According to H. P. Grice, the intelligibility of the conversation is ruled by “a rough general principle which participants will be expected (ceteris paribus) to observe”, namely:

‘Make your conversational contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged’. One might label this the Cooperative Principle. (1975, p. 45; capitalized in the text).

This “Principle of Cooperation”, is specified under four forms, “Quantity, Quality, Relationship and Manner” (ibid.).

— Quantity: “I expect your contribution to be neither more nor less than is required” (ibid.).
— Quality: “I expect your contribution to be genuine and not spurious” (ibid.). This can be compared to the requirement of accuracy mentioned in the pragma-dialectical rule 8; the same concern is also found in Hedge’s Rule 1 “For an honorable controversy”, S. Rules.
— Relation: “I expect a partner’s contribution to be appropriate to immediate needs at each stage of the transaction” (ibid.). This concerns in particular the relevance of the turn in relation to the present topic of dialogue and action. Grice recognizes the difficulty of identifying what is relevant in an exchange. The pragma-dialectical “Relevance rule” deals with this same requirement (van Eemeren, Grootendorst (2004, p. 192). S. Relevance; Rules.
— Manner: “I expect a partner to make it clear what contribution he is making” (ibid.). This entry can cover the refusal of the obscurity of expression and action; of ambiguity (the first of the Aristotelian fallacies); of the unnecessary prolixity, corresponding to the fallacy of verbiage.

Grice holds that his principles capture the rational character of conversation:

One of my avowed aims is to see talking as a special case or variety of purposive, indeed rational behavior. (Id., p. 47)

as well as its reasonable character: Respecting these principles is not merely “something that all or most do IN FACT follow, but as something that it is REASONABLE for us to follow, that we should not abandon” (id., p. 48; capitalized in the text).

These four principles can be compared with those advanced by normative theories of argument, S. Rules.

A statement violating Grice’s principles is not eliminated as fallacious, but is understood as an indirect speech act. When a participant notes that something is not in conformity with a conversational rule, the reaction is not to accuse the partner of making an irrelevant or irrational contribution, but to engage in an interpretive process to identify why he or she has flouted the conversational rule. The analysis of fallacies reverts to this interpretive orientation whenever it adds to its logic pragmatic considerations taking into account the contextual conditions of the exchange.

In an argumentative situation, the concept of cooperation is a strategic issue redefined by the participants, who are not necessarily willing to cooperate, for example in their own refutation. There is nothing scandalous or irrational about this, insofar as partners are aware of being in such an intentionally opaque context, S. Politeness. Rational, reasonable, as well as honorable rules for discussion are intended to reintroduce or strengthen cooperation in such antagonistic contexts.

1. Logic

In logic, two propositions are converse (in a relation of conversion) if they swap their subjects and their predicates. “As are Bs” and “Bs are As” are converse propositions. The converse of a true proposition is not necessarily true, S. Proposition.

2. Grammar and argumentation

In grammar, conversion can apply to any binary structures. Restructuring an expression of the opponent, that is, playing with his or her words, can be instrumental in reversing the global orientation of his or her discourse, according to the mechanisms of the antimetabole, S. Orientation Reversal.

Well, you know, this talk about the so-called pleasures of retirement is just empty talk to mask the retirement of pleasures.
Personally, I’d prefer a frightful end to this endless fright.
González, on Kohl ‘He fought for a European Germany, never again a German Europe.” (El País, 07-01-2017)

One can radically counter-argue a proposition by emphatically supporting its converse, S. Causality (II); Analogy:

S1 —     A is the cause of B;
A is like B; A mimics, copies B.

S2 —     Not at all! B is the cause A!
B is like A; B copies A.

In the same way, a sweeping defense strategy consists in converting the roles of accuser and accused, first by applying the reciprocity principle, “it takes one to know one”:

You blame me (for X), I blame you (for Y)
You filed a complaint against me (for X), I file a complaint against you (for Y).

and, second, by converting the position about the same criminal offense:

You are the culprit, you did it, you, who accuse me!

The child’s reply “he who says it did it” converts the accusation, , and justifies the counter-accusation:

S1 — You stole the orange!
S2 — No, you stole it, who says it did it!

The fact that S1 accuses S2 is used by S2 as an argument to accuse S1. S. Reciprocity; Stasis.

The conclusion of an argumentation is usually expressed in a single statement, possibly expanded in a brief conclusive speech, S. Argument – Conclusion. The argument part, supporting and sometimes surrounding the conclusion, can be considerably developed along quite different lines, referred to as:

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

— Linked argumentation, also called coordinate argumentation is composed of several statements combining into an argument.

— Serial argumentation, also called subordinate argumentation is composed of a succession of argumentations, such as the conclusion of the first one is taken as argument to support a second one and so on, S. Sorite.

Convergence is a basic mode of organization of complex discourse supporting 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 convergent or co-oriented, and the argumentation is called convergent or multiple.
“Two reasons are better than one”: in a convergent argumentation, a claim is defended on the basis of several arguments which, considered separately, can be relatively inconclusive, but, considered as a whole, combine to make a stronger case: “My computer is beginning to age, there are discounts on the price of my favorite brand, I’ve just got a bonus, I will buy one! ”.

 

In the above diagram, 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. S. Script.

This open structure defines the argumentative net, as opposed to the demonstrative chain. In the demonstrative chain, each step is necessary and sufficient; if one step is invalid, the constituent parts, and, in turn, the whole construction collapses. In the case of the argumentative net, if one link in the mesh breaks, 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 strength, a weak argument alongside a strong argument risks damaging the whole argumentation, especially if this 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 discourse general organization (Lat. dispositio) discussed the supposed 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

The argument 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 upon argument(s), they present them as if each one was actually sufficient for the conclusion, and are adduced 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 considers that each argument brings in a part of truth, and that these parts can be arithmetically added to create one big decisive discourse. Speech activity theory considers that by nature, an argument is presented as sufficient, and that their addition actually obeys the logic of commercial display for consumers (the audience), that is to say the speaker offers the audience a range of equally satisfying and self-sufficient arguments.

Case-by-case argument  — To refute the conclusion of a convergent argument, each of the arguments supporting this conclusion must be discarded. Thus, a convergent argument is countered by a case-by-case refutation, limited to cases that have been advanced by the proponent.

1. Definition

In logic, the square of oppositions links the affirmative and negative propositions, universal and particular, according to a set of immediate inferences, among them the relations of contradiction and contrariety, S. Proposition.

— Two propositions P and Q are contradictory when they cannot be simultaneously true or simultaneously false; that is, one of them is true, and the other is false, as shown in the truth-table below.
— Two propositions P and Q are contrary when they cannot be simultaneously true, but can be simultaneously false, S. Contrary and contradictory.

These terms can be easily mixed up. The easiest way to avoid confusion is to refer the relations of contrariety and contradiction to two kinds of universes, defining two kinds of opposites. Let U be a universe including a series of individuals.

(i) Contradictories — In the case of contradiction, the opposition is within a bi-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 system of genres, they are contrary propositions.

U is a two dimensional universe; two properties P₁ and P₂ are defined upon this universe, such as:

— Any members of this universe possess either the property P₁ or the property P₂:
— None possess both properties P₁ and P₂: neither is both (P₁ & P₂). This is noted (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 are in a relation of contradiction.

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

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

— Any members of this universe possess one of these properties, P_j; that is, is either a P₁ , … or a P_i, … or a P_n.
— None possess two or more properties P₁ , … P_i, … P_n, that is, none is both (P_k & P_l).
— P₁ , … P_i, … P_n are contraries; they are in a relation of contrariety.

To sum up, semantically connected predicates, or properties, are opposite if they divide exhaustively their universe of reference 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.

	Two-dimensions opposition: the two opposite properties are contradictories
Opposites
	More than two-dimensions opposition: the more-than-two opposite properties are 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 summarizing the positions of the Vatican about “modernist” ideas. Considered as retrograde, the Syllabus is 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 proposal, condemned as false, erroneous and even heretical, does not necessarily imply the assertion of the contrary, that could be another mistake, but only its contradictory. The contradictory proposition is the one that simply excludes the condemned proposition. The contrary is the one that goes beyond the simple exclusion.
Well! It is this common rule that they apparently have not even suspected in the inconceivable interpretation of the Encyclical and the Syllabus 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 bend under the will of princes. This is jumping to the last end of the contrary, and attributing 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 liberties, the holy freedom of souls. And that’s what they claim the Pope said!
Félix Dupanloup, Bishop of Orleans, [The September 15th Convention, and the December 8^th [1864] Encyclical], 1865[1].

Is the universe of the Encyclical binary or multidimensional? Let’s consider a position X.

— If it comes in a binary opposition, “allowed vs. forbidden”, then the proposals “it is permitted (to refuse obedience)” / “it is forbidden (to refuse obedience)” are contradictory contraries: 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”, otherwise said: “we must always bow our heads under the will of the princes.”

Thus, for Dupanloup, the malevolent “modernists” substitute contradictories for contraries, what he describes as “jumping to the last end of the contrary”, which is a proper designation of the contradictories.

He accuses his opponents of reframing the Pope’s position, using a strategy of absurdification (an exaggeration up to the absurd, S Exaggeration.

— If the position X enters a three dimensional universe, as “prescribed / permitted (indifferent) / forbidden” then the proposals “it is allowed / 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 inference “If X is not fought, X is required” is not valid. If we condemn “it is permitted to refuse obedience to legitimate princes” then we can only conclude one or the other of these opposites:

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

As 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.”

— If two or more additional options, “encouraged” and “discouraged” are introduced, we get a five dimensional universe “prescribed / advised / permitted (indifferent) / recommended / forbidden”. The interpretation “encouraged” is hardly possible, for reasons previously seen; “discouraged” could correspond to the intention of the Syllabus, such as interpreted by Dupanloup. One then wonders why this sentence seems so solemn : if we admit that something which is not recommended is something that we do not do without good reason, it is obvious that one does not disobey the legitimate prince without some good reason.

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[1] Quoted after 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.

1. In dialogue, a contradiction emerges when a first speech turn is not ratified by the partner’s following turn. The contradiction is open when the two parties produce anti-oriented speech turns. When the opposition is thematized and ratified by both participants, it gives rise to an argumentative situation.

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

Contradictions can be solved on the spot by a series of adjustments and arrangements, by playing on the margins of indeterminacy and windows of opportunity left by ordinary language and actions.

2. Non-Contradiction^@principle; Ad hominem; Consistency.

3. Contradiction as a relation between opposite terms, S. Opposites

4. Contradiction as a relation between propositions: S. Contrary and contradictory; Absurd.

The fundamental expression of argumentative coherence or consistency is non-contradiction.

S. Non-contradiction; Absurd; Ad hominem.

The consistency requirement is of special importance in systems of regulations of human behavior, religion, law, as well as ordinary institutional or familial rulings.

The consistency requirement is expressed a contrario in the refutation strategy mentioned in Aristotle’s Rhetoric, topic n° 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

The topos ≠5, “On consideration of time” appeals to consistency. This topic is not explicitly stated, but presented through 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) accomplishes a feat (maybe an impulsive heroic act)
   2. After this, he asks for a reward.
   3. Argument: if he had asked before, they would have agreed on a reward.

The hero considers 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, you’ll receive…
L2:   — I have done and done well, so give me…

This topic explains the disappointment of one who reports the found wallet and receives no reward.

2. Human (in)consistency

Consistency may be the rule, but inconsistency is a fact of life. This is what topic n° 18 expresses:

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 topic materializes in the following enthymeme:

When we were exiles, we fought in order to return; now 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, exiles fought to return home, and they returned; in the current situation, they are suspected of refusing to fight, and preferring exile. They deny the charge by this enthymeme, which is a claim of consistency, as in:

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

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 to not be chased out of it!”
Those accusing them reply that “Men do not always make the same choice, etc.”

The opposing party argues from an opposing vision 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 system of laws
and stability of the objects of the law

Lat. arg. a cohærentia, de cohærentia, “formation into 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 argumentative line can therefore be rejected if it leads to the view that two laws are contradictory; this is a form of argumentation from the absurd.

In practice, this principle excludes the possibility of the same case being settled in two different ways by the courts.

By applying this principle, if two laws contradict one another, they are said to do so only in appearance, and, as a consequence, they must be interpreted so 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 solve conflicts of standards. To prevent this kind of conflict, the legal system provides for adages, which are interpretative meta-principles, such as “the most recent law takes precedence over the oldest”. These adages are interpretative meta-principles, coming from Roman law and sometimes expressed in Latin: “lex posterior derogat legi priori”.

3.2 Principle of 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 substantial form of consistency. It requires that a law be understood in the context of other laws having the same goal or relating to the same subjects, that is to say the same beings (persons, things, acts) or the same topic.

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. S. Classification; A pari.
This principle of consistency prompts 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 might be questioned. Anti-terrorism laws, for example, are a package of different statutory provisions, for which it is necessary to ensure that the definition of “terrorism” remains the same in each of the passages that uses the term. If this is not the case, these laws need to be made consistent, which implies that they themselves must be underpinned by 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 substantial topoi, “human conduct is, or should be consistent” and “human conduct is inconsistent”.

4. Argument from narrative inconsistency

As a particular case of ad hominem argumentation, showing inconsistencies in the accusatory narrative can rebut a charge:

S1:    — you are the heir, you benefit from the crime, you killed to inherit!
S2:    — if so, I should have murdered the other legatees 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 put forward by the prosecution to show that the actions of the suspect do not fit in the proposed scenario; the accusatory narrative contains flaws or contradictions.

 

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

You say I’m the murderer. But it has been proven that just before the crime, I spent an hour at the cafe in front of the victim’s home, everyone saw me. It is not coherent conduct on the part of a murderer to show himself at the scene of his crime.

Any weakness identified in the prosecution scenario can then be used to clear 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 mingled with argumentation are vulnerable to this type 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 n° 22, 25 and 27 and probably 18 (cf. supra) of the Rhetoric are relevant to this discussion (Aristotle, Rhet., II, 23), S. Collections (2).

The word consequence can mean:

— Effect, referring to a causal, cause / effect relationship S. Causality.
— Consequent, referring to a logical, antecedent / consequent relationship, S. Connectives, §Implication

1. Effect-to-cause argumentation

The effect to cause argument goes back from the consequences to the cause. Given data is considered the effect of a hypothetical cause that can be reconstructed on the basis of this data combined with a known causal relationship between these type of facts and their cause. Other expressions can also be used, such as argument by the effect, or from the effect to the cause.

You have a temperature, so 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 categorization process.
— Causal Law: There is a causal law linking I facts “having an infection” to T facts, “having a temperature”
— Conclusion: t has a type T cause, an infection, and the patient should be treated accordingly.

This corresponds to the diagnostic process; one could speak of diagnostic reasoning.

The effect (the temperature) is the natural sign of the cause. Such natural, palpable, effects provide endless basis for argument by natural signs:

See! The cinders are still hot, there was a recent fire (… they cannot be very far)

In the area of ​​socio-political decision, the argument by the consequences corresponds to the pragmatic argument, transferring upon the measure itself the positive or negative evaluation of the effects of a proposed measure.
The pathetic argument scheme is a special kind of pragmatic argument.

The argument from the consequences is sometimes referred to in Latin as argumentation quia “because” in opposition to the arguments by the cause or propter quid “because of which”.
S. A priori, a posteriori.

2. Arguments by the identity of the consequences

The same kind of argumentation applies to deductions made from the implied meaning of words, as an appeal to the sense of semantic coherence or logical consecution:

Scheme: “Another topic consists in 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. p. 313-315).

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

3. Refutation by contradictory consequences

The refutation by contradictory consequences is a kind of ad hominem*, used in dialectic:

Peter says “S is P”.
S has the consequence Q: the fact is known and accepted by the opponent.
P and Q are incompatible
So Peter says incompatible things about S.

Example:

Pierre says that power is good.
Yet, everyone agrees that power corrupts (consequence)
Corruption is an evil.
The good is incompatible with the evil; to be good, power should exclude corruption.
Peter says contradictory things.

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1. Consensus as agreement

S. 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 everybody thinks or does this, and that other proposals should be rejected. It implies that by flouting the existing consensus, the proponent of a new measure, that is the opponent to consensus, is on the verge of being excluded from this community, S. Burden of proof. These arguments have the general form:

We always thought, desired, did … like that; so buy (please, do…) like that.
Everybody loves the product So-and-So.
Everybody puts Such and Such ketchup on their burger!

The universal consensus argument claims that “all men in all times have thought so and things have always been done that way”. The existence of God has been argued upon the universal consensus argument.

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

The majority / many people … think, desire, do … X.
Three million Syldavians have already adopted it!
My book sold better than yours.
He is a well-known actor.

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

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

Populist argument is based on a kind of consensus among the people (or attributed to it), S. Ad Populum.

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

A connective word is a function word that combines several propositions, simple or complex, into a new, integrated, (more) complex proposition.

1. Connectives in propositional calculus

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 of construction of 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 opposition with 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 used 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 specific logical connective is defined by the kind of combination it accepts between the truth-values of the component proposition.

1.1   The truth tables 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 three-column, five-line table.

— 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 falsehood are said of propositions, “P is True”, “P is False”; so the corresponding abbreviating letters use a different typographic character.

P	Q	P connec Q
T	T	(depends on the connective)
T	F	(depends on the connective)
F	T	(depends on the connective)
F	F	(depends on the connective)

— 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 take into account, P, Q and “P connec Q”.
The four following lines express the truth-values of these propositions. As each proposition can be T or F, four combinations must be considered, each corresponding to one line.

1.1.1 Conjunction “&”

By definition, the conjunction “P & Q”

— is true when P and Q are simultaneously true: line 2
— is false when one of the two is false: line 3 and 4; both are false: line 5.

This is expressed in the following truth table:

P	Q	P & Q
T	T	T
T	F	F
F	T	F
F	F	F

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 ”

1.1.2 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:

P	Q	P ↔ Q
T	T	T
T	F	F
F	T	F
F	F	T

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

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

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

Truth table of the inclusive disjunction:

P	Q	P V Q
T	T	T
T	F	T
F	T	T
F	F	F

 

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

Truth table of the exclusive disjunction:

P	Q	P W Q
T	T	F
T	F	T
F	T	T
F	F	F

1.1.4 Implication “→”

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

Truth table of logical implication:

P	Q	P → Q
T	T	T
T	F	F
F	T	T
F	F	T

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, anything can follow from a false assertion, a truth as well as a falsehood.

The equivalence, conjunction, inclusive disjunction and exclusive disjunction connectives are symmetrical, 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 condition:

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

Causal relation may 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, necessarily, the road is wet when it rains.
The implication thus defined is called material implication; it has nothing to do with the substantial logic of Toulmin.

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 erase 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 taken 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.

1.2 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.

1.2.1 “Laws of thought”

Binary connectors combine in equivalences known as De Morgan’s laws, considered to be laws of thought. For example, the connectives “&” and “V” enter in 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 upon inclusive disjunction.

1.2.2 Hypothetical (or conditional) syllogism

S. Deduction

1.1.3 Conjunctive syllogism

The following statement expresses a logical law:

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

[¬(P & Q) & P] → ¬Q

The corresponding three-steps deduction is known as 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:

Nobody 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 suspect; that his interest is to hide that he was actually in Bordeaux, and that the witness is more reliable than the suspect, we may conclude that Peter lied when he pretended to be in London yesterday at 6:30pm.

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 speech, where their self-evidence ensures that they go unnoticed. It would be a mistake not to take them into account on the pretext that, since these arguments are valid, they are not arguments.

2. Connectives in logic and in language

Introductory logic courses make a consistent use of ordinary language to illustrate both the capacities and specificities of logical languages. Generally speaking, logic can be “applied to the usual language” (Kleene 1967: p. 67-73) as an instrument for expressing, analyzing and evaluating ordinary arguments as valid or invalid reasoning. These translation exercises run 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 for example, this reasoning is then tested for validity, and declared valid.

In order to identify similarities and differences, natural language components and properties can be compared with their counterpart in a logical language. This enables us to better understand both kinds of languages. Such exercises are helpful when it comes to gaining a better understanding of logical or linguistic systems, and may also be of benefit when it comes to argumentation education. Nonetheless, there are some additional facts which should be taken into one consideration when using this methodology.

(i) The preceding exercise did not focus on the correct combination of the truth-values of semantically independent propositions such as in the logical talk about the moon and Napoleon (cf. supra §1.4). The exercise introduces a strong condition on semantic coherence between the linked propositions, which belong to the same domain of action, in this case, TV installation.

(ii) Natural language connectives do not connect propositions in the way 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 a whole narration, with various twists and turns:

Thus, the prince married the princess — The End —

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

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

Nonetheless, in:

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

yet introduces a more complex scenario, and the preceding example is not a complete discourse. Yet announces that more indications are to come specifying the weak points of the assessed task.

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

(iv) Logical reasoning does not cover all ordinary reasoning:

I have eaten three apples and two oranges, so I have had my five fruits diet today

First, this apparently 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 upon an addition that is easier to solve in arithmetic than in a logical language. Toulmin’s layout would meet this condition by adding a warrant-backing system referring 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 are 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, classical logical theory does not have adequate concepts to deal with phenomena of argumentative orientation, and imposes no obligation in this respect. Quine’s argumentative strategy consists in minimizing the problem and delegating it to rhetoric, seen as a refuse site for problems left unsolved by logical analysis.

And carries with it subtle semantic conditions, for example, a sensibility to temporal succession. 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 given to this word:

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

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

3. No subordination, but bilateral relations

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

Logic is an autonomous mathematical language, that can be constructed from the suggestions of some chosen segments of ordinary language. From the very beginning, the teaching of logic may draw more or less heavily on the resources of ordinary language. The same applies to the teaching of everyday argument in relation to the resources provided by 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 for the evaluation of educational methods.