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)
| P | Q | P contradictory with Q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
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.
| P | Q | P contrary with Q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
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, … Pi, … Pn.
— Every member of this universe has one of these properties, Pj; that is, is either a P1 , … or a Pi, … or a Pn.
— No one has two or more properties P1 , … Pi, … Pn, that is, no one is both (Pk & Pl).
— P1 , … Pi, … Pn 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.
| Two-dimensional 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 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.