11.2 Tricks and Shortcuts

There are a number of tricks or shortcuts to repairing arguments in order to make them valid. These tricks involve employing many of the rules of inference.

Table of Contents

• 11.2  Tricks and Shortcuts
  • 11.2.1 Disjunctive Strategies
    • The Process of Elimination (DS)
    • Linking Disjunction (DS)
    • Same Result Either Way (vE)
  • 11.2.2 Conditional Strategies
    • The Linking Conditional (MP)
    • A Chain of Conditionals (HS)
  • 11.2.3 Categorical Strategies
    • Linking Universal (CS or UI)
  • Submodule 11.2 Quiz

11.2.1 Disjunctive Strategies

Limit the options with a disjunction, and then use a process of elimination.

The Process of Elimination (DS)

There are two simple strategies for making arguments valid that involve using the disjunctive rules we studied earlier. The first is a strategy you have probably heard of before, the “process of elimination”, where a number of options are laid out, and then each option is eliminated until only one is left standing. The rule which corresponds to this pattern of reasoning is Disjunctive Syllogism. For instance, consider the following argument: 

“Money doesn’t grow on trees. Either we raise taxes, or we cut spending, or else the national debt is going to grow each year. The Republicans won’t let us raise taxes, and the Democrats won’t let us cut spending. So, increasing the national debt is the only option.” 

We might symbolize the claims in this way. Notice how we have simplified the language: 

T = We raise taxes. 

S = We cut spending 

N = The national debt increases. 

Our first stab at symbolizing the argument might look like this: 

1. ~T 
2. ~S 
C. N 

Now, to make the argument valid, we need to add a premise that would allow us to use disjunctive syllogism: we put the conclusion at one end of the disjunction, and then add the negative premises to it: 

1. T v (S v N) 
2. ~T 
3. ~S 
C. N 

Now, we can make use of disjunctive syllogism to prove the argument is valid. 

1. T v (S v N) 
2. ~T 
3. ~S 
4. S v N (1, 2 DS) 
C. N (3, 4 DS)

Lastly, we translate back into English:

1. Either we raise taxes, or else we either cut spending or the national debt increases. (Basic)_ 
2. We don’t raise taxes. (Basic) 
3. We don’t cut spending. (Basic) 
4. Either we cut spending or the national debt increases (1, 2 DS) 
5. The national debt increases (3, 4 DS) 

Linking Disjunction (DS)

This procedure can be applied to any argument you want to repair, even if it doesn’t intuitively involve a process of elimination. At any point, if you have a negative premise (~P), and you want to figure out how to make the conclusion (C) follow from that premise, then add a premise with a linking disjunction, P v C:

1. ~P 
C. C 

This becomes: 

1. ~P 
2. P v C 
C. C (1, 2 DS) 

By adding an extra step with the rule of double negation (DN), the same procedure can apply where the premise is affirmative rather than negative, by adding a disjunction containing the negation of the premise and the conclusion. 

1. P 
C. C 

This becomes: 

1. P 
2. ~P v C 
3. ~~P (1 DN) 
C. C (2, 3 DS) 

Same Result Either Way (vE)

A different strategy involving disjunctions uses the rule of disjunction elimination. It corresponds to an argument that, while we might not know which of two possibilities might happen, the same thing will be true either way, so it really doesn’t matter which way it goes. For instance, suppose somebody argues that it doesn’t matter which party wins office, either way the national debt will increase. We can model that argument in this way: 

R = Republicans win 

D = Democrats win 

N = The national debt increases 

Here is a first stab at the argument: 

1. R v D 
C. N 

Now, to make it valid, we add two conditionals, both with the conclusion of the argument in the consequent: 

1. R v D 
2. R => N 
3. D => N 
C. N

Lastly, we use vE and MP to prove the argument valid: 

1. R v D 
2. R => N 
3. D => N 
4. | R (assume) 
5. | N (2, 4 MP) 
6. | D (assume) 
7. | N (3, 6 MP) 
8. N (1, 4-5, 6-7 vE) 

Translating back into English:

1. Either Republicans or Democrats win (Basic) 
2. If Republicans win, then the national debt increases. (Basic) 
3. If Democrats win, then the national debt increases. (Basic) 
4. | Republicans win. (Assume) 
5. | The national debt increases (3, 4 MP) 
6. | Democrats win (Assume) 
7. | The national debt increases (3, 6 MP) 
8. The national debt increases. (1, 4-5, 6-7 vE) 
   

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11.2.2 Conditional Strategies

A conditional can link up the premises with the conclusion.

The Linking Conditional (MP)

The simplest way to link up a premise with a conclusion is to create a conditional sentence, with the premise in the first part and the conclusion in the second part of the sentence: IF [premise], THEN [conclusion]. It is then possible to use modus ponens to derive the conclusion from the premise. We’ve already used this strategy before in the class, because it is so very easy to use, and it is probably the easiest way to make an argument valid when you are stuck. We take an argument like this:

1. P 

C. C 

And we turn it into this: 

1. P 

2. P => C 

C. C (1, 2 MP) 

For instance, suppose you want to extract an argument from this paragraph: 

Life is great. I can’t believe that people ever feel down about their lives. Seriously, what is wrong with them? Get lost, sad people, and stop bringing us down! I like my life, I have a great pet fish and I eat the best pork chops ever, every night. People, just think about how good you have it! Especially compared to dead people. Do you realize how very not happy dead people are? Dead people don’t get massages. Anyway, I also like the taste of orange soda, it’s the best. Hurray! I don’t know how anybody can deny that.

This paragraph might not make it easy to extract an argument, because most of the sentences in the paragraph are either not statements, or they are statements about the author’s own feelings. Nonetheless, it has a clear conclusion:

C. Life is great. 

We can also charitably summarize the author’s reasons for believing this suggested by the paragraph, and their thoughts about orange soda, massages, and pork chops, to create a first premise: 

1. There are many good things in life. (Basic)  

C. Life is great. 

Now, we can make the argument valid by simply adding a linking conditional: 

1. There are many good things in life. (Basic) 

2. If there are many good things in life, then life is great. (Basic) 

C. Life is great. (1, 2 MP) 

Now, the argument is valid; somebody who disagrees with the conclusion would have to reject either premise 1 or premise 2. 

A Chain of Conditionals (HS)

Quite often, an argument involves not just one conditional, and one use of modus ponens, but a long string of conditionals which are supposed to link together to reach the conclusion. We can use the rule of hypothetical syllogism to represent the reasoning in this arguments. For instance, suppose somebody makes this argument:

“Now that we’ve decided to hire Horace, I guess we’re going to have to put up with his eccentric footwear in the office. So, out of fairness, we’ll have to put up with Hypatia’s crazy political views too. Once that happens, Brad will start smoking in the office, and that means Brett will go back to wearing the Guy Fawkes mask every Friday. You see where this is leading: Gina is going to quit.”

The conclusion of this argument is clear: 

C. Gina is going to quit.”

The first premise isn’t hard to identify either: 

1. We are hiring Horace. (Basic) 

C. Gina is going to quit. 

The remainder of the paragraph gives a long string of conditions, each of which is supposed to follow from the others. We can rephrase these as conditionals in If… Then… format: 

1. We are hiring Horace. (Basic) 

2. If we are hiring Horace, then we put up with eccentric footwear. (Basic) 

3. If we put up with eccentric footwear, then we put up with Hypatia’s political views. (Basic) 

4. If we put up with Hypatia’s political views, then Brad starts smoking in the office. (Basic) 

5. If Brad starts smoking in the office, then Brett wears a Guy Fawkes mask every Friday. (Basic) 

6. If Brett wears a Guy Fawkes mask every Friday, then Gina is going to quit. (Basic) 

C. Gina is going to quit. 

Now, we can apply the rule of Hypothetical Syllogism, as well as modus ponens, to show the conclusion is valid. 

1. We are hiring Horace. (Basic) 

2. If we are hiring Horace, then we put up with eccentric footwear. (Basic) 
3. If we put up with eccentric footwear, then we put up with Hypatia’s political views. (Basic)
4. If we put up with Hypatia’s political views, then Brad starts smoking in the office. (Basic) 
5. If Brad starts smoking in the office, then Brett wears a Guy Fawkes mask every Friday. (Basic) 
6. If Brett wears a Guy Fawkes mask every Friday, then Gina is going to quit. (Basic) 
7. If we are hiring Horace, then we put up with Hypatia’s political views. (2, 3 HS) 
8. If we are hiring Horace, then Brad starts smoking in the office. (4, 7 HS)
9. If we are hiring Horace, then Brett wears a Guy Fawkes mask every Friday (5, 8 HS) 
10. If we are hiring Horace, then Gina is going to quit. (6, 9 HS) 
C. Gina is going to quit. (1, 10 MP) 

Failing to Meet Expectations (MT)

One last strategy for extracting an argument using a conditional involves applying the rule of modus tollens. One way to reason to a conclusion is to consider what one would expect if the conclusion were false, and to show that what one would expect hasn’t happened. For instance: 

“Darla isn’t really Dana’s friend. She never shares anything that Dana posts online.” 

The conclusion of this argument is: 

C. Darla is not Dana’s friend. 

We could represent the premise in this way: 

1. Darla does not share things that Dana posts online. (Basic) 

We could link up the premises by adding a conditional that says that, if Darla is Dana’s friend, then Darla shares things that Dana posts online. The rule of modus tollens would then make the argument valid: 

1. Darla does not share things that Dana posts online. (Basic) 

2. If Darla is Dana’s friend, then Darla shares things that Dana posts online. (Basic) 

C. Darla is not Dana’s friend. (1, 2 MT) 

This is a very easy, and also very applicable strategy. It is especially useful for modeling arguments made on the basis of a scientific study which fails to demonstrate some expected result. For instance, the following pair of statements form an argument. 

“There was no statistically significant change in flu symptoms among subjects who took Elderberry, so it does not cure the flu.” 

The argument could be represented in this way: 

1. If Elderberry cures the flu, then subjects taking Elderberry have change in flu symptoms. (Basic) 
2. Subjects taking Elderberry do not have change in flu symptoms. (Basic) 
C. Elderberry does not cure the flu.

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11.2.3 Categorical Strategies

A universal claim is true of everything.

Linking Universal (CS or UI)

A final strategy for making extracted arguments valid is to apply one of the fifteen valid forms of Categorical Syllogism, or the rule of Universal Instantiation. So long as both the first premise and the conclusion share at least one term in common, a second premise can be added which would create a valid categorical syllogism. Most commonly, the premise added is a universal rather than a particular claim, a “linking universal”. The easiest way to demonstrate this is through examples. For instance:

1. Some whales are intelligent. (Basic) 

C. Some whales can communicate. 

This has the form: 

1. Some S are M 

C. Some S are P 

So, we need to add a premise that says: 

2. All M are P 

Thus, we can make it valid in this way: 

1. Some whales are intelligent. (Basic) 

2. Everything which is intelligent can communicate. (Basic) 

C. Some whales can communicate. (1, 2 CS) 

The linking universal could also be a negative claim. For instance, suppose somebody is arguing that some whales do not have rights, because they are held in captivity: 

1. Some whales are captive animals. (Basic) 

C. Some whales do not have rights. 

Here, adding a universal negative can make the argument valid: 

1. Some whales are captive animals. (Basic) 

2. No captive animals have rights. (Basic) 

C. Some whales do not have rights. (1, 2 CS) 

The rule of universal instantiation can be used when the claim is about a particular thing, rather than about members of a category. For instance: 

1. Moby Dick is intelligent. (Basic) 

C. Moby Dick can communicate. 

This becomes: 

1. Moby Dick is intelligent. (Basic) 

2. Everything which is intelligent can communicate. (Basic) 

C. Moby Dick can communicate. (1, 2 UI) 

Taken together, the disjunctive, conditional, and categorical strategies should allow you to repair almost any argument in order to make it valid.

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Submodule 11.2 Quiz

Licenses and Attributions

Key Sources:

• Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).

Next Page: 11.3  Evaluating Arguments