Oppositional Inference (Logic Slide 7)
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OPPOSITIONAL INFERENCE Opposite or Opposed Propositions Are propositions that cannot be simultaneously true or that cannot be simultaneously false, or that cannot be either simultaneously true or simultaneously false. This impossibility of being simultaneously true, or false, or either true or false is the essential note of logical opposition. Propositions are opposed if they have the same subject and predicate but differ from one another in quality or quantity, or both in quality and quantity.
Transcript of Oppositional Inference (Logic Slide 7)
OPPOSITIONAL INFERENCE
Opposite or Opposed Propositions
Are propositions that cannot be simultaneously true or that cannot be simultaneously false, or that cannot be either simultaneously true or simultaneously false. This impossibility of being simultaneously true, or false, or either true or false is the essential note of logical opposition.
Propositions are opposed if they have the same subject and predicate but differ from one another in quality or quantity, or both in quality and quantity.
Every man is seated.
A E
No man is seated.
Some man is seated.
ISome man is not seated.
O
CONTRARIE
S
subcontraries
(Superior)SubalternantSubalternate(Subaltern)(Inferi
or)
(Superior)SubalternantSubalternate(Subaltern)(Inferi
or)
Contradictory OppositionTwo propositions are contradictories if one is the denial or negation of the other, that is, if they cannot be both true and they cannot be both false.It is the opposition of a pair of propositions so related to one another that they cannot be either simultaneously true or simultaneously false. The truth of one excludes the truth of the other, and falsity of one exclude the falsity of the other.Quantified attributive propositions having the same subject and predicate but differing in both quality and quantity (A & O, E & I) are contradictories
The Rules for Contradictories:
1.) If one of two contradictory propositions is true, the other is false.
2.) If one is false, the other is true.
Contrary Opposition
Two propositions are said to be contraries if they cannot be both true, that is, if the truth of either one entails that the other is false.The opposition of a pair of propositions so related to one another that they cannot be simultaneously true but they can be simultaneously false. The truth of one excludes the truth of the other, but the falsity of one does not exclude the falsity of the other.Universal attributive, or categorical propositions having the same subject and predicate but differing in quality (A & E) are contraries. The Rules for Contraries:
1.) If one of two contraries is true, the other is false.2.) If one is false, the other is doubtful.
Subcontrary Opposition
Two propositions are said to be subcontraries if they cannot both be false, although they may both be true.
Subcontrary opposition is the opposition of two propositions that cannot be simultaneously false but can be simultaneously true: if one is false, the other must be true; but both of them can be true.
Particular propositions having the same subject and predicate but differing in quality (I & O) are subcontraries. The Rules for Subcontraries:
1.) If one of two subcontraries is false, the other is true.2.) If one is true, the other is doubtful.
Subalternate
Two propositions that have the same subjects and the same predicate terms, and agree in quality but differ only in quantity are subalterns.Subalterns are not, strictly speaking, opposites at all because neither the truth nor the falsity of either of them excludes the truth or falsity of the other. Both of them can be true and both of them can be false.Propositions having the same subject, predicate and quality but differing in quantity (A & I, E & O) are subalternates.The Rules for Subalternates:1.) If the universal is true, the particular is true; but
if the universal is false, the particular is doubtful.
2.) If the particular is true, the universal is doubtful; but if the particular is false, the universal is false.
Exercises:
Give the contradictory, contrary, subcontrary and subaltern of each of the following (if it has one); and supposing that the original proposition is true, state whether each of the opposites is true, false or doubtful.1. All men are mortal
2. No cat is a dog
3. Some house is white.
4. Some house is not white.
5. John is a very wealthy man.
6. If the sun is shining, it is day.
7. John says he is not going.
8. He went to the hospital because he was sick.
Exercises:
Supposing that the propositions after each of the numbers are true or false as indicated, state whether the propositions following them are true, false or doubtful, and tell what kind of opposition is illustrated by each example.1. If it were false that all women are human beings,
That no women are human beings would be…
2. If it were true that every cat is a dog,
That no cat is a dog would be…
That some women are human beings would be…
That some women are not human beings would be…
That some cat is a dog would be…
That some cat is not a dog would be…
3. If “some aliens are seditious” is true,
That all aliens are seditious would be…
4. If it is true that all horses can’t jump,
That all horses can jump would be…
That no aliens are seditious would be…
That some aliens are not seditious would be…
That some horses can jump would be…
That some horses can’t jump would be…
State the kind of opposition illustrated in each example; and, whenever an inference has been made, state whenever it is valid or invalid.
1. All hounds are dogs; therefore some hounds are dogs.
4. It is false that all men are angels; therefore it is true that no men are angels.
5. If it is true that all jewelers have jewels to sell, it is false that no jeweler has jewel to sell.
2. Some dogs are hounds; therefore some dogs are not hounds.
3. Some dogs are not hounds; therefore some dogs are hounds.
6. If it were true that no jewelers sold jewels, it would be false that some jeweler did not sell them.7. It is false that no jeweler sells jewels; therefore it is true that some jeweler does sell jewels.
8. I always tell my salesmen not to be discouraged if the first couple of customers on a block give them trouble. They know that there’ll be at least few soft touches near the end.9. “I told the workers that since some labor unions are not honestly managed, it follows that some are honestly managed, and that ours was one of these.”10. “Every single American is desirous to improve the living conditions of the lower classes. America they say, will then be a true democracy, with all the citizens living, if not on similar levels of luxury, wealth, and importance, at least on an equal basis of freedom. But unfortunately several big businessmen, controlling monopolies, and thus able to ruin thousands of other businessmen, do not desire to help the poor. Therefore, probably little improvements will be carried out until the power of this minority is overcome.”11. If it is false that he neither plays the saxophone nor the clarinet, it must be true that he plays either a saxophone, or a clarinet, or both a saxophone and a clarinet.
12. If it is false that beggars should be choosers, it follows that beggars should not be choosers.
13. Suppose it is false that John should drink milk; it follows that John should not drink it.
14. If the speaker is not telling the truth, he must be lying.15. “It is a fallacy to argue from the variety of system and the extravagant character of certain philosophies to the impotence of all metaphysics. If the system of one philosopher is to great extent the expression of his personal temperament, it does not necessary follow that alike judgment can be passed on the system of another philosopher: if some of the arguments of a given philosopher are sophistical, it does not follow that all his arguments are sophistical.”
