Immediate Inference

Immediate inference provides tools useful for determining whether a proposition is true, false or indeterminate after a number of manipulations have been performed on it and given its initial truth or falsity. Immediate inference is somewhat similar to an argument with one premise and one conclusion. Distribution is not a kind of immediate inference, but will prove useful in understanding immediate inference.

Distribution

A term is distributed if its proposition makes a claim about each and every member of that category. With the A proposition "All men are pigs." for example, a claim is being made about each and every man. But the same is not true for the predicate term: pigs. Other pigs, in addition to those referred to, may exist. All A propositions are the same in this regard: the subject term is distributed and the predicate term is not distributed. Similarly, all other categorical propositions are consistent about which terms they do or do not distribute. E propositions distribute both the subject and the predicate terms. I propositions do not distribute either term. O propositions are a little contrary to intuition. Obviously O propositions do not  refer to all members of the subject class. But they are said to refer to all members of the predicate class. The idea is that, from the few of the subject class referred to, the entire predicate class is excluded. For example, in "Some cows are not clean." each and every member of the clean things is excluded from the subgroup of cows referred to by "some cows".

Two Rules:

Conversion

 Conversion is the manipulation which simply switches the subject and predicate terms.

A Propositions

The converse of "All cars are automobiles." would be "All automobiles are cars." From the truth of the former one cannot tell whether the latter is true or false so it is indeterminate. Someone might object that the latter is clearly false, since trucks are automobiles. But one knows that from outside information rather than from logical inference. To see that one cannot rely on the converse of a true A proposition being false consider: "All bachelors are unmarried males." The converse is "All unmarried males are bachelors." which is happens to be true. Similarly, if one starts with a false A proposition such as "All crows are pink." the converse "All pink things are crows." is indeterminate. Again this particular example might seem to produce a false, but consider that "All mammals are cows." which is false converts into "All cows are mammals." which is true. In sum, conversion is not legitimate for A propositions.

E Propositions

The converse of the true proposition "No bachelors are married." is the also true statement "No married people are bachelors." The converse of the false "No crows are birds." is the false "No birds are crows." This result is not a feature of the examples we happened to choose or our having outside information. If all of one group is excluded from another, then obviously all of the other group must equally be excluded from the one. Conversion is legitimate for E propositions.

I Propositions

The converse of the true "Some crows are albinos." is the true "Some albinos are crows." and the converse of the false "Some cows are purple." is the equally false "Some purple things are cows." As with the E proposition, this is always the case. If some members of one kind of thing are members of another kind then, since the two kinds must have members in common in order for this to be true, some members of the other kind must be members of the first kind. And in order for an I proposition to be false,  no members can be common between the two kinds, which will equally make the converse of the original I proposition false. Conversion is legitimate for I propositions.

O Propositions

The converse of the true "Some cows are not brown." is the indeterminate "Some brown things are not cows." As in the case of the A proposition, if you think the converse is true, consider the true "Some cows are not pregnant holsteins." which becomes "Some pregnant holsteins are not cows." which is false. The converse of the false "Some cows are not female." is the indeterminate "Some females are not cows." Again if this converse looks true, consider "Some holstein cows are not female holsteins" which becomes the false "Some female holsteins are not holstein cows." (Assuming that cow properly refers to only female bovines.)

Note on Distribution

The fact that E and I are symmetrical in distributing or not distributing their terms is significant here. In the case of the unsymmetrical A propositions where only the subject is distributed, converting results in a claim about every member of the former predicate class. But the basis of the new claim is supposed to be the original claim which did not make a claim about every member of the predicate class. Thus, about that class we end up claiming more than we were given as true (or false). Nothing justifies the stronger claim about that class.

Partial Conversion (Conversion by Limitation)

A propositions may be legitimately partially converted. Consider the true "All ducks are birds." The subaltern will also be true: "Some ducks are birds." which can be legitimately converted into "Some birds are ducks." Partial conversion, then, involves taking the subaltern first and then converting. Partial conversion only is legitimate for true A propositions.

Warning: Web Page Notes are not intended as a substitute for attending lectures.

Obversion

Obversion is a manipulation involving two changes: the quality is changed (either from affirmative to negative or negative to affirmative) and the complement of the predicate class is substituted for the predicate term. The complement of a class is the class of everything not in the original class. The class of non-dogs is the complement of the class of dogs. Equally the class of dogs is the complement of the class of non-dogs. We will take the prefix "non" to be trustworthy in picking out the complement of a class. Other prefixes are not trustworthy. The class of immature things is not the complement of the class of mature things, because some things are neither. Obversion is always legitimate.

A Propositions

The obverse of "All believers are heaven bound." is "No believers are non-heaven-bound." If the proposition is true so is the obverse and if the proposition is false so is the obverse. The obverse of an A proposition is always an E.

E Propositions

The obverse of "No reptiles are birds." is "All reptiles are non-birds." If the proposition is true so is the obverse and if the proposition is false so is the obverse. The obverse of an E proposition is always an A.

I Propositions

The obverse of "Some cars are Fords." is "Some cars are not non-Fords." If the proposition is true so is the obverse and if the proposition is false so is the obverse. Notice that the word "not" and the prefix "non" perform different roles logically. The "not" is part of the copula and determines the quality of the proposition. The "non" is part of the predicate term and helps identify the predicate class. The obverse of an I proposition is always an O.

O propositions

The obverse of "Some cars are not Fords." is "Some cars are non-Fords." If the proposition is true so is the obverse and if the proposition is false so is the obverse. Notice that the word "not" and the prefix "non" perform different roles logically. The "not" is part of the copula and determines the quality of the proposition. The "non" is part of the predicate term and helps identify the predicate class. The obverse of an O proposition is always an I.

Contraposition

Contraposition is also a manipulation involving two changes: both terms are replaced by their complements, and the terms are switched (as in conversion). Contraposition is legitimate for A propositions and O, but not for E and I. One may use the other forms of immediate inference to derive the contrapositive over three steps: first obvert, second convert and third obvert again. Performing these steps will confirm that contraposition is legitimate for A and O, but not legitimate for E and I. The reason is the middle step of conversion that is not always legitimate.

Summary

In the following table 'S' stands for the Subject term in the original proposition, and 'P' for the Predicate term in the original proposition. Except where marked Not Legitimate if the original proposition is true, the resulting proposition is true, and if the original proposition is false, then the resulting proposition is false. (Note that Partial Conversion is not included in this table.)
 
A
E
I
O
Proposition All S are P No S are P Some S are P Some S are not P
Converse All P are S 
Not Legitimate		 No P are S Some P are S Some P are not S 
Not Legitimate
Obverse No S is non-P All S is non-P Some S are not non-P Some S are non-P
Contrapositive All non-P are non-S No non-P are non-S 
Not Legitimate		 Some non-P are non-S 
Not Legitimate		 Some non-P are not non-S

For more information see Garth Kemerling's Web Page on Immediate Inference.