The formal language of SL

The key to studying argument is to set aside the subject being argued about and to focus on the way it is argued for. If we say an argument is good, then the same kind of argument applied to a different topic will also be good. If we say an argument is good for solving murders, we will also say that the same kind of argument is good for deciding where to eat, what kind of disease is destroying your crops, or who to vote for. Because logic sets aside what an argument is about, and just looks at how it works rationally, logic is said to have content neutrality. In formal logic we get content neutrality by replacing parts of the argument we are studying with abstract symbols. In this chapter, we begin our study of formal logic by learning about a symbolic language called $SL$

§ 1 What is meant by 'formal'?

Consider the two arguments below:

$P_{1}$   1: Socrates is a person.

$P_{2}$   2. All persons are mortal.

$\therefore$   C: Socrates is mortal.

$P_{1}$   1: Socrates is a person.

$P_{2}$   2. All persons are carrots.

$\therefore$   C: Socrates is carrot.

These arguments are both valid. In each case, if the premises were true, the conclusion would have to be true. (In the case of the first argument, the premises are actually true, so the argument is sound, but that is not what we are concerned with right now.) What makes these arguments valid is that they are put together the right way. Another way of thinking about this is to say that they have the same logical form. Both arguments can be written like this:

$P_{1}$   1: S is M

$P_{2}$   2.All M are P.

$\therefore$   C: S is P

In both arguments $S$ stands for Socrates and $M$ stands for person. In the first argument, $P$ stands for mortal; in the second, $P$ stands for carrot. The letters S, M, and P are variables. They are just like the variables you may have learned about in algebra class. In algebra, you had equations like $y = 2x + 3$, where $x$ and $y$ were variables that could stand for any number. Just as $x$ could stand for any number in algebra, `S' can stand for any name in logic. In fact, this is one of the original uses of variables.

The importance of the variable for the history of mathematics is obvious. But it was also incredibly important in one of its original fields of application, logic. For one thing, it allows logicians to be more content neutral. We can set aside any associations we have with people, or carrots, or whatever, when we are analyzing an argument. More importantly, once we set aside content in this way, we discover that something incredibly powerful is left over, the logical structure of the sentence itself. This is what we investigate when we study formal logic. In the case of the two arguments above, identifying the logical structure of statements reveals not only that the two arguments have the same logical form, but they have an impeccable logical form. Both arguments are valid, and any other arguments that have this form will be valid.

As formal logic evolved, however, the idea of being 'formal' would take on an additional meaning. Despite its historical importance, Aristotelean logic has largely been superseded. Starting in the 19th century people learned to do more than simply replace categories with variables. They learned to replicate the whole structure of sentences with a formal system that brought out all sorts of features of the logical form of arguments. The result was the creation of entire artificial languages. An artificial_language is a language that was consciously developed by identifiable individuals for some purpose.

Artificial languages contrast with natural language, which develop spontaneously and are learned by infants as their first language. Natural languages include all the well-known languages spoken around the world, like English or Japanese or Arabic. The languages developed by logicians are artificial, not natural. When the languages first started being developed in the late 19th and early 20th centuries, the goal was, in fact, to have a logically pure language, free of the irrationalities the plague natural languages. More specifically, they had two distinct goals: first, remove all ambiguity and vagueness, and second, to make the logical structure of the language immediately apparent, so that the language wore its logical structure on its face, as it were. If such a language could be developed, it would help us solve all kinds of problems. The logician and philosopher Rudolf Carnap, for instance, felt that the right artificial language could simply make philosophical problems disappear. (The few quotes that randomly occur on the front page of this website encapsulate this sentiment.)

For the purposes of this textbook, we will say that the core idea of a formal_language is that it is an artificial language designed to bring out the logical structure of ideas and remove all the ambiguity and vagueness that plague natural languages like English. Creating formal languages always involves all kinds of trade offs. On the one hand, we are trying to create a language that makes a logical structure clear and obvious. This will require simplifying things, removing excess baggage from the language. On the other hand, we want to make the language perfectly precise, free of vagueness and ambiguity. This will mean adding complexity to the language. The other thing was that it was very important for the people developing these languages that you be able to prove the all the truths of mathematics in them. This meant that the languages had to have a certain scope.

This was a trade off no logician was ever able to get perfectly correct, because, as it turns out, a logically pure language is impossible. No formal language can do everything that a natural language can do. Logicians became convinced of this, naturally enough, because of a pair of logical proofs. In 1931, the logician Kurt Goedel showed that you couldn't do all of mathematics in a consistent logical system, which was enough to persuade most of the logicians engaged in the project to drop it. There is a more general problem with the idea of a purely logical language, though, which is that that many of the features logicians were trying to remove from language were actually necessary to make it function. Arika Okrent puts the point quite well. For Okrent, the failure of artificial languages is precisely what illuminates the virtues of natural language.

§ 2 Introducing $SL$

This section introduces a logical language called SL, which is a version of sentence logic, because the basic units of the language will represent statements, and a statement is usually given by a complete sentence in English.In SL, capital letters, called sentence letter are used to represent simple statements. Considered only as a symbol of SL, the letter $A$ could mean any statement. So when translating from English into SL, it is important to provide a symbolization key, or dictionary. The symbolization_key provides an English language sentence for each sentence letter used in the symbolization.

Consider this argument:

$P_{1}$   There is an apple on the desk.

$P_{2}$   If there is an apple on the desk, then Jenny made it to class.

$\therefore$   Jenny made it to class

This is obviously a valid argument in English. In symbolizing it, we want to preserve the structure of the argument that makes it valid. What happens if we replace each sentence with a letter? Our symbolization key would look like this:

A: There is an apple on the desk.

B: Jenny made it to class.

We would then symbolize the argument in this way:

$P_{1}$   A

$P_{2}$   If A, then B.

$\therefore$   B

There is no necessary connection between some sentence $A$, which could be any statement, and some other sentences $B$, which could also be anything. The important thing about the argument is that the second premise is not merely any statement, logically divorced from the other statement in the argument. The second premise contains the first premise and the conclusion as parts. This is why our symbolization key for the argument only needs to include meanings for $A$ and $B$, and we can build the second premise from those pieces.

§ 3 The Building Blocks of $SL$

There is a structure behind symbolizing English into $SL$. The basic idea is that each sentence should express one self-standing idea and we should explicate logical relations using other symbols. For instance, the sentence 'sky is blue and water is wet' contains two ideas that can be expressed independently. So they each should be assign to a difference letter.

Atomic Sentences

The individual sentence letters in SL are called atomic sentences, because they are the basic building blocks out of which more complex sentences can be built. We can identify atomic sentences in English as well. Anatomic sentence is one that cannot be broken into parts that are themselves sentences. 'There is an apple on the desk' is an atomic sentence in English, because you can't find any proper part of it that forms a complete sentence. For instance 'an apple on the desk' is a noun phrase, not a complete sentence. Similarly 'on the desk' is a prepositional phrase, and not a sentence, and 'is an' is not any kind of phrase at all. This is what you will find no matter how you divide 'There is an apple on the desk.' On the other hand you can find two proper parts of 'If there is an apple on the desk, then Jenny made it to class' that are complete sentences: 'There is an apple on the desk' and 'Jenny made it to class.' As a general rule, we will want to use atomic sentences in SL (that is, the sentence letters) to represent atomic sentences in English. Otherwise, we will lose some of the logical structure of the English sentence, as we have just seen.

Not just anything in English can be translated into $SL$, however. Logic in general only concerns itself with $statements$, in a technical sense: a unit of language that can be true or false. Statements, as philosophers would say, declarative - they make claims about the world. 'Gress is green' and 'Logic is awesome' are declarative. Things like commands, exclamation, and questions are not declarative and cannot be captured by $SL$.

Complex Sentences and Logical Connectives

Logical connectives are used to build complex sentences from atomic components. In SL, our logical connectives are called sentential connective because they connect sentence letters. There are five sentential connectives in SL. In this section we will go over three of those.


Suppose we want to translate the sentence:

$$\text{Mary is not in Barcelona}$$

It might be tempting to think of this sentence as being atomic, but logically we should think of this as

$$\text{It is not the case that Mary is in Barcelona}$$

It's a mouthful, but it makes its logical structure explicit: this statement is a denial of a certain fact, that Mary is in Barcelona. To symbolize statements that involve denial, we use the negation symbol $\neg$. So suppose we symbolize 'Mary is in Barcelona' as $B$. To symbolize the complex sentence, we write

$$\neg B$$

Note that a statement can express a denial without using the word 'not'. For instance, 'Mary is nowhere near Barcelona' can be translated as $\neg B$.


Suppose we want to translate the sentence:

$$\text{Mary is not in Barcelona and Bill is in Hong Kong.}$$

We already have half the statement translated. What we need is a way to connect $\neg B$ to a symbol that says 'Bill is in Hong Kong.' Let's symbolize the latter as H. To symbolize the idea of and, we introduce the symbol $\wedge$, so

$$(\neg B \wedge H)$$

is a symbolization of the original statement in $SL$ The logical connective $\wedge$ is called the conjunction. $\neg B$ and $H$ are called conjuncts.

Again, like negation, there is a wide range of English statements that might have identical SL translation. For instance, 'Both A and B' would have the same translation as 'A and B'. Perhaps slightly more surprising are sentences with words like 'but' and 'although' - in SL they are nothing but conjunction. So 'A but B' would be translated as $(A \wedge B)$.


Another way in which two statements can be connected is through disjunction, such as

$$\text{Either Amy is a logician or a basketball player.}$$

To translate statements with is the connective 'or', we introduce the disjunction symbol $\vee$.

$$(A \vee B)$$

A and B are called disjuncts. Sometimes in English, the word 'or' excludes the possibility that both disjuncts are true. This is called an exclusive disjunction. An exclusive disjunction is clearly intended when a restaurant menu says, 'Entrees come with either soup or salad.' You may have soup; you may have salad; but, if you want both soup and salad, then you will have to pay extra.

At other times, the word 'or' allows for the possibility that both disjuncts might be true. If you are at a dinner party at a friend's house and she says to you 'would you like some more wine or food?' We can reasonably infer that she is offering not just one of them. This would be an instance of an inclusive disjunction The point is that the use of disjunction is ambiguous in English, and that is something we have to fix in a formal language.

In response, logicians have chosen to have $\vee$ to denote disjunction in the inclusive sense: $(A \vee B)$ means that at least one of the disjuncts, A or B, must be true. They can be both true, but they can't be both false. There is nothing sacred about this - it's simple a choice we have to make in our language. This is not to say that the inclusive sense of disjunction is the meaning of disjunction. In fact, we can symbolize an exclusive or in SL. We just need more than one connective to do it. We can break the sentence into two parts. The first part says that you get one or the other. We translate this as $(A \vee B)$. The second part says that you do not get both. We can paraphrase this as 'It is not the case that A and B.' How would we symbolize this?

Reading Exercise: English/SL Translation

For this reading exercise, you will be given either a sentence in SL or in English, and you are asked to pick out the corresponding translation from the options. Do 10 of these to complete the exercise. Click 'next' to begin.

§ 4 Truth tables and the precise 'meaning' of logical connectives

So far we rely on our intuitive understanding of 'and,' 'or,' and 'not' to give meanings to their corresponding logical symbols. To make this more precise, we make use a formal tool called a 'truth table,' which is a very effective to show the semantics of the connectives,i.e., how complex sentences are to be interpreted, in terms of whether or not it is true. This is the tables for negation:

Each column represents the possible values that the sentence can hold. $T$ and $F$ are called truth values. This table captures this idea: for any sentence $A$: If $A$ is true, then $\neg A$ is false. If$\neg A$ is true, then $A$ is false. So on the row that $A$ gets T, it means that $A$ is true. This is why for any row where $A$ has a $T$, $\neg A$ has a $F$. Note that A here is just a placeholder - it stands for any grammatically correct SL sentence. It's a metavariable that can represent any sentence in the formal language. (Technically we are supposed to use special symbol like $\alpha$ for metavariable, but we will not concern ourselves with that here.)

Truth table is supposed to represent all possible combinations of truth values, so the number of rows needed for a table is dependent on how many letters we are dealing with. Thus, $A\wedge B$, since it has two letters, will have $2^2 = 4$ rows.

For any sentences $A$ and $B$, $A \wedge B$ is true if and only if both $A$ and $B$ are true. Conjunction is symmetrical because we can swap the conjuncts without changing the truth value of the sentence. That is, $A\wedge B$ and $B \wedge A$ have the same values.

Lastly, this is the table for disjunction

Since we interpret $\vee$ as being inclusive, the only situation where $A \vee B$ is false is when they are both false.

§ 5 Truth Functions

Any nonatomic sentence of SL is composed of atomic sentences with sentential connectives. The truth value of the complex sentence depends only on the truth value of the atomic sentences that it comprises. In order to know the truth value of $(D\vee E)$, for instance, you only need to know the truth value of $D$ and the truth value of $E$. Connectives that work in this way are called truth functional. A truth function is a rule that tell us how to determine the truth of a sentence by looking at its combination of true sentences and connectives. We define a truth-functional connective as an operator that builds larger sentences out of smaller ones, and fixes the truth value of the resulting sentence based only on the truth value of the component sentences.

Because all of the logical symbols in SL are truth functional, the only aspect of meaning we need to worry about in studying the semantics of SL is truth and falsity. If we want to know about the truth of the sentence $A \wedge B$, the only thing we need to know is whether $A$ and $B$ are true. It doesn't actually matter what else they mean. So if $A$ is false, then $A \wedge B$ is false no matter what false sentence $A$ is used to represent. It could be I am the Pope' or 'Pi is equal to 3.19.' The larger sentence $A \wedge B$ is still false. So to give an interpretation of sentences in SL, all we need to do is create a truth assignment, which is a function that maps the sentence letters in SL onto our two truth values. In other words, we just need to assign Ts and Fs to all our sentence letters.

§ 6 Interpreting Complex Sentences

The truth value of sentences that contain only one connective is given by the truth table definition for that connective. The truth table definition for conjunction, for example, gives the truth conditions for any sentence of the form $(A\wedge B)$. Even if the conjuncts A and B are long, complicated sentences, the conjunction is true if and only if both A and B are true. Consider the sentence $(H\wedge I)\vee \neg H$. We consider all the possible combinations of true and false for $H$ and $I$, which gives us four rows. We then copy the truth values for the sentence letters and write them underneath the letters in the sentence.

Now consider the subsentence $H\wedge I$. This is a conjunction $A\wedge B$ with $H$ as A and with $I$ as B. $H$ and $I$ are both true on the first row. Since a conjunction is true when both conjuncts are true, we write a T underneath the conjunction symbol. We continue for the other three rows and get this:

We do the same for $\neg H$. Based on the truth table definition, $\neg H$ is true when $H$ is false, so:

The entire sentence is a disjunction $(A \vee B)$ with $(H \wedge I)$ as A and with $\neg H$ as B. On the second row, for example, $(H\wedge I)$ is false and $\neg H$ is false. Since for a disjunction to be true at least one of the disjunct has to be true, we write a F in the second row underneath the disjunction symbol. We continue for the other three rows and get this:

Of course, this is rather messy. Most of the columns underneath the sentence are only there for bookkeeping purposes. Normally we do not write out every single truth value like that. As a matter of fact, when we deal with longer sentences it becomes practically unfeasible to write them all out. Normally, we only write out the truth values for the main connective like this:

When you become more adept with truth tables, you will probably no longer need to copy over the columns for each of the sentence letters.

A sentence that contains only one sentence letter requires only two rows, as in the truth table definition for negation. This is true even if the same letter is repeated many times, as in this sentence: $$[(C\wedge C) \vee C] \wedge \neg(C \vee C)$$ The complete truth table requires only two lines because there are only two possibilities: $C$ can be true, or it can be false. A single sentence letter can never be marked both T and F on the same row.

A sentence that contains three sentence letters requires eight lines. For example

We call a table that gives all the possible interpretations for a sentence or set of sentences in SL a complete truth table. In order to fill in the columns of a complete truth table, begin with the right-most sentence letter and alternate Ts and Fs. In the next column to the left, write two Ts, write two Fs, and repeat. For the third sentence letter, write four Ts followed by four Fs. This yields an eight line truth table like the one above. For a 16 line truth table, the next column of sentence letters should have eight Ts followed by eight Fs. For a 32 line table, the next column would have 16 Ts followed by 16 Fs. And so on.

§ 7 Chapter Definition Review

Finish this chapter by completing the the definition review quiz below.