method of deduction

8/12/2019 Method of Deduction

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Method ofDeduction

Formal Proof of Validity

The Rule of Replacement

Proof of Invalidity

Inconsistency


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FORMAL PROOF OF VALIDITY


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A way of evaluating the validity of anargument. We can establish validity of anargument more efficiently by a sequence of

elementary arguments, each of which is knownto be valid. Such a step-by-step procedure iscalled a formal proof.

We shall use rules of inference to assemblelists of true statements, called proofs. A

proof is a way of showing how a conclusionfollows from a collection of premises.

Rules of Inference provide for the standardpatterns to which a valid argument must

conform with.

A rule of inference allows you to deduce acertain sentence from one or two others.


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Modus Ponens MP)"the way that affirms

by affirming

When the antecedent

of a conditionalstatement (first

premise) is accepted

in the second

premise, the

consequent can be

validly inferred in

the conclusion

Form:

P Q ---premise

P ---premise

Q ---conclusion

It can be summarized

as "P implies Q; P is

asserted to be true,so therefore Q must

be true."


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Modus Ponens (MP) examples

If roses are red

and violets areblue, then sugar

is sweet and so

are you.

Roses are red and

violets are blue.

Therefore, sugar

is sweet and so

are you.

If it's raining,

I'll meet you atthe movie

theater.

It's raining.

Therefore, I'll

meet you at the

movie theater.


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Modus Ponens (MP) examples

If the cake is made

with sugar, then the

cake is sweet.

The cake is made with

sugar.

Therefore, the cake is

sweet.

If rats are quadrupeds

then they are four-

legged creatures;

But, rats are

quadrupeds.

Ergo, they are four-

legged creatures.


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Modus Tollens MT)denying theconsequentWhen the consequent

of a conditionalstatement isrejected in thesecond premise, wecan validly infer

the negation of theantecedent in theconclusion.

Form:

P Q

Q

P

If P, then Q.

If not Q , then not

P.


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Modus Tollens (MT) examples

If the watch-dog

detects an intruder,

the dog will bark.

The dog did not bark

Therefore, no intruder

was detected by the

watch-dog.

If I am the axemurderer, then I

can use an axe.

I cannot use anaxe.

Therefore, I am

not the axemurderer.


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Modus Tollens (MT) examples

If a patient is

suffering from life-threatening illness

then his health is

very serious,

The patients health

is not very serious,

Ergo he is not

suffering from alife-threatening

illness.

If Sam was born in

Canada, then he isCanadian.

Sam is not Canadian.

Therefore, Sam was

not born in Canada.


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Hypothetical Syllogism (HS) examples

If I do not wake up,

then I cannot go towork.

If I cannot go to

work, then I will not

get paid.

Therefore, if I do not

wake up, then I will

not get paid.

If it rains, we will

not have a picnic.

If we don't have a

picnic, we won't need

a picnic basket.

Therefore, if it

rains, we won't need a

picnic basket


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Hypothetical Syllogism (HS) examples

If the Montreal

Canadiens win theStanley Cup, I'll owe

my dad some money.

If I owe my dad some

money, I'll need to go

to the bank.

So, if the Montreal

Canadiens win the

Stanley Cup, I'll need

to go to the bank.

If Disneyland is a

carnival then it is an

amusement place.

If it is an amusement

place then it is a fun

place.

Ergo, if Disneyland id a

carnival then it is a fun

place.


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Disjunctive Syllogism DS)When one of the

disjuncts is

rejected in the

second premise,

then the other

disjunct shall be

validly accepted in

the conclusion.

Form:

P v Q

PQ

If either P or Q is

true and P isfalse, then Q is

true.


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Disjunctive Syllogism (DS) examples

Either the breach is a

safety violation, or it

is not subject to fines.

The breach is not a

safety violation.

Therefore, it is not

subject to fines.

The cake has either

chocolate or vanillafrosting.

The cake does not have

vanilla frosting.

Therefore, the cake

has chocolate

frosting.


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Either I will choose

soup or I will choosesalad.

I will not choose

soup.

Therefore, I will

choose saladIt is either red or

blue.

It is not blue.

Therefore, it is

red.


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Either dolphin is a

fish or it is a

mammal.

Dolphin is not a

fish.

Ergo, it is a mammal.

Either the Sun orbits

the Earth, or the

Earth orbits the Sun.

The Sun does not

orbit the Earth.

Therefore, the Earth

orbits the Sun.


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Conjunction (Conj) example

Whale is a huge

sea creature.

It is a mammal.

Ergo, whale is a

huge sea creatureand it is a

mammal.


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Simplification SP)

A conjunct can be

validly inferredfrom a conjunction

Form:

P Q

P


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Simplification (SP) examples

It's raining andit's pouring.

Therefore it'sraining.

A doctor is a

professional and amedical expert.

Ergo, a doctor isprofessional.


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Addition Add)Adding any

statement to the

original statement

validly forms theconclusion by

disjunction.

Form:

P P v Q


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Addition (Add) example

Gold is a

treasure.

Ergo, either gold

is a treasure or

it is something

else.


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Absorption AB)From a conditionalstatement, we canvalidly form anotherconditional statementin the conclusion

whose antecedent is aconjunction of theoriginal antecedentwith itself, andwhose consequent is aconjunction of theoriginal consequentand the originalantecedent

Form:

P Q

P ( P Q )

If P implies Q, thenP implies P and Q


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Absorption (AB) example

If it will rain, then

I will wear my coat.

Therefore, if it will

rain then it will

rain and I will wearmy coat.

If earth is abundant

with living thingsthen it is inhabited.

Ergo, if Earth is

abundant with livingtings then the earth

is abundant with

living things and is

inhabited.


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Constructive Dilemma (CD) examples

If I win a milliondollars, I will donate

it to an orphanage.If my friend wins amillion dollars, hewill donate it to awildlife fund.

Either I win a milliondollars, or my friendwins a milliondollars.

Therefore, either anorphanage will get a

million dollars, or awildlife fund will geta million dollars.

If it's gold then I'mrich and if it's

pyrite then I'm poor.

It iseither gold or pyrite.

Therefore, either I'm

rich or I'm poor.


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Constructive Dilemma (CD) examples

If some countries are

poor then theireconomy isfluctuating and ifsome countries arerich then theireconomy is stable.

Either some countriesare poor or rich.

Ergo, either theireconomy isfluctuating orstable.


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There are many valid truth-functional arguments whose

validity cannot be proved using only the nine rules

of inference given thus far. Therefore, an additional

principle of inference is the rule of replacement.

It permits us to infer from any statement the result

of replacing any component of that statement by anyother statement logically equivalent to the componentreplaced

The Rules of Replacement permit a statement to be

substituted by its logical equivalence.

A rule of replacement allows you to substitute oneformula or sentence for another. This replacement is

made possible by the fact that the two formulas are

equivalent (the formula being replaced and the

formula replacing it).


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RULES OF REPLACEMENT Double Negation (DN)

Association (Assoc)

Commutation (Com)

Tautology (Taut)

Transposition (Trans)

Exportation (Exp)

DeMorgans Theorem (DeM)

Material Implication (Impl)

Material Equivalence (Equiv)

Distribution (Dist)


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Double Negation DN)

Any proposition is

equivalent to the

negation of its

negation.

Form:

P P

If a statement is

true, then it is

not the case that

the statement is

not true."


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Double Negation (DN) examples

Today is a cold

day.

It is not true

that today is nota cold day.

The play was

fascinating.

It is not true

that the play was

not fascinating.


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Double Negation (DN) examples

Christmas is ourfavorite holiday.

It is not true thatChristmas is notour favorite

holiday.


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Association Assoc)This permits

modification of

the parentheticalgrouping of

certain

statements.

When applied toDISJUNCTION, it hasthe form:

[P v (Q v R)] [(P v Q) v R]

When applied toCONJUNCTION, it hasthe form:

[P (Q R)] [(P Q) R]


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Commutation Com)It shows that the

order of the

component parts of

a conjunction or a

disjunction can be

reversed.

Forms:

(P Q) (Q P)

(P v Q) (Q v P)


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Tautology Taut)Any statement may

be replaced by

another statement

which is simplythe conjunction or

disjunction of the

original statement

with itself

Forms:

P (P P)

P (P v P)


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Exportation Exp)A conditional statementwhose antecedent is a

conjunction of two simple

statements is equivalent

to another conditional

statement whose antecedent

is the first conjunct of

the original statements

antecedent, and whose

consequent is likewise a

conditional statement

composed of the second

conjunct and of the

original consequent,

Form:

[(PQ) R]

[P (Q R)]


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Exportation Exp) exampleIt rains and the

sun shines implies

that there is a

rainbow.

Thus, if it rains,

then the sun

shines implies

that there is arainbow.


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DeMorgans Theoren DeM)The negateddisjunction is

equivalent to a

conjunction of the

negation of the

disjuncts

Likewise, the negated

conjunction isequivalent to a

disjunction of the

negation of the

conjuncts

Form:

(P v Q) ( P Q)

"not P or Q)" is thesame as "not P) andnot Q)"

(P Q)

(

P v

Q)"not P and Q)" isthe same as "not P)or not P)"


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Material Implication Impl)A conditional

statement is

equivalent to a

disjunction itsparts provided

that the

antecedent is

negated

Form:

(P Q) (P v Q)


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Material Implication Impl) example

If it is a bear,

then it can swim.

Thus, it is not a

bear or it can

swim.


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Material Equivalence Equiv)A biconditionalstatement is equivalent

to the conjunction of

two conditionals where

the antecedent and

consequent of the firstconjunct are

interchanged in the

second conjunct.

(PQ) [(P Q)(Q P)]

If you have PQ, then you

have P Q and Q P

A biconditionalstatement is equivalent

to the disjunction of

its parts such that the

first disjunct is a

conjunction of thecomponent statements of

the biconditional and

the second disjunct is a

conjunction of the

negation of thecomponent parts

(PQ) [(PQ) v (P

Q)]


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Material Equivalence (Equiv) example

If Lisa is

in France, then

she is in Europe.

If Lisa is not in

Europe, then she

is not in France.


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Distribution Dist)A conjunction made

of a simple

statement and a

disjunction is

equivalent to thedisjunction of two

conjunctions.

[P(QvR)]

[(PQ)v(PR)]

A disjunction of a

single statement

with a conjunction

is equivalent to a

conjunction ofdisjunctions.

[P v (QR)]

[(PvQ)(PvR)]

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