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.
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, |
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(B) |
Sam wins - gets no kiss, |
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(C) |
Sam loses - gets a kiss, |
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(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.
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example 1
Di at home, studying |
Let p, q, r, and s be the statements
p : Di is home |
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r : Di is sleeping , |
q : Di is studying |
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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) |
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b. (r Ú ~p) → q |
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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) |
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b. |
(F Ú ~T) → T |
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c. |
(T → F) Ú (~T → ~T) |
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T → F |
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(F Ú F) → T |
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F Ú (F → F) |
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F |
F → T |
F Ú T |
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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.
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If it rains, I take my umbrella. |
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If it rains, then I take my umbrella. |
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All kangaroos hop. |
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If it is a kangaroo, then it hops. |
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You will pass the course if you study hard. |
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If you study hard then you will pass the course. |
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No pig can fly. |
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If it is a pig, then it cannot fly. |
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I only sleep when the room is dark. |
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If I sleep, then the room is dark. |
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I sleep whenever the room is dark. |
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If the room is dark, then I sleep. |
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The canary sings only if it is happy. |
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If the canary sings, then it is happy. |
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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 |
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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 |
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If Alice is here, then she came with Bill. |
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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. |
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The gate is open if the horse is gone. |
The horse is gone and the gate is not open. |
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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 |
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q : you are healthy . |
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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. |
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If the rhino does not charge, it is not angry. |
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If you brush your teeth, you do not get cavities. |
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If you get cavities, then you do not brush your teeth. |
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If George is not at work, then he is home. |
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If George is not home, then he is at work. |
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If you are not careless, then you do not have accidents. |
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If you have accidents, then you are careless. |
EXERCISES 3C
p : Fido barks |
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q : Felix jumps |
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r : Mickey runs |
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s : Tweety sings . |
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a. p → q |
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b. q → ~s |
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c. r → (p Ù q) |
d. (s Ú r) → q |
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e. ~p → s |
f. s → (r Ù q Ù p) |
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g. ~p → (~r Ù s) |
h. (~s Ù ~r) → p |
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i. (r Ù q) → (p Ù ~s) |
j. (q Ù ~r) → (p Ú s) |
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k. If Tweety sings then Fido barks. |
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l. If Felix does not jump, then Fido does not bark. |
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m. If Felix jumps, then Mickey runs and Tweety does not sing. |
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n. Mickey runs only if Felix jumps. |
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o. Mickey does not run and Fido does not bark, if Tweety sings. |
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p. Whenever Felix jumps, Mickey runs or Fido barks. |
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q. If Fido barks and Mickey runs and Tweety sings, then Felix jumps. |
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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. |
a. p → s |
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c. (p Ú r) → s |
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e. (~s → q) Ù (r → ~p) |
b. ~(s → q) |
d. ~r → (q Ú s) |
f. [(p → ~q) Ú (~s Ù q)] → (r → ~p) |
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