[ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

The purpose of this collegium is to establish a group for those interested in ancient philosophy and a place where philosophical discussion and study may take place. Join at: http://romanrepublic.org/civitas/joint_ ... sophiae/42

Moderators: Marca Marcia, Gaius Flavius Aetius, Paullus Aemilius Gallus, Aula Flavia Philippa

[ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

Postby Gaius Florius Lupus » Sun Jun 25, 2017 5:07 pm

Salvete, philosophi!

Today I would like to talk about a very dry and boring subject, which is nevertheless essential for philosophy and therefore cannot be simply skipped. It is about syllogistic logic.
What I am trying to demonstrate here, is that logic is a science just as precise as mathematics, which is actually only a sub-category of logic. It has exact rules that have been elaborated by classical philosophers, first of all Aristotle.
In Stoicism logic is one part of the tripartite division of philosophy as described by Zeno and Chrysippus:
  1. Logic
  2. Physics
  3. Ethics

The Stoic Egg
stoic-egg-diagram.jpg
stoic-egg-diagram.jpg (9.64 KiB) Viewed 2639 times

The egg is a typical metaphor used by the Stoics. Logic is the shell, ethics the egg white and physics the yolk. Some philosophers had the order changed and ethics as the yolk. However all agreed that logic had to be passed first in order to get to the nucleus of philosophy.

Logic itself has at least two distinct approaches:
  • Deductive reasoning - conclusions about particular instances from general principles
  • Inductive reasoning - establishment of general principles from observation of particular instances

image005.gif
image005.gif (3.28 KiB) Viewed 2639 times


Both methods of reasoning work in a circular way. By inductive reasoning we establish general principles based on particular observations. This is what the scientific method is based on.Its conclusions are not absolute and they are based on probability.
From these general principles we can then make predictions about particular instances using deductive reasoning. The deductive method is more precise, but the truth value of its conclusions depends on its premises, the general principles.
If the predictions made from this conclusions are correct, then this strengthens the general principle. This confirmation is again made by inductive reasoning.
By each passing through the circle of deduction and induction our established general principles become better.

Deduction is what most people associate with logic, induction on the on the other hand is what the scientific method is based on. It is an empirical science. Deduction however is not empirical. Its methods are always valid. This is what we call a priori valid, valid even prior to empiric experience. The opposite is a posteriori, valid only after (post) empirical observation.

Syllogistic Logic

Aristotle established the first systematic approach to deductive reasoning in his Organon. This is what we call Syllogistic Logic or Term Logic or simply Aristotelian Logic.
The following might be pretty boring, and it would be unrealistic to expect that anybody learns all these formulas. However it is important to understand how they work and where to look them up. In the Middle Ages these formulas, when logic was a subject in schools and universities, these tables and formulas were indeed learned by the students, but for our purpose it is enough that you take the message with you that there is a precise scientific way of syllogistic reasoning, that it is universally valid and therefore agreeable to everyone who is willing to use the logos as his guide in life.

Let us go in medias res.

A syllogism according to Aristotle consists of three and exactly three elements:
  1. The Major Premise - propositio
  2. The Minor Premise - adsumptio
  3. The Conclusion
Each of this proposition has two terms, a subject and a predicate. We see how our grammar exactly reflects the logic principles. The entire syllogism has three terms.
  1. Major term (P)
  2. Middle term (M)
  3. Minor term (S)
The Middle term is shared by both premises. The major premise also includes the major term, the minor premise includes the minor term. And the conclusion has the minor term as subject and the major term as predicate.

Exempli gratia:

Major Premise: All men are mortal.
Minor Premise: Aristotle is a man.
Conclusion: Aristotle is mortal.

Major term (P): mortal
Middle term (M): man
Minor term: (S): Aristotle

In a proposition a term is called distributed, if it refers to the entire category. It is called undistributed, if it refers to only some members of the category.
Exempli gratia in the proposition "All men are mortal" men is distributed, because the proposition refers to all men. Mortal is undistributed, because those mortals who are men are not all mortals that exist. Animals are mortals too, but not men.
If a term is distributed it is indicated by the universal quantifier Ɐ (reversed A). If a term is undistributed, it is indicated by the existential quantifier Ǝ (reversed E).

The following symbols will be used in all the formulas here:
Affirmation: ∈ "is"/"is element of"
Negation:∉ "is not"/"is not element of"
Universal Quantifier: Ɐ "all"
Existential Quantifier: Ǝ "some"
Attachments
logica syllogistica.pdf
(340.99 KiB) Downloaded 92 times
User avatar
Gaius Florius Lupus
 
Posts: 590
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna

Re: [ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

Postby Gaius Florius Lupus » Sun Jun 25, 2017 6:14 pm

There are four types of syllogisms (moods), characterized by the letters A E I O:

A:
All S are P.
Ɐ S ∈ P
Affirmation - Universal Affirmative
Example: All men are mortal.
Subject - distributed; predicate - undistributed

E:
No S are P.
Ɐ S ∉ P
Exclusion - Universal Negative
Example: No men are immortal.
Subject – distributed; predicate – distributed

I:
Some S are P.
Ǝ S ∈ P
Inclusion - Particular Affirmative
Example: Some men are philosophers.
Subject – undistributed; predicate – undistributed

O:
Some S are not P
Ǝ S ∉ P
Omission - Particular Negative
Example: Some men are not philosophers.
Subject – undistributed; predicate – distributed

These syllogisms can be arranged in any of these four figures:
M - Middle Term Variable
S - Subject of the Conclusion - Minor Term Variable
P - Predicate of the Conclusion - Major Term Variable

figures.jpg
figures.jpg (8.25 KiB) Viewed 2637 times


Valid Categorical Schemata

There are a total of 256 possible syllogisms. Only those categorical schemata that fulfill the following rules are valid.

Rules of Validity
  1. A syllogism must contain exactly three terms.
  2. The Middle Term must be distributed in at least one premise.
  3. No term can be distributed in the conclusion, which is not distributed in the premises.
  4. No syllogism can have two negative premises.
  5. If either premise is negative, the conclusion must be negative.

Valid Syllogisms

syllogisms.jpg
syllogisms.jpg (37.45 KiB) Viewed 2637 times


As you can see the syllogisms have Latin names. So students of logic in the Middle Ages formed a rhyme to memorize these valid syllogisms:

BARBARA, CELARENT, DARII, FERIOque prioris;
CESARE, CAMESTRES, FESTINO, BAROCO secundae;
Tertia DARAPTI, DISAMIS, DATISI, FELAPTON, BOCARDO, FERISON habet;
Quarta insuper addit BRAMANTIP, CAMENES, DIMARIS, FESAPO, FRESISON.

Each vowel in the words represents the mood of the propositions.
Exempli gatia: The syllogism CELARENT means the major premise is a universal negation (E), the minor premise a universal affirmation (A) and the conclusion a universal negation (E).
Major Premise: All M are not P or No M are P
Minor Premise: All S are M
Conclusion: All S are not P or No S is P.

Some syllogisms are only conditionally valid:
The minor term must have elements:
Barbari AAI-1
Celaront EAO-1
Cesaro EAO-2
Camestros AEO-2
Calemos AEO-4

The middle term must have elements:
Darapti AAI-3
Felapton EAO-3
Fesapo EAO-4

The major term must have elements:
Bramantip AAI-4

All syllogism that are not included in the table above are logical fallacies (parasyllogisms). This is called non sequitur, it means that a conclusion does not follow from the premises.

The following formal syllogistic fallacies result from non-adherence to the rules of validity above:

Ambiguous Middle Term (equivocation) - Common ambiguity in syllogisms in which the middle term can have two meanings, which are equivocated resulting in a quaternio terminorum fallacy.

Existential Fallacy (existential instantiation) - The assumption that a class has members (existential import), if it is not necessarily the case. Example AAI-3 syllogism: All M ∈ P; All M ∈ S; Some S ∈ P. Incorrect, if M has no member. A particular conclusion implies existence, which is not implied in a universal premise.

Fallacy of Exclusive Premises - A categorical syllogism that is invalid because both of its premises are negative.

Fallacy of the Undistributed Middle - A categorical syllogism where the middle term is not distributed in at least one of the premises.

Illicit Affirmative
(negative conclusion from affirmative premises) - A categorical syllogism with a negative conclusion but affirmative premises.

Illicit Major - A categorical syllogism that is invalid because its major term is not distributed in the major premise but distributed in the conclusion.

Illicit Minor - A categorical syllogism that is invalid because its minor term is not distributed in the minor premise but distributed in the conclusion.

Illicit Negative (affirmative conclusion from a negative premise) - A categorical syllogism with a positive conclusion, but at least one negative premise.

Quaternio Terminorum
(fallacy of four terms) - A categorical syllogism that has four terms.


Now after all this boring stuff, we are going to put these rules of syllogistic logic to some use.
Today's politics provides lots of examples for logical fallacies (non sequitur).
Let us examine one of them, which is typical for our times.

Major Premise: Many Trump voters have only a lower education.
Minor Premise: Peter voted for Trump.
Conclusion: Peter has a lower education.

This is apparently a non sequitur. The conclusion might be true, but it does not logically follow from the premises.
Our major term is "people with lower education". Our minor term is "Peter". And our middle term is "Trump voters".

So let us re-write the syllogism:
Ǝ "Trump voters" (M) are "people with lower education" (P) - Mood I
Ɐ "Peter" (S) is "Trump voter" (M) - Mood A
Therefore: "Peter" (S) is "person with lower education" (P). - Mood A

We can see that it is a Figure III syllogism. The syllogism is therefore type IAA-1.
As we can see above, IAA-1 is not in the list of valid syllogisms.

So which rule of validity was violated?
Answer: The middle term "Trump voters" was nor distributed in either premise. In the major premise there was only a reference to "some Trump voters" 8i.e. undistributed), and in the minor premise it was the predicate of a universal affirmation, which is also undistributed.
The corresponding fallacy is called Fallacy of the Undistributed Middle (Non Distributio Medii).

We have seen today that syllogistic logic is an exact science, and it has applications in today's life.
This is the message that Aristotle has for us today. And it is just as relevant and important today, as it was in his time.
It would be nice, if this post has helped to familiarize some of us with the technical terms of syllogistic logic.

Keywords: Deduction; term logic; Aristotelian logic; syllogism; parasyllogism; non sequitur, minor and major premise; minor, major and middle term, distributed and undistributed terms, quantifiers; universal and particular affirmations; universal and particular negations; moods; figures

If anybody thinks, he has found an exception to syllogistic logic, a case where Aristotle's rules do not apply, it would be very interesting to discuss it in this thread.

Valete!
User avatar
Gaius Florius Lupus
 
Posts: 590
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna

Re: [ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

Postby Gaius Curtius Philo » Sun Jun 25, 2017 9:25 pm

Salve amice!

Very nice. I'll have to write all this down for future reference. I shall try and use each form in different examples, including the fallacies, to see if I have properly grasped the details of it.

I think this is a fascinating subject and one of the most important and essential of them all, because it is the ground work through which all Natural Truth can be unlocked.
"Ignis aurum probat" - Seneca
C. Curtius L. f. Vot. Philo Aurelianus
User avatar
Gaius Curtius Philo
 
Posts: 1591
Joined: Sun Feb 14, 2016 3:56 pm
Location: Praia Grande, São Paulo, Brazil

Re: [ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

Postby Gaius Florius Lupus » Mon Jun 26, 2017 9:39 am

Salve amice,

One of the attachments, the PDF file is already a summary. It is in Latin, but due to the heavy use of Latin technical terms in logic even in English texts, its meaning is obvious.

Some post-classical and even classical philosophers have come up with pradoxa that seem to contradict Aristotle's rules, but they were not convincing. The existential fallacies were originally not considered by Aristotle, but as described above, there are syllogisms, which are only conditionally valid.
Exempli gratia:
Major premise: All unicorns have only one horn.
Minor premise: All unicorns are mammals.
Conclusion: Some mammals have only one horn.
This is a Darapti (AAI-3) syllogism. The problem is that the middle term (unicorns) has no members, i.e. unicorns do not exist. This leads to a false conclusion.
One might add a sixth rule of validity:
6.A particular conclusion cannot have two universal premises.
The reason is that a particular proposition implies existence, a universal one does not.
If this is taken into consideration, it is a consistent system.

Vale!
User avatar
Gaius Florius Lupus
 
Posts: 590
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna

Re: [ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

Postby Gaius Curtius Philo » Mon Jun 26, 2017 11:59 am

How does one express uniqueness in this formula? As in, for example:

ONLY humans are rational
Pandas are not rational
Therefore pandas are not humans.

I came with that phrase but noticed there was no way to express the concept of Only in your text.
"Ignis aurum probat" - Seneca
C. Curtius L. f. Vot. Philo Aurelianus
User avatar
Gaius Curtius Philo
 
Posts: 1591
Joined: Sun Feb 14, 2016 3:56 pm
Location: Praia Grande, São Paulo, Brazil

Re: [ΛΟΓΟΣ] De Logica Syllogistica (Aristotelica)

Postby Gaius Florius Lupus » Tue Jun 27, 2017 1:59 pm

Syllogistic logic deals with categories, so Aristotle does not need the concept of "only". The predicate of a proposition means that the subject category is a subgroup of the predicate category. S is an element/subcategory of P. If only S is P, then S and P would be identical.
Only humans are rational would mean that the category of humans would be identical with the category of rational beings.
We will find this concept of equivalence in the Propositional Logic of the Stoics, which will be my next topic. Propositional logic as developed by Chrysippus might look more familiar for the modern reader.
The concept of uniqueness is indeed used in other modern systems and means, a term has only one member. It is indicated by the quantifier Ǝ! (reverse E with exclamation mark), but has no use in Aristotelian logic.
User avatar
Gaius Florius Lupus
 
Posts: 590
Joined: Tue Feb 16, 2016 11:33 am
Location: Africa Magna


Return to Collegium Philosophicum

cron