We
use either propositional
symbolization with single letters representing complete statements or
we use predicate symbolization in which a capital letter represents a
characteristic and a small letter represents a name of some sort.
P:
The princess is
walking. Pp:
Patricia is a
princess. Pp
& Wp: Patricia is a
princess who is walking.
We negate a letter or an expression with a curl (~), and we represent
how different things are connected with four special signs:
The princess is walking and the queen is knitting: P &
Q (conjunction can be expressed also as
"but"
or "although" or any equivalent)
Either the princess is walking or the queen is knitting: P v
Q (disjunction can also be expressed with "unless")
If the princess is walking then the queen is knitting: P ->
Q (we read the idea of the princess walking as
a
sufficient condition for us to know the queen is knitting)
The queen is knitting only if the princess is walking: Q ->
P (we read the idea of the princess walking as
a
necessary condition for us to know the queen is knitting)
The princess is walking if and only if the queen is knitting: P
<-> Q (here we have the queen knitting as both a
sufficient
and a necessary condition)
As we develop more complex expression we use the punctuation of
parentheses and brackets:
If the princess is walking or the queen is knitting then the king is
riding, but the princess is not walking: [(P v Q) -> K] &
~P
We use quantifiers to talk about all or some of a group. When we
are talking about characteristics we also think in terms of conditional
for universal quantifiers or conjunctions for existential quantifiers.
(x)Px:
Everyone is a
princess. (Ex)Kx:
There are kings. (x)(Px
->
Wx): All the princesses are walking. (Ex)(Kx &
Rx): Some of the kings are riding.
(x)~Qx or ~(Ex)Qx: There is no
queen. (x)(Rx ->
Kx): Only the kings are riding.
We
use the curl to negate our ideas, but here we have to be careful:
(x)(Px
-> ~Wx): No princess is
walking. (x)(~Px
-> Wx): Anyone not a
princess is walking. ~(x)(Px -> Wx):
Not all the princesses are walking.
EVALUATING
EXPRESSIONS FOR THEIR TRUTH VALUE
Every
expression can be said to be true (1) or false (0). Whenever we
have a connective or operator (including our quantifers), its truth
value depends on the elements making it up. The key rule is that
both elements have to be true for a conjunction to be true, both false
for a disjunction to be false, and a conditional is always true with
the one exception of its left-hand part (the antecedent) being true and
its right-hand part (the consequent) being false.
When we
have parentheses or brackets we work from the inside out. For
instance, given that P and R are true while Q is false, we would say
that (P & Q) -> R
is true while (P & R) -> Q
is false.
The
truth value of an entire expression is
determined by what is the
main connective. This is true for quantifed expressions also, and
a
quantifier is itself a connective.
For instance, in the
following examples we are starting with the idea
that P and R and Sa are true while Q and S and Tb are false
(P
& R) -> (S v Q) is false (0) since P
& R together are true while S v Q together are false, and the arrow
is the main connective.
P & [S -> (R v Q)]
is true (1) since P alone is true while the group that follows is a
case of false -> true, which is true, and the ampersand is the main
connective
(x)~Sx -> ~(x)Tx is true (1)
since knowing Sa is true tells us that (x)~Sx is false and knowing that
Tb is false tells us that ~(x)Tx is true
THE
MOST COMMON PROBLEMS
Not
representing conditional relationships correctly. We
cannot just go by the word order in a sentence, Instead, we need
to rethink the kind of condition being expressed.
If logic is easy then it is
fun
Logic is fun if it is easy.
Being easy makes logic fun.
It is enough for logic to be
easy for it to be fun.
In all of these we hear being easy as a
sufficient condition for us to know that logic is fun, and so we
symbolize the thought that logic is easy to the left of the arrow.
E
-> F
Logic is interesting only if
it is fun.
Logic has to be fun in order
for it to be interesting.
Only if logic is interesting
will it be fun.
In all of these we hear being fun as a
necessary condition for logic to be interesting, and so we symbolize
the thought that logic is interesting to the right of the arrow.
I
-> F
Not seeing whether to use the universal
quantifier (x) or the existential quantifier (Ex), and then not seeing
how to symbolize what follows, especially if there some type of
negation involved.
Everyone is happy. (x)Hx
All are happy. (x)Hx
No one is failing.
(x)~Fx
Not everyone is
failing. ~(x)Fx
note how the thought here is not the
same as the thought in the previous sentence
Everyone is a student. (x)Sx
Everyone is a happy
student. (x)(Sx &
Hx) look at the difference between this
statement and the next
Every student is
happy. (x)(Sx ->
Hx)
Anyone not failing is
happy. (x)(~Fx
-> Hx)
Someone failing is
unhappy. (x)(Fx ->
~Hx) this is intended as a generalization
without saying anyone actually is failing
Only someone passing is
happy. (x)(Hx ->
Px) note the parallel with how we expressed
necessary conditions above
Students who are passing are
happy. (x)[(Sx &
Px) -> Hx]
No one failing is
happy. (x)(Fx ->
~Hx)
There are no failing students
who are happy. (x)[(Sx
& Fx) -> ~Hx]
Some individuals are
happy. (Ex)Hx
There are happy
students. (Ex)(Sx & Hx)
the order in the parenthesis does not
make a difference
Someone is not
passing. (Ex)~Px
A few are not
passing. (Ex)~Px
Many are not
passing. (Ex)~Px
Most are not
passing, (Ex)~Px
we use the same
existential quantifier since what matter is that we are not talking
about everyone
There
are failing students who are unhappy. (Ex)[(Fx & Sx) & ~Hx]
Failing to recognize what is the main
connective and have this seen in the grouping.
Everyone is happy although some are not
passing. (x)Hx &
(Ex)~Px
Everyone is happy although not
passing. (x)(Hx & ~Px)
If every student is passing
then all the teachers are happy. (x)(Sx -> Px) -> (x)(Tx -> Hx)
Students are happy only if
they are passing. (x)[Sx
-> (Hx -> Px)]
