Chapter 5: Formal and Informal Logic

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–1

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–2

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 Chapter 5: Formal and Informal Logic 5–3

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–4

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 Chapter 5: Formal and Informal Logic 5–5

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 Chapter 5: Formal and Informal Logic 5–6

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–7

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–8

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 ~p ⊃ ~q
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–9

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 Chapter 5: Formal and Informal Logic 5–10

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")
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–11

(1) The goal of a thing is its perfection True
Death is the goal of life False
Therefore, death is the perfection of life False / Valid
(2) The termination of a thing is its perfection False
Death is the termination of life True
Therefore, death is the perfection of life False / Valid
(3) The goal of a thing is its perfection True
Death 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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–12

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–13

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 Chapter 5: Formal and Informal Logic 5–14

Other Paradoxes
• The Postcard Paradox
• The Heterological Paradox
• The Barber Paradox
• The Protagoras Paradox
• The Russell Paradox
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–15

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
ARGUMENT: Critical Thinking, Logic and the Fallacies Chapter 5: Formal and Informal Logic 5–16