Chapter 5: Formal and Informal Logic

1. Logical Form Exercise 5.1

2. Formal Logic 3. Equivocation and Amphiboly Exercise 5.2

4. The Paradox of the Liar Exercise 5.3

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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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Logical Form – Disjunctive Syllogism •

Bill is in New York or Bill is in London It is not the case that Bill is in New York Therefore, Bill is in London

•

Sue went to the movies or Sue left town It is not the case that Sue went to the movies Therefore, Sue left town

•

(P1) p ∨ q (P2) ∼p (C)

q ARGUMENT: Critical Thinking, Logic and the Fallacies

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Grammatical versus Logical Form

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

is the structure of the proposition (or argument) as indicated by the 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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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 the box) ARGUMENT: Critical Thinking, Logic and the Fallacies

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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. "To be able to see Nobody! And at that distance too! Why, it's as much as 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) ARGUMENT: Critical Thinking, Logic and the Fallacies

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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 using logical form alone

•

Anti-formalists claim that not all logical properties can be explained 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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Uniform Substitution Instances

From logical forms to propositions •

Given a logical form, any number of arguments may be produced by uniformly substituting (atomic or molecular) propositions for propositional variables

From propositions to logical forms •

Given a proposition, a finite number of logical forms may be produced by uniformly substituting propositional variables for propositions

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Example — Uniform Substitution Instances

Find all of the propositional forms for which the following proposition is a uniform substitution instance: Proposition •

~A ⊃ ~B

Propositional forms •

p

p⊃q

~p ⊃ q

p ⊃ ~q

ARGUMENT: Critical Thinking, Logic and the Fallacies

~p ⊃ ~q

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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. But we know that the law of gravity and other scientific laws have not been made by any human law maker. So it follows that there must be a 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") ARGUMENT: Critical Thinking, Logic and the Fallacies

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Example – Equivocation

The end of a thing is its perfection Death 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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(1) The goal of a thing is its perfection Death is the goal of life

True False

Therefore, death is the perfection of life (2) The termination of a thing is its perfection Death is the termination of life Therefore, death is the perfection of life (3) The goal of a thing is its perfection Death is the termination of life

False True False / Valid True True

Therefore, death is the perfection of life (4) The termination of a thing is its perfection Death is the goal of life Therefore, death is the perfection of life ARGUMENT: Critical Thinking, Logic and the Fallacies

False / Valid

False / Invalid False False False / Invalid

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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 argument

Example •

Thrifty people save old cardboard boxes and waste paper Therefore, thrifty people waste paper p∧q

p∧q

q

r

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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 will be either true or false • If it is true, then it is true that it is false; so it must be both true and false • 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 ARGUMENT: Critical Thinking, Logic and the Fallacies

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Other Paradoxes

• • • • •

The Postcard Paradox The Heterological Paradox The Barber Paradox The Protagoras Paradox The Russell Paradox

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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

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