1.1 Implication

We will start our examination of implication by giving a precise definiton of the term.

Put another way, p implies q whenever it is impossible for p to be true and q to be false.

Sentences of this type are called implications, or sometimes 'conditionals', 'if then statements', 'hypotheticals' or 'entailments'.

Let us try to visualize what the definition for implication is saying. To do so we create a table that has the letters p and q at the top. These letters stand for statements and are called statement variables. Since we could have p true but q false, or q false when p is true, and so forth, the first thing we do is list all four of these possible combinations in our table.

 p q T T T F F T F F

Notice that the first row represents the case where p and q are both true, while on the third row, for example, the table tells us that p is false while q is true.

Now we need to add a third column which indicates the truth value of the statement, "p implies q" which will depend of the values of p and q as given in each row of the table.

From the second part of the definition we see that if the statement "p implies q" is true then it is impossible for p to be true and q to be false, hence if p is true and q is false (the combination given in the second row) then the statement "p implies q" must also be false, since that is what the definition tells us. The following table represents the above information.

 p q "p implies q" T T T F F F T F F

The next question is a bit tricky. We don't want to leave the other rows under the heading "p implies q" blank. Recall if we put a "T" in that column for a specific row, it indicates that the statement "p implies q" can be true for the specific combination of truth values given on that row.

As it turns out, the definition of implication only forbids one combination of p and q (the one on the second row) and says nothing about the others, hence for now we will add the value of T to all of the other rows. At this point we will not state why we can do this, but indeed that turns out to be the correct truth value in the other rows, and we will justify this completely when we look at formal logic in Part 2.

So to summarize: What the definition of implication says is that "p implies q" is a false statement only if it is possible for p to be true and q to be false.

 p q "p implies q" T T T T F F F T T F F T

The above observations lead to the following two step method we can use to determine whether a proposition p does not imply another proposition q.

Note - this is a test to determine whether p does NOT imply q.

 The Two Step Method to determine whether a proposition p does NOT imply another proposition q.   ☐ It is possible for the proposition p to be true (by this we mean that one can imagine p being true, even if we know it is not actually true). ☐ Under the possibility given above, we can imagine q being false.   If both of these conditions (boxes) can be checked, then p does NOT imply q. On the other hand, if one or more of these conditions (boxes) can't be checked, then p does imply q.

Example 1.1.1

Let p be, "Mary knows all the capitals of the United States", and let q be, "Mary knows the capital of Kentucky".

We use the two step method to see whether p does not imply q.

☐ It is possible for the proposition p to be true (by this we mean that one can imagine p being true, even if we know it is not actually true).

☐ Under the possibility given above, we can imagine q being false.

We can clearly imagine that someone named Mary does know all of the capitals of the United states, hence we can place a check mark in the first box.

☑ It is possible for the proposition p to be true (by this we mean that one can imagine p being true, even if we know it is not actually true).

However, if Mary really does known all of the capitals of the United States, then the statement that "Mary knows the capital of Kentucky" can not be false. So we can't place a check mark in the second box.

☒ Under the possibility given above, we can imagine q being false.

Hence we can not check both boxes, hence p implies q.

Example 1.1.2

Let p be, "Everyone in the race ran the mile in under 5 minutes and John was a runner in the race", and let q be, "John ran the mile in less than 5 minutes", then q implies p.

Again we use the two step method to see whether p does not imply q.

☐ It is possible for the proposition p to be true (by this we mean that one can imagine p being true, even if we know it is not actually true).

☐ Under the possibility given above, we can imagine q being false.

Since it is possible for p to be true, we only need to check if, under this imaginary scenario, q could be false. But if indeed everyone in the race ran the mile in under 5 minutes, then the statement that John ran the mile in less than five minutes can't be false.

Hence while we can put a check in box one, we can't put a check in box two. So p implies q.

Example 1.1.3

Let p be, "The questionnaire had a total of 20 questions, and Mary answered only 13", and let q be the statement, "The questionnaire answered by Mary had 7 unanswered questions".

Again, we can imagine p being true, so the real question is whether under that possibility q can be false. But if the questionnaire really has a total of 20 questions and Mary only answered 13 (we are imagining this to be true), then q can't be false. Hence we can't check both boxes, so p implies q.

Example 1.1.4

Let p be, "The winning ticket starts with 3-7-9 and Mary's ticket starts with 3-7-9", and let q be the statement, "Mary's ticket is the winning ticket".

We can imagine p being true, so we can check the first box.

☑ It is possible for the proposition p to be true (by this we mean that one can imagine p being true, even if we know it is not actually true).

But we can also imagine that q is false, since even if p is true, we can imagine the winning ticket being 3-7-9-4 where Mary's ticket is 3-7-9-0. Hence we can check the second box also.

☑ Under the possibility given above, we can imagine q being false.

Hence we have checked both boxes, so p does NOT imply q in this case.

Exercise 1.1.5

Suppose p is the proposition, "Today is Monday", give a proposition q such that p implies q.

Solution: We want to find a proposition q which has to be true whenever "Today is Monday" is true. There are many possibilities, but we will choose an easy one. Let q be the proposition "Tomorrow is Tuesday", then if "Today is Monday" is a true statement, then by definition of weekdays, "Tomorrow is Tuesday" must also be a true statement.

Exercise 1.1.6

Suppose p is the proposition, "Today is Monday", give a proposition q such that p does not imply q.

Solution: We want to find a proposition q which can be false whenever "Today is Monday" is true. There are many possibilities, but we will choose an easy one. Let q be the proposition "Tomorrow is July 4th", then it is possible for p to be true and q to be false, hence we have chosen a q such that p does not imply q.

Exercise 1.1.7

Suppose q is the proposition, "All circles are perfect squares", find a proposition p such that p implies q.

Solution: This one is a bit harder than the previous two exercises, and will require some careful thinking. First note that the proposition q is always false. This seems to make the choice of p impossible, since if p can be true, then it is possible to have p true and q false (since q is always false), which would mean our choice of p does not imply q. Does that mean that the question is impossible to answer? Think carefully about the question first and then check your answer by clicking on the box below.

Is it possible to choose a p such that p implies q when q is always false?

Before we consider the second relation between statements called relevance, test your understanding by taking the following quiz.

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