Lesson

Logic is a tool for establishing sound arguments and drawing unshakable conclusions. Mathematicians use logic to prove their concepts. Lawyers use logic to sway judges and juries. Computer engineers use logic to design complex systems. You might even use logic to convince your friends to see a movie with you! Let's learn some of the basics of logic so that we can build solid arguments in the future.

We'll start with the most basic element of logic, the conjecture. A conjecture is a statement expressing a concept, and they are always either true or false.

Here are some examples of conjectures, and examples of things that are not conjectures:

Conjectures | Not conjectures |
---|---|

"Dogs have tails." | "Dogs." |

"Birds eat elephants." | "Birds and elephants." |

"I am sitting in a school." | "Hold the elevator!" |

"Aliens exist." | "Always be yourself." |

A conjecture can be true sometimes and false at other times, and we may not know whether a conjecture is true right now, but it must be either true or false in the end.

A hypothesis (plural: **hypotheses**) is a conjecture made as a starting point for reasoning, without needing to know whether or not it is true. We might wonder about hypothetical situations, and use these conjectures:

*"If dogs have tails..."
"If birds eat elephants..."
"If Aliens exist..."*

All of these are hypotheses. After we make a hypothesis, we often follow it naturally with a conclusion. A **conclusion** is a conjecture that might be true if the hypothesis is true:

*"...then my dog has a tail."
"...then elephants are scared of birds."
"...then other planets must have life."*

When we pair a hypothesis with its conclusion, we have a conditional statement. A conditional statement is a sentence expressing a hypothetical situation and its consequence. Each of the statements above come in matching pairs, but you can join them together however you like - the conjecture

*"If dogs have tails, then other planets must have life."*

... is a conditional statement! It also happens to be false.

Here are some more examples of conditional statements:

- If there are no more cookies left in the jar, then someone ate them all.
- If social media didn't exist, then we'd spend a lot less time on computers.
- If you don't clean your room tonight, then you're going to be grounded this weekend.

Look for the pattern in these statements - it's easy to identify the hypothesis and conclusion of a conditional statement when the words "if" and "then" are used. The word "if" is used for the hypothesis. It allows us to consider something, imagining a world where it might be true. The word "then" communicates the consequence of our hypothesis in a conclusion statement, what such a world might look like.

However, these words aren't always used. English, like all other languages, has many different ways of expressing the same thing. Here are some other examples of conditional statements that do not use either "if" or "then":

- An apple a day keeps the doctor away.
- Buy one, get one free!
- Eating chocolate every day is healthy.

The key to recognizing these as conditional statements is that each one imagines something happening, then describes what happens after. An important step will often be to rewrite these statements using "if" and "then" to clearly separate the hypothesis from the conclusion:

- If someone eats an apple a day, then they keep the doctor away.
- If you buy one, then you get one free!
- If you eat chocolate every day, then you are being healthy.

Conjectures, hypotheses, conclusions

A conjecture is a statement that can be either true or false.

A hypothesis is a statement that is assumed to be true, and a conclusion is a statement that is a result of the assumption. Together these two statements form a conditional statement (that can, as a whole, be either true or false).

Which of the following are conjectures? Select all that apply.

Colorless green ideas furious.

AWhat is your favorite color?

BDon't sit too close to the screen.

CWelcome to all the new residents!

DApples are made out of sunshine.

ESpider silk is stronger than steel.

FColorless green ideas furious.

AWhat is your favorite color?

BDon't sit too close to the screen.

CWelcome to all the new residents!

DApples are made out of sunshine.

ESpider silk is stronger than steel.

F

What is the negation of the statement "Some turtles do not have claws"?

All turtles have claws.

ASome turtles have brains.

BMore than one type of turtle does not have claws.

CAll creatures that have claws are turtles.

DAll turtles have claws.

ASome turtles have brains.

BMore than one type of turtle does not have claws.

CAll creatures that have claws are turtles.

D

In this question we will identify different kinds of statements.

Which of the following are conjectures? Select all that apply.

What does this button do?

AIf he replies to me, then we won't be late.

BYesterday when it all happened.

CThe music is loud and the room is crowded.

DThe moon orbits the earth.

EWhat does this button do?

AIf he replies to me, then we won't be late.

BYesterday when it all happened.

CThe music is loud and the room is crowded.

DThe moon orbits the earth.

EWhich is a conditional statement?

The moon orbits the earth.

AThe music is loud and the room is crowded.

BIf he replies to me, then we won't be late.

CThe moon orbits the earth.

AThe music is loud and the room is crowded.

BIf he replies to me, then we won't be late.

CWhich statement is the conclusion?

We won't be late.

AHe replies to me.

BWe won't be late.

AHe replies to me.

B

Now let's look at a few other ways to relate, combine, and change conjectures.

A conjecture has a truth value - it can be either true or false (but never both). A statement that is opposite in truth value to a conjecture is its negation.

Consider the conjecture: *"Birds eat elephants."* We know this is **false**.

The opposite of this statement is *"Birds do not eat elephants."* This is the **negation** of the original conjecture, and has the opposite truth value - it is **true**.

Whenever we abbreviate conjectures with a capital letter, like $P$`P`, we can represent its negation as $\sim P$~`P`, which you can say out loud as *"not* $P$`P`*"*.

The conditional statement formed by reversing the hypothesis and conclusion of another conditional statement is called the converse.

Consider the conditional statement, *"If an animal is a dog, then it has four legs."* Is this the same as saying, *"If an animal has four legs, then it is a dog"*?

We'll soon see how the converse of a true statement isn't always true itself - a horse has four legs, but a horse is not a dog!

We say that a conditional statement and its converse are logically independent - one, both, or neither can be true, depending on what kind of statement it is. If the original conditional statement was represented as *"If *$P$`P`* then *$Q$`Q`*"*, the converse is represented as *"If *$Q$`Q`* then *$P$`P`*"*.

We might change a conditional statement in another way by negating the hypothesis and conclusion but keeping the order the same. This statement is called the inverse.

Using the same conditional statement as before, *"If an animal is a dog, then it has four legs"*, the inverse statement would be *"If an animal is not a dog, then it does not have four legs"*.

Just like the converse, the inverse of a true conditional statement isn't always true itself - there are many animals that have four legs that are not dogs!

If the original conditional statement was represented as *"If *$P$`P`* then *$Q$`Q`*"*, the inverse is represented as *"If *$\sim P$~`P`* then *$\sim Q$~`Q`*"*.

When we negate both parts of a conditional statement (like we did for the inverse), and also switch the order (like we did for the converse), we form the contrapositive.

Let's keep going with the same conditional statement, "If an animal is a dog, then it has four legs". If we take the negation of the hypothesis and the negation of the conclusion, and swap the order they appear in, we get the statement *"If an animal does not have four legs, then it is not a dog"*.

Think about this statement for a moment - does it seem true to you?

In fact, the contrapositive is logically equivalent to the original conditional statement - if either are true, then the other one must be too! This will prove very useful in the future.

Using symbols, If the original conditional statement was represented as *"If *$P$`P`* then *$Q$`Q`*"*, the converse is represented as *"If *$\sim Q$~`Q`* then *$\sim P$~`P`*"*.

Let's summarize what we've learned.

Derived logical statements

The negation of $P$`P` is $\sim P$~`P`, and has opposite truth value to $P$`P`.

The converse of "if $P$`P` then $Q$`Q`" is "if $Q$`Q` then $P$`P`", and is logically independent of the original.

The inverse of "if $P$`P` then $Q$`Q`" is "if $\sim P$~`P` then $\sim Q$~`Q`", and is logically independent of the original.

The contrapositive of "if $P$`P` then $Q$`Q`" is "if $\sim Q$~`Q` then $\sim P$~`P`", and is logically equivalent to the original.

Let's see what these statements look like in a more mathematical setting.

Find the converse, inverse, and contrapositive of the following statement:

*"The square root of an even square number is also an even number."*

**Think:** We recognize this as a conditional statement since we are imagining something (in this case, taking the square root of an even square number) and describing what must follow (in this case, that we get an even number as a result). Let's start by rewriting the statement using "if" and "then":

**Original:*** "If *$x^2$`x`2* is even, then *$x$`x`* is even."*

Now it will be easier to find these related statements. The opposite of "even" is "odd", which will come in handy when we find them.

**Do:** To find the converse, we just swap the hypothesis and conclusion:

**Converse:*** "If *$x$`x`* is even, then *$x^2$`x`2* is even."*

To find the inverse, we take the negation of each of the parts without swapping:

**Inverse:*** "If *$x^2$`x`2* is odd, then *$x$`x`* is odd."*

To find the contrapositive, we both take the negation of each part, **and** swap their order:

**Contrapositive:** *"If *$x$`x`* is odd, then *$x^2$`x`2* is odd."*

**Reflect:** Which of these are true? Is it always this way?

If we think a conditional statement is the same as its converse or its inverse, or we think that it is different from its contrapositive, we might be the victim of a logical fallacy - believing an argument is valid when it really isn't!

Consider the conditional

If $C$`C` then $D$`D`,

Where $C$`C` and $D$`D` are both conjectures.

Which of the following represents the converse of the statement?

If $D$

`D`then $C$`C`.AIf $\sim$~$C$

`C`then $\sim$~$D$`D`.BIf $C$

`C`then $D$`D`.CIf $\sim$~$D$

`D`then $\sim$~$C$`C`.DIf $D$

`D`then $C$`C`.AIf $\sim$~$C$

`C`then $\sim$~$D$`D`.BIf $C$

`C`then $D$`D`.CIf $\sim$~$D$

`D`then $\sim$~$C$`C`.D

Consider the conditional

If $B$`B` then $D$`D`,

Where $B$`B` and $D$`D` are both conjectures.

Which of the following represents the inverse of the statement?

If $\sim$~$B$

`B`then $\sim$~$D$`D`.AIf $\sim$~$D$

`D`then $\sim$~$B$`B`.BIf $B$

`B`then $D$`D`.CIf $D$

`D`then $B$`B`.DIf $\sim$~$B$

`B`then $\sim$~$D$`D`.AIf $\sim$~$D$

`D`then $\sim$~$B$`B`.BIf $B$

`B`then $D$`D`.CIf $D$

`D`then $B$`B`.D

Consider the conditional

If a number ends in zero, then it is divisible by ten .

Which of the following represents the contrapositive of the statement?

If a number does not end in zero, then it is not divisible by ten.

AIf a number is not divisible by ten, then it does not end in zero.

BIf a number ends in zero, then it is divisible by ten.

CIf a number is divisible by ten, then it ends in zero.

DIf a number does not end in zero, then it is not divisible by ten.

AIf a number is not divisible by ten, then it does not end in zero.

BIf a number ends in zero, then it is divisible by ten.

CIf a number is divisible by ten, then it ends in zero.

D