Part II, Lesson FourPart II, Lesson FourThe Opposition of PropositionsThe Opposition of PropositionsThe Rules of Truth and FalsityThe Rules of Truth and Falsity
IntroductionIntroductionOpposition between propositions occurs when werelate two propositions to each other.We have already seen how to distinguish the partsof a proposition (subject, predicate, and copula),as well as the use of words within theproposition (supposition and distribution).Now we will consider the ways of relating oneproposition to another.
Opposition of PropositionsOpposition of PropositionsIn general, opposition between two propositionsoccurs when one affirms and the other deniesthe same predicate of the same subject.Example:All dogs are cats.No dogs are cats.These two propositions are said to be opposedbecause one affirms and the other denies “cats”of “dogs”.
If a different predicate is used (or a differentsubject), then there is no opposition between thetwo propositions; they are merely different.Example: All dogs are carnivorous.No dogs are rational.Because we are not affirming and denying thesame subject of the same predicate, thesepropositions are not opposed to each other inany way.
Furthermore, in order to have opposition betweentwo propositions, not only must the samesubject and same predicate be used in each, butalso they must have the same meaning and thesame supposition. Nor is it permissible to useequivocal or analogous words.
Kinds of OppositionKinds of OppositionThere are different ways of affirming and denyingthe same predicate of the same subject, whichgives rise to different kinds of oppositionbetween propositions.The distinction of the kinds of opposition has todo with both the quality (affirmative ornegative) and the quantity (universal, particular,indefinite, or singular) of the two propositions.
1. Contradictory Opposition1. Contradictory OppositionWhen there is contradictory opposition betweentwo propositions, one denies absolutelyeverything that the other affirms. They are asopposed as can be.Example: All men are honest.Some men are not honest.
At first glance, we might be tempted to think thatthe contradictory of “All men are honest” is“No men are honest”, as it seems that they aremore opposed than the two mentionedpreviously. Yet this is not so; in order to refutethe truth of the proposition “All men arehonest”, it would be enough to show that somemen are not honest, or even that one man is nothonest. One exception would disprove the truthof the universal affirmative proposition.
Thus, in order to contradict the proposition,“All apples are red”, all we need to show is that“Some apples are not red”, or even “This appleis not red”.The contradictory proposition of a universalaffirmative proposition is a particular negativeproposition (using the same subject and thesame predicate, of course.)
The same applies in the case of a universalnegative proposition, whose contradictory willbe a particular (or singular) affirmativeproposition that uses the same subject and thesame predicate.Example: No exam is difficult.will be contradicted bySome exams are difficult.or even byThis exam is difficult.
Contradictory opposition is opposition in truthand falsity.This means that whenever we know that one ofthe two propositions with this kind ofopposition is true, the other must necessarily befalse. It is impossible that both be true or thatboth be false.
All men are honest.Some men are not honest.If the first proposition is false, the second mustnecessarily be true.
No exam is difficult.Some exams are difficult.This exam is difficult.
2. Contrary Opposition2. Contrary OppositionContrary opposition exists between twopropositions when both have universal quantitybut one affirms and the other denies itspredicate of the subject.Example: All men are honest.No men are honest.
At first glance, it might appear that this is a moreradical type of opposition than contradictoryopposition because “all” and “none” areextremes.However, contrary opposition is in fact not asgreat as contradictory opposition because thecontraries are opposed only in truth.That is, it is impossible for both propositions to betrue, but both may be false.
To say that contrary opposition betweenpropositions is an opposition only in truth is tosay that when one of the contrary propositionsis true, its contrary must necessarily be false.But if we only know that one of the two contrarypropositions is false, we cannot by that fact aloneknow that its contrary is true; it could be true or itmight also be false.
Sometimes when one contrary is false, the other istrue.Example:No man is rational. (False)All men are rational. (True)
But other times, the contrary of a false propositionis also false.Example:All men are honest.No man is honest.(False)(False)
In other words, when one of the contrarypropositions is false, the other may be true or itmay be false. In this case its truth is unknown.
Contrary propositions do not have as absolute anopposition as is found between contradictorypropositions.Contradictory propositions are opposed in truthand in falsity, but contrary propositions are onlyopposed in truth.Also, contrary propositions are both universal.With contradictory propositions, one is universaland the other is particular or singular. Thus,contradictory propositions differ in quality andquantity, whereas contrary propositions onlydiffer in quality.
3. Sub-contrary Opposition3. Sub-contrary OppositionTwo propositions are in sub-contrary oppositionwhen they differ in quality but are bothparticular.Example: Some dogs are black.Some dogs are not black.
Propositions in sub-contrary opposition areopposed in falsity only. That is, if one is false,the other is necessarily true.However, it may be that both are true, as in theexample just given.Example: Some dog is black.Some dog is not black.
Example:Some dogs are cats. (False)Some dogs are not cats. (True)
Summary of OppositionSummary of Opposition1. Contradictory Opposition:One proposition denies the other absolutely.Opposition in truth and falsity.The propositions differ in both quality andquantity.To have opposition between two propositions,they must use the same subject and the samepredicate.
2. Contrary OppositionOpposition in truth only.Both propositions are universal (one isaffirmative, the other negative.)
3. Sub-contrary OppositionOpposition in falsity only.Both propositions are particular (one isaffirmative, the other is negative).
The Relation of Sub-alternationThe Relation of Sub-alternationThere is another possible relation between twopropositions that use the same subject and thesame predicate, but this is not a relation ofopposition.This relation, called sub-alternation, occurs whenthe propositions differ in quantity but not inquality (which is why there is no oppositionbetween them.)
Example: All men are brave.Some men are brave.
When the universal proposition is true, itssubalternate must also be true.If all we know is that the particular is true, this tellsus nothing about the truth of the universal.But if the particular is false, the universal must alsobe false.
Since subalternation is not a kind of opposition,there is no opposition in truth or falsity.Yet we can conclude from the truth of theuniversal to the truth of the particular, or fromthe falsity of the particular to the falsity of theuniversal.
Example:All strawberries are sweet.Some strawberries are sweet.If the universal is true, the particular is necessarilytrue as well.
And if it is false thatSome children are not human beings.then it must necessarily be false thatNo children are human beings.
The Square of OppositionThe Square of OppositionThe Square of Opposition is a very useful visualaid to understanding the consequences of thevarious relations of opposition and sub-alternation of propositions using the samesubject and the same predicate.
It uses vowels to represent the main types ofpropositions:A stands for the universal affirmative.E stands for the universal negative.I stands for the particular affirmative.O stands for the particular negative.
The Square of OppositionThe Square of OppositionAll men are honest.All men are honest. No men are honest.No men are honest.AA EEII OOSome men are honest.Some men are honest. Some men are not honest.Some men are not honest.
The lines of the Square represent the three typesof opposition and the relation of subalternation.AO and EI (the diagonals) represent thepropositions in contradictory opposition.AE represents the propositions in contraryopposition.IO represents the propositions in sub-contraryopposition.AI and EO represent the relation of sub-alternation.
The Rules of Truth and Falsity inThe Rules of Truth and Falsity inthe Square of Oppositionthe Square of OppositionTo use the Square of Opposition, the propositionsmust use the same subject and the samepredicate with the same meaning, the samesupposition (personal or simple) and mustrespect the difference between true universalnames and collective names.
1. Two propositions in contradictoryopposition cannot simultaneously be true,nor simultaneously false.If one is true, the other will be false, and if oneis false, the other will be true.
2. Two propositions in contrary oppositioncannot be simultaneously true.When one is true, the other will be false, but ifone is false, the other will be unknown.
3. Two propositions in sub-contraryopposition cannot be simultaneously false.If one is false, the other must be true, but ifone is true, the other is unknown.
4. In the relation of subalternation, when theuniversal is true, the particular must alsobe true, and when the particular is false,the universal must also be false.If the particular is known to be true, this tellsus nothing about the truth of the universal (itstruth is unknown.)Similarly, when the universal is known to befalse, the particular is unknown.This is sometimes summarized by saying that wecan descend with truth and rise withfalsehood.
In this entire discussion, we have been examiningwhat can be concluded from the formalrelationship between propositions. To have astarting point (to know that a proposition is trueor false), we need knowledge from some scienceoutside of Logic. Logic can help us arrive atvaluable consequences from the formalrelationship between propositions once we havethat starting point.We can be quite sure that these consequencesfollow from the mere fact that propositions arerelated in this way, no matter what the subjectmatter being discussed.
We are always forced to distinguish between thematter and form of the propositions we use,between the subject matter and the form we useto express our knowledge of it. ElementaryLogic is necessarily a consideration of the formof our expressions. Knowledge of the subjectmatter comes from other branches ofknowledge.
Whenever we use words we are necessarilyconsidering the “subject matter” of theproposition, that is, what the proposition meansas well as what form it is expressed in.In order to avoid being distracted unnecessarily bythe content of propositions, we could merely useletters in place of actual subjects and predicates,to bring out more clearly the formal aspects ofthe propositions.For example, we could avoid considering thespecific subject matter by using expressions suchas “All S is P” or “Some S is not P.”
All S is P.All S is P. No S is P.No S is P.AA EEII OOSome S is P.Some S is P. Some S is not P.Some S is not P.