Logic Review and Exercises

Logic is broadly speaking the study of arguments, or more strictly of inference:

When can a statement be inferred from another statement or set of statements? or:
What statement/s are implied by some statement or set of statements? or:
What statement/s follow from some statement or set of statements?

Logic develops ways of answering these questions.

Key concepts:

Argument: a set of statements in which the truth of one is intended to be supported or demonstrated by the others.
Conclusion: the statement being supported or demonstrated.
Premises: the statement/s offered in support of the conclusion (reasons).
Inference: the step of reasoning from the premises to the conclusion.

How do you tell if a set of statements is an argument? The key question is: is there an inference? a step of reasoning? Sometimes that’s indicated by inference indicators—words like: thus, therefore (conclusion indicators) or because or since (premise indicators).

Examples:

Last week my friend and I rode our bikes through the Santa Ynez Valley. We left from Buellton, rode to Los Alamos, then to Los Olivos, then back to Buellton. It was beautiful and not too difficult. No argument here, because there’s no inference.

My friend will win the next cycling race he’s in. He’s been training hard. He bought a lighter bike. The course is perfect for his strengths. And his strongest competitor is injured. This is an argument because there is an inference. The first statement is being supported by the following four statements.

But watch out, words that sometimes indicate an inference can be use to indicate an explanation: The Warriors won because they rebounded well, is an explanation. The difference between an argument and an explanation is that in an argument the conclusion is in question and the premises (hopefully) are not. The premises are offered as a reason to believe something you were uncertain of. In an explanation the thing being explained (the warriors won) is not in question. We’re just looking for how that happened.

Which of these are arguments and which are not?

Each of Julia Roberts’ last ten films grossed over $100,000. So it’s likely that the next one will too.

Julia Roberts must have a husband who does long distance bike races. She was waiting at the finish line of a recent race in Solvang.

Julia Roberts has made films with some of the best actors of the last half-century. But she never made a film with Robert Redford.

No humans have gills, but Nemo has gills. So Nemo isn’t a human

I have never seen a person taller than 7 foot 8 inches, so no people are taller than 7 foot 8 inches.

People over 7 feet tall sometimes have mobility problems as they age. They need higher blood pressure, which means they get winded more easily.

We’ll either go surfing today or do our homework. If the waves are flat, we won’t surf. The waves are flat. So we’ll do our homework.

There are different kinds of argument, or more precisely, different kinds of inferences.

Deductive arguments are those whose premises are meant to give full support to the conclusion.
Inductive arguments are those whose premises are meant to give a high degree of support to the conclusion.

The basic difference is the relationship between the information in the premises and the conclusion:

In deductive arguments the information contained in the conclusion is already implicitly contained in the premises. The conclusion is intended just to make explicit information that is already contained implicitly in the premises. The inference is explicative.
In inductive arguments the information contained in the conclusion goes beyond that contained in the premises. That means the premises can never make the conclusion absolutely certain. The inference is ampliative.

But again, be careful. Sometimes deductive arguments fail. Their premises do not offer the full support for the conclusion that the arguer intended. That does not make them inductive arguments. Here’s an example:

If the warriors rebound well, they win. They did win. So they must have rebounded well.

That argument is deductive, but the conclusion does not follow from the premises. It is a failed deductive argument, not an inductive argument.

Common forms of inductive arguments are generalizations and predictions

In a generalization one observes some characteristic in a limited number of some kind of thing and infers that all of that kind of thing has that characteristic. Every swan I’ve ever seen is white, therefore all swans are white.

In a prediction one observes some characteristic in a limited number of some kind of thing and infers that some unobserved instance of that kind of thing has that characteristic:

I’ve never seen my cousin’s pet swan, but every swan I’ve ever seen is white, therefore my cousin’s pet swan is white.

Statistical reasoning is a common form of inductive reasoning.

Which of these arguments are inductive and which deductive?

Each of Julia Roberts’ last ten films grossed over $100,000. So it’s likely that the next one will too.

No humans have gills, but Nemo has gills. So Nemo isn’t a human

I have never seen a person taller than 7 foot 8 inches, so no people are taller than 7 foot eight inches.

We’ll either go surfing today or do our homework. If the waves are flat, we won’t surf. The waves are flat. So we’ll do our homework.

It has been raining here for a week straight. So the skiing conditions in Big Bear are going to be perfect.

Evaluating reasoning

When we evaluate an argument we’re basically asking if we should accept its conclusion. There are two questions to ask in evaluating an argument:

Are the premises true?
Is the inference good?

If the premises aren’t true, then, even if the conclusion follows from them, they don’t give us a reason to think the conclusion is true

If the inference isn’t good, then even if the premises are true, they don’t give us a reason to think the conclusion is true. They’re not related to the conclusion in a supportive way.

Deductive and inductive inferences are evaluated differently since they have different ambitions. In a deductive argument the premises are meant to give absolute support to the conclusion, but in an inductive argument can never do that by its very nature.

Key concepts:

Deductive Validity: A deductive inference is valid if, given the truth of the premises it is impossible for the conclusion to be false.

If a deductive argument has a valid inference and its premises are true, it is sound.

Inductive Strength: An inductive inference is strong if, given the truth of the premises, it is highly likely that the conclusion is true.

If an inductive argument has a strong inference and its premises are true, it is cogent.

Logicians typically leave inductive reasoning to the statisticians, and concern themselves with just deductive reasoning.

But before we leave inductive reasoning, let me give a couple of key concepts regarding inductive generalizations:

Inductive generalizations start from the observation of some feature in a limited number of some kind of thing and infer that all of that kind of thing have that feature. The limited number of the kind of thing observed is called the sample. The whole group about which you’re drawing a conclusion is called the target population

The key to the strength of an inductive generalization is whether the sample is like the target population is the relevant ways. Is it representative? The more representative, the stronger the inference.

Representativeness is hard to ensure directly, so statisticians use a substitute that gives approximate representativeness. They try to construct a random sample. In a random sample, each member of the target population must have the same chance of being selected in the sample.

Logicians also leave the question of the truth of an argument’s premises to the relevant experts.

They concern themselves just with the validity of deductive inferences, and in particular with ways of determining whether a deductive inference is valid or invalid.

The crucial idea in deductive logic is that the validity of an argument is a matter of the argument’s form.

If an argument is valid, all arguments that have its form will be valid. It will have a valid argument from.

But discerning the form of an argument takes a little work.

There have been two different systems developed to discern the form of an argument. Sometimes arguments are about categories of things. All presidents have been men.All men have inflated egos. So all presidents have had inflated egos. Here we might capture the form by seeing how the categories are related to each other. The category “presidents” is part of the larger category “men”. The category “men” is part of the larger category “those with inflated egos.” So the category “presidents” is part of the category “those with inflated egos.” This way of thinking about the form of arguments is called categorical logic, and it’s very powerful. But it doesn’t capture all natural language arguments.

An alternate system, the one we’ll study, takes sentences as the basic building block of arguments, rather than categories. It’s called sentential logic. It recognizes that some statements are made up of more basic statements that are being operated on by some word or phrase. These are called complex statements. Statements that aren’t made up of simpler components are called simple statements.

Logicians identify five kinds of complex statements distinguished by the word or phrase that is operating on the components (the logical operator).

Negation (It is not the case that…)
Conjunction (both… and…)
Disjunction (either… or…)
Conditional (if… then…)
Biconditional (… if and only if…)

Note that negations are the odd case of a one-component complex statement. Note too that English has many, many ways of expressing each of these kinds of statements.

Each logical operator has a symbol:

Negation (It is not the case that…)~
Conjunction (both… and…)•
Disjunction (either… or…)v
Conditional (if… then…)-> (typically a right-facing horseshoe)
Biconditional (… if and only if…)= (that’s supposed to be three horizontal bars)

That give us five standard forms of complex statements

~pnegation
p • qconjunction
p v qdisjunction
p -> qconditional
p=qbiconditional

Note that the components of conjunctions, disjunctions and conditionals have special names: conjunctions have conjuncts, disjunctions have disjuncts and conditionals have antecedents (before the arrow) and consequents (after the arrow). More on antecedents and consequents below.

Are the following statements simple or complex? If they’re complex what is their form?

If the Lakers win, they will make the playoffs.
President Obama will either sign this legislation or the Republicans will win the next election.
President Obama won’t yield on cutting defense spending.
My dog will either sleep all day or bark at the neighbor’s cat.

More challenging—these are complex statements with complex components. What is the form of the overall statement?

If the Lakers don’t win, they won’t make the playoffs.
President Obama will either sign this legislation or the Democrats won’t win the next election.
My dog will take a long walk and either sleep all day or bark at the neighbor’s cat.
President Obama won’t yield on cutting defense spending and if he doesn’t, the Democrats won’t win the next election.

A key element in logic is translating statements from a natural language, like English, to the artificial language that allows us to see just the form of statements.

The steps to translating are simple, but you have to be very precise. A crucial rule is to work from the outside in. Discern the form of the overall statement first, then of its components, then of their components, and so forth.

1. Is the statement simple or complex?

1a. If simple, assign it a letter (B = Jim is a bachelor) and write the letter. You’re done.

1b. If complex, write parentheses (with plenty of space between).

2. What kind of complex statement is it? (negation, conjunction, disjunction, conditional, biconditional). Write the symbol for that kind of complex statement between the parentheses.

3. Look at the first component of the complex statement (if a negation, there will be only one). Is it simple or complex?

3a. If simple, assign it a letter, write the letter in the appropriate place and go to step 1 for the next component. If there are no more components, you’re done.

3b. If complex, write parentheses.

4. What kind of complex statement is the component? (negation, conjunction, disjunction, conditional, biconditional). Write the symbol for that kind of complex statement between the parentheses.

Go back to 3 and repeat 3 and 4 until you have symbolized every component down to its simple statements and logical operators.

If I don’t get married, I will travel to Europe and buy a motorcycle.

Is the overall statement simple or complex? Complex!

1b. ( )

What kind of complex statement is the overall statement? Hmm…Conditional!

( -> )

3. Is the first component (I don’t get married) simple or complex?Complex!

3b. Write parentheses where that component goes

( ( ) -> )

4. What kind of complex statement is that component? Negation! Write the symbol for that kind of complex statement inside the parenthesis (toward the left in the case of negation).

( (~ ) -> )

Repeat 3 for the component being negated (I get married). Is it simple or complex? Simple!

3a. M = I get married

( (~ M) -> )

You’re done with the antecedent of the overall conditional.

Go on to the next consequent, I will travel to Europe and buy a motorcycle.

Repeat step 3. Is it simple or complex? Complex!

3a. Write parentheses where that component goes:

( (~ M) -> ( ) )

Repeat 4 for this component. What kind of complex statement is it? Conjunction!

Write the symbol for conjunction inside the parentheses.

((~ M) -> ( • ) )

Repeat 3 for the first conjunct (I will travel to Europe). Is it simple or complex? Simple!

3a. E = I will travel to Europe

((~ M) -> ( E • ) )

Repeat 3 for the second conjunct (I will buy a motorcycle). Is it simple or complex? Simple!

3a. B = I will buy a motorcycle

((~ M) -> ( E • B ) )

We’re done! We’ve seen that,If I don’t get married, I will travel to Europe and buy a motorcycle. is a conditional whose antecedent is the negation of a simple statement and whose consequent is the conjunction of two simple statements.

Try these:

If the Lakers don’t win, they won’t make the playoffs.

President Obama will either sign this legislation or the Democrats won’t win the next election.

My dog will take a long walk and either sleep all day or bark at the neighbor’s cat.

We can distinguish different statement forms by how the world would have to be in order for them to be true (their truth conditions). Take the statement:

I won’t stay up past 10:30.

You can see it’s a negation whose component is I stay up past 10:30. What has to be the case about that component in order for the negation I won’t stay up past 10:30. to be true? The component has to be false. And if the component is true (I do stay up past 10:30) the original statement will be false.

We can put this in a table (called a truth table):

Let S = I stay up past 10:30

S | ~S

t | f

f | t

The column on the left represents all the ways the world can be with respect to the statement’s component, S. S could be true (t) or false (f). These “ways the world can be with respect to S” are called valuations. The column on the right shows the truth or falsehood (the truth value) of the complex statement ~S for each valuation. When S is true, ~S is false; when S is false, ~S is true.

If we let a lower case italicized letter, say p, stand for any statement we can make a truth table for negation:

p | ~p

t | f

f | t

And we can do the same for all other statement forms.

Conjunctions are true when both conjuncts are true; otherwise they’re false.
Disjunctions are true if either or both of their disjuncts is true; if both are false, the disjunction is false.
Conditionals are false only if the antecedent is true and the consequent is false; otherwise they’re true.
Biconditionals are true if both their components are true or both are false; otherwise they’re false.

You can see this on the tables. (Memorize these—except for biconditional—you’ll need them later!)

Negation

p | ~p

t | f

f | t

Conjunction

p q | p • q

t t | t

t f | f

f t | f

f f | f

Disjunction

p q | p v q

t t | t

t f | t

f t | t

f f | f

Conditional

p q | p -> q

t t | t

t f | f

f t | t

f f | t

Biconditional

p q | p <-> q(Double arrow replaces three bars.)

t t | t

t f | f

f t | f

f f | t

These statement forms allow us to see the form of arguments. All we have to do is translate the premises and conclusion in English (or whatever language we speak) into our symbolic language.

Try these:

If I get married, I’ll be happy the rest of my life. I’ll get married. So I’ll be happy the rest of my life.
If I get married, I won’t be happy the rest of my life. I won’t get married. So I’ll be happy the rest of my life.
I’m either going to cook dinner or mow the lawn. I’m not going to mow the lawn. So I’ll cook dinner.
If I get married, I’ll buy a motorcycle with my spouse and we’ll ride across the country. I’m getting married. So my spouse and I will ride across the country.

Some deductive argument forms are so common that they’ve been given standard names and you can just memorize whether they’re valid or invalid.

If the Lakers win, they will make the playoffs. They will win, so they will make the playoffs.

The first premise of this argument is a conditional. The second premise affirms the antecedent of the conditional. The conclusion infers the consequent. The form is:

p – q

p_____

That form is called Modus Ponens (Affirming the Antecedent) and, by the way, it is valid:

Here are some other valid forms:

Modus Tollens (Denying the Consequent):

p – q

~q_____

Hypothetical Syllogism (Conditional Chain)

p –> q

q –> r__

p –> r

Disjunctive Syllogism

p v qp v q

~p___ or~q___

And here are some invalid forms:

Denying the Antecedent

p – q

~p_____

Affirming the Consequent

p – q

q_____

What is the form of the following arguments? Are they valid or invalid?
If I get married, I’ll be happy the rest of my life. I’ll get married. So I’ll be happy the rest of my life.
I’m either going to cook dinner or mow the lawn. I’m not going to mow the lawn. So I’ll cook dinner.
If I get married, I’ll buy a motorcycle. If I buy a motorcycle, my spouse and I will ride across the country. So if I get married, my spouse and I will ride across the country.
If I stay single, I’ll be happy the rest of my life. I won’t be happy the rest of my life. So I won’t stay single.
If I get married, I’ll buy a motorcycle. I’m not getting married. So I’m not going to buy a motorcycle.

A little harder: