The last type of derivation that we will consider is an indirect derivation. An indirect derivation is, generally speaking, a derivation used in order to show that something is not true. How do we ordinarily know that something is not true? One thing that we do is to reject statements that conflict with things that we already know to be true.
You get a call from the elementary school, saying that your little brother was absent. Later, you ask: where were you this afternoon? Your brother says: I was at school. You then know that what your bother says is false, since it contradicts what you know to be true.
As a materials engineer, you know for a fact that a certain type of building material cannot handle humid climates. A contractor tells you that the best material to use in your new house, to be built in a humid climate, is this particular building material. In this case, you know that the contractor is not saying something true, because what he says contradicts what you know for sure.
You happen to know that, to ski down Mount Terror, the suspect would have needed to buy some skiing gear. You also happen to know that if the suspect spent the day skiing down Mount Terror, then the suspect had no time to buy any skiing gear. You ask the suspect: “Where were you today?” They say “why, I was skiing down Mount Terror!”. Then you know the suspect is lying to you.
Let’s take a closer look at this last case. How do we know the suspect is lying? Well, suppose the suspect did go skiing down mount terror. Then given what we know, they’d have to have bought some skis. But given what we know, they also could not have bought some skis. So, under the assumption that the suspect did go skiing down mount terror, they both did and did not buy skis. And that is surely impossible.
Given what we know, if the suspect did go skiing, they both bought and did not buy skis. But that simply is not possible. So we can safely conclude on the basis of what we know that the suspect did not, in fact, go skiing down Mount Terror.
We can represent the abstract structure of this sort of reasoning as a (simple) Indirect Derivation
- Indirect Derivation
A (simple) indirect derivation is a sequence of assertions
Aimed at showing a statement φ is false, or equivalently that ∼φ is true,
beginning with an assumption that φ is true,
in which every assertion other than the first justified, either because it is a premise or because it is the conclusion of a rule of direct inference with previous lines as premises,
To use an indirect derivation to show something, we’ll need to do many of the same things that we needed to do to show something with a conditional derivation. First of all, we will need to keep track of what we are trying to show, so that others can know what we are intending to do. Second of all, we will need to keep track of what we are assuming, and what we are actually justifying on the basis of other things. Third of all, we will need to keep track of what we have already shown, since each new step must be based on previous steps or premises. Fourth, we will need to keep track of the justification for each step that we are making, so that we, and others, can easily verify the correctness of each step in our reasoning.
As before, we will write “Show” to indicate what we are showing, we will number our lines, and indicate where the premises to rules MP,MT, and DN are coming from by using line numbers.
For simple indirect derivations, we will aim only to show negative statements of the form ¬φ. Second, we will begin each derivation by assuming the statement φ. As usual, we will write :AS to justify our assumption. Third, we will consider ourselves finished when we manage to assert both some statement ψ and the negation ∼ψ of ψ. At this point, we will write the numbers of the line where we asserted these two statements, together with :ID, to mean “I have shown the negative statement by means of a indirect derivation ending with these lines.
The result will look like this:
1. Show ¬φ 2. φ :AS 3. ASSERTION :JUSTIFICATION 4. ASSERTION :JUSTIFICATION 5. ψ :JUSTIFICATION 6. ASSERTION :JUSTIFICATION 7. ASSERTION :JUSTIFICATION 8. ¬ψ :JUSTIFICATION 9. :ID 5 8
The particular numbers do not matter, of course, and neither does whether ψ or ¬ψ comes first in the derivation.
Here’s an example of how you might use an indirect derivation, to prove the validity of an argument like this one: P → ¬P ⊢ ¬P
1. Show: ~P 2. P :AS 3. P->~P :PR 4. ~P :MP 2 3 5. :ID 2 4
Here’s another example, this time showing how to prove P → Q, P → ¬Q ⊢ ¬P
1. Show: ~P 2. P :AS 3. P->Q :PR 4. Q :MP 2 3 5. P->~Q :PR 6. ~Q :MP 2 5 7. :ID 4 5
Notice something important about this example: the lines that you apply :ID to do not have to be the same as the lines that you assume to begin the indirect derivation. As long as your derivation reaches a contradiction, that’s enough for you to have shown the negation of the assumption that began your derivation—it doesn’t matter whether or not the assumption is actually part of the contradiction.
Nested Indirect Derivations
In the same way that conditional and direct derivations can be nested inside of one another, indirect derivations can be nested inside of other types of derivations, and can also have other types of derivations nested inside of them.
Here’s the official definition.
- Nested Indirect Derivation
A nested indirect derivation derivation is a sequence of assertions, aimed at showing ∼φ, beginning with the assumption that φ, each of which after the first is justified because either
it is a premise of the argument under consideration;
it is the conclusion of a rule of direct derivation applied to previous assertions which are available from the line it occupies; or
it is listed next to a completed show-line (a show line that has been paired with a QED line), meaning that this assertion has been shown to be true by a derivation.
Where by an available line here, we mean the same thing as in the previous chapter—a line is available at a certain point if it is not part of a subproof that has already been completed (because it is between an earlier show line and a QED line corresponding to that show line). Here are some simple examples of indirect derivations.
Here’s an example of how you might use an indirect derivation nested inside a conditional derivation, to prove the validity of an argument like this one: P → Q ⊢ (P → ¬Q)→¬P
1. Show: (P -> ~Q) -> ~P 2. P->~Q :AS 3. Show: ~P 4. P :AS 5. P->Q :PR 6. Q :MP 5 4 7. ~Q :MP 2 4 8. :ID 6 7 9. :CD 3
Problem Set 7
Abbreviations are the same as in previous chapters. When the argument turns a light green color, then you can press the “submit” button to submit your work.
Construct indirect derivations (no nesting will be necessary) to show the validity of the listed arguments.
Construct nested derivations, which will include some indirect derivations, to show the validity of this second set of arguments. If you’re not sure how to start, here’s a hint: look at the main connective of the conclusion you’re trying to show. Which of ID and CD would help show that conclusion?