9.4 Argumentative Strategies

This module offers you some strategies when trying to figure out how to move forward with a proof when it isn’t obvious how the conclusion follows.

First, it discusses some strategies within propositional logic for each of the three two-place main connectives: disjunction, conjunction, and conditionals. Second, it discusses some strategies within categorical logic for creating a valid categorical syllogism when the sentences available do not fit into the rules required by standard form. Lastly, it presents the idea of “extracting an argument”, which is taking an informal argument and reconstructing it as a formal argument.

Table of Contents

• 9.4 Argumentative Strategies
  • 9.4.1 Choosing Propositional Rules
    • Making the Premises Useful
  • 9.4.2 Choosing Categorical Rules
    • Getting an Argument into Standard Form
    • Obversion and Contraposition
    • Quantifier Negation
  • 9.4.3 Preview of Extracting Arguments
    • Extractions
  • Submodule 9.4 Quiz

9.4.1 Choosing Propositional Rules

Pick the tools which are most likely to be useful.

Making the Premises Useful

One valuable strategy when you are completing a proof is to think first about what information is given in the premises, and then think about how you might use that information to get to the conclusion.

For instance, if you have a conjunction in the premises (P & Q) that isn’t in the conclusion, then using &E to get each of P and Q on a separate line is a good first step towards getting to the conclusion. On their own P and Q are more likely to be useful:

1. P & Q
2. P (1 &E)
3. Q (1 &E)

On the other hand, if you have a conjunction in the conclusion (P & Q) that isn’t in the premises, then you will likely want to first get P, and then get Q, and then use &I to put the two of them together:

1. P
2. Q
3. P & Q (1, 2 &I)

If you have a disjunction in the premises (P v Q) that isn’t in the conclusion, then you will likely be using DS (disjunctive syllogism), finding either ~P or ~Q on a separate line, and using that to derive either Q or P respectively. If you have a disjunction in the premises that isn’t in the conclusion, look for how you might get the negation of one of the disjuncts:

1. P v Q
2. ~Q
3. P (1, 2 DS)

If you have a disjunction in the conclusion (P v Q) that isn’t in the premises, then you will need to use one of the rules that allows you to create a disjunction. Either you will be using IMPL to get P v Q from (~P => Q):

1. ~P => Q
2. ~~P v Q (1 IMPL)

3 P v Q (2 DN)

… or else you will be using vI by getting either P by itself or Q by itself on a single line, and then using that to generate P v Q:

1. Q
2. P v Q (1 vI)

If you have a conditional in the premises (P => Q) which is not in the conclusion, then you will likely need to get either P on a line by itself, so that you can use MP to get Q:

1. P => Q
2. P
3. Q (1, 2 MP)

… or else you will need to get ~Q on a line by itself, so that you can use MT to get ~P:

1. P => Q
2. ~Q
3. ~P (1, 2 MT)

The other propositional rules we have learned, such as DN, DEM, NEGCON, and DISTRIB are most useful when you are stuck at a dead end in a proof and not sure how to move on.

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9.4.2 Choosing Categorical Rules

Some arguments must be trimmed up a bit to get them into Standard Form.

Getting an Argument into Standard Form

When you can tell that the conclusion of an argument should be something you could derive from a Categorical Syllogism, but the premises are not currently in the form of a Categorical Syllogism, your goal should be to use the rules that apply to Universals and Existentials, and to rephrase the argument, in order to get the argument into the form of a Categorical Syllogism. Remember that this form requires, for a conclusion of the form S are P:

• A Major premise of the form M are P, or P are M, which is an A, E, I or O sentence 
• A Minor premise of the form M are S, or S are P, which is an A, E, I, or O sentence 
• A mood and figure that falls into one of the fifteen valid argument forms. 

Let’s look at some examples of how to use other rules to get an argument into standard form.

Obversion and Contraposition

Recall the rules of obversion and contraposition for categoricals:

Sentence Form	OBV	CTRA-CG
All S are P (A)	No S are not P	All non-P are non-S
No S are P (E)	All S are non-P	INVALID
Some S are P (I)	Some S are not non-P	INVALID
Some S are not P (O)	Some S are non-P	Some non-P are not non-S

If you have a sentence which is not in the A, E, I or O form, you can use Obversion or Contraposition to get it into an A, E, I or O form. For instance, suppose we have this argument:

1. No cats are not animals. 

2. Some non-pets are not non-cats. 

C. Some animals are not pets.

The premises aren’t currently in standard form for categorical syllogism. We can use these rules to put the argument into standard form, however:

1. No cats are not animals. 

2. Some non-pets are not non-cats. 

3. Some cats are not pets (2, CONTRA-CG) 

4. All cats are animals (1, OBV) 

C. Some animals are not pets (3, 4 CG) 

Lines 3-C now give us a valid syllogism of the form OAO-3

Quantifier Negation

Another rule we can use to put an argument into standard form is Quantifier Negation: 

Not All S are P ≡ Some S are not P 

In other words, the negation of an A sentence is equivalent to an O sentence: 

Not all babies are cute ≡ some babies are not cute 

For example, suppose we have this argument: 

1. All non-silent people are non-dead people. 

2. Jessica is a philosopher. 

3. Jessica is a talkative person. 

4. All talkative people are non-silent people. 

C. Not all philosophers are dead people. 

We can now use our other rules to get the premises into standard form.  

1. All non-silent people are non-dead people. 

2. Jessica is a philosopher. 

3. Jessica is a talkative person. 

4. All talkative people are non-silent people. 

5. Jessica is a non-silent person. (3, 4 UI) 

6. Jessica is a philosopher and Jessica is a non-silent person (2, 5 &I) 

7. Some philosophers are non-silent people (6 EG) 

8. Some philosophers are not silent people (7 OBV) 

9. All dead people are silent people (1 CONTRA-CG) 

10. Some philosophers are not silent people (8 Reit.) 

This allows us to conclude through an AOO-2 syllogism: 

11. Some philosophers are not dead people (9, 10 CG) 

And now we can use Quantifier Negation rule to get to the conclusion: 

12. Not all philosophers are dead people (11 QN)

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9.4.3 Preview of Extracting Arguments

A formal argument can be extracted from an informal argument.

Extractions

Extracting an argument means taking an informal argument expressed in a paragraph, a page, a speech, a discussion, or simply in your own thoughts, and making it into a valid formal argument that follows rules of inference. Having the ability to extract an argument for yourself is one of the end goals of the course, and it requires you to apply all of the skills which you have studied so far.

For example, suppose that you want to write an letter to persuade someone that the city needs more bike paths, or that you are the best qualified candidate for a job. Extracting your argument formally first, before attempting to write the argument informally, will be useful for ensuring that you are practicing sound reasoning. Even if your reasons “make sense” to you, extracting the argument formally before you begin writing the letter will ensure your reasoning is sound, and will help you see which premises you need to address potential objections to. For instance:

1. Anyone who has years of experience in the area is qualified for this position. 

2. I have years of experience in the area. 

C. I am qualified for this position (1, 2 AUI)

Alternatively, suppose that you’ve read some marketing materials trying to persuade you that you should do something, like that you should major in business, or that you should vote for a proposition to require more renewable energy. The argument you read was probably presented informally, without logical symbols or appeal to modus ponens, but it was an argument nonetheless. You can use the extraction method to try to represent the reasoning in the argument to determine which premises it depends upon, and whether the reasoning is sound, or how it can be challenged.

1. All things which would increase renewable energy are things we should do. 

2. Voting for this proposition would increase renewable energy. 

C. Voting for this proposition is a thing we should do. (1, 2 CS)

There a few more rules in logic which we need to study, but soon we will turn from studying the rules to applying them by learning how to extract arguments.

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Submodule 9.4 Quiz

Licenses and Attributions

Key Sources:

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

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