s intentions, presupposes a double argumentative address, the real intentions can be captured only by a careful reader, while they remain unknown to the hasty reader, who appreciates the weak refutation because it can be easily understood, absorbed and repeated, S. Strategy.

3. Refutation and counter-discourse

The concept of refutation is defined at the very general level of the challenged argumentation. The counter-discourse approach specifies the possible rebuttal strategies according to the specific argument pattern (testimony, authority, definition, induction, causal claim, etc.). The argument pattern is flanked by a counter-pattern, an integral part of the form and substance of the argument considered.

In the Skeptical philosophical style, such a counter-discourses can also be directed at the argumentative type itself, as a general discourse, “against authority, analogy, causality, etc.”, which rejects a priori all forms of argument from authority, etc.

Herrings turn red when smoked; red herrings were used by fugitives to set dogs on a false trail. The expression is used figuratively in argumentation, where the so-called “red herring” strategy is referred to as a diversion strategy, where a distractor is used to eschew the issue under discussion, and deflect the discussion towards an irrelevant issue, S. Relevance.

In mathematics, the relation of reciprocity corresponds to the relation of symmetry: a relation R is symmetric in the domain in which it is defined, if for all the pairs of elements a and b both aRb and bRa hold. The relation “being the friend of” is symmetrical:

Peter is the friend of Paul = Paul is the friend of Peter = Peter and Paul are friends.

1. Returning and anticipating the good

In natural language, the reciprocal relation is defined on the basis of a set of actions which binds two persons. That is to say that if a does something positive to/for b, then b must reciprocate, doing the same thing to/for a. This is the principle of returning favors. The individuals a and b are equal in this relationship. The principle of reciprocity acts as a constraint:

If you treat me to dinner, I must treat you to dinner.

As a form of natural morality, the imperative of reciprocity is expressed by the principle:

Do to others as you would have them do to you. (Luke Gospel, 6:31)
Do not do to others what you would not have them do to you.

This principle is applied in the argument:

I’m polite to you, so be polite to me.

The speaker defines him/herself and defines his or her partner as members of the same category, who must be treated in the same way, S. Rule of Justice.

1. Returning and anticipating the wrong:
Reciprocity as retaliation

Eye for eye, tooth for tooth
If your disappointed lover disfigured you with vitriol, the court grants you the right to treat him likewise.

The law of retaliation is a primitive rule of justice that if A has wronged B in some way, then B can legitimately do the same wrong to A. In contemporary times, we might consider nuclear deterrence, based on the certainty of reciprocal destruction, as a concrete application of this principle. This theory corresponds to a particular case of the, “You too!” argument.

Reciprocity as a legal principle allows different states to assert their equal international dignity, and possibly to justify a retaliatory measure:

If country A requires a visa for the nationals of country B, it is right that country B also requires that nationals of country A acquire a visa.

The concept of argumentative question originates in the notion of stasis, developed primarily by the rhetorical theory of judicial interaction.

The concepts of an argumentative question and an argumentative situation are interdependent. An argumentative situation emerges when two speeches concerning the same topic begin to diverge to some extent. The contact can be made during a remote or face-to-face, oral or written, interaction. Such potentially argumentative situation may evolve into an actual argumentative situation when the divergence is topicalized and ratified by a participant. All these necessary developments delimit an argumentative space, defining what is argumentation, before the appearance of arguments strictly speaking (discursive segments supporting a conclusion).

The existence of a question is at the origin of the paradoxes of argumentation.

1. Proposition, opposition, doubt: A question

The following example, constructed around the recurring question “Should we legalize drugs?” shows how the question assigns argumentative roles, on the basis of the three fundamental argumentative speech acts, to propose, to oppose and to doubt.

• The current state of the law
  

In Syldavia 2022, drug production, importation, exportation, trade, possession, and consumption are forbidden.

This statement corresponds to the state of Syldavian legislation, generally backed by “dominant opinion”, perceived as a matter of course, so needing no argument.

• A proposition

Another discourse is oriented towards a proposition opposed to this prohibition:

P: — The consumption of soft drugs should be legalized, or at least tolerated.

Speaker P steps into the argumentative role of proponent, and opens the debate. All speakers aligned with this proposal serve as allies.

• An opposition

Other speakers oppose the proposal:

O: — That’s staggering!

The speaker O plays the argumentative role of opponent. Speakers willing to hold this type of rejection discourse with respect to the proposition are allies.

• Doubt and question: emergence of the argumentative question

Some participants refuse to align with either position. They are in the position of third parties, synthetizing the proposition vs. opposition relation into an argumentative question, and transforming the discourse confrontation into a full argumentative situation:

TP: — All this is quite perplexing. Should the prohibition of all these drugs they call soft be maintained or not?

The argumentative question is thus generated by the contradiction “discourse / counter-discourse”, hence the schema:

Proposition vs. Opposition → Argumentative Question (AQ)

2. The conclusion as an answer to the argumentative question

When discourse develops into a confrontation, good reasons are needed and quickly provided. The proponent bears the burden of proof and, in order to meet this requirement, must put forward arguments, for example by re-categorizing soft drugs in the same category as alcohol or anxiolytics:

P — Soft drugs are not more dangerous than alcohol or anxiolytics; alcohol is not subject to any general prohibition, and anxiolytics are subject to medical prescription.

This argument supports the slogan:

Yes! We should have at least a more tolerant approach to soft drugs!

Produced under the general scope of the argumentative question, this conclusion gives an answer to this question.

The opponent must show that the proponent’s speech is untenable. First, he rejects the arguments of the proponent:

O: — No! Alcohol has nothing to do with drugs. We know how to drink in this country; alcohol is part of our culture, drugs are not. And if you legalize soft drugs next you’ll have to tolerate hard drugs!

O: — In Syldavia, they tried to legalize drugs, and the experience failed. Enough with social experimentation detrimental to young people!

Conclusion:

— Let us reject this crazy new proposal of legalization!

Secondly, O presents a counter-argument in favor of another position. This may correspond to maintaining the status quo:

— Honest citizens live peacefully thanks to the prohibition; the situation is under control as it is

Under the standard regime, the doxa “goes without saying”; but once the argumentative situation has been opened, it requires justification.

Argumentative questions are distinct from informative questions. The latter permits direct, unequivocal relevant answers:

S0: — When did you arrive? In which hotel are you staying?
S1: — Yesterday, and I stay in Grand Brand Hotel.
S0_2: — Oh, that’s wonderful! And what are you doing tonight?

Whereas the answer to the former necessitates an argument:

S0 — Does the fight against terrorism authorize restrictions upon freedom of expression?
S1 — Yes.
S0_2 — Oh, that’s wonderful. Now, let’s turn to the next question.

3. Argumentative situation: form and structuring rules

3.1 Representation

In a stabilized argumentative situation, proponents and opponents are also called upon to make positive arguments and to refute the antagonistic position. This situation can be roughly represented as follows:

Argument is seen as a mode of constructing answers to a question for which incompatible answers have been given.

Under the coherence assumption, all the semiotic acts produced in this situation are oriented towards the consolidation of the Answer-Conclusion.

The argumentative question is essentially open; the legitimacy (interest, respectability…) of the pro and contra interventions is acknowledged, at least factually. Sometimes the participants agree on a mutually satisfactory answer – conclusion, other times they don’t.

In many cases, an element of doubt remains attached to the surviving, ratified, answer, and the question may re-emerge. In other words, the answer is provisional; it cannot be completely separated from the question and the set of pro and counter-arguments that generated it. The answer is therefore an answer by default; an unstable answer, which may be subject to revision.

Centrality of Third parties

Considering that third parties play decisional roles, it follows that:
1) The development of the exchange will generally alter the original positions as expressed in the opening sequence. The final conclusion will not be identical with one of the positions as expressed at the opening sequence of the interaction.

2) A well-executed, successful argumentative exchange may conclude without a winner and a loser.

3) The loser is not compelled to relinquish their doubts.

Question and relevance
The question sets the relevance principle for argumentative contributions: relevance of the arguments for the conclusion, relevance of the conclusions as answers to the question.

The question, and consequently the relevance of interventions, may themselves  be challenged during the debate. It may be rejected on the basis of being flawed, poorly formulated, or irrelevant in consideration with “deeper” issues. S. Relevance; Refutation.

Burden of proof
The preceding graphic sought to represent the asymmetry between discourse and counter-discourse, established by the burden of proof resting on the proponent. This allocation may change with the participants and the kind of forum where the discussion takes place.

3.2 A Double constraint

Arguments are built under a double constraint; on the one hand, they are oriented by a question, and, on the other hand they are under the pressure of the counter-discourse. This situation is characterized by macro-discursive phenomena, such as the following ones:

 Bipolarization of discourse
Followers are attracted by the question; they identify themselves with the speakers involved; they adjust their language to reflect the words and practices of the lead speakers; in contrast, they exclude speakers and supporters of the opposing discourse (we vs. them).

Crystallization of discourse
Emergence of fixed lexical collocation, of antonymic pairs, tendency to stereotype the positions, especially stabilization of ready-made argument scripts.

Resistance to refutation
Appearance of mechanisms of resistance to refutation. Presentation of arguments in the form of self-argued claims, mimicking analyticity.

3.3 Changing mind, language and roles

Not only at the end of the discussion, but also during the exchange, participants can be persuaded to change their mind, alter their opinion and language, shifting from one role to another.

4. Monologization of the “Question — AnswerS” game

The vision of argumentation as a discussion between incompatible points of views about the same object is operative in both monologue and dialogs.

4.1 Dialogs can be monologized in two different ways

4.1.1 Monologal, non polyphonic interventions

An argumentative intervention developing a series of co-oriented arguments towards a conclusion, the arguer voices just one position, and assumes a demonstrative “no alternative” rhetoric. The monologue is monophonic.

Monophonic interventions ignore the speeches and positions of the opponents. This means that their practical study will necessitate the construction of a corpus bringing together the various interventions supporting the different answers. The plea for P is best understood when referred to some contestation, or neglect of P.

4.1.1 Monologal, polyphonic intervention

In another kind of monologue, the arguer adopts different positions, ands put forward several hypotheses about the same argumentative issue, without advocating any of them in particular. The discourse stages several voices, especially the main competing voice that of the oppoonent. Such a monologue is polyphonic, S. Interaction, Dialogue, Polyphony.

Polyphonic interventions contain a representation of the speech of the other participants. They take over, under various polyphonic modalities, the set of situational discursive data, the question and the opponent’s speech and position, which are re-framed under different discursive regimes, corresponding to different images allocated to the interlocutor and different self-allocated ethos. As a result, the assertion is introduced under an interrogative veil.

These strategies of polyphonic monologization of the question have been clearly identified in ancient rhetoric, where they are considered to be figures of speech, interrogation (interrogatio), subjection (subjection) and dubitation (dubitatio) (Lausberg, [1960], § 766-779).

(i) The question is framed as having one self-evident answer (interrogatio)

This is the case of the interrogatio, or “rhetorical question” defined classically as a question having an obvious answer.

Now, can such a person make a better president than our candidate? Certainly not.

The speaker takes possession of the argumentative question and gives an answer presented as the only possible, self-evident answer. This operation “disambiguates” the question, by imposing one sole response, S. Ambiguity.

The speaker takes the position of “the one who knows” and embeds the answer in the question. Third parties are framed in the position of allies who also know and applaud; opponents are challenged by a form of reasoning through ignorance. The purpose of the interrogatio strategy is to suggest that “there is no problem with this issue”.

(ii) The question is framed as having one justified answer (subjectio)

Lat. subjectio, “put before, under the eyes”; here “submit to” the audience)

The question is presented as requiring clarification rather than argumentation, as explanatory rather than argumentative, S. Explanation. The speaker takes the place of the investigator or the teacher who asks the right question and resolves it objectively. The interlocutor is framed as the pupil or the judge, sharing the direct question and admitting the proposed answers according to the logic of pedagogical co-construction.

Here is the situation, here is the question, and here are the data. One can think of three different answers, solutions, possibilities… Solution (a) is a variant of solution (b), as we will show. For such and such a good reason, solution (c) must be preferred to solution (b). So, the correct answer is (c).

Doctoral dissertations might approximate this strategy. During the defense, a member of the jury will possibly re-dialectize the monologue, expressing differently solution (a), and reversing the evaluation of (c) over (b)

(iii) The Question is framed as an open question, and the speech builds the answer in real time (dubitatio)

The speaker now takes the place of the third party, the ignorant party who has his or her doubts. In a kind of reversal of roles, the interlocutor is put in the high position of an assistant or counselor. The construction of the solution is now attributed to the interlocutor-counselor, not to the speaker-investigator.

In the three cases, the monologization of the argumentative situation plays heavily upon the preference for agreement. It does not leave the floor to other participants, and can channel their voices towards the speaker’s conclusion.

Perelman & Olbrechts-Tyteca introduce the class of quasi-logical arguments as the first of the three categories of “association schemes” ([1958], p. 191), that is argument schemes. Quasi-logical arguments can be understood “by bringing them closer to formal thought, logical or mathematical. But a quasi-logical argument differs from a formal deduction in that it always presupposes adherence to non-formal theses, which alone allows the application of the argument” (Perelman 1977, p. 65)

Six schemes are more precisely analyzed, and these bear the same name as their logical counterparts:

Among the quasi-logical arguments, we shall first analyze those which depend on logical relations — contradiction, total or partial identity, transitivity; we shall then analyze those which depend on mathematical relations — the connection between the part and the whole, the smaller and the larger, and frequency. Many other relations could obviously be examined. (Perelman & Olbrechts-Tyteca [1958], p. 194)

Definitions are “typical of quasi-logical argumentation” (id., p. 214):

When they are not part of a formal system, and when they nevertheless claim to identify the definiens and the definiendum, we shall consider them a form of quasi-logical argumentation” (id., p. 210).

The “quasi-logical” label is symptomatic of the method of the Treatise, rejecting “logic” but constantly using it a contrario to define argumentation in general and in particular to characterize the “quasi-logical” super-category of argument schemes. The category includes all the argumentative strategies involving phenomena such as negation, scales, relations and definitional stereotypes. In fact, it is the system of language that is considered to be a quasi-logic.

The arguments in this category are defined by a common characteristic:

[Quasi-logical arguments] lay claim to a certain power of conviction, in the degree that they claim to be similar to the formal reasoning of logic or mathematics. Submitting these arguments to analysis, however, immediately reveals the differences between them and formal demonstrations, for only an effort of reduction or specification of a non-formal character makes it possible for these arguments to appear demonstrative. This is why we call them quasi-logical. (Id., p. 193)

According to the traditional definition, a fallacy is an argument that looks like a valid argument but is not. There is a striking similarity between this, and the definition given in the Treatise: quasi-logical argumentation “claim[s] to be similar” to formal reasoning, but is not.

S. Fallacies; Logic; Collections (III).

The word proposition may be a synonym of proposal, “the point to be discussed” or “demonstrated” (MW, art. Proposition). A proposition may be developed in a complex argumentative discourse, justifying the briefly expressed concrete proposition itself. S. Argument – Conclusion.

In classical logic, a proposition is an autonomous statement. Propositional logic considers concatenations of unanalyzed propositions P, Q, R…. Predicate logic considers a proposition analyzed in two terms, the subject and the predicate, “S is P”.

1. Term

In logic, a distinction is made between categorematic and syncategorematic terms. Categorematic terms function as subject names or concept names (predicates). Used without further clarification, the word term refers to a categorematic term.
Syncategorematic terms include negation, binary logical connectives (“&”, and, etc.) and quantifiers (“∀”, all, etc.). They cannot function as subject or concept names, they appear only in combination. They have no independent meaning; their meaning being defined by specific contribution they make to the meaning of the terms or proposition they combine with.

In grammar, a parallel distinction contrasts the so-called full words, having a full semantic content (verbs, substantives, adjectives, adverbs) and the so-called empty or grammatical words (such as linking words, discursive particles, auxiliaries…)

2. Predicate, variable, constant

A sentence may be represented by its pivotal element, the verb, accompanied by variables representing its complements. Variables are denoted ‘x’, ‘y’, or simply as empty places, “—”.

— Paul sleeps: To sleep is a one place predicate, written “— sleeps » or “x sleeps”. :
— Paul eats an apple: To eat is a two-place predicate, written “— eats —” or “x eats y”:
— Paul gave the apple to the lady in black: To give is a three-place predicate, written “— gives — to —” or “x gives y to z”.

The same object can be attached to an infinite number of predicates, for example “— is a car”; “— is a means of transport”; “— is an object that can be bought”; « — is a cause of pollution”… Discourse constantly creates new predicates, according to the interests of the speakers, as “— was ​​carried out on 10 June 2017”; “— is a car available for next Saturday’s trip”.

In the case of a predicate admitting several variables, one or more empty places may be occupied by a constant. The predicate is then said to be partially saturated, which corresponds to a new predicate, for example, where “Paul gives y (something) to z (somebody)”, ​​“x (somebody) gives y (something) to John”, “Peter gives y (something) to John”.

In ordinary language, variables are expressed by indefinite phrases and pronouns: any, all, some, a (person)…”.

Constants are denoted ‘a’, ‘b’; in natural language, they are expressed by referring terms or phrases:

— Proper names (Peter), permanently attached to individuals.
— Pronouns (this the other, the next one). Their referential anchoring is based both on deictic maneuvers and on definite descriptions whose reference can be retrieved from the context. S. Object of discourse
— Definite descriptions, or denoting phrases (the man with the green hat). The noun phrase can be complexified at will: the seated man, the man who pretends to look elsewhere.

This simple notation renders explicit the skeleton of the sentence and is the basis of a more detailed semantic analysis of both its internal structure and external position in the broader discourse to which it belongs. Argument schemes are currently expressed in such a semi-symbolic notation.

3. Proposition

In classical logic, a proposition is a judgment, which can take only two values, true (T) or false (F); a proposition cannot be “more or less” true or false. A proposition is only a way of telling the truth or the false, without any consideration upon its meaning and conditions of use.

A proposition is unanalyzed if no information on its internal structure is available. Logical connectives and the laws of their combination are defined on the basis of such unanalyzed propositions.

A proposition is analyzed if its internal structure is taken into consideration. Classical logic considers that the analytic structure of a logical proposition is basically “Subject is Predicate”, “S is P”:

— The subject refers specifically (if a constant) or generally (if a variable) to the elements of a universe of reference.
— The predicate says something about these elements.
— The proposition categorically (without condition) affirms or denies that the predicate accepts the subject.

Capital letters A, B, C… P, Q, R… are used to denote both unanalyzed propositions and the subject and predicate in analyzed propositions.

3.1 Quality and quantity of a proposition

The quality of a proposition refers to its two possible dimensions, affirmative or negative.

The quantity of the proposition varies according to whether the subject refers to a being, certain beings or all beings of the universe of reference.

Quantifiers express the quantity. The quantifiers such as all (all N), or some (some N) express quantities. According to their quantity, propositions are:

Universals: all poets
Particular: a poet; some poets

Particular does not refer to a constant, a specific, known, individual. In its traditional form, logic does not deal with propositions predicating something from a determined individual, such as “Peter” or “this poet« ; S. Syllogism:

The combination of quantity and quality produces four kinds of propositions:

A     universal affirmative All S are P.
E     universal negative No S is P.
I      particular affirmative Some S are P.
O     particular negative Some S are not P

Traditionally, affirmatives are denoted by the letters A and I (two first vowels of the Latin verb AffIrmo “I affirm”) and the negatives by the letters E and O (nEgO, “I deny”).

3.2 Converse propositions

The converse proposition of a given proposition is obtained by swapping subject and predicate. The subject of the original proposition is the predicate of its converse proposition and the predicate of the original proposition is the subject of its converse proposition.

The quality (affirmative or negative) of the two propositions is the same.

The negative universal E and its converse are equivalent (they have the same truth conditions, cf. infra §4.2, Logical Square):

No P is Q ↔ no Q is P.

The positive universal E and its converse are not equivalent

all P are Q ≠  all Q are P.

3.3 Distribution of a term

A term is distributed if it says something of all the individuals belonging to the reference set. If not, the term is undistributed.
The terms preceded by the quantifier all are distributed. The terms quantified by some, many, almost all … are undistributed.
For example, in a universal affirmative proposition A, “All Athenians are poets”:

— The subject term, Athenians, is distributed.
— The predicate term, poet, is undistributed; the proposition only says that “some poets are Athenians”.

The notion of distribution is used by the rules of evaluation of the syllogism, S. Paralogism.

3.4 The presupposition of existence

Some expressions such as “the unicorn”, “the present king of France”, “real-life dragons”, are misleading, insofar as they appear to be referring expressions despite the fact they do not refer to any existing being. This being the case, when such phrases are used as subjects of a proposition, this proposition cannot be said to be true or false, the present King of France is neither bald nor hairy. To avoid such perplexities, it is assumed that the universe of reference of the subject term is assumed not to be empty. S. Presupposition.

4. Immediate inference and logical square

4.1 Immediate inference

An immediate inference is a one-premise argument, inferring from one proposition to another:

All the A are B, so some B are A

The two terms of this single premise are found in the conclusion, only the quantity of the proposition changes. While syllogistic inference requires a medium term (middle term), “im-mediate” inference does not need such a transition term. It is debatable whether immediate inference is a kind of reasoning.

Immediate inference is an inference, not a reformulation. The reformulation relation presupposes the identity of meaning between the two utterances it links:

Some A are B, so some B are A (conversion, see §3.2).
All the A are B, so some B are A (subalternation, see infra).

In the first case, the immediate inference corresponds to an equivalence. This is not true, however, of the second.

4.2 Logical square

The logical square expresses the set of immediate inferences between analyzed propositions of the subject-predicate form according to their quality, affirmative or negative, and the quantity of their subject (A, E, I, O, see above).

These four propositions are linked by the following relations.

— Contrariety, between the affirmative universal A and the negative universal E. A and E are not simultaneously true, but may be simultaneously false. In terms of immediate inference, if one is true, then the other is false.

— Subcontrariety, between the particular affirmative I and the negative particular O. At least one of the two propositions I and O is true. They may be simultaneously true, but cannot be simultaneously false. In terms of immediate inference, if one is false, then the other is true

— Contradiction, between:

The universal negative E and the particular affirmative I.
The universal affirmative A and the particular negative O.

E and I cannot be simultaneously true or simultaneously false (only one of them is true). The same will be true for A and O. In terms of immediate inference, the truth of one immediately implies the falsity of the other, and vice versa.

­— Subalternation, between:

A and I, the universal affirmative and the particular affirmative.
E and O, the negative universal and the negative particular.

If the superaltern is true, its subaltern is true. Immediate inference:

Every S is P, so some S are P.

If the subaltern is false, its superaltern is false. Immediate inference:

It is false that some S are P, so it is false that every S is P.

The subaltern may be true and the superaltern false.

Moreover, propositions E and I are convertible; cf. supra, §3.2.

5. Immediate inference, quantifiers and terms

Immediate inference is an inference from a single premise. The two terms of the single premise are found in the conclusion (examples above). In the case of the syllogism^@, the inference proceeds from two premises and three terms. The middle term functions as a “mediator”, an intermediary, between the major term and the minor term. In the case of immediate inference, the conclusion is “not-mediated” by a middle term.

From a cognitive point of view, argumentation by definition assigns to an individual the properties characterizing the class to which it belongs. From a linguistic point of view, argumentation by definition assigns to an individual designated by a name, all the elements of the linguistic definition of this term. Argumentation by definition is therefore an immediate, substantial, semantic inference, on the meaning of the terms. Immediate inferences are formal; they are not made on the basis of full words, but on the basis of quantifiers. Both kinds of inference function as semantic reflexes in ordinary discourse, linking natural statements, according to ordinary semantic intuition combined with contextual references based on the laws of discourse and the cooperative principle.

Because of their seeming obviousness, the way we handle such these inferences often goes unnoticed. This does not mean, however, that the process is always error free. Taking the correct approach to such inferences is part of the argumentative competence.

The Greek word [analogia] means “proportionality”, as the Latin proportio.

1. Proportion

A relation is a stable connection between two things:

shell : fish                       old age : life
glove : hand                    pilot : ship

A proportion is an analogy between at least two relations (not between individuals, as in categorical analogy); it implies at least four terms. Two pairs of beings are in a relation of proportion if, in their respective fields, they are bound by the same, or a similar relation.

shell : fish ~ feather : bird                   — cover the body of —
glove : hand ~ shoe : foot                   — protect the —
leader : group ~ captain : ship             — guide the —
old age : life ~ evening : day                — last moment of the —

The analogy of proportions is expressed through parallel syntactic structures:

(Since) a ship needs a pilot, any group needs a leader!

In arithmetic, a proportion is defined as the relation between two numbers, such as ‘17 / 27’. The same proportional relation binds two pairs of numbers a/b and d/d if they obey the following rule:

3/2 = 9/6, same proportion 1.5
a/b = c/d <=> ad = bc (a = bc/d, etc.)

The analogy between proportions corresponds to the linear equation with one unknown, that is to say, to the “rule of three”:

a / b = c / x where ax = bc, ax-bc = 0; and x = bc / a

In geometry, two similar figures have the same shape and different dimensions. Two congruent triangles have equal angles and proportional sides.

The process of understanding is the same in the case of mathematics as it is in argumentation. The reasoning by which the value of ‘x’ is mathematically extracted from the arithmetical proportion is the same as the argument which extracts the necessity of a leader from the analogy of proportion between a ship’s crew and a group of people more generally.

The analogy of proportion is at the basis of a specific kind of metaphor:

old age, evening of life.

The analogy of proportion is open to ironic self-refutation:

A woman without a man is like a fish without a bicycle.

2. Proportionality

Lat. ad modum argument, Lat. modus, “measure, “just measure”
NB: Besides “moderation”, the Lat. temperentia can mean “just measure, fair proportion”.

The argument of proportionality justifies a provision or an action by claiming that it is well proportioned to the facts, gradual. It is invoked a contrario in routine press releases such as:

(The association, the trade union, the government…) X condemns the disproportionate use of force used against…

Let us consider a situation of unrest, described by the current government as a seditious uprising, led by a handful of terrorists. The government organizes a large military presence to “show strength not to have to use it”. This strategy of psychological war may have perverse effects. In reality, the argument of proportionality allows calculations that defeat the desired effect:

The deployment of strength, far from minimizing the enemy, made it stronger. (Pierre Miquel, [The Algeria War], 1993[1])

This conclusion is based on the topos, “one does not use cannons to shoot flies”. A strong refutation of a (declared) weak position entails the same kind of paradox.

The argument of proportionality is a form of argument on the right measure, which can also be defined as the intermediate measure, S. Moderation.

The proportionality strategy can be used to avoid the risks posed by the escalation strategy.

[1] Pierre Miquel, La Guerre d’Algérie. Paris: Fayard, 1993, p. 190.

The proper name argument scheme corresponds to topic n° 28 of Aristotle’s Rhetoric, “another topic is derived from the meaning of the name. For instance, Sophocles says, ‘Certainly thou art iron, like thy name’.” (Rhet., II. 23, 1400b29, Freese, p. 323). The example is a pun on the proper name of the hero and the word meaning “iron”.

Unlike the nickname, which claims to refer to a characteristic of the individual, the proper name is not motivated; it does not mean its bearer. When the proper name (first name or last name) of a person is homonymous with a common name, the topic of the name attributes to the person the characteristics of the homonymous thing; he or she is re-categorized as a non-human being, which may be less-than-flattering. The name functions as an index from which truths about the person might be inferred. Aggravated by the infinite resources of paronomasia, and rhyme, proper names can be the basis for all kinds of derivations, particularly, although not exclusively in the school playground:

You are Peter [Lat. Petrus], and on this rock [Lat. petram] I will build my church.
(Matthew, 16:18)

June will be the end of May[1]

Being named Peter, and thus being like a rock, is being apt to be a foundation: the name is an aptonym, the character and destiny of the person are pre-inscribed in his or her name. Mr. Child is of course a pediatrician, or a teacher or perhaps he has a childish character; the aptonym reinforces the person’s suitability for his task or confirms the attribution of a trait of character. Referring to John R. Searle as Sarl, (French acronym for “Limited Liability Company”, Inc.) Derrida re-casts, fairly or not, Searle’s work as a kind of business.[2]. S. True Meaning of the Word; Ambiguity.

[1] The slogan appeared during the campaign for general elections to be hold in June 2017, Mrs Theresa May serving as Prime Minister of the United Kingdom since 2016.

[2] Jacques Derrida, Limited Inc. Evanston, IL: Northwestern University Press, 1988.

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.

Situation:

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.

The speaker may choose to connect his or her own argumentative line to a counter-discourse that he or she knows or anticipates and, in any case, rejects. The prolepsis steals the argument from the mouth of the (real or fictitious) opponent, “I know (perhaps better than you) what you are going to say”. The counter-discourse is resumed with an indefinite degree of distortion, from a literal referenced quotation to a sketchy evocation of a possible objection, which may be framed as a self-refuting scarecrow, S. Speech Resumption At the very least, the quoted speech is extracted and re-adjusted in view of the new discursive environment, and its ethotic force is kept at bay. Through the magic of quotation, an intended refutation becomes a mere objection.

The degree to which the counter-discourse is rejected is itself variable. The counter-discourse may be radically rejected; dismissed as absurd (“do we intend to ruin all small savers? No, quite the contrary, and for many reasons…”), or maintained in full force, until further information becomes available. In this sense, the Modal-Rebuttal component of argumentation is a special case of prolepsis, S. Layout of argument.

The proleptic structure covers not only coordinated or subordinated pairs of statements but any discourse pattern whose configuration corresponds to the staging of two anti-oriented discourses, the speaker taking responsibility for one of them; it represents the maximum development of monological argumentation, S. Connective; Destruction; Concession; Refutation.

Several rhetorical terms refer to this same structure:

— The anteoccupation refers to a refutative structure, composed of a prolepsis, which evokes the position of the opponent, followed by an hypobole, which refutes this position or expresses the position supported by the speaker (Molinié 1992, [Anteoccupation]). Lausberg ([1963], § 855) terms this same strategy preoccupation (Latin prefixes pre-, ante- “in advance”).

— The procatalepsis and the metathesis refer to a discursive configuration by which the speaker “reminds listeners of past events, presents to them the facts to come, foresees objections” (Larousse, quoted in Dupriez 1984, p. 290; Metathesis has another quite distinct meaning, “swapping two sounds or letters of a word”).