Chapter3Standard Form and Validity

3.1 Logical Arguments

Defining “arguments” as public social exchanges is too broad for our purposes here. We are specifically concerned with logical arguments. In making a logical argument, a claim is put together with a presentation of reasons that support the truth of that claim. The objective of a logical argument is to show some statement or position to be true or reasonable. We call what is to be shown the conclusion of the argument. Typically, the conclusion is based on its logical relationship to certain statements or sentences in the body of the argument; we call these the premises. One way to visualize an argument is by thinking of the premises as the legs of a table and the surface as the conclusion. The premises provide support for the conclusion.

Often there are certain intermediate steps, which may add no new information to the premises but show the logical relationships between the premises and the conclusion. We will call these intermediate steps. The intermediate steps are not essential to the logical quality of the argument; they are simply devices for helping one see the relationship between the premises and the conclusion. So we may define a logical argument for our purposes as a sequence of sentences or statements (P1, P2, . . . Pn, C), where the sentences or statements P1 to Pn are the premises and C is the conclusion.

In order to make clear the structure of a logical argument, we put it in a standard form. To do this, we identify the premises and the conclusion, list the premises in a vertical stack with a line below it, and place the conclusion below the line.

In order to differentiate the premises from the conclusion, it helps to ask “why” and find the “because.” We might ask, “Why the conclusion?” Because the premises. Consider: “I got sick last time I drank coffee, so I will pass on coffee today.” Try both directions on the “why” question to identify the conclusion: Why did you pass on coffee today? Because you got sick last time. It is quite different to say, “Why did you get sick on coffee last time? Because you passed on coffee today?” This second version doesn’t work. But if you try the reversal test using “why” and “because,” it will help to identify the premises from the conclusion. Keep in mind that people will use words like “why” and “because” improperly in everyday language, so when you are looking for logic, you should not necessarily trust the way the statements are presented at first blush.

Pulling arguments out of paragraphs can be difficult. One reliable but imperfect shortcut can be to try to identify the use of indicator words. Indicator words are words that signal the logical relationship between claims. We use them all the time! Consider words and phrases such as “because,” “therefore,” “since,” “so,” “thus,” “and,” “but,” “or,” “in conclusion,” “consequently,” and so on. However, please note that sometimes an argument doesn’t use any indicator words!

These words occur in language, but they are not always used correctly. When determining if a claim is a conclusion or a premise, we can use indicator words as guides but not guarantees. We still need to be asking ourselves what the speaker intends to be in support of the conclusion (the premises) and what the speaker intends to convince their audience of (the conclusion).

3.2 Deductive Versus Inductive Arguments

There are two major types of arguments: deductive arguments and inductive arguments. The conclusion of a good deductive argument may be quite independent of what is true because these arguments depend on their logical structure. Before we understand what makes a good deductive or inductive argument, we need to identify whether an argument is inductive or deductive in the first place. We focus on deductive arguments because we can study their forms without paying attention to the specific facts. We don’t have to worry about new information with a deductive argument, but with an inductive argument, the conclusion can be overturned by new information about the world.

Arguments try to prove or convince. Just as indicator words can help identify a logical structure, the kinds of words used in an argument can help you identify whether it is inductive or deductive. Inductive arguments will have words like “likely” or “is probable” within their conclusions. A deductive argument will be more conclusive. It will contain words indicating a necessary conclusion (one that cannot be overturned by new evidence) using words such as “this proves that” and “necessarily.”

Inductive arguments depend in complex ways on empirical facts about the world. Inductive arguments take facts about the world and “induce” or bring forth a conclusion that is likely, often introducing new information. Inductive arguments provide reason to think a conclusion probable or likely, and a strong inductive argument is one in which, given the assumption that the premises are true, the truth of the conclusion is very probable or highly likely. Like coffee, inductive arguments come in different strengths, and depending on the context, an inductive argument can be a good argument even if it is rather weak. Inductive reasoning is harder to study than deductive reasoning because it is messier, but the vast majority of our ordinary inferences are inductive, and most of our knowledge of the world—whether scientific or common sense—is merely probable rather than demonstratively certain.

A deductive argument does something different. It uses the information provided by the premises to conclude something about their logical relationship. If it is a deductive argument, the conclusion doesn’t introduce any new information.

One way to make sense of the difference between the two kinds of arguments is to suppose you wanted to disagree with the conclusion. Ask yourself, Can you overturn the conclusion by showing one of the premises is false? Or can you only overturn the conclusion by adding a premise? Consider an example:

What happens if we say premise 2 is false? It weakens the conclusion, but it doesn’t overturn it. But, what if we added a premise 4: “Kristin was googling burnout and vitamins while logged into her browser”? This would overturn the conclusion because it makes the conclusion far more unlikely (her phone isn’t necessarily a hot mic, but her sponsored ads could be due to her searching behaviour). So, this argument is inductive. The premises support the conclusion, but not without vulnerability to new evidence—it is contingent on what the world is like. Consider a different example:

What happens if we say premise 1 is false? Does it affect the conclusion? Yes, it would overturn the conclusion. The conclusion depends on the combination of premises 1 and 2. This identifies the argument as deductive in form.

3.3 Inductive Strength and Probability

When we evaluate inductive arguments, we are interested in two central features of their conclusion: their likelihood or probability and their reliability, which has to do with their causal structure. Inductive arguments are typically based on probabilities in the sense that we support an inductive argument by gathering empirical evidence—such as above when Kristin was gathering evidence that her phone is listening to her. Usually, gathering evidence takes the form of collecting data points, counting the frequency of outcomes of differing types, or polling individuals. The evidence is then processed with appropriate statistical methods and expressed as a probability.

These arguments usually are used for calculating risk or giving evidence for causal hypothesis. For example, it is common to hear that smoking increases your risk of dying from lung cancer by about 80 percent. And we can say cancers linked to tobacco use account for about 40 percent of all cancer deaths. Here we use “linked” to mean very likely the cause and “increases your risk” to mean there’s a higher probability that something will happen. The important point in evaluating if an inductive argument is strong is to use appropriate, explicit, and precise statistical methods.

The simplest form of inductive inference is called enumerative induction. It argues from a set of premises about members of a group to a generalization about the entire group. Almost all of our beliefs about the world are about the unobserved, but we cannot help but assume the unobserved will largely be like the observed, so we use past experiences to give us guidance in the future. Consider the claim that “the sun will rise tomorrow.” What would be conclusive proof this is true? Not only do we know there is a day that this won’t be true (five billion years or so from now), but there could be an unforeseen catastrophe that undermines the claim. All we can do is generalize based on positive past instances where the sun did rise. We make this inference based on enumerative induction. Here are two traditional examples of enumerative induction that differ on the basis of whether there is an indefinite or definite number of possible observations on which to base a conclusion:

The intuition is that as n (the number of observations) gets larger, the truth of the conclusion becomes more likely and thus more reasonable to believe. The irony of example 1 is that if you imagine the strength of the conclusion from the perspective of someone who lived somewhere that only had white swans (a British person in the seventeenth century), you might think you have conclusive proof only because they have never happened upon a black swan (because the swans are native to what is now known as Australia). How many white swans would a seventeenth century British person need to observe to conclusively verify that “all swans are white”? The point here is that the universal claim about swans is much harder to support than the strength of the disproof by the single black swan. This brings us to the value of falsifying hypotheses.

Karl Popper, the Austrian philosopher of science, argued that science should not pursue an inductivist account of scientific method aimed at showing which theories are true but instead should concentrate on crucial experiments aimed at falsifying hypotheses. This would mean designing experiments aimed at falsifying universal law hypotheses rather than designing experiments meant to verify them. So we would not be aiming to prove a conclusion true for all time, but we would be taking hypotheses and subjecting them to rigorous and harsh attempts to refute them as the next best thing. Just because we cannot conclusively verify an inductive universal generalization doesn’t mean that science is a useless project!

However, the swan example is very different than when we are working with a definite number of things. For the marbles, imagine pulling them out one at a time but not looking at them until after you’ve take them out. You can feel the marbles and have a rough sense of how many there are, and you can stir them up and pull out a black marble. You reach in and stir vigorously from the bottom and pull out another black marble, repeating this a number of times. You wonder, “Are all the marbles black?” “Are a majority of them black?” Unlike with the swan example, your intuitions have more to work with. First of all, it is, in principle, possible to take out all of the marbles and observe them. If you took out one hundred marbles and they were all black, your conclusion would be strong—so strong it would be deductively certain (one hundred black marbles, therefore all marbles are black). Secondly, since you did a good job stirring up the marbles and reaching to the bottom, you made it more likely that you randomly selected a representative sample (or a non-biased sample—imagine if the marbles were in coloured layers and black was just the top). You kept drawing marbles, increasing the size of the sample, making it more likely to be sufficiently large. Lastly, you had helpful background beliefs about how the marbles got there and their patterning (which may or may not be well supported, but they contributed to your inductive conclusion). You had probably made some assumptions about how the marbles got into the box and whether the colours of the marbles would exhibit a regular pattern. And, in addition, other background beliefs you already have will provide some support for the assumptions you bring to this problem. Let’s change the example:

In example 3, you have reason to think that all the marbles are probably one colour, or that there are two colours of marbles in a box, half and half, and at least that there will be a regular proportion of marbles of different colours. If you make such an assumption, then you will be testing for the relative probability of a small number of alternative hypotheses given your evidence. This is important because if there were ninety-nine black marbles and only one red one in the box, you could pull out a very large number of marbles before picking the red one—you couldn’t easily rule out that hypothesis on inductive grounds alone; after all, if the box is from your grandfather’s toy chest in the attic, it might well contain his favourite red agate shooter and ninety-nine black marbles. If you think about this case for a moment, you will see that the probability of the conclusion of an inductive argument doesn’t depend on the evidence alone but upon the alternative possible conclusions that are compatible with the evidence. Part of what made the swan example weak was that we have no difficulty envisioning different breeds of swans that have different colours (since this is so common among other species), and so even testing all the swans in England does nothing to rule out a differently coloured swan in Australia.

To recap, there are two major types of arguments: deductive, or formal, arguments and inductive, or informal arguments. In a good deductive argument, the truth of the conclusion follows inescapably from the truth of the premises. In a good inductive argument, the truth of the premises makes the conclusion highly probable or likely. Good deductive arguments depend upon their logical structure alone. Inductive arguments, however, depend in complex ways on empirical facts about the world, the strength of evidence, our observations, and other conditions. Accordingly, they cannot be studied in abstraction from the facts as we believe them to be.

3.4 Validity

For the first two-thirds of this text, we will deal almost exclusively with deductive arguments, so until further notice, we will use the word “argument” to mean “deductive argument.”

We have just said that in a good deductive argument, the conclusion follows inescapably from the premises; this can be made more precise. The “goodness” of a good deductive argument is called validity. A valid deductive argument has the property that in any situation in which the premises are true, the conclusion must also be true. Invalidity is the failure of this relation. In an invalid argument, it is possible for the premises to be true and the conclusion to be false.

Validity and invalidity deal with the formal relationship between premises and the conclusion. If an argument is valid and, in addition, it has true premises, it is called a sound argument.

Starting from the outside of figure 3.1, many deductive arguments are invalid. They do not have the correct logical structure. Validity is larger than soundness to represent that there are valid arguments that are not sound. Soundness is validity plus truth: Not all valid arguments are sound. Validity is a formal property of arguments whereby if the premises are true, then the conclusion must also be true. Valid arguments can have false premises, which is why we differentiate validity and soundness.

To more easily identify validity, when we put an argument into standard form—we put the conclusion at the bottom. You will often have to rearrange the order of statements to present the logical structure. Sometimes what you need to do is find that support relationship—to return to our earlier metaphor, you will have to separate the legs from the table surface.

Consider these three sentences, which are all versions of the same argument:

1. 1. You are tired, and tired people should sleep, so you should sleep.
2. 2. You should sleep, because you are tired, and tired people should sleep.
3. 3. Tired people should sleep, so you should sleep, because you are tired.

In 2, the conclusion comes first, and the word “because” signals that the other claims made support it; in 1, the conclusion comes at the end, and the word “so” functions to connect it to the other claims as following from or being supported by them. In order to make the structure of a logical argument clear, we put it in a standard form. To do this we identify all the premises (making implicit premises visible) and identify the conclusion, list the premises in a vertical stack with a line below it, and place the conclusion below the line.

Here is the standard form of 1–3 above:

What makes “You should sleep” the conclusion is not where it appears in the argument but the logical relation it has to the other parts of the argument. An important part of recognizing an argument is seeing the relationships of support and dependence that the component claims have on each other; these relations give the argument its force by showing reasons that the conclusion should be accepted.

Philosophers have identified specific deductive patterns that are always valid or invalid. Many everyday arguments can fit into these patterns, and thus we would have an easy way of telling whether they are truth-preserving. We will present five valid forms and two invalid forms.

3.5 Five Valid Deductive Argument Patterns

Logical arguments usually occur in characteristic patterns. These patterns represent relationships of premises to each other that make them support a conclusion. We will look at a large number of argument patterns, some informal and some highly structured, with the aim of making you a better reasoner. Deductive arguments are important for critical thinking because the correctness of a deductive argument is purely a matter of its form or the argument pattern it exemplifies.

Modus ponens is the first form or syllogism we will discuss. “Modus ponens” is Latin for “method of affirming.” This is because the second premise affirms what the first premise hypothetically affirms. This means the argument states a version of “if this thing happens—and it does!—then this other thing happens.” See the form here:

Here, the letters P and Q each stand for their own sentence. The logical relationship is a conditional statement (premise 1) and an assertion (premise 2), then a conclusion that is the result of the combination of premises 1 and 2.

Here, “We live in Saskatoon” is P, and “We live in Saskatchewan” is Q. Modus ponens is a valid argument pattern because if the premises (1 and 2) are true, then the conclusion must also be true. It is true that if you live in Saskatoon, you live in Saskatchewan. The current relationship between cities and provinces is such that Saskatoon is contained within Saskatchewan, which is much larger than just one city. Premise 2 is different. It could be true; it could be false. Let’s assume it is true, which is what we need to do to evaluate the structure of the argument. Notice, though, that premise 1 says, “If.” This “if _____, then _____” relationship means that when you have P, you automatically get Q. Premise 2 asserts P, and then the conclusion says, “Therefore Q.”

Validity means that if the premises are true, then the conclusion must also be true. This means that validity can apply to an argument with false premises. Consider this example:

Premise 2 begins a kind of “chain reaction” between premises 1 and 2 and the conclusion. Just like above, premise 2 triggers the “if _____, then _____” in premise 1, and thus the conclusion follows inescapably. But, revisiting figure 3.1, remember that an argument can be valid without being sound. This same chain reaction can happen in the next syllogism, which is called a hypothetical syllogism. This argument form is as follows:

This deductive pattern has three terms and it does something a bit different than modus ponens. It asks you to combine premises 1 and 2 (as usual), but it more specifically looks at how Q is in the middle of P and R, which allows you to cut the middle out and also assert (in the conclusion) that “if P, then R.” We will deal with this relation in part 2 when we talk about transitivity. Looking at the form, ask yourself if the conclusion is true. On the basis of the two premises, can you see how if you had P you would also be able to deduce R, and thus the conclusion is true? Let’s look at an example:

Here, “If we drink too much” is P, and “We fall down a lot” is Q. R is “We miss Eric’s exciting lecture.” Falling down a lot doesn’t have to be mentioned in the conclusion, because it is already asserted in the premises that P gets you Q, which always leads to R, so you can know for sure that if you have P, then you will always get R, which is the conclusion.

The next syllogism we will cover is modus tollens. This name in Latin means “method of denying” or “method of lifting out or removing.” This is because it is similar to modus ponens, but it uses a negation. The argument form is as follows:

Modus tollens has a very different premise 2. It essentially affirms that the second part of premise 1, Q, does not happen. What happens to the chain reaction when we negate a term? Does anything result from the second part of premise 1 not happening? What can be deduced from that?

This is a good place to introduce a bit of language about how the “if _____, then _____” statements we’ve been using work. In “If P, then Q,” there are two parts: the antecedent (what comes before) P and the consequent (or result) of that condition, which is Q. In other words, “If the antecedent, then the consequent.” This statement is hypothetical as the name “hypothetical syllogism” suggests, since the condition comes into effect “if” something else happens—it requires that the “if” happens; otherwise, it remains hypothetical. But in the case of modus tollens, it is very, very important that the difference between the antecedent and the consequent are clear. Modus tollens denies the consequent. There is an invalid form of argument we will discuss below, which is “denying the antecedent.” Let’s look at an example of modus tollens:

Assuming we know that Toronto is in Ontario, then if we don’t live in Ontario at all, there’s no way that we could live in Toronto. “We live in Ontario” is the consequent that is denied here; it is denied in premise 2, which essentially says, “Not Q.” You should be able to see that this argument is valid, since not living in Ontario should mean you can’t live in Toronto. But think back to modus ponens here. If premise 2 said, “We live in Toronto,” we could deduce that “we live in Ontario,” which would also be valid.

The next syllogism we cover is a disjunctive syllogism. The root of “disjunction” is to separate something, whereas the root of “conjunction” is to bring something together. In the sentence “P or Q,” P is one disjunct and Q is the other disjunct. Instead of having “if _____, then _____” as the logical connection between terms, the connection looks like “_____ or _____.” The way we understand disjunction in the disjunctive syllogism is that one of the two disjuncts must be true. They cannot both be false, and they cannot both be true. Think of “you can’t be a little bit pregnant—you are either pregnant, or you are not.” That is how our “or” operates. It operates exclusively. There are forms of “or” that are inclusive because they allow for both disjuncts to be true. We think of it as pancake house “or” because you can have toast or hashbrowns or both. But we are using an exclusive “or.” They cannot both be true, and they cannot both be false. So when one is false, the other is automatically true. The argument form is as follows:

In the disjunctive syllogism, we are asserting in premise 1 that either P or Q happens or is true. Something being true could also mean it is the state of affairs or that it happens. If premise 2 says that P does not happen, then it must be the case that Q does. This is forced, since premise 1 clearly states it is one or the other. Let’s look at an example:

In this syllogism, it is set out that either the coffee is black or it isn’t (has something in it). Since those two options must be exclusive, the denial in premise 2 makes the conclusion automatically true. This means that disjunctive syllogism is valid.

Let us assume the argument is about us, and we don’t happen to live in either place, therefore premise 1 is false. Nevertheless, premise 1 asserts that we either live in Saskatoon or Toronto, so the form dictates that as given. Premise 2 rules out Toronto, so we have to conclude Saskatoon. Again, this is valid but not sound (in our case, anyway!).

The last syllogism puts together what we’ve seen above.

Here you see that the first two premises are “if _____, then _____” premises where the first terms (antecedents) of both are contained in premise 3 as an “_____ or _____” (disjunction). So if either of those antecedents (P or R) is true, then it allows us to draw either of the consequents of premises 1 or 2. Thus, the conclusion asserts the disjunction “Q or S” (the consequents of premises 1 and 2).

Here, premise 3 really should grab our attention. It asserts that either the antecedent in 1 or the antecedent in 2 occur. Thus, we are able to say that either the consequent of 1 or the consequent of 2 occur. The conclusion is a disjunction. The soundness of this argument depends on the truth of the premises. Premises 1 and 2 are questionable, but it is unlikely that premise 3 is true (there are more options, we’re guessing, for us all at any time), meaning that this example is valid but unsound.

3.6 Two Invalid Deductive Argument Patterns

In each of these previous examples, the argument is valid by virtue of their form. In each case, the premises cannot all be true without the conclusion also being true. The truth of the conclusion is necessitated by the truth of the premises. Of course, not all argument patterns are valid. There are many invalid deductive argument patterns. We will look at two invalid deductive argument patterns, both of which are impersonating modus ponens and modus tollens, but they violate the conditional form the of “if _____, then _____.” Here is the first of the two invalid forms we will cover:

Denying the antecedent means that P, the antecedent in the conditional phrase that is premise 1, is denied. From the denial of P, the arguer concludes that Q, the consequent, absolutely does not occur. What is wrong with this pattern? It helps to think of an example with true premises that leads to a false conclusion.

When evaluating if an argument form is valid or invalid, you can ask yourself, Is there a possibility in which the premises are all true but the conclusion is false? In this instance, it might be true that “if Muffin is a poodle, then Muffin is a dog.” Premise 2 rules out the possibility that Muffin is a poodle, so we know the conditional in premise 1 does not get triggered. Then how can the conclusion be that Muffin is not a dog? All we know is that Muffin is not a poodle. Couldn’t she be a Doberman? By virtue of the premises, we cannot conclude she is not a dog whatsoever. Because the conclusion is possibly false even if the premises are true, this is an invalid form. Let’s look at another example:

Hopefully everyone who has ever lived in or visited Saskatchewan can attest to the fact that there is plenty of Saskatchewan beyond Saskatoon’s outskirts. Imagine a situation in which the premises are true and the conclusion false: You live in Canoe Lake Cree Nation, Saskatchewan. Thus, we can declare denying the antecedent to be an invalid argument form. It might help to think of denying the antecedent as trying to trick you into thinking it is modus tollens but it is backwards. Remember that modus tollens denies the consequent, not the antecedent. Let’s look at the second invalid form we will be covering:

Affirming the consequent is a common mistake where an arguer goes the wrong way with the conditional—it masquerades as modus ponens, but it is invalid. The conditional is always triggered on the antecedent condition, not the consequent. Recall that modus ponens affirms the antecedent.

Again, all these statements could be true, but we live in Canoe Lake Cree Nation, Saskatchewan. Of anyone, presumably, it is true that if they live in Regina, they live in Saskatchewan. But affirming they are in Saskatchewan doesn’t tell us they live in Regina whatsoever. One way to think of affirming the consequent is that in some logical symbolizations, the “if _____, then _____” relationship is symbolized with an arrow: P → Q. In affirming the consequent, a person reasons the wrong way up the arrow. The arrow points from the antecedent to the consequent, not the other way around.