Chains of Inference

Kareem Khalifa

Department of Philosophy

Middlebury College

Overview

• Why this matters

• The general idea

• The basic process

• How this relates to paraphrasing

• Exercises

Why this matters

• Breaking up complex chains of arguments is the key to reading comprehension.

• Failure to break up complex chains of arguments is one of the largest hurdles to clear writing and thinking.

• This is the “ordinary language” way of priming you for proofs in formal language.

The general idea

• People process shorter arguments better than longer ones.– Typical “audience-friendly” arguments have no more

than four premises in them.• People process arguments involving basic

inferences better than those involving fancier inferences.– It’s easier to understand an argument involving

modus ponens than, e.g., ~p v ~~q, p v ~~p├ q• So deriving intermediate conclusions using basic

forms of inference helps to make your argument audience-friendly.

Basic Process

• First step: Identify the main claims. This also means identifying the premises, intermediate conclusions, and the ultimate conclusion.

• Second step: Fit parts of the argument into interconnected common argument patterns.

Example

1. Al stays mum or snitches, but not both.2. Bob stays mum or snitches, but not both.3. If Al stays mum and Bob snitches, then Al receives a 10

year sentence and Bob goes free.4. If Al and Bob both stay mum, then both receive a 1 year

sentence.5. If Al snitches and Bob stays mum, then Al goes free and

Bob receives a 10 year sentence.6. If Al and Bob both snitch, then they both receive a 5

year sentence.7. Al does not receive a 10 year sentence. Either Al snitches or Bob stays mum.

Applying your paraphrasing skillsArgument A

7. Al does not receive a 10 year sentence.

8. It’s not the case that Al receives a 10 year sentence and Bob goes free.

Argument B

3. If Al stays mum and Bob snitches, then Al receives a 10 year sentence and Bob goes free.

8.

9. It’s not the case that Al stays mum and Bob snitches.

Argument C

9.

10.Either Al does not stay mum or Bob does not snitch.

Argument D

1. Al stays mum or snitches, but not both.

2. Bob stays mum or snitches, but not both.

10.

11.Al snitches or Bob stays mum.

A useful diagram7

8 3

9

10 1 2

11

Argument A

Argument B

Argument C

Argument D

Analysis of Example1. Al stays mum or snitches, but not both.2. Bob stays mum or snitches, but not both.3. If Al stays mum and Bob snitches, then Al receives a 10 year

sentence and Bob goes free.4. If Al and Bob both stay mum, then both receive a 1 year sentence.5. If Al snitches and Bob stays mum, then Al goes free and Bob receives

a 10 year sentence.6. If Al and Bob both snitch, then they both receive a 5 year sentence.7. Al does not receive a 10 year sentence.8. By 7, it’s not the case that Al receives a 10 year sentence and

Bob goes free. (ARGUMENT A)9. By 3 and 8, it’s not the case that Al stays mum and Bob snitches.

(ARGUMENT B)10.By 9, either Al does not stay mum or Bob does not snitch.

(ARGUMENT C)11.By 1, 2, and 10, either Al snitched or Bob stayed mum.

(ARGUMENT D).

Nolt, Exercise 11. If the person exists after death, then the

person is not a living body.A

2. The person is not a dead body. A

3. Any body is either alive or dead. A

4. The person exists after death. A

5. The person is not a living body. 1, 4

The person is not a body. 2, 3, 5

Nolt, Exercise 21)x is an odd number A

2) x +y = 25 A

3) x > 3 A

4) 30/x is a whole number A

5) x <10 A

6) x = {5, 7, 9} 1, 3, 5

7) x = 5 4, 6

8) y = 20 2, 7

Nolt, Exercise 31)You will graduate this semester. A

2) In order to graduate this semester, you must fulfill the Humanities requirement this semester.

A

3) You fulfill humanities requirement when and only when you have passed 2 courses in literature and a single course in either philosophy or art; or 2 courses in philosophy and a single course in either literature or art.

A

4) You have taken and passed one art course but no philosophy or literature courses.

A

5) You have time for only 2 courses this semester. A

6) Among the philosophy courses, only one fits your schedule. A

7) You will fulfill the humanities requirement this semester 1, 2

8) You will take 2 courses in literature and a single course in either philosophy or art; or 2 courses in philosophy and a single course in either literature or art in order to graduate.

3, 7

9) You will take either 2 courses in literature or 2 in philosophy in order to graduate

4, 8

10) You will take 2 literature courses this semester. 5, 6, 9

In Class Exercise

“Research shows lengthy sentences do nothing to improve public safety. But these long sentences are turning prisons into geriatric centers where the cost of care is prohibitively high. The practice of routinely locking up people forever — especially young people — also ignores the potential for rehabilitation. The whole trend is deeply counterproductive. States need to encourage more rational sentencing, restore the use of executive clemency and bring parole back into the corrections process.” “Writing Off Lives” by THE NY TIMES EDITORIAL BOARD Published: September 29, 2013

First step: Identify the main claims

1. Lengthy sentences do not improve public safety.

2. Lengthy sentences are too costly.

3. Lengthy sentences ignore the potential for rehabilitation.

4. Lengthy sentences are counterproductive.

5. States need to shorten sentencing, restore executive clemency, and reintroduce parole into the corrections process.

Second step: Fit into common argument patterns

1. If lengthy sentences do not improve public safety, are too costly, and ignore the potential for rehabilitation, then they are counterproductive.

2. Lengthy sentences do not improve public safety, are too costly, and ignore the potential for rehabilitation.

3. Lengthy sentences are counterproductive.

4. If lengthy sentences are counterproductive, then states need to shorten sentencing, restore executive clemency, and reintroduce parole into the corrections process.

5. States need to shorten sentencing, restore executive clemency, and reintroduce parole into the corrections process.

An even leaner paraphrase?

1. If lengthy sentences do not improve public safety, are too costly, and ignore the potential for rehabilitation, then states need to shorten sentencing, restore executive clemency, and reintroduce parole into the corrections process.

2. Lengthy sentences do not improve public safety, are too costly, and ignore the potential for rehabilitation.

3. States need to shorten sentencing, restore executive clemency, and reintroduce parole into the corrections process.