The Logic of Compound Statements Section 12 13 Modus Tollens Conditional and Valid amp Invalid Arguments Conditional Statements A conditional statement is a sentence of the form if ID: 143723
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Slide1
Chapter 1
The Logic of Compound StatementsSlide2
Section 1.2 –
1.3 (Modus
Tollens)
Conditional and Valid & Invalid ArgumentsSlide3
Conditional Statements
A conditional statement is a sentence of the form “if
p
then
q
” or p -> q (p implies q).p is the hypothesisq is the conclusionSlide4
Example
“If
you show up for work Monday morning, then you will get the job
.”
p
= You show up for work Monday Morning.q = You will get the job.p -> q
When is this statement false?Slide5
Example ->
p
v
~
q -> ~pOrder of precedence: 1. ~, 2. ^,v, 3. ->, <->p v ~q -> ~p
(
pv~q
) -> (~p) Slide6
Logical Equivalence ->
p
q
->
r (p ->r) ^ (q ->r)Slide7
Equivalence -> & or
p
->
q
~p v qExample~p v
q
= “Either you get to work on time or you are fired.”
~
p = You get to work on time.q
= You are fired.
p
= You do not get to work on time.
p
->
q
= “If you do not get to work on time, then you are fired.”Slide8
Negation of Conditional
Negation of if
p
then
q
“p and not q”~(p -> q)
p
^ ~
q
Derivation from Theorem 1.1.1~(p ->
q
)
~(~
p
v
q
)
~(~
p
) ^ (~
q
) by
DeMorgan’s
p
^ ~
q
by the double
neg
law
Example
If Karl lives in Wilmington, then he lives in
NC.
Karl lives in Wilmington and he does not live in
NC.Slide9
Contrapositive of a Conditional
The
contrapositive
of
p
-> q is ~q -> ~p.Conditional is logically equivalent to its contrapositive: p -> q ~
q
-> ~
p
p
q
~
p
~
q
p
->
q
~
q
-> ~
p
T
T
F
F
T
T
T
F
F
T
F
F
F
T
T
F
T
T
F
F
T
T
T
TSlide10
Example
Conditional
p
->
q
If Howard can swim across the lake, then Howard can swim to the island.p = “Howard can swim across the lake.”q = “Howard can swim to the island.”Contrapositive ~q -> ~p
If Howard cannot swim to the island, then Howard cannot swim across the lake.Slide11
Converse of Conditional
Converse of conditional “if
p
then
q
” (p->q) is “if q then p” (q->p)Converse is not logically equivalent to the
conditional.
Example
(conditional) If today is Easter, then tomorrow is Monday.
(converse) If tomorrow is Monday, then today is Easter.Slide12
Inverse of Conditional
Inverse of conditional “if
p
then
q
” (p->q) is “if ~p then ~q” (~p->q)Inverse is not logically equivalent to the
conditional.
Example
(conditional) If today is Easter, then tomorrow is Monday.
(inverse) If today is not Easter, then tomorrow is not Monday.
However, the converse and inverse are logically equivalent
.
p
q
~
p
~
q
p
->
q
q
->
p
~
p
->~
q
T
T
F
F
T
T
T
T
F
F
T
F
T
T
F
T
T
F
T
F
F
F
F
T
T
T
T
TSlide13
Biconditional
Biconditional is “
p
if, and only if
q
”.Biconditional is T when both p and q have the same logic value and F otherwise. Symbolically – p <-> qSlide14
Biconditional Truth TableSlide15
Necessary & Sufficient Conditions
For statements
r
and
s
,r is a sufficient condition for s (if r then s) means “the occurrence of r is sufficient to guarantee the occurrence of s”.
r
is a necessary condition for
s
(if not
r then not s) means “if
r
does not occur, then
s
cannot occur”.Slide16
Valid & Invalid Arguments
An argument is a sequence of statements.
All statements in an argument, except for the final one, is the premises (hypotheses).
The final statement is the
conclusion.
Valid argument occurs when the premises are TRUE, which results in a TRUE conclusion.Slide17
Testing Argument Form
Identify the premises and conclusion of the argument form.
Construct a truth table showing the truth values of all the premises and the conclusion.
If the truth table reveals all TRUE premises and a FALSE conclusion, then the argument from is invalid. Otherwise, when all premises are TRUE and the conclusion is TRUE, then the argument is valid.Slide18
Example
If Socrates is a man, then Socrates is mortal.
Socrates is a
man
.
:. Socrates is mortal.Syllogism is an argument form with two premises and a conclusion. Example Modus Ponens form:If p then q.p:. qSlide19
Example Valid Form
p
v
(
q v r)~r:. p v qSlide20
Example Invalid Form
p
->
q
v ~rq -> p ^ r:. p -> rSlide21
Modus Tollens
If
p
then
q
.~q:. ~pProves it case with “proof by contradiction”Example:if Zeus is human, then Zeus is mortal.
Zeus is not mortal.
:. Zeus is
not human.