Chapter 1: The Subject Matter of Logic

Arguments and Reasoning

In this class, we are going to learn about reasoning.

What is reasoning?

Reasoning is giving reasons for beliefs.

When a lawyer approaches a jury and says

You must find my client innocent, and here’s why…

that lawyer is giving reasons for beliefs. When an engineer says

If we follow these construction plans, the bridge we build will fall down, and here’s why…

the engineer is giving reasons for beliefs. When a politician says

I think the death penalty is barbaric, and here’s why…

the politician is giving reasons for beliefs.

Reasoning, like the air we breathe, can be found anywhere that we find human beings. Like the air, reasoning is often invisible to us until we focus our attention on it. Once we do focus our attention on this pervasive activity, we may become curious about it. What makes something a piece of reasoning?

What all the examples above seem to have in common is a certain structure. The structure involves, first, proposing a belief. The lawyer proposes that the client is innocent. The engineer proposes that the bridge is structurally unsound. The politician proposes that the death penalty is barbaric. The structure then requires the proposer to support their belief. So some evidence is given for adopting the belief. In each case above that is what we left out, as “…”.

If we are to begin to study the nature of reasoning, we will need to find some words for these parts of reasoning. So let us call the belief proposed a conclusion and let us call the evidence for accepting the conclusion some premises. We can call the whole unit of reasoning, the premises and conclusion taken together, an argument.

Here are some examples of arguments

My house is quite safe from earthquakes, since it is located far from any major fault lines.

There are no cottonmouths around here—cottonmouths are the same as water moccasins, and water moccasins can be found only in southeast Kansas.

My client is innocent by reason of insanity, since he ate a very bad diet in the months leading up to the shooting, and such a diet can cause a person to become unable to control themselves.

There’s no fastest motion, because any motion can be made faster by applying more force.

It is common practice among those who study arguments to write an argument with the premises in a list, above the conclusion, like so:

1. My house is far from fault lines.
My house is safe from earthquakes.

1. Cottonmouths are water moccasins.
2. Water moccasins can be found only in Southeast Kansas.
There are no cottonmouths outside of Southeast Kansas.

1. Any motion can be made faster.
There’s no fastest motion

1. My Client ate a bad diet.
2. A bad diet can cause a lack of self-control.
My client is innocent by reason of insanity.

This allows us to conveniently see the structure of the arguments.

Good and Bad Arguments

Not all of the arguments above are entirely foolproof. For example, a bad diet might cause a lack of self-control without causing enough lack of self-control to merit a finding of innocent by reasoning by insanity. And some arguments are downright bad; politicians often say misleading things in an attempt to elicit beliefs among their constituents. Engineers make mistakes. Lawyers sometimes persuade a jury to believe one thing, when they really ought to believe another.

When arguments are bad like this, it is not always because they are not persuasive. Often, people are persuaded by arguments that aren’t very good. For example, it is not uncommon to hear people say that because they’ve had a loosing streak in some game of chance, they are more likely to win in the future. But having a losing streak has no effect on your future chances of winning, even though people tend to feel like it does, perhaps because they feel like they deserve to win, after losing for a while. In other cases, bad arguments may be decorated with emotionally loaded or smart-sounding language, in order to cause the audience to feel like the conclusion is true.

What we want to know is not which arguments feel like they lead us to the truth. Which is not to say that this is not an interesting question. But it is a question for a psychology or English composition class, not for a logic class. We want to know which arguments actually do lead us to the truth. From this point of view, neither our ordinary habits of reasoning, nor the emotional impact of an argument (whether it seems intimidatingly smart or heartwrenchingly moving) count for anything at all. We want to know when an argument really gives a good reason to believe. If we knew this, we could be better voting citizens, better jury members, and better managers of engineering projects, amongst other things. We could also be better lawyers, politicians, and engineers, insofar as we could try to craft arguments that give our audiences good reasons to believe.

In some cases, finding out whether an argument gives us good reason to believe something might require doing careful research. We might ask a geologist, for example, whether the house really is far from faultlines. If it’s next to a giant fault, then you shouldn’t listen to someone who says that it is safe because it is far from faults. But in some cases, knowing the facts is not enough. We may be sure about the truth of all the premises of an argument, but remain unsure about whether the argument supports its conclusion.

For example, consider these two arguments:

1. Nine out of ten stop and frisk subjects are not charged with a crime.
NYC Police are unreliable judges of whether someone has committed a crime.

1. Unless soil is deep and sandy, it is not suitable for carrots.
2. Soil is either suitable for carrots or tomatoes, but not both.
 Soil suitable for tomatoes is not both deep and sandy.

It is far from immediately obvious, but the first argument does not give reason to believe its conclusion. That is in spite of the fact that we may have little doubt about the first premise. If we consider the second argument, we’ll also find ourselves unsure about the conclusion. If a master gardener tells us that the first two premises are true and a gardener of unknown skill remarks to us “Ah, and so soil suitable for tomatoes is not both deep and sandy”, we may be quite confident of the premises, but unsure of whether they support the conclusion being drawn.

The concern of logic is not to answer questions like whether houses are far from fault lines, or whether the same soil can support both carrots and tomatoes. Instead, logic aims to tell us which arguments describe a real connection between premises and conclusion, and which arguments merely appear to do so.

Validity

When an argument fails to lead us to the truth, it is because the premises are true and the conclusion false. The arguments we are after are the ones that cannot fail to lead us to the truth in this way. So we are after arguments in which it is impossible for the premises to be true while the conclusion is at the same time false. We call these arguments deductively valid

Deductively Valid

An argument is deductively valid if either

  1. It is impossible for its premises to be true and its conclusion false.
  2. The truth of the premises guarantees the truth of the conclusion.

If we have a deductively valid argument for some conclusion, then as far as logic goes, all is right with the world. All that remains to be shown is that the premises are true. Once that is established, the conclusion is guaranteed. This is a fairly high standard. For example, consider the following argument pair:

1. If the mouse isn’t in the trap, the cat ate it.
2. The mouse is not in the trap.
The cat ate the mouse.

1. If the mouse isn’t in the trap, the cat ate it.
2. The mouse is not in the trap.
The mouse is dead.

The first argument is deductively valid. The second is not—while the premises make the conclusion very likely, it is still possible for them to be true and the conclusion false. This might happen if, for example, the cat ate the mouse in one bite and then coughed it up fully intact a moment later.

Once we begin to think about deductive validity, we may notice something interesting. The cat/trap argument is valid, but so is this one:

1. If the moose isn’t in the trap, the dinosaur ate it.
2. The moose is not in the trap.
The dinosaur ate the moose.

and so is this one:

1. If the moose isn’t an albino, the dinosaur ate it.
2. The moose is not an albino.
The dinosaur ate the moose.

If you were somehow to find out that the premises of the latter argument were true, even if you did not know anything about albino moose and dinosaurs, you could still responsibly infer that the dinosaur must have eaten the moose. What this suggests is that the validity of the argument sketched here does not depend on the specific content of the premises. Rather, since we could vary the content of the premises without changing the validity of the argument, the validity of this argument must depend on something that all three arguments have in common, something that we did not vary. We call this thing the logical form of the argument.

Logical Form

Logical form is what similarly structured arguments have in common.

One way of depicting the form of the three arguments above is as follows:

If P, then Q.
P.
Q.

This logical form is called modus ponens. The result of replacing the letters P and Q with actual assertions will be a deductively valid argument.

When the deductive validity of an argument depends only on the argument’s logical form, we say that the argument is formally valid.

Formally Valid

An argument is formally valid if

  1. It is deductively valid because of its logical form.
  2. Every argument with the same logical form is deductively valid.

The basic idea of formal logic is to try to find and classify all of the logical forms that make arguments deductively valid. If we could do this, then we would have a good test for the deductive validity of arguments: we could just figure out their logical forms, and then check the form against our classification.

Formal Languages

Ambiguity

The project of classifying arguments by their logical form is simple enough. But if we try to push it very far, we will meet a serious challenge. Here is an example. The following argument looks quite compelling.

1. A silver medal is better than a bronze medal.
2. A gold medal is better than a silver medal.
A gold medal is better than a bronze medal.

Moreover, it seems to be good because of its form. For example, the argument

1. A field goal is better than a safety.
2. A touchdown is better than a field goal.
A field goal is better than a safety.

is deductively valid as well. This might lead us to believe that the following logical form confers validity on the arguments that have it:

1. B is better than C.
2. A is better than B.
A is better than C

But what about this argument?

1. Nothing is better than world peace.
2. Cold pizza is better than nothing.
Cold pizza is better than world peace.

The conclusion is certainly false, but the premises are plausibly true. What is going on here?

The trouble is that English (and indeed all natural languages) contains many ambiguous sentences.

Ambiguous

An ambiguous sentence is a sentence that has more than one possible meaning.

And logical form is determined not by the sentences that make up our arguments, but by their meanings. When the sentences that make up an argument have more than one meaning, the argument can have more than one logical form. And if an argument has more than one logical form, then the answer to the question “is it formally valid?” will generally be “it depends”. This is an unsatisfying state of affairs.

Here are some examples of arguments whose validity depends on how we interpret ambiguous sentences. It is a good exercise to try to spot how one reading makes the argument deductively valid and the other reading does not.

1. Salvatore brought a hat from Italy.
Salvatore has been to Italy.

1. Bill and Barb are married.
Bill is Barb’s husband.

1. Charlotte’s Web is a children’s novel about a pig named Wilbur who is saved from being slaughtered by an intelligent spider named Charlotte.
In C.W., Charlotte saves Wilbur.

1. John saw the man on the mountain with a telescope.
The man on the mountain has a telescope.

If we are to get very far with formal logic, we will need to have a way of dealing with ambiguity. The key idea here come from a mathematician-philosopher named Gottlob Frege. Frege’s idea was this. Because natural languages contain ambiguous sentences, we need a special artificial language for the purpose of studying logical consequence. Such a language should be free of ambiguity. Each sentence should have exactly one meaning, and exactly one logical form. If we could devise such a language, then we could say clearly and systematically which arguments are formally valid. Such a language is called a formal language.

In order to design an unambiguous formal language language, we need to get some grip on the sources of ambiguity in natural language. That way, we can make sure to prevent those sources from including ambiguity in our formal language. What are the sources of ambiguity? Let’s consider another example.

“I shot an elephant in my pajamas” See The Marx Brothers Video.

What makes this sentence ambiguous is that it is not clear which words are meant to modify which other words. The sentence might be read in one of two ways, either as:

Or alternatively, as

The meaning of the sentence depends on how we read it. Of course, we can eliminate the ambiguity by specifying which of the trees above we intend. But this is pretty awkward. A better way to eliminate the ambiguity is to use parentheses to stick together the units that are supposed to go together—the units that we do not unpack until further down the tree.

So we might express the first reading of the sentence by writing “I ((shot an elephant) in my pajamas)”, and the second by writing “I (shot (an elephant in my pajamas))”. This idea may seem unfamiliar, but it is actually something that you have been doing for a long time. If you know the difference between “(2 + 2)×2” and “2 + (2 × 2)” then you know how to disambiguate language by using parentheses.

Our formal language will make use of parentheses for the same purpose.

Our Formal Language

It will pay to start simple. So we will begin by building our formal language out of just two types of ingredients: assertions, and connectives. Assertions are simple true or false statements of fact. For example “The atomic number of gold is 79” is an assertion. So is “The atomic number of gold is 32”, and “the air conditioner is noisy”. Some things that are not assertions are “Do your homework!” and “Who’s the number one marching band drummer of all time?”. Connectives are expressions that can be used to combine assertions into complex assertions. Some examples of connectives are “and”, “or” “if … then …”, and “if and only if”. We can use them to stick together sentences like this:

  1. If I go to work then I will pass by Detroit.
  2. If I dance a jig then I will amuse onlookers.
  3. There’s a storm outside and hail is beginning to come down.
  4. You’re in or you’re out.

Another connective is “it is not the case that”. This connective is a bit odd, because it does not serve to connect two sentences. But it is, nevertheless, useful for forming new assertions out of old ones. For example, we can take “there’s a cat on the roof”, and get “It’s not the case that there’s a cat on the roof”.

Even with this small set of ingredients, however, ambiguity is possible. Consider:

“Jack R. was in town and the night watchman saw him or nobody saw Jack R.”

Does this sentence imply that Jack R. was in town? It depends on which of the following two ways we read it:

  1. “(Jack R. was in town and the night watchman saw him) or nobody saw Jack R.”
  2. “Jack R. was in town and (the night watchman saw him or nobody saw Jack R.)”

The first doesn’t imply that Jack R. was in town. The second does. If we are to avoid this kind of unclarity, we are also going to need to include parentheses in our language, in order to make sure that the structure of each assertion of our language has exactly one clear meaning.

The last consideration is efficiency. We want to be able to write down our sentences quickly, and to see their structure at a glance. So it will be helpful to be rather minimalistic about what we use to represent the assertions and connectives of our language.

Let us use, to represent assertions in our language, single capital letters P, Q, R, ..., Z. If we need more, let’s make them by adding a little subscript to letters that we already have: P1, P2, .... That way, we will have as many assertion letters as we need. Let’s also abbreviate the connectives of our language. Let’s use just the five connectives from before, and write

  1. ” for “and”
  2. ” for “or”
  3. ” for “if … then …”
  4. ” for “if and only if”
  5. ¬” for “it is not the case that”

This tells us what the ingredients in our language will be: assertion letters, connective symbols, and parentheses. But how should we put them together? Here, we need to be careful. It must be clear how to extract the structure of a sentence from how we write it. In order to be sure that this is possible, we must adhere to the following two rules in building sentences:

  1. Every sentence letter “P, Q, R, ..., Z, P1, Q1, R1, ..., Z1, P2, ...” is a grammatical assertion.
  2. We may build a grammatical assertion:
    1. by taking two grammatical assertions we already have, putting an ∧, ∨, → or between, and wrapping the result in parentheses;

      For example, (P ∧ R) is a grammatical assertion.

    2. by taking one grammatical assertions we already have, and putting a ¬ in front.

      For example, ¬P is a grammatical assertion, as is ¬(P ∧ R). Notice that adding ¬ does not require us to add new parentheses.

Every grammatical assertion in our language must be constructed according to these two rules. No other assertions will count as grammatical. we will eventually allow for some abbreviations—but for an abbreviated sentence to be grammatical will just mean that it is an abbreviation of a sentence constructed according to the two rules above We call those assertions which can be built by merely repeatedly applying the two rules above sentences in official notation.

Official Notation

A sentence is in official notation when it is built by merely repeatedly applying rules one and two above.

We can, notice, easily recover the tree that should be associated with a sentence by thinking about the order in which that sentence must have been created. For example, consider the sentence ((P ∨ Q)∨R). This must have been built by first taking a P and Q and sticking them together, then taking an R and sticking that on. So it has the tree:

In general we can recover the tree of a sentence by finding the last connective which was added, breaking it up into the one or two sentences that it was made out of, and repeating. To take another example: in the sentence (((P ∨ Q)∨R)→S), the connective must have been added last (since it is wrapped in only one set of parentheses). So we can break the sentence up as

In (((P ∨ Q)∨R), the must have been added last (since it is wrapped only once), so we can break as:

Finally, in (P ∨ Q), was obviously added last. So we have:

We call the last connective to be added the main connective of the assertion.

Main Connective

The main connective of a sentence in official notation is the last connective to be added in the process of building that sentence up.

Sentences in official notation are very clear—it is easy to find their structure. They can sometimes seem a bit cluttered, though. For example, do we really need the outer parentheses in the sentence “(P ∧ Q)”? Would “P ∧ Q” be any less clear? The answer is obviously no. So, next week, we will introduce some simplification rules—kind of like PEMDAS—that will let us know when we can afford to leave out the extra parentheses.

Exercises

To practice some of these concepts, try the following exercises: what you need to do is enter the main connective of the highlighted sentence. To do this, type in the main connective and then press return (don’t press the submit button until you’re done with the problem—when you are, the display will say “success!”).

Once you’ve entered that connective, the sentence will be broken down into its parts—keep going, entering the main connective for each highlighted sentence. When you reach a sentence that has no connectives, press return to advance to the next branch.

When you’re done, and the sentence has been completely broken down, so that the display reads “success!”, press the submit button to save your work. You’ll then be able to see the completed problem in your user page.

The way to type in the different connectives is given in this table:

ConnectiveKeyboard Shorthand
/\
\/
->
<->
¬~

Problem Set 1

Here’s your first set of problems:

exercise 1.1
1.1 P_1 /\ P_2
exercise 1.2
1.2 P_1 \/ P_2
exercise 1.3
1.3 P_1 /\ (P_2 -> P_3)
exercise 1.4
1.4 P_3 <-> (P_2 \/ ~P_1)
exercise 1.5
1.5 (P_2 /\ P_3) <-> (P_1 \/ (P_5 /\ ~P_2))
exercise 1.6
1.6 (P_1 -> P_2) \/ (P_2 -> P_1)
exercise 1.7
1.7 (P_1 -> (P_2 -> P_3)) -> ((P_1 -> P_2)->(P_1 -> P_3))
exercise 1.8
1.8 (P_1 \/ P_2) -> (P_3<-> ~P_1\/P_2)
exercise 1.9
1.9 (P_1 /\ (P_2 /\ (P_3 /\ (P_4 /\ P_5))))
exercise 1.10
1.10 (P_1<->P_2) \/ (P_1/\ -P_2) \/ (-P_1 /\ P_2)