﻿ 2.3 Immediate inferences

### 2.3 Immediate inferences

Inferences with one premise (and one conclusion) are called immediate in traditional logic.

• Subalternation as immediate inference

We saw in the previous section that according to traditional logic general sentences logically entail their subalterns, i.e. from a universal affirmative or universal negative sentence we may validly infer the particular affirmative or particular negative sentence with same subject and predicate, respectively. Here are two examples:

 All dolphins are mammals. No humans are sinless. Some dolphins are mammals. Some humans are not sinless.

Symbolically:

 SaP SeP SiP SoP

Such conclusions are immediate inferences as they have one premise. We will not consider the conclusions based on the other relations in the square of opposition as immediate inferences as they represent transitions form the truth of a sentence to the falsehood of another or vice versa – not from the truth of a sentence to the truth of another.

Conversion is an immediate inference whose conclusion is obtained from the premise by interchanging the subject and the predicate – the subject becomes predicate and the predicate becomes subject. Such interchange not always represents a valid inference. It is valid for the E and I sentences, valid by limitation for the A sentences (we will see what that means) and invalid for the O sentences.

If the premise is a universal negative (E) sentence, conversion is logically valid.

 No S are Р. SeР Example: No humans are gods. No Р are S. РeS No gods are humans.

We say of the two sentences that they are converses of one another. The converse of an E sentence is the same sentence with the subject and predicate interchanged. To see why conversion is logically valid for E sentences, look at the following diagram showing all possible relations between the extensions of the subject and the predicate of a categorical sentence:

The only possible case in which the premise “No S are P” is true is 4). In that case, the sentence “No P are S” is also true. Thus, it is not possible for the conclusion to be false when the premise is true, which means that the inference is valid.

The conclusion is the converse of the premise but also the premise is the converse of the conclusion, therefore the premise also follows logically form the conclusion, the two sentences are actually logically equivalent: we have SePPeS (not only SePPeS).

If the premise is a particular affirmative (I) sentence, conversion is also logically valid:

 Some S are Р. SiР Example: Some women are drivers. Some Р are S. РiS Some drivers are women.

Again, we can verify the validity of conversion for I sentences by the above diagram. The premise “Some S are P” is true in cases 1), 2), 3, and 5). In all of them “Some P are S” is also true.

As with E-sentences, since “Some S are P” and “Some P are S” are related symmetrically, the logical inference is in both directions: SiPPiS (not only SiPPiS).

The (simple) converse of a universal affirmative sentence “All S are P” is “All P are S” and the latter cannot be validly inferred from the former. For example, it is true that all trees are plants but false that all plants are trees. The problematic case is when the extension of S is a proper subset of the extension of P. Then “All S are P” is true but “All P are S” is false. However, in that case we can validly infer the sentence “Some P are S”. For example, form “All trees are plants” we can infer that some plants are trees. Generally, the validity of the inference scheme SaPPiS is seen as follows. By subalternation, from “All S are P” (SaP) we may validly infer “Some S are P” (SiP) and, as we have seen, the last sentence entails by conversion the sentence “Some P are S” (PiS).1

The inference of “Some P are S” from “All S are P” is called conversion by limitation. For convenience we will consider “Some P are S” to be the converse of “All S are P” dropping the addition “by limitation” but we have to keep in mind that “All S are P” is not the converse of “Some P are S”. The convers of the latter is “Some S are P”. In addition, unlike with E and I sentences, the converse (by limitation) of “All S are P” is not logically equivalent to it. The sentence “All S are P” entails “Some P are S” but the latter does not entail the former. For example, “Some cats are animals” does not entail “All animals are cats”.

 All S are Р. SaР Example: All trees are plants. Some Р are S. РiS Some plants are trees.

If the premise is a particular negative (O) sentence conversion is not logically valid, i.e. “Some S are not P” does not entail “Some P are not S”. This can be seen through an example. “Some plants are not trees” is a true sentence but its converse “Some trees are not plants” is false. Generally, to see the invalidity of the inference scheme SoPPoS, we may use the diagram above. The problematic case, in which the premise “Some S are not P” (SoP) is true but the conclusion “Some P are not S” (PoS) is false is 5) – when the extension of P is a proper subset of the extension of S. In that case, there are some S that are not P but all P are S, which makes the sentence “Some P are not S” false.

The table below summarizes the immediate inference of conversion.

 Type Premise Converse Symbolically A All S are P. Some P are S. (by limitation) SaP ⇒ PiS E No S are P. No P are S. SeP ⟺ PeS I Some S are P. Some P are S. SiP ⟺ PiS O Some S are not P. (Conversion is not valid.)

In addition to conversion (and subalternation), there are two other types of immediate inferences that traditional logic addresses: obversion and contraposition. They use the concept of complement, so let us briefly dwell on it.

• complement

The primary use of the notion is related to classes. The class Y is the complement of a class X if and only if Y’s elements are all and only those things that are not elements of X. For example, if X is the class of all cats, its complement (or complementary class) is the class of all non-cats. The latter consists of everything that is not a cat (all non-living things included). The class of cats is the extension of the term “cat” and its complement is the extension of the term “non-cat”. So, starting from an arbitrary term S, whose extension is the class X, we can always form a second term “non-S”, whose extension is the complement of X (“cat” – “non-cat”, “human” – “non-human”, etc.). Therefore, as а secondary meaning, the concept of complement relates not to classes, but to terms formed from other terms by the negative word “non”. For example, we may refer to the class of all non-cats as the complement of the class of all cats but also we may refer to the term “non-cat” as the complement (or the complementary term) of the term “cat”.

Note that the complement of the complement of a class is the class itself – a non-non-cat is simply a cat. Therefore, (usually) the complement of a complement of a term is replaceable by the term itself.

We should be careful not to mistake complementary terms for contrary terms. The complement of the term “beautiful” is not the term “ugly” (which is the contrary term) but the term “non-beautiful”. In addition to things that are ugly, under the second term fall things that are neither beautiful, nor ugly. If a term is the complement of another term, each thing falls under one or the other term. We may use this as a test whether a term is a complement of a term. Each thing is either a cat or a non-cat but there are things that are neither ugly nor beautiful.

We turn now to the immediate inference of obversion. Here is an example:

 All narcotics are addictive. No narcotics are non-addictive.

To obvert a sentence, we replace the predicate with its complement and change the quality – to negative if affirmative, and vice versa. The obtained sentence is called the obverse of the initial one. In the above example, the predicate “addictive” is replaced by its complement “non-addictive” and the type of the sentence is changed from universal affirmative to universal negative – the quantity has remained the same (universal) but the quality has been switched to negative from affirmative. This is the defining pattern of obversion.

Unlike conversion, obversion is logically valid for all four types of categorical sentences and it is such without reservations (there is no things like “by limitation” in it). A sentence and its obverse are actually logically equivalent. The logical inference is in both directions, because replacing the predicate with its complement and changing the quality compensate for each other so that in the end the sentences have the same meaning. Let us go through the four type of categorical sentences and see how obversion will look like with them.

The obverse of a universal affirmative (A) sentence “All S are P” (SaP) is the E-sentence “No S are non-P” (S e non-P). For example, “All cats are intelligent” is obverted to “No cats are non-intelligent”. We may use again the above diagram to convince ourselves of the logical validity of the inference scheme SaPS e non-P. The premise “All S are P” is true in cases 1) and 2). The class of all non-P is represented in the diagram by the whole area outside the circle of P. In the two cases there is no overlapping between this area and the circle of S, so no S is a non-P when all S are P.

 All S are Р. SaР Example: All cats are intelligent. No S are non-P. S e non-P No cats are non-intelligent.

A universal negative (E) sentence “No S are P” (SeP) is obverted to the A-sentence “All S are non-P” (S a non-P). For example, from “No cats are intelligent” we may validly infer “All cats are non-intelligent”. “No S are P” is true in case 4) of the diagram. In that case, the circle of S is completely included into the area outside the circle of P (representing the class of all non-P). So, all S are non-P when no S are P.

 No S are P. SeР Example: No cats are intelligent. All S are non-P. S a non-P All cats are non-intelligent.

A particular affirmative (I) sentence “Some S are P” (SiP) has as its obverse the O-sentence “Some S are not non-P” (S o non-P). For example, “Some cats are intelligent” is obverted to “Some cats are not non-intelligent”. Taking into account that the set of things that are not non-P is exactly the set of P, the equivalence of the two sentences is obvious.

 Some S are P. SiР Example: Some cats are intelligent. Some S are not non-P. S o non-P Some cats are not non-intelligent.

The obverse of a particular negative (0) sentence “Some S are not P” (SoP) is the I-sentence “Some S are non-P” (S i non-P). For example, “Some cats are not intelligent” is obverted to “Some cats are non-intelligent”. The equivalence of the sentences is even more obvious here than in the previous case.

 Some S are not P. SoР Example: Some cats are not intelligent. Some S are non-P. S i non-P Some cats are non-intelligent.

We showed that a sentence logically entails its obverse. That the sentences are actually logically equivalent (that the obverse entails the initial sentence, too) is seen by the fact that the obverse of the obverse is the initial sentence. For example, the obverse of “All S are P” (SaP) is “No S are non-P”. Obverting the latter sentence, we get “All S are non-non-P”, in which the two “non” can be dropped. This shows that “All S are P” follows logically from “No S are non-P” (its obverse) by obversion (which we have already shown to be valid). The situation is the same with the other three types of categorical sentences.

The table below summarizes the immediate inference of obversion:

 Type Premise Obverse Symbolically A All S are P. No S are non-P. SaP ⟺ S e non-P E No S are P. All S are non-P. SeP ⟺ S a non-P I Some S are P. Some S are not non-P. SiP ⟺ S o non-P O Some S are not P. Some S are non-P. SoP ⟺ S i non-P

Terms change their places under conversion and are negated under obversion. Under the immediate inference of contraposition, both things happen – terms are negated and change their places. Here is an example:

 All narcotics are addictive. No non-addictive substances are narcotics.

In contraposition, the complement of the predicate becomes a subject. We should look at that immediate inference as constructed by sequential applications of first obversion and then conversion:

 obversion ⇨ conversion

As we have seen, obversion is always valid but conversion is invalid for O-sentences and valid only by limitation for A sentences. Therefore, the validity of contraposition will depend on the second step of conversion – if it is valid, contraposition will be valid, too. Let us go through the four types of categorical sentences and see what contraposition will look like for them.

A universal affirmative (A) sentence “All S are P” (SaP) has as its contrapositive the equivalent sentence “No non-P are S” (non-P e S). For example, from “All cats are intelligent” we may validly infer by contraposition “No non-intelligent beings are cats”. That the two types of sentences are equivalent is seen from the fact that the second can be obtained from first as follows. Obverting “All S are P” (e.g. “All cats are intelligent”), we get the sentence “No S are non-P” (“No cats are non-intelligent”). As we know, obversion always yields an equivalent sentence. Then, applying conversion to the last sentence, we get “No non-P are S” (“No non-intelligent beings are cats”). Conversion does not always yield an equivalent sentence but it does when the premise is an E-sentence. As the two steps are equivalent transformations, an A-sentence and its contrapositive are logical consequences of each other.

 All S are P. SaР Example: All cats are intelligent. No non-P are S. non-P e S No non-intelligent beings are cats.

A particular negative (O) sentence “Some S are not P” (SoP) has as its contrapositive the equivalent sentence “Some non-P are S” (non-P i S). For example, from “Some cats are not intelligent” we may validly infer “Some non-intelligent beings are cats”. The validity of the inference is shown through the same sequence of transformations – obversion and then conversion. “Some S are not P” (SoP) is obverted to the “Some S are non-P” (S i non-P), whose converse is the equivalent sentence “Some non-P are S” (non-P i S).

 Some S are not P. SoР Example: Some cats are not intelligent. Some non-P are S. non-P i S Some non-intelligent beings are cats.

For universal negative (E) sentences, contraposition is valid only by limitation, similarly to conversion for A-sentences. That the sentence “No S are P” does not entail “All non-P are S” may be seen by example. “No cats are dogs” is true but its contrapositive “All non-dogs are cats” is false – humans are non-dogs that are not cats. The reason for the invalidity of the (simple) contraposition in this case is that by the first step of obversion we get an A-sentence – “All S are non-P” (“All cats are non-dogs”), which cannot be inverted simpliciter but only by limitation. “All cats are non-dogs” entails “Some non-dogs are cats” rather that “All non-dogs are cats”. Therefore, the sentence obtained through the obversion should be converted by limitation to “Some non-P are S”. The result is that the contrapositive (by limitation) of a universal negative sentence “No S are P” (SeP) is “Some non-P are S” (non-P i S) – the same as when the premise is an O-sentence.

 No S are P. SeР Example: No cats are intelligent. Some non-P are S. non-P i S Some non-intelligent beings are cats.

Contraposition is invalid for particular affirmative (I) sentences. The reason is that after obverting “Some S are P” we get the O-sentence “Some S are not non-P” and, as we know, conversion in invalid for O-sentences. For example, assuming that each thing is identical to itself, if we start with the true I-sentence “Some women are identical to themselves” (SiP), after the obversion we will get “Some women are not non-identical to themselves”, which is the same as “Some women are not different from themselves”. Obviously, the latter sentence cannot be validly converted to the false sentence “Some entities that are different from themselves are not women”. It is false as there are no entities that are different from (not identical to) themselves.

The table below summarizes the immediate inference of contraposition:

 Type Premise Contrapositive Symbolically A All S are P. No non-P are S. SaP ⟺ non-P e S E No S are P. Some non-P are S. (by limitation) SeP ⇒ non-P i S I Some S are P. (Contraposition is not valid.) O Some S are not P. Some non-P are S. SoP ⟺ non-P i S

We may use sequences of immediate inferences to prove that (or to find out if) a sentence logically entails another one – something that may not be immediately evident. The use of symbolic representations rather than the sentences themselves makes the proofs considerably easier to find and to follow. As an example, we will show that the sentence “All socialists are pacifists” logically entails the sentence “Some non-socialists are not pacifists”.

We symbolize the premise with “SaP” (S for “socialist” and P for “pacifist”). Then the conclusion has to be symbolized by “non-S o P”. Here is a proof of the validity of that inference:

 1. SaP / non-S o P 2. S e non-P 1, obversion 3. non-P e S 2, conversion 4. non-P a non-S 3, obversion 5. non-S i non-P 4, conversion 6. non-S o non-non-P 5, obversion 7. non-S o P 6, “non-non-” dropped