1.1: Logic

People are trying to get you to believe things all the time. Whether it is commercials during your favorite show, politicians trying to get your vote, or someone trying to convince you to go out with them for a cup of coffee, people want you to believe what they think will benefit them. But what benefits you is believing what is most likely to be true.

Humans are psychological beings, we have minds that are wired funny. There are ways to get us to believe things that are not true…but we will be absolutely certain that they are. These tricks can lead us to think and do things that are not the best for us, those we care about, or the world as a whole. Logic is the study that allows us to cut through these rhetorical tactics, courtroom tricks, and cognitive biases.

In philosophy, logic is the study of rational argumentation. Notice what just happened – we took one word “logic” that we didn’t know the meaning of and we defined it in terms of two words, “rational” and “argumentation” that we don’t know the meanings of. In philosophy, we call that progress.

A belief is rational if we have good reason to believe it is at least probably true. There are lots of ways to acquire beliefs. Some of them are rational and some of them are not. One may believe something because of coercion. “Logicians are cool – believe this or I will fail you this semester.” Effective (perhaps), but not rational. What we need is not incentive to believe (or, at least to act as if we do), we need actual support for the claim. For a belief to be rational is to have evidence that it is probably the case. This is where arguments come in.

Again, an argument is a set of sentences such that one sentence, the conclusion, is claimed to follow from the other sentences, the premises. Arguments thus have two parts, a conclusion and premises. The conclusion is the point of the argument. It is the thing being argued for, that which we are trying to convince ourselves or others of. We give arguments in order to provide legitimate reasons to believe the conclusion. The premises are those reasons. The premises are the grounds that are being proposed to support rational belief in the conclusion.

Every argument has one and only one conclusion. Conclusions are like noses – everyone has one. More than one conclusion, more than one argument. No conclusion, 7 no argument. Premises, on the other hand, can come in any number – one, two, eighteen…. There are mathematical arguments with an infinite number of premises and weird logical arguments that have no premises. Conclusion – one and only one, premises – any number.

Consider the old chestnut: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. In this case, the conclusion is “Socrates is mortal.” That is what the argument is trying to convince us of. Why should we believe that Socrates is mortal? Because he is a man and all men are mortal. The first thing you do when you approach an argument is to find the conclusion and then set out the premises. It is crucially important that you do this correctly.

Consider what would happen if we misidentified the conclusion in this case – if it is true that all men are mortal and that Socrates is one of these guys, then it turns out that it is absolutely the case that Socrates is mortal. But suppose we wrongly thought that the conclusion is “all men are mortal” and that the premises are “Socrates is a man” and “Socrates is mortal.” Just because one man is mortal, it doesn’t necessary follow that they all are. By misidentifying the conclusion and premises, we have taken a good argument, one that gives us good reason to believe something and turned it into a flawed argument that does not.

So, the first tasks that are necessary for us to develop are (1) figuring out when we have an argument, and (2) determining what the conclusion is and what the premises are. We often have help with these tasks, what we call indicator words. There are certain words we use to point out conclusion and there are certain words we use to point out premises.

Think about what a conclusion is supposed to do, it is supposed to be the thing that is established by the argument, so we use words that indicate this. The most obvious one is “therefore,” but we use other words for this function as well: thus, hence, and so.

We do need to be careful, though, in that not every use of these words indicates a conclusion. Take the word “so.” Sometimes, it does function this way – I haven’t eaten today, so I am hungry. That I have not yet eaten gives you good reason to believe that I am, in fact, hungry. But in the sentence “I am so hungry,” now “so” is not an indicator 8 word. “Hungry” is not a conclusion. This is a different meaning of “so,” instead of synonymous with “therefore,” it is synonymous with “very.”

We have premise indicator words as well. We use words and phrases like “because,” “since,” and “given that.” Again, not every use of these words is an indication of a premise in an argument. “I am hungry since I haven’t eaten all day” is a case where “since” points out a reason for belief. Why should you believe I am hungry, because I haven’t eaten today. But in the sentence “I haven’t eaten since yesterday,” we see since used just to signal time, not as an indicator word. “Yesterday” is not a premise.

Indicator words are the easiest way to determine whether we have an argument and if so, what the conclusion is and what the premises are. But we don’t always have indicator words. How then do we determine if we are looking at an argument? The easiest way is to try to insert your own indicator words. I prefer “therefore” and “because.” If you look at a passage and the word “therefore” can be naturally inserted in a way that maintains the meaning of the passage, you are probably looking at an argument and what immediately follows “therefore” is your conclusion. Similarly, if you can insert “because” into a passage without changing the meaning, you are likely looking at an argument and what comes right after your inserted “because” is probably a premise.

Arguments have two parts – a conclusion that we are trying to be convinced of and premises which are trying to do the convincing, that is, trying to provide good reason for belief. Recall that we defined an argument as “a set of sentences such that one sentence, the conclusion, is claimed to follow from the other sentences, the premises.” Notice that we do not need the conclusion to actually follow from the premises to have an argument, we just need the claim that it does. This claim is that there is an inference between the premise set and the conclusion, that is, that the premises do logically lead to reason to believe the conclusion. All that is necessary for the existence of an argument is the claim of an inference. Good arguments will have an actual inference and bad arguments will lack an inference despite the claim. How do we determine which is which? That is the central question of logic.

An argument has two parts, a conclusion and premises, and that there is the claim of an inference from the premises to the conclusion. Not all texts contain arguments, that is, not all communication – even those that are trying to convince us of 9 something – will contain a claim of an inference, that is, independent reason why we should believe what is trying to be conveyed. So, the first skill we will need to develop is to spot when there is an argument and when there is not. The second is to find the conclusion and lay out the premises.

Consider the following: You are dehydrated. You need to drink more water. The human body does not function properly when it is not appropriately hydrated and you need to maintain full function. Argument? Yes. What is the conclusion? Where does the word “therefore” fit naturally in a way that maintains the meaning and where does “because” fit naturally to maintain the meaning?

1. You are dehydrated.
2. The human body does not function properly when it is not appropriately hydrated.
3. You need to maintain full function.

Conclusion: You need to drink more water.

How about the following: “I did turn in my homework. I swear. Believe me, I’m telling the truth.” Argument? Clearly, there is someone trying to convince someone of something. But is this person giving independent reason for thinking it is true? No. The student is simply pleading for the person to believe the claim, not giving evidence for it. This is not an argument.

Determining if something is an argument and finding the parts are the first two steps for us. The third step is determining what kind of argument it is. Arguments primarily come in two types: deductive and inductive. An argument is deductive if and only if the conclusion does not contain information that is not already contained in the premises. (Logicians call this property being “non-ampliative”.) A deductive argument argues from broad to narrow, that is, the content of its conclusion does not outrun the content of its premises. The following argument is deductive:

1. All men are mortal.
2. Socrates is a man.

Conclusion: Socrates is mortal.

The premises talk about all men, but the conclusion only mentions a subset, one guy.

The other sort of argument we’ll consider is inductive. An argument is inductive if and only if its conclusion does contain information not contained in its premises. An inductive argument argues from narrow to broad, that is, the content of its conclusion does outrun the content of its premise set. (Logicians call this being “ampliative.”) The following argument is inductive:

1. The first paper I submitted in this class got an A.
2. The second paper I submitted in class got an A.
3. The third paper I submitted in this class got an A.
4. There are five papers I have to write for this class.

Conclusion: All my papers in this class will get an A.

The evidence is six of the ten papers and the conclusion is all ten. The conclusion is broader than the premises, that is, not fully contained in the premises and that makes the argument inductive.

This is the form of induction called an “inductive generalization,” but we could also make an inductive argument to a single, as of yet unobserved instance. This is called “inductive analogy.” It would look like this:

1. The first paper I submitted in this class got an A.
2. The second paper I submitted in class got an A.
3. The third paper I submitted in this class got an A.
4. There are five papers I have to write for this class.

Conclusion: The fourth paper I am about to submit will get an A.

This is inductive because the premise set discusses six papers and the seventh paper is not one of them. The inference goes beyond the data on which it is based. Deductive arguments milk content out of their premises, while inductive arguments extend their scope beyond their premises.

So, we now know to (1) determine if it is an argument, (2) find the conclusion and premises, and (3) determine the type of the argument. The last step is the most important – evaluating the argument. Does the argument contain a legitimate inference? Does it provide us good reason to believe its conclusion?

Consider the following three arguments and determine which are good arguments:

I. 1. All men are mortal

2. Socrates is a man

Conclusion: Socrates is mortal.

II. 1. All politicians are Martians.

2. Lyndon Baines Johnson is a politician

Conclusion: Lyndon Baines Johnson is a Martian.

III. 1. All Presidents of the United States have been human

2. This newborn baby is human

Conclusion This newborn baby is President of the United States

Which of the above provide you with good reason to believe the conclusion? Only the first one. What is wrong with the other two?

The second argument should strike you as good in some way, but surely we don’t have reason to believe its conclusion. Similarly, the third argument does seem to have a virtue. Its premises are true. All Presidents of the United States have been humans and all newborn children – given that by children, we mean the offspring of humans – are also humans. But, again, surely the conclusion is false. So, what is right about the first example, and what goes wrong with the second two?

Validity concerns the form of the argument. An argument is valid if and only if, assuming the truth of the premises for the sake argument, the conclusion follows from them. The important thing to notice here is that we are assuming the premises are true for the sake of argument. Maybe they are true, maybe they are false, we don’t care. Validity does not concern the content of the premises. All we are looking at is whether the premises, IF true would lead you to the conclusion. Validity is not about the content of the argument, but about the form of the argument. Validity looks at the skeleton of the argument and determines if it is strong enough to support the weight of the conclusion.

Let’s look at the first argument. If we take it as true that all men are mortal, and if we take it as true that Socrates is one of those men, then it necessarily follows that Socrates is mortal. This is a valid argument.

Similarly, with the second argument. If we assume it to be true that all politicians are Martians, and if we take it as true that LBJ was a politician, then again it necessarily must follow that LBJ was a Martian. This, too, is a valid argument.

Indeed, from a validity standpoint, it is the same argument as the first. Remember that validity only cares about the structure of the argument, not the content. Both of the first two arguments have the same form: All A’s are B, C is an A, therefore, C is a B. All arguments of this form will be valid. Let’s play Madlibs. If I ask for a plural noun, a proper noun, and an adjective and put them into this form, we get a valid argument. All cats are purple. Pee Wee Herman is a cat. Therefore, Pee Wee Herman is purple. Valid argument. Validity is a function of form – any conclusion of that form will be true if the premises of that form are also true.

Consider the third argument. This one is invalid. It is true that newborn babies are human and it is true that all Presidents of the United States are and have been humans, but just because they are true, it does not follow that all newborn children are Presidents of the United States. All A’s are B and all C’s are B, but that does not mean that all A’s are C. The truth of the premises does not lead to the truth of the conclusion. The problem is not with the premises, but with the form of the argument.

problem is not with the premises, but with the form of the argument. That we are assuming the truth of the premises in our first criterion should bother you a little bit. After all, that is a huge assumption to make. What justifies our ability to make such an assumption? The answer is our second criterion – well-groundedness. An argument is well-grounded if and only if all of its premises are true. Well-grounded arguments have true premises. Maybe the conclusion is true, maybe it is false, but what is important for us in looking at the well-groundedness of an argument is just the truth or falsity of the premises.

Our first argument? Well-grounded. All men are, in fact, mortal and Socrates is (or at least was) a man.

The third argument. Also well-grounded. All newborn children are, in fact, human and all Presidents of the United States have actually been human.

But the middle one? Not well-grounded. LBJ was indeed a politician, but not all politicians are Martians (certainly, no more than half). It is a valid argument; that is, if the premises were true, the conclusion would be, but not all of the premises are true. As such, we have no good reason to believe the truth of the conclusion.

An argument that satisfies both of our criteria, that is, an argument that is both valid and well-grounded, is called sound. (valid + well-grounded = sound) A sound argument gives us good reason to believe its conclusion. What we want are sound arguments.

If an argument is sound, then the conclusion is supported. But how strongly it is supported depends on the type of argument. If you have a deductive argument that is both valid and well-grounded, then the conclusion must be true – 100%, absolutely, no doubt about it true. If, on the other hand, you have an inductive argument that is sound, then the conclusion is probably true and gives us a degree of belief in the truth of the conclusion. Stronger inductive arguments give us reason to think the conclusion is more probably true, whereas sound, but weak inductive arguments give us reason to think the conclusion is probably true, but does not give us the same degree of belief.

But suppose the argument is not sound. If the argument – deductive or inductive – is flawed, that is, if it is invalid or not well-grounded, then we know nothing of the truth or falsity of the conclusion. There are always bad arguments that can be built for true conclusions. Consider the following:

1. The moon is made of green cheese.

2. Only people twelve feet tall can join the Boy Scouts..

Conclusion: 1+1=2

It is, of course, absolutely true that 1+1=2, but not for the reasons that are given in the premises – premises that are, obviously, false. So, if an argument satisfies both criteria, the conclusion should be believed; but if either or both criteria are not satisfied, we have no reason to believe anything about the truth or falsity of the conclusion.

In order to determine which arguments are sound, we need to develop tests for validity and well-groundedness. Validity looks at the structural elements of the argument and its study is called “formal logic,” not because you need to dress up to do the work, but because it is an examination of the form of arguments. Validity for deductive and inductive arguments are completely different matters and we need different tools – like standard and metric wrenches. We will delve into both of these in turn. Well-groundedness concerns look at the acceptability of the argument other than the form and are called “informal logic” or “critical thinking.”

Exercise \(\PageIndex{1}\)

Do the following passages contain arguments? If so, identify the conclusion and premises.

1. Jokes are effective if the audience laughs at them. You can’t have two emotions at the same time. Amused and insulted are different emotions. Your joke insults the audience. So, your joke is not effective.

2. That joke is not funny. I get it, I just don’t think it works.

3. “Gumpy’s” is funnier than “Frumpy’s” as the name of a potato chip because the sound the letter g makes is funnier than the sound the letter f makes.

Are the following arguments deductive or inductive?

4. Roberta loves everything Aparna Nancherla does. Her last special came out last month. Roberta loved it.

5. Roberta has loved everything Aparna Nancherla has done. Her next special comes out next month. Roberta will love it. Are the following arguments valid?

6. If you find a joke funny, you have a working brain. You found that joke funny. So, you must have a working brain.

7. If you find a joke funny, you have a working brain. You didn’t find that joke funny. So, you must have a problem with your brain function.

8. If you find a joke funny, you have a working brain. You have a working brain. So, you find that joke funny.

Answer

1. This is an argument. The conclusion indicator word “so” picks out the conclusion, “That joke is not funny.” The other sentences are the premises.

2. This is not an argument. A person is stating an opinion about a joke, but is not providing reasons why one should believe the opinion is true.

3. This is an argument. The indicator word “because” picks out the premise “the sound the letter g makes is funnier than the sound the letter f makes” which is offered in support of the conclusion “‘Gumpy’s’ is funnier than ‘Frumpy’s’ as the name of a potato chip.”

4. This is a deductive argument. The premises are “Roberta loves everything Aparna Nancherla does” and “Her last special came out last month.” The conclusion is “Roberta loved it.” The first premise is a universal claim, that Roberta has loved all of Aparna Nancherla’s work, while the conclusion refers to one special, that is, a part of the whole. As such, the argument proceeds from broad to narrow and is therefore deductive.

5. This is an inductive argument. The premises are “Roberta has loved everything Aparna Nancherla has done” and “Her next special comes out next month.” The conclusion is “Roberta will love it.” The first premise only includes Roberta’s opinion about the work that has already been done, whereas the conclusion makes a claim about the upcoming special which is not covered by that premise. That means that the argument is ampliative, the content of the conclusion goes beyond the content of the premises.

6. This is a valid argument. The premises are “If you find a joke funny, you have a working brain” and “You found that joke funny.” The conclusion is “You have a working brain.” The first premise states that finding a joke funny is sufficient to indicate a working brain, that is, if you are capable of processing a joke and determining it to be funny, such cognitive processes would indicate a working brain. You did find this joke funny, that is, you did all the neurological work necessary for the outcome we deemed sufficient to indicate a working brain. So, it does follow that the conclusion must be true, if the premises are. In other words, if the premises are true, the conclusion cannot be false. The truth of the conclusion does follow from the truth of the premises making this a valid argument

7. This argument is invalid. The premises are “If you find a joke funny, you have a working brain” and “You didn’t find that joke funny.” The conclusion is “You must have a problem with your brain function.” The first premise states that finding a joke funny is sufficient to indicate a working brain, that is, if you are capable of processing a joke and determining it to be funny, such cognitive processes would indicate a working brain. In this case, you did not find this joke funny, that is, you still did all the neurological work necessary to determine 16 whether a joke was funny or not, but you found it to not be funny. The first premise tells you what is sufficient, that is, what would be enough by itself, to demonstrate a working brain; but it does not say it is necessary, that is, something that must be the case to demonstrate a working brain. It is perfectly possible for the premises to be true and the conclusion to be false. Therefore, the argument is invalid.

8. This argument is invalid. The premises are “If you find a joke funny, you have a working brain” and “You have a working brain.” The conclusion is “You find that joke funny.” The first premise states that, if you find a joke funny, that by itself is enough to indicate that your brain is working; but the converse does not follow, that if your brain is working then you will find some joke funny. It is perfectly possible for both of the premises to be true and the conclusion to be false. So, this is an invalid argument.