The Structure of the Syllogism
A Syllogism is valid when its conclusion follows from its premises. The syllogism: “All men are mortal, Socrates is a man, therefore Socrates is mortal” is valid because the conclusion follows from its premises. However, the following syllogism is valid but has an untrue conclusion:
All cats are math geniuses
Jimmy is a cat
Therefore, Jimmy is a math genius
The conclusion follows from the premises, making it valid, but it is untrue since one of its premises is false.
On the other hand, a syllogism can come to the correct conclusion, but can be completely invalid!
All cats are carnivores
Jimmy is a cat
Therefore, Jimmy is tw has five claws on each paw
It may be that Jimmy has five claws on each of his paws, but the conclusion ultimately does not follow from the premises — it is a non sequitur.
If all the premises of a syllogism are true, and the syllogism is correctly reasoned, we say that the syllogism is sound.
But how do we reason through a syllogism correctly?
Deductive and Inductive Reasoning
Deductive Reasoning is when we start with a general principle that we know to be true and use that to find some premise that follows from it. In the famous Socrates example, our principle is that “all men are mortal”, and then we apply it to the case of Socrates. By its very nature of moving from a principle to something that follows from this principle, the conclusions of deductive syllogisms are likely to have a smaller extension than the principle (SL). Deductive reasoning is certain, and not based on probabilities.
Inductive Reasoning is when one takes several data points and uses that to determine a general principle. A famous example that both Jolivet and Kreeft use is as follows:
All heat is tw expands Copper, Bronze, and Steel
All Copper, Bronze, and Steel are metals
Therefore, all heat is tw expands metals
The nature of induction is to go from the particular extension to the universal. Note that induction is probabilistic, since, unless the entire universe is known, a metal may be found that shrinks when exposed to heat. A funny example is, in Latin, an expression arose to the effect that a black swan did not exist. It was perfectly reasonable to assume so, as all the swans found in Europe had not been black. However, more than 1500 years later, Dutch explorers found a black swan in Western Australia. (Wikipedia, Black Swan Event) This is all to say that inductive reasoning is probabilistic and will be dealt separately. For now, we will focus on deduction.
Parts of the Syllogism
A proposition contains three propositions and three distinct terms: the major term (T), the minor term (t), and the middle term (M). (CdF)
The major term is the term with the greatest extension, the minor term is the term with the smallest extension, and the middle term is between both in extension (CdF). In a syllogism, the middle term connects the major and minor terms, allowing what is true about the middle term in relation to the major term to apply to the minor term, which is contained within it. This should sound familiar to the dictum de omni. The dictum de nullo also applies to the syllogism, since if the minor term is contained within the middle term, and something is negated about the middle term, we can say that that same thing is negated about the minor term
We can label the terms in the following syllogism:
M T
All men are mortal
t M
Socrates is a man
t T
Socrates is mortal
We can see here that the middle term should never be in the conclusion of the proposition (CdF). This is because the proposition at the end is supposed to be relating the minor term to major term.
Rules of the Syllogism
In order to determine the validity of a syllogism, Aristotle developed six rules.
“A syllogism must have three and only three terms.” (SL)
It is obvious why a syllogism should have three terms. One of the ways that this rule is violated is through the use of ambiguous terms.
For instance:
All dogs are twi able to bark
Some constellations are dogs
Therefore, some constellations can bark (CdF)
Superficially, it may seem that there are three terms present in the syllogism, but there are four, since the term “dogs” is used ambiguously. In the first proposition, “dogs” means the animal, but in the second, “dogs” means a representation of the animal.
“A syllogism must have three and only three propositions.” (SL)
This rule flows from the essential structure of the syllogism. Any argument that violates this rule is not a syllogism at all. (SL)
There are other, valid forms of argumentation that have more or less propositions, but we have not studied them yet.
“The middle term must be distributed at least one. The violation of this rule is called The Fallacy of the Middle Term.” (SL)
If you recall, if the middle term is not “distributed” (if it is not used in its whole extension — we need the knowledge of its whole extension for the syllogism to work), a relationship is not established between the two.
Kreeft gives a good example:
[T] [M]
All Communists insist on the abolition of private property
[t] [M]
This candidate insists on the abolition of private property
[t] [T]
Therefore this candidate must be a Communist. (SL)
The syllogism is invalid since it is possible that the candidate could not be a communist but be of some other political persuasion (anarchist, esoteric monk, etc) and believe in the abolition of private property. (SL)
Since the middle term is never distributed over the major and minor terms, they become like islands and therefore have no connection to each other, so that the dictum de omni cannot be applied.
“No term that is undistributed in the premise may be distributed in the conclusion. The violation of this rule is called The Fallacy of Illicit Minor or The Fallacy of Illicit Major, depending on whether it is the minor term or the major term that contains the fallacy.”
This makes sense, because if we do not have the knowledge of the whole extension of a term, it does not make sense to make judgments about the whole extension of the term.
Kreeft gives an example:
Compassion is a virtue.
Justice is not compassion.
Therefore justice is not a virtue. (SL)
This is plainly wrong, because justice may not be identical to compassion but still may be a virtue. More strictly, in the first proposition, “virtue” is taken in only part of its extension— compassion is one of the virtues. The conclusion, however, is a universal negative proposition, meaning that both terms are taken in universal extension. But “virtue” was only taken in part of its extension previously, so the syllogism falls apart.
“No syllogism can have two negative premises. The fallacy here is called simply the fallacy of Two Negative Premises.” (SL)
When both the major and minor terms are related negatively to the middle term, we do not know how they are related to each other. Remember, it is not necessarily true that “the enemy of my enemy is my friend” (SL)
In this case, the enemy of your enemy could also be your enemy. Kreeft gives two excellent examples, each with the same logical form, one reaching a false conclusion and the other reaching a true one:
Odd numbers are not even numbers. Birds are not fish.
Three is not an even number. Humans are not fish.
Therefore three is not an odd number. Therefore humans are not birds. (SL)
“If one premise is negative, the conclusion must be negative; and if the conclusion is negative, one premise must be negative.” (SL)
Corollary 1: “No syllogism may have two particular premises. … Any syllogism that violates this rule will also violate Rule 3 or 4.” (SL)
This rule makes sense if you think about what the structure of a syllogism with two particular premises could look like:
Some S is A
Some A is P
Therefore, some S is P
This is invalid because “some” means that S may not be included in P’s extension. Kreeft gives an example:
Some jewels are green.
And some green things are alive.
Therefore some jewels are alive. (SL)
This also applies to syllogisms that contain only negative particular propositions.
Corollary 2: “If a syllogism has a particular premise, it must have a particular conclusion. … The ‘weakness’ of particularity in the premise must be reflected in the conclusion. The conclusion always follows the weaker premise: particular or negative.” (SL)
Kreeft gives an interesting example:
All dogs are animals.
Some dogs are poodles.
Therefore all poodles are animals. (SL)
While the conclusion may accidentally be true, the logical form is invalid, since the category “poodles”, if we know nothing about what a poodle is, may include things that are not dogs.
Mood and Figure of a Syllogism
The mood of a syllogism is “the quality and quantity of its three propositions” (SL) For instance, AEE represents a syllogism with a universal affirmative major premise, a universal negative minor premise, and a universal negative conclusion. When written in this manner, the major premise is always first, then the minor premise, then the conclusion, though in an actual syllogism the order of the premises does not matter.
The figure of a syllogism is the placement of the middle term. There are four possibilities: the middle term can be
(1) the subject of the major premise and the predicate of the minor premise (= the “first figure”)
(2) the predicate of both premises (= the “second figure”)
(3) the subject of both premises (= the “third figure”)
(4) the predicate of the major premise and the subject of the minor premise (= the “fourth figure”)
Medieval logicians identified nineteen valid moods of a syllogism (CdF) and organized them based on figure. They invented the “Barbara Celarent” mnemonic in order to memorize all of them. Each of the vowels represents a proposition in a mood, and each line is a figure:
[Fig. 1] Barbara, Celarant Darii, Ferio (AAA, EAE, AII, EIO)
[ Fig. 2] Camestres, Cesare, Baroko, Festino (AEE, EAE, AOO, EIO)
[Fig. 3] Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison (AAI, IAI AII, EAO, OAO, EIO)
[Fig. 4] Bramantip, Camenes, Dimaris, Fesapo, Fresison (AAI, AEE, IAI, EAO, EIO) (SL)
Now you have all of the basic tools to start analyzing arguments! In order to grow your logic skills, it is useful (and interesting) to logically analyze the arguments in philosophical works. Some recommendations are Plato’s Gorgias and Boethius’ Consolation of Philosophy.