# 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 , … Pi, … Pn.

— Any members of this universe possess one of these properties, Pj; that is, is either a P1 , … or a Pi, … or a Pn.
— None possess two or more properties P1 , … Pi, … Pn, that is, none is both (Pk & Pl).
— P1 , … Pi, … Pn 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 8th [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.

[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.