As men at first made use of the instruments supplied by nature to accomplish very easy pieces of workmanship, laboriously and imperfectly, and then, when these were finished, wrought other things more difficult with less labour and greater perfection; and so gradually mounted from the simplest operations to the making of tools, and from the making of tools to the making of more complex tools, and fresh feats of workmanship, till they arrived at making, complicated mechanisms which they now possess. So, in like manner, the intellect, by its native strength, makes for itself intellectual instruments, whereby it acquires strength for performing other intellectual operations, and from these operations again fresh instruments, or the power of pushing its investigations further, and thus gradually proceeds till it reaches the summit of wisdom. Spinoza, Treatise on the Emendation of the Intellect
In this chapter, we will introduce a technique that makes proving things easier. The technique is roughly, putting together several different inferential moves into a single step, called a derived rule. By combing a sequence of moves into one, we can remove a lot of boilerplate from our proofs. In every place where we would repeat a certain common sequence of moves, we can instead using our new derived rule to complete the same task in just one inference. In addition to shortening our proofs, we save ourselves the mental effort of “rediscovering”, over and over, how to complete certain simple tasks in our proofs. We only need to figure it out once to create a derived rule, but once the derived rule is created, we can complete the task as many times as we want without any additional effort.
Here is the idea. Well-formed proofs establish the formal validity Remember, a formally valid argument is an argument that is valid because of its logical form. For more discussion of this idea, take another look back at chapter 1. of arguments like this one:
- P, P → Q ⊢ Q
But when we know an argument is formally valid, then we know that any argument that results from replacing the sentence letters of that argument with actual assertions will be valid as well.
Notice, however, that any argument we can get by replacing the sentence letters in an argument form like this one:
- R ∨ S, R ∨ S → Q ⊢ Q
we can also get by replacing the sentence letters in P, P → Q ⊢ Q—all we have to do is replace P in the first example above with the sentence that would have replaced R ∨ S with in the second example.
Here’s the crucial observation to make at this point. Since P, P → Q ⊢ Q is formally valid, any argument we get by replacing its sentence letters with actual assertions is deductively valid. Hence, any argument that we get by replacing the sentence letters of our R ∨ S, R ∨ S → Q ⊢ Q with actual assertions is also deductively valid (since we could have gotten that argument from P, P → Q ⊢ Q). Hence, R ∨ S, R ∨ S → Q ⊢ Q is also formally valid.
The general lesson here is that, whenever we have a deductively valid argument, every argument form that results from replacing the sentence letters of that argument with other sentences in our formal language will also be deductively valid. So, for example, because P, P → Q ⊢ Q is formally valid, so is every instance of the scheme ϕ, ϕ → ψ ⊢ ψ.
Of course, you already knew that. The scheme ϕ, ϕ → ψ ⊢ ψ is just modus ponens. We use it all the time. What’s new is that, it turns out, whenever you prove that an argument is formally valid, you learn that a new rule is correct. Just like modus ponens expresses that, given three sentences which have the same shape as P → Q, P and Q, you can infer the third from the first two, when you prove an argument is formally valid, you learn that you can use whatever rule expresses that you are allowed to infer, from sentences that have the same shape as the premises of your deductively valid argument, conclusions that have the same shape as the conclusion of your deductively valid argument.
In the next few sections, we’ll look at several new inference rules that can be introduced in this way. In order to use these rules, you’ll need to prove the corresponding argument using the rule builder in your index of rules. Problem set eight is distributed through the different sections of this chapter, in order to help you see which problems might be easier to solve with new rules.
Let us begin with a very simple derived inference rule. The rule is traditionally called hypothetical syllogism—syllogism, because it is a very argument with two premises, and hypothetical because it involves conditionals. It looks like this:
- Hypothetical Syllogism
Hypothetical syllogism is the argument form
ϕ → ψ, ψ → χ ⊢ ϕ → χ
It’s up to you what you want to name the inference rule corresponding to hypothetical syllogism, but I suggest the name HS. If this is what you call it, then you will cite this rule in proofs by writing :D-HS, followed by the line numbers of the premises you wish to use.
The intuitive idea behind a hypothetical syllogism is simple. If we know that one thing leads to a second, and the second to a third, then we know that the first thing leads to the third. For example, If Sam is a dog or a cat, then Sam is a mammal, and if Sam is a mammal, then Sam is a vertebrate. Therefore, if Sam is a dog or a cat, then Sam is a vertebrate.
Here are some problems to work on that are greatly simplified by use of the hypothetical syllogism.
The Material Conditional Rules
The next set of derived rules make explicit something that we’ve already observed. The rules are these:
- The Material Conditional Rules
The material conditional rules are the following two argument forms
- ψ ⊢ ϕ → ψ
- ¬ϕ ⊢ ϕ → ψ
Each of these rules tells us something important about how the if-then connective → works.
Let’s focus on the first of these. What it says is that, when you know that the right hand side of a conditional is true, then you know that the whole conditional is true. This strikes some people as strange. After all, can’t I know that the door is shut without knowing that if the house were burning down, then the door would be shut? It seems like, if that were to happen, the door might well pop open.
The key thing to notice here is that there is a difference between a sentence like
- “If the house were burning down, then the door would be shut.”
- “If the house is burning down then the door is shut.”
The difference is that in sentence (4), “the house is burning down” and “the door is shut” are both complete assertions that can stand on their own. In sentence (3), we have an if-then connecting two strings of words—namely “the house were burning down” and “the door would be shut”—neither of which is a complete sentence on its own.
This signals the difference between the two sentences. Sentence (4) involves what is traditionally called a material conditional—an if-then connective that expresses what we known about the actual truth or falsity of the sentences that it connects. Sentence (3) expresses what is traditionally called a subjective, or counterfactual conditional, which is a conditional that expresses what we know about how the two conditions it connects are related to one another in merely possible (counterfactual, or contrary to fact) situations.
In introductory logic courses, we always use the material conditional. Logic has plenty to say about counterfactual conditionals, and about other kinds of conditionals as well. But the material conditional is the simplest kind of conditional. So, we save the study of those other conditionals for later, and build the material conditional into our inference rules.
The kinds of cases where we use material conditionals usually involve what we know about what is actually true. For example, if you are unsure of the exact place you left your keys, you might still believe “if my keys are not in the car, then they’re on the bedstand”. If your keys turn out to be on the bedstand, then your belief was correct. It’s only if they turn out to not be in the car and not be on the bedstand that you were wrong. Here is another place you use material conditionals, perhaps without realizing it. It turns out that hypothetical syllogism requires you to use a material conditional. It is not a good argument form when you are dealing with subjunctive conditionals. For example, the argument “If I was dolphin, I would be able to swim at 20 km/h., and if I could swim at 20 km/h, I would be an Olympic champion, therefore, if I was a dolphin, I would be an Olympic champion.” is not valid. Dolphins are not eligible to compete in the Olympics.
Here are some arguments that are greatly simplified by using the material conditional rules.
There are several ways of rearranging an if-then statement. Suppose you know that P → Q is true. Then which of the following must also be true?
- Q → P (this is sometimes called the converse of P → Q)
- ¬P → ¬Q (this is sometimes called the inverse of P → Q)
- ¬Q → ¬P (this is sometimes called the contrapositive of P → Q)
Well, if P is “Sam is a cat” and Q is “Sam is a mammal”, then P → Q must true. But Q → P (if Sam is a mammal, Sam is a cat) need not be true, because Sam could be, for example, a human. And ¬P → ¬Q (if Sam is not a cat, then Sam is not a mammal) need not be true either. Even if Sam is not a cat, Sam could be some other kind of mammal. However, ¬Q → ¬P doesn’t seem to have any obvious counterexamples. If Sam is not a mammal, Sam certainly is not a cat.
It turns out that a given conditional ϕ → ψ and its contrapositive ¬ψ → ¬ϕ are logically equivalent, just like ϕ is logically equivalent to ¬¬ϕ. You can infer the first statement from the second, and the second from the first. So we have the following two rules
- The Contrapositive Rules
The contrapositive rules are the following two argument forms
- ϕ → ψ ⊢ ¬ψ → ¬ϕ
- ¬ϕ → ¬ψ ⊢ ψ → ϕ
Here are a few problems in which the contrapositive rules are useful
You’re in a dilemma when you’re damned if you do, and damned if you don’t. You’re in a logical dilemma when both assuming some statement is true and assuming that the statement is false leads to the same conclusion. In such a situation, even if you don’t know whether the statement is true, you can still safely infer that the conclusion (the one that you’re led to in both cases) is true.
The Dilemma rule is the following
ϕ → ψ, ¬ϕ → ψ ⊢ ψ.
Here’s an example of an argument where constructing a logical dilemma can be useful (you may find hypothetical syllogism and contraposition helpful here too):
Suppose that a statement is “self-contradictory”—it implies its own denial. No reasonable person could believe such a statement, since believing it would mean either failing to accept its consequences (which would be unreasonable), or believing both the statement and its denial (which would be very unreasonable).
Now, imagine a statement which is “undeniable”, for something like the same reason. In order to deny the statement, you must accept it. For example, some people have observed that in order to deny that epistemology (the study of what we can and can’t know) will ever reach any conclusions, you need to accept that it is possible to know something about what we can and can’t know (specifically, you must think that we can’t know anything about what we can know).
In a case like this, what you set out to deny is presupposed by your denial, and that makes it the denial of this “undeniable” statement just as unreasonable as the assertion of a self-contradictory statement.
This logical observation is quite old. The rule is called consequentia mirabilis (meaning “admirable consequence”), or sometimes “Clavius’s law”, after Christopher Clavius. Christopher Clavius is a renaissance mathematician who did a great deal to reinvigorate the study of geometry and mathematics after the end of the middle ages. Isaac Newton and Gottfried Leibniz are both believed to have studied from his textbooks.
It can be written formally as follows:
- Consequentia Mirabilis
The rule of consequentia mirabilis is the following:
¬ϕ → ϕ ⊢ ϕ.
Consequentia Mirabilis is easy to prove using a logical dilemma: P is true if we assume P is true, obviously, so if P is also true when we assume that ¬P is true (which is given, by the premise that ¬P → P), then P is true no matter what.
Here’s a problem for which consequentia mirabilis (and some of our other rules above) might come in handy:
Ex Falso Quodlibet
The Arabic philosopher Avicenna, in a moment of great frustration with someone who kept contradicting themselves, quipped that
Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.
This is perhaps a bit much. But to violate the law of non-contradiction by both asserting and denying the same thing carries a logical penalty that is almost as bad as being burned and beaten. Actually, maybe it’s worse. Plenty of very admirable people have preferred to be burned or beaten, rather than denying the things that they most deeply believe, or assenting to things that they abhor. Poetically, a person who wishes to assert too much ends up logically committed to asserting everything.
In particular, we have the following logical rule:
- Ex Falso Quodlibet
The rule of ex falso quodlibet is the following:
ϕ, ¬ϕ ⊢ ψ
The name “ex falso quodlibet” means roughly, “from what’s ridiculously false, you can infer whatever conclusion you want”.
Here’s a problem where ex falso quodlibet might come in handy:
Here’s a final interesting statement to try to prove by combining the rules you’ve learned.
If you get stuck, here’s a hint: Try to construct a dilemma, by showing that P → ((P → Q)→P)→P) using the material conditional rules, and that ¬P → ((P → Q)→P)→P) by using the material conditional rules and our ordinary strategies for dealing with conditionals.