rules of inference and replacement

8/10/2019 Rules of Inference and Replacement

1/4

RULES OF INFERENCE

RULE OF INFERENCE

(VALID ARGUMENTS)PRINCIPLE

CORRESPONDING FALLACY

(INVALID ARGUMENTS)PRINCIPLE

MODUS PONENS

P Q

P

Q

Example:

If I own a company, then I am

rich.

I own a company.

Therefore, I am rich.

P implies Q; P is asserted

to be true. Therefore, Q

must be true.

NOTE:

Affirming the antecedent

MODUS PONENS

AFFIRMING THE CONSEQUENT

P Q

Q

P

Example:

If I own a company, then I am rich.

I am rich.

Therefore, I own a company.

(INVALID)

Invalid because since

P was never asserted

as the ONLY sufficient

condition for Q, other

factors could accountfor Q (while P could be

false).

MODUS TOLLENS

P Q~Q

~P

Example:


rich.I am not rich.

Therefore, I dont own acompany.

P implies Q; When you

deny the consequent (Q),

it can be logically

concluded that it is not

the case that P.

NOTE:Denying the consequent

MODUS TOLLENS

DENYING THE ANTECEDENT

P Q~P

~Q

Example:


I dont own a company.Therefore, I am not rich. (INVALID)

DETACHING THE CONSEQUENT

P Q Q

Example:


Therefore, I am rich. (INVALID)

Invalid because if the

antecedent (P) is not

true, you cant

conclude that

consequent (Q) is not

true.

Invalid since you

cannot infer the

consequent because

the consequent and

antecedent of the

conditional can both

be false yet the

conditional can still be

true. (FF=T)

HYPOTHETICAL SYLLOGISM

P QQ R

P R

Example:


rich.

If I am rich, I can afford fancy

dinner.

Therefore, if I own a company,

then I can afford a fancy dinner.

P implies Q and Q implies

R; then P can implicate R

subsequently

FALLACY OF MISPLACED MIDDLE

P QR Q

P R

Example:


If I can afford fancy dinner, then I m

rich.

Therefore, if I own a company, I can

afford a fancy dinner.

Invalid. Although

conclusion could be

right, the conclusion

did not follow the

formal logic of

deducting. When

common element is

the consequent of both

premises you commit

a fallacy.

DISJUNCTIVE SYLLOGISM

P Q

~P

Q

Example:

Either the Defensor or Trillanes

will run for office.

Defensor will not run for office.

Therefore, Trillanes will run for

office.

If we are told that at least

one of the two

statements is true; and

also told that it is not the

former that is true, we

can infer that it has to be

the latter that is true.

If either P or Q is true

and P is false, then Q s

true.

DENYING AN ALTERNATIVE

P Q P Q

Q P

~P ~Q

Example:

Either Defensor or Trillanes will run

for office.

Trillanes will run for office.

Therefore, Defensor will not run for

office. (INVALID)

Invalid because even if

you know that the

other premise is true,

you cannot infer that

the 2ndstatement is

false because it can

still be true.

NOTE:

An alternation is true

when both alternants

are true.


2/4

DESTRUCTIVE DILEMMA

P QR S

~Q ~S ~P~R

Example:

If my friend wins the lottery,

then my friend will donate to a

school.If I win the lottery, I will donate

to an orphanage.

Neither an orphanage nor school

received a donation.

Therefore, neither I nor my

friend won the lottery.

Valid because it denies

the consequent of both

conditionals and

therefore you can validly

infer the denial of the

antecedent in the

conclusion.

NOTE:Destructive dilemma is a

combination of two

modus tollens.

DESTRUCTIVE - take

note of the negation

PSEUDO DILEMMA I

P QR S

~P ~R ~Q ~S

Example:

If my friend wins the lottery, then

my friend will donate to a school.

If I win the lottery, I will donate toan orphanage.

Neither I nor my friend won the

lottery. Therefore, neither an

orphanage nor school will receive a

donation.

(Might appear valid, but still

INVALID)

Invalid because it

denies the antecedent

of the conditional

premises. And so it

cannot conclude the

denial of both the

consequents.

RULE:

Either AFFIRM theANTECEDENT, or

DENY the

CONSEQUENT to get a

valid argument.

CONSTRUCTIVE DILEMMA

P Q

R S

P R Q S

Example:

If my friend wins the lottery,

then my friend will donate to a

school.

If I win the lottery, I will donate

to an orphanage.

Either my friend or I win the

lottery.

Therefore, an orphanage or

school will receive a donation.

Valid because it affirms

the antecedent P and R of

both conditionals and

therefore, you can validly

infer the consequent Q S in the conclusion.

NOTE:

Constructive dilemma is

a combination of two

modus ponens.

PSEUDO DILEMMA II

P Q

R SQ S

PR

Example:

If my friend wins the lottery, then

my friend will donate to a school.

If I win the lottery, I will donate to

an orphanage.

Either an orphanage or school will

receive a donation.

Therefore, my friend or I will win

the lottery.

(Might appear valid, but still

INVALID)

Invalid because it

affirms the consequent

of both conditional

statements and

therefore cannot

affirm the antecedent.

RULE:

Either AFFIRM the

ANTECEDENT, or

DENY the

CONSEQUENT to get a

valid argument.


3/4

10 RULES OF REPLACEMENT

RULES OF REPLACEMENT LOGICAL STRUCTURE/FORM

1. Double Negation DN)

Principle: No matter what simple or compound

statement we substitute for p, the same

statement with ~~in front will have exactlythe same truth value as original statement

~ ~

Example:

Alan is clever. p

It is not the case that Alan is not clever. ~~p

2. De Morgans Theorem DeM.)

Principle: Establishes systematic relationship

between statements and statements byproviding a significant insight into the truth-

conditions for the negations of both

conjunctions and disjunctions.

In other words, you REVERSE the connector

when distributing negation over a conjunct or

disjunct.

~( q) (~ ~q)

Example:

It is not the case that I am both bald and fat. ~(p q)Either I am not bald or I am not fat. (~p ~q)

~( q) (~ ~q)

Example:

I am neither bald nor fat. ~(p q)

I am not bald and I am not fat. (~p ~q)

3. Material Implication Impl.)

Principle: Logically defines connector in

terms of and ~. Since expressions ofthese two forms are logically equivalent, we

could make conditional assertions without

using at all, though compound statementwould be a bit more complicated

( q) (~ q)

Example:

If it rains, then we cancel the picnic. (pq)Either it doesnt rain or we cancel the picnic. (~p q)

4. Material Equivalence Equiv.)

Principle: Provides alternatice definition for

connective by first defining in terms of

two , justifying the use of termbiconditional, and lastly by combining these

two conditionals with a conjunct .

The second form defines by pointing outits basic truth conditions. Meaning [(p q) (~p ~q)]will have the same exact truth

value as pq.

[ q] [( q) (q )]

Example:

We ski if and only if it snows. [pq]

If we ski, then it snows. And if it snows, then we ski. [(pq) (qp)]

[ q] [( q) (~ ~q)]

Example:

We ski if and only if it snows. [pq]Either we ski and it snows, or we don t ski and it doesnt snow. [(p q)

(~p ~q)]

5. Transposition Trans.)Principle: The logical equivalence of any statement is equivalent to switching the

antecedent and consequent AND NEGATING

BOTH.

FALLACY OF CONVERTING THE CONDITIONAL

Switching the antecedent and consequent

without negating creates an INVALID

argument.

FALLACY OF NEGATING THE ANTECEDENT &

CONSEQUENT

Negating both elements without switching is

INVALID.

( q) (~q ~ )Example:

If it produces pleasure, then it is right. (p q)If it isnt right, then it doesnt produce pleasure. (~q ~p)

6. Commutation Comm.)Principle: Statements in conjunction or

disjunction form can simply be reversed.

( q) (q )Example:

Either Uncle Sam is an American or Juan is Filipino. (p q)

Either Juan is Filipino or Uncle Sam is an American. (q p)

( q) (q )

Example:

Uncle Sam is an American and Juan is a Filipino. (p q)Juan is a Filipino and Uncle Sam is an American. (q p)

7. Association Assoc.)

Principle: Permits modification of the

parenthetical grouping of certain statements.

[ (qr)] [( q) r]

Example:

Harold is 21 or either Jane or Kelly is. [p (qr)]

Either Harold or Jane is 21, or Kelly is. [(pq) r]


4/4

[ (qr)] [( q) r]

Example:

Harold is over 21, and so are Jane and Kelly. [p (qr)]Harold and Jane are over 21, and so is Kelly. [(pq) r]

8. Distribution Dist.)

Principle: 1stform A conjunct is distributedover a disjunction. 2ndform a disjunct isdistributed over a conjunction.

[ (qr)][( q) ( r)]

Example:

Paul is tall, and so is either Susan or James.

Either Paul and Susan are tall or Paul and James are.

[ (qr)][( q) ( r)]Example:

Either Paul is tall or both Susan and James are. [p (qr)]Paul or Susan is tall and so is either Paul and James. [(pq) (pr)]

9. Exportation Exp.)

Principle: Allows conditional statements

having conjunctive antecedents to be replaced

by statements having conditional consequents

and vice-versa.

[( q)r)][ (qr)]

Example:

If Harry is tall and quick, then he plays well. [(pq)r)]

If Harry is tall, and the if he is quick, then he plays well. [p(qr)]

10. Tautology Taut.)

Principle: Permits replacement of any

statement by (or with) another statement that

is simply the disjunction or conjunction of the

original statement with itself.

Although its ordinary language use invariably

seems pointless and redundant, this pattern of

reasoning is useful in propositional calculus.

( )

Example:

Harry is tall or Harry is tall. (p p)

( )Example:

Harry is tall or Harry is tall. (p p)

rules of inference and replacement

Documents