Appendix A — Logic
Throughout this book, I present arguments, defend their premises, and then claim that the conclusions of those arguments follow from the premises. In this appendix, I’ll explain what it means for a conclusion to follow from some premises and how you can tell when a conclusion follows from some premises. In section 1, I introduce the notion of a valid argument, that is, an argument whose conclusion follows from its premises. Then, in section 2, I identify four types of valid arguments. Finally, in section 3—because nothing is sacred in philosophy—I show how even claims about which types of arguments are valid can be called into question.
A.1 Valid Arguments
Suppose you and I have gotten our hands on a live chicken. I want to keep it as a pet, and I’ve already even given it a name: ‘Camilla’. You want to slaughter it and eat it. I’m trying to convince you that we shouldn’t eat Camilla, and I give you the following two arguments:
You probably aren’t convinced by either argument. Why not?
It’s easy to say what goes wrong with the Cuteness Argument. You might say the first premise is false, because chickens are ugly. Or you might deny the second premise, saying that just because something is cute doesn’t mean it’s wrong to eat it. Or maybe you’ll deny both. Either way, the problem with the argument is that its premises aren’t true.
But what about the Feathers Argument? You probably don’t find it any more convincing than the Cuteness Argument. But both of its premises are true. So, what is the problem with the Feathers Argument? The problem is that the conclusion doesn’t follow from the premises. Or, as philosophers like to say, the argument is not valid.
A valid argument is an argument whose conclusion is a logical consequence of its premises. When an argument is valid, the premises guarantee the truth of the conclusion; it’s impossible for the premises to be true without the conclusion being true. You would be contradicting yourself if you accepted all the premises but denied the conclusion.
- Validity
- An argument is valid just in case the premises guarantee the truth of the conclusion: it’s impossible for the premises to be true without the conclusion being true.
The problem with the Feathers Argument is that it’s invalid: it doesn’t follow from Camilla’s having feathers and feathers’ being soft that it’s wrong to eat her. There’s no contradiction in accepting the premises of that argument while denying its conclusion. The Cuteness Argument, by contrast, is valid: the claim that it’s wrong to eat Camilla is a logical consequence of the claim that Camilla is cute and the claim that it’s wrong to eat cute things. Anyone who accepts the premises of the Cuteness Argument is logically required to accept the conclusion as well, on pain of contradicting themselves.
You might be surprised that I just called the Cuteness Argument ‘valid’. But look again at my definition of ‘valid’. That definition doesn’t require the premises of a valid argument to be true, or even plausible. All that’s required is that if the premises are true, then the conclusion is guaranteed to be true as well. An argument can be valid and still be a pretty bad argument, like the Cuteness Argument, because its premises are implausible. (Philosophers have a different word for arguments that are valid and all of whose premises are true. We call them sound arguments.) Also, as defined above, validity can only ever be a feature of arguments. So, at least in philosophical discussions, it’s best to avoid calling premises or points ‘valid’. Only arguments should be described as valid or invalid.
One more word of warning: don’t confuse following and following from. To see what I have in mind, consider this argument from Chapter 4:
It’s true that FD2 follows FD1. That is, it comes immediately after FD1. But FD2 does not follow from FD1. To say that it follows from FD1 is to say that there’s a valid argument whose conclusion is FD2 and whose only premise is FD1. That, in turn, implies that you would be contradicting yourself if you accepted FD1 while at the same time denying FD2. But notice that this isn’t at all contradictory. You can agree that you cease to be conscious when you die (FD1), and yet reject FD2 on the grounds that you don’t have to be consciously aware of bad things in order for them to be bad for you. What is true is that FD3 follows from FD1 and FD2. But FD2 doesn’t itself follow from FD1.
A.2 How to Check for Validity
Many of the arguments in this book have conclusions you won’t like. If the arguments were invalid then, as with the Feathers Argument, you could just reject the conclusion without having to find a premise to reject. But since the arguments are all valid—I made sure of it!—rejecting the conclusion of any one of them always requires finding some premise to deny.
But what did I do to ensure that the arguments were all valid? How can you tell if an argument is valid? One way is to eyeball it: look at the premises, and check whether it seems like the conclusion follows from them. But we can do better than that. We can identify certain recurring forms or patterns whose presence guarantees that an argument is valid, regardless of what the argument is about. Accordingly, another way to check for validity is to see if the argument has one of these forms. If it does, then it’s valid. I’ll give four examples.
A.2.1 Modus Ponens
To see what I have in mind by a “form” of argument, compare these two arguments:
In some ways, the arguments are pretty different: one is about Kristina and drinking, the other is about God and morality. But there’s also something they have in common, something structural.
To see what they have in common, let’s recall some vocabulary that we learned in Section 3 of the Introduction. Claims of the form ‘if… then…’, like DK2 and MA2, are called conditionals. The bit that comes between the ‘if’ and the ‘then’ is the antecedent of the conditional, and the bit that comes after the ‘then’ is the consequent of the conditional.
What the Drinking Age Argument and the Moral Argument have in common is that each contains one premise that’s a conditional, another premise that’s the same as the antecedent of that conditional, and a conclusion that’s the same as the consequent of that conditional. In other words, they both have the following form:
Arguments with this form are called modus ponens arguments. (‘Modus ponens’ is Latin for method of affirming: you reach the conclusion by taking a conditional premise and combining it with a premise that affirms its antecedent.) Every modus ponens argument is a valid argument.
Here are two things to note about modus ponens arguments. First, it doesn’t matter whether the conditional premise comes first or second. For example, this is also a modus ponens argument:
That said, you do have to “mind your Ps and Qs” and how they’re distributed in the argument. This, for instance, is not a modus ponens argument:
This one doesn’t have the form “P, if P then Q, so Q” but rather “P, if Q then P, so Q.” This other argument form is called ‘affirming the consequent’, and is clearly invalid. Think about it. You can consistently accept MD1 and MD2 while denying MD3, for instance if you thought Jean Blanc was 18 years old. (You’d still accept MD2, that if he’s twenty, he’s still not allowed to buy alcohol.) By contrast, you can’t consistently accept RD1 and RD2 while denying RD3. That’s because the argument for RD3 is valid, whereas the argument for MD3 is invalid.
A.2.2 Modus Tollens
Another form that guarantees the validity of an argument is what’s called modus tollens, Latin for method of denying. A modus tollens argument is an argument with one premise that’s a conditional, another premise that’s a denial of the consequent of that conditional, and whose conclusion is the denial of the conditional’s antecedent. Using the ‘~’ symbol to symbolize denial, we can display the form of modus tollens arguments as follows:
Here are some examples of modus tollens arguments:
Again, the arguments are about entirely different topics but share a common structure. Also, as with modus ponens arguments, the order of the premises doesn’t matter: it would still be a modus tollens argument if WF2 came first and WF1 came second. But the order within the premises does matter. You’ve got to have the denial of the conditional’s consequent as a premise and a denial of its antecedent as the conclusion, not vice versa.
One other thing to notice here is that the same basic line of thought can be presented either as a modus ponens or as a modus tollens argument. The Moral Argument (from section 2.1) and the Flipped Moral Argument (just above) are really just two ways of packaging one and the same idea: that God must exist because objective morality presupposes the existence of God.
With these two types of valid arguments in hand, one can also construct more complicated arguments that involve both. For instance:
This argument combines a modus ponens argument and a modus tollens argument. The subconclusion FK3 follows, by modus ponens, from FK1 and FK2. And the conclusion FK5 follows, by modus tollens, from FK3 and FK4. Looking back at the Against Fearing Death argument in section 1, you can see that that argument combines two instances of modus ponens: a modus ponens argument from FD1 and FD2 to FD3, and another modus ponens argument from FD3 and FD4 to FD5.
A.2.3 Chained Conditionals
Here is a third type of valid argument, which I’ll call a chained conditional, since the conclusion chains together the antecedent of one conditional premise with the consequent of another conditional premise.
This form of argument is especially useful when you want to argue for a conditional claim, that is, when you want to give an argument that has a whole conditional as its conclusion.
Here are two examples of arguments with this form:
A.2.4 Universal Instantiation
I’ll mention one more form that a valid argument can have. This one is called universal instantiation, since it involves a “universal” premise claiming that everything belonging to one category also belongs to some second category. Together with an additional premise that one or more particular things belong to the first category, what follows is that those particular things also belong to the second category. Here it is schematically:
To get a valid argument of this form, you plug in some category for ‘F’, some second category for ‘G’, and a person or object for ‘o’. (This makes it unlike the previous three types of valid arguments, where you plug in whole sentences for the variables ‘P’, ‘Q’, and ‘R’.)
Here’s an example of an argument by universal instantiation:
The argument is valid, and what makes the argument valid is not the truth or the plausibility of the premises, but rather that the conclusion is a logical consequence of the premises. If you affirm the premises and yet deny the conclusion, you’ve contradicted yourself.
Universal instantiation arguments don’t always wear their form right on their sleeve. Take the Cuteness Argument:
TODO: repeated argument
Superficially, this doesn’t match the form of a universal instantiation argument, specified above. But with just a bit of rewording and rearranging, we can see that it’s a universal instantiation in disguise:
A.3 Challenging Modus Ponens and Modus Tollens
We have now seen four types of valid arguments: modus ponens arguments, modus tollens arguments, chained conditionals, and universal instantiations. These are not the only types of valid argument, and there’s some controversy (in the philosophy of logic) about what would go on a complete list of valid forms of argument. But when you’re constructing arguments of your own, so long as they have one of these four forms—or combine together arguments of these forms in the way suggested in section 2.2—you can be confident that your own argument is valid.
That said, because I apparently cannot go ten pages without arguing for some outrageous conclusion, I’m now going to argue—contrary to what virtually every philosopher and logician will tell you—that modus ponens and modus tollens arguments are not always valid.
Let’s start with modus tollens. Consider the following case:
Now, consider the following argument, which looks to be a counterexample to the thesis that all modus tollens arguments are valid:
This does appear to be a modus tollens argument: the first premise is a conditional, the second is a denial of its consequent, and the conclusion is a denial of its antecedent. Moreover, the premises are both true. DT1 is true because Chicago is in Illinois, so Olivia can’t very well be in Chicago without being in Illinois. DT2 is true too. If someone were to say “she must be in Illinois,” I could rightly respond: no, she might still be in New York. So DT2 rightly denies that she must be in Illinois.
But surely the argument is not valid. If it were, then DT3 would follow from those premises, and I would be able to use this argument to figure out where she is: she isn’t in Chicago, so she must be in New York. Clearly, though, I can’t know that Olivia is not in Chicago by using this argument. So, the argument must not be valid. In other words, this looks to be a counterexample to the claim that all modus tollens arguments are valid.
Now for modus ponens. Consider the following case:
Now consider the following argument, which looks to be a counterexample to the thesis that all modus ponens arguments are valid.
This is a modus ponens argument. One premise is a conditional (albeit one that has a whole conditional as its consequent); another premise affirms the antecedent of that conditional; and the conclusion is the consequent of the first conditional. Moreover, the premises are both true. Esmée is clearly going to win, and she is a woman. So DP1 is true. DP2 is true as well. If a woman wins and it isn’t Esmée then it has to be Celeste, since she is the only other woman still in the running. But DP3 is false: if Esmée loses, then it’s Grant who’s going to win. Celeste’s performance was a disaster, so if Esmée lost, it would certainly be because a majority of the judges voted for Grant, not because they voted for Celeste.
If the argument were valid, then the truth of the premises would guarantee the truth of the conclusion. But since the premises are true and the conclusion is false, the premises clearly don’t guarantee the truth of the conclusion. So, the argument isn’t valid. Thus, not all modus ponens arguments are valid.
I’ll leave it to you to figure out what (if anything) goes wrong in these arguments against the validity of modus ponens and modus tollens.
Sources and Resources
The argument against modus tollens is drawn from Kolodny and MacFarlane (2010). The argument against modus ponens is drawn from McGee (1985). For more on the philosophy of logic, see Haack (2007) or Sainsbury (2001).