Conditionals and Negations

Chapter 1 introduces a fragment of Sentential Logic, focusing just on the logic of ‘if’ and ‘not’. The rest of Sentential Logic—including the logic of ‘and’, ‘or’, and ‘if and only if’—is introduced in Chapter 2. We start with just this fragment to keep things simple.

You just read the first two sections of Chapter 1. The first section introduces a new symbolic notation—two new symbols, ‘\({\mathord{\sim}}\)’ and ‘\({\mathbin{\rightarrow}}\)’—together with some rules for combining those symbols with sentence letters—capital letters \(P\) through \(Z\)—and parentheses to produce symbolic sentences.

The second section explains what the new symbols mean. ‘\({\mathord{\sim}}\)’ is the negation sign, and we use it to symbolize ‘it is not the case that’. ‘\({\mathbin{\rightarrow}}\)’ is the conditional sign, and we use it to symbolize ‘if…then…’.

In this supplement, I am going to work through the same material in a slightly different order. I begin with a review of how we use the negation sign and the conditional sign to symbolize negations and conditionals, and end with a review of the rules for combining our new symbols together to create well-formed symbolic sentences.

Negations

We will use the sentence letters—capital letters \(P\) through \(Z\)—to represent simple sentences. This is an arbitrary choice: other logic textbooks might use other letters, or lowercase letters.¹ We will use ‘\({\mathord{\sim}}\)’ to symbolize negation. This is again an arbitrary choice. Some textbooks use ‘\(\neg\)’ instead.

A negation is a sentence that is true just in case the sentence it negates is false. So, for example, suppose we are at a party, and you ask for a beer, and I say,

• There are some beers in the refrigerator.

You go look, but you don’t find any beer. So you say,

• There are not any beers in the refrigerator.

At this point, we would probably send somebody on a beer run. But maybe the party is a logic party, so we decide to translate our exchange into symbols instead.

We can represent my sentence with a single sentence letter,

• \(R\): There are some beers in the refrigerator.

Your sentence is the negation of my sentence. What you said was true just case what I said was false, and vice versa. So, using our negation sign, we can represent your sentence as,

• \({\mathord{\sim}}R\): It is not the case that there are some beers in the refrigerator.

Here are some other examples of sentences and their negations, together with appropriate symbolizations:

\(S\): Snow is white
\({\mathord{\sim}}S\): Snow is not white
\(T\): Frozen is the greatest musical of all time
\({\mathord{\sim}}T\): Frozen isn’t the greatest musical of all time
\(R\): The NSA respects the rights of US citizens
\({\mathord{\sim}}R\): The NSA fails to respect the rights of US citizens

As these examples show, in English we tend to express negations by including a ‘not’ somewhere in the middle of the sentence, but we can also use contractions or expressions like ‘fails to’.

Not all sentences that involve negativity are negations. Consider:

1. Canada is the best!
2. Canada is the worst!

(2) is not the negation of (1), because (2) is not true just in case (1) is false. Maybe Canada is neither the best nor the worst, but just so-so. The negation of (1) is a sentence that is true just in case (1) is false, like (3):

3. Canada is not the best.

Not all sentences that differ just by a ‘not’ are negations. Consider:

1. I want to eat a cupcake.
2. I want to not eat a cupcake.

(2) is not the negation of (1). For (2) to be true, I must want to avoid eating cupcakes, but maybe (1) is false because I am simply indifferent to cupcakes. The negation of (1), then, is

3. I don’t want to eat a cupcake.

So, let’s work through some examples.

Suppose \(Z\) stands for ‘The level of zooplankton in the Great Lakes is dangerously high.’ What English sentence does \({\mathord{\sim}}Z\) symbolize?

What English sentence expresses the negation of ‘Satan is evil’?

How would you symbolize, ‘It is not the case that the levels of zooplankton in the Great Lakes are not dangerously high’?

Suppose \(P\) represents the sentence ‘Petunia is a purple pig’. What are three ways of expressing, in English, \({\mathord{\sim}}P\)?

Conditionals

We use the conditional sign, ‘\({\mathbin{\rightarrow}}\)’, to represent conditionals, that is, if…then… statements. This is again an arbitrary choice. Some textbooks use ‘\(\supset\)’ instead.

A conditional is an if…then… statement, like

1. If you eat all the popcorn, then I will be sad.

The antecedent of the conditional is the “if” part, so, in this case,

• \(P\): You eat all the popcorn.

The consequent is the “then” part, so,

• \(S\): I will be sad.

This terminology is important, because it gives us a precise way to refer to the different parts of a conditional.

‘Ante’ means before. Hence ‘ante up’ in poker, and the antebellum period, before the Civil War.

We can represent (1) as

2. \((P{\mathbin{\rightarrow}}S)\)

Now suppose I want to deny (1): my happiness does not depend on your eating habits! I can do this by expressing the negation of the conditional,

3. \({\mathord{\sim}}(P{\mathbin{\rightarrow}}S)\)

or, in English,

4. It is not the case that if you eat all the popcorn, then I will be sad.

The parentheses are important. Consider yet another way my happiness and your eating habits might be related: maybe my happiness does depend on your eating habits, but not in the way that (1) suggests. Maybe I want you to eat all the popcorn, and if you don’t, I am going to be sad. That is,

5. If you don’t eat all the popcorn, then I will be sad.

(5) is a conditional. Its antecedent is,

• \({\mathord{\sim}}P\): You don’t eat all the popcorn,

and its consequent is,

• \(S\): I will be sad.

So we can symbolize (5) as,

6. \(({\mathord{\sim}}P{\mathbin{\rightarrow}}S)\)

Study the difference between (3) and (6), and the corresponding difference between (4) and (5). In (3), the negation sign is inside the parentheses. In (6), it is outside. Can you see how that captures the difference between (4)—the negation of the conditional—and (5)—a conditional with a negation as its antecedent? If we didn’t have parentheses, we could not capture this distinction.

So, what is the antecedent of this conditional: ‘If you can’t love the one you want, love the one you’re with’?

What is the consequent of this conditional: \(((P{\mathbin{\rightarrow}}Q){\mathbin{\rightarrow}}R)\)

Ambiguity

Sentences in English can be ambiguous—that is, they can have multiple distinct interpretations. Sometimes this is because the sentence contains a word that is ambiguous, as in

• Meet me down at the bank.

Are we meeting at a financial institution, or beside the river?

But sometimes, the ambiguity has to do with the structure of the sentence. Consider the following ambiguous newspaper headlines:

• Dr. Ruth to Talk about Sex with Newspaper Editors
• Squad Helps Dog Bite Victim
• Stolen Painting Found by Tree
• Enraged Cow Injures Farmer with Ax

Each of these sentences can be interpreted in two different ways, and the truth or falsehood of the sentence depends on how you interpret it.

We don’t want this sort of ambiguity in our symbolic language. Parentheses are the tool we use to avoid it.

The sentence, ‘If you are in the class you should do the reading if you want to succeed’, is ambiguous. What are the two readings?

Formal Syntax

The discussion so far has provided an informal introduction to our symbolic language. But part of the point of logic is to be as clear and rigorous as possible. So we will now provide the formal syntax for the language.

The symbols of our new language are:

• the connectives: the negation sign, \({\mathord{\sim}}\), and the conditional sign, \({\mathbin{\rightarrow}}\)²
• the left and right parentheses
• the sentence letters: \(P\) through \(Z\)³

This is a recursive definition. That means you can apply the clauses of the definition to generate some symbolic sentences, and then apply them again to the sentences you just generated to generate yet more symbolic sentences, and then apply them again to the sentences you just generated to generate yet more symbolic sentences, and so on.

The first clause provides us with a foundation of simple symbolic sentences—we call them the atomic sentences. For example, according to the first clause,

• \(P\) and \(Q\) a both symbolic sentences, because they are sentence letters.

The second two clauses allow us to generate more complex sentences out of simpler sentences. We call these complex sentences molecular sentences.

So, for example, according to the second clause, if \(P\) a symbolic sentence, so is \({\mathord{\sim}}P\). And we’ve already established that \(P\) is a symbolic sentence, so

• \({\mathord{\sim}}P\) is a symbolic sentence.

And, again, according to the second clause, if \({\mathord{\sim}}P\) is a symbolic sentence, so is \({\mathord{\sim}}{\mathord{\sim}}P\). So

• \({\mathord{\sim}}{\mathord{\sim}}P\) is a symbolic sentence.

In this way, we can quickly generate infinitely many symbolic sentences just by repeated application of the second clause.

The same is true of the third clause. Since both \(P\) and \(Q\) are symbolic sentences, so is

• \((P{\mathbin{\rightarrow}}Q)\)

And since \(P\) and \((P{\mathbin{\rightarrow}}Q)\) are symbolic sentences, so are both

• \((P{\mathbin{\rightarrow}}(P{\mathbin{\rightarrow}}Q))\)
• \(((P{\mathbin{\rightarrow}}Q){\mathbin{\rightarrow}}P)\)

and since these both are, so is

• \(((P{\mathbin{\rightarrow}}(P{\mathbin{\rightarrow}}Q)){\mathbin{\rightarrow}}((P{\mathbin{\rightarrow}}Q){\mathbin{\rightarrow}}P))\),

and so on.

Why isn’t \(A\) a symbolic sentence?

\({\mathord{\sim}}(Q{\mathbin{\rightarrow}}P)\) is a symbolic sentence. Can you see how it is generated?

\(((P{\mathbin{\rightarrow}}{\mathord{\sim}}Q){\mathbin{\rightarrow}}R)\) is a symbolic sentence. Can you generate this sentence by repeated application of the three clauses of our recursive definition?

Main Connectives

The main connective of a symbolic sentence is the connective that governs the whole sentence. If you think about how the sentence was built up, by successive applications of the three clauses of the definition, the main connective is the last connective added. For example, in \({\mathord{\sim}}{\mathord{\sim}}P\), the main connective is the first ‘\({\mathord{\sim}}\)’, which was added to ‘\({\mathord{\sim}}P\)’ in accordance with clause (2).

So, what is the main connective in \({\mathord{\sim}}(P{\mathbin{\rightarrow}}Q)\)?

What is the main connective in \(({\mathord{\sim}}P{\mathbin{\rightarrow}}{\mathord{\sim}}Q)\)?

What is the main connective in \((P{\mathbin{\rightarrow}}(Q{\mathbin{\rightarrow}}R))\)?

Parsing

It is useful to have a way of representing the syntactic structure of sentence. We can do this by using a syntactic tree—a tree that shows how the sentence was generated from atomic sentences, by successive application of the clauses of the recursive definition.

For example, we can parse \((P\rightarrow Q)\) as follows:

Informal Notation

Parentheses are really important! Without them, our symbolic sentences would be ambiguous. But they can make our sentences cluttered and hard to read. So we allow ourselves to write sentences in an informal notation, which means we get to leave some of the parentheses out.

But informal notation does not mean anything goes. We have strict rules concerning when parentheses can be omitted. For now, we only have one rule:

• If the symbolic sentence, in official notation, begins with a left parenthesis and ends with a right parenthesis, you can omit those parentheses.

So, a sentence like \((P{\mathbin{\rightarrow}}Q)\) can be written as \(P{\mathbin{\rightarrow}}Q\). But you cannot omit any of the parentheses from a sentence like \({\mathord{\sim}}(P{\mathbin{\rightarrow}}Q)\), because it does not begin with a parenthesis—it begins with a negation sign.

Also, please note that there is no convention or rule—formal or informal—that lets you add parentheses willy-nilly. \((P{\mathbin{\rightarrow}}Q)\) is a symbolic sentence; \(((P{\mathbin{\rightarrow}}Q))\) is not.

Is \((P)\) a sentence in official notation, informal notation, or is it not well-formed?

Is \(P{\mathbin{\rightarrow}}{\mathord{\sim}}{\mathord{\sim}}Z\) a sentence in official notation, informal notation, or is it not well-formed?

\({\mathord{\sim}}(P{\mathbin{\rightarrow}}Q)\) is a sentence in official notation. Can we drop any of the parentheses, if we want to write it in informal notation?

\({\mathord{\sim}}P{\mathbin{\rightarrow}}Q\) is a sentence in informal notation. How is it written in official notation?

Suppose a sentence contains 3 occurrences of \({\mathbin{\rightarrow}}\), and is written in official notation. How many pairs of parentheses does it contain?