# Which practical applications of the fourth figure of syllogism exist?

There are four figures of simple categorical syllogism, well-known since Aristotles.

I started to wonder about practical applications of each figure, so I've addressed different literature, obtaining the following knowledge about the three:

• 1st figure is commonly used to prove that a particular case is a valid part of general case.

• 2nd figure is commonly used to prove that a particular case is not a valid part of general case.

• 3rd figure is used to prove the falsity of a general case by providing particular cases contrary to it.

However, I was unable to find or derive such practical application for the last, 4th figure. All I know is that it's rarely used in thought process and that it can be transformed into the 1st figure in some cases.

So what are practical application for the 4th figure of simple categorical syllogism?

• See Syllogism: "Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and Jean Buridan—reject the fourth figure as a figure distinct from the first." Jun 16 at 7:54
• @MauroALLEGRANZA Thus, the fourth figure is just an indirect case of a first figure, hence it has the same practical application as the first? Jun 16 at 7:57
• I’d like to post something answer, but I do not understand the descriptions of the 2d and 3rd figures. Jun 16 at 17:12
• Where did you get those motions about the mood & figures from? Did you get those ideas from outside of Philosophy? Were you reading math, physics, psychology, computer science, etc. The notion particular case & general case stuff is typically not from legit philosophers. Phrases such as those are not used typically which is why I ask what I ask. Your terminology is already off. Jun 16 at 18:09

Well, this question sent me back to my college text in logic: Barker, The elements of logic (McGraw-Hill 1965).

The first figure of a syllogism takes the form: All M are P, All S are M; thus All S are P. Here, S and P are the subject and predicate, and M is the middle term. AAA in the first figure is the only syllogism where an A statement (all are) is a valid conclusion. So AAA-1 does indeed show that a particular case is valid part of a general case,

The second figure has the form: All P are M; All S are M; therefore All S are P. The main problem with the second figure is the undistributed middle term. So the valid syllogisms in the second figure all have premises where both the subject and the predicate are distributed (E statement; No are) or only the predicate is distributed (O statement; Some are not). The premises cure the distribution problem. So I suppose that the second figure can be used to show that a particular case is not part of a general case, although I have not seen an example of this.

The third figure takes the form: All M are P; All M are S; thus All S are P. I don’t see a general use for the third figure as a tool to falsify a general case. However, it is clear that the premises of valid third-figure syllogisms tend to include I statements (Some are), suggesting that this reasoning tends to limit the general case in some manner.

That brings us to the fourth figure: All P are M; All M are S; thus, All S are P. The practical application of this figure seems to be the same limiting function as the third figure. In general, among the fourth-figure valid syllogisms, there is one universal premise, one particular premise, and a particular conclusion.

Thanks for asking. This question was a real mind-bender.

• I noticed that you did not address the fact if the fourth figure was viewed as another case of the first figure as Aristotle did. In this way Aristotle emphasized only three figures & not four figures. One can always transform a fourth figure into a first figure in the mood AAA where all propositions are universal for instance. Mentioning this will help in some understanding. The OP could ask what is the practical use of syllogisms overall but most likely he believes it is an IF . . . THEN . . . Statement because of the popularity of mathematical logic these days. Jun 17 at 21:29
• Far be it from me to disagree with Aristotle, but I do not understand how the fourth figure is merely another example of the first. The positions of the middle terms are completely different in the two. Jun 18 at 0:10
• Yes that is clearly the visual point! Change the order of the premises and guess what position the middle terms are in! Aristotle saw this which is why he did not acknowledge a fourth figure. QED. [The QEDis a joke by the way.] Jun 18 at 1:34
• I am at a loss. I do not understand the point you are making. Jun 18 at 2:13
• @Speakpigeon. The answer generally describes the 4th figure correctly; the positions of the middle term are correct. The specific example is invalid. An example of a valid 4th-figure syllogism is IAI: Some P are M; All M are S; thus Some S are P. As you noted, AAI is also valid. Jun 18 at 20:10

Suppose someone is denying that there are vertebrates that are bats. One possible counter-argument might go like this:

All bats are mammals;

All mammals are vertebrates;

So, some vertebrate is a bat.

This is called "Bramantip" and one form of the 4th figure:

BaA, CaB; so, AiC.

Seems to work fine.

There are other forms. They can all be used in real-life arguments.

Another example:

All rational numbers are p/q numbers where p and q are integers and q is not null;

No p/q number where p and q are integers and q is not null is a number which is the product of an odd number by the square root of 2;

So, no number which is the product of an odd number by the square root of 2 is a rational numbers.

This is Camenes, also of the 4th figure:

BaA, CeB; so, AeC.

• I think your last line should be: BaA, AeC; so, CeB. Jun 17 at 22:05
• @Bumble No, BaA, AeC; so, CeB is not valid. C may be B. Jun 18 at 9:58
• C may not be B if all Bs are As but no A is a C. But your "BaA, CeB; so, AeC" is definitely not valid, since a counterexample would be where all Bs are As, some or all Cs are As, but no C is a B. Let B be "is a bird", A be "has wings" and C be "is a bat". Then you would have true premises, All birds have wings, and No bat is bird, and a false conclusion, No winged thing is a bat. Jun 18 at 13:45
• @Bumble That's what I thought, you're misreading the code: BaA does not mean "All B's are A". It means "B belongs to all A's", which was Aristotle's rather quirky expression. It means predicate B belongs to all A's, or "All A's are B", not your "All B's are A". This is also apparent in the fact that we start with B (BaA), not A, which signals an inversion from the syllogism "All A's are B". Same for the rest. Jun 18 at 15:12

Aristotelian logic NOW has 4 figures due to medevial philosophers.

Aristotle only recognized 3 figures & 14 valid figures with moods. Medevial philosophers added the 4th figure and raised the valid number of figures with moods to 19.

With the invention of Mathematical logic the idea of existential import added 5 more valid figures with mood for a total of 24 valid figures with moods.

The fourth figure was just a variation of the first figure to Aristotle. The fourth figure is formally invalid following the rules of categorical syllogisms, BUT can be transformed always to the first figure so after all the fourth figure too must be valid.

The subject - predicate rule is broken literally for the fourth figure: the subject of the conclusion must come from the minor premise not the major premise. The fourth figure has the subject of the conclusion coming from the major premise if you want to write it down with valid form. The predicate of a categorical syllogism must be from the major premise but in the fourth figure to be written down as valid the predicate must come from minor premise. The process is also known by transposing the premises into the first figure.

If you were to use the standard categorical syllogism rules you would create an invalid argument. That is if the subject comes from the minor premise & the predicate comes from the major premise that literally is invalid. You would have true premises while the conclusion could be false.

One must swap subject & predicate to make the conclusion true while the premises are also true. To convert the fourth figure to the first all one must do is change the order of the premises: make the minor premise first & the second premise the major premise to write the syllogism in the first figure.

All syllogisms in the first figure are VALID by form alone regardless of the argument content. One can have all three propositions false & the argument would still be valid.

This would not be practical in reality if you don't already know the content of the argument. In Aristotelian logic both form & content matter whereas in Mathematical logic form alone is considered. Aristotelian logic is concerned with sound arguments while Mathematical logic is concerned with just validity alone.

***I edited this to clarify what is a fourth figure syllogism. It has the schematic as follows:

Premise 1: All P are M

Premise 2: All M are S

Therefore All P are S.

Notice the M stands for the Middle term, S stands for the Subject term & P stands for the predicate term.

If one changes the order of the premises you should see the first figure. That is to say, move premise 2 above premise 1 and you should see this:

All M are S

ALL P are M

Therefore All P are S.

This last syllogism is "now magically" in the first figure with the mood AAA. [there was an attempt at humor there.] This specific syllogism was deemed "the perfect figure" because it is always valid regardless the topic or content of the premises. Aristotle knew he can just transpose the premises (switch their positions) to make the argument valid so Aristotle did not emphasize a 4th figure. He emphasized only three figures. Aristotle tried to reduce all valid arguments to the first figure AAA. That was what the following Latin mnemonic was for:

Barbara celarent darii ferio baralipton

Celantes dabitis fapesmo frisesomorum

Cesare camestres festino baroco

Darapti felapton disamis datisi bocardo ferison

This reduced syllogisms to valid forms to show validity. The vowels represent the MOOD. The constants S, P, M, C stood for the method to reduce the syllogism to show validity. All other constants were said to be space fillers. The lines of the poem were the figures. The idea was to show every valid syllogism can be made into the first figure by the method in the poem. This poem was not Aristotle by the way. Medieval logicians added to Aristotelian logic after the death of Aristotle all the way to around 1845 when Mathematical logic was invented. Aristotelian logic predates Mathematical logic by thousands of years. Mathematical logic goes by several pseudonyms: modern logic, symbolic logic, predicate logic, etc. Prior to any of that all logic was classified under a branch of Philosophy alone. To this day there are distinction between how math teaches so called "logic" compared to how philosophers taught logic. [Peter Smith makes a blog Logic Matters where he once described the distinctions he recognized.] The concepts can be contrary at times because the same terms are used but the context are different between the two fields.

• If readers are going to give minus marks then there should also be a justification for such marks. There should be no spiteful actions which seems to be the case here. If something is inaccurate you should be able to point errors out specifically instead of being spiteful readers. Jun 16 at 16:37
• I agree. Apparently I got dinged a point for my comment to the original post, which only asks for clarification. Jun 16 at 17:55
• +1) I just upvoted since you've made such a non-trivial effort with relevant historical knowledge which not many people know any more these days... Maybe you could generalize and formulate a practical, interesting and perhaps deep philosophical question out of this as for the necessity to give justification for both boo (downvote) AND approve (upvote), as Arendt famously claimed Nobody Has the Right to Obey, otherwise it may bring dangerous consequences... Jun 17 at 3:04