What is the name of "syllogisms" having conclusions with major and minor terms reversed? Can such "syllogisms" be valid?

Question

It seems we can write a correct and incorrect version of any syllogistic form, depending on the order of the major and minor terms in the conclusion. Take, for example, AAA-4.

Correct version: All P is M; All M is S; All S is P. In this version, the minor term forms the subject of the conclusion, while the major term forms the predicate of the conclusion.

Incorrect version: All P is M; All M is S; All P is S. In this version, the minor term forms the predicate of the conclusion, while the major term forms the subject of the conclusion.

Question 1: Is there a name for these incorrect "syllogisms"?

Question 2: Is the incorrect version of AAA-4 above valid? It certainly looks valid when I test it using a Venn diagram.

With many thanks.

Answer 1

Disclaim, from On the Fourth Figure of the Syllogism (1949):

Perhaps the strangest controversy in the history of logic is that over the fourth figure of the syllogism. There was never any argument as to what syllogisms are valid, but merely as to how they should be arranged.

There are many exhaustive overviews, like e.g. Lei Ma, The essential and the derivative moods of Aristotelian syllogism (2017); see Table page 5 and discussion.

The First figure has the basic mode Barbara: "PaM, MaS, therefore: PaS", where the symbols read: PaS is "P belongs to all S" ("Every S is P").

The three terms a called: major, minor, and middle term. The major is the predicate and the minor is the subject of the conclusion. The middle (that is not present in the conclusion) is what joins the two premises.

See An.Pr, 25b32-26a2:

Whenever three terms are so related to one another that the last is in the middle as in a whole, and the middle is either in, or not in, the first as in a whole, the extremes must be related by a perfect deduction. I call that term middle which both is itself in another and contains another in itself: in position also this comes in the middle. By extremes I mean both that term which is itself in another and that in which another is contained.

If A is predicated of every B, and B of every C, A must be predicated of every C.

This is the prefect syllogism (called Barbara in the Middle Ages).

Note. If we stay with this convention, the conclusion of every syllogism must be written as PxS: the Predicate belongs (in some way) to the Subject.

We have to be consistent in reading the symbolic formula: if we write PaS for the conclusion of Barbara, this is: "P (predicate) belongs to all S (subject)", that in modern words maps to: "Every S is P".

So, when we write the ambiguous " All P is S" we have to clarify that we are meaning "Every P is S", because in this case P is the subject and S the predicate (the predicate "belongs" to the subject).

In modern literature, the reading "Every S is P" is more usual: thus, the conclusion of the syllogism is written as: SaP (see e.g. Henle's and Ma's articles, as well as D.Hadgopoulos, THE PRINCIPLE OF THE DIVISION INTO FOUR FIGURES (1979) and M.Kelikli, The Fourth Figure in Aristotle (2018).

In what follows, I will use the more traditional one: PaS, i.e. "P belongs to every S".

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Having said that, Ma's review deals with many cases of Fourth figure, like Barbaraˇ (AAA④) with schema: MaP-SaM-SaP,

This type of moods has a common feature, i.e. the change of the premises has no effect on the conclusion, whose judgment object and its scope will not be changed accordingly. It seemed to Aristotle, of course, that there is no distinction between so-called major and minor premises since the two premises have equal status. However, the syllogism obtained from changing the premises of a syllogism is still regarded as a new one, because the position of the middle term which determine the structure of a syllogism is determined by the order of the two premises and the order of the subject and predicate of a premise.

The syllogism so formed can be called counterpart mood, because a mood and its original mood has a perfect symmetry relationship, which means the two moods can equivalently be obtained from one another. I use ④ to mark the counterpart moods[emphasis added].

In conclusion, if we read Barbara (AAA①) as "Every M is P. Every S is M. Therefore: Every S is P", the "counterpart" mode: Barbaraˇ (AAA④) "can be obtained from interchanging S and P."

Personally, I do not think that this is relevant.

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The discussion about the Fourth figure dates from the Medieval Tradition, that used the mnemonic codes : Baralipton (aka: Bramantip), and so on.

We have to make the assumption that Fourth figure is defined by the rule: the middle term is the predicate of the major premiss and is the subject of the minor.

If we consider Bramantip, we have the conclusion PiS ("P belongs to some S"). Thus, following the rule, we have that the two premises must be: MaP (middle term predicate of the major) and SaM (middle term subject of the minor).

The mode is a "derivative" one, because from MaP and SaM we have SaP (Barbara) and by Conversion: PiS (because conversion is inferring from a proposition another which has the subject and predicate interchanged. AaB converts to BiA).

An hint to the Fourth figure is in An.Pr, 29a19–29:

It is evident also that in all the figures, whenever a deduction does not result, if both the terms are affirmative or negative nothing necessary follows at all, but if one is affirmative, the other negative, and if the negative is assumed universally, a deduction always results relating the minor to the major term, e.g. if A belongs to every or some B, and B belongs to no C; for if the propositions are converted it is necessary that C does not belong to some A. Similarly also in the other figures; a deduction always results by means of conversion.

In conclusion: if Barbara is "P belongs to all M; M belongs to all S. Therefore: P belongs to all S", the syllogism "having conclusion with major and minor terms reversed" is of form Bramantip, with conclusion: "S belongs to some P", and not SaP ("S belongs to all P").

Answer 2

You asked the question: What is the name of "syllogisms" having conclusions with major and minor terms reversed? Can such "syllogisms" be valid?

Well here is a short direct answer. The figure of a Standard Form Categorical Syllogism depends on if you are reading English or the Latin versions. The old Latin version is backwards to those of us who read English. So the first figure in Latin is what readers in English call the Fourth Figure. Aristotle only had three figures. He did not claim four figures; he acknowledged the First Figure has a revere premise formation just like you asked in your question but he did not name it directly.

If you look in modern text in English you will see the first figure similar to the following:

All A are B.

All C are A.

Therefore All C Are B. [Where A is the middle term position].

Aristotle would have wrote the same example as this:

Every B belongs to all A.

Every A belongs to all C.

Therefore Every B belongs to all C.

[Notice the A is still the middle term in a different position from the English version].

Notice what happened there in Aristotle's version. The term that represents the lager category or set is always to the LEFT hand side. The term that represents the smaller category or set is always to the RIGHT hand side. Finally, the conclusion --in this version-- has the subject come from the first listed premise; the other term in the conclusion is from the second listed premise. So everything is basically BACKWARDS to English.

Look at them side by side and you will see this:

All dogs are mammals. versus Every Mammal belongs to all dogs.

All beagles are dogs. versus Every Dog belong to all beagles.

All beagles are mammals. versus Every Mammal belongs to all beagles.

We are kind of playing semantics here. The arguments express the exact same thing. However, in English, what we call the FOURTH figure is NOT VALID as written.

this is the fourth figure in English:

All beagles are dogs.

All dogs are mammals.

All mammals are beagles.

[NOTICE the correct conclusion here based on the fact the subject term in the conclusion always comes from the minor premise in a standard form categorical syllogism; the predicate of the conclusion always comes from the major premise in the syllogism.]

As the syllogism is written in the FOURTH figure in English the above example always commits the fallacy of illicit minor. Again if the syllogism is written in English it always commits a fallacy. So what slick people do to avoid the fallacy is swap the subject and predicate terms to make the conclusion true as if no one will notice the switch they made. You are not allowed to take the subject of the conclusion from the first premise! If they did not make the swap of subject predicate they may end up with a blatant false conclusion and make them look bad.

If you have not noticed yet the difference is the ORDER of the premises makes on figure VALID and the other INVALID in English. All one has to do in either case is swap the order of the premises and you get the same argument like this:

All dogs are mammals.

All beagles are dogs.

All beagles are mammals.

versus just switching the order of premises

All beagles are dogs.

All dogs are mammals.

All dogs are beagles [which is blatantly false so we swap the subject and predicate (to all beagles are dogs) and hope no one catches the violation of a syllogism rule].

So all this is just changing the order of the premises if you want the original Aristotle used or if you want to go from the Latin to the modern English. Either way you will arrive at one of the first figures by the transposition of the premises.