After a Twitter exchange with someone who was particularly challenged with regard to logic, I promised to post some basic lessons in logic (I’ve been teaching the subject for many years).

Part I began with a basic discussion of the nature of argumentation, which is one of the primary ways in which reasoning operates: it makes connections between statements. In this post, I talk about the criteria for evaluating arguments.

(And apologies for getting into teacher mode here…)

Jeremy Brett as Sherlock Holmes
Jeremy Brett as Sherlock Holmes. “Elementary…”


There are two criteria for evaluating arguments:

A) Whether the premises support the conclusion;
B) Whether the premises are true.


Because there are two different types of arguments (see Part I), there are two different ways of talking about the first criterion.

I. With FORMAL ARGUMENTS, we talk about VALIDITY: In a valid (formal) argument, the premises really do establish the conclusion. If the premises are true, the conclusion has to be true.


These are VALID formal arguments:

1) “Socrates is a man, and all men are mortal, so Socrates is mortal.”

2) “If it’s raining, then my car is wet. My car’s not wet, so it’s not raining.”

In both cases, if the premises are true, the conclusion is guaranteed to be true.


This is an INVALID formal argument:

“If this is Tuesday, then we have logic class; it’s not Tuesday, so we don’t have logic class.”

The conclusion doesn’t follow from these premises. (Think for a minute about how this differs from the similar argument above about my car being wet.)


II. With INFORMAL ARGUMENTS, we talk about LOGICAL STRENGTH: A logically strong (informal) argument is one in which the premises tend to support the conclusion; they tend to establish the conclusion with a high degree of probability.


“Jones has been polling well, he has a clear message, he knows how to connect to his constituents, and his opponent has made some serious gaffs, so Jones will likely win the election in November.”

These premises provide support for the conclusion that Jones will win the election, but they certainly don’t guarantee that conclusion. The premises could all turn out to be true, and Jones could still lose.


Please note that the two criteria above (whether the premises support the conclusion, and whether the premises are true) are two very different things: Untrue or even absurd premises can provide very good support for a conclusion; and clearly true premises can fail to establish the conclusion.


“If this is Tuesday, then the moon is made of green cheese. This is Tuesday, so the moon is made of green cheese.”

Note that this argument is valid: if the premises are true, then the conclusion has to be true. But the first premise is absurd.

“New York is comprised of five boroughs, it has a rich and varied history, and the mayor’s name is Bill, so it’s an ideal place to start a business.”

In this case, the premises are true, but they have no bearing on the conclusion. This is an example of a non sequitur fallacy (which I’ll discuss in the next post).


I’ll break this down into three parts:

I. From the logician’s point of view.

Logicians are concerned with the formal structure of arguments; that is, they want to know what kinds of statements follow from what other kinds of statements. So they’re not concerned about whether the statements themselves are true. That’s why in considering arguments, logicians symbolize the arguments. They take out the natural language.

Take the above example:

“If this is Tuesday, then the moon is made of green cheese. This is Tuesday, so the moon is made of green cheese.”

Let P = “This is Tuesday” and Q = “The moon is made of green cheese.”

Then the argument form would be (the line separates premises from conclusion):

If P, then Q
Therefore Q

We can then devise a test to determine whether this argument is valid (and it clearly is). So, interestingly, whatever propositions we plug in for P and Q, the conclusion will always follow.

Also take the invalid example above:

“If this is Tuesday, then we have logic class; it’s not Tuesday, so we don’t have logic class.”

This time let P = “This is Tuesday,” and Q = “We have logic class.”

Then the form of the argument would be:

If P, then Q
Not P
Therefore Not Q

This is an example of a formal fallacy, so no matter what we fill in for P and Q, the conclusion just doesn’t follow. (If we ONLY had logic class on Tuesday, then the argument would work; but of course we can have logic class on other days as well, so just because it’s not Tuesday, it doesn’t follow that we don’t have logic class.)

Only half Vulcan, but oh so logical.
Only half Vulcan, but oh so logical.

II. From an everyday point of view.

Premises come in varying degrees of concreteness or abstraction. Something like “New York is comprised of five boroughs” is a very concrete statement of fact and is easily verified (we could just look at a map or call City Hall).

Other premises are more abstract, those dealing with political, religious, or philosophical matters, for instance. These are likely to be more controversial, and some of them may need to be established via argumentation.

Claims like “democracy is the best form of government,” “God is the cause of everything,” or “human actions are determined” are more abstract and more difficult to establish. (That doesn’t mean that they’re merely matters of opinion; claims, even very abstract ones, can be better and worse supported by argument and evidence.)

So, in general, we all know how to verify concrete statements; and the more abstract a statement, the more likely it will be that it will need to be supported with further argumentation if we wish to use it in an argument.

III. From a philosophical point of view.

I only mention this in passing, but the branch of philosophy that’s concerned with knowledge, belief, truth, etc., is epistemology. And from a philosophical point of view, the question of how we know certain propositions to be true—how it is that we’re acquainted with facts about the world—is of great importance and of equally great difficulty.

This gets us into very involved and interesting questions about how our minds work, how perception works, and how we understand and perceive the world and facts about the world.

After all, we could be trapped in The Matrix…


Identify which of the following is a formal and which is an informal argument. Can you tell which of the formal arguments are valid and which are invalid?

1. Mary is sulking, and she skipped dinner. I bet she’s depressed again.

2. Frogs are amphibians, and amphibians are vertebrates, so frogs are vertebrates.

3. All Marxists are socialists, so all socialists are Marxists.

4. The trashcan is turned over, the garbage is everywhere, and I see paw prints. The dog must’ve gotten into the trash.

5. If it rains, Ronny takes the bus, and I know he took the bus, so it must be raining.

6. Rex is a dog, and no dogs can solve logic problems, so Rex can’t solve logic problems.

7. If it rains, Ronny takes the bus, and it’s not raining, so he must not be taking the bus.

8. ‘Neo’ is the hacker alias of Thomas Anderson, and ‘Neo’ is an anagram of ‘One.’ An anagram is a word that’s produced by rearranging the letters of a different word.

9. George can’t be that serious about Sally. If he were, he’d quit messing around with Angela, and I saw the two of them heading into the Motel 6 last night.

10. Ethel is a monstrously big woman, and she does have a beard, so she really ought to join the circus.

(Answers to follow)

2 thoughts on “Logic Part II: Evaluating Arguments

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