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Chapter One, Tutorial Four
Further Concepts for Deductive Logic

Deductive logic is concerned with more than arguments and their validity and soundness. This final tutorial for chapter one takes up further concepts defined in terms of (logical) possibility.

We begin with a notion related to that of meaning: logical equivalence. Very roughly, the idea here is that two sentences are logically equivalent when they both express the same idea.

For example, the following two sentences say the same thing in different ways:

Neither Sandy nor Tim passed the exam.
Both Tim and Sandy failed the exam.

So, we will say they are logically equivalent; they express the same idea.

But, again, this rough definition is vague. We need something more precise. And as a first step towards this exactness, we define "logical equivalence" in terms of possibility.

Think about the Sandy and Tim example. These two sentences can be seen to say the same thing because if one of the two is true, then the other must also be true. (It's inescapable that one be false if the other is true!) So, we can give a definition like that for validity in terms of what is possible and not possible:

The two members of a pair of sentences are logically equivalent if and only if it is not possible for one of the pair to be true while the other is false.

Now, test this idea of logical equivalence. Which of the following sentences is logically equivalent to
        "No student doesn't like logic"?
Click on the answer:

1. All students get 'A's in logic.
2. No students like logic.
3. All students like logic.

Try one more. Which of the following is logically equivalent to "Beth and Jerry did not both receive a diploma":

1. At least one of the two did not receive a diploma.
2. Exactly one of the two did not receive a diploma.
3. Neither of the two received a diploma.

Yes!  "It's not the case that both received a diploma" means "either one received a diploma or neither did". Choice 1 is the way to say this.

One final point. We see from these examples one benefit of logic. We start to get very clear on meaning and thus practice at being very precise in our thinking.

So far we have developed concepts that apply to whole arguments (validity, soundness) and one concept applying to a pair of sentences (logical equivalence.) Be careful here: we will not say that an argument is, or that two arguments are, logically equivalent. Nor will we say that the members of a pair of sentences are valid. Such claims don't make sense: logical equivalence applies only to sentence pairs and validity applies only to arguments.

Now, let's consider concepts that apply only to single sentences. Consider the following fact about me:

Halpin is either male or he's not.

Is this true (it's about the author of the Logic Cafe)? Well of course it is. One doesn't have to know me to see that I'm male or I'm not. Roughly, we might say that this sentence is true because of the logic or meaning of the words "or" and "not".

Similarly,

No women are bachelors.

is true, surely, because of the meaning of the words involved. (You didn't have to do a survey to see that it's true!)

The idea, here, is that certain sentences must be true in virtue of their logic. This brings up the notion of logical possibility again. And we can give the following definition.

A sentence is logically true if and only if it could not possibly be false.

Now, think about a different sort of example:

No women are women.

There is something wrong with this sentence, of course. We will call it logically false. This time, you provide the definition.

Erase the three question marks, fill in the correct word, then press the SUBMIT button.

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So, our definition should be:

A sentence is logically false if and only if it could not possibly be true.

One more concept relating to a single sentence is required. The two concepts just defined for a single sentence, logically true and logically false, apply to sentences whose truth value is determined simply from the form or meaning of the words used in the sentence. Thus their truth value is determined by logic.

Interesting sentences usually are not like this at all. Whether they are true or not depends on how the world is. Examples are numerous and easy to come up with:

There are eight planets in the solar system.

Pluto is a big asteroid not a planet.

Everyone in class likes logic.

Most people do not know what logic is.

I have a lover but won't tell anyone who she is.

And so forth. Almost anything you are likely to say or deny falls into this category. Notice that meaning plays a role in all these sentences. But so does the way the world is.

We will call sentences like these "logically indeterminate". This simply means that the world, and not logic or language alone, is what determines the truth value of each of these sentences. One way to precisely define this notion is to say that a logically indeterminate sentence is neither logically true nor logically false. But let me ask you to come up with a slightly more perspicuous definition:

Again, replace the question marks to complete the definition then click the submit button.

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So, our definition should be:

A sentence is logically indeterminate if and only if it is possible that it be true and possible that it be false.

Finally, we need to introduce one concept relating to sets of sentences. A set of sentences is just a group or collection of sentences. Suppose I say (er, said) all of the following things to my mom:

"No mom, I didn't do it. I didn't go anywhere near Mr. Smith's swimming pool. No, my hair's not wet. I don't know...anyway, Bobby made me go in."

I clearly lied. The things said clearly don't hang together. They contradict each other. To say at one time that I didn't go in Smith's pool and later that Bobby made me go in is to contradict myself.

We will call such a set of sentences "logically inconsistent". And we will be very strict about what counts as inconsistent in this sense:

A set of sentences is logically inconsistent if and only if it is impossible for all members of the set to be true together.

Sometimes in common speech we will say that someone's expressions are inconsistent when they are merely incoherent but not literally contradictory. For example, it is often said that a politician's pronouncements are inconsistent when it is hard to see what political principles may underlie them all. (Someone might be accused of inconsistency for advocating the rights of ethnic minorities while holding that gays have no specific rights as gays.) But in logic, "inconsistent" means more than incoherent, not fitting together: It means not logically possible.

Finally, we should define what it is for a set of sentences to be consistent.

A set of sentences is logically consistent if and only if it is possible for all members of the set to be true together.

Typographical aside: Sometimes we give a set of sentences a name. The capital sigma is often used. But you'll need logic "font" (or typeface) to see it on screen. Download it from the home page and then install it into your fonts folder. Anyway, sigma looks like this on your computer: "\". If that symbol looks like a slash, then you haven't yet gotten the Logic font installed properly!

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Is the following set of sentences logically consistent or not? Click on the appropriate button below to give your answer.

{ "All brave warriors deserve respect." ,
"Some who deserve no respect are brave warriors."}

(Note the standard set brackets '{' and '}'. The members of the set are listed within the brackets.)

Now we can apply some of these concepts in the informal proofs we first presented in the last chapter. For instance, it's easy to see that

(***) If a set of sentences includes one which is logically false, then that set is inconsistent.

How do we prove this? Well, we argue as before (using the sigma '\' as name for a set of sentences; the symbol here will appear as a slash if one is without the Logic font):

Whew. The simple idea behind this proof gets a bit obscured behind all the verbiage. There's a better way. We'll look at a new example using this "indirect" method; then let you improve the above proof in the exercises.

(****) Any unsound but valid argument has at least one false premise

This last demonstration provides an example of an "indirect" method of proof. One makes an assumption in order to prove it couldn't be true!

In our example, we assumed that our unsound but valid argument A had no false premises. We showed that this would mean that A is sound. But, from the way we defined it, A is not sound.

The upshot is that the assumption in red must be mistaken. We'd better take it back if we don't want to contradict ourselves!

By the way, this contradiction is sometimes called an "absurdity". And this indirect method of proof is sometimes called "reductio ad absurdum". It reduces the mistaken assumption to an absurdity!

So, this method has a funny name and is sometimes a little hard to get used to, but it can be a powerful technique in thinking. Here's the synopsis:

To use the indirect or "reductio ad absurdum" method to prove some proposition P, start by assuming that P is false and finish by showing that this assumption leads to contradiction. (Thus, because assuming P false turns out to be absurd, P must be true.)