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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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Disjunctive Syllogism (DS) examples
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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Disjunctive Syllogism (DS) examples
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)]