1. “Ignorance of refutation”, a fallacy of method

Lat. ignoratio elenchi. The Greek word [elenkhos] means: “1. Argument to refute … 2. Proof in general” (Bailly, [elenkhos])”. The Latin title of Aristotle’s Sophistical Refutations is De Sophisticis elenchi (Hamblin 1970, 305).

The fallacy of “ignorance of refutation” (ignoratio elenchi) is defined in the context of the dialectical game, where a participant, the Respondent (or Proponent), is committed to a statement, and the partner, the Questioner (or Opponent), tries to lead the Respondent to a contradiction, to thus refute the statement he or she (the Respondent) had previously accepted. The dialectical game considers only contradictory propositions (one and only one of them is true). The opponent must conform to the rules of the method in order to truly refute (and not in appearance) the primitive affirmation, S. Dialectic. The fallacy of ignorance of refutation is independent of language, it occurs “because the terms ‘proof’ or ‘refutation’ have not been defined, and because something is left out in their definition”. (Aristotle, R. S., 167a20, §5), S. Fallacy: Aristotle. In other words, the misconception of refutation is a general term covering all methodological errors occurring in a dialectical game.

This concept may be extended to any argumentative language game: “the arguer argues and does not know how to argue; thinks something is being proven or successfully refuted, when this is not the case; his or her practical concept of argument is flawed, etc.” This basically occurs when the argument does not respect the principles of relevance: on the one hand, the argument must be relevant to the conclusion (internal relevance) and, on the other hand, the conclusion must be relevant as a reply to the question (external relevance), S. Argumentative question.

2. Relevance of the argument for the conclusion

In the context of a dialectic game, the Respondent asserts P. Starting from P, the Questioner deductively constructs a chain of propositions ending with proposition not-P. So, the Questioner claims that this chain proves proposition not-P. Apparently, the Respondent has been refuted, and the Questioner has won the game. But the Respondent claims that the chain of proofs backing not-P is not valid because the arguments put forward do not actually support this conclusion; so, the Respondent claims that the Questioner actually failed to demonstrate not-P.

This schematizes the general situation when an arguer claims to have refuted the opponent ex datis, that is using only beliefs and modes of inference supposedly admitted by the opponent. In the same way, in an ex datis or ad hominem procedure, the opponent can resist the refutation by breaking the inference chain leading to the conclusive step s/he is supposedly compelled to concede. In other words, he or she argues that the arguments are not relevant to the conclusion. This issue actually involves all the program of criticism of argumentation.

3. Relevance of the conclusion as a reply to the Question

In the general case, the proponent commits himself to P, the opponent constructs from P a chain of propositions, at the end of which the proposition Q is reached. The proponent therefore claims that “Q = Not-P”. The proponent argues that proposition Q is not contradictory to P, and that, accordingly, it has not been rebutted. The arguments may be relevant to the conclusion, but the conclusion does not disprove the original thesis.

To argue that an intervention is externally irrelevant is to argue that it misses the point, is off-topic, etc. It may also be denounced as an attempt to put the adversary on a false trail, S. Red herring; the accusation of paralogism is reinforced by a suspicion of sophistry.

Criticisms of internal relevance and external relevance are cumulative. They invalidate a speech by saying that it does not back its conclusion, and that, besides, the conclusion has nothing to do with the issue.

4. The question is not relevant to the “real debate”

The dialectical framework is binary, the proposition to discuss is expressed in a simple and explicit proposition, and the methodology of a refutative discussion is well defined. Since the question is “P or not-P?”, claiming that the opponent’s conclusion does not logically contradict P, is to claim that it is not relevant to the debate.

The situation can be equally clear in an ordinary discussion. A student contests, that is, wants to “refute” the grade he has received: “if you don’t up my grade, I’ll fail the exam; please, I badly need just three extra points!”. The argumentation by the consequences is quite valid, but the negative consequences of the bad grade are irrelevant to the determination of the grade (according to the classical scientific and educational regimes at least). The student’s conclusion is irrelevant, failing to acknowledge the real issue: “what mark does my assignment deserve in itself?”. The student’s question is different from the teacher’s question, and the teacher is the master of the question.

Things may be more complicated. When the proponent refutes the rebuttal by saying, “what you disagree with has nothing to do with what I am saying”, what he actually says can be difficult to pin down, and may be constantly reformulated and reinterpreted S. Resumption of speech. On the other hand, even when the original claim and its intended rebuttal have been previously set down in writing, the link between the two does not necessarily have the clarity of the binary contradiction. For example, does S2 refute S1, or merely show that the issue is complex:

S1: — Speculators buy raw material in advance just to speculate on future price variations. Such operations on raw material should be banned by law.
S2: — Nonetheless, it is essential for companies to purchase in advance the raw materials they need, to cover themselves against price fluctuations.

Finally, in ordinary argumentation, the issue itself may be controversial. When none of the participants is the (natural or conventional) master of the question, each key participant will be tempted to give a definition of the question, and, will, accordingly reject the opponent’s answer as irrelevant to the real issue:

S1: — That’s not the question!
S2: — This is my answer to the problems that really arise. You’re not asking the right question.

The accusation of fallacy of conclusion irrelevant to the question under debate can be answered by a counter accusation of a fallacious, wrongly framed question, irrelevant to the “real” debate.

The function of the participating third party, be it the judge, the (universal) audience or the informed participants, is to construct, manage and decide upon the question, and accordingly, to determine what is or is not relevant in the debate.

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A relation is a two-place predicate R associating two objects, a and b, denoted by “aRb”. Relations are characterized by three general properties, symmetry, transitivity, and reflexivity.

— Symmetry, or Reciprocity: The same relationship holds between “a and b” and “b and a”.
— Reflexivity: The relationship connects an object to itself.
— Transitivity: The relationship connecting a to b and b to c also connects a to c.

1. Symmetry, or reciprocity

A relation is symmetric or reciprocal if it relates both a to b and b to a. In other words, both “aRb” and “bRa” hold. If a loves b, b does not necessarily love a: a love relationship is not symmetrical. “Meeting” is a symmetric relationship. The following argument is neither more nor less logical than any other, but it would make a valid point in any detective novel; it can only be rejected by accusing Peter of lying:

If Peter confessed to having met Paul at the bar, we must assume that Paul met Peter. Paul cannot deny the obvious.

2. Reflexivity

A reflexive relation relates a being to itself, noted “aRa”. “— being contemporary of —” is a reflexive relationship: a is its own strict contemporary. For the average person, the causal relationship is not reflexive; only God is causa sui, his own cause.

The reflexive relation can be used ad hominem. The principle “charity begins at home” for example forces the reflexivity of the relationship “a makes charity to b”; all the same, the love of others can be used to encourage self-care:

You who love the whole of humanity, you should try to love yourself as well!

The competence of an adviser can be challenged by inciting him to make a reflexive application of his talents:

Physician, heal thyself!

Such replies correspond to the ad hominem variety setting up practices against words.

3. Transitivity

A relation is transitive if, when it relates a to b and b to c, it also connects a to c; in other words, “aRb and bRc” imply that “aRc”.

If a loves b, and if b loves c, then a does not necessarily love c; a relationship of love is thus not transitive. The relation “— be the father of —” is not transitive, but “— being an ancestor of —” is transitive. If a is an ancestor of b and if b is an ancestor of c, then a is an ancestor of c.

Inferences based on the transitivity of a predicate apply whenever at least three objects are positioned on a graduated scale:

If a is bigger, older, richer … than b
and b larger, older, richer … than c,
Then a is bigger, older, richer … than c.

Inferences based on these properties are part of the unnoticed evidences exploited by everyday reasoning and argument. They are sometimes considered to be “quasi-logical”, S. Quasi-logic; but being sound and valid does not preclude being an argument.

4. Conversion

S. Conversion

Lat. A conjugata argument, Lat. conjugatus, “related, of the same family”

Three types of argument are based on the fact that two words are materially related, depending on the nature of their relationship:

An etymological link S. True Meaning of the world.
A morphological link, S. Derived Words.
A phonic or graphic resemblance, S. Ambiguity.

Synonymy and antonymy are semantic relations.

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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.

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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.

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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.

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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.

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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.

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

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