Chapter 2 The scope of logic

In chapter 1, we talked about arguments, i.e., collections of sentences (the premises), followed by a single sentence (the conclusion). We said that some words, such as ‘therefore’, indicate which sentence is supposed to be the conclusion. ‘Therefore’, of course, suggests that there is a connection between the premises and the conclusion, namely that the conclusion follows from, or is a consequence of, the premises.

This notion of consequence is one of the primary things logic is concerned with. One might even say that logic is the science of what follows from what. Logic develops theories and tools that tell us when a sentence follows from some others.

What about the main argument discussed in chapter 1?

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  Either the butler or the gardener did it.

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  The butler didn’t do it.

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  The gardener did it.

We don’t have any context for what the sentences in this argument refer to. Perhaps you suspect that “did it” here means “was the perpetrator” of some unspecified crime. You might imagine that the argument occurs in a mystery novel or TV show, perhaps spoken by a detective working through the evidence. But even without having any of this information, you probably agree that the argument is a good one in the sense that whatever the premises refer to, if they are both true, the conclusion cannot but be true as well. If the first premise is true, i.e., it’s true that “the butler did it or the gardener did it”, then at least one of them “did it”, whatever that means. And if the second premise is true, then the butler did not “do it.” That leaves only one option: “the gardener did it” must be true. Here, the conclusion follows from the premises. We call arguments that have this property valid .

By way of contrast, consider the following argument:

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  If the driver did it, the maid didn’t do it.

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  The maid didn’t do it.

• ∴
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  The driver did it.

We still have no idea what is being talked about here. But, again, you probably agree that this argument is different from the previous one in an important respect. If the premises are true, it is not guaranteed that the conclusion is also true. The premises of this argument do not rule out, by themselves, that someone other than the maid or the driver “did it.” So there is a case where both premises are true, and yet the driver didn’t do it, i.e., the conclusion is not true. In this second argument, the conclusion does not follow from the premises. If, like in this argument, the conclusion does not follow from the premises, we say it is invalid .

How did we determine that the second argument is invalid? We pointed to a case in which the premises are true and in which the conclusion is not. This was the scenario where neither the driver nor the maid did it, but some third person did. Let’s call such a case a counterexample to the argument. If there is a counterexample to an argument, the conclusion cannot be a consequence of the premises. For the conclusion to be a consequence of the premises, the truth of the premises must guarantee the truth of the conclusion. If there is a counterexample, the truth of the premises does not guarantee the truth of the conclusion.

As logicians, we want to be able to determine when the conclusion of an argument follows from the premises. And the conclusion is a consequence of the premises if there is no counterexample—no case where the premises are all true but the conclusion is not. This motivates a definition:

A sentence $A$ is a consequence of sentences $B_1$, …, $B_n$ if and only if there is no case where $B_1$, …, $B_n$ are all true and $A$ is not true. (We then also say that $A$ follows from $B_1$, …, $B_n$ or that $B_1$, …, $B_n$ entail  $A$.)

This “definition” is incomplete: it does not tell us what a “case” is or what it means to be “true in a case.” So far we’ve only seen an example: a hypothetical scenario involving three people. Of the three people in the scenario—a driver, a maid, and some third person—the driver and maid didn’t do it, but the third person did. In this scenario, as described, the driver didn’t do it, and so it is a case in which the sentence “the driver did it” is not true. The premises of our second argument are true, but the conclusion is not true: the scenario is a counterexample.

We said that arguments where the conclusion is a consequence of the premises are called valid, and those where the conclusion isn’t a consequence of the premises are invalid. Since we now have at least a first stab at a definition of “consequence”, we’ll record this:

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  An argument is valid if and only if the conclusion is a consequence of the premises.

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  An argument is invalid if and only if it is not valid, i.e., it has a counterexample.

Logicians are in the business of making the notion of “case” more precise, and investigating which arguments are valid when “case” is made precise in one way or another. If we take “case” to mean “hypothetical scenario” like the counterexample to the second argument, it’s clear that the first argument counts as valid. If we imagine a scenario in which either the butler or the gardener did it, and also the butler didn’t do it, we are automatically imagining a scenario in which the gardener did it. So any hypothetical scenario in which the premises of our first argument are true automatically makes the conclusion of our first argument true. This makes the first argument valid.

Making “case” more specific by interpreting it as “hypothetical scenario” is an advance. But it is not the end of the story. The first problem is that we don’t know what to count as a hypothetical scenario. Are they limited by the laws of physics? By what is conceivable, in a very general sense? What answers we give to these questions determine which arguments we count as valid.

Suppose the answer to the first question is “yes.” Consider the following argument:

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  The spaceship Rocinante took six hours to reach Jupiter from Tycho space station.

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  The distance between Tycho space station and Jupiter is less than 14 billion kilometers.

A counterexample to this argument would be a scenario in which the Rocinante makes a trip of over 14 billion kilometers in 6 hours, exceeding the speed of light. Since such a scenario is incompatible with the laws of physics, there is no such scenario if hypothetical scenarios have to conform to the laws of physics. If hypothetical scenarios are not limited by the laws of physics, however, there is a counterexample: a scenario where the Rocinante travels faster than the speed of light.

Suppose the answer to the second question is “yes”, and consider another argument:

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  Priya is an ophthalmologist.

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  Priya is an eye doctor.

If we’re allowing only conceivable scenarios, this is also a valid argument. If you imagine Priya being an ophthalmologist, you thereby imagine Priya being an eye doctor. That’s just what “ophthalmologist” and “eye doctor” mean. A scenario where Priya is an ophthalmologist but not an eye doctor is ruled out by the conceptual connection between these words.

Depending on what kinds of cases we consider as potential counterexamples, then, we arrive at different notions of consequence and validity. We might call an argument nomologically valid if there are no counterexamples that don’t violate the laws of nature, and an argument conceptually valid if there are no counterexamples that don’t violate conceptual connections between words. For both of these notions of validity, aspects of the world (e.g., what the laws of nature are) and aspects of the meaning of the sentences in the argument (e.g., that “ophthalmologist” just means a kind of eye doctor) figure into whether an argument is valid.

One distinguishing feature of logical consequence, however, is that it should not depend on the content of the premises and conclusion, but only on their logical form. In other words, as logicians we want to develop a theory that can make finer-grained distinctions still. For instance, both

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  Priya is either an ophthalmologist or a dentist.

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  Priya isn’t a dentist.

• ∴
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  Priya is an eye doctor.

and

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  Priya is either an ophthalmologist or a dentist.

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  Priya isn’t a dentist.

• ∴
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  Priya is an ophthalmologist.

are valid arguments. But while the validity of the first depends on the content (i.e., the meaning of “ophthalmologist” and “eye doctor”), the second does not. The second argument is formally valid . We can describe the “form” of this argument as a pattern, something like this:

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  $A$ is either an $X$ or a $Y$.

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  $A$ isn’t a $Y$.

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  $A$ is an $X$.

Here, $A$, $X$, and $Y$ are placeholders for appropriate expressions that, when substituted for $A$, $X$, and $Y$, turn the pattern into an argument consisting of sentences. For instance,

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  Mei is either a mathematician or a botanist.

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  Mei isn’t a botanist.

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  Mei is a mathematician.

is an argument of the same form, but the first argument above is not: we would have to replace $Y$ by different expressions (once by “ophthalmologist” and once by “eye doctor”) to obtain it from the pattern.

Moreover, the first argument is not formally valid. Its form is this:

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  $A$ is either an $X$ or a $Y$.

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  $A$ isn’t a $Y$.

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  $A$ is a $Z$.

In this pattern we can replace $X$ by “ophthalmologist” and $Z$ by “eye doctor” to obtain the original argument. But here is another argument of the same form:

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  Mei is either a mathematician or a botanist.

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  Mei isn’t a botanist.

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  Mei is an acrobat.

This argument is clearly not valid, since we can imagine a mathematician named Mei who is not an acrobat.

Our strategy as logicians will be to come up with a notion of “case” on which an argument turns out to be valid only if it is formally valid. Clearly such a notion of “case” will have to violate not just some laws of nature but some laws of English. Since the first argument is invalid in this sense, we must allow as counterexample a case where Priya is an ophthalmologist but not an eye doctor. That case is not a conceivable situation: it is ruled out by the meanings of “ophthalmologist” and “eye doctor.”

When we consider cases of various kinds in order to evaluate the validity of an argument, we will make a few assumptions. The first assumption is that every case makes every sentence true or not true—at least, every sentence in the argument under consideration. That means first of all that any imagined scenario which leaves it undetermined if a sentence in our argument is true will not be considered as a potential counterexample. For instance, a scenario where Priya is a dentist but not an ophthalmologist will count as a case to be considered in the first few arguments in this section, but not as a case to be considered in the last two: it doesn’t tell us if Mei is a mathematician, a botanist, or an acrobat. If a case doesn’t make a sentence true, we say it makes it false . We’ll thus assume that cases make sentences true or false but never both.¹¹ 1 Even if these assumptions seem common-sensical to you, they are controversial among philosophers of logic. First of all, there are logicians who want to consider cases where sentences are neither true nor false, but have some kind of intermediate level of truth. More controversially, some philosophers think we should allow for the possibility of sentences to be both true and false at the same time. There are systems of logic in which sentences can be neither true nor false, or both, but we will not discuss them in this book.

Before we go on and execute this strategy, a few clarifications. Arguments in our sense, as conclusions which (supposedly) follow from premises, are of course used all the time in everyday and scientific discourse. When they are, arguments are given to support or even prove their conclusions. Now, if an argument is valid, it will support its conclusion, but only if its premises are all true. Validity rules out the possibility that the premises are true and the conclusion is not true at the same time. It does not, by itself, rule out the possibility that the conclusion is not true, period. In other words, it is perfectly possibly for a valid argument to have a conclusion that isn’t true!

Consider this example:

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  Oranges are either fruit or musical instruments.

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  Oranges are not fruit.

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  Oranges are musical instruments.

The conclusion of this argument is ridiculous. Nevertheless, it follows from the premises. If both premises are true, then the conclusion just has to be true. So the argument is valid.

Conversely, having true premises and a true conclusion is not enough to make an argument valid. Consider this example:

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  London is in England.

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  Beijing is in China.

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  Paris is in France.

The premises and conclusion of this argument are, as a matter of fact, all true, but the argument is invalid. If Paris were to declare independence from the rest of France, then the conclusion would no longer be true, even though both of the premises would remain true. Thus, there is a case where the premises of this argument are true without the conclusion being true. So the argument is invalid.

The important thing to remember is that validity is not about the actual truth or falsity of the sentences in the argument. It is about whether it is possible for all the premises to be true and the conclusion to be not true at the same time (in some hypothetical case). What is in fact the case has no special role to play; and what the facts are does not determine whether an argument is valid or not.²² 2 Well, there is one case where it does: if the premises are in fact true and the conclusion is in fact not true, then we live in a counterexample; so the argument is invalid. Nothing about the way things are can by itself determine if an argument is valid. It is often said that logic doesn’t care about feelings. Actually, it doesn’t care about facts, either.

When we use an argument to prove that its conclusion is true, then, we need two things. First, we need the argument to be valid; i.e., we need the conclusion to follow from the premises. But we also need the premises to be true. We will say that an argument is sound if and only if it is both valid and all of its premises are true.

The flip side of this is that when you want to rebut an argument, you have two options: you can show that (one or more of) the premises are not true, or you can show that the argument is not valid. Logic, however, will only help you with the latter!

Many good arguments are invalid. Consider this one:

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  Every winter so far, it has snowed in Calgary.

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  It will snow in Calgary this coming winter.

This argument generalises from observations about many (past) cases to a conclusion about all (future) cases. Such arguments are called inductive arguments. Nevertheless, the argument is invalid. Even if it has snowed in Calgary every winter thus far, it remains possible that Calgary will stay dry all through the coming winter. In fact, even if it will henceforth snow every winter in Calgary, we could still imagine a case in which this year is the first year it doesn’t snow all winter. And that hypothetical scenario is a case where the premises of the argument are true but the conclusion is not, making the argument invalid.

The point of all this is that inductive arguments—even good inductive arguments—are not (deductively) valid. They are not watertight. Unlikely though it might be, it is possible for their conclusion to be false, even when all of their premises are true. In this book, we will set aside (entirely) the question of what makes for a good inductive argument. Our interest is simply in sorting the (deductively) valid arguments from the invalid ones.

So: we are interested in whether or not a conclusion follows from some premises. Don’t, though, say that the premises infer the conclusion. Entailment is a relation between premises and conclusions; inference is something we do. So if you want to mention inference when the conclusion follows from the premises, you could say that one may infer the conclusion from the premises.