What's the CONTRAPOSITIVE?

[L and G] —> Z

/Z —> [/L OR /G]

When contraposing a statement that has "and", switch both sides, negate all concepts, and switch the AND to an OR.

What's the CONTRAPOSITIVE?

/R —> [W and Z]

[/W OR /Z] —> R

When contraposing a statement that has "and", switch both sides, negate all concepts, and switch the AND to an OR.

What's the CONTRAPOSITIVE?

[V or M] —> R

/R —> [/V AND /M]

When contraposing a statement that has "or", switch both sides, negate all concepts, and switch the OR to an AND.

What's the CONTRAPOSITIVE?

W —> [/H or /O]

[H AND O] —> /W

When contraposing a statement that has "or", switch both sides, negate all concepts, and switch the OR to an AND.

TRUE OR FALSE:

[W or T] —> B

is equivalent to

W —> B

AND

T —> B

TRUE

The initial statement means that either W or T would be sufficient to guarantee B. So this means that W, by itself, is sufficient for B, and also, T, by itself, is sufficient to guarantee B.

The rule:

When OR is on the LEFT side of a conditional, you CAN SPLIT the OR into two separate statements.

TRUE OR FALSE:

[Q and Z] —> E

is equivalent to

Q —> E

AND

Z —> E

FALSE

The initial statement means that having both Q and Z will guarantee E. But it doesn't mean that having Q by itself, or that having Z by itself, will guarantee E.

The rule:

When AND is on the LEFT side of a conditional, you CANNOT SPLIT the AND.

TRUE OR FALSE:

Y —> [O and C]

is equivalent to

Y —> O

AND

Y —> C

TRUE

The initial statement means that Y is enough to guarantee both O and C. So if Y is true, O must be true. And, if Y is true, C must also be true.

The rule:

When AND is on the RIGHT side of a conditional, you CAN SPLIT the AND into two statements

TRUE OR FALSE:

B —> [S or D]

is equivalent to

B —> S

AND

B —> D

FALSE

The initial statement means that B is enough to guarantee *at least one* of S or D. But we don't know that both of them have to follow. So we can't say that B will guarantee S. And we can't say that B will guarantee D.

The rule:

When OR is on the RIGHT side of a conditional, you CANNOT SPLIT the OR.

DIAGRAM this statement and the CONTRAPOSITIVE:

The park rangers will find the man-eating bear only if they follow the bloody trail and stay very quiet.

Find bear —> [Follow bloody trail AND stay very quiet]

[Do NOT follow bloody trail OR do NOT stay very quiet] —> NOT find bear

**ONLY IF **introduces the **NECESSARY** condition.

When contraposing a statement with "and", switch both sides, negate all terms, and transform AND to OR.

DIAGRAM this statement and the CONTRAPOSITIVE:

No student who procrastinates will feel good about their paper or get a lot of sleep.

Procrastinates —> [NOT feel good about paper AND NOT lot of sleep]

[Feel good about paper OR Lot of sleep] —> NOT procrastinate

**NO **introduces **SUFFICIENT**, but **NEGATE THE OTHER CONCEPT**. Since the other concept used OR ("feel good about their paper OR get a lot of sleep"), to negate it we turned OR into AND and negated both terms.

Procrastinates —> [NOT feel good about paper AND NOT lot of sleep]

To take the contrapositive, we switched both sides of the arrow, negated every concept, and turned the OR back into an AND.

[Feel good about paper OR Lot of sleep] —> NOT procrastinate

What is the NEGATION of this statement?

If A, then B.

A can be true even if B is NOT true.

The negation of a conditional statement is the idea that the SUFFICIENT can be true even though the NECESSARY is NOT true.

What's the NEGATION of this statement?

X can occur only if G occurs.

X can occur even if G does NOT occur.

The negation of a conditional statement is the idea that the SUFFICIENT can be true even though the NECESSARY is NOT true. The original statement was asserting that X was sufficient for G. The negation means that X can be true without G being true.

When can we connect one conditional statement to another?

When the SUFFICIENT CONDITION of one statement MATCHES the NECESSARY CONDITION of the *other* statement.

A —> **B**

**B **—> C

Thes statements above can be combined:

A —> B —> C

(Note that it also must be true that A —> C. You don't always need to write out the complete connection.)

Write out the conditional connections possible from the statements below:

X —> V

/X —> L

/L —> X —> V

/V —> /X —> L

The contrapositive of the second statement is /L —> X, which allows us to connect to the first statement. (You can also do the contrapositive of the first statement, /V —> /X, to make the connection.)

Write out the conditional connections possible from the statements below:

B —> F

T —> L

L —> B

T —> L —> B —> F

/F —> /B —> /L —> /T

The second statement connects to the third, which then connects to the first.

Write out the conditional connections possible from the statements below:

R —> T

Y —> T

Q —> Y

Q —> Y —> T

/T —> /Y —> /Q

The third statement connects to the second statement. Note that the first statement does NOT connect to the second, since the shared term, T, is in both necessary conditions. In order to make a connection, the shared term has to be in the SUFFICIENT condition of one statement AND the NECESSARY condition of another.

Write out the conditional connections possible from the statements below:

We cannot hire more teachers with science degrees without increasing teacher pay. In order to improve our country's performance on science tests, we must hire more teachers with science degrees.

Improve peformance on science test —> Hire more teachers —> Increase teacher pay

NOT increase teacher pay —> NOT hire more teachers —> NOT improve performance on science test

To make the connection above, we have to take the contrapositive of one of the initial statements. The first statement can be diagrammed as "NOT increase teacher pay —> NOT hire more teachers". The contrapositive is "Hire more teachers —> increase teacher pay", which connects to the necessary condition of the second statement.

Write out the conditional connections possible from the statements below:

The wild badgers overrunning the town can be stopped only if the citizens of the town work together. But an end to the violent conflict between the Smiths and the Changs is required in order for the citizens of the town to work together. A sizable payment to each family would be sufficient to end the conflict between the Smiths and the Changs.

Wild badgers stopped —> Citizens work together —> Violent conflict ends

Violent conflict does NOT end —> Citizens do NOT work together —> Wild badgers NOT stopped

The first statement connects to the second. But the last statement does NOT connect to ther others. Even if the payment is sufficient to end the conflict, that doesn't mean the payment is necessary to end the conflict.

DIAGRAM as a conditional:

The gym is closed or the mall is closed.

If gym is NOT closed —> Mall is closed

If mall is NOT closed —> Gym is closed

An OR statement can be diagrammed as a conditional by negating one of the terms and making it the sufficient condition. This makes sense because OR means *at least one of these things is true*. So if one of them is NOT true...the other one must be true.

This is mostly useful in logic games rules. In logical reasoning, diagramming OR like this might be more confusing than it's worth.

DIAGRAM as a conditional:

A person cannot be both an athlete and a lawyer.

Athlete —> NOT lawyer

Lawyer —> NOT athlete

"CANNOT BE BOTH" can be diagrammed as a conditional by making one of the terms the sufficient condition, negating the other term and making it the necessary condition. This makes sense because NOT BOTH means that the options are mutually exclusive: if one of the option is true then the other option cannot be true.

Diagramming NOT BOTH is mostly useful in logic games.

DIAGRAM:

If you pass the test, then you will be selected for the honors program, if you apply before June 1.

[Pass test AND apply before June 1] —> Selected for honors program

NOT selected for honors program —> [NOT pass test OR NOT apply before June 1]

Here's another way to diagram the statement:

Pass test —> [Apply before June 1 —> Selected for honors program]

In this version, the NECESSARY condition of the overall statement is *itself* a conditional statement (in brackets). But what this means is if you pass the test AND you apply before June 1, then you'll be selected for the honors program.

DIAGRAM:

Unless Y is true, J only if Z.

[/Y AND J] —> Z

/Z —> [Y OR /J]

This is another diagram that means the same thing:

/Y —> [J —> Z]

[J AND /Z] —> Y

VALID OR INVALID?

Premise: [X or Y] —> Z

Premise: T —> Y

Conclusion: T —> Z

VALID.

Premise 2 tells us that T guarantees Y. Premise 1 tells us that either X or Y (or both) would guarantee Z. Since T leads to Y, and Y leads to Z, we can conclude that T guarantees Z.

VALID OR INVALID?

Premise: [Y and R] —> E

Premise: L —> R

Conclusion: L —> E

INVALID

Premise 1 tells us that having BOTH Y and R is sufficient for E. But having R by itself is not enough to prove E. So even though Premise 2 tells us that L guarantees R, do still don't know anything about Y, which means we cannot conclude that L is sufficient for E.

VALID OR INVALID?

Premise: /T —> R or D.

Premise: R —> Z

Premise: /Z —> /D

Conclusion: /Z —> T

VALID

Premise 1 tells us that NOT T leads to R or D. But the next two premises tell us that R leads to Z and D also leads to Z. (Note that you haev to do the contrapositive of the third premise to see that D leads to Z.) Since each of R and D leads to Z, that means that NOT T does ultimately lead to Z, no matter which part of the OR is true. The conclusion here is just the contrapositive of "NOT T —> Z", so it's valid.