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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 51

Chapter 5: Formal and Informal Logic

1. Logical FormExercise 5.1

2. Formal Logic3. Equivocation and Amphiboly

Exercise 5.2

4. The Paradox of the Liar

Exercise 5.3

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 52

Logical Form Repetition

Bill has $5 in his pocket

Therefore, Bill has $5 in his pocket

Sue has visited California

Therefore, Sue has visited California

(P1) p

(C) p

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 53

Logical Form Disjunctive Syllogism

Bill is in New York or Bill is in LondonIt is not the case that Bill is in New York

Therefore, Bill is in London

Sue went to the movies or Sue left townIt is not the case that Sue went to the movies

Therefore, Sue left town

(P1) p q (P2) p (C) q

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 54

Grammatical versus Logical Form

The grammatical form of a proposition (or of an argument)

is the structure of the proposition (or argument) as indicated bythe surface grammar of its natural language

The logical form of a proposition (or of an argument)

is the logically effective structure of the proposition (or argument)as indicated by the meanings of the logical terms it contains

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 55

Example Grammatical versus Logical Form

"Tom, Dick and Harry lifted the box"

Grammatical form

(Tom, Dick, Harry) lifted the box

Potential logical forms

(Tom, Dick, Harry) lifted the box

(Tom lifted the box) and (Dick lifted the box) and (Harry lifted thebox)

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 56

Example Grammatical versus Logical Form

"I see nobody on the road," said Alice.

"I only wish I had such eyes," the King remarked in a fretful tone. "Tobe able to see Nobody! And at that distance too! Why, it's as muchas I can do to see real people, by this light!"

Grammatical forms

I see somebody on the road I see nobody on the road

Logical forms

I see somebody on the road It is not the case that (I see somebody on the road)

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 57

Material Content versus Logical Form

Is validity always a function of an argument's logical form?

Formalists claim that all logical properties can be explained usinglogical form alone

Anti-formalists claim that not all logical properties can beexplained using logical form alone

Example Socrates is a father

Socrates is a father [All fathers are male] Therefore, Socrates is male Therefore, Socrates is male

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 58

Uniform Substitution Instances

From logical forms to propositions

Given a logical form, any number of arguments may be producedby uniformly substituting (atomic or molecular) propositions forpropositional variables

From propositions to logical forms

Given a proposition, a finite number of logical forms may beproduced by uniformly substituting propositional variables forpropositions

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 59

Example Uniform Substitution Instances

Find all of the propositional forms for which the following proposition isa uniform substitution instance:

Proposition

~A ~B

Propositional forms

p p q ~p q p ~q ~p ~q

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 510

The Fallacy of Equivocation

The fallacy of equivocation occurs whenever an argument depends inappropriately on a

semantic ambiguity occurs whenever a semantic ambiguity plays a significant but

inappropriate role in an argument

Example "The existence of a law means that there must be a law maker. Butwe know that the law of gravity and other scientific laws have notbeen made by any human law maker. So it follows that there must bea non-human law maker, God." Here the equivocation is on "law" (i.e. "a prescriptive claim

enacted by a government" or "a descriptive regularity in nature")

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 511

Example Equivocation

The end of a thing is its perfectionDeath is the end of life

Therefore, death is the perfection of life

Here the equivocation is on the word "end" (i.e. "goal" or"termination")

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 512

(1) The goal of a thing is its perfection TrueDeath is the goal of life False

Therefore, death is the perfection of life False / Valid

(2) The termination of a thing is its perfection FalseDeath is the termination of life True

Therefore, death is the perfection of life False / Valid

(3) The goal of a thing is its perfection TrueDeath is the termination of life True

Therefore, death is the perfection of life False / Invalid

(4) The termination of a thing is its perfection False

Death is the goal of life False Therefore, death is the perfection of life False / Invalid

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 513

The Fallacy of Amphiboly

The fallacy of amphiboly occurs whenever an argument depends inappropriately on a

grammatical, rather than a purely semantic, ambiguity occurs whenever a grammatical ambiguity plays a significant but

inappropriate role in an argumentExample Thrifty people save old cardboard boxes and waste paper

Therefore, thrifty people waste paper

p q p q

q r

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 514

The Paradox of the Liar

Is the following proposition true or false?

This proposition is false

If every proposition is either true or false then this proposition willbe either true or false

If it is true, then it is true that it is false; so it must be both true andfalse

If it is false, then it is false that it is false; so it must be true; so it

must be both true and false So in both cases it is both true and false, which is impossible

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 515

Other Paradoxes

The Postcard Paradox

The Heterological Paradox

The Barber Paradox

The Protagoras Paradox

The Russell Paradox

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ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 516

Object-language versus Meta-language

A meta-language is

any language used to talk about a (usually separate) language

An object language is

any language being talked about