10.4 Applications of Conditional and Indirect Proof
This module discusses how to use conditional proof and indirect proof to model arguments, and some proof strategies.
Table of Contents
10.4.1 Modeling Arguments as Conditional Proofs
An argument is like an arrow, with the premises pointing the way towards the conclusion.
Reconstructing a Conditional Argument
We have now finished our study of the rules of inference in formal logic. It is time to turn to the process of “extracting” or “reconstructing” valid arguments using these rules. There are a few simple strategies that tend to be effective when reconstructing arguments. One of them is to use conditional proof.
Sometimes what people are arguing for, what they want to conclude, is conditional. “If we increase border security, then drug use will decrease,” they might say. Or, “If we legalize drugs, then crime will decrease.” Or, “If we raise the minimum wage, then there will not be as many jobs.
Representing someone’s argument using formal logic helps ensure that the argument is valid, so that we know what premises have to be defended in order to show the conclusion is true. When the conclusion of someone’s argument is a conditional, then conditional proof is the best way to model the argument.
Begin by identifying the conclusion and symbolizing it as a conditional. For instance, if someone wants to show, “If we raise the minimum wage, then there will not be as many jobs,” we can symbolize this as:
C. R => ~J
Where:
R = we raise the minimum wage
J = there will be as many jobs.
We know our proof should look something like this, where we don’t know the line numbers, marked ‘?’:
? (basic premise)
? (basic premise)
…
? | R (assumption)
… | …
? | ~J
C. R => ~J (?-? CP)
The next step is to figure out how R and ~J are supposed to be connected. What basic premises are going to allow us to link up R and J within the proof? Here, we have to think about the person’s reasoning. Perhaps they think that, if the minimum wage goes up, then it will be more expensive to hire people, and there will be as many jobs only if it is not more expensive to hire people. We write these as basic premises. Let:
E = it will be more expensive to hire people.
The argument begins:
1. R => E (basic)
2. J => ~E (basic)
3. | R (assumption)
… | …
? | ~J
C. R => ~J (3-? CP)
The last step is to link up R and ~J with what we have in the basic premises. Here is how it works:
1. R => E (basic)
2. J => ~E (basic)
3. | R (assumption)
4. | E (1, 3 MP)
5. | ~~E (4 DN)
6. | ~J (2, 5 MT)
C. R => ~J (3-6 CP)
Now, take a shot at practicing reconstructing an argument using conditional proof.
10.4.2 Modeling Arguments as Indirect Proofs
One way to object is that an opponent is being inconsistent.
Reconstructing a Proof by Contradiction
Indirect proof, the rules of ~E and ~I, provides another way of easily modeling somebody’s reasoning in order to make it valid. Many arguments can be reconstructed as indirect proofs.
The first step is, again, to identify the conclusion. Unlike with conditional proof, the conclusion of an argument by indirect proof can have any structure. For instance, suppose that somebody wants to argue for the conclusion that Child Labor should be legalized.
Let:
L = Child labor should be legalized.
We already know the structure of the argument. It is surely not a theorem of logic that child labor will be legalized, so there will be some basic premises which the person making this argument will have in mind and will want to defend. Then, we’ll assume ~L, and then we’ll show some contradiction with the basic premises, which allows us to conclude that L by ~E:
? (basic premise)
? (basic premise)
? | ~L (assumption)
… | …
? | (contradiction)
C. L (?-? ~E)
Suppose the person who is arguing for the legalization of child labor has the following premises. First, we should do whatever maximizes economic growth. Second, children should either be educated in public schools or child labor should be legalized. Third, public schools cost money:
E = We should do whatever maximizes economic growth.
P = children should be educated in public schools
M = public schools cost money.
So, we have these premises:
1. E (basic)
2. P v L (basic)
3. M (basic)
These premises are not yet going to be enough for our argument, however. They don’t in themselves generate a contradiction. The argument the person is giving for child labor is not yet valid; they need to add some other premise or premises. For instance, perhaps they need to add that, if children should be educated in public schools and public schools cost money, then we should not do whatever maximizes economic growth:
1. E (basic)
2. P v L (basic)
3. M (basic)
4. (P & M) => ~E (basic)
Now we can put together a proof by contradiction:
1. E (basic)
2. P v L (basic)
3. M (basic)
4. (P & M) => ~E (basic)
5. | ~L (assumption)
6. | P (2, 5 DS)
7. | P & M (3, 6 &I)
8. | ~E (4, 7 MP)
9. | E (1 Reit.)
10. L (5-9 ~E)
Of course, the argument rests on four premises, and premise 4 is going to be difficult to defend. But by representing the argument in a valid way, using a valid form and structure and valid rules of inference, we’ve helped reveal exactly what those premises are, and thus what needs to be defended.
10.4.3 Three Proof Strategies
Think ahead to what your next move needs to be.
Completing Proofs
Completing a proof requires, given a set of basic premises and a goal for the conclusion, figuring out how, step by step, to reach the conclusion from those basic premises. The rules which we have studied in this module provide us with three strategies for completing proofs.
The CP Strategy
The CP strategy works best when the conclusion is a conditional. Suppose you are asked to complete this proof:
1. V / C. ~(G v H) => V
At first this might seem daunting. Where do G and H come from? But instead of trying to figure it out in advance, work with what you know. You know the conclusion is a conditional. So, use CP and start by assuming the antecedent of that conditional:
2. | ~(G v H) assumption
We now know we need to get V somewhere under the assumption line. The proof turns out to be quite easy, because we already have V on line 1, so we can simply reiterate:
3. | V (1 Reit.)
Now, we use CP:
4. ~(G v H) => V (2-3 CP)
The IP Strategy
The indirect proof strategy works in all sorts of cases. Simply assume the negation of whatever you are tasked with proving, and often the rest will fall into place. Suppose you are asked to complete this proof:
1. ~(H => F) / C. ~F v W
Again, don’t try to figure it all out in your head. Instead, begin with the simple strategy of assuming the negation of what you want to prove:
2. | ~(~F v W) (assumption)
Use what you know to simplify lines 1 and 2:
3. | ~~F & ~W (2 DEM)
4. | H & ~F (1 NEGCON)
Now, use &E and DN to take pieces apart
5. | ~~F (3 &E)
6. | F (5 DN)
7. | ~F (4 &E)
And there we have a contradiction!
8. ~F v W (2-7 ~E)
There are, of course, other ways to do this proof without ~E. We might have used just NEGCON, &E, and vI (try it on your own if you like). But often hunting for a contradiction is much easier than trying to figure out a clear path to the conclusion. So, while ~E and ~I often make proofs longer, they also make it cognitively easier for us humans to think through them.
The vE Strategy
Lastly, the vE strategy works best when one of the premises is a disjunction. Suppose that you are asked to complete this proof:
1. B => E (Basic)
2. A v B (Basic)
3. ~(A & ~F) (Basic) / C. E v F
Don’t try to figure out each step of the proof from the start. Instead, step by step, work through the proof going off of what information you currently have. Knowing that premise 2 is a disjunction, one strategy will be to assume the left disjunct, proof what you want to prove, and then assume the right disjunct to do the same thing.
4. | A (assumption)
We can then look at what other information we have in the premises relevant to A. Premise 3 for instance. We know that premise 3 can be modified using DeMorgan’s law:
5. | ~A v ~~F (3 DEM)
We now see how double negation and disjunctive syllogism could get us F, which is part of the conclusion:
6. | ~~A (4 DN)
7. | ~~F (5, 6 DS)
8. | F (7 DN)
The remainder of the conclusion comes by vI:
9. | E v F (8 vI)
10. A => (E v F)
We’re halfway there! We still need to prove the other half, from the disjunct B.
11. | B Assumption
12. | E (1, 11 MP)
13. | E v F (12, vI)
14. B => (E v F)
Now, having shown that A and B both lead to E v F, we can conclude:
15. E v F (2, 10, 14, vE)
Concluding Advice
In the author’s experience, at least one of these three strategies will work 90% of the time, even when other approaches fail. The recurring theme, which you’ve probably noticed, is not to let yourself be overwhelmed by proof or to try to solve it in advance in your head, which will most likely make the puzzle too difficult, but instead to walk through, step by step, following these strategies and applying what you know, while the proof gradually completes itself.
Submodule 10.4 Quiz
Licenses and Attributions
Key Sources:
- Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).
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