The basic form of a syllogism By Timo Schmitz, Philosopher

The basic form of a syllogism By Timo Schmitz, Philosopher In my article What is logic? (02 April 2017), I pointed out that an apophantic sentence is always a proposition. To find out whether the formal statement (not the content!) is true or false, one can use the law of noncontradiction, which according to Wilholt 40 says No proposition can be right and wrong at the same time! ( Keine Aussage ist zugleich wahr und falsch! ). Thus, if one takes the proposition and assumes its negation, there must be a contradiction. A syllogism is a form of logical argument which is made up of more than one proposition. Traditionally, there are two propositions and one conclusion. Geithe shows it very clearly in her handout: at first there is a first premise, followed by a second premise, finally followed by a conclusion. It follows the form SaP, thus all humans are mortal is a valid sentence, which we can take as first premise. The term humans serves as middle term, and has to be in both premises, however, the middle term has to keep its meaning. In the second premise, the subject humans which has a certain quality (here mortal connected by the copula be ), becomes the predicate. Thus, the sentence all Greeks are humans is a valid logical proposition, which follows the form SaP, where the predicate humans describes the Greek. However, the predicate mortal already describes the humans, which means that the Greeks who are belonging to the class of humans, must also belong to the class of mortals. The conclusion must be: All Greeks are mortal. Geithe points out that at least one of the premises has to be affirmative. If both premises are affirmative, the conclusion must be affirmative, too. If one premise is negative, then the conclusion must be negative as well. Wilholt introduces the syllogism in 8, 9, 10, Tugendhat/Wolf introduces syllogisms in Chapter 5. Why are syllogims so important? At first, because syllogisms are a way to find arguments. An argument always has to have a conclusion, and thus at least one premise (Wilholt, 2). The argument wants to prove the thesis. As pointed out in my article What is logic?, logic does not try find truths, it just tries to prove what was said. Thus, the sentence all elephants have four wheels Timbuktu is an elephant Timbuktu has four wheels is a valid conclusion. We have proven that if the premises are correctly, then the conclusion must be correctly, too (as of the law of noncontradiction). However, the content of the sentence is not true, but that is a question of semantics, not of logic, though both interact a lot, as we need semantics to

Timo Schmitz: The basic form of a syllogism -2- recognise an apophantic sentence as such, since we do not say whether the form is right or wrong, but we look at the content (see Tugendhat/Wolf, Chap. 2&3). To resume what was said until now, we can say: On vient de le dire, un syllogisme, c est un raisonnement comportant deux prémisses, et une conclusion, non pas simplement une prémisse et une conclusion. Les deux prémisses comme la conclusion peuvent être du type A, E, I ou O. (Gandon, 3.11) So a syllogism must have at least two premises and the two premises must be either affirmative or negative, and either universal or particular (remember the square of opposition from my article What is Logic? ). The example which was introduced above is called the type Barbara and follows the form MaP/ SaM/ SaP. Thus, the first subject is the middle term (M), it is the mediator between the subject in the second premise and the predicate in the first premise to make an SaP conclusion. Anyways, a premise might be universally negative as well, and thus MeP. As a result, the conclusion must be negative, as quoted through Geithe above. We can build a syllogism with the form MeP/ SaM/ SeP then, and call the type Celarent. Or to decypher the formula in words: No M is P, all S are M, so no S is P: No animal can write Latin letters, all cats are animals, thus no cat can write Latin letters. Here, animals must be M, since it mediates between the subject cats and can write Latin letters which is the predicate. To connect the subject cats with the predicate can write Latin letters, both premises must have the same middle term. As it is the case, the conclusion can be done. However, it must be negating. Thus, the form SeP instead of SaP is chosen. If the second premise is just particular, then the conclusion can just be adopted to a particular group, as in the form Darii : MaP/ SiM/ SiP. Example: All elephants are tall, Some animals are elephants, Some animals are tall.

Timo Schmitz: The basic form of a syllogism -3- Here, the middle term elephants is refered to the predicate which is universal to all elephants, however, not all animals are elephants, thus the subject and the middle term are brought into a particular relation. As a result, the predicate can just refer to some subjects, and not all of them. The same scheme can be used with a particular negation, such as in the figure Ferioque which goes MeP/ SiM/ SoP. Example: No man can give birth, Some human-beings are men, Some human-beings cannot give birth. Here the first premise is universally negative, and as result there must be a negation in the conclusion. However, the second premise shows that this negation just applies to some, which means that the group from the first category (here men ) is part of a larger category (here human-being ) and just those who belong to the category man are affected by the negation not: can give birth. As there are 4 types of truths (all, some, no, some not), there are 64 combinations for a syllogism, since there are two premises (4x4) and again 4 possibilities for the conclusion, so total 4x4x4 (see Gerand, 3.22; Tugendhat/ Wolf, Chap. 5). Examples for the different categories or classes, see the sheet by Wheeler. Why do I need these syllogisms now? As can be seen above, we can give use reasoning in this way, as we use the deductive method for our argumentation. Deduction means, to go from the general to the individual, thus if all men are mortal (GENERAL), then Socrates must be mortal, too (INDIVIDUAL), if he is a man. We can find information about a single thing, and prove its validity through the general thing. Of course, the semantic validity is only given, if the premises are valid, too. However, the formal validity is given here under any circumstance, no matter which nonsense one fills in. Therefore, I have to point out again that the syllogism just tries to prove that a certain conclusion must be true if its premises are true. The syllogism however does not find a truth. As Wheeler points out: There is a difference between asserting that a premise is untrue, and asserting that the logic of the argument is faulty. All dogs can fly. Fido is a dog. Fido can

Timo Schmitz: The basic form of a syllogism -4- fly. That is a perfectly valid argument in terms of logic, but this flawless logic is based on an untrue premise. If a person accepts the major and minor premises of an argument, the conclusion follows undeniably by the sheer force of reason. However, if we have complex matters, where we cannot simply see a truth, but know that the premises are true, we can put the premises in the scheme of a syllogism and we get a conclusion, which in result must be true, too. Memory marker: Now some people use to ask me How the hell do you remember how to handle a syllogism? It makes my brain explode! Here I want to give you a memory help, however, please be aware that it is only a help, and as can be seen in the fun fact, in reality, the memory help and the syllogism have nothing to do with each other. Maybe you remember how you added sums as a small child in primary school when you had 123 + 159 and had to sum it up below, e.g: 123 + 159 = 282 It says If you take 123, and you add 159, so must have 282 in the end. A syllogism can be remembered like this: You sum up two things and then put them together, just like an addition, just that the two premises have to be apophantic sentences, and either universal or particular and either affirming or negating. However, while in primary school mathematics 123+159 and 159+123, are interchangable, due to the commutative law; premises in syllogism have a strict order, as we use the deductive method. Therefore, we talk of a major premise and a minor premise. We have to keep the rule that the major premise comes at first, and is then followed by the minor premise. Fun fact: The reason why the order is interchangable in the commutative law but not in a syllogism can be seen in the formalisation. The commutative law can be formalised like this:

Timo Schmitz: The basic form of a syllogism -5- It says: A builds a union with B, thus B must have union with A. (both are equal) Or in other words: You have a circle with an infinite number of ones, and a second circle with an infinite number of ones. Now take 123 ones from the first circle and 159 ones from the second circle and the union of both circles with ones is 282, thus you have 282x1. It plays no role which circle you take at first, since the number of ones does not change. However, a syllogism is formalised like this: It says: p implies q AND q implies r which follows p implies r. And implication means that something can be concluded through something else (deductive reasoning). Sources: : Formula Sheet, University of Central Florida (Department of Computer Science), http://www.cs.ucf.edu/files/foundation_exam/formulasheet.pdf, retrieved on 03 April 2017 Gandon, Sébastien: Théorie du syllogisme et logique stoïcienne, Université Blaise Pascal Clermont-Ferrand, http://lettres.univ-bpclermont.fr/sites/lettres.univbpclermont.fr/img/pdf_theorie_du_syllogisme.pdf, retrieved on 02 April 2017 Geithe, Laura: Syllogismus (Handout), Universität Leipzig, 2012, http://www.math.unileipzig.de/~waack/veranstaltungen_files/syllogismus_handout.pdf, retrieved on 02 April 2017 Schmitz, Timo: What is logic?, self-published online article, 02 April 2017, https://schmitztimo.files.wordpress.com/2017/04/what-is-logic.pdf, retrieved on 03 April Tugendhat, Ernst; Wolf, Ursula: Logisch-semantische Propädeutik, Reclam: Stuttgart, 1993 Wheeler, L.K.: Syllogisms Deductive Reasoning, Carson-Newman University, https://web.cn.edu/kwheeler/documents/syllogisms.pdf, retrieved on 03 April 2017

Timo Schmitz: The basic form of a syllogism -6- Wilholt, Thorsten: Logik und Argumentation, Leibniz-Universität Hannover, 2014 (online download: https://www.philos.unihannover. de/fileadmin/institut_fuer_philosophie/personen/wilholt/logik.pdf, 02 April 2017) Timo Schmitz. Published on 3 April 2017 http://schmitztimo.wordpress.com Suggestion for citation: Schmitz, Timo: The basic form of a syllogism, self-published online article, 3 April 2017, http://schmitztimo.wordpress.com