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Chapter 5: Formal and Informal Logic 1. Logical Form Exercise 5.1

2. Formal Logic 3. Begging the Question Exercise 5.2

4. Equivocation and Amphiboly Exercise 5.3

5. The Paradox of the Liar Exercise 5.4

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

(97-98)

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

(99)

Logical Form – Modus Ponens 

If Bill is in Ottawa then Bill is in Canada Bill is in Ottawa Therefore, Bill is in Canada



If Sue has read Shakespeare then Sue will pass her test Sue has read Shakespeare Therefore, Sue will pass her test



(P1) p  q (P2) p (C)

q

Grammatical vs Logical Form

(98)

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

Example — Grammatical vs Logical Form

(98)

"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)

Example — Grammatical vs Logical Form

(98)

"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!" Surface Grammatical form 

I see nobody (i.e. some object, nobody, is seen) on the road

Logical forms  

I see nobody (i.e. some object, nobody, is seen) on the road It is not the case that I see somebody on the road

Material Content vs Logical Form

(100)

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

Uniform Substitution Instances

(101)

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

Example — Uniform Substitution Instances Find some propositions that are uniform substitution instances of the following propositional form: Propositional form 

pq

Possible propositions 

AB

If the kids behave, then we’ll go the party



A  (B  C)

If the kids behave, then we’ll go the party or the beach



(A  D)  ~(B  C)

If the tires are flat or low, then we won't be able to swim or play

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

Hint: Ask whether propositions can be found that, if substituted into the form, would result in the creation of the original proposition

Example — Uniform Substitution Instances Find all of the propositional forms for which the following proposition is a uniform substitution instance: Proposition 

A  ~A

Propositional forms 

p

Not:

pq

p  ~p

p  ~q

~p  p even though (A  ~A)  (~A  A)

Compare: 7 + 7 and the forms x + x and x + y

Formal vs Informal Logic

(102)

Formal logic 

studies the formal (or structural) attributes of propositions that affect validity and other logical properties



distinguishes between logical (or topic-neutral) and non-logical terms



obtains a proposition’s logical form by uniformly replacing its nonlogical terms with variables

Informal logic 

studies the informal attributes of propositions that affect validity and other logical properties

Begging the Question

(106)

Begging the question 

is a type of argument in the broad sense



occurs whenever an arguer uses as a premise of his argument any proposition that his opponent presently rejects



is also called the fallacy of petitio principii

Moral: One does not defeat an opponent simply by mouthing propositions he already disagrees with

Example – Begging the Question

Student: You just can’t give me a C! Prof: Oh, I thought your paper sort of suggested the opposite… Why do you think so? Student: Why, I’m an A student!



To show that the student deserves a better grade, he or she needs to offer more evidence than simply the claim that he or she is always supposed to get better grades



Student’s claim begs the question against the Prof

Arguing in a Circle

(106-107)

Circular arguments 

are a type of argument in the narrow sense



occur whenever an argument’s conclusion simply repeats a premise, or asserts a proposition contained within or that is equivalent to, a premise

Note: Because an opponent is always likely to reject a premise that simply assumes (or presupposes) the very proposition that is supposed to be proved, arguing in a circle is one (main) way of begging the question

Example – Arguing in a Circle Sue: Natural selection, roughly, is a theory that only the “fittest survive”. Bill:

Yes, that’s what I often hear. But I don’t really understand what the predicate “fittest” mean. How do you define the individuals who are the “fittest”?

Sue: Well, clearly, these are the ones that leave the most offspring. Bill:



Hold on a second! Doesn’t “leave the most offspring” mean exactly the same as those who survive? Sue’s reply assumes that the fittest individuals leave the most offspring, but she defines the fittest individuals as those that leave the most offspring

The Fallacy of Equivocation

(113)

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 – Equivocation

>

Criminal actions are illegal, and all murder trials are criminal actions, thus all murder trials are illegal. 

Here the term "criminal actions" is used with two different meanings

Example – Equivocation

(113) >

The end of a thing is its perfection Death is the end of life Therefore, death is the perfection of life 

Here the equivocation on the word "end" (i.e. "goal" versus "termination"), making four possible interpretations



All four interpretations, in fact, turn out to be unsound

Example – Equivocation

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

(113) >

True False False / Valid False True False / Valid

Example – Equivocation

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

(113)

True True False / Invalid False False False / Invalid

The Fallacy of Amphiboly

(114)

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

The Paradox of the Liar

(117)

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...