Major premise: All asteroids are made of rock.
Minor premise: Ceres is an asteroid.
Conclusion: Ceres is made of rock.

Major premise: All nontrivial Perl programs contain variables.
Minor premise: All versions of majordomo are nontrivial Perl programs.
Conclusion: All versions of majordomo contain variables.

Major premise: No cat enjoys the company of snakes.
Minor premise: Fluffy enjoys the company of snakes.
Conclusion: Fluffy is not a cat.

Major premise: I will not eat anything that smells like feet.
Minor premise: Some kinds of cheese smell like feet.
Conclusion: There are some kinds of cheese I will not eat.

A form of deduction logic that connects two ideas through a third one: "All men are mortal; Socrates is a man; Socrates is mortal" (this form is called Darii), "All coal is black: some rocks are not black; some rocks are not coal" (this form is called Baroko)
The syllogism is composed of major premise; minor premise: conclusion.
The premises and the conclusion are traditionally affirmations or negations of some property of a set of objects. If the set of object is qualified by "all" ("all dogs", "mammals", "all the black things"), the premise is called universal.
If the premise is existential ("some grey things", "there exist some animals that...") it is called particular.

Valid syllogisms are classified according to four figures:

  1. first figure: M P; S M: S P All men are mortal; Socrates is a man; Socrates is mortal (in this Darii M="all men", P="mortal entities" and S="Socrates")
  2. second figure: P M; S M; S P All cats are mammals; some animals are not mammals: some animals are not cats
  3. third figure: M P; M S; S P No snake lives in Sardinia; All snakes are reptiles; some reptiles do not live in Sardinia (this is a Felapton)
  4. fourth figure: P M; M S; S P Some bows are wood objects; all wood objects are organic: some bows are organic (this is a Dimaris)
What changes here is the position of M the middle term.

Syllogisms can be constructed in the four figures according to these rules:

  1. There are only three terms in a syllogism (by definition).
  2. The middle term is not in the conclusion (by definition).
  3. The quantity of a term cannot become greater in the conclusion
  4. The middle term must be universally quantified in at least one premise - you cannot deduct anything from particular observations.
  5. At least one premise must be affirmative.
  6. If one premise is negative, the conclusion is negative.
  7. If both premises are affirmative, the conclusion is affirmative.
  8. At least one premise must be universal.
  9. If one premise is particular, the conclusion is particular.

The syllogism appears in Aristotle's works, but it was formalized in the Middle Ages by the Scholastic philosophers according to an elegant scheme.
The five hexameter verses that follow contain the symbolic names (called moods) of all the different syllogistic figures:

“Barbara, Celarent, Darii, Ferioque, prioris.
Cesare, Camestres, Festino, Baroko, secundae.
Tertia, Darapti, Disamis, Datisi, Felapton,
Bokardo, Ferison, habet. Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The parts in italics are there just to make the verses work; they are not part of the names of the figures. The vowels mean the following:

  • A universal affirmative.
  • E universal negative.
  • I particular affirmative.
  • O particular negative.

For example, Darii is DARII, that's to say

universal affirmative + particular affermative = particular affirmative
All cats are mammals + Beppo is a cat = Beppo is a mammal

Another example, Bokardo is BOKARDO, that's to say

particular negative + universal affirmative = particular negative
some men are not white + all men are mammals; some mammals are not white

What follows is the detailed listing of all the valid syllogism. Notice that you can very well create invalid figures. For example IEI or IEA would be invalid ("Some foos are bars; no bars are quux: ??? "... no conclusion is allowed).
Notice also that the order of the parts inside each member of the syllogism is important; Darii differs from Datisi precisely in that respect.

Barbara all M is P; all S is M: all S is P
Celarent no M is P; all S is M: no S is P
Darii all M is P; some S is M: some S is P
Ferio no M is P; some S is M: some S is not P

Cesare no P is M; all S is M: no S is P
Camestres all P is m; no S is M: no S is P
Festino no P is M; some S is M: some S is not P
Fakofo all P is M; some s is not M: some S is not P
Baroko all P is M; some s is not M: some S is not P

Darapti all M is P; all M is S: some S is P
Disamis some M is P; all M is S: some S is P
Datisi all M is P; some M is S: some S is P
Felapton no M is P; all M is S: some S is not P
Dokamok some M is not P; all M is S: some S is not P
Bokardo some M is not P; all M is S: some S is not P
Ferison no M is P: some M is S: some S is not P

Bramantip all P is M; all M is S: some S is P
Camenes all P is M; no M is S: no S is P
Dimaris some P is M; all M is S: some S is P
Fesapo no P is M; all M is S: some S is not P
Fresison no P is M; some M is S: some S is not P

In greater and painful detail, notice that the initial of all the words shows to which of the first forms (Barbara, Celarent, Darii, Ferio) the syllogism should be reduced. For example, Bokardo (also written Bocardo) should be reduced to Barbara.

How does the reduction take place? This is where the other letters come into play.

  • S stands for simpliciter, "simply". This means "invert the subject and the predicate and don't touch the quantifier". Only E (universal negative) and I (particular positive) can be converted simpliciter.
    For example "No cats are reptiles" converts to "No reptiles are cats", and "Some cats are female" converts to "Some females are cats". Datisi converts to Darii simpliciter.
  • P means per accidens, "special case". Only A (universal positive) and E (universal negative) can be converted this way. For example "All cats are mammals" converts to "Some mammals are cats" (notice that we are going from "For all X that are cats, X is a mammal" to "There exists one mammal M, for which M is a cat") via a quantifier change.
    For example Felapton converts to Ferio per accidens (refer to the above table), by converting the universal affirmative to a particular affirmative in the second member.
  • M means "exchange the minor and major premise"
  • C is to per absurdum

There is an issue here with modern logic, about deriving particular truths from two universal truths. In particular consider Bramantip: "all P is M; all M is S: some S is P". For example, "All Romans are Italians; all Italians are Europeans; some Europeans are Romans". This is all good until nonexistent things enter the scene... for example All unicorns are furry; all furry things are cute; some cute things are unicorns appears to prove the existence of unicorns (since we know that cute things do exist).
Read about this in the excellent article "In defense of Bramantip" at http://www.friesian.com/syllog.htm. The idea is that you cannot really use empty sets as terms in a syllogism; and the syllogism has been around much longer than the idea of the empty set.

A good resource for playing around is the JavaScript Syllogistic Machine at http://home.swipnet.se/~w-33039/Syllog.machine.html.

For bonus points, try to work out which moods do Frater's example belong to.


taken in part from The Dictionary of Phrase and Fable (publ. 1894)

Syllogisms Made Easy

baffo's writeup does a very good job of explaining the intricacies of formal syllogisms, but I'd like to tackle the same topic in a way that is perhaps a bit more accessible to those not versed in formal logic. In my opinion, it is tragic that the study of logic is so steeped in symbols and math-like structure that those who need it the most (i.e. those that are not logically or mathematically inclined) are those least likely to be attracted to it. It is my feeling that most people could benefit from just a basic understand of some the common logical arguments and fallacies.

A good starting point for an understanding of logic is the syllogism, as it is both easy to comprehend and used often (in various forms) in everyday arguments. With this in mind, what follows is my attempt to explain the syllogism to somebody who has no previous concept of formal logic:

A syllogism is a very common argument (in the context of logic, the word argument means "line of reasoning" rather than "point of contention" as it often does in everyday speech). The classic syllogism has three parts: two premises and a conclusion drawn from the premises. A syllogism is a way to compare or contrast groups or categories of things.

The Universal Syllogism

The easiest example of a syllogism to follow is that of the universal syllogism. This is a type of argument that compares the three categories of things. An example:

All poodles are dogs.
All dogs are animals.
Therefore, all poodles are animals.

The example is clear-cut; most syllogisms that you'll come across are not. Consider the following hypothetical passage:

"We all know that geeks are much smarter than the rest of the population. And common sense tells us that smart people make more money. This is why geeks make so much money."

Here we have a syllogism, though it may not be obvious at first. Break it down into its components, though, and you get:

All geeks are smarter than the rest of the population.
Smart people make more money (than average).
Therefore, geeks make more money than the average person."

The two examples of syllogisms we've looked at follow a specific form. They are examples of a universal syllogism. What this means is that the two premises and the conclusion make claims about all the members of a group.

Something worth noting is that we can switch the order of our two premises without changing the meaning of the syllogism. In other words:

All dogs are animals.
All poodles are dogs.
Therefore, all poodles are animals.

Is just as deductively valid as our first example. The term deductively valid (which we'll shorten to just "valid") means that, if we accept the premises as true, the conclusion must also be true. If all dogs are animals and all poodles are dogs, then all poodles must be animals. We'll learn how to evaluate syllogisms (determine if they're valid or not) in a little bit.

One thing you cannot change in a universal syllogism is the order of the claims within the premises and the conclusion. "All animals are dogs" is a very different statement than "All dogs are animals." The first category of a claim is called the subject and the second category is called the predicate. In the claim "All dogs are animals," the subject category is "dogs" and the predicate category is "animals." It is important to remember these two terms and to not mix them up while evaluating syllogisms.

The Universal-to-Particular Syllogism

Let's take a look at another type of syllogism: the universal-to-particular syllogism. This type of syllogism has as its first premise a universal claim, then makes a claim about a limited number of the members of a group. The conclusion then draws some relation between the two premises. These are a bit more complicated than the universal syllogism but are probably more common. An example:

All dogs are animals.
Some dogs are poodles.
Therefore, some animals are poodles.

This is also a valid syllogism. Like the universal syllogism, it doesn't matter which order the premises occur in (the "Some dogs are poodles" premise could come before the "All dogs are animals" premise without changing the validity). There is a weird catch when dealing with particular claims, though. Consider our previous example, now slightly altered:

All dogs are animals.
Some poodles are dogs.
Therefore, some animals are poodles.

Did you spot the difference? We've flipped the order of the claims in the particular premise. While this may seem weird at first, we can switch the subject and predicate categories in a particular claim without changing its meaning. "Some snakes are cold-blooded animals" is the logical equivalent of "Some cold-blooded animals are snakes." This is obviously not true of a universal claim (remember "All snakes are animals" is not the same as "All animals are snakes")!

Because of this, syllogisms with particular claims get tricky to recognize and evaluate. We will discuss how to evaluate them in a moment. It is worth mentioning that there is no way to construct a valid syllogism that only has particular claims and the reason for this will become obvious when we learn how to evaluate them. Before we get to that, let's recap:

  • A syllogism consists of three parts: two premises (the first of which is traditionally called the "major premise" while the second is the "minor premise") and a conclusion.
  • A universal claim is a claim about all members of a group or category and usually starts with the word "All," though it can just as easily start with a word like "Every."
  • A particular claim is a claim about a particular set of members within a larger group or category and usually starts with a the word "Some" but can also start with words like "Most" or "Many."
  • A universal syllogism is a syllogism in which both premises make universal claims. A universal-to-particular syllogism is one which has a universal and a particular claim as premises.
  • Changing the order of premises in either type of syllogism does not affect the syllogism at all.
  • Changing the order of the subject and predicate categories in a universal claim does affect the statement. This is not true of particular claims.

Evaluating Universal Syllogisms

We understand the basic structure of two common syllogisms (there are more complex forms which will be dealt with later). How do we determine if a syllogism is valid or invalid?

The universal syllogism is easy to evaluate. All valid universal syllogisms follow the following form:

All A is B.
All B is C.
Therefore, All A is C.

The only tricky thing to remember is that we can switch the order of the two premises ("All B is C, All A is B, Therefore all A is C") without changing the validity of the syllogism. For the syllogism to be valid, just make sure that the predicate category in one of the premises is the subject category in the other and that the conclusion draws a relationship between the other two categories.

A common counterfeit universal syllogism follows this form:

All A is B.
All A is C.
Therefore, all B is C.

An example of this sort of syllogism:

All dogs are animals.
All dogs are furry.
Therefore, all animals are furry.

It's easy to see, using that example, how silly this reasoning is. Saying "All dogs are animals" does not imply that "All animals are dogs" and thus we cannot say that because all dogs are furry that all animals must be. For all we know, dogs could be the only furry animals. That dogs are a subset of a larger group (animals) does not imply anything about other members of that group.

This counterfeit can be harder to spot than you might think, especially if they're not clearly set up as syllogisms. Look at the following example:

"Obviously, all gun-owners are nutjobs who want to overthrow the government! Consider that everyone who's a Republican owns a gun. It becomes clear when you realize that Republicans secretly want to overthrow the government."

The reasoning is slightly more muddled, and the conclusion is stated first, but the argument is a syllogism:

All Republicans own guns.
All Republicans want to overthrow the government.
Therefore, all gun-owners want to overthrow the government.

Again, we see that just because the group "Republican" intersects with the group "gun-owner" it doesn't mean that all gun-owners are Republican. This can be very tricky to watch out for, especially on issues that you have strong personal feelings about (and as such are less likely to think critically about).

The easiest way to determine validity for a universal syllogism when you identify one in print or speech is to simplify the argument, write it clearly as a syllogism and ensure it follows the proper form. So long as the syllogism follows the correct form, it must be valid (remember that just because it is valid does not mean that the conclusion is true).

Evaluating Universal-to-Particular Syllogisms

The universal-to-particular syllogism can be a bit harder to evaluate than the universal syllogism. This is because the form is less clear. Generally speaking, the valid form of the syllogism looks like this:

All A are B.
Some A are C.
Therefore, some B is C.

The above is a perfectly valid argument. What's confusing is that the following is also valid:

All A are B.
Some C are A.
Therefore, some B is C.

Remember that we can switch the order of the subject and predicate categories in a particular claim (that's the second part of the universal-to-particular syllogism) without adjusting its meaning! This throws a monkey-wrench into our evaluation process: this adds a new layer of complexity that means you would have to memorize a bunch of forms to check for validity.

Fortunately, there's a way to simply evaluate these syllogisms without memorizing forms. Here's the key: for a universal-to-particular syllogism to be valid, the subject category of the universal claim must appear in the particular claim. IT CAN BE EITHER THE SUBJECT OR THE PREDICATE CATEGORY, SO LONG AS IT APPEARS.

This seems kind of weird, at first, but it makes sense once you try constructing a few universal-to-particular syllogisms and swap the subject and predicate in the particular claim. As an example:

All E2 users are smart.
Some E2 users are good-looking.
Therefore, some smart people are good-looking.

The above example (valid, though I'd question its soundness) is equivalent to saying the following:

All E2 users are smart.
Some good-looking people are E2 users.
Therefore, some smart people are good-looking.

The two examples differ only in their minor premise, and all we've done is to switch the subject with the predicate. But they still both mean the same thing. If this is unclear, try building a few example syllogisms of your own (the more outrageous the better) and I suspect you will understand what's going on.

Here is a common counterfeit universal-to-particular syllogism:

All A is B.
Some B is C.
Therefore, Some A is C.

You should recognize immediately why this is invalid: the subject category in the universal premise is not present in the particular premise. An example may be helpful:

All E2 users are smart.
Some smart people are astronauts.
Therefore, some E2 users are astronauts.

While there may be an astronaut on E2 (you never know), the conclusion certainly doesn't follow from the premises. Again, I invite the reader to try and construct a valid universal-to-particular syllogism that doesn't have the subject category from the universal premise somewhere in the particular premise. It just doesn't work.

So evaluating these types of syllogisms is pretty simple: merely check to make sure the subject category of the universal claim is present in the particular claim and it should be clear whether or not the syllogism is valid. A final example, first how you might read the argument in the real world, then in a clear syllogism form:

"Every player in the NFL is rich! A lot of those rich people got that way by screwing over the poor. Obviously, some of those pro-football players have been ripping off poor people."

In the simple form:

All pro-football players are rich.
Some rich people steal from poor people.
Therefore, some pro-football players steal from poor people.

Odds, Ends & Exercises

This writeup is growing quite lengthy, so I will only briefly touch on a few other points:

There are quite a few other types of syllogisms you should be aware of. There is the hypothetical syllogism which is valid in this form:

If A, then B.
If B, then C.
Therefore, if A, then C.

You can add multiple premises, so long as they always follow each other ("If A, then B. If B, then C. If C, then D. If D, then E. Therefore, if A then E."). And of course you can switch the order of the premises (though that makes things quite messy).

Another very common syllogism is the disjunctive syllogism which works as follows:

Either A or B.
Not B.
Therefore, A.

The validity of this syllogism should be self-evident. Those still interested in logic would do well to read up on conditional statements, particularly the forms Modus Ponens and Modus Tollens. Evaluating simple conditionals is very similar to evaluating syllogisms in that both require only a basic understanding of their forms.

The hard part about logic is recognizing these forms when they appear in real life. It is uncommon that, upon hearing a bad argument, someone says, "Ah, but that's a counterfeit universal syllogism!" or, "That can't be right -- you just denied the antecedent!"

With that in mind, I leave the still-interested reader with a few exercises in recognizing and then evaluating syllogisms. I am confident that this writeup contains the tools needed to solve the exercises without too much difficulty. Answers are at the bottom of the writeup, but I strongly suggest you work them out for yourself -- this often reveals flaws in your reasoning that you wouldn't catch if you just looked at solutions.

Exercises: Determine what sort of syllogism each argument is. Write it out in a simplified form. Determine if each argument is valid or invalid (remember that an argument can be valid even if its conclusion is untrue). Some of them may have words, phrases or sentences that aren't directly related to the argument -- this isn't sneaky, it's realistic!

  1. "You say that sappy movies are always bad? I can prove that isn't true. For instance, Steven Spielberg always makes great movies! But some of his movies are sappy. So, some sappy movies are also great."
  2. "Since smoking is so bad for you, anybody who smokes must be unhealthy. Also, unhealthy people don't live as long as normal people. That means all smokers are going to die sooner than the average person!"
  3. " History has shown that wars are always expensive for the country that starts them. Not only that, but wars are bad. It's clear that anything expensive for a country is bad."
  4. "I hate most computers. They're all complicated! Since a lot of complicated things are a pain to use, most computers are a pain to use."


Source:

Most of my information for this WU is pretty basic logic (and was introduced to me in a beginning logic class at Palomar College). I did, however, get several of the terms from "Open Minds and Everyday Reasoning," a tremendous introductory logic text by Zachary Seech (who just so happened to be the professor for the class).

Answers for the exercises below (if you can't read them, try pasting them into a text editor such as Notepad):

  1. Valid universal-to-particular syllogism
  2. Valid universal syllogism
  3. Invalid counterfeit universal syllogism
  4. Invalid counterfeit universal-to-particular syllogism

Syl"lo*gism (?), n. [OE. silogisme, OF. silogime, sillogisme, F. syllogisme, L. syllogismus, Gr. syllogismo`s a reckoning all together, a reasoning, syllogism, fr. syllogi`zesqai to reckon all together, to bring at once before the mind, to infer, conclude; sy`n with, together + logi`zesqai to reckon, to conclude by reasoning. See Syn-, and Logistic, Logic.] Logic

The regular logical form of every argument, consisting of three propositions, of which the first two are called the premises, and the last, the conclusion. The conclusion necessarily follows from the premises; so that, if these are true, the conclusion must be true, and the argument amounts to demonstration

; as in the following example:

Every virtue is laudable; Kindness is a virtue; Therefore kindness is laudable.

These propositions are denominated respectively the major premise, the minor premise, and the conclusion.

⇒ If the premises are not true and the syllogism is regular, the reasoning is valid, and the conclusion, whether true or false, is correctly derived.

 

© Webster 1913.

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