The Ides of Science II: The Building Blocks of Science – Syllogistic Logic

We have probably heard those jokes that are framed around the simple logical arguments. One of my favorites is from the brilliant BBC comedy series Yes, Prime Minister.

 

It goes as follows:

Sir Arnold: It’s the old logical fallacy. All cats have four legs. My dog has four legs…

Sir Humphrey: Therefore my dog is a cat.

And the audience laughs at the absurdity of the proposition.

 

Have you ever wondered why such a formulation is a fallacy? On its face it appears to follow a reasonable process of logical deduction, right? Could this be cited as a reason for a failure of logic? Could logic present failures? If so, what good is it to us?

 

I am reminded of a scene from Star Trek Voyager (Season 1: Prime Factors).

 

The relevant quotes are as follows:

Janeway: You can use logic to justify almost anything. That’s its power. And its flaw.

Tuvok: My logic was not in error. But I was.

 

The Tuvok character is correct here. Logic is not in error when it is correctly applied. When it is incorrectly applied we call it a logical fallacy. Logical fallacies are dangerous things and it is important to learn to catch them early and often. And also, I might add, to not let those who commit them go unchecked.

 

The application of logical fallacies is a common debating trick, often applied when one is poorly prepared for a particular argument. I have noticed that scientists can fall into the logical fallacy traps in their publications and in debates and discussions. Often, those applying the fallacy are not even aware that they are doing so. This might be because few people, including scientists, are trained in epistemology or in the foundations of logic, and yet it is from these foundations that the building blocks of philosophy, and ultimately science, emerge.

 

I would like to discuss the logical fallacies that I have encountered throughout my career as a scientist, but before I do so I think it would be appropriate to get us moving down the correct line of thought so that, hopefully, the fallacies would be easier to see when they appear. So, for this edition of the Ides of Science, I would like to discuss the building blocks of science – syllogistic logic.

 

The Framework of Syllogistic Logic

Yes Prime Minister’s Sir Arnold was correct when he said that it is an old logical fallacy; syllogism goes back to the time of Aristotle. In its basic form, a syllogism assumes the structure of a general statement, called the major premise, accompanied by a specific statement, called the minor premise, from which a logical conclusion is deduced. Let’s stick with the one above:

 

Major Premise: All cats have four legs.

Minor Premise: My dog has four legs.

Conclusion: My dog is a cat.

 

The structure contains three terms: “cats”, “my dog” and “four legs”, and each premise has a term that is in common with the conclusion. We would call “cats” the major term in this structure, since it is within the major premise, and “my dog” would be the minor term for the obvious reason. The term that they have in common, called the middle term, would in this case be “four legs”. This is actually an incorrectly structured syllogism known as the undistributed middle, but we’ll get to that.

 

Types of Syllogisms

We might be able to see that a three-line structure like the one given by Sir Arnold could take many forms. For the major premise we could, for example, replace the “All” with “Some” or “No”, or we could replace “have” with “do not have”. We can therefore arrive at a logical conclusion from a variety of pathways: 256 of them to be precise. Of these, only 24 of them are considered valid syllogisms. The others can be called syllogistic fallacies. It need not be said, but the statements presented in the syllogism need to be true.

 

Sticking to the terms used by Aristotle, premises can take the following forms:

(1) All A is B;

(2) No A is B;

(3) Some A is B;

(4) Some A is not B.

 

We can see that, assuming that A and B can be either the major, minor or middle term and that their order is not interchangeable, we can construct 256 different combinations (which will need a third term, C, worked into the three-line structure as well). However, the vast majority of them do not lead to a valid conclusion.

 

Let’s go through some examples.

Major Premise: All A are B.

Minor Premise: All C are A.

Conclusion: All C are B.

This syllogism uses Form #1 in the major and minor premise and is valid, so long as the two premises are universal.

 

Let’s apply Form #2 to both premises:

Major Premise: No A are B.

Minor Premise: No C are A.

Conclusion: No C are B.

Another valid syllogism, once again so long as the premises are universal.

 

Can we mix them up?

Major Premise: All A are B.

Minor Premise: Some C are A.

Conclusion: Some C are B.

This is valid, whereas:

 

Major Premise: All A are B.

Minor Premise: Some C are A.

Conclusion: All C are B.

is not. This is invalid because the minor premise implies that there are some C that may not be A, and since the major premise says that all A are B, then we can conclude that some C may not be B either.

 

Another example is the following:

Major Premise: No A are B.

Minor Premise: Some C are not A.

Conclusion: Some C are B.

This one doesn’t work because the minor premise does not require any C to be B and all we have from the major premise is that none of the A are B. In other words, the opposite conclusion that no C are B can also be drawn from these premises.

 

Going through each of the 256 combinations with the same logical processes we find that there are only 24 valid syllogisms. These 24 and some good examples of each can be found on the Wikipedia entry.

 

Where is Sir Arnold’s Fallacy?

Let’s revisit the syllogism presented by Sir Arnold above:

All cats have four legs.

My dog has four legs.

Therefore my dog is a cat.

 

Let’s put this into syllogistic form:

Major Premise: All A (cats) are B (things with four legs).

Minor Premise: Some C (dogs) are B (things with four legs).

Conclusion: Some C (dogs) are A (cats).

 

Note that we have broadened the “my dog” category to be “dogs” in general. The conclusion, then, is that some dogs are cats.

 

At first glance, this appears to be logically sound. In syllogistic terminology, it appears to follow the Darii (AII) form and would therefore be a valid syllogism. There is, however, a small difference which is most important. We have replaced the middle term with the major term in the minor premise.

 

This is correct:

All A are B.

Some C are A.

Therefore some C are B.

 

This is not:

All A are B.

Some C are B.

Therefore some C are A.

 

Notice how the A has been replaced with B in the minor premise? This particular fallacy is known as the undistributed middle.

 

So let’s reconstruct Sir Arnold’s logical form with one that correctly follows the Darii syllogistic structure. It would need to be worded as follows:

All cats have four legs.

My dog is a cat.

Therefore my dog has four legs.

 

We can see how the only way for this to work is for our original conclusion to be the minor premise. We need go no further since that statement is not true, even though the conclusion drawn from this syllogism happens to be true.

 

Why is the Order of Things So Important?

It all comes down to groups and subgroups. To form a logically valid syllogism it is important to keep track of your subsets. It is perfectly valid for dogs and cats to be subgroups within a larger group called “things with four legs”, yet still not contain subsets of each other. It is also valid for two larger groups to contain subgroups with four legs and not overlap each other. For example, cats are a subset of a larger carnivore group that has four legs, and dogs are a subset of a larger omnivore group that has four legs, but that need not mean that cats and dogs are the same thing. The phrase “All cats have four legs” is NOT the same thing as “All things with four legs are cats.”

Building Blocks of Science - Syllogistic Logic: Groups and subgroups
Syllogistic logic considers groups and subgroups. This demonstrates how one statement does not imply its inverse.

What is the Point of All of This?

I know it sounds a little silly to spend all of this time describing what should be instinctively obvious, but it is remarkable how the careless usage of language can quickly lead down some treacherous paths. I have shown, using the rules of syllogistic logic, how something as simple as a misplaced predicate term can lead to the “logical” conclusion that my dog is actually a cat. This kind of thing occurs more commonly and more subtly than you might think. Here are some examples of syllogistic fallacies that I have encountered in the real and fictional worlds:

 

* “All animals are equal, but some animals are more equal than others” from George Orwell’s Animal Farm. Implied conclusion: “Some animals are more important than others.”

* Arguing with a relative: “It is required by law for every citizen of Switzerland to carry a firearm. Switzerland has the lowest crime rate in Europe.” Implied conclusion: “Switzerland has a low crime rate because its citizens all carry firearms.”

* Similar argument from (then candidate for US president now the US Secretary of Housing) Ben Carson at a GOP Town Hall: “We have had guns for years, and we have been free for years. There is a correlation.” Implied conclusion: “We have been free because we’ve had guns.”

* “EarthGov has promised a job to anyone that wants one. So, if someone doesn’t have a job, they must not want one.” Julie Musante talking with John Sheridan in the TV series Babylon 5.

* Mistaking the statement of “I do not believe that God exists” with “I believe that God does not exist”. A common hurdle in discussions between theists and atheists.

* “Small government is a good thing. If bad things happen because of small government, it must be because the government is not small enough.” Paraphrasing the general mantra from free market capitalists.

* “If you plead guilty you are guilty.” Favourite line by Judge Judy. The implied conclusion is that there must be no other reason for why an individual would plead guilty in court.

* Footballer Mario Gomez once forgot to sing to the national anthem in the youth team, he scored a goal in that match and hasn’t sung along to the anthem since. See other footballer’s superstitions here.

 

So there we are. We’ve all been using syllogistic logic all the time but many of us may not thought about it in that way. These are the building blocks of basic logic and follow a very precise set of rules. Knowing these rules should allow us to be sensitive to those who break them to falsely prove a point or make an argument.

 

Know your syllogisms, know their fallacies, and learn to recognize their application. And of course, be vigilant in avoiding the syllogistic traps, subtle though they may be.

 

For those interested in learning more The Wikipedia entry for Syllogism is very good and I recommend it as a starting point. The Syllogism solving machine is also quite fun.

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