If . . . then? An Introduction to Logic
Relations between statements. Implication, Relevance and Independence

Introduction

Logic, like mathematics, psychology, history, physics, and many other academic fields is a broad subject with many specialized areas and sub-disciplines. The aim of the first part of this text is to introduce you to the main ideas behind what is called philosophical logic.

In general, philosophical logic is the study of the principles of correct reasoning. This study includes all the tools one can use in deciding how well a claim is supported by a set of reasons given to support the truth of that claim. As it turns out, this study naturally branches to other important topic, all of which can be traced back directly to questions related to how well claims are supported by premises or how claim can be proven, meaning that if the reasons given for the claim are true, then the conclusion must be true.

Even though this is an introductory course, this examination will cover a variety of topics, which include:

• Relations between propositions
• Arguments and argument classifications
• Truth table methods applied to propositional forms and argument forms
• The method of natural deduction
• Sets and basic set operations
• Basic concepts of probability and probabilistic inference

How to do well in this course

Like so many formal subjects, such as mathematics, physics, biology, etc., logic requires careful attention to the definitions of terms. Every correct answer will follow from the correct definition(s) of the term(s) involved and basic reasoning skills which you bring with you to a logic class (such as the process of elimination). Also note that the logical meanings of many terms will be different than the meanings of those same terms outside of logic! This signifies that if you think you know what a term means, it is a good idea to always check to see if your meaning is exactly the same of the one given in the reading.

WARNING: the previous paragraph should be taken seriously - most errors in logic can be traced back to the student not recalling the precise definition of a term, and defaulting to some 'vague' meaning of the term they acquired before the course.

Logic is also a subject that must be learned actively, with penciled and paper in hand to attempt to reason through the questions given in the body of the text, and take notes on points that you do not understand and ask questions and consider different examples until you do understand.

Section 1

The Roll of two dice.

To start us down that road, consider the following question given below.

Was your answer correct? If not, do you understand why?

The process of looking at information, and drawing conclusions from that information invites us to examine relations between statements, and our primary question for the next few sections will be to learn how to determine whether the truth of one statement affects the truth of another statement, and when it does, what can be said about that relationship.

1 Relations between statements

Let's consider another example:

1. I have a dime, a quarter, and 11 pennies in my pocket.

2. I have some change in my pocket.

For simplicity's sake, let's call the first statement p and the second statement q. Now we pose our first question related to logical analysis:

Does the truth of p have anything to do with the truth of q?

By this, we simply mean, in the broadest of terms, whether the truth of the statement, "I have a dime, a quarter and 11 pennies in my pocket" lends credence to the statement, "I have some change in my pocket"?

Clearly the answer is yes, for if the first is true, then certainly the second must be true. Why is this so? The answer to this question has to do with the connection between the terms, "dime", quarter" and "pennies" and the word "change" when used in this context and the many possible ways one can have change in one's pocket.

Now let's turn the question around and ask:

Does the truth of q have anything to do with the truth of p, or more plainly, does the truth of, "I have some change in my pocket" increase the probability that, "I have a dime, a quarter, and 11 pennies in my pocket" is true?

To make this as clear as possible, what we are asking is whether the truth of q gives us any reason whatsoever to suspect that p might be true?

Again, the answer is yes, as one possible way to have change in one's pocket is indeed to have a dime, a quarter and 11 pennies, but as this is just one of many combinations of coins which makes the statement true, the truth of q does not guarantee the truth of p. Let's consider another example:

1. Today is October 1.
2. The word, "ostentatious" has 12 letters.

As above we will refer to the first sentence as p and the second as q , and ask the same questions:

Does the truth of p have anything to do with the truth of q , and similarly

Does the truth of q have anything to do with the truth of p?

If you are worried that statement p is not actually true, (since today is probably not October 1), don't be. To answer this question, we just assume that it is true - just like we assumed in the previous case that we had a dime, a quarter and 11 pennies in our pocket.

In both cases, we are inclined to say that the truth of p has nothing to do with the truth of q, and also that the truth of q has nothing to do with the truth of p.

The supposition that today is the first of October is unrelated to the fact that the word "ostentatious" has 12 letters, and the fact that the word "ostentatious" has 12 letters as nothing to do with today's date. As above, the reason is related to the connection between the meanings of the terms, but for now we will put aside that connection and simply record our observations.

Given two statements, p and q , then we have at least the following three cases:

1. The truth of one statement guarantees the truth of another statement.
2. The truth of one statement gives some support to the truth of another statement but does not guarantee it.
3. The truth of one statement has nothing to do with the truth of another statement.

Before we continue, it might be helpful to point out that our use of the letters p and q is arbitrary. We could just as well use k , m , #, © or any other symbol, as long as it is understood what these symbols stand for. In our case, our symbols are just going to stand for sentences which have the property of being either true or false,where false is understood to mean the same thing as "not true". Sentences that have this property are known as propositions. Propositions are so important in our study of logic that we will enter the term in our list of formal definitions.

 Definition: A sentence which has the property of being either true or false is called a proposition. Propositions are also called statements.   Are all sentences propositions?

As it turns out, using the letters t, f, or v to stand for propositions might cause confusion, so we will not use them in this textbook.

Let us now consider the first type of relationship, where the truth of one statement guarantees the truth of another statement. This type of relationship is known as implication.

1.1 Implication

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

 Definition: If the truth of a statement p guarantees that another statement q must be true, then we say that p implies q, or that q is implied by p.   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 doe not explicitly say anything 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 and to use this test we must be given a specific proposition p and another specific proposition 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.   NOTE: This is a test on whether p does NOT imply q, if the test fails, 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".

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.

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

###### Test your understanding 1: Propositions and Implication

1.2 Relevance

Let's consider Example 1.1.4 one more time: 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". It is easy to see that p does not imply q, but knowing the truth of p does give one reason to hope that Mary has won. This is a case where the truth of one statement gives support to the truth of another statement but falls short of guaranteeing it. This relation between the statements p and q gives rise to our next important concept, that of relevance.

 Definition: Let p and q be two propositions, then p is said to be relevant to q if the truth of p increases the probability that q is true but p does not imply q.

The first thing we should note is that implication is not a form of relevance, since the truth of p guarantees the truth of q whenever p implies q. For this textbook we have included in the definition of relevance the condition that p does not guarantee q. Some textbooks which discuss relevance do not include this clause, so if reading other sources, always check their definition. We include this clause so as to be able to use these concepts later when we discuss different types of arguments - which are classified easily by the relationship between the argument's premises and conclusion.

Perhaps the best way to understand relevance is to consider several examples.

Example 1.2.1

Let p be, "Jane knows the capital of Arizona"and let q be, "Jane knows all the capitals of the United States" , then the truth of p is relevant to the truth of q.

Here q could be made true by the conjunction of several cases; Jane knows the capital of California, and Jane knows the capital of Nevada, and Jane knows the capital of and so forth, and clearly Jane knowing the capital of Arizona is one such case. In this case the relevance results in the fact that p is one of the many ways which together make q true. Note that on one hand p is relevant to q, but on the other hand q implies p. This observation tells us that order is important, since p can be relevant to q, but q may not be relevant to p because q implies p. This is one possibility. Many times p is relevant to q and q is relevant to p. The next few examples illustrate this.

Example 1.2.2

Let p be, "John ate uncooked meat" and let q be, "John will develop an e-coli infection", then p is relevant to q.

Clearly p alone does not establish q, but the fact that John ate uncooked meat and then came down with a case of e-coli is some evidence which supports the truth of q. In this case, knowing that John developed an e-coli infection is also some support for the proposition that John ate uncooked meat, hence in this case q is also relevant to p.

Example 1.2.3

Let p be, "Mr. Smith is a millionaire" and let q be, "The Smiths live in a mansion", then p is relevant to q, since being a millionaire increases the probability that q is true, but does not guarantee it.

In all of the above cases, we have two statements, p and q, where the truth of p is relevant to the truth of q. In other words, knowing that p is true lends some credence to the truth of q, but unlike implication, the truth of p statement does not guarantee the truth of q.

We might think of relevance in terms of degrees of evidence. When we want to establish that a certain statement is true, many times we offer evidence from various sources. Each single piece of evidence is related to the truth of the statement we want to establish (is relevant to it), but in many cases no single piece of evidence by itself firmly establishes what we wish to show.

It is now time to test your understanding of relevance by taking the following quiz where we explore some interesting cases concerning when p is relevant to q.

###### Test your understanding 2: relevance

Question [h] in the above quiz introduces us to the concept of context when determining if p is relevant to q.

In general this means that the judgment of p being relevant to the truth of q depends on the assumption of other factors in such a way that one assumption means p is very relevant to q (in the sense that the truth of p means q is likely) whereas another assumption means that p is not very relevant to q. Recall that for p to be relevant to q, we only require that the truth of p gives us some reason to believe that the probability of q being true has increased without guaranteeing its truth.

Here is another example:

Let p be, "Mary is over 18" and let q be the statement, "Mary has a license to drive". In the United States, the truth of p gives some reason to accept the truth of q, so p is relevant to q. However it could be that is some cultures women never drive, or if Mary is a person who lived in the past before cars were invented. So depending on the context the degree of relevancy between p and q will vary, or in some cases make p not relevant to q, since the truth of p guarantees the falsity of q.

So how does one treat context in determining whether p is relevant to q? The answer is pretty simple - if some context is known to affect the relevancy of p to q, then just state the contextual dependence. In our above example we could state:

If Mary lives in the United States, then p is relevant to q, but if Mary lives in a culture where women never drive, then p is not relevant to q or the degree of relevance is very low.

The reason for introducing contextual relevance is to help you start thinking about degrees of relevance. Since the bar is pretty low for p to be relevant to q, it is important to consider to what degree does the truth of p support the truth of q?

That is what we will do next.

Degrees of relevance

When considering the definition of relevance, we noted that we only require some connection between p being true and the determination of q being true. This is a very low bar, and it is natural to make distinctions between degrees of relevance. To that end, we introduce two more definitions which we will use in this course:

 Our Definition: Suppose that p is relevant to q and that the truth of p means the truth of q is more probable than not, then we say p is strongly relevant to q.

 Our Definition: Suppose that p is relevant to q and that the truth of p means the truth of q is less probable than probable, then we say p is weakly relevant to q.

Read the above definitions carefully! Pay special attention to the phrases, "more probable than not" and "less probable than probable".

What are we to make of such statements?

In general, when determining the probability of q being true given the fact that p is true, if that probability is greater than 0.5 (or 50%), then we say that p is strongly relevant to q. On the other hand, if the probability of q being true given the fact that p is true, is less than 0.5 (or 50%), then we say that p is weakly relevant to q.

If the probability of p is .50 given that q is true, we will say that p is as equally probable as not given that q is true, in which case we can use the term 'equally relevant'.

In the above division of weak versus strong relevance, the choice of 0.5 was chosen to conform to probabilistic uses of phrases like 'p is more probable than not'. This choice is not universal and is used here only to start the thinking process concerning degrees of relevance. In other disciplines, such as statistics, it is not uncommon to expect much higher levels of probability than what is given here. In other cases it may not even make sense to talk about the probability of an event, or it is far too difficult to actually calculate the probability of an event. In those cases we can still talk about weak and strong relevance but we need to justify why we think p is strongly relevant or weakly relevant to q by giving additional reasons, some of which may appeal to what is known as the principle of induction which we will discuss later. Thus the above definitions are given as a start to thinking about degrees of relevance, and should not be taken as standard or universal definitions.

What this does tell however is that a more complete understanding of weak and strong relevance requires an examination of probability, and in particular what is known as conditional probability. We examine this in Part 4 of this text and only make informal use of the terms now. A more complete treatment of this topic takes us away from logic to the mathematical study of probability and statistics.

Guess the game show!

Contextually Weak or Strong Relevance

As it turns out, the determination of weak and strong relevance may also depend on context - which means that additional information about the statements p and q may affect whether p is strongly or weakly relevant to q. To illustrate this consider the following example:

Let p be the statement, "Smoking doubles your risk of getting lung cancer and 90% of the inhabitants of Warren County Kentucky have smoked since young adulthood" and let q be the statement, " About 90% of the inhabitants of Warren County will contract lung cancer".

Suppose it is known that only 1 in 10,000 people get lung cancer (this is a figure which is entirely made up). Then doubling your risk means 2 in 10,000 people get lung cancer, which is still a small chance. Knowing this means that p does not support q, but rather supports its negation (90% of the inhabitants of Warren Country will probably not contract lung cancer). On the other hand, suppose that the risk of getting lung cancer were 1 out of 2 people. Then doubling that chance (by smoking) results in 1 out of 1 people contracting lung cancer. Since 90% of the inhabitants of Warren County smoke, this mean about 90% will get lung cancer, so in this case p is relevant to q.

The above example is just one example of what are often termed statistical fallacies. We will treat fallacies in more detail later, but at this point a fallacy can be thought of as a case where p and q are weakly related but they are wrongly thought to be strongly related. Thus statistical fallacies are errors in reasoning related to misuse or misunderstanding of probabilistic and statistical statements.

What contextual relevance tells us in general is that when a relation between two statements is not one of implication new information can change the strength or weakness of an inference. While not obvious at this point we will note that if a relation between two statements is one of implication new information can not change the nature of the inference.

1.3 Independence

After considering the concept of relevance, it is natural to ask about cases where p does not imply q and where p is also not relevant to q. For example, if p is the statement, "There are 8.4 million inhabitants of New York City" and q the statement, "Dogs are mammals", the truth of p is not relevant to the truth of q, and also the truth of q is not relevant to the truth of p, nor does p imply q or q imply p. In these cases we say that p is independent of q. We will formally define this notion, but before we do so we will need two preliminary definitions.

 Definition: A proposition p is said to be contingent if it is not always true or not always false. In other words, contingent propositions are true under some conditions and false under others.

 Definition. Let p and q be contingent propositions, then if the truth or falsity of a statement p counts toward the truth or falsity of another statement q and if the truth or falsity of q counts toward the truth or falsity of p, then we say that p and q are truth functionally related.

 Definition: If propositions p and q are not truth functionally related, then we say p and q are independent.

Notice that the two-way nature of truth functional relations means that if p is independent of q, then q is also independent of p.

The notion of truth independence will play a central role in formal logic, but for now we will just examine some examples to get a feel for what it means for two propositions

to be be independent.

Example 1.3.1. Let p be, "Tom's first child was born October 9, 2010", and let q be, "Pianos have 88 keys", then the truth of q is independent of the truth p.

Example 1.3.2. Let p be, "At normal pressure water freezes at 0 degrees centigrade", and let q be, "February has 28 days except for leap years", then q is independent of p.

Example 1.3.3. Let p be, "Columbus sailed for the Americas in 1492", and let q be, "The concert ended at 10:15pm", then q is independent of p.

Sometimes it is difficult to say whether two statements are logically independent or relevant, as relevance allows for the truth of two propositions to be connected even in remote ways. This possibility poses no problem for the logician who is primarily interested in hypothetical cases where the question, "If. . .then" is of more interest many times than the question of "is actually". As a result, when questions concerning relevance versus independence arise the logician can just consider what follows if p is relevant to q, and similarly what follows if q is independent of p. A similar problem arises between the relations of implication versus relevance. Exploring these different possibilities has been one of the most fruitful areas of logical investigation, and in a sense has defined the direction of such investigations since at least Aristotle's time. In particular the question of implication versus very strong relevance characterizes what is known as the problem of induction, which we will encounter in some detail in Part 1 Section 3.

Summary Section 1

In this section we have introduced the concepts of propositions and explored when two propositions, which we have arbitrarily called p and q, are related to each other in terms of truth.

We defined formally the following concepts:

1. A sentence which has the property of being either true or false is called a proposition.
2. Statement variables are letters used to stand for propositions, just like letters are used to stand for numbers in algebra.
3. If the truth of a statement p guarantees that another statement q must be true, then we say that p implies q, or that q is implied by p.
4. A proposition p is said to be relevant to another proposition q if the truth of p counts for the truth of q but does not guarantee it.
5. Suppose that p is relevant to q and that the truth of p means the truth of q is more probable than not, then we say p is strongly relevant to q.
6. Suppose that p is relevant to q and that the truth of p means the truth of q is less probable than probable, then we say p is weakly relevant to q.
7. A proposition p is said to be contingent if it is not always true or not always false. In other words, contingent propositions are true under some conditions and false under others.
8. Let p and q be contingent propositions, then if the truth or falsity of a statement p counts toward the truth or falsity of another statement q and if the truth or falsity of q counts toward the truth or falsity of p, then we say that p and q are truth functionally related.
9. If propositions p and q are not truth functionally related, then we say p and q are independent.

When a relation between two statements is not one of implication new information can change the strength or weakness of the relationship, but if a relation between two statements is one of implication new information can not change the nature of the relationship.

The relations of implication, relevance, and independence are mutually exclusive if we restrict the scope of our propositions to those propositions which are contingent.

There are other relations between statements than the ones discussed in this section which will be covered in future sections.