CONVERGENT, LINKED, SERIAL Argumentation

The conclusion of an argumentation is usually expressed in a single statement, possibly expanded in a short closing speech, see Argument – Conclusion.
The arguments, i.e., the part of the argumentative discourse that supports and sometimes surrounds the conclusion, can be considerably developed along quite different lines:

— Convergent argumentation, also called multiple argumentation, combines several co-oriented arguments.

— Linked argumentation, also called coordinate argumentation consists of several statements that combine to form an argument.

— Serial argumentation, also called subordinate argumentation consist of a sequence of argumentations, such as the conclusion of the first one is taken as an argument to support a second one and so on, see sorite.

One caveat: these categories are logical categories; they assume that arguments correspond to clean-cut, continuous explicit linguistic segments, just like premises in a logical reasoning. This is not the case in ordinary language, where arguments can be intertwined, an argument can contain another argument,  and episodes of expository language can take a distinctly argumentative turn. See Tagging.

CONVERGENT argumentation

Convergence is a basic mode of organization of complex discourse to support a conclusion, S. Convergent, Linked, Serial.
Two or more arguments are convergent when they independently support the same conclusion. The arguments are said to be co-oriented, and the argumentation is said to be convergent or multiple.
“Two reasons are better than one”: In a convergent argumentation, a claim is defended on the basis of several arguments that, taken separately, may be relatively weak, but, taken together, combine to make a stronger case: “My computer is getting old, there are discounts on the price of my favorite brand, I’ve just got a bonus, I’m going to buy one! ”.

The above diagram shows each argument is represented as a whole. The following diagram spells out the transition laws according to Toulmin’s proposal, S. Layout; compare with linked argumentation:

As well as pro-arguments, counter-arguments can converge to refute a claim, see Script.

This open structure defines the argumentative network, in contrast to the demonstrative chain. In the demonstrative chain, each step is necessary and sufficient; if one step is invalid, the whole construction collapses. In the case of the argumentative net, if one link in the mesh is broken, the net can still be used to catch fish, at least the biggest ones.

In a convergent argumentation, the organization of the sequence of arguments is relevant. If the arguments are of a very different strengths, a ridiculous or a weak argument next to a strong one risks damaging the whole argumentation, especially if that argument ends the enumeration:

He’s a great hunter, he killed two deer, three wild boars and a rabbit.

In classical rhetoric, the theory of the general organization of discourse (Lat. dispositio) discussed the supposedly different persuasive effects of the various possible textual arrangements of converging arguments of different strength, S. Rhetoric.

Convergent arguments can be merely listed (paratactic disposition):

Arg, Arg and Arg, so Concl

They can be connected by any listing or additive connective:

first, Arg1; second, Arg2; third, Arg3; so Concl.
Additionally, also, in addition, let alone, moreover, not only, 
besides

Connectives such as besides, not only, in addition, let alone, not to mention… not only add argument(s) upon argument(s), they present them as if each one was actually sufficient for the conclusion, and only added just “for good measure” (Ducrot & al. 1980, pp. 193-232):

No, Peter will not come on Sunday, he has work, as usual, besides his car broke down.

The additive approach holds that each argument contributes a piece of truth, and that these pieces can be arithmetically added together, to form a large, conclusive discourse. Speech activity theory holds that, by default, an argument is presented as sufficient, and that the addition of other arguments actually follows the logic of commercial presentation to consumers (the audience), i.e. the speaker offers the audience a series of arguments he considers equally satisfying and self-sufficient.

Case-by-case argument  — To refute the conclusion of a convergent argumentation, each of the arguments supporting that conclusion must be refuted. Thus, a convergent argument is countered by a case-by-case rebuttal, limited to the cases presented by the proponent.

CONTRARY and CONTRADICTORY propositions

1. Definition

In logic, the « square of oppositions » connects the affirmative and negative propositions, the universal and particular propositions, according to a set of immediate inferences, among which are the relations of contradiction and contrariety, see Proposition §4

— Two propositions P and Q are contradictory if they cannot be simultaneously true or simultaneously false; that is, one of them is true, and the other is false, as shown in the  following truth table (see Logical connectives)

In the logical square, the proposition « All M are N » and « some M are not N » cannot be simultaneously false true of false; they are contradictory propositions.
A proposition and its negation are contradictory proposition.

— Two propositions P and Q are contrary when they cannot be simultaneously true, but can be simultaneously false.

In the logical square, the proposition « All M are N » and « No M is N » can be simultaneously false when « some M are N« ; they are contrary propositions.

These terms can easily be confused. The easiest way to avoid confusion is to relate the relations of contrariety and contradiction to two kinds of universes, thus defining two kinds of opposites. Let U be a universe containing a number of individuals.

(i) Contradictories — In the case of contradiction, the opposition is within a two-dimensional universe, such as the traditional system of genre: “— is a man” and “— is a woman” are contradictory predicates in this system. In a non-traditional genre system, they are contrary propositions.

U is a two dimensional universe; two properties P₁ and P₂ are defined upon this universe, such as:
— Every member of this universe possesses either the property P₁ or the property P₂:
— No one possesses both properties P₁ and P₂: no one is both (P₁ & P₂). This is noted as (P₁ W P₂), with the symbol ‘W’ for “disjunctive or”.

P₁ and P₂ are complementary properties; they divide the universe U into two complementary (non-overlapping) sets.
— P₁ and P₂ are contradictories (opposites); they stand in a relation of contradiction.

(ii) Contraries — In the case of contrariety, the opposition is within a multidimensional universe such as the universe of colors. “— has white hair” and “— has red hair” are contrary predicates: a person cannot have both white and red hair (notwithstanding the case of badly dyed hair roots); and he may have brown hair.

U is an n-dimensional (more than two dimensions) universe: P₁, … P_i, … P_n.

— Every member of this universe has one of these properties, P_j; that is, is either a P₁ , … or a P_i, … or a P_n.
— No one has two or more properties P₁ , … P_i, … P_n, that is, no one is both (P_k & P_l).
— P₁ , … P_i, … P_n are contraries; they are in a relation of contrariety.

To sum up, semantically related predicates, or properties, are opposite if they exhaustively divide their reference universe into a series of non-overlapping sets. If there are just two such properties, they are said to be contradictory properties; if there are more than two, they are said to be contrary properties. So, contradictories are the limit case of contraries.

2. Refutation by substitution of contrariety to contradiction

It follows that an assertion based on a contradiction can be refuted by showing that the universe under discussion should not be considered as two-dimensional, but multi-dimensional. This seems to be the case in the following example.

In 1864, Pope Pius IX published the Syllabus, that is, a collection or a catalog of the Vatican’s positions on “modernist” ideas. Considered retrograde, the Syllabus was strongly attacked by “the modernists. In 1865, Mgr. Dupanloup, defended the Syllabus in the following terms; “they” refers to the modernists.
It is an elementary rule of interpretation that the condemnation of a proposition, condemned as false, erroneous and even heretical, does not necessarily imply the assertion of its contrary, which could be another error, but only of its contradictory. The contradictory proposition is the one that simply excludes the condemned proposition. The contrary proposition is the one that goes beyond the simple exclusion.
Now! It is this general rule that they have apparently not even suspected in the unthinkable interpretation of the Encyclical and the Syllabus that they have been giving us for the past three weeks. The Pope condemns this proposition: “It is permitted to refuse obedience to legitimate princes” (Prop. 63).
They claim that, according to the Pope, disobedience is never permitted, and that it is always necessary to submit to the will of princes. This is a leap to the extreme of the contrary, and ascribes to the Vicar of Jesus Christ, the most brutal despotism, and slavish obedience to all the whims of the kings. This is the extinction of the noblest of all liberties, the holy liberty of souls. And that’s what they claim the Pope said!
Félix Dupanloup, Bishop of Orleans, [The Convention of September 15, and the Encyclical of December 8 [1864] ] (1865) [1].

Reasoning on the content
There are several possible responses for someone who receives an order from a civil authority (« the prince »). Let’s look at the following three:

a. Obey
b. Disobey = « refuse to obey »
d. Appeal against the order, to a higher authority than the one who gave the order: the latter is not legitimate; they abused their power, etc.

We do not mention the case of the interpretation (§3) of the order, which is probably too specific. Reasons such as the conscience clause are not considered at this time.

1. « The Pope condemns this statement: ‘It is permissible to refuse obedience to legitimate princes‘« .

<NEG (refuse obedience)> is contradictory to, i.e., exludes <refuse obedience>

The syllabus excludes option b. « refuse to obey », but leaves open all other opposites of obey. In other words, excluding refusal to obey is not imposing obedience.
« The contradictory proposition is the one which simply excludes the condemned proposition« . It does not say whether one must obey the command or protest against it.
Similarly, <NEG having blonde hair> is contradictory to, that is, excludes <having blonde hair>, but it does not say that the hair in question is brown or chestnut.

2. »The contradiction is that which goes beyond this simple exclusion » of the possibility of disobedience. It claims that this exclusion is tantamount to a prohibition of disobedience.
Modernists « leap to the extreme opposite« , misinterpreting what is a contradiction (three possibilities) as an opposition between two exclusive possibilities. They consider only two cases: either obey or disobey, they omit the case of appeal against the command.

Reasoning on the modality
Is the universe of the Syllabus binary or multidimensional? Let’s consider a position X.

— If it is a binary opposition, “allowed vs. forbidden”, then the propositions “it is permitted (to refuse obedience)” / “it is forbidden (to refuse obedience)” are contradictory: only one of these propositions is true. If we condemn the proposition “it is permitted to refuse obedience to legitimate princes”, then we have to conclude that the contradictory is true, that is to say, “it is forbidden to refuse obedience to legitimate princes”, or, in other words: “we must always bow our heads under the will of the princes.”
Thus, for Dupanloup, the malevolent “modernists” substitute contradictories for contraries, which he describes as “jumping to the last end of the contrary”, I understand a leap to the (binary) contradiction, which is the limit of (multidimensional) contrariety.
He accuses the modernists of reframing the Pope’s position, using a strategy of absurdification (an exaggeration to the point of absurdity), see exaggeration.

— If the position X enters a three-dimensional universe, as “required / permitted (indifferent) / forbidden” then the propositions “It is permitted / it is forbidden” (to refuse obedience) are not contradictories but contraries: they are not simultaneously true, but they can be simultaneously false, e.g. if X is indifferent. The conclusion “If X is not opposed, X is demanded” is not valid. If we condemn “It is permissible to refuse obedience to legitimate princes” then we can only conclude one or the other of these opposites:

It is obligatory to refuse obedience to legitimate princes.
It is forbidden to refuse obedience to legitimate princes.

Since it would be difficult to admit that Pius IX, or anyone else, prescribes a systematic duty of disobedience to the legitimate rulers, we are left with the other member of the disjunction, that is, “X is forbidden.

One could also put in parenthesis the alternative obey/desobey, neither obey nor disobey, but file an appeal against the order, arguing that the prince is not legitimate, or not empowered to issue this kind of order, or that the order is harmful to the common good, etc.
This might be worth a try, if the appeal is not suspensive, and if the prince is interested in discussing his policies with the people he orders.

[1] Quoted from Félix Dupanloup, La Convention du 15 Septembre et l’Encyclique du 8 décembre [1864]. In Pius IX, Quanta Cura and the Syllabus. Paris: Pauvert, 1967. P. 104-105.

[2] https://www.nd-chretiente.com/dossiers/pdf/articles/2010_la%20vertu%20d%27obeissance.pdf St Gregory sets the following limits to obedience:
No one is obliged to obey men in everything. The limit of obedience is the abuse of power.
Resistance to an abusive command is justified when its execution would cause certain harm to the common good.
Such an abuse may occur when the order comes from an authority that is not legitimate, or when the order comes from a legitimate authority but encroaches on a sphere that is not its own. There is also an abuse of power when the order of a legitimate superior, who commands within the limits of his authority, is contrary to the order of a higher superior: this establishes the duty to resist an order or law that is contrary to natural law or a formal order of God.

CONTRADICTION

1. In dialogue, a contradiction emerges when a first turn  of speech is not ratified by the partner’s next turn.
Emerging contradictions can be resolved on the spot through a series of adjustments and arrangements, playing with the margins of indeterminacy and windows of opportunity left by ordinary language and actions.
The contradiction is open when both parties produce oppositional (anti-oriented) turns of speech. When the opposition is thematized and ratified by both participants, an argumentative situation arises.

See Disagreement; Argumentative Question; Stasis;
Denying; Refutation; Counter-argumentation.

2. Contradiction in speech: ;Ad hominem; Consistency;  Absurd.

3. Contradiction between terms, see opposites.

4. Logical treatment of contradiction: Non-contradiction principle;
Propositions contrary and contradictory.

CONSISTENCY

The basic expression of argumentative coherence or consistency is non-contradiction,  see non-contradiction; absurd; ad hominem.

The consistency requirement is particularly important in systems of regulation of human behavior, religion, law, and  ordinary institutional or family rules.

The consistency requirement is expressed a contrario in the refutation strategy mentioned in Aristotle’s Rhetoric, topic # 22:

Another line of argument is to refute your opponent’s case by noting any contrast or contradiction of dates, acts or words that it anywhere displays. (1400a15; RR p. 373).

1. After the event as before

Topos ≠5, “On the consideration of time” appeals to consistency. This theme is not explicitly stated, but is presented by two examples:

If before doing the deed I had bargained that, if I did it, I should have a statue, you would have given me one. Will you not give me one now that I have done the deed? (Rhet, II, 23, 5; RR, p. 361).

The situation is this:

1. 1. X (asks nothing and) performs a deed (perhaps an impulsive heroic act)
   2. Afterwards, he asks for a reward.
   3. Argument: if he had asked before, a reward would have been agreed upon
      

The hero feels that all feats must be paid for as such. It is as if the definition of the word feat includes the characteristic “deserves a reward”:

L1:   — If you do it, you’ll get…
L2:   — I have done it, and done well, so give me …

This argument scheme express the disappointment of someone who reports finding a wallet and does not receive a reward.

2. Human (in)consistency

Consistency may be the rule, but inconsistency is a fact of life. This is what the argument scheme #18 says:

Men do not always make the same choice on a later and on an earlier occasion, but reverse their previous choice. (Rhet, II, 23, 18; RR, p. 371)

This argument scheme is materialized in the following enthymeme:

When we were exiles, we fought in order to return; now that we have returned, it would be strange to choose exile in order not to have to fight. (ibid.)

The enthymeme seems to assume the following situation. In the past, the exiles fought to return home, and they returned. In the present situation, they are suspected of refusing to fight, and preferring exile. They deny the charge with this enthymeme, which is a claim of consistency, as in:

You fought for this position, now you can’t accept being thrown out like this!

This is a kind of positive ad hominem argument; it may presuppose an a fortiori: “We fought to return to our homeland, a fortiori we will fight not to be thrown out of it!”
The accusers reply “Men do not always make the same choice, etc.”

The opposing party argues from a contrary view of human nature; the two opinions “men are constant / inconstant”, are equally probable (see ibid I, 2, 14; p. 25). They can thus be the basis for two antagonistic conclusions.

S. Ad hominem; A fortiori.

3. Consistency of the legal system and stability of the objects of the law

Lat. arg. a cohærentia, de cohærentia, “to form a compact whole”.

3.1 Principle of coherence of laws, a cohærentia

This principle requires that, within a legal system, one norm cannot conflict with another; the system does not allow antinomies. An argument can therefore be rejected if it leads to the view that two laws are contradictory; this is a form of argument from the absurd.
In practice, this principle excludes the possibility of the same case being decided in two different ways by the courts.

According to this principle, if two laws contradict each other, they are only seemingly contradictory, and, consequently, they must be interpreted in such a way as to eliminate the contradiction. If one of these laws is obscure, it must be clarified by reference to a less doubtful one.

The argument a cohærentia is used to resolve conflicts of norms. To prevent such conflicts, the legal system provides for adages, which are meta-principles of interpretation, such as “the most recent law takes precedence over the oldest”. These adages are interpretive meta-principles, coming from Roman law and sometimes expressed in Latin: “lex posterior derogat legi priori”.

3.2 Principle of the stability of the object of the law, in pari materia

Lat. in pari materia: lat. par, “equal, like”; materia, “topic, subject” argument “in a similar case, on the same subject”.

The argument a cohærentia deals with the formal non-contradiction of laws in a legal system. The argument in pari materia, or argument “on the same subject”, expresses a substantive form of consistency. It requires that a law be understood in the context of other laws that have the same goal or refer to the same beings, that is to say the same beings (persons, things, actions) or the same subject.

The given definition of the subject of the law must be stable and consistent. The application of the argumentation a pari presupposes the stability of the legal categories. see Classification; A pari.
This principle of consistency leads the legislator to harmonize the system of laws on the same subject. What constitutes the same subject and the set of laws on the same subject may be questioned. For example, anti-terrorist laws, are a package of different legal provisions, for which it is necessary to ensure that the definition of “terrorism” remains the same in each of the passages that use the term. If this is not the case, these laws need to be made consistent, which means that they themselves need to be underpinned by a consistent policy.

The two topoi discussed in the two following paragraphs are taken from Aristotle’s Rhetoric. They are based on the two incompatible, but equally recognized substantive topoi, “human conduct is, or ought to be consistent” and “human conduct is inconsistent”.

4. Argument from narrative inconsistency

As a special case of ad hominem argumentation, showing inconsistencies in the accusatory narrative can refute an accusation:

S1:    — You are the heir, you benefit from the crime, you killed to inherit!
S2:    —Then, I should have killed the other legatee too.

The prosecution will have to prove that S2 also intended to murder the other heir, or otherwise find an alternative motive. The defense starts from the hypothesis proposed by the prosecution to show that the actions of the suspect do not fit into the proposed scenario; the accusatory narrative contains flaws or contradictions.

The incoherent accusation argument exploits a basic principle of practical rationality: the suspect’s actions must be consistent with his or her claimed goal. The defendant can refute the accusatory narrative by showing that, according to hat narrative, he acted inconsistently:

You say I’m the killer. But it has been proven that just before the crime, I spent an hour in the cafe in front of the victim’s house, everyone saw me. It is not consistent behavior for a murderer to show himself at the scene of the crime.

Any weakness found in the prosecution’s scenario can then be used to exonerate the defendant.

The principle of consistency of laws and the principle of stability of the subject of the law concern the coherence of the legal system. The argument from the inconsistency of the narrative exploits the resources of narrative rationality: all the narratives offered as excuses, all the narratives mixed with argumentation are vulnerable to this kind of refutation.
Conversely, the argument seems plausible and reasonable because the story is so, and because the speaker knows how to tell it.

The strategies described in the topoi # 22, 25 and 27 and probably 18 (see above) of the Rhetoric are relevant to this discussion (Aristotle, Rhet., II, 23), see Collections 2.

EFFECT-TO-CAUSE Arg.

The word consequence can mean:

— Effect, referring to a causal, cause-effect relationship, see causality.
— Consequent, referring to a logical, antecedent-consequent relationship, see connectives (§Implication)

1. Effect-to-Cause Argument

Other expressions can also be used as well, such as argument by the effect, or from the effect to the cause, see also a priori, a posteriori.
The effect-to-cause argument works backwards from the effect to its cause. Data is considered to be the effect of a hypothetical cause, which that can be reconstructed based on the data combined with a known causal relationship between this type of fact and its cause.

You have a fever, therefore you have an infection

— Argument: A confirmed fact t, the patient’s temperature. This fact t belongs to the category of facts or events T, “having a temperature”, as defined by medicine. This is a categorisation process.
— Causal law: There is a known causal law linking I-facts “having an infection” to T-facts, “having a temperature.”
— Conclusion: t has a I-type cause, an infection, and the patient should be treated accordingly.

This corresponds to the diagnostic process. One might speak of diagnostic reasoning, a type of abduction.

The effect (the temperature) is the natural sign of the cause. These natural, palpable, effects provide a basis for argument from natural signs:

Look! The ashes are still hot, th a fire must have occurred recently. They cannot be very far away.

In the field of ​​socio-political decision-making, the argument by consequences corresponds to the pragmatic argumentation, which transfers the positive or negative evaluation of the effects of a proposed measure to the measure itself.

The pathetic fallacy is a type of pragmatic argument that goes from the premise that « Rain would ruin our party » to the conclusion, « So, it won’t rain”, as if one’s wishes could influence the natural course of events.

2. Arguments by the Identity of the Consequences

The same type of argument applies to deductions made from the implied meaning of words, as an appeal to the sense of semantic coherence or logical consistency:

Topos: “Another topic consists of concluding the identity of precedents from the identity of results.”
Instance: “There is as much impiety in asserting that the gods are born as in saying that they die; for either way the result is that at some time or other they did not exist” (Aristotle, Rhet. II, 23, 1399b5; F. pp. 313-315).

 If the reason for banning marijuana is that it causes a loss of control, then all substances that cause a loss of control must also be banned, including alcohol for example.
If something is condemned because it mechanically involves something negative, then it automatically creates a category of causes “having that kind of negative consequences”, which must also be condemned.

3. Refutation by Contradictory Consequences

Refutation by contradictory consequences is a type of ad hominem, used in dialectic:

Peter says “S is P”.
The fact that S has the consequence Q: the fact is acknowledged by Peter.
However, P and Q are incompatible.
Therefore Peter is saying incompatible things about S.

Example:

Peter says that power is good.
However, everyone agrees that power corrupts.
Corruption is evil.
Since good and evil are incompatible, power should exclude corruption to be good.
Peter says contradictory things.

CONSENSUS

1. Consensus as agreement

See Agreement; Persuasion

2 Argument from consensus

The label argument from consensus, appeal to consensus, covers a family of arguments claiming that a belief is true or that things must be done in such and such a way on the basis that everyone thinks or does so, and that other proposals should be rejected.

We have always thought, wished, done … so; so, buy (please, do…) so.
Everybody loves so-and-so product.
Everybody puts so-and-so ketchup on their burgers!

It implies that by disregarding the existing consensus, the proponent of a new action or a new idea, that is the opponent of the consensus, risks being excluded from that community, see burden of proof.

The universal consensus argument claims that “all people in all times have thought this way things have always been done this way”.
The existence of God has been argued on the basis of the universal consensus argument.

The argument from the relative (partial) consensus includes the argument from majority, the argument from number (Latin ad numerum; numerus, “number”) and related expressions:

The majority / many people … think, wish, do … X.
Three million Syldavians have already adopted it!
My book is selling better than yours.
He is a famous actor.

Common Sense — The argument of consensus is the kind of authority generously granted to traditional wisdom or to common sense, S. Authority.

I know that all true Syldavians agree with this decision
Only the extremes attack me, all people of common sense will agree with me.

The populist argument is based on a real or supposed consensus among (or attributed to) the people, see Ad Populum.

Bandwagon argument and fallacy — The bandwagon argument is a special case of the consensus argument. The bandwagon is the decorated chariot that leads the orchestra through town, the bandwagon argument adds joy and enthusiasm to the dry consensus argument. To get on the bandwagon is to follow the popular movement, to participate in a popular “emotion” in the etymological sense of “a public upheaval”. Joining a party to have fun and sing along should not be condemned as systematically fallacious; but, from the perspective of any opposing party, climbing on the bandwagon can be seen as fallacious, as a follow-the-group or follow-my-leader attitude, sheepish behavior, as uncritically adopting the views of the most vocal or visible group.

Logical CONNECTIVES

Logical connectives articulate simple or complex well-formed propositions so as to construct well-formed complex propositions, or formulas. Propositional calculus studies logical syntax, that is the rules for constructing well-formed formulas. It determines, among these formulas, which are valid formulas (logical laws, tautologies).

Propositions are denoted by the capital letters P, Q, R… They are said to be unanalyzed, that is, taken as a whole, in contrast to the analyzed propositions “[Subject] is [Predicate]” considered in the predicate calculus.

A binary logical connective combines two propositions (simple or complex) P and Q into a new complex proposition “P [connective] Q”. Logical connectives (or connectors) are also called functors, function words or logical operators

The most common connectives are denoted and read as follows:

↔           equivalence, “P is equivalent to Q”,
→           implication, “if P then Q”
&             conjunction, “P and Q”
V           disjunction, “P or Q”
W           exclusive disjunction, “either P or Q (not both)”

Logical connectives are defined on the basis of the possible truth values given to the propositions they combine. A particular logical connective is defined by the kind of combination it accepts between the truth values of the component propositions.

1.   The truth table approach to binary connectives

A logical connective is defined by its associated truth table. The truth table of a “P connec Q” binary connective is a table with three columns and five rows.

— The letters P, Q … denote the propositions; the letters T and F denote their truth values: true (T) or false (F). P and Q are propositions, while truth and falsity are said of propositions, “P is True”, “P is False”; so, the corresponding abbreviating letters use a different typographical character.

— Columns:

The truth values ​​of the proposition P are expressed in the first column
The truth values ​​of the proposition Q are expressed in the second column
The corresponding truth values ​​of the complex formula “P connec Q” are expressed in the third column.

— Lines:

The first line mentions all the propositions to be considered, P, Q and “P connec Q”.
The next four lines express the truth values of these propositions. Since each proposition can be T or F, there are four combinations to consider, each corresponding to a line.

1.1 Conjunction “&”

By definition, the conjunction “P & Q”

— is true if both P and Q are simultaneously true: line 2
— is false if one of them is false: line 3, 4; or both of them: 5.

This is expressed in the following truth table:

This truth table reads:

line 2: “when P is true and Q is true, then ‘P & Q’ is true”
line 3: “when P is true and Q is false, then ‘P & Q’ is false”
line 4: “when P is false and Q is true, then ‘P & Q’ is false ”
line 5: “when P is false and Q is false, then ‘P & Q’ is false ”

1.2 Equivalence, “ ↔ ”

The logical equivalence “P ↔ Q” reads “P is equivalent to Q”. This resulting proposition is true if and only if the original propositions have the same truth values.

Truth table of logical equivalence:

By this definition, all true propositions are mutually equivalent, and all false propositions are mutually equivalent, regardless of their meaning.

1.3 Disjunctions: Inclusive “V”; Exclusive, “W”

The inclusive disjunction “P ∨ Q” is false if and only if P and Q are simultaneously false; otherwise, it is true.

Truth table of the inclusive disjunction:

The exclusive disjunction <P W Q> is true if and only if only one of the two propositions it connects is true. In all other cases, it is false.

Truth table of the exclusive disjunction:

1.4 Implication: “→”

The logical implication symbol “→” reads “P implies Q”. P is the antecedent of the implication and Q, its consequent.

Truth table of the logical implication:

This table reads:

line 2:       The true implies the true
line 3:       The true does not imply the false
line 4:       The false implies the true
line 5:       The false implies the false

Only truth can be logically derived from truth (line 1), whereas, everything can follow from a false proposition, a truth as well as a falsehood.

The connectives for equivalence, conjunction, inclusive disjunction and exclusive disjunction are symmetric, that is, for these connectives, “P connective Q” and “Q connective P are equivalent (convertible):

P ↔ Q         ↔             Q ↔ P
P & Q          ↔             Q & P
P V Q         ↔             Q ∨ P
P W Q          ↔             Q W P

The implication connective is not convertible; that is, “P → Q” and “Q → P” have different truth tables.

The laws of implication express the notions of necessary and sufficient conditions:

A  → B (is true)
A is a sufficient condition for B
B is a necessary condition for A

A causal relationship can be expressed as an implication. To say that if it rains, the road is wet, means that rain is a sufficient condition for the road to be wet, and that the road is necessarily wet when it rains.
The implication thus defined is called a material implication; it has nothing to do with Toulmin’s substantial logic.

The implication “P → Q” is false only when P is true and Q false (line 2). In other words, “P → Q” is true if and only if “not-(P & not-Q)” is true.

Line (3) asserts the truth of the implication “If the moon is a soft cheese (false proposition), then Napoleon died in St. Helena (true proposition)”. Like the other logical connectives, the implication is indifferent to the meaning of the propositions it connects. It takes into consideration only their truth-values.​​ The strict implication of Lewis tries to elminate this paradox by requiring that for “P → Q” to be true, Q must be deducible from P. This new definition introduces semantic conditions, in addition to the truth values. This explains why the word “implication” is sometimes used in the sense of “deductive inference”.

Systems of natural deduction are defined in logic (Vax 1982, Deduction). They have nothing to do with Grize’s Natural Logic.

2 Logical laws

Using connectors and simple or complex propositions, one is able to construct complex propositional expressions, for example “(P & Q) → R”. The truth value of such a complex expression is only a function of the truth of its component propositions. Truth tables can be used to evaluate these expressions. Some of them are always true, they correspond to logical laws.

2.1 “Laws of thought”

Binary connectors combine in equivalences known as De Morgan’s laws, which are considered to be laws of thought. For example, the connectives “&” and “V” enter into the following equivalences:

The negation of an inclusive disjunction is equivalent to the conjunction of the negations of its components:
¬ (P V Q) ↔ (¬P & ¬Q)

The negation of a conjunction is equivalent to the disjunction of the negations of its components:
¬ (P & Q) ↔ (¬P V ¬Q)

Case-by-case argumentation is based on inclusive disjunction.

2.2 Hypothetical (or conditional) syllogism S. Deduction

2.3 Conjunctive syllogism

The following statement expresses a logical law:

If a conjunction is false and one of its components is true, then the other component is false

[¬(P & Q) & P] → ¬Q

The corresponding three-step deduction is called a conjunctive syllogism:

¬(P & Q)           the major proposition denies a conjunction
P                      the minor affirms one of the two propositions
————
¬Q                   the conclusion excludes the other

An adaptation to ordinary reasoning:

No one can be in two places at the same time
Peter was seen in Bordeaux yesterday at 6:30pm (UT)
So, he was not in London yesterday at 6:30 pm. (UT)

Knowing that Peter is a suspect; that his interest is to hide that he was really in Bordeaux, and that the witness is more reliable than the suspect, we can conclude that Peter lied when he pretended to be in London at 6:30pm yesterday.

In the following example, the major of the disjunctive syllogism is the negation of an exclusive disjunction:

¬(P W Q)          a candidate cannot be admitted and rejected
¬P                   my name is not on the list of successful candidates
————
¬Q                   I am rejected

All these deductions are common in ordinary language, where their self-evidence ensures that they go unnoticed. It would be a mistake to ignore them on the pretext that, since these arguments are valid, they are not arguments.

3. Connectives in logic and in language

Introductory courses in logic make consistent use of ordinary language to illustrate both the capabilities and the peculiarities of logical languages. In general, logic can be “applied to ordinary language” (Kleene 1967: p. 67-73) as a tool for expressing, analyzing and evaluating ordinary arguments as valid or invalid reasoning. These translation exercises are as follows (id. p. 59):

I will only pay you for your TV installation only if it works  — translated as P → W
Your installation does not work  — translated as ¬W
So I will not pay you — translated as  ¬P

Using the truth table method this reasoning is then tested for validity, and declared valid.

To identify similarities and differences, natural language components and properties can be compared with their logical language counterparts. This allows us to better understand both types of languages. Such exercises are helpful in gaining a better understanding of logical or linguistic systems, and may also be useful in teaching argumentation. However, there are some additional facts which should be taken into one consideration when using this methodology.

(i) The previous exercise did not focus on the correct combination of the truth values of semantically independent propositions, as was the case of the logical argument about the moon and Napoleon (see §1.4 supra). The exercise introduces a strong condition on semantic coherence between the linked propositions, they belong to the same domain of practical action, in this case, TV installation.

(ii) Natural language connectives do not connect propositions in the way that logical connectives do. The former can be said to be between the two propositions, whereas the latter are syntactically attached to the second proposition. Logical connectives and natural language connectives have two different syntaxes.

As a consequence, the right scope of a linguistic connective is essentially defined by the sentence to which it belongs, whereas its left scope can be much larger, and may include an entire narrative, with various twists and turns:

So, the prince married the princess — The End —

Connectors are classically thought of as connecting two statements in a complete discourse, such as yet in:

the path was dark, yet I slowly found my way (google)

Still, in:

 It is good, yet it could be improved (d.c, Yet)

yet introduces a more complex scenario, and the previous example is not a complete discourse. Yet announces that more hints will follow, specifying the weaknesses of the task being evaluated.

(iii) In many cases, the logical reconstruction of ordinary reasoning must introduce new propositions that are said to be present but are left implicit in the discursive string under consideration. This string is then said to contain an “incomplete argument”, see Enthymeme.

(iv) Elementary logical reasoning does not cover all ordinary reasoning:

I ate three apples and two oranges, so I have had my five-fruit diet today

First, this seemingly crystal-clear reasoning is loaded with implicit knowledge, such as “apples are fruits,” “oranges are fruits,” and that “no orange is an apple”. “Three citrus fruits and two oranges” sum up as five fruits only if none of the mentioned three citrus fruits is an orange.

Second, the critical fact here is that the conclusion is based on an addition that is easier to solve in arithmetic than in a logical language. Toulmin’s layout would satisfy this condition by adding a warrant-backing system that refers to the laws of arithmetic.

(v) Logical connectives capture only a small part of the linguistic role played by natural language connectives. The connector “&”requires only that the conjoined clauses be true. This property is common to many ordinary words, and, but, yet … and to all concessive words:

The circumstances which render the compound true are always the same, viz. joint truth of the two components, regardless of whether ‘and’, ‘but’ or ‘although’ is used. Use of one of these words rather than another may make a difference in naturalness of idiom and may also provide some incidental evidence to what is going on in the speaker’s mind, but it is incapable of making the difference between truth and falsehood of the compound. The difference in meaning between ‘and’, ‘but’, and ‘although’ is rhetorical, not logical. Logical notation, unconcerned with rhetorical distinctions, expresses conjunction uniformly. (Quine 1959, p. 40-41)

In other words, elementary logical theory has no adequate concepts to deal with the phenomena of argumentative orientation, and imposes no obligation in this respect. Quine’s argumentative strategy is to minimize the problem and delegate it to rhetoric, which is seen as a dumping ground for problems left unsolved by logical analysis.

The word « and » carries with it subtle semantic conditions, such as a sensitivity to temporal sequence. If “P & Q” is true, then “Q & P” is true. But these two statements do not contain the same information, and this is no longer a matter of rhetoric, whatever the meaning of that word means:

They married and had many children.
They had many children and married.

One might think that, under certain conditions, this logical analysis introduces a third proposition “The events succeeded in this order”. For other conditions affecting the use of and, see Composition and division.

3. No subordination, but bilateral relations

There is no ideal way to imagine the relationship between logical language and natural language; everything depends on the theoretical and practical goals of the researcher, whether he is building a conversational robot, or developing a formal syntax for ordinary language, or teaching second-level argumentation courses.

Logic is an autonomous mathematical language, that can be constructed from some chosen segments of ordinary language. From the beginning, the teaching of logic can draw more or less heavily on the resources of ordinary language. The same is true for the teaching of everyday argumentation in relation to the resources provided by the logical language. The teacher is free to make pedagogical choices, and possible alternative approaches should be judged by their results, according to the standard methods used to evaluate educational methods.

CONDUCTIVE ARGUMENT

Conductive arguments are defined by Wellman as third type of argument, parallel to deduction and induction. Using the following examples (my numbering), he notes that, “it is tempting, therefore, to define a conductive argument as any argument that is neither deductive nor inductive” (1971, p. 51):

(1) You have to take your son to the circus because you promised.
(2) This is a good book because it is interesting and thought provoking.
(3) Although he is tactless and nonconformist, he is still a morally good man because of his underlying kindness and real integrity. (Ibid.)

Wellman distinguishes between three types of conductive arguments

(i) “A single reason is given for the conclusion” (id. p. 55), as in

(4) You ought to help him because he has been very kind to you.
(5) That was a good play because the characters were so well drawn. (Ibid.)

(ii) “In the second pattern of conduction, several reasons are given for the conclusion” (id., p. 56), as in:

(6) You ought to take your son to the movie, because you promised to do so, it is a good movie, and you have nothing better to do this afternoon.
(7) This is not a good book, because it fails to hold one’s interest, is full of vague description, and has a very implausible plot. (Ibid.)

(iii) “The third pattern of conduction is that form of argument in which some conclusion is drawn from both positive and negative considerations. In this pattern, reasons against the conclusion are included as well as reasons for it” (id., p. 57), as in

(8) In spite of a certain dissonance, that piece of music is beautiful because of its dynamic quality and its final conclusion.
(9) Although your lawn needs cutting, you ought to take your son to the movie because the picture is ideal for children and will be gone by tomorrow. (Ibid.)

The key feature of conductive reasoning seems to be condition (3), where, depending on the speaker, and given the same reasons, the pros may outweigh the cons or vice versa (Blair 2011). Given the same data, another speaker might reach the opposite conclusion.

(8.1) In spite of a certain dynamic quality and its final conclusion, that piece of music is ugly because of its dissonance.

The adjective certain seems to be attached to the connective in spite of, indicating that the speaker will not argue on the basis of this argument (will not identify with this voice), S. Interaction, Dialogue, Polyphony.

A conductive argument does not seem to be amenable to default reasoning. Their conditions of refutation are different. Default reasoning might be updated or changed as new information is accessed, while conductive reasoning does not depend on information as such. A conductive argument is typically concerned with values, either moral or aesthetic. The specific problem of conduction is the hierarchization, or balancing of values. While some pairs of values will be very difficult, if not impossible, to balance, others will be quite plausibly balanced. for example, sentence (8) can be plausibly transformed into (8.1), because the three implied values cannot, in my view, be hierarchized, while (9) invokes values that seem easier to balance:

(9.1) I know, the movie is ideal for children and won’t be showing in the cinema after tomorrow, but you ought to cut your lawn!

Cutting the lawn seems to be a task that can easily be postponed, in view of the children’s education and their legitimate satisfaction, which could be prioritized. So, in the case of (9), the consensus would be that the pros clearly outweigh the cons.

In any case, more complex interactional data might provide some indications of how dissenting speakers fare when dealing with competing values.

CONDITIONS OF DISCUSSION

The Treatise on Argumentation insists on the necessity and variety of “prior agreements” between participants to develop an argumentation — that is, an argument₁; no prior agreements are necessary to engage in an argument₂:

For argumentation to exist, an effective community of minds must be realized at a given moment. First, there must be agreement in principle about the formation of this intellectual community, and then about the fact of debating a particular question together: now, this does not happen automatically. (Perelman & Olbrechts-Tyteca [1958], p. 14)

Two different kinds of agreements are mentioned here, and, as the text points out, neither of them can be taken for granted.

1. Formation of speech communities

This first type is concerned with the realization of an “effective community of minds”, constituted on the basis of the free decision of the participants. It can be considered as an ideal form of argumentative communication. Its closest approximation may be philosophical or scientific friendly encounters.

Not all argumentative practices depend on the production of such a community. The court is a prototypical argumentative venue, and no prior voluntary agreement with criminals is required to ensure their timely appearance; if necessary, legal coercion may be used. Institutions that define specific forums, problems and rules of interaction determine the social and legal conventions that govern argumentative communities. The existence of these social infrastructures makes it possible to avoid the previous cumbersome negotiations between speech communities.

2. Agreement on the issue

To discuss an issue, must we first “agree to discuss this issue together”? As in the case of the types of agreements described immediately above, the different legal systems determine who has the legal right to determine the charges that will lead to the appearance of a particular party; the defendant does not necessarily agree to discuss the matter, but is summoned by the judge.

ln institutionally structured communities, preliminary discussions may be useful in determining the items to be discussed at a given meeting. But the agenda is not necessarily determined by the prospective participants in the discussion; it may be the prerogative of an individual in charge of the organization. On the other hand, the topic itself, may be reformulated during the meeting.

Intellectual communities are also social communities, even when they deal with questions of human existence in general. The disputability of a question is itself an argumentative exercise, in the same way as is the process of discussing the question itself. There are two distinct subquestions to be considered, first, a central one, the conditions for the “disputability” of the issue properly speaking, and second, if all the potential partners agree to discuss such and such an issue, a practical question has to be settled, the material conditions for the discussion itself – where, when, who will chair the discussion, etc. — not to mention the shape of the table.

The dispute over maximizing vs. minimizing the right to discuss defines what may be called the stasis of stasis.

2.1 Maximizing the right to discuss

In terms of content, one can either emphasize the principle of radical freedom of expression according to which any point of view can be affirmed and challenged, or one can emphasize the pragmatic conditions of such discussion. The first of the “Ten Commandments for Reasonable Discussants” states that,

Commandment 1, Freedom rule: Discussants may not prevent each other from advancing standpoints or calling standpoints into question. (van Eemeren, Grootendorst, 2004, p. 190)
S. Rules.

This is also the position taken by Stuart Mill:

If all mankind minus one, were of one opinion, and only one person were of the contrary opinion, mankind would be no more justified in silencing that one person, than he, if he had the power, would be justified in silencing mankind. (Mill, [1859], p. 76)

2.2 Conditioning the rights to discuss

Absolute freedom of speech would give free rein to racist speech, hate speech, collective verbal and non-verbal harassment of the individual chosen as the scapegoat of a group, etc., forms of speech that many would find unacceptable. If individuals are free to discuss anything privately, provided they can find partners willing to do so, actual speech communities impose conditions on social discussion. For example, the res judicata principle prevents the reopening of an issue that has already been decided, unless a new fact is to be considered.

Moreover, the proper functioning of a speech community must take into account the fact that it is not possible to discuss anything (condition on the subject, on the agenda), with anyone (condition on the participants), anywhere and anytime (material conditions on place and time), no matter how (according to what procedure), see Manipulation:

Some Truths Are Not for Common Ears. It is lawful to speak the truth; it is not expedient to speak the truth to everybody at every time and in every way.
Erasmus, [1524], On the Freedom of the Will. (no pag.) [1]

The Treatise is very sensitive to the “anybody” condition:

There are beings with whom any contact may seem superfluous or undesirable. There are some with whom one cannot be bothered to speak. There are also others with whom one does not wish to discuss things, but to whom one merely gives orders.
(Perelman & Olbrechts-Tyteca [1958], p. 15)

Aristotle limits the subjects of legitimate discussion to the endoxa, and rejects debates that question “everything”, that is, affirmations that in practice no one doubts:

Not every problem, nor every thesis, should be examined, but only
 one which might puzzle one of those who need argument, not punishment or perception. For those who are puzzled as to know whether one ought 
to honor the gods and love one’s parents or not need punishment, while those who are puzzled to know whether snow is white or not need perception. (Top., 11)

The uncontroversial refers to three kinds of evidence: sensory evidence, “snow is white”; religious evidence, “we must honor the gods”; and social evidence, “we must love our parents”; these statements are uncontroversial because it is inconceivable that anyone would argue otherwise — in Aristotle’s Athenian society of course. For an opinion to be worthy of doubt, it must, on the one hand, fall within the scope of the doxa. That is, it must be part of the defining beliefs of the community, or seriously held by some of its honorable members or a subset, see Doxa.

On the other hand, the doubt must be serious, that is motivated. Since argument is a costly activity, one must have a good reason to doubt. In other words, the person who wishes to question an accepted proposition bears the burden of proof.

In the same spirit, the theory of stasis categorizes as uncontroversial (a-stasic) misplaced, poorly formulated or intractable questions, or, conversely, questions whose answer is obvious, S. Dialectic; Self-evidence; Stasis; Argumentative question.

On the legitimizing effects of debate, see Paradoxes.

3. Agreements on what counts as an argument

Agreements on the community of speech and on the issue must be supplemented by agreements on beings, facts, rules and values ​​(Perelman, Olbrechts-Tyteca [1958], II, 1). Agreements here should determine what counts as an argument: the condition of truth; the relevance of the true statement to the defended conclusion; the relevance of the conclusion (defended by a true and relevant statement) for the debate itself, see Relevance.

When it is impossible to determine whether a statement is true, or relevant to a conclusion, or relevant to the debate itself, a general system of acceptance or tacit agreement is invoked. In serious global disagreements, partial agreements are difficult to reach; the disputants anticipate their opponent’s conclusion, knowing well that once the argument is accepted, the conclusion will quickly follow, hence the tendency to postulate disagreement as a governing principle, even upon what should be considered facts, see Politeness; Dissensus; Disagreement.

This “appeal to agreement” is actually based on an argument of perverse effects, considering that the absence of agreement would condemn the debate to an undesirable state of deepening disagreement, which might even lead to a collapse of the discussion (Doury 1997). In practice, two facts have to be considered. First, points of agreement and disagreement can be negotiated on the spot, during the discussion. Second, the lack of agreement does not preclude argumentation, it is sufficient for third parties to take the reins of the discussion. The judge’s decision, and more generally that of the third party, is often made on the basis of an argument rejected or ignored by one or both parties, see Roles. Judicial organizations intervene precisely when no agreement can be reached between the parties; as representatives of the ruling power, they dispense with agreements — not with arguments.

In general, if one agrees on the data and rules, the conclusion follows automatically; argumentation becomes demonstrative. But argumentation is a linguistic way of dealing with differences in a system of generalized disagreement and uncertainty. There is a crucial incompatibility between the material interests at stake: one can indeed divide the cake, but what is eaten by any one person cannot be shared with the other. Serious, deep, insoluble … disagreement between the parties, proponents and opponents, should be considered to be the basic condition of argumentation; that is why third parties play a key role in argumentative devices.

[1] Quoted from Desiderius Erasmus, On the Freedom of the Will. Trans. by E. Gordon Rupp (no pag., no date). www.sjsu.edu/people/james.lindahl/courses/Hum1B/s3/Erasmus-and-Luther-on-Free-Will-and-Salvation.pdf (05-23-17).