“Topic from the opposite » (Freese & Rhys Roberts); « from the contrary » (Ryan).
As the topos plays on two pairs of opposites, the plural « topos of opposites » can be used.
1. Topos of the opposites
Cicero recognizes the enthymeme based on opposites as the archetypal enthymeme, S. Enthymeme. The topos of the opposite is the first on Aristotle’s list of rhetorical topoi:
One line of positive proof is based upon consideration of the opposite of the thing in question. Observe whether that opposite has the opposite quality. If it has not, you refute the original position. If it has, you establish it. (Rhet., II, 23; RR, p. 355)
If courage is a virtue, cowardice is a vice
Ryan reformulates the Aristotelian topic as:
“1A — If A is the contrary of B, and C the contrary of D,
then if C is not predicated of A, then D is not predicated of B
1B — If A is the contrary of B, and C the contrary of D,
then if C is predicated of A, then D is predicated of B” (1984, p. 97)
We follow Freese and Rhys Roberts and use the label “topic from the opposite”. Ryan uses the equivalent word “contrary” in his discussion of the topic.
The clause “— is not predicated” can be read as “is not true, acceptable, possible…”. Applied to the logical implication, “P implies Q”, the topic validates the conclusion “not-P implies not-Q”; this conclusion is not quasi-logical, but plainly false, as a case of a negation of the antecedent (modus tollens), S. Deduction.
The problem here is that logical negation applies to a predicate saying something about its subject, but not to a name. A bottle and a gloomy thought equally qualify as non-cows. As the argument from the opposite is formulated in ordinary language in a given situation, the application of negation to any word is open-ended and debatable. But whoever wants to discuss that point becomes vulnerable to the accusation of “trying to pick up a senseless quarrel over semantics”.
2 A dialectical resource
The topos of opposites is a dialectical resource, used to test claims such as A is B, “courage is a virtue”. If the proponent holds that “A is B”, then the opponent can examine what is going on with the opposites of A and B. In a dialogue format:
The test procedure using the method of opposites develops as follows
Question: Is courage a virtue?
Topos of the opposite:
Opposite of courage: cowardice
Opposite of “— being a virtue”: “— being a vice”
Let’s predicate the contrary of virtue upon the contrary of courage:
“cowardice is a vice”
This proposition seems undisputable. So, let’s conclude that courage is indeed a virtue.
Argumentation: “Courage is (indeed) a virtue, since cowardice is certainly a vice”.
Let’s submit the claim “pleasant things are (intrinsically) good” to the same test.
Question: “are pleasant things good?”
Topos of the opposite:
Opposite of pleasant: unpleasant
Opposite of “— being good”: “— being bad”
Derived claim: “Unpleasant things are bad”
New question: “Are unpleasant things always bad?”
The answer is no, because cod liver oil is quite unpleasant to drink (in its natural state) and nonetheless good for the health.
Conclusion: (not all) pleasant things are not (intrinsically) good.
“pleasant things are not intrinsically good, since unpleasant things can also be good”
The topos of opposites can also be used to suggest practical actions:
Inhaling black coal dust made the miners sick, they will recover their health if they drink white milk
If the cold rain has given him a cold, a hot tea will do him good.
2.3 Generic linguistic form and logical form of the topos of opposites
The topos of opposites is expressed by Aristotle in a language that is both ordinary in its construction and technical in its use of a specialized vocabulary: rhetorical terms like topos or enthymeme, or grammatical ontological terms like subject or predicate. These terms are indeterminate, that is, taken in their broadest intension: « a subject (a being), a property (a predicate) ». This corresponds to a generic formulation of topos.
Since the topos expresses a structure common to a set of enthymemes, it can be defined as their common logical form. The logical form of the topos of opposites is very simple which is not the case for all topoi.
According to Ryan’s formulation (1984, p. 97, cf. supra), it is written:
1A – If A is the opposite of B, and C the opposite of D,
Then, if C is not predicated of A, then D is not predicated of B.
1B – If A is the opposite of B, and C the opposite of D,
then if C is predicated of A, then D is predicated of B.
The clause “— is not predicated” can be read as “is not true, acceptable, possible …”.
According to the formulation of Walton & al. (2008, p 107) the argument « from opposites » argument has two forms:
The opposite of the subject S has the property P
Therefore, S has the property non-P (the opposite of the property P)
The opposite of the subject S has the property non-P
Therefore, S has the property P (the opposite of the property non-P)
In practice, the « logical form » is obtained by replacing the indefinite terms (the variables), by letters. The initial proposition is noted in the standard form of the analyzed propositions « A is C » (Ryan), or « S is P » (Walton). This writing abbreviation, very useful because it avoids the tortuous formulations sometimes necessary to correctly express the coreference.
A « logical form » in the strong sense would be a form capable of being used in a logical calculation. But the only operation here is the actualization of the generic form (topos) into a specified form (enthymeme).
1.3 Is the topos of opposites intrinsically fallacious?
1.3.1The topos of opposites is logically invalid
Applied to the logical implication, « A implies B », the topos validates the conclusion « not-P implies not-Q ». A sufficient condition is taken as necessary and sufficient. The conclusion is not « quasi-logical », but simply false, as a case of a negation of the antecedent (modus tollens), S. Deduction.
Logical negation applies to a predicate insofar as it asserts or not something about a subject, but does not apply to a noun. A bottle and a gloomy thought equally qualify as non-cows.
Moreover, since the topos occurs in natural speech and situation, the true meaning of negation applied to one of the components of a sentence is open-ended and debatable. But whoever who would venture to discuss that point would be dismissed as “trying to pick up a meaningless quarrel over semantics”.
1.3.2 The topos of opposites is conditionally valid
Let us consider a non-transparent box; we know that1) It contains two kinds of objects, cubes and balls; 2) These objects are red or green (or exclusive); 3) Objects of the same shape are of the same color.
The observer draws an object from the box, for example a ball, and sees that it is green. In this case, a ball is a non-cube; and the non-green is red. We can safely conclude that the balls are green; and that the non-balls (cubes) are non-green (i.e., blue).
1.4 Topos of the opposites in literature
Corresponding to a semantic rule, the topos of the opposites can be found in literary passages, where it serves poetic oratory amplification without losing its argumentative value of confirmation.
Satan leads the war against the angels, and has just undergone a cruel defeat. He calls “His potentates to council”, and explains to their assembly how a new weapon of his invention — powder and gun — will permit them to take their revenge.
He ended, and his words their drooping cheer
Enlighten’d, and their languish’d hope reviv’d
Th’invention all admir’d, and each how he
To be th’inventor mifs’d; so easy’ it feemed
Once found, which yet unfound moft would have thought
Milton, Paradise Lost, , Book VI, 498-501; (My italics) .
The same conclusion applies to Columbus’ egg: « what seemed impossible before seems easy after« .
2. How the topos applies
In the preceding cases, the topos is quite easy to apply, because it operates on an elementary linguistic structure “A is B” (as in previous cases), easily transformed into “Not-A is not-B”.
The topos is also easy to identify, when the final formulation of the argumentation “A is B, then non-A is non-B”, leaves the topical relation transparent.
In other cases, the topos is more deeply embedded in the discourse, and its perception and reconstruction are more complex. In all cases, simple or complex, an argument is needed to show that such and such a passage corresponds to such and such a type of argument, S. Waste; Type of argument. For example, how can we decide if the following passage is structured by the topos of opposites (corresponds to an occurrence of the topos of the opposites)?
It took billions of years and ideal conditions before humans appeared on the planet, maybe one global warming will be enough to make it disappear
This is clearly an inferential structure, progressing from a categorical assertion about the past to a restricted assertion about the future:
E1, maybe E2
The corresponding Toulminian structure is « Data, SO, Modal, Claim« . The two connected statements have the same structure, and express consecutions. This parallelism augurs well for an occurrence of the topos of the opposites.
The structure to be considered for the operation is not the simple grammatical structure « S is P », but the consecutive structure:
« Conditions, Result », « C resulted in R », « C (resultative) R »
Do these C and R contain opposite predications on opposite subjects?
“It took billions of years and ideal conditions before humans appeared on the planet”
it took B before A = B has been necessary for A
billions of years and ideal conditions result humans appeared on the planet
[condition C1] billions of years and ideal conditions [result R1] humans appeared on the planet
“may be one global warming will be enough to make it disappear”
may be W will be enough for D
one global warming result [makes] it disappear
[condition C2] one global warming [result R2] [makes] it disappear.
The contraries are to be looked for not on a simple predicative structure, but on the two parallel structures “C [results in] R”. The results R1 and R2 are clearly opposites:
humans appeared on the planet
to make [humanity] disappear
Are their respective conditions in the same relationship? Condition C2 “one global warming” cannot be self-evidently opposed to condition C1 “it took billions of years and ideal conditions”. Nonetheless, their argumentative orientations are clearly opposed.
(i) C1, it took billions of years and ideal conditions before …
— billions of years is oriented towards conclusions like “that’s a long time”
— ideal conditions is oriented towards conclusions like “it’s rare, difficult to obtain”
— The construction “it takes X to Y” is oriented towards “it’s a lot”.
These three orientations converge to give rise to the global inference “this is a very complex process”.
(ii) Conversely, C2 is oriented towards a class of conclusions of the type: “this is a very simple process”:
— the determiner “one” is oriented towards unicity, “just one”, and simplicity;
— will be enough is oriented towards a limitation “no more than”, maybe “less than expected”, for such and such accomplishment.
If this reconstruction is acceptable, then the following argumentative structure is attributed to the discourse:
It has been really complicated to produce R
so, maybe, it will be very easy for R to disappear.
Such examples also suggest that the classical Aristotelian formulation of the topos may be oversimplified.
3. Trivial and non-trivial conclusions delivered by the topos
The application of the topos of opposites is a semantic reflex. Reasoning from opposites is a basic way of thought, in much the same way as causal reasoning, or reasoning by analogy or by definition. Reasoning from opposites may seem to deliver commonplace conclusions, empty because analytical reformulations of the original sentence when both terms are equally obvious.
Nonetheless, even in this case the topos does help to clarify the meaning of the words, which is no less necessary in philosophy than in general disputes:
Temperance is beneficial; for licentiousness is hurtful. (Aristotle, Rhet., II, 23; RR, p. 355)
There are, however, cases where the “opposite reflex” may, or must, be inhibited: If a prayer says “Peace to the people who love you”, should we apply the topos and conclude something like “War on those who don’t”?
Let us consider the following argumentations based on the opposites:
If war is the cause of our present troubles, peace is what we need to put things right again. (Ibid.)
Those who sank the country into the crisis are perhaps not the best suited to get us out of the mess.
We cannot trust the same failed market mechanisms to successfully steer the country out of this crisis (after Linguee, 25-10-2015)
These conclusions are met with the argument that « we have failed for lack of determination and radicalism »:
If we are in trouble, it is because we just have waged a limited war; this limited war is the cause of our present troubles, an all-out war is what we need to put things right again; only an outright victory will bring us peace.
Our policy did not fail, you prevented us from actually implementing it
The conclusion of the following example is not trivial:
For even not evil-doers should / Anger us if they meant not what they did / Then can we owe no gratitude to such / As were constrained to do the good they did us. (Aristotle, Rhet., II, 23; RR, p. 355)
The following one is also quite suggestive:
Since in this world liars may win belief, / Be sure of the opposite likewise — that this world / Hears many a true word and believes it not (id., p. 357).
The a contrario reflex is a typical example of how argumentation leads us to contemplate things from a different perspective, under a different wordings; or, as Grize would say, in a different light, S. Schematization.
4. A transcultural topos
The application of the topos from opposites is a semantic reflex that combines well with analogical reasoning. Like the topos a fortiori, the topos from opposites has cross-cultural validity. The following two examples come from the Chinese tradition.
Wang Chung, Four Things to be Avoided. 
There are four things which, according to public opinion, must be avoided by all means. The first is to build an annex to a building on the west side, for such an annex is held to be inauspicious, and being so, is followed by a case of death. Owing to this apprehension, nobody in the world would dare to build facing the west. This prohibition dates from days of yore. […]
On all the four sides of a house there is earth; how is it that three sides are not looked upon as of ill omen, and only an annex in the west is said to be unpropitious? How could such an annex be injurious to the body of earth. or hurtful to the spirit of the house? In case an annex in the west be unpropitious, would a demolition there be a good augury? Or, if an annex in the west be inauspicious, would it be a lucky omen in the east? For if there be something inauspicious, there must also be something auspicious, as bad luck has good luck as its correlate. […]
Han Fei Tzu. “Precautions within the palace”. 
Moreover, whether one is ruler of a state of ten thousand chariots or of a thousand only, it is quite likely that his consort, his concubines, or the son he has designated as heir to his throne will wish for his early death. How do I know this is so? A wife is not bound to her husband by any ties of blood. If he loves her, she remains close to him; if not she becomes estranged. The saying goes, “if the mother is favored, the son will be embraced”. But if this is so, then the opposite must be, “if the mother is despised, the son will be cast away.”
5. A contrario
Lat. Contrarius, “contrary”. Two constructions might be used to refer to the argument, with the same meaning:
— the Latin proposition a: argument a contrario sensu, “by opposite meaning”
— or, less commonly, the Latin preposition ex: “complecti ex contrario” “conclude on the basis of the opposite meaning” (Cicero, quoted in Dicolat, art. Complector).
S. Latin labels
The label “argument a contrario” can be used with the meaning of “inversion”, to refer to the various kinds of argumentations which draw on contradiction, S. Contradiction.
Argumentation from the opposite corresponds to one kind of argumentation a contrario. In law, an a contrario argument is defined as:
A discursive process according to which a legal proposition being given which asserts an obligation (or other normative qualification) of a subject (or a class of subjects), for want of any other express provision, we must exclude the validity of a different legal proposition, which asserts this same obligation (or other normative qualification) with respect to any other subject (or class of subjects)” (Tarello 1972, p. 104). Thus, if a provision obliges all young men, who have attained the age of 20, to perform their military service, it will be concluded, a contrario, that young girls are not subject to the same obligation. (Perelman 1979, p. 55)
If a rule explicitly concerns a category of things, then it does not apply to the things that are not part of this category. The rule applies only in the defined area, to all the specified things, and only to them. This is an application of Grice’s Rule of Quantity, stipulating that the speaker must provide just the necessary amount of information, no more and no less.
This rule assumes that the system of law is well made and stable. In a period of social change and revision of the law, the argumentation a pari will be opposed to argumentation on the opposites. Women engaged in a battle for gender equality will refuse to oppose their status to men’s status, and will demand that laws be applied a pari, be it beneficial (right to vote) or quite possibly less attractive (military service).
There is no paradox in the fact that a pari / a contrario argumentation can apply to the same material situation. Political issues are not unanimous, and cannot be solved by an automatic application of an algorithm; their discussion brings in historical considerations, values and affects.
 Edinburgh: Donaldson.
 Wang Chung, Four Things to be Avoided. In Lun-hêng, “Balanced discussions”, Book XXIII, Chap. III, 68. Translation and notes by Alfred Forke., Leipzig, 1906. Reprint by Paragon Book Gallery, New York, 1962. (p. 793-794)
Quoted after http://classiques.uqac.ca/classiques/wang_chung/lunheng/wangchung_lunheng.pdf
 Han Fei Tzu. Basic Writings. Section 17, “Precautions within the palace”. Translated by Burton Watson. New York, London, Columbia University Press, 1964. P. 84-85.