FALLIBLE FOX

​Deductive arguments

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In deductive arguments the author intends to prove their conclusion using inescapable logic. That is, it's the author's intention for the conclusion to follow by necessity from the premises. The intention part is more important than you might at first realize—regardless of whether the conclusion does in fact follow from the premises, the author's intention for the conclusion to follow from the premises is what makes the argument deductive.

 

Soon, we'll discuss deductive arguments in which a problem with the logical relationship between premises and conclusion means that the conclusion does not follow. For now, we'll seek to pin down the basic nature of deductive arguments. Here are some examples of deductive arguments:

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All people are mortal (premise 1)

Kanye West is a person (premise 2)

Therefore, Kanye West is mortal (conclusion)

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Halifax is in Canada. (premise 1)

I am in Halifax. (premise 2)

Therefore, I am in Canada (conclusion)
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In the above examples, you can see that if we are to accept the premises as true (regardless of whether they are true), the logic is—and is intended by the author to be—inescapable. In other words, in the above arguments, if the premises are true, the conclusion must follow. Stated differently, from the premises (if taken to be true), there's no way around arriving at the conclusion—if we're being logical, we're forced to accept it. 

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Try this: based on the basic description of deductive arguments in the two paragraphs above, are the following deductive arguments?

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When Jim goes to bed late, he has time to watch a full movie.

Last night, Jim went to bed late.

Therefore, last night Jim had time to watch a full movie.

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This year, 95% of the Halloween candy in my neighbourhood is chocolate.

My son and daughter collected their candy exclusively in our neighbourhood.

Therefore, most of the candy my kids collected will be chocolate.

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Both of the above arguments have two premises and one conclusion. They also both have an indicator word—"therefore"—showing you where the conclusion is (remember from a previous section that, although this indicator word is helpful, it need not be present). Only one of these two arguments, however, is a deductive argument.

 

It's the first one. Why? In the first one, if we accept the premises as true, we have no choice but to accept the conclusion. There's no way around it—Jim had time to watch a movie last night! That doesn't mean he did watch a movie, but if the premises are both true he definitely had time. 

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By contrast, we shouldn't take the second of the two arguments to be deductive. The premises render the conclusion likely, but not certain. There's a good chance that the conclusion is true (and I'd certainly bet on it being true), but it's not certain. Even if the kids went to random houses throughout the neighbourhood, there's a very small chance that they visited the houses in the 5% without chocolate at a greater frequency than the 95% that did have chocolate. For instance, it's possible that the kids went to only three houses before they got cold and at least two of those three houses happened to be among the 5% without chocolate.

 

In short, given true premises it's still possible that the conclusion is false. The conclusion in the second argument is about probability, not certainty. We're not logically compelled to certainty about the conclusion, but that conclusion remains likely. It's a fine argument, but not a deductive one.  Rather, it's an inductive argument, which we'll get to later. For now we'll stick to deductive arguments.

 

Let's have a look at what it means for a deductive argument to be valid. This is when a deductive argument's structure renders it impossible to have a false conclusion given true premises. 

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Valid and invalid deductive arguments

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Here's a deductive argument:

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All people are mortal (premise 1)

Mitsy is a person (premise 2)

Therefore, Mitsy is mortal (conclusion)

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In the argument above, if we are to assume both premises to be true, we must accept the conclusion. There's no way around it. You can see that the premises are logically tied to the conclusion. In this case, it's impossible to have true premises and a false conclusion. That means this is a valid deductive argument. We can say a deductive argument is valid when, given true premises, it would be impossible for the conclusion to be false.

 

Validity in the context of deductive arguments is only about the logic in the relationship between premises and conclusions. Validity means the conclusion logically and inescapably follows from the premises. Importantly, to assess validity, put aside whether the premises are in fact true or false—validity has nothing to do with actual truth or falsity of premises. That is, a deductive argument can still be valid when one or more of the premises is false.

 

For instance, if it turns out that Mitsy is a cat, not a person, the above argument is still a valid deductive argument. Again, that's simply because, putting aside truth of premises, the argument's internal logic works perfectly well—the conclusion follows logically from the premises.

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Here's an alteration that renders the argument invalid:

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All people are mortal (premise 1)

Mitsy is in Argentina (premise 2)

Therefore, Mitsy is mortal (conclusion)

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The above example is an invalid deductive argument. This occurs when there's an error in the logic linking the premises and conclusion. In this example, although the author seems to be intending to prove Mitsy is mortal, the relationship between the premises and conclusion of the argument doesn't make sense.

 

To accept the conclusion, we need to know whether Mitsy is a person and the argument, unfortunately, does not speak to this concern. Given the content of this argument alone, it's possible, that Mitsy is a mountain range, a city in Argentina, or the sound ketchup makes when it comes out of the bottle. There's nothing in the argument suggesting otherwise.

 

Further, we don't need to know anything at all about where Mitsy is positioned geographically. So, premise 2 is totally irrelevant to the conclusion—there's no connection there at all. Therefore, we cannot conclude from these premises that Mitsy is mortal. Obviously, an invalid argument like this is a poor argument.

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The following is another example of an invalid deductive argument. The reason it's invalid may be quite a bit harder to grasp, so take a minute with it.

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All living things need water. (premise 1)

Roses need water. (premise 2)

Therefore, roses are living things. (conclusion)

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In the above argument, even if we assume the premises are true, they do not provide evidence for the conclusion. Specifically, the premises do not say anything about whether roses are living things. Roses may indeed need water, but that doesn't tell us whether they belong to the category of living things. For instance, there are non-living things that need water, such as swimming pools and lemonade. If we replace the word "roses" with "swimming pools", but keep the argument's structure the same, we can see the problem: 

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All living things need water. (premise 1)

Swimming pools need water. (premise 2)

Therefore, swimming pools are living things. (conclusion)

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Simply replacing roses with pools renders the argument logically ridiculous, but the logic is exactly the same as for the first of the two arguments. You can't say swimming pools are alive just because they need water, and the same goes for roses! If you're still stuck on this one, don't just skip forward—give it a think until you see the problem with the logic.

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By the way, upon first reading the roses argument above, many university students think it makes logical sense and is a valid argument. It shouldn't be a huge concern if you, too, find this one tough. It should, however, suggest to you how difficult it can be to evaluate the logic of even simple arguments.

 

Why is it so hard to see the problem with the roses argument? To help us, let's bring back some of the content from the metacognition module. It might be because we're processing the argument in "type 1" thinking mode, relying too much on our intuition and, therefore, jumping to conclusions. For example, a person might intuitively and automatically do only half the required work, evaluating the truth of the statements alone (they're all true!) but ignoring the logic in the links between the statements. When we ponder arguments, we have to consider both (a) the truth of each premise and  (b) the logic of argument. Those are two completely different considerations. This often requires that we slow down our thinking.

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Another way your intuition might lead you astray is if you're relying too much on what you already know instead of thinking solely about the internal logic of the argument itself. That is, you already know that the conclusion is true: "roses are living things". But I did not ask you if the conclusion is true. I ask if the conclusion that roses are living things follows from the premises. In other words, although the conclusion is true it's not this argument that renders it so.  

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Again, validity of arguments hinges only on the logical connection between premises and conclusion. When you're assessing validity, forget about truth—focus on the argument's structural logic alone.

 

Below, we'll get to soundness, which adds truth of premises to the mix.

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Sound and unsound deductive arguments

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Let's return to an example from above:

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All people are mortal (premise 1)

Kanye West is a person (premise 2)

Therefore, Kanye West is mortal (conclusion)

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Here, again, the premises together work to make the conclusion certain (if the premises happen to be true). That makes the argument valid. What's important now is that the premises are true: we know it's correct to say that (1) all people are mortal (i.e., all people will one day die) and (2) Kanye is a person. These statements are true.

 

Now, because (a) the conclusion logically follows from the premises and (b) the premises are true, there's no way around your acceptance of the conclusion.

 

When both of those conditions are met in a deductive argument, we refer to it as a sound deductive argument. That is, a sound deductive argument is (a) valid (the conclusion logically follows from the premises) and (b) the premises are true. A sound argument is a good deductive argument.   

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This means that validity is a precondition for soundness—an argument cannot be sound if it is invalid. So, if the logic of the argument doesn't work, it doesn't matter if the premises are true. It's both invalid and unsound.

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Let's revisit another argument from above:

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Halifax is in Canada. (premise 1)

I am in Halifax. (premise 2)

Therefore, I am in Canada. (conclusion)

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In this argument, again, there is a clear logical connection between the premises and conclusion. It is, therefore, a valid deductive argument. That is, if we can accept the premises as true, we must also accept the conclusion. However, it happens to be the case that one of the premises is not true: I am currently not in Halifax. If one or more of the premises is false, the argument is not sound—we'll call it an unsound deductive argument. So, this one is valid but unsound.

 

Get it? It might be a good idea to draw yourself a diagram of valid vs. invalid and sound vs. unsound deductive arguments. While I've placed one near the top of this chapter for you, constructing one yourself as you read can be helpful, too. We're starting to juggle a lot of new concepts.

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Before we leave this basic look at deductive arguments, note that the above arguments we've considered are all of the same 3-line structure, with two premises and a conclusion. The two above arguments take the form of a syllogism, which is generally represented in three lines, with two premises and a conclusion. However, deductive arguments need not take this form—they may contain only one or more than two premises. I just use these so-called syllogisms here to keep things simple (Although not essential for moving forward with this section, you can learn more about the specific nature of syllogisms here and different types of syllogisms, with several examples, here).

 

Anyway, keep in mind that deductive arguments need not appear in this 3-line form. What makes an argument deductive is simply that the author is trying to prove that a conclusion is true with what they intend to be inescapable logic.

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Inductive arguments

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While deductive arguments aim to provide proof for a conclusion, other arguments instead offer evidence in support of a conclusion (i.e., they render the conclusions likely or probable). These are inductive arguments. In a good inductive argument, the conclusion is likely rather than certain to follow from a set of true premises.

 

85% of St. Joseph's students complete their homework on time. (premise 1)

Dave is a student at  St. Joseph's school. (premise 1)

Dave probably completes his homework on time. (conclusion)

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This sort of argument is inductive because the statistical information found in the premises render the conclusion that Dave completes his homework on time probable but not certain. Dave may well be among that 15% of students that do not complete their homework on time, but it's quite a bit more likely that he's among the other 85%.

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Here's a different sort of example that doesn't deal with statistics:

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Mike just unexpectedly broke off his engagement with Julie. (premise 1)

Julie was in love with and wanted to marry Mike. (premise 2)

Julie is probably upset. (conclusion)

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Here, assuming the premises are true, although it might seem very likely that Julie is upset, it doesn't necessarily follow that Julie is upset. There are a number of reasons Julie wouldn't be upset. For example, as unlikely as it might be, it's possible that Julie found out other news that kept her afloat. Maybe Julie somehow misheard or misinterpreted what Mike said when he meant to break of the engagement. It's also possible that Julie has a condition that renders it impossible to experience negative emotions.

 

In short, we don't have enough information here to know that Julie would be upset given those two premises. However, given this information alone, it would be reasonable to conclude that Julie is probably upset. It's inductive; it's likely or probable, not certain or definite that she's upset.

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In the Julie example above, the word, "probably" acts as an indicator word suggesting an inductive argument; however, even if that word were absent we'd generally still be better to take this sort of argument as inductive rather than deductive. 

 

In short, indicator words like "probably", "likely", and "odds are" can be useful clues that you're seeing an inductive argument, but they don't always appear in the argument so it's best to not depend on them alone to tip you off. 

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​​​Strong versus weak inductive arguments

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Like deductive arguments, inductive arguments vary in their effectiveness. Once again, this has to do with (a) whether there is a logical connection between the premises and conclusion (i.e., does the conclusion follow from the premises?) and (b) whether the premises are true. One of the following inductive arguments is better than the other. As yourself which one it is and why. Make sure to consider the wording carefully.

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1. In a well-executed, representative survey of 500 professors across Canada, 80% expressed non-belief in a higher power (premise).

Therefore, if we take a random professor at a Canadian university, it's likely that they will express non-belief in a higher power (conclusion).

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2. In a well-executed, representative survey of 500 professors across Canada, 80% expressed non-belief in a higher power (premise).

Therefore, if we take a random Canadian, it's likely that they will express non-belief in a higher power (conclusion).

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In the first argument above, the premise and conclusion are tightly linked. Though the survey findings cannot provide proof for the conclusion, we should be able to draw reasonable inferences about the majority of Canadian professors' beliefs from a well-done, representative survey of Canadian professors' beliefs. This is what is referred to as a strong inductive argument—the premise(s) provide good evidence (again, not proof) in support of the conclusion.

 

In a weak inductive argument the premises don't sufficiently support the conclusion. The premise of the second argument, for example, does not provide good evidence for the argument's conclusion. I would not be convinced by an argument like this and you shouldn't either—a survey of professors, is not representative of the Canadian population as a whole, so inferences about the general population cannot reasonably be drawn from such a sample. In short, the premise provides insufficient evidence for the conclusion. This is a weak deductive argument and should not convince us of the conclusion (to foreshadow a later section on logical fallacies, this argument is representative of fallacious reasoning—in particular, it commits the fallacy of hasty generalization).

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Importantly, note that whether or not the premises in the two arguments above are true does not matter in the least for determining whether these arguments are strong or weak. Strength of inductive arguments is only concerned with the connection between the premise(s) and conclusion.

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A new term is going to be needed for the when we consider truth of premises: cogency.

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Cogent and uncogent inductive arguments

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We can't forget that the truth of the premises matters in inductive arguments just as it does for deductive arguments. To think about this, we'll stick with the first argument from the section above for a moment longer. Remember, this is a strong inductive argument. Here it is again:

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In a well-executed, representative survey of 500 professors across Canada, 80% expressed non-belief in a higher power (premise).

Therefore, if we take a random professor at a Canadian university, it's likely that they will express non-belief in a higher power (conclusion).

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If the premise is true—that a real survey of 500 professors across Canada, did in fact reveal that 80% of professors express non-belief in a higher power—then we can call this a cogent inductive argument. This means that (a) there is a logical connection between the premises and conclusion and (b) the premise(s) are true.

 

Perhaps, however, the premise was revealed to be false. Maybe the arguer lied about the study having been conducted. Maybe the arguer thought this was a real  study but it was faked! Alternatively, perhaps they misinterpreted the findings of the research (unfortunately, that happens a lot).

 

In any case, established falsity of this premise would render this an uncogent inductive argument. An uncogent argument is found where at least one of the premises in an inductive argument is false and/or when an inductive argument is weak. Again, this is because, for an argument to be cogent both of the following must be the case: (a) the premises are true and (b) the premises provide good enough support for the conclusion to be likely.

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It may at first be confusing to note that, even if the premise of this argument were false—say, for example, the noted research was faked—the argument would still be labeled a strong inductive argument. This is because inductive strength vs. weakness deals solely with whether there is a logical connection between the premises and conclusion. That is, the truth of premises has no bearing on the degree of argument strength. When we're interested in evaluating truth of premises and the connection between the premises and conclusion, we must speak of cogency. That's a bit of terminology it may be tough to grasp, so spend some time with it.

 

As I noted in the section above on deduction, it may be useful to draw a figure or a table to help you map these terms out.​​​

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Categorizing Ambiguous Arguments Using the Principle of Charity

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It's not always easy to judge whether an argument is meant to be inductive vs. deductive. This is because we have to depend in part on (1) understanding the author or speaker's intent (which can be difficult, if not impossible) and (2) what we can extract from the wording and subject matter. That is, does the wording suggest an attempt to prove a conclusion or an attempt to support the conclusion's likelihood? If it's the former, it's a deductive argument. If the latter, it's inductive.​

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Here's a good rule of thumb about sorting more ambiguous arguments: if (a) the premises don't seem to prove the conclusion and (b) you can reasonably make the case that the argument is inductive rather than deductive, it's better to take the argument as inductive.

 

Use the principle of charity. The principle of charity, in this context, asks us to interpreting the argument in the best possible light within the bounds of reason. By angling toward interpreting the argument as inductive, you're giving the author the benefit of the doubt—you're assuming, if possible, that the author is aware that they're dealing with likelihoods rather than certainties. 

 

For instance, imagine an argument put forward by a competent scientist appearing on a podcast. She says:

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Most research studies support the idea that students learn more from in-person compared to online instructions (premise).

Therefore, in-person instruction is better for students' learning (conclusion).

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Even if she sounds quite certain, we probably better to assume her argument is inductive. ​​​Although a given speaker might be saying with certainty that this body of research proves that learning in person is better, we know that this speaker is a scientist. A scientifically literate person would know that this body of research cannot prove the conclusion. This is because science doesn't generally deal with certainties but, rather, probabilities. A series of studies of this sort can give evidence in favour of a conclusion but cannot prove it. 

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So, even though it might sound deductive, it would be more charitable to the speaker if we take it as inductive. That is, the argument is much better if taken as an inductive argument (dealing with likelihood) than it is if we assume it's deductive (dealing with certainty). In other words, it's easier to find fault in or knock down this argument if we assume the speaker is truly certain about the conclusion. ​
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Similarly, it's more charitable for us to assume that an argument is inductive when the conclusion is referring to some future event, such as the outcome of an upcoming election.

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The majority of polls tell us that candidate X has a strong lead over candidate Y (premise).

Therefore, candidate X will win the presidency (conclusion).

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No poll and no set of polls can tell us with certainty what the future holds. The speaker should have said "candidate X will probably win the presidency." Although the speaker should have been clearer, it's also your job as a discerning member of the audience to use clues to figure out what they really mean—are they really perfectly certain about that conclusion? If this speaker is a pollster or knowledgeable journalist, they are likely aware that polls deal with probability. In such a case, it's more charitable to assume this person is making a probabilistic argument (inductive), even though it might sound like they think the polls are proof of an upcoming victory.

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Why is it important to treat others' arguments charitably? Gregory Bassham and colleagues (p. 62) write: 

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The principle of charity serves two important goals in critical thinking. First, it fosters civility and mutual understanding in argument by demanding that we treat the arguments of others with the same generous and respectful spirit that we would like others to treat our own arguments. Equally important, it promotes the discovery of truth by insisting that we confront arguments that we ourselves admit to be the strongest and most plausible versions of those arguments.

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In practice, when evaluating real-world arguments, there's more to consider than just the wording. If it's important to get author or speaker's intentions right, take the responsibility to do some digging into the context and source of the argument as it could reveal important information about things left unsaid such as hidden intentions behind the argument that render it more likely to be inductive than deductive. 

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