18. Implication
19. Other Forms of the Implication
20. Contrapositive
21. Negation of an Implication
22. Converse
23. Equivalence Statements
24. If and Only If
25. Law of Detachment
26. Law of Detachment not Usable
27. Law of Syllogism
28. Law of Syllogism not Usable
If the river is narrow, then we can cross it quickly.
It may be written in symbols as follows:
p implies q,
where
p = The river is narrow
q = We can cross the river quickly.
In logic, this implication means that when p is true, q must also be true. But when p is false we cannot tell from the implication whether q is true or false. In other words, there are two possibilities: (1) q is true because p is true, and (2) p is false. These facts are summarized in the following definition of the implication:
p implies q = not-p or q.For example, the statement
If the river is narrow, then we can cross it quickly
has the same meaning as
The river is not narrow, or we can cross it quickly.
If the food has been cooked, then we can have lunch.
If the river is narrow, then we can cross it quickly
are as follows:
If the food has been cooked, then we can have lunch
in the other ways given above.
p implies q = not-p or q.
From the commutative law we have:
not-p or q = q or not-p.
Therefore
p implies q = q or not-p.
From the double negative law we have:
q = not-(not-q).
Therefore
p implies q = not-(not-q) or not-p.
Again, from the definition of an implication we have:
not-(not-q) or not-p = not-q implies not-p.
Therefore
p implies q = not-q implies not-p.This is called the contrapositive law. The statement not-p implies not-q is called the contrapositive of the statement p implies q.
For example, let
p = The river is narrow
q = We can cross the river quickly.
Then p implies q is the implication:
If the river is narrow, then we can cross it quickly,
and the contrapositive not-q implies not-p is:
If we cannot cross the river quickly, then the river is not narrow.
These two implications have the same meaning.
If the tank is full, then we have enough water.
p implies q = not-p or q.
Therefore, taking the negation of each side of the equation, we have:
not-(p implies q) = not-(not-p or q).
From the negation law for logical OR we have:
not-(not-p or q) = not-(not-p) and not-q.
Therefore
not-(p implies q) = not-(not-p) and not-q.
From the double negative law we have:
not-(not-p) = p.
Therefore
not-(p implies q) = p and not-q.This is the law for the negation of an implication.
For example, the negation of the statement
If the river is narrow, then we can cross it quickly
is the statement
The river is narrow and we cannot cross it quickly.
If the tank is full, then we have enough water.
p implies q
has a converse statement
q implies p.
These two implications do not have the same meaning. For example, the implication
If she is inside the house, then she cannot see the moon
does not have the same meaning as its converse statement
If she cannot see the moon, then she is inside the house.
If the tank is full, then we have enough water.
(p implies q) and (q implies p).
For example, let
p = Her left shoe is black
q = Her right shoe is black.
Then
(p implies q) and (q implies p)
means
(a) If her left shoe is black then her right shoe is black, and if her right shoe is black then her left shoe is black.
This kind of statement is often written in the shorter form:
(b) Her left shoe is black if and only if her right shoe is black.
Another important way of writing the same statement is:
(c) For her left shoe to be black it is necessary and sufficient that her right shoe should be black.
Statements such as these are called equivalence statements.
p = The post office is open
q = Today is a weekday.
Use the patterns (a), (b), and (c) given above.
p iff q = (p implies q) and (q implies p).Here the first implication means that when p is true, q must be true, and we cannot have p true and q false. The second implication means that when q is true, p must be true, and we cannot have q true and p false.
Therefore, in an equivalence statement the only possibilities are: (1) p and q both true, and (2) p and q both false. Then p and q are said to be equivalent.
For example, let
p = This month is May
q = Next month is June.
Here it is true to say: If this month is May, then next month is June, and if next month is June, then this month is May. Therefore the statements p and q are equivalent.
As another example, let
p = This month is May
q = Next month is later than April.
Here it is true to say: If this month is May, then next month is later than April. But it is not true to say: If next month is later than April, then this month is May. (Because q is true and p is false when, for example, this month is June.) Therefore, in this example, the statements p and q are not equivalent.
p and (p implies q).
By the definition of the implication, this is:
p and (not-p or q).
By the distributive law, we now have:
(p and not-p) or (p and q).
By the law of contradiction p and not-p is false, so p and q must be true. This shows we may conclude that q is true, because we have already supposed that p is true.
We may summarize this result as follows:
From p and (p implies q) we conclude q.This is called the law of detachment.
For example, let
p = This machine is a bicycle
q = This machine has two wheels.
Then
p implies q = If this machine is a bicycle, then it has two wheels.
If p is true and p implies q is always true, then we conclude that q is always true. In other words we conclude: This machine has two wheels.
The quality of this material is good
If the quality of this material is good, then we shall buy it.
For example, suppose we have the statements
If the animal has eaten poisoned food, then it is dead (p implies q)
and
The animal is dead (q).
We cannot conclude that the animal has eaten poisoned food; it may have died from another cause.
(p implies q) and (q implies r).
The first implication means that when p is true, q must also be true and we cannot have p true and q false. The second implication means that when q is true r must also be true, and we cannot have q true and r false. These results show that when p is true r must also be true, and we cannot have p true and r false. In other words: p implies r.
We may summarize this result as follows:
From (p implies q) and (q implies r) we conclude (p implies r).This is called the law of syllogism.
For example, let
p = The canal is open
q = The ship can go through the canal
r = The company can transport the goods.
Then, from the two statements
If the canal is open, then the ship can go through (p implies q)
and
If the ship can go through the canal, then the company can transport the goods (q implies r)
we conclude
If the canal is open, then the company can transport the goods (p implies r).
If the electric power is cut, then the refrigerator does not work.
If the refrigerator does not work, then the food is spoiled.
For example suppose we have the statements
If the calculator is broken, then it is not working (p implies r)
and
If the calculator has no battery, then it is not working (q implies r).
We cannot conclude anything. When the calculator is not working we cannot conclude that it is broken, or that it has no battery. There may be another reason why it is not working (the battery may be old, or the calculator may be switched off).
Backward links: Statements I
Forward links: Quantifiers I Quantifiers II Sets I Sets II Relations I Relations II
By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.
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