# A valid syllogism can have wrong premises

## Syllogism saying

### From MosaPedia

The Syllogism saying is a recurring logical-philosophical statement in MOSAIK and its secondary universes. So far, three characters have pronounced it in four different situations.

###  Definition of a syllogism

A syllogism is a certain form of logical statement about three concepts A, B and C, in which two basic assumptions (the Premises) to an inference (the Conclusion) is closed. The two premises link two of these terms with each other, whereby one of the terms appears in both premises, but not in the conclusion, and the other two terms, which appear in only one premise, are ultimately linked in the conclusion:

1. First premise: A ~ B
2. Second premise: B ~ C
3. Conclusion: A ~ C

The possible combinations of two terms A and B in the premises and the conclusion consist of four types: A applies to all B; A does not apply to any B; there is A for which B holds; there is A for which B does not hold. (The first two links / statements are called accordingly generally, the last two particular.) Under these prerequisites, one arrives at a total of 256 possible syllogisms, i.e. 256 variants in which the three starting terms can be combined. Of these, only 24 syllogisms are logically valid, i.e. if both premises are true, the conclusion is also true; the other 232 are invalid, i.e. a logical conclusion from the two premises on the statement in the conclusion is not possible. It should be noted that the validity of a syllogism does not mean that the conclusion is actually true - if one of the premises is false, the conclusion is false even with valid syllogisms; conversely, a true statement in the conclusion does not mean that it must be based on a valid syllogism.

 Two examples of valid syllogisms: First premise: All Abrafaxe are giants. Second premise: Abrax is an Abrafax. Conclusion: Abrax is a giant. First premise: All beings with square heads are called Tetrapax. Second premise: No Abrafax is called Tetrapax. Conclusion: No Abrafax has a square head. Two examples of invalid syllogisms: First premise: there is an Abrafax who has no red hair. Second premise: Brabax is one of the Abrafaxes. Conclusion: Brabax has no red hair. First premise: All beings with square heads are called Tetrapax. Second premise: no Abrafax has a square head. Conclusion: No Abrafax is called Tetrapax.

The first syllogism delivers a wrong conclusion because the first premise about the gigantic Abrafaxe is already wrong. Nonetheless, it is logically valid. Compare the two Tetrapax syllogisms with each other: The second is invalid because the first premise leaves open whether it is also called beings Tetrapax give the none have a square head, so that a statement about the names of the individual Abrafaxe is not possible, although according to the second premise none of them have a square head. That the stated conclusion is true - it actually means none of the Abrafaxe Tetrapax -, is in this case not a sign that the syllogism is valid. The first Tetrapax syllogism, on the other hand, is valid because would be If there is an Abrafax with a square head, this would have to be according to the first premise Tetrapax are called, which is excluded according to the second premise.

###  Basic form of the saying

The basic form of the saying, which is used in MOSAIK in a more or less modified form, is roughly:

 If the conclusion is based on a syllogism, it is erroneous, since it appeals to premisses that are too free.

Above all, the "premisses that are too free" are puzzling, because such a category does not exist in syllogisms. The reason is as follows: Author Jens-Uwe Schubert got the saying from the novel At swim-two-birds (German Two birds swimming or On swimming-two-birds - Interestingly, in the first issue of this saying there is a MOSAIC figure of the same name) by Flann O'Brien, where it is pronounced as follows in the original in a pub scene:

 The conclusion of your syllogism, I said lightly, is fallacious, being based on licensed premises.

The last translation by Harry Rowohlt in 2002 reads as follows (the earlier translation by Lore Fiedler is very similar):

 The end of your syllogism, I said lightly, is erroneous, since it is based on premisses that are too free and, moreover, is not economic, but pub-economic.

The original is a play on words by the character in the novel. English premise means both "premise" and "local", and as licensed premises a bar with a license to serve alcohol is called a pub. The joke can hardly be reproduced in the German translation, which is why you had to help yourself with an additional subordinate clause.

The puzzling formulation with the "premisses too free" is thus also clarified: It is originally an attempt to reproduce an English pun, and not a logically correct reproduction of a syllogistic argument. Interestingly, however, all cases in MOSAIK that are identified with the saying and about which something is known in terms of content can be traced back to a certain invalid syllogism type in which both premises each name a subset of a total set and in the conclusion of which it is finally claimed that one of these Subsets are in turn a subset of the other subset. Since both subsets mentioned in the premises are independent of each other, the formulation with the "premisses too free" fits quite well.

The type mentioned, to which the invalid syllogisms in MOSAIC belong, can also be broken down as follows:

1. First premise: term A belongs to term B.
2. Second premise: term C belongs to term B.
3. Conclusion: Term A belongs to term C.

Here is a fine example of a false syllogism of this type:

1. First premise: The Abrafaxe are goblins.
2. Second premise: The Digedags are goblins.
3. Conclusion: The Abrafaxe belong to the Digedags.

Here is another example of this type, which happens to offer a true conclusion, but still remains logically inadmissible:

1. First premise: Dig is a leprechaun.
2. Second premise: The Digedags are goblins.
3. Conclusion: Dig is one of the Digedags.

In those cases in MOSAIK in which the syllogism saying is used, only the inadmissibility of the conclusion is declared - regardless of whether the statement reached is actually true or not.

###  Mr. Copperplate

The officers of the Golden Hind advise in issue 278 how to deal with the mutiny of the occupation. You get tangled up in procedural and philosophical questions. This is how Mr. Copperplate accuses the first officer of Lord Kenterbury:

 With all due respect, Lord Kenterbury: If your last conclusion is based on a syllogism, it is erroneous, since it appeals to too free premises!

Since neither the previous allegation of the Lord nor his reaction to Copperplate's accusation is known, it is not clear what this refers to. The whole situation is - from the point of view of the MOSAIK author - only to illustrate that the officers are unable to achieve anything other than aloof speech.

###  Brabax for the first

Although Copperplate's saying in the above situation was still derogatory with "They talk and talk!" comments, Brabax argues later in issue 359 to Califax in the same way. On their ride through the Orly forest together with Hugo von Payens, the Abrafaxe discuss the question of whether there are dragons. Brabax does not believe in dragons, while Abrax can imagine the existence of dragons but does not like to believe in gangs of pygmy robbers. Califax is open to anything. When they are actually captured by a gang of dwarf robbers shortly afterwards, he concludes to Brabax that there must therefore also be dragons. His logic, which he now executes, is the following:

 You said there were no dragons and you didn't believe in dwarves either. But you were wrong! And since the dwarfs obviously exist, that must be true with the dragons too.

Brabax agrees with Califax's first sentence, does not contradict the second sentence, but defends himself against the conclusion in the third sentence with the slightly shortened syllogism saying and a deliberate meanness ad hominem:

 If it weren't for you, Califax, I would reply, "Your syllogism is erroneous because it invokes premisses that are too free". But in your case, I just roll my eyes.

He translates Califax's argumentation into the following invalid syllogism:

1. First premise: The assumption that dragons do not exist is one of Brabax's assumptions.
2. Second premise: There are false assumptions among the Brabax assumptions.
3. Conclusion: So the assumption that dragons do not exist is a false assumption.

Or to put it another way: Brabax complains that not all of his assumptions have to be wrong just because there are obviously some for which this applies.

A very similar argument to that of Califax can be found in the novel, by the way Baudolino by Umberto Eco. However, there is no indication that this is a faulty syllogism.

###  Leibniz

They come across in the English Channel Comète by Jean Bart and the Zuidersee (with Brabax and Leibniz on board) each other. After a surprising maneuver by Brabax, he was pleased to see that the sails of the other ship were being reefed. Obviously they gave up! Leibniz contradicts with a minimal variant of the syllogism saying:

 The conclusion is erroneous because it relies on premises that are too free, Mr. Secretary.

Leibniz does not explicitly mention a syllogism through the use of the term Premises but implicitly assumed. Brabax's invalid syllogism can therefore be reconstructed as follows:

1. First premise: The occupation of the Comète reefs the sails.
2. Second premise: if you capitulate, you raise your sails.
3. Conclusion: So she has Comète-Crew surrender.

In other words: although sailing meetings can be a sign of small surrenders, it can also have completely different reasons - as Brabax discovered shortly afterwards. It is unusual that Brabax, who otherwise likes to point out false syllogisms to other people, falls into this trap himself and has to be corrected by Leibniz like a schoolboy.

###  Brabax for the second

In the album Emperor, warrior, lion hunter At the Völkerschlachtdenkmal the Abrafaxe experience three successive time leaps without changing location with the help of a time telescope: from the present to the year 1813 in the middle of the battle of the nations (whereby the monument "disappears" of course), from the year 1813 to the year 1913 for the inauguration of the monument and from the year 1913 "back to the present". The second jump was "too short" because the Abrafaxe actually wanted to get from 1813 to the present. Of course, Califax does not notice this immediately, but is initially happy that the jump was successful, because the Völkerschlachdenkmal is back:

 The memorial is back - we did it!

Brabax comments on this with a long variant of the syllogism saying:

 If your conclusion is based on a syllogism, it is erroneous because it appeals to premisses that are too free. [...] Little joke.

The erroneous syllogism this time would be:

1. First premise: We are at a time when the Monument to the Battle of the Nations exists.
2. Second premise: The Völkerschlachtdenkmal exists in our original present.
3. Conclusion: So we are in our original present right now.

In other words, although the Völkerschlachtdenkmal undoubtedly exists in the present, it also exists at other times. After a few adventures, the Abrafaxes finally manage to make the third leap in time, with which they finally land back in their presence.

###  Syllogism without saying

Another syllogism is discovered by Odo von Biscuit in the little village of Erythros on the turtle island (booklet 371). First he describes the arguments of the erythreos:

 From the fact that I am not an erythroid, you conclude that I am leuconite. You assume that there are only Erythreos and Leukonites on the island.

This conclusion can be expressed as follows:

1. First premise: All people on the island who are not Erythreos are Leukonites.
2. Second premise: Odo is one of those people on the island who are not Erythreos.
3. Conclusion: Odo is leuconite.

Without using the corresponding terms, the argumentation of the Erythreians is a syllogism. In contrast to all the syllogisms treated above, this is even valid, i.e. if both premises are true, the conclusion is also true. Odo therefore has to prove to his listeners that the first premise Not true is. He succeeds in showing the Erythreians with the help of little Romanos that since the Red Galley was stranded there are other people on the island besides the Leukonites who are not Erythreos.

###  Saying without syllogism

The two Aboriginal observers in volume 449 cannot make sense of the strange behavior of the pale noses Stuart Bingley and Califax, who are taking care of the diseased merino sheep. Burnum suspects that "knocking over" the "multi-haired four-legged friends" is a religious ritual. Gelar rebukes him with the following words:

 Your conclusion is erroneous because it is based on false observation, my friend.

So this is a modification of the syllogism saying without a syllogism at all in play.