PROPOSITION in LOGIC

In classical logic, a proposition is an elementary 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

Logic distinguishes between categorematical and syncategorematical terms. Categorematical terms function as subject names or concept names (predicates). Used without further clarification, the word term refers to a categorematical term.
Syncategorematical 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 is defined by the specific contribution they make to the meaning of the terms or the propositions they articulate

In grammar, a parallel distinction is made between the so-called full words, with full semantic content (verbs, nouns, adjectives, adverbs) and the so-called empty or grammatical words (such as linking words, discursive particles, auxiliaries…)

2. Predicate, variable, constant

A sentence can be represented by its central element, the verb, accompanied by variables representing its complements. Variables are written ‘x’, ‘y’, or simply as empty places, “-”.

– Paul is sleeping: To sleep is a one-place predicate, written “– sleeps » or “x sleeps”. :
– Paul is eating 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, e.g. “– is a car”; “- is a means of transportation”; “- is an object that can be bought”; « – is a cause of pollution”… The discourse constantly creates new predicates, according to the interests of the speakers, e.g. “ was ​​carried out on June 10, 2017”; “- is a car available for next Saturday’s trip”.

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

In ordinary language, variables are preceded by quantifiers: any x, all y, some z, one w.
Constants are denoted by ‘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, see object of discourse
— Definite descriptions, or denoting phrases (the man with the green hat). The noun phrase can be complexified at will: the sitting man, the man pretending to look elsewhere.

This simple notation makes the skeleton of the sentence explicit, and is the basis for a more detailed semantic analysis of both its internal structure and its 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, that 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 signifying the true or the false, without any consideration of its meaning and conditions of use.

A proposition is unanalyzed if there is no information about its internal structure. Logical connectives and the laws of their combination are defined on the basis of such unanalyzed propositions.
A proposition is analyzed when its internal structure is taken into account. In classical logic, the analytic structure of a logical proposition is basically “Subject is Predicate”, “S is P”:

— The subject refers specifically (if it is a constant) or generally (if it is a variable) to the elements of a reference universe.
— The predicate says something about those elements.
— The proposition categorically unconditionally) affirms or denies that the predicate accepts the subject.

The 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, all beings or no beings of the universe of reference.
Quantifiers express the quantity. The quantifiers like 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 that predicate something from a determined individual, such as “Peter” or “this poet« ; see syllogism:

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

A     universal affirmative All S are P.
E     universal negative No S are 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 the subject and the 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. Otherwise, the term is not distributed.
The terms preceded by the quantifier all are distributed. The terms quantified by some, many, almost all … are not distributed.
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, see paralogism.

3.4 The Presupposition of Existence

Some expressions such as “the unicorn”, “the present king of France”, “real dragons”, are misleading, in that they seem to be referential expressions although they do not refer to any existing being. This being the case, when such expressions are used as subjects of a proposition, that proposition cannot be said to be true or false, the present king of France is neither bald nor hairy. To avoid such puzzles, it is assumed that the reference universe of the subject term is not empty. see presupposition.

4. Immediate inference and logical square

4.1 Immediate inference

An immediate inference is a one-premise argument. It concludes 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. 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. In the second case, it is not.

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 connected by the following relations.

– Contrariety, between the affirmative universal A and the negative universal E. A and E are not simultaneously true, but can 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 is 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, the propositions E and I are convertible; cf. above, §3.2.

5. Argumentation by Definition, Immediate Inference

An 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 acts 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 that characterize 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 that term. Argumentation by definition is therefore an immediate, substantial, semantic inference, about the meaning of the terms. Immediate inferences are formal; they are made not on the basis of full words, but on the basis of their quantifiers.
Both types 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 deal with such inferences often goes unnoticed. This does not mean, however, that the process is always flawless. Handling the  such inferences correctly is part of the argumentative competence.

---

 

The argument of PROPORTION

1. Argument based on proportion

1.1 Proportion

A relation is a stable association between two things:

shell : fish
glove : hand
leader : group
old age : life

There is a relation of proportionality between two pairs of things if these two pairs are connected by the same or a similar relation in their respective fields.
A proportion [1] is an equality (an equivalence) between relations; it implies at least four terms.

Domain A	Domain B	SAME RELATION
shell : fish glove : hand leader : group old age : life	feather : bird shoe : foot captain : ship evening : day	A covers the body of B A protects B A guides B A is the last moment of B

The relationship A between shell and fish, is identical (similar) to the relationship B between feather and bird.
Shells are to fish what feathers are to birds.

In arithmetic a proportion is a relationship between two numbers. 3/6, 1/2, 50/100, etc. express the same proportion.
The equality of proportion corresponds to a first degree equation, with one unknown. This equation corresponds to the « rule of three »: a/b = x/c, from which ac = bx and x = ac/b – Three eggs cost €1.20, how much do four eggs cost?

In geometry, we talk about similarity of shape. Two similar figures have the same shape but different sizes. Two similar triangles have the same angles and proportional sides.

1.2 Proportional argument

This argument uses the analogy of proportion through mechanisms of parallelism:

(Since) a ship needs a pilot, every group needs a leader!
Pilot: Ship = X : Group:
Who is the pilot in a group? — The leader!

The process of understanding is the same in the case of mathematics as it is in argumentation. The reasoning that mathematically extracts the value of ‘x’ from the arithmetic proportion is the same as the argument that extracts the necessity of a leader for a group from the analogy between a ship’s crew and a group of people.

1.3 Metaphor and analogy of proportion

In the Poetics, Aristotle defines metaphor as the application to a thing of a name foreign to it, by a shift from genus to species, from species to genus, from species to genus, or according to a relation of analogy. (Trans. Magnien, p. 139). The « relation of analogy » is defined using examples of proportional metaphors:

A cup has the same relation to Dionysus as a shield has to Ares. So we would say that the cup is « the shield of Dionysus » and that the shield is « the cup of Ares.
Or again, old age has the same relation to life as the evening has to the day, so that we can say that the evening is « the old age of the day » and old age […] is « the evening of life » or « the twilight of life ». (Id., p. 140)

1.4 Destruction of the Proportional Analogy

The basic form « An A without a B is like an X without a Y » can be used to destroy the argument based on this proportional analogy:

L1 – A group without a leader is like a ship without a pilot.
L2 – Yes, and a woman without a man is like a fish without a bicycle (a feminist slogan).

2. Proportionate measure

The idea of proportionate measure has two Latin names:
– Argument ad modum, from modus « measure »
– Arg. ad temperentiam, from temperentia, « just measure, just proportion ».

The argument of proportionate measure justifies a rule by affirming that it is reasonable, balanced, and modifiable according to the evolution of its object.
This argument assumes that there is a graduated scale of the severity of the riots (domain A) and a corresponding graduated scale of the severity of the repression (domain B), depending on the severity of the riots. The idea of proportion corresponds to the covariance on these two scales.

The more/less the demonstration « endangers the security of the state, the citizens, their property… ».
The more/less repression is to be expected.

A judicial system that is not proportional would apply the same punishment to all offenders.
A judicial system that is not proportionate would not take into account the circumstances of the criminal act.

The argument of proportionality is invoked a contrario in the recurring press release:

The association,( the union, the government…) condemns the disproportionate use of force.

Flexing one’s muscles means announcing severe repression and, by applying the law of proportionality, proclaiming the strength of the enemy.
Consider a situation of unrest described as the work of a few isolated troublemakers. According to the principle of proportionate repression, it is expected that ordinary repressive measures will suffice: a relatively harmless demonstration: light repression.
However, the authorities decided to organize a large military show to « impress the enemy » and « reassure the population ».

The argument of proportionate action allows for a calculation that defeats this psychological strategy:

Far from diminishing the enemy, the show of force made him appear larger.
Pierre Miquel, La guerre d’Algérie, 1993[2]

The conclusion is based on the topos: « You don’t fire a cannon at flies« ; if we were really dealing with a few isolated hotheads, we wouldn’t be positioning tanks in front of official buildings. So this is a real popular uprising.
This paradox can be found in the case of a strong refutation of a position declared to be weak.

The proportionate measure is a form of argument about the just measure, which can also be defined as the middle measure.

---

[1] Latin proportio, « proportion; analogy ». The. Latin word translates the Greek analogia [ἀναλογία] « 1. Mathematical proportion 2. Correspondence, analogy » (Bailly ἀναλογία)

[2] Paris, Fayard, p. 190.

PROPER NAME

1. Conventionalism and Realism

The proper name is given by convention; it contains no information about its bearer. That is, all the animals called cow belong to the category of cows and share common properties; proper names don’t correspond to categories, and don’t share common properties.
Proper names are conventionally ascribed, in the sense that they do not correspond to a description of the person they designate.
It follows that we cannot make any inferences about proper names in the way that we can about names of natural categories.

For example, a cow is called « cow » and we recognize it as a cow when we meet it. This is because of its main distinguishing features combined with a family resemblance. When I meet a stranger I have no way of deducing his proper name from considerations about his person; if I know that he is called Smith, I know nothing about him (but see infra).

2. Proper Names as Indices

Since a proper name is a conventional social designation, it cannot be exploited by arguments from definition, but it can be used as an argument based on indexical evidence.

Indication of origin – The proper name can be associated with particular groups of people who generally bear that kind of name, by an argumentat that uses the proper name as an index. If I am to meet Mr. Martin-Dupont, I can only think that he is most likely of French origin, — unless . . . see lay out of argument.

Lineage Indicator – In general, the identity of the proper name may be a sign of family relationship, which may or may not be flattering. Having the name of the wrong person or the guilty party is extremely burdensome. If John Doe is unanimously condemned and stigmatized as an incestuous anti-Semitic pedophile, then at the height of the media storm, the Doe family can be interviewed in order to take a position on the John Doe case. Clarifications appear: “Alan Doe is not related to John Doe”.

3. Onomastic contamination

Onomastic contamination can occur between the name of a person at the center of the news and the name of another person

Pablo₁ Iglesias₁ […] is old-fashioned, obsolete, passé. His last name, for starters. And the best proof of this is that there was once a Pablo₂ Iglesias₂, a left-wing politician, in the 1920s. A century ago! What’s more, the Spanish call his₁ political doctrine as “Pablism_a”. But there was once a Pablism_b, named after a Trotskyist dinosaur (= Pablo₃ Iglesias₃)!
Marc Crapez Divisions of the Spanish Left: How Íñigo Errejón made Pablo Iglesias uncool, 2019 [2] (my notes)

I can’t find a more charitable reading of these two converging arguments than the following.
– The first line of argument attaches the “nerdy” character to the name « Pablo Iglesias » and to all those who have the misfortune to be called Pablo Iglesias, especially Pablo₂₀₁₉ Iglesias₂₀₁₉. A certain Pablo₁ Iglesias₁ lived “a century ago!”; the link between the two Iglesias is their name, Iglesias. The argument is “PI₂₀₁₉ is obsolete, since PI_{XXth century} is obsolete. »

– In the second convergent argument, the argument is based on the name of the doctrine.  The political doctrine of Pablo₂₀₁₉ Iglesias₂₀₁₉ (= « his_PI2019 political doctrine ») is Pablism, derived from his₂₀₁₉ first name Pablo₂₀₁₉, that is, « Pablism₂₀₁₉« , This doctrine is apparently homonymous with another political doctrine, Pablism_ancient, which the author does not like. The argument is “Pablism₂₀₁₉ is outdated, because Pablism_ancient is outdated ». Here, the inferring license is the derivation: « Pablism_ancient is a political dinosaur, so Pablism₂₀₁₉ is also a political dinosaur ».
The argument is topped by a clear appearance of the argument of the novelty argument (a contrario)

A vague suspicion that they might belong to the Landru family hangs over all people named Landru (a famous serial killer of women). When a person’s name is the same as that of a famous person who is elevated to the status of a paragon, the character of the paragon is attributed to them by antonomasia.

4. Naturalization of proper names

Some conceptions of proper names reactivate the realist conception of proper names, according to which the name and the person actually share properties in common; the proper name John, refers to a category grouping all persons named John and that these John’s do share some interesting properties.

The following argument attributes to a person with a particular name the characteristics of other people with the same name. The popular science of proper names attributes to them a “character” that is not unlike the common name of a natural category:

Characteristics of the first name Fleury
Fleury tends to have an amiable character. … he is also positive. He is a person close to his family. But sometimes he can be too charming… [1]

We are on the way to making the first name a natural species name: The toadstools are poisonous, and the Fleury’s live like a big family. The name announces the character, and we can apply the argument by definition; if his name is Fleury, he is nice, it is normal, natural.

The situation is different with nicknames. His friends call him « the bull » because of his particular way of doing things. The argument of the proper name gives the proper name the meaning of the corresponding common name; the proper name signifies its bearer.
Mr. Bull can be the most sensitive person imaginable, his friends think he has something of a bull about him.

4. Aptonyms: Topos of the name

4.1 Aptonym

A proper name is an aptonym 1) if it is homonymous with a common name, and 2) if something in his life can be closely associated with the bearer of the proper name, especially his profession.

Because someone is called Mr. / Ms. Children, we infer that he has some essential relationship with children, and therefore it is normal for him to become a pediatrician, a teacher, etc., or that he has a childlike character: the name is an aptonym, that reinforces the person’s suitability for his job, confirming the attribution of a character trait.

Phrases like “It’s not for nothing that he’s called …”, “With a name like that!”, “He lives up to his name” turn the first name into an aptonym. Because the opponent is called Black, we infer that he has a black soul, and we suspect him of black designs; if he is involved in a suspicious business, we assume that his name predestined him for it; as if the person were trying to reach his signifier through his actions. The first name functions as an appropriate nickname,

4.2 Topos of the Name

The topos of the proper name is based on aptonymy:

Another [place] is derived from the name; for example, as Sophocles does, Having the hardness of iron, you bear your name well (Aristotle, Rhet., II, 23, 1400b18; Dufour, p. 126).

The note specifies that it is a play on the Greek proper name Sidero and the Greek noun meaning “iron, iron instrument”: « He is inflexible, his name is iron ».

In June of 2017, a general election was held in the United Kingdom, called by the Conservative Prime Minister Theresa May. According to a slogan of the Labor Party, the defeat of the Prime Minister was written in her name:

June will be the end of May.

In the Latin Gospel, Christ chose Peter to be the first head of the Church in this way:

You are Peter (Latin Petrus), and upon this rock (Latin petram) I will build my church.

This construction is a special case of naming, the repetition of the same word in a statement with two different meanings. In the general case, the property of the thing is directly predicated on the proper name.

Stigmatization by Proper Names

To infuriate and humiliate the other person by distorting his name is a playground practice. But no one, not even the greatest minds, can resist turning his opponent’s proper name against him. In the course of a controversy, the philosopher Jacques Derrida renamed his colleague J. Ronald Searle “Sarl”, that is, the acronym SARL with the corresponding insinuation that it was an aptonym. [3]

In response to Michelle Loi‘s vision of Maoist China, Simon Leys, published a pamphlet titled L’oie et sa farce (The Goose and its Farce) (Wikipedia, Simon Leys) In French, Loi and l’oie are perfect homophones.

The process of stigmatization by distortion of the proper name can serve anti-Semitic and political hatred:

[In this press sympathetic to the extreme right] we find proper names manhandled, proper names reworked: André Glucksmann becomes “André Glücksmann”, Simone Veil becomes “Shimone Veil”, Robert Hue becomes “Hue-coco” (*)
(*) Former First Secretary of the French Communist Party active against anti-Semitism; « coco » for « communist ».
Krieg 1999, p. 12 [4]

Stigmatization by proper name is one of the most intolerable instruments of personal attacks, harassment and racism.

---

[1] https://www.parents.fr/prenoms/fleury-40932#Caract%C3%A8re-du-pr%C3%A9nom-Fleury

[2] Le Figaro, www.lefigaro.fr/vox/societe/divisions-de-la-gauche-espagnole-comment-inigo-errejon-a-ringardise-pablo-iglesias-20191011 (13-01-2020)

[3] Jacques Derrida, Limited Inc. Evanston, IL: Northwestern University Press, 1988. Limited Inc. means « SARL, Société à responsabilité limitée” in French.

[5] Krieg, Alice, 1999. Vacance argumentative : l’usage de (sic) dans la presse d’extrême droite contemporaine. Mots 58, pp. 11-34. https://www.persee.fr/doc/mots_0243-6450_1999_num_58_1_2523

 

PROOF and the ARTS OF PROOF

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

1. Vocabulary of the Arts of Proof

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

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

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

Agent Names – Some names are related to their root verbs with the meaning “person who (verb)”; so are arguer; reasoner; prover. Demonstrator derives from to demonstrate_2 “to show other people how something is used or done”.
This can be interpreted as a sign of subjective involvement in the mechanism of proving, arguing, reasoning.

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

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

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

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

Positional Markers – These terms which may be considered as quasi-synonyms in some contexts, can clearly appear as markers of argumentative positions in the context of a debate. In the legal arena, the judge hears the statements and arguments of the parties; each party presents (what it considers to be)  evidence and rejects the evidence presented by the opponent  as quibbling and sophistry. We are no longer dealing with synonyms, but with strongly opposed antonyms. The difference between evidence/proof, argument, and quibble becomes a simple matter of perspective. Proof and evidence are now nothing more than the positive value I assign to my argument and deny to my opponent’s.
A polite but decisive rebuttal is proposed as a mere objection and a simple argument. Argument is then a “lexical softener” for proof, its use implies a distance, a lesser commitment of the speaker to the claim.

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

2. Proof Between Fact and Discourse

Proofs are expressed in a language, natural or formal, and presented in a discourse. Formal proof brought about by a hypothetical-deductive demonstration is often considered the archetypal proof. Its counterpart in ordinary language would be the argument based on essentialist definition used in philosophy and theology. In other fields of activity, the discourse of proof requires a reference to the world, in which case, proof is now seen as evidence and fact. The proof-evidence is constructed through a series of experiences and calculations, as suggested by the concrete metaphors used to talk about evidence: to produce, to provide evidence, to bring a proof, to make a demonstration. This connection to reality is what distinguishes proof and argumentation on the one hand from formal demonstration on the other.

The concept of proof as fact invokes non-discursive evidence of material realities, that can be seen and touched. The proof that I did not murder Peter is that he is alive and standing in front of you. Such situations would seem to render language superfluous. But facts can become evidence through discourse alone. Evidence is relative to a problem, and discourse frames the situation in which the evidence solves the problem. Evidence can be silently presented to the appropriate judges. If some facts “speak for themselves”, at other times they are not so “eloquent”, or even “silent” to many. They need to be spoken for, and discourse is needed to make the material evidence visible. The cruel experience of Semmelweiss has certainly shown us that the de facto existence of seemingly indisputable facts does not predict their acceptance, see proof, facts and belief.

3. Functional Heterogeneity of the Discourse of the Proof

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

— Alethic: It establishes the truth of a fact.

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

— Explanatory: It accounts for facts that are not self-evident, by their integrating them into a coherent discourse in the correct language, be it a demonstration, or a story that explains what happened.

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

— Psychological: It removes doubt and inspires confidence.

— Rhetorical: It is persuasive.

— Dialectical: It removes the challenge, and closes the discussion.

— Social: It builds consensus, reassures the community affected by the problem, and strengthens their confidence in their technical ability to produce evidence especially, but not only, in the social and legal domains.

— Conversely, evidence excludes: Those who accept proof believe that those who resist it proof must be crazy, weak-minded, or carried away by their passions.

4. Unity of the Arts of Proof

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

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

— An intention: Like any discourse, the flow of demonstrative, argumentative, probative, and inferential discourse is organized by an objective, that is, an intention.

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

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

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

— Domain dependence. As argumentation, demonstration and proof are domain-dependent. The modes of production of proofs differ according to the domain, the kind of technical language used and the kind of experimental method employed in that domain. The establishment of large classes of scientific proofs is the task of epistemologists. Natural language argumentation is characterized by its unique ability to combine a large variety of heterogeneous proofs, corresponding to the different argument schemes with technical proofs coming from technical domains.

5. Argumentation Among the Arts of Proof

Perelman & Olbrechts-Tyteca’s New Rhetoric contrasts “argumentation” with “calculation”:

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

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

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

---

 

PROLEPSIS

The speaker may choose to link his own  line of argument 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 indeterminate degree of distortion, ranging from a literal quotation, together with its references,  to a sketchy evocation of a possible objection, which may be framed as a self-refuting scarecrow, see speech resumption At the very least, the quoted speech is extracted and readjusted 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 can be radically rejected; it can be dismissed as absurd.

Do we want to ruin all small savers? No, on the contrary, and for many reasons…

It can also be maintained in full force, until further information becomes available. In this sense, the modal-rebuttal component of Toulmin’s layout of argument is a prolepsis.

The proleptic structure includes not only coordinated or subordinated pairs of statements but any discourse pattern whose configuration corresponds to the staging of two opposing (anti-oriented) discourses, the speaker taking responsibility for one of them; it represents the maximum development of monological argumentation, see connective; destruction; Concession; refutation.

Several rhetorical terms refer to the same structure:

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

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

PROGRESS argument

1. Argument of Progress

By definition, “progress moves forward”; the progress argument evaluates the most recent as the best. If F1 and F2 are in the same category, and if F2 comes after F1, then F2 is preferable to F1.

The progress argument rejects the authority of elders and their practices, which are considered outdated; the contemporary practices that follow their model are dismissed as regressive, even repulsive.

Cats are no longer burned on cathedral forecourts, animal fights were outlawed in 1833, owls are no longer nailed to barn doors, and rats are no longer crucified as targets for darts. Whatever may be said in bullfighting circles, bullfighting with killing is doomed. (Le Monde, Sept. 21-22, 1986)

The argument is organized around the following operations.
– First, bullfighting is classified as a case of animal abuse, which puts it in the same category as burning cats, organizing cockfights, nailing owls to doors and crucifying rats.
– In a second step, the practices belonging to this category are listed in the chronological order in which they disappeared.
– This line of facts is then extrapolated to conclude that bullfighting should also be condemned in the face of social progress — and the sooner the better.

2. Argument of Novelty

Latin ad novitatem; novitas, “novelty; condition of a man who, as the first of his family, attains an eminent position (senator)” (Gaffiot [1934], Novitas). Novitas is opposed to nobilitas. Its argumentative orientation can be positive (the dynamic of the novitas is opposed to the decadent nobilitas), or negative: the homo novus, the “new man” who comes from nowhere, is suspected.

2.1 Traditional Orientation 

The argument of progress reverses the traditional view of the higher esteem accorded to the ancients, especially in the religious sphere: “the novitas is the index of heresy” (Le Brun 2011, §1). The argumentative orientation of the judgment “this is a novelty!” is reversed.

The argument of progress is opposed to the argument of the decline of civilizations, which attributes all virtues to the ancients.

2.2 Contemporary Orientation

The contemporary interpretation combines the argument of novelty with the argument of progress: “What just came out” is “super” exciting, and “déjà vu” is of little value. This argument values ​​innovation over routine, and the new over the old. It underlies the call:

Be the first to adopt it!

According to this rule, the recently published manual would necessarily be better than its predecessors, and, in politics, the newest candidate is already seen as the much-needed savior.

The syzygy is a different vision of progress, as a passage from an imperfect world to a perfect and unchanging one.

3. Ancients and Moderns

The argument of progress structures the eternal dispute between Ancients and Moderns. In its radical form, the argument affirms the absolute superiority of the latter over the former, both in the of arts and culture and in the sciences. Ultimately, this superiority would be that of the modern individual over his or her ancestors. In a relativized form, the argument of progress is compatible with the individual superiority of the ancients, “we are dwarfs on the shoulders of the giants”, not taller, but able to see further. This is classically refuted by the objection that the lice on the head of the giant’s head cannot see any further than the giant.

---

 

PRESUPPOSITION

The concept of presupposition can be approached as a logical or as a linguistic problem.

1. A Logical Problem

The problem of presupposition was first addressed in the field of logic, see Proposition The logic of analyzed propositions postulates that propositions such as “all As are Bs” can have two truth values, the true and the false. The problem arises when the reference of A and/or B is null (there is neither A nor B), as in “unicorns can fly (are flying creatures)” or “no unicorn is a dragon”. In such a case, is the proposition “all As are Bs” true or false? Consider the proposition “The king of France is bald”, said in 1905. It is impossible to assign a truth value to this statement, since in 1905, and still today, the French Republic does not have a king (Russell 1905).

From the point of view of logical technique, it is sufficient to add the premises “there are As”, and “there are Bs”, or “there is one, and only one king of France”. An apparently mono-propositional statement such as “the king of France is bald” is then translated into logical language by the conjunction of three propositions, each of which has its own truth value:

 “there is a king of France” & “there is only one king of France” & “he is bald”.

In 1905 as now, the first of the three propositions is false. It follows that the conjunction of logical propositions representing the statement “the king of France is bald” is simply false. This analysis has been criticized for failing to reflect the linguistic intuition of the ordinary speaker, for whom the statements “there is a king of France” and “this king is bald” do not have the same status in the original sentence: the first is asserted, the second is presupposed. This is true, but the objection is irrelevant, since formal logic does not aim to represent linguistic intuition, but wants to solve a technical problem, and this is what it does.

2. A Linguistic Problem

Ordinary statements can articulate different judgments, which have different semantic and discursive statuses, as is clearly shown in the case of statements which presuppose another statement
Imposing a presuppositional judgment violates the logical principle that a proposition expresses a single judgment (if it contains multiple judgments, each must be asserted separately). Therefore, it violates the dialectical rule that each proposition must be explicitly accepted or rejected by the respondent. Thus, S1 could therefore ask S2 « why P? » only if S1 and S2 had previously agreed on the existence of P. From a Perelmanian perspective, the question of presuppositions should be settled within the framework of prior agreements, see conditions of discussion.

2.1 The Layered Structure of Meaning

The presupposition is defined as an element of the semantic content of the utterance that resists negation and interrogation. The statement “Peter no longer smokes” presupposes that “Peter used to smoke”, and posits that “Peter no longer smokes”. The negative statement “Peter has not stopped smoking” and the interrogative “Has Peter stopped smoking?” share this presupposed content, “Peter used to smoke”. Negation and interrogation deal with the posed content (“Peter smokes now”), and do not change the presupposed content.

This layered structure of sentences is one of the main features that distinguishes statements made in ordinary language from logical propositions.

2.2 Presupposition as a speech act and the “many questions” problem

Ducrot redefines presupposition as a strategic act (an illocutionary act) performed with the aim of influencing, i.e. restricting the interlocutor’s possibilities to speak. The act of presupposition is a discursive power grab by which the speaker performs

An act of legal value [1], and thus an illocutionary act […] This act] transforms the interlocutor’s possibilities of speech, […] It modifies the interlocutor’s right to speak” (1972, p. 91).

Consider the following question:

Interviewer — What are you going to do to fight corruption in your own party?

The question assumes that “there are corrupt people and practices in your party. The respondent is given a choice:

(i) Either he or she accepts the presupposed statement and gives a response within the range of pre-formatted, expected responses, such as:

Respondent — I’m going to suspend all the members under investigation.

This answer respects the linguistic orientation of the question. It falls perfectly within the framework of the dialogue as established by the first speaker. The respondent submits to the interviewer.

(ii) He could also deny the presupposed statement:

Interviewee — To my knowledge, there is no (proven) case of corruption within my party.

This second response reframes the routine consensual dialogue; the interviewee rejects the claim made by the interviewer, and the dialogue takes on an uncompromising and polemical character, opening an argumentative₂ situation structured by the question “are there (proven) cases of corruption in the party?” The refusal of the assumption

“[is] always considered aggressive: it personalizes the debate, which becomes a quarrel. […] To attack the opponent’s presupposition is to attack the opponent himself” (Ducrot 1972, p. 92).

The presupposition seeks to impose an “ideological framework” (ibid., p. 97) on the ensuing dialogue, that is, to direct the partner’s speech. S. Many Questions; Conditions of Discussion; Persuasion.

Of course, the phenomena of presuppositions are not limited to dialogue, but, as always, dialogue serves to clarify the problem. A monologue that did not respect its own presuppositions would be inconsistent, while, in dialogue, the rejection of a presupposition is controversial. In reality, dialogue (i) develops under the same conditions as a monologue.

The case of presupposition is by no means the only content that can be exploited by a multiple question. In ordinary language all statements are more or less « loaded » not only by their orientation, but also by their implicit contents of various kinds, some of which are derived from the orientation of the words. In reality, it is always possible to extract controversial implicit judgments or to infer propositional judements from a statement and to hold the interlocutor responsible for them; any proposal is loaded with its probale consequences.
Consider a conversation between a banker and a customer who is trying to get a better interest rate:

S1₁: – I went to the bank opposite my house and they immediately offered me a loan at a lower rate than the one you proposed!
S2: – That’s because they wanted you as a customer.
S1₂: – Because you do not want to keep me as a customer?

S1₂ extracts or infers from S2’s intervention an implicit content that S2 certainly rejects, but nevertheless shows the banker that more explanations are needed. This move can be seen as a special straw man maneuver (de Saussure 2015).

---

[1] Based on a valid linguistic convention.

---

 

PRECEDENT

The argument from precedent corresponds to the topic n° 11 of Aristotle’s Rhetoric:

Another line of argument is founded upon some decision already pronounced, whether on the same subject or on one like it or contrary to it. (Rhet., II, 23, 11; RR, p. 365)

“Judgment” refers not only to the sentence of a court but to any judgment or decision made in the past, in ordinary life as well as in the political or legal sphere. A precedent in any domain is an example to be follow, a model.
If the matter has not been decided in a formal assembly, it may have been decided by such authorities as well-known fables, parables, proverbs or celebrated verses (Lausberg [1960], § 426).

Judgments are made in the context of previous judgments on cases “of the same kind”, i.e., belonging to the same category. Precedents correspond to a requirement of continuity and consistency between decisions made in the past and the decision to be made The structural coherence of the discursive field in question is thus strengthened, and protected against any ad hominem accusation directed against the institution, see ad hominem.

Like the argument ab exemplo, the argument from precedent motivates a decision or interpretation by reference to data or examples drawn from tradition. It is a conservative principle, that limits innovation in all areas in which it is applied. As such, it combines well with arguments that appeal to “the wisdom of our forefathers” (Bentham, 1824; see political argument; authority; progress.

The precedent principle proceeds in the following stages:

(i) A problem, P1, a case to be decided.
(ii) Research into similar problems and cases, leading to
(iii) A categorization: this case is similar to a previous case P0; it falls into the same category as P0, see Analogy (II); Categorization.
(iv) The decision, judgment, evaluation … E has been made about P0;
(v) By application of the rule of justice, a similar judgment must be made about P1. “Similar” here means the same judgment, a judgment that is proportional, or opposite; or, more simply, a judgment that is consistent with E.

The appeal to precedent can be blocked at the second stage, where it can be argued that there are substantial differences between P1 and the previous case P0.

The problem of judgment is automatically solved as soon as an analogy is drawn between the problematic fact and an established fact. Precedents save time and effort.

Past Historical Facts as Precedents

Not all past historical facts can serve as precedents; to function as such, they must serve as a common reference in a given present situation. In other words, past historical facts as precedents must be socially ratified, i.e., have a social presence, see schematization.

Historical memory is full of such historical moments that serve as guides for action, either positively, as models, or negatively, as anti-models.

The extermination of Jews, Gypsies and homosexuals during the Second World War is now the exemplary characterization of what the world should not be, and a founding example for political thought.
For the United States before the Iraq war, Vietnam was the basic analogical resource that was called to the rescue when it came to opposing military intervention abroad.

Such events produce figures that are a source of antonomasia. Antonomasia is the figure of speech by which a member of a category is designated by the name of the paragon of that category. In 1938, French  Prime Minister Paul Daladier and British Prime Minister Neville Chamberlain surrendered to Hitler at Munich. Now, their names are common name: A Daladier, a Chamberlain is a politician who capitulates to a dictator, the incarnation of what a politician should not do. By the same mechanism, to call a political conference « a new Munich » is to assume that it resulted in a capitulation. And of course, the paragon of all anti-models is Hitler.

 

---

 

PATHETIC ARGUMENT

1. Pathetic argument

Pathetic as evaluative
A participant may dismiss an argument that he totally rejects as “a pathetic, pitiful argument” because he finds it childish, empty or desperate.
In this use, the label pathetic argument is evaluative and can be applied to any kind of argument scheme. One can say “I find this argument pathetic” (evaluation), but not “I find this argument a pari”, only “in my view, this is an argument a pari” (description).

Pathetic as descriptive
The label “pathetic argument” can be applied descriptively to a variety of arguments based on negative or positive consequences. characteristic of wishful thinking. The conclusion is regarded as impossible and rejected because it would frustrate the arguer; or it is taken for granted because it is agreeable to him.

I fear that P, therefore not-P.
I wish that P, so P

– It can’t rain on Sunday, that would ruin our picnic!
– That’s not possible, we couldn’t handle the consequences!
– If this criticism were true, what would become of our discipline?
– Such pollution is unthinkable, it would claim thousands of victims.

This last case may be a purely pathetic argument, or a step in an argument that ends with a call to action: « urgent action must be taken. »

The pathetic argument applies to the realm of knowledge a style of argument that is quite common in the field of practical action:

I want P, so I strive for P, I pray for P, I try to bring about P.
I fear P, so I try to avoid P, to prevent P…

But wishing for P is different from striving to achieve P. This kind of  argument can be systematically be classified as pathetic i.e., “naive and desperate”.
Here, the label « pathetic argument » corresponds to a particular kind of appeal to consequences, with a feedback loop from effect to cause, as in the pragmatic argument. But the pragmatic argument depends on human action, while the pathetic argument invokes a magical action.

Pathemic argument and pathetic argument
A pathemic argument is not a pathetic argument. Pathemic derives from pathos; one can speak of a pathemic arguments to refer to ad passiones arguments, that is, to any emotion-based argument, such as appeal to anger, enthusiasm, pity, etc

Pathetic arguments and pathemi c arguments. 

2. “Pathetic fallacy”

The label “pathetic fallacy” refers to the anthropomorphic attribution of human emotions to non-human, inanimate beings. The label is evaluative, condemning the use of the rhetorical figure of personification.
The term was coined by John Ruskin:

I want to examine the nature of the other error, that which the mind admits when affected strongly by emotion. Thus, for instance, in Alton Locke,
They rowed her in across the rolling foam
The cruel, crawling foam.
The foam is not cruel, neither does it crawl. The state of mind which attributes to it these characters of a living creature is one in which the reason is unhinged by grief. All violent feelings have the same effect. They produce in us a falseness in all our impressions of external things, which I would generally characterize as the ‘pathetic fallacy’.
John Ruskin, Of the pathetic fallacy, [1856][1]

The label pathetic argument is consistent with the label pathetic fallacy. The pathetic fallacy condemns the personification of the natural world, while the pathetic argument suggests that the natural world is subservient to human desires; both movements blur the boundaries between the human and the natural worlds.

---

[1] In Modern Painters, vol. III, part IV, London: Smith Elder, p. 160. Alton Locke is a novel by Charles Kingsley (1850).

PARALOGISM

In classical Aristotelian logic, a paralogism is defined as an invalid syllogism. These paralogisms of deduction are “arguments of traditional syllogistic form which break one or another of a well-known set of rules” (Hamblin 1970, p. 44).

1. Syllogism Rules

Traditional logic has established the following rules for identifying and eliminating invalid syllogisms.
The following syllogisms respect all the rules of the syllogism; they are valid.

A syllogism contains three terms.

Nothing can be deduced from two negative premises,

no M is P
no S is M
no conclusion

If one premise is negative, then the conclusion must also be negative.

no M is P                             the major premise is negative.
some S are M,
so some S are not P           the conclusion is negative

In a valid syllogism, the middle term must be distributed at least once.

no M is P                       M is distributed (universal).
all S are M,
so no S is P                     the conclusion is valid.

If a premise is particular, then the conclusion is particular

no M is P
some S are M                  the minor premise is particular.
So, some S are not P          the conclusion is particular.

2. Paralogisms

A paralogism is a syllogism that does not respect one or more of the preceding rules. Of the 256 possible modes of the syllogism, only 19 modes are valid, meaning a syllogism can be fallacious in 237 different ways.
Whether it “seems” conclusive or not is irrelevant. The term paralogism  refers to a miscalculation.

The main forms of syllogistic paralogisms are as follows. The first form corresponds to the paralogism of homonymy, the others correspond to an inadequate distribution of qualities and quantities.

(1) Paralogism of four terms.
(2) Paralogism of two negative premises.
(3) Paralogism of a positive conclusion drawn from a negative premise.
(4) Paralogism of the undistributed middle term.
(5) Paralogism of the universal conclusion from a particular major.
(6) Paralogism of a universal conclusion drawn from a particular minor.

Examples

— The following paralogism consists of four terms:

Metals are simple bodies.
Bronze is a metal.
* Therefore bronze is a simple body.

But bronze is not a simple body but an alloy. In the minor premise, bronze is said to be a metal because it resembles an authentic metal such as iron, it can be melted and formed. In the major premise, metal is used in its strict sense. However, Metal is homonymous, and the syllogism actually has four terms, see ambiguity.

— The following paralogism concludes from two negative premises:

Some B are not C              some rich are not arrogant
No A is B                              no poet is rich.
* Therefore No A is C         *no poet is arrogant.

— The following paralogism concludes universally from a particular major premise.

all A are B                            all men are mortal
no C is A                               no dog is man
* Therefore No C is B          * no dog is mortal.

In the major premise, “all men are mortal”, the major term, mortal, is not distributed: this premise says nothing about all mortals, but only about certain mortals, namely, that “they are men”. Yet the conclusion “No dog is mortal” claims something of all mortals: “no mortal is a dog”. The major term is distributed in the conclusion and not in the major premise. Thus, the conclusion affirms more than the premise, which is impossible.

3. Evaluation Using the Rules of the Syllogism

Traditionally, syllogisms are evaluated using a system of rules (§1), in a step-by-step process.

— Check the number of terms, and propositions.
— Identify the middle term, the major term, and the minor term.
— Determine the quantity and quality of the premises and the conclusion.
— Identify the distribution of terms.
— Check the organization of the distribution of terms: check that the middle term is distributed at least once. If the major term or the minor term is distributed in the conclusion, ensure that it is also distributed in the premises and so on.

This cumbersome method is based on the unintuitive notion of the quantity of predicates. It shifts the analyst’s attention from what the syllogism asserts, that is, from the understanding of the structure and articulation of the syllogism, to the fragmented application of a system of rules.While it may develop the ability to apply an algorithm, but it is far from an everyday critical thinking process.

4. Evaluation with Venn Diagrams

The use of Venn diagrams provides a clearer, more intuitive basis for evaluating syllogisms. Three intersecting circles represent the three sets corresponding to the three terms. The assertion made by each premise is carried to the corresponding circle.
If a premise asserts that a set (consisting of a circle or a portion of a circle) contains no elements, that circle or the portion of a circle is blacked out (striped).

If a premise asserts that a set (id.) contains one or more elements, a cross is placed in the circle or portion of a circle.
Thus, a part of a circle is either black (i.e., has a cross), or remains white. If it is white, nothing can be said about it.

Once the data from the premises is plotted on the diagram, the result can be compared with what the conclusion. The diagram shows whether the syllogism is or is not valid.

Consider the following syllogism:

Some rich people are not arrogant.
No poet is rich.
* No poet is arrogant.

The three intersecting circles represent the rich (R), the poets (P) and the arrogant (A), respectively.

– “Some rich are not arrogant”: look at the circle of the rich and the circle of the arrogant; put a cross outside of their intersection: there are some people within this zone.
– “No poet is rich”: look at the circles of poets and of the rich people. Blacken their intersection: there is nobody within this zone.
– Finally, look at the circles of poets and of arrogant people. The conclusion asserts that the intersection of the circle of poets and the circle of arrogant people is black; but we see that this is not the case; it is partly white.
Conclusion: This syllogism is a paralogism.

Consider the syllogism:

No M is P
All S is M
Therefore No S is P

The three intersecting circles represent the set M, S and P.

— “No M is P”: the intersection of the circles M and P is black (empty).
— “Every S is M”: the non-intersecting zone of the circles S and P is black (empty).
— Looking at the intersection of the  S and P circles, we can see that the intersection is black (empty); this is precisely what the conclusion claims, “No S is P”. This syllogism is valid.

5. Paralogism of the permutation of the quantifiers

By generalization, the word paralogism can refer to any error in applying the rules of formal logic.
For example, the paralogism of quantification is an error committed when the existential quantifier and the universal quantifier are permuted:

All humans have a father; so they have the same father

For every human H, there is a human F, such that F is the father of H
* Therefore there is a human F such that for every human H, F is the father of H.

The following passage may contain such a paralogism:

And all the geniuses of science, including Copernicus, Kepler, Galileo, Descartes, Leibnitz, Buler, Clarke, Cauchy, speak like [Newton]. They all lived in true worship of the harmony of the worlds and of the all-powerful hand that threw them into space and sustained them.
And this conviction is not based on impulses, like poets. Figures, theorems of geometry give it its necessary basis. And their reasoning is so simple that children would follow it. First, they state that matter is essentially inert. It follows that, if any material element is in motion, it is because another has constrained it; for every motion of matter is necessarily a communicated motion. Thus they assert that if there is an immense motion in the heavens, which in the infinite deserts carries away  billions of suns with a weight that crushes the imagination, it is because there is an omnipotent engine.
Secondly, they state that this movement of the heavens presupposes the solution of the problems of calculation, which have required thirty years of study, etc.
Ém. Bougaud, [Christianity and the Present Times], 1883. [2]

---

[1] (My italics).
[2] Em. Bougaud, Le Christianisme et le temps présent, t. I. Paris: Poussielgue Frères, 5th ed., 1883.

---