3C  Implications


An implication is a statement having the form “if p then q”. Examples are


1)  If it rains then I will stay home.

2)  If you get a degree then you can get a job.

3)  If the car is gone then Lisa has left.


It can be confusing as to when an implication should be considered true and when it should be considered false. To resolve this question we discuss a hypothetical example in some detail.

Sam and Sue

Who won ?

Let us suppose that Sam will play tennis with his girl friend Sue, and that Sue, wanting to motivate Sam a little, entices him with the statement

“If you win then I will give you a kiss.”

What does Sue mean? If Sam wins, then obviously he will get a kiss, but Sue did not commit herself one way or the other in the event that Sam loses. There are four possible outcomes at the end of the match - namely,


(A)

 

Sam wins - gets a kiss,

(B)

 

Sam wins - gets no kiss,

(C)

 

Sam loses - gets a kiss,

(D)

 

Sam loses - gets no kiss.


However, Sue's statement rules out (B). She did not mention (C) or (D), so if Sam loses Sue is free to kiss him or not. In effect, what Sue means in her statement to Sam is that outcomes (A), (C), and (D) might happen, but that (B) will not. Sue will be caught in a lie only if outcome (B) occurs - in the other three cases she will have spoken the truth.

To write Sue's statement symbolically we define simpler statements,

p : you win       ,        q : I will give you a kiss .

The logic symbol for implication is “→”, and is read “implies”. We point this arrow from p to q to form Sue's compound statement,

p → q : If you win, then I will give you a kiss.

Our analysis concludes that this implication is false only when p is true and q is false, in which event Sam wins but sadly receives no kiss; in all other outcomes the statement is true. Accordingly, our truth table for implication winds up looking as shown; the corresponding logic equations for implication are listed at the right of the table.


Truth Table
for Implication

p

q

p → q

T

T

T

T

F

F

F

T

T

F

F

T

 

 

T → T = T

T → F = F

F → T = T

F → F = T


example 1

di  

Di at home, studying

Let p, q, r, and s be the statements


p : Di is home

,

r : Di is sleeping ,

q : Di is studying

,

s : Di is sick .


Below are some symbolic statements involving these simple statements, followed by their interpretations into words.


  p → r  

  If Di is home then she is sleeping.

  s → ~p  

  If Di is sick then she is not home.

  ~q → (s Ù r)  

  If Di is not studying, then she is sick and sleeping.

  (p Ú r) → ~q  

  If Di is home or sleeping, then she is not studying.

  (~r Ù ~s) → (q Ú ~p)  

  If Di is not sleeping and not sick, then she is studying or not at home.




example 2

Given that statements p and q are true, but that r is false, we determine the truth value of

a.   p → (q Ù r)

 

b.   (r Ú ~p) → q

 

c.   (q → r) Ú (~p → ~q) .

We substitute T for p, T for q, and F for r, and proceed as indicated:


a.

T → (T Ù F)

 

b.

(F Ú ~T) → T

 

c.

(T → F) Ú (~T → ~T)

 

T → F

 

(F Ú F) → T

 

F Ú (F → F)

F

F → T

F Ú T

 

T

T


Thus, the statement of part a is false, while those of parts b and c are true.




In everyday language there are many ways we can state implications without using the exact “if p then q” format; it is the intended meaning that determines an implication - not the precise language. Whenever we express in some way or another that one thing leads to something else, then we communicate an implication. Following are some implications stated in various ways, with accompanying translations into the “if p then q” format. As the examples perhaps demonstrate, it is not always immediately obvious how to make this translation.


kangaroo

 

If it rains, I take my umbrella.

If it rains, then I take my umbrella.


All kangaroos hop.

If it is a kangaroo, then it hops.

 

You will pass the course if you study hard.

If you study hard then you will pass the course.

 

No pig can fly.

canary

If it is a pig, then it cannot fly.

 

I only sleep when the room is dark.

If I sleep, then the room is dark.

 

I sleep whenever the room is dark.

If the room is dark, then I sleep.

 

The canary sings only if it is happy.

If the canary sings, then it is happy.


 

little girl

How do we negate an implication? Suppose that a mother tells a little girl, “If you clean your room, you may watch TV”. We can write the mother's statement as p → q, where p and q are the simple statements

p : you clean your room

,

q : you may watch TV .

This statement will be found to be false if the little girl does indeed clean her room but then is not allowed to watch TV - that is, if the statement

p Ù ~q

is true. These observations suggest that the negation of the implication p → q is the conjunction p Ù ~q. The corresponding formula for negating an implication then is


~(p → q) ≡ p Ù ~q .


We can verify this formula with a truth table. We build up both sides of the formula in the table, and observe that the columns corresponding to the two sides of the equation are identical:


p

q

p → q

~(p → q)

~q

p Ù ~q

T

T

T

F

F

F

T

F

F

T

T

T

F

T

T

F

F

F

F

F

T

F

T

F


example 3

On the left are some implications, and on the right their negations :


IMPLICATION


NEGATION

If Alice is here, then she came with Bill.

 

Alice is here and she did not come with Bill.

If the cat's away, the mice do play.

The cat's away and the mice do not play.

The gate is open if the horse is gone.

The horse is gone and the gate is not open.

Only if he is alone is Al homesick.

Al is homesick and he is not alone.




Given an implication p → q, there are three other compound statements naturally associated with this implication - the converse, the inverse, and the contrapositive of the implication. The table below describes the relationship between these four statements.


 Implication

 p → q 

 Converse

 q → p 

 Inverse

 ~p → ~q 

 Contrapositive   

 ~q → ~p 


Consider for example the implication “if you jog, you are healthy”. This statement has the form p → q, where p and q are specified as

p : you jog

,

q : you are healthy .

jogger

 

The four associated statements in this instance are


Implication : If you jog, you are healthy.

Converse : If you are healthy, you jog.

Inverse : If you do not jog, you are not healthy.

Contrapositive : If you are not healthy, you do not jog.


The truth table below exhibits the truth values of these four associated statements.


p

q

~p

~q

p → q

q → p

~p → ~q

~q → ~p

T

T

F

F

T

T

T

T

T

F

F

T

F

T

T

F

F

T

T

F

T

F

F

T

F

F

T

T

T

T

T

T


The implication and its contrapositive, having identical columns, are equivalent; therefore,


p → q ≡ ~q → ~p .


Likewise, the converse and the inverse have identical columns and are equivalent - but this is nothing different, as the inverse is the contrapositive of the converse. Also, the “contrapositive of the contrapositive” is the original implication. (If you negate p and q and switch the direction of the arrow, and then repeat the process, negating ~p and ~q and again switching the direction of the arrow, you are back to where you started.)


example 4

In each pair of statements below, the two statements are contrapositives of each other and thus logically equivalent - that is, logically they mean the same thing.


If the rhino is angry, it charges.

 

rhino

If the rhino does not charge, it is not angry.

 

If you brush your teeth, you do not get cavities.

If you get cavities, then you do not brush your teeth.

 

If George is not at work, then he is home.

If George is not home, then he is at work.

 

If you are not careless, then you do not have accidents.

If you have accidents, then you are careless.





EXERCISES 3C


  1. Let p, q, r, and s be the statements

    p : Fido barks

    ,

    q : Felix jumps

    ,

    r : Mickey runs

    ,

    s : Tweety sings .

    Change the symbolic statements into ordinary sentences :
    Fido

     

    a.  p → q

     

    b.  q → ~s

     

    Felix

    c.  r → (p Ù q)

    d.  (s Ú r) → q

    e.  ~p → s

    f.  s → (r Ù q Ù p)

    g.  ~p → (~r Ù s)

    h.  (~s Ù ~r) → p

    i.  (r Ù q) → (p Ù ~s)

    j.  (q Ù ~r) → (p Ú s)

    Write the sentences symbolically :
    Mickey

     

    k.  If Tweety sings then Fido barks.

     

    Tweety

    l.  If Felix does not jump, then Fido does not bark.

    m.  If Felix jumps, then Mickey runs and Tweety does not sing.

    n.  Mickey runs only if Felix jumps.

    o.  Mickey does not run and Fido does not bark, if Tweety sings.

    p.  Whenever Felix jumps, Mickey runs or Fido barks.

    q.  If Fido barks and Mickey runs and Tweety sings, then Felix jumps.

    r.  Persuant to Tweety's failure to sing and Fido's refusal to bark, either Felix is obliged not to jump or Mickey discovers an irresistable urge to ambulate swiftly.


  2. Given that statements p and q are true, and r and s are false, determine the truth value of each of the following :

    a.  p → s

     

    c.  (p Ú r) → s

     

    e.  (~s → q) Ù (r → ~p)

    b.  ~(s → q)

    d.  ~r → (q Ú s)

    f.  [(p → ~q) Ú (~s Ù q)] → (r → ~p)


  3. Mr. Chen said to his wife, “If it does not rain today, I will cut the grass.” Mrs. Chen came home in the evening and found water everywhere but the grass nicely trimmed. Did Mr. Chen tell the truth?

  4. Change each of the implications into the format “if p then q” :
    1. I feel relaxed if I listen to soothing music.
    2. Evelyn buys clothes only when Ala Moana is having a sale.
    3. A green apple tastes sour.
    4. 3x + 1 = 7 implies x = 2.

       

      riding a chicken
    5. Eating greasy foods causes you to gain weight.
    6. No one in his right mind will wrestle a gorilla.
    7. A rise in interest rates leads to a decline in disposable income.
    8. Drinking coffee in the evening makes me lose sleep at night.
    9. You get a free dessert when you order a deluxe dinner.
    10. You don't qualify for a scholarship without a B average.
    11. A wild and reckless life is inevitably accompanied by premature aging.
    12. You cannot ride a chicken without a saddle.

  5. Use a truth table to verify that the statements p → q and ~p Ú q are logically equivalent.

  6. Use the result of the previous exercise to write a disjunction equivalent to the given implication :

     

    goat

     

    1. If you turn your back, the goat will butt you.
    2. If the milk was left outside the refrigerator, then it is spoiled.
    3. If Angie has not left already, then she will be late.
    4. If I do not drink coffee in the morning, then I get sleepy.
    5. If you do not study hard, then you make bad grades.

  7. Give a negation for each implication :
    1. If the pig has wings, it can fly.
    2. If wages are up, then unemployment is down.
    3. If the window is open, then it rained inside.
    4. If the car won't start, it is out of gas.
    5. If Brad were not shy, he would ask Fay for a date.
    6. The spaghetti is done if it sticks to the wall.
    7. The door is open only if the professor is having office hours.
    8. Mrs. Ramos is unhappy whenever Mr. Ramos is late for dinner.

     

    flying pig

  8. For each implication, state the converse, inverse, and contrapositive :

     

    peacock
    1. If the leaves have fallen, then autumn is here.
    2. If it is six o'clock, then the news is on TV.
    3. If the surf is not up, then the beach is empty.
    4. If the light is off, then you cannot read.
    5. If the match is wet, it will not light.
    6. If you do not read the paper, you are not informed.
    7. The cars proceed only if the light is green.
    8. The peacock spreads his feathers if he wants his picture taken.