If . . . then? An Introduction to Logic
Types and Properties of Arguments

# Types and Properties of Arguments

## The Argument

Suppose someone wants to assert a claim that something is true. Following convention we will call that claim q (recall that our letters stand for propositions, sentences which are either true or false). Now if q is not obviously true, it seems appropriate for the person asserting the claim to support it with some type of evidence, or at the very least provide reasons for claiming q is true. Since evidence comes in the form of propositions which are either true or false, then we can also denote each piece of evidence as p, r, s, etc. When we do this, we are saying that because of p, r, s, etc. it is reasonable to accept the claim q as true.

One of the major goals of philosophical logic is classifying arguments and discovering whether an argument really gives good reasons for accepting its main claim. This process of reflection and analysis is called reasoning, and as a matter of fact, it is of such importance that many textbooks define philosophical logic as the process of distinguishing between good and bad reasoning! Let us now informally define the argument.

 Definition: An argument consists of a claim, which is called a conclusion, together with a set of propositions, called premises, which are given to support the truth of the conclusion.

It is important to note that an argument consists of two parts, a conclusion and at least one premise given to support the conclusion. Informally, an argument is similar to the following abstract claim: "because p is true, then it is reasonable to conclude q is true". In this case, p is the premise or set of premises, and q is the conclusion.

In real life arguments are far more complex than this simple abstract example, and we will examine many in this section. In particular most arguments have more than one premise and many times more than one conclusion. When this happens, we can try to identify the main conclusion and consider the other conclusions as acting as premises (in context of the entire argument) which are intended to support the main conclusion. Let's illustrate this process with some examples:

Example 2.0.1

Since there was heavy rush-hour traffic, I feel certain that Tom missed his flight.

In this case, our conclusion (claim) is, "Tom missed his flight", and the reason given for this claim is the single premise, "there was heavy rush hour traffic". In this simple case, it is easy to see which part of the argument is the conclusion and which is the premise. The ignored phrase, "I feel certain" is not really part of the argument, but rather acts as an indicator that the person making the arguments feels strongly that the stated conclusion is actually true. The word "since" is an important indicator that what follows immediately after this word is a premise, rather than a conclusion.

Example 2.0.2

If Henry is the person who damaged the rental car, then he must have been in San Diego during Spring Break. If Henry were in San Diego during Spring Break, then he could not have been in Tucson at the same time. But we know Henry was in Tucson during Spring Break, so Henry is not the person who damaged the rental car.

In this example, the conclusion is that Henry is not the person who damaged the rental car. The premise that Henry was in Tucson during Spring Break supports the claim that Henry was not in San Diego, which in turn leads directly to the conclusion that Henry was not responsible for the damaged rental car, since after all, the very first premise asserts that, "If Henry is the person who damaged the rental car, then he must have been in San Diego during Spring Break."

This is an example of a chain of reasoning, where one premise leads to another conclusion which together with another premise leads to the final conclusion. Chains of reasoning are also called hypothetical syllogisms, which we will study more in Part 2.

Example 2.0.3

It is a good idea to make sure you have working fire alarms in your house. Just look at the family whose house burned down on Christmas. They lost everything in their house, and almost lost their daughter who nearly died of smoke inhalation. Their house did not have working fire alarms, and for that reason, the fire itself went unnoticed while the family slept. They were saved only by the chance occurrence of a neighbor's teenage son arriving home late from a Christmas-eve date who noticed the fire and woke the family up.

Example 2.0.3 is a bit more complicated than the previous examples but it is more like arguments we encounter in our daily lives and merits a more detailed analysis.

In order to start our analysis of the argument we break it down into its parts, which means finding its premises and the principle conclusion. To find the main conclusion, consider each sentence in turn, and ask, "Is this the main claim all the other sentences taken together aim to establish?" Another way to detect the main conclusion is to look for key words and phrases which often are stated before the conclusion (or could be stated before the conclusion without change in meaning). A list of such words and phrases would include:

• therefore
• so
• hence
• it follows that
• we can see
• thus
• we conclude

While it is not always true, many times the conclusion is the first sentence of an argument or the last sentence of an argument. Sometimes a general conclusion is stated first and a more specific conclusion is stated in the last line of the argument.

Using the criteria above, we see that the main conclusion is stated at the very start of the argument, namely, "Its a good idea to make sure you have working fire alarms in your house." The remainder of the sentences basically give reasons that support this conclusion.

Since premises are reasons given to support the main conclusion, let's turn to those reasons now and examine how they support the conclusion. In doing so, we may have to identify some unwritten assumptions that the author has made, and re-write some sentences so that they take the form of propositions (sentences which are either true or false). To help with this process, let's enumerate each sentence of the argument and consider each in turn:

1. It is a good idea to make sure your have working fire alarms in your house.
2. Just look at the family whose house burned down on Christmas.
3. They lost everything in their house, and almost lost their daughter who nearly died of smoke inhalation.
4. Their house did not have working fire alarms, and for that reason, the fire itself went unnoticed while the family slept.
5. They were saved only by the chance occurrence of a neighbor's teenage soon arriving home late from a Christmas-eve date who noticed the fire and woke the family up.

Considering each sentence we discover that 2 - 5 support 1, which is just saying that 1 is the main conclusion and 2 - 5 are premises that support that conclusion.

However, a really attentive reader might point out that according to our definition of an argument sentence number 2 does not seem to be part of the argument! Why?

According to the definition of an argument, premises are propositions, and propositions are statements which are either true or false. In this case, sentence number 2 is a command, not a proposition. Commands are neither true or false (whether or not you obey the command is a proposition, but the command itself is not). According to our definition, 2 can not form part of our argument. But that sentence is connected in some way to the to the argument as a whole and should not be ignored. How do we resolve the problem? Of course, we could just simply re-define the term "argument" to include cases like these, or just throw out sentence 2, but there is a less drastic option which logicians employ constantly; we simply re-state the essence of the sentence in such a form as to keep its connection to the main conclusion, while turning it into a proposition. Such a restatement of 2 may be: "A family's house burned down on Christmas." Now clearly, this statement is either true or false, and is important to the conclusion and the remaining sentences, and this re-writing of 2 fully preserves the original's connection to the whole, hence logicians consider this an acceptable change. After we have made this change, we see the remaining sentences are indeed propositions and support the main conclusion.

But what is the nature of their support for the main conclusion? To answer this question the concept of unstated assumptions, or unstated premises, made by the argument is helpful. What are some of these assumptions which make the premises relevant to the conclusion?

• It is possible that houses burn down.
• It is better that family members live than die in house fires.
• One can die from house fires which go unnoticed.
• Many things go unnoticed while sleeping.
• A working fire detector can sound the alarm, which, at the very least will alert even sleeping people to a fire which might otherwise go unnoticed.
• A fire detector can monitor one's house at all times, especially late at night when people are asleep.
• It is better not to leave the fire alarm alert to chance.
• Fire alarms are easy to buy and do not cost much, especially when compared to the possible cost of not having a fire alarm.

With these assumptions in mind, we can re-read Example 2.0.3 and see that each sentence is connected to each of the above assumptions, and the natural conclusion is that it is better to have a working fire alarm than none at all. It is, of course, a legitimate question as to whether the person making the above argument should include the above assumptions as additional premises, but the point of example 2.0.3 was to examine an argument that is more like one encountered in our daily lives. Notice that we have not stated anything about the strength of the above argument, or just how relevant the premises are to the conclusion, whether any single premise or conjunction of premises implies the conclusion nor have we said anything about the likelihood of the conclusion being true, or in general, whether a given argument presents a good case for its specific conclusion(s). These concepts will be the detailed subject of later investigations.

To get us started down that road, we now first consider how to combine propositions, and then turn to the very important concept of inference where we will use that concept to classify arguments in terms of what can be said about the likelihood of a conclusion given the assumption that all of the premises are true.

## Combining Propositions

In our analysis in previous sections we allowed ourselves to refer to specific propositions with single letters, such as p, q, and r and so forth,with the restriction that we do not use t, f, or v. We will continue to use this convention throughout this text, indeed, when we start our formal analysis of arguments such single letters will dominate our exposition, together with other symbols which do not stand for propositions but instead words like, "and", "or", "if. . .then" and others.

To anticipate such usage, let us informally stipulate that two propositions p and q are equivalent only if p and q are true (or false) under the exact same conditions or circumstances. For example, if p stands for, "Exactly 24 hours have passed since we put the petri dish in the incubator", and if q stands for, "Exactly one day has passed since we put the petri dish in this incubator", then p is equivalent to q since one day is 24 hours by definition.

Keeping this in mind, suppose we have two propositions, p and q, which are not equivalent, can we form another proposition that implies both p and q but not equivalent to either alone? The answer is yes!

To illustrate how, consider two specific propositions: let p be, "John is a student at Pima Community College" and let q be, "Mary is a student at the University of Arizona". Suppose that John and Mary are not connected in any way, (in other words p and q are independent propositions). We want to create another proposition, which we will call R, which implies both p and q.

The way we do this is to join p and q together with the conjunction, "and" and in doing so create another propositions R which implies both. A specific example will be helpful: using the p and q given above we create R by connecting p and q with the word "and", hence R becomes, "John is a student at Pima Community College and Mary is a student at the University of Arizona" . Symbolically we might say that R = "p and q". Clearly R is not equivalent to p or to q alone since if just one of p or q is false then R is false (which means R could be false while p is true making it impossible for the two to be equivalent). But the truth of R implies the truth of both p and q individually, since R is true only when both p and q are true (if you have become lost with the p's and q's at this point, just go back and re-read the above replacing p, q and R with the given propositions).

The process of joining two propositions with the word "and" is called conjunction. In the above example, we restricted the use of conjunction to two propositions which are not logically equivalent. This restriction assured us that the resulting proposition R was not equivalent to either of the original two. In general, we can ignore this restriction and can use conjunction freely between any number of propositions of any type to create another proposition whose truth implies the truth of each of the individual propositions.

To summarize, we can take any number of propositions, join them all together with multiple uses of the word "and" to form another proposition which is the conjunction of all of the individual ones with the result that the truth of the new proposition implies that each individual proposition be true.

Example 2.1.1

Let R be "John is 28 and John is a student at San Antonio College and John is a philosophy major", then R is true only if the individual propositions, "John is 28", "John is a student at San Antonio College" and "John is a philosophy major" are all true.

Example 2.1.2

Let p, q, s, m and n all be premises to an argument whose conclusion we denote as c. Let R be the proposition created by the conjunction of p, q, s, n and m. In other words, R is equivalent to the proposition "p and q and s and n and m".

We will learn more about the process of conjunction when we study formally the truth table definitions where words like "and' will play an important role.

For now we introduce the concept of inference which will take up a long way in our goal to classify arguments in the sense of deciding which arguments give good reasons to accept the truth of their conclusions.

## Inference, Validity and Invalidity

Recall the primary goal of philosophical logic is to be able to distinguish between correct and incorrect reasoning. As it turns out, reasoning is exemplified by arguments, in other words, when one creates an argument one is participating in the process of reasoning.

Suppose we have an argument. Then by definition, we have a set of premises and a main conclusion. Let P be the conjunction of all of the premises (meaning we just take each individual premise, connect them each with the word "and" and form a larger proposition which we are calling P), and let C be the conclusion. We will call the truth functional relationship between P and C the argument's inference.

 Definition: The truth functional relationship between the conjunction of an argument's premises and the argument's conclusion is called the argument's inference.

In Section 1 we explored three distinct and types of relations between statements, namely implication, relevance and independence. Since an argument is something created by humans (for now at least) then we can ignore any cases where all of the premises of an argument are actually independent of the conclusion. Since an argument's inference is the relationship between the argument's premises and its conclusion, we will classify arguments by whether the argument's inference is that of implication or relevance.

Let's illustrate by starting with an example.

Example 2.2.1

Suppose Emily notices the following pattern:

1 = 12

1+3 = 4 = 22

1+3+5 = 9 = 32

1+3+5+7 = 16 = 42

1+3+5+7+9 = 25 = 52

Emily points out that the pattern suggests that the sum of 1 odd number is 12 and the sum of the first 2 consecutive odd number is 22 and the sum of the first 3 consecutive odd numbers is 32 and so forth. Emily then tells you that she has a proof showing that the sum of the first n odd numbers is n2 for any whole number n. She then gives you that proof, which consists of a set of premises that lead to the conclusion that the sum of the first n odd numbers is n2. What does Emily mean when she says she has a proof?

In logic and mathematics (and systems which use formal reasoning) to say one has a proof of some proposition C is to say that if the premises one presents in the proof are true, then the conclusion C of the proof must be true. In other words, to prove a claim C is to offer reasons (premises) such that if those premises are true then the conclusion C must be true. To recast the notion of proof in terms of inference and relations between statements we introduce a key term used in logic which is more common than the term "proof".

 Definition: Let P be the conjunction of each premise for a given argument, and let C be the argument's conclusion. If P implies C, then we say the argument is valid. Equivalently, if an argument has the property that it is impossible to have all true premises and a false conclusion, then the argument is said to be valid. In other words a valid argument is an argument whose inference between the conjunction of its premises and the conclusion is one of implication. When the inference of an argument is one of implication, then the inference is said to have the validity property and the argument is said to be valid.

Note that an argument's inference is valid or not. We will use this fact as our first step in classifying arguments. As a result our first step in analyzing arguments will be to decide whether the inference of an argument is valid or not. We represent this step visually the following way:

The general question as to how one determines whether an argument's inference is valid is a question of such importance that it is worth thinking about for a few seconds before continuing to read. Check your thoughts on how this might be done by answering the question below.

The answer really should not be all that surprising. After all we already have a two-step method of checking whether any proposition p does not imply another proposition q, and validity is defined in terms implication. Hence we can extend the two step method we used to check to see if a proposition p implies another proposition q to check to see if an argument is valid or not.

 The two step method for determining whether an argument is not valid. To determine whether an argument is valid or not, many times it is easier to check to see if it is not valid. An argument is not valid only if both of the conditions [a] and [b] are true:   [a] It is possible for the argument to have all true premises, or in other words it could have been or could be the case that all of the premises are true. [b] Under the possibility above, the argument can have a false conclusion.   An argument is not valid only if both of these conditions hold. The failure of one or both of these conditions means the argument is valid.

Think about conditions [a] and [b] as check boxes. To be able to place checks in both boxes means the argument's inference is not valid. If one can not place a check mark in both boxes, then the argument's inference is valid.

Let's see how this works by considering some exercises.

Exercise 2.2.2 Determine whether the following arguments have valid inferences:

(a)

If John makes the free throw, then the U of A will win the game.

Hence the U of A won the game.

(b)

If John makes the free throw, then the U of A will win the game.

John did not make the free throw

Hence the U of A lost the game.

(c)

If John makes the free throw, then the U of A will win the game.

John did not make the free throw

Hence the Thursday follows Wednesday in week-day ordering.

Solution:

For each argument we check first to see if we can think of some possible way both premises can be true. If so, we have checked condition [a]. If not, we stop and declare the argument to be valid (make sure you understand why we make this declaration, if you need to, re-read the previous paragraphs).

Second we check to see if under the possibility where all premises are true (condition [a]), it is possible to have a false conclusion. If so, then we have checked conditions [a] and [b] and hence conclude the inference is not valid. Otherwise we declare the argument's inference valid.

Now apply the above criteria and check your answers by completing the quiz below:

## Common Misunderstandings

Let us now consider some of the most common mistakes made with respect to the concept of validity. Consider the following questions and then take the true/false quiz.

 Common mistakes people make when thinking about the concept of validity or invalidity. A valid argument must have a true conclusion. If all premises of a valid argument are false, then the conclusion must be false. A valid argument is an argument with all true premises and a true conclusion. The conjunction of all the premises in a valid argument must be true.

The first statement omits a very key part of the definition of validity that states "If all premises are true". Missing or forgetting key parts of definitions is perhaps the number one reason for making logical errors.

The second statement is not the definition of a valid inference or a statement implied by that definition. Review the visual representation of implication for now, paying attention to the last row. We will prove this later in Part 2.

The third statement is also incorrect. Re-read the definition of valid and take note of the differences between this statement and the definition.

The fourth statement is a bit more difficult. So we will demonstrate its falsity by example. Consider the following argument:

The Cullinan diamond weighs more than 100 grams.

The diamond was worn by Queen Victoria when she was crowned in 1964.

Therefore the Cullinan diamond weighs more than a 2 gram feather.

Clearly, this argument is valid (it is impossible to have a false conclusion with all true premises), but since Queen Victoria was not crowned in 1964, the conjunction of the premises can't be true (recall this is another way of saying at least one of the premises is false). The argument is valid because the truth of the first premise alone implies the truth of the conclusion, which means it is impossible to have a false conclusion and all true premises.

We can now formally define arguments which are not valid. As you may have guessed these arguments are called "invalid".

 Definition: If it is possible for an argument to have all true premises but still have a false conclusion, then the argument is invalid, which is to say it is not valid. Equivalently, if the inference from the premises to the conclusion of an argument is not that of implication, then we call the argument invalid.

We use the term "invalid" to characterize the inference between the conjunction of the premises of an argument and the conclusion. As noted in the side-bar to this page, some logic books restrict this term to refer to arguments which claim to be proofs. In this textbook we simply call this restricted use by the name, "erroneous proof" and generalize the terms valid or invalid to apply to any argument.

Again, be careful not to add anything to the definition of an invalid argument that is not stated in the definition. By our classification, any argument where the premises do not imply the conclusion is an invalid argument. Invalid does not mean "false" or even "bad reasoning" - it means only that the possibility is open (no matter how remote) for the conclusion to be false even if all of the premises are true. This means that arguments with premises which are strongly relevant to the conclusion are still classified as being invalid.

All arguments can be classified under our classification scheme as being either valid or invalid. To help you keep the difference between these two classifications in mind, we have gathered together what this classification means with respect to the conclusion for arguments which are valid and invalid and when the premises are all true or when they are not all true.

 Premises Conclusion Argument is valid Argument is invalid All true Must be true Need not be true (could be true or false) Not all true (at least one is false) Need not be true (could be true or false) Need not be true (could be true or false)

Note that the table is divided into what is true of all premises and the corresponding result for the truth of the conclusion based on the inference type of the argument (valid or not). In every case except one, the conclusion need not be true. Keep this is mind when thinking about the implications of an argument being valid or not.

## Truth versus Inference and the concept of Soundness

The concepts of validity and invalidity are so central to the study of logic that it is worthwhile discussing them more at some length, as long experience has shown that the different use of these terms outside of a logic course lingers in the minds of many learning philosophical logic. This can cause confusion and wrong inferences based on those different meanings.

Perhaps the most persistent confusion concerning the concepts of validity and invalidity is the tendency to think they are synonymous with the concepts of truth and falsehood, namely that valid is equivalent to true, and invalid is equivalent to false. This is not the case, and great effort should be taken to learn the difference between these concepts. Truth and falsity are properties of propositions, whereas validity and invalidity are properties of inference (or, as we often say, arguments). The inference of an argument is either valid or invalid, whereas the individual components of the argument - its premises and conclusion - are either true or false. The non equivalence of these terms leads to the possibility that all of the components which make up an argument (the premises and conclusion) might be false, whereas the inference made by the argument is valid. To illustrate this possibility consider the following argument:

Example 2.3.1

Today is Wednesday and the day that follows Wednesday is Thursday.

Therefore tomorrow is Thursday.

If you are reading this on any day other than Wednesday, both the premise and conclusion are false, but the inference is valid - since if the conclusion were false (tomorrow is not Thursday) it is impossible for the premise, "Today is Wednesday and the day that follows Wednesday is Thursday" to be true. If the above example still seems odd, review again the formal definition of validity.

Another common misconception is the incorrect view that invalid arguments are somehow bad arguments, in the sense that either the conclusions to invalid arguments must be false, or somehow any reasoning which is invalid is poor reasoning. This too is incorrect. Consider the following invalid argument:

Example 2.3.2

I parked my car in the west parking lot this afternoon, as I do every time I drive to school.

My car has always been exactly where I parked it at the end of the school day.

Therefore, after classes, my car will be parked in the west parking lot.

In this case, it is possible for the conclusion to be false (your car will be missing), even though the premises are true. What this means is that the argument is invalid, but the premises are strongly relevant to the conclusion (imagine drawing the opposite conclusion from the same premises, that your car will not be parked in the west parking lot). Again, this illustrates the difference between invalidity and poor reasoning. An inference may be invalid, but still present good reasons to accept the conclusion - the difference is that the conclusion to an invalid argument is not guaranteed to be true whenever all of the premises turn out to be true, whereas with a valid argument we have such a guarantee.

### Two trivial cases.

Recall the two-step test for invalidity:

 The two step method for determining whether an argument is not valid. To determine whether an argument is valid or not, many times it is easier to check to see if it is not valid. An argument is not valid only if both of the conditions [a] and [b] are true:   [a] It is possible for the argument to have all true premises. [b] Under the possibility above, the argument can have a false conclusion.   An argument is not valid only if both of these conditions hold. The failure of one or both of these conditions means the argument is valid.

Now consider the following two arguments.

I.

Today is Monday Jan. 5 and Sunday Feb. 3.

Hence I will win the lottery.

II.

If the afternoon traffic is heavy, then I may not make it home in time to see the game.

Hence tomorrow I will go to work or tomorrow I will not go to work.

Note that for argument I, the only premise is never true, hence condition [a] can't be checked, and hence the argument is valid by our two step method. For argument II, we can't check box [b] since the conclusion is always true. Hence argument II is also valid. Clearly this type of validity is a special case, which we denote by the term 'trivially valid".

 Definition: An argument that is valid by virtue of the fact that it is impossible to have all true premises or because the conclusion is always true, is called trivially valid.

Clearly arguments which are trivially valid are only of passing interest. What we really care about are arguments whose premises can be sometimes true and sometimes false, but when they are all true then the conclusion must be true. And among the subclass of all such arguments, we really care about those which actually have all true premises. These valid argument are so important they have a special classification.

 Definition: An argument is sound if it is valid and has all true premises.

Notice that for an argument to be sound, two conditions must be meet:

1. The argument must be valid
2. The argument must actually have all true premises.

Think about what this means for a second, and check your understanding so far by answering the following question:

Hopefully, your reasoning went something like this: Since the argument is sound, then it is both valid and actually has all true premises, so the conclusion must be true, by definition of validity.

The following is an example of a sound argument:

Example 2.3.3

If a number is greater than 7 it is greater than 3.

8 is greater than 7.

Therefore 8 is greater than 3.

First verify that the argument is valid (check to see it is impossible to have all true premises and a false conclusion), and then check to make sure the premises are all true. Once this is done the truth of the conclusion follows by logical necessity.

Notice that sound arguments could still be trivially valid, since the conclusion of an argument might be a statement which is always true and the premises could be statements such that each one could be actually true. Indeed there have been attempts (which are still current) to avoid the issue of trivial validity by restricting the type of premises and conclusion an argument can have. These efforts have not really caught on, perhaps in part since it is so easy to just label the case as an example of a trivially valid argument (or a trivially sound argument) and move on - as there is nothing more here of interest.

We complete one branch of our argument classification scheme related to validity by introducing one last (easy) definition.

 Definition: If an argument is valid, but has at least one premise which is false, then the argument is valid but not sound.   Equivalently, if a valid argument has a conclusion which is actually false, then the argument is valid but not sound.

We have actually encountered an argument which was valid but not sound already, which we repeat here as an example:

Example 2.3.4

The Cullinan diamond weighs more than 100 grams.

The diamond was worn by Queen Victoria when she was crowned in 1964.

Therefore the Cullinan diamond weighs more than a 2 gram feather.

As noted before, Queen Victoria was not crowned in 1964, so the argument has at least one false premise. However, the argument is valid since if both premises were true then the conclusion must be true. This means the argument is valid but not sound.

Before moving on to the concepts of weak and strong arguments, let's review what we have so far.

Our first step is to classify the inference of an argument as either valid or not. If valid, then we check to see if the argument has all true premises. If the answer is yes, then the argument is classified as sound. If the answer is no, then the argument is classified as valid but not sound. The following diagram illustrates these possibilities.

Test your understanding by taking the following quiz which asks you to classify arguments according to the most specific classification above (if an argument is valid and sound, then classify it as sound, in other words move to the right as far as possible in the classification tree).

## Strong, Cogent and Weak Arguments

Let's turn now to Emily from example 2.2.1, who has left the role of mathematician and now assumes the role of biologist.

Example 2.4.1

Emily, who lives in the United States, is classifying swans and notes that every swan she has seen is white. She has seen many swans, as they are her main subject of study. So she forms an argument that goes something like this:

Premise 1: The first swan I observed was white.

Premise 2: The second swan I observed was also white.

.

.

.

Premise 1003: The 1003th swan I observed was also white.

Conclusion: Hence all swans are white.

Now let P be the conjunction of all 1003 premises, and C be the conclusion as noted above. What is the inference type between P and C?

Upon reflection the inference between P and C is not implication, since it is possible that when she sees more swans she will discover a swan which is not white. This means that it is possible that all of the premises to her argument are true, but the conclusion is false. Hence in our classification the argument is invalid. However, it does seem that the fact that all 1003 swans observed as being white is strongly relevant to the conclusion that all swans are white, especially if we also assume the swans she has observed are were randomly selected of the total population. Whenever this is the case, we classify the argument as being strong.

 Definition: When the inference between the premises of an argument and the conclusion is judged to be relevant and in addition the judgment is that of strong relevance, then the argument is known as a strong argument.   Additionally when an argument is considered to be strong, and actually has all true premises, then the argument is called cogent. On the other hand if the argument is considered strong, but at least one premise is false, then the argument is classified as being strong but not cogent.

So similar to the classifications of valid - sound - valid but not sound we have the classification of strong - cogent - strong but not cogent. Note this is a similarity in classification, but the characteristics of each classification have important differences. The following example illustrates this.

Example 2.4.2

According to the 2010 census, the population of the 5 largest cities is over 1.5 million.

Phoenix Arizona is the 6th largest city.

Hence Phoenix has a population at least over 1 million.

If one deems the above argument to be strong, then it is an example of a cogent argument since the premises are true according to the 2010 census. While it turns out that Phoenix has a population of 1,445,632 according to the 2010 census, this fact is not important - it could have been otherwise. Cogent arguments are just strong arguments with all true premises. Strong arguments are just arguments whose inference between the premises and conclusion is considered more probable than not. Be careful to keep this in mind, cogent arguments, unlike sound arguments, can have a false conclusion!

To create an example of a strong argument that is not a cogent argument, just change the first premise of the above argument to, "According to the 2010 census, the population of the 5 largest cities is over 3 million". Since this sentence is false, the resulting argument becomes strong but not cogent, since it was created from a cogent argument with one premise changed to be false.

Not all relevant inferences are strong. For example, suppose that Emily, now a budding biologist, has only observed two swans and both were white and from this data concludes that all swans are white. As above, her conclusion could be false, so the inference between the premises and conclusion is only not that of implication, and since she is basing the conclusion on only a few observed cases the inference between the premise and conclusion is only weak.

 Definition: When the premises of an argument are only relevant to the conclusion, and in addition they are judged to be weakly relevant, then the argument is called a weak argument.

Example 2.4.3

I once ate at that restaurant on the corner, and become ill the next day.

Today Sally told me she ate there and did not feel good either.

Hence the restaurant at the corner has cooks who do not wash their hands.

Even if we assumed that 2 cases of people eating at a restaurant and getting sick were good evidence that the restaurant prepared bad food, it surely does not give good evidence to come to the more specific conclusion that the cooks do not wash their hands, so Example 2.4.3 is an example of a weak argument.

We now turn to our last classification of arguments based on inference type.

Suppose John is trying to prove some conclusion - in other words John is trying to create an argument whose inference is valid. However he makes a mistake in that proof and his proof turns out to be erroneous (perhaps John is in an advanced mathematics class or logic class where he is asked to prove a theorem and his proof is wrong). These cases will be called erroneous proofs in our classification system, but it should be noted in the traditional classification system based on induction vs deduction these arguments are the only arguments classified as being 'invalid'.

We will examine such arguments in more detail in our next section but for now we will just define the concept and give one example.

 Definition: An argument whose inference is invalid but which claims or is intended to be valid is said to be an erroneous proof, or an invalid deductive argument.

Example 2.4.4

Suppose x2 = 25, prove x = 5.

Proof:

x2 is mathematical shorthand for the product of any real number x times itself.

We are given that x2= 25.

But (5)(5) = 52 = 25.

Therefore x = 5.

Since the above is a mathematical argument, we can easily say it is intended to prove its conclusion. However all the premises can be true but the conclusion false, since the value of x could be -5 rather than 5, so the above is an example of an erroneous proof.

We can now list a diagram that shows all of the possible argument classifications based on the inference of an argument.

Exercise 2.4.5 Classify the following arguments using the most specific classification possible (go as far to the right in the diagram as you can). For example if the inference of an argument is valid and the premises can be shown to be true, then classify the argument as sound rather than just valid.

(a)

The Empire State Building is taller than the Eiffel Tower.

The Eiffel Tower is taller than the Statue of Liberty.

Therefore the Empire State Building is taller than the Statue of Liberty.

(b)

This is a large hat.

Someone is the owner of this hat.

The owners of large hats are people with large heads.

People with large heads have large brains.

Hence the owner of this large hat has a large brain.

(c)

No one was at the front desk of the office.

Over 30 people work at the office.

Therefore everyone was probably on break..

For argument (a), we first need to discover whether the inference is valid or not. We try to show it is not by coming up with some possibility where all premises are true and the conclusion is false - but find that is impossible (think it through)- so the inference is valid.

Next we check to see whether the premises are actually true (if this check is impossible we stop at the classification of valid). A quick check gives the following heights:

• Empire State building: 1,250 feet
• Statue of Liberty: 305 feet
• Eiffel Tower: 984 feet

So all premises are true, hence the argument is sound.

For (b) we check to see if the inference is invalid the same way we do for (a), and discover that if all of the premises were true, the conclusion would have to be true, so again the inference is valid, since it is impossible to have all true premises and a false conclusion. Can we go further? Yes! In general it is not true that owners of large hats are people with large heads, or that people with large heads have large brains. Hence at least one premise is false, so this argument is valid but not sound.

For (c) the check of whether it is possible to have all true premises and a false conclusion turns out to be true. So we know the argument's inference is invalid. The argument is not mathematical or logical in nature and does not show any evidence of being a proof of the conclusion. Experience tells us that there are many reasons why no one might be at the front desk of an office, everyone being on break being just one possibility, so we classify the argument as being weak.

Check your understanding now by taking the following quiz.

## Summary

An argument consists of a claim, which is called a conclusion, together with a set of propositions, called premises, which are given to support the truth of the conclusion.

The relationship between the conjunction of an argument's premises and the argument's conclusion is called the argument's inference.

Let P be the set of premises for a given argument, and let C be the argument's conclusion. If P implies C, then we say the argument is valid. In other words a valid argument is an argument whose inference is one of implication. Equivalently, if an argument has the property that it is impossible to have all true premises and a false conclusion, then the argument is said to be valid. When the inference of an argument is one of implication, then the inference is said to have the validity property and the argument is said to be valid.

An argument that is valid by virtue of the fact that it is impossible to have all true premises or because the conclusion is always true, is called trivially valid.

If it is possible for an argument to have all true premises but still have a false conclusion, then the argument is invalid, which is to say it is not valid. Equivalently, if the inference from the premises to the conclusion of an argument is not one of implication, then we call the argument invalid.

An argument is sound if it is valid and has all true premises.

If an argument is valid, but has at least one premise which is false, then the argument is valid but not sound.

When the inference between the premises of an argument and the conclusion is judged to be relevant and in addition the judgment is that of strong relevance, then the argument is known as a strong argument.

Additionally when an argument is considered to be strong, and actually has all true premises, then the argument is called cogent. On the other hand if the argument is considered strong, but at least one premise is false, then the argument is classified as being strong but not cogent.

When the premises of an argument are only relevant to the conclusion, and in addition they are judged to be weakly relevant, then the argument is called a weak argument.

An argument whose inference is invalid but which claims or is intended to be valid is said to be an erroneous proof, or an invalid deductive argument.

We considered how to combine propositions into larger propositions using the word 'and' and termed the process of combining any set of propositions 'conjunction', where we noted that the truth of a conjunction implies the truth of all of the individual propositions combined using the term 'and'.

Since our goal was to classify arguments - to aide us in that goal we introduced term inference as the relationship between the conjunction of all the premises of an argument and the conclusion and stated that our classification of arguments would be based on the argument's inference.

Finally we introduced the concepts of validity, invalidity, soundness, valid but not sound, relevance, strong and weak arguments, cogent and strong but not cogent and erroneous proof as a list of argument classifications based on an argument's inference.