As we will soon see, the key to what happens in symbolic logic
is found in a solid understanding of conditional relationships.
These are the connections we make when we say that one thing being true
is enough for us to know that something else has to be true (what we
mean when we talk about X being a sufficient condition for Y)
or when we say that something being true is a requirement for something
else to be true (what we mean when we talk about Y being a necessary condition for
X). For example, a necessary condition for getting a grade in
your logic course at Pierce is that you are enrolled in it while a
sufficient condition for not getting a grade is not being enrolled in
Do keep in mind that conditionals often express causal relationships
(one thing in fact makes another happen or is required for it to
happen) but logically conditions are not the same as causes. In
fact, determining when we should say that X causes Y is something quite
difficult and is discussed in courses on inductive logic or discussions
of scientific method. Let's imagine that it is
always true that in very cold weather students taking an exam are
dressed warmly, sweaters and jackets and all that. Let's say that
in a particular class all these bundled up students do pass their
exam. Since we never find an exception we could say truthfully
enough that if a student dresses warmly she will pass her exam (this
would make dressing warmly a sufficient condition) but we know better
than to say being dressed warmly actually is the reason she
passes. Formal logic is no help at all in learning what makes
things happen--or, to put it somewhat differently, correlation (what we
have when X and Y are always found together) does not of itself prove
We typically express a sufficient condition by using the word "if" and
we symbolize the idea that P
is a sufficient condition for Q
by writing this as P->Q.
We typically write a necessary condition by using the phrase "only if"
and we symbolize the idea that Q
is a necessary for P by writing this as P->Q.
Look again at what I have above and you will see that the expressions
are identical. A key here is that whatever is the left of the
arrow (we call this an antecedent)
does express a sufficient condition so that what is to the right is the
result and whatever is to the right of the arrow (we call this a consequent) expresses a
necessary condition so that what is to the left is the result.
Remember that conditions do not of themselves indicate anything about
time (one thing having to happen before another) and so do not carry
over some of our ideas about cause and effect.
Supposing we have E->F.
We could use this to translate either "if logic is easy then it is fun"
or "logic is easy only if it is fun." In everyday conversation
these two sentences suggest different things but as far as formal logic
is concerned they express the same relationship. For a parallel,
think of how the sentences "Jack is Jill's brother" and "Jill is Jack's
sister" are different grammatically but express exactly the same
relationship between Jack and Jill.
What does matter, however, is that we have the right position for the
letters. With our other connectives, as we will see, it does not
matter which we have on the left and which on the right. E & F tells us exactly the
same thing as F & E,
and the same is true when we have E
v F and F v E.
But E->F and F->E give us different
stories: one might be true while the other is not (as when logic being
easy would make it fun but it does not really have to be easy in order
for it be fun). This is why understanding the exact relationship
intended in a sentence is crucial, since Engish word order alone is not
relationships, then, a key thing is to decide whether we have one
thing as a
sufficient condition for another (X being true makes Y true) or whether
it is a necessary condition (X has to be true in order for Y to be
true). Being in the water, for instance, is a necessary condition
for a person to swim but it is certainly not enough. Again, let's look
at examples, keeping in mind that the English word order is not itself
enough for us to go by. Pay attention to different ways we can
express the idea of something being either a sufficient or a necessary
Lee has to study in order to do well.
study is a
necessary condition for doing
Study is enough in order for Lee
to do well. study
sufficient condition for doing well S->W
Logic being easy and fun means Lee
studies and does well. (E & F) -> (S & W)
Lee does well only if either logic
or easy or she does study. W -> (E v S)
Not studying implies Lee will not
do well. ~S -> ~W
If logic is not easy or she does
not study then Lee will not do well. (~E v ~S) -> ~W
Once again, make
sure you see the importance of recognizing what is the main connective
when we are using parentheses, as we just had to do in the last set of
P -> (Q
& R) here we have the idea that P implies both Q
and R (the arrow is the main connective) (P
-> Q) &
R here we have the idea that P implies
Q and also that R is true (the ampsersand is the main connective) P
~(Q v R) here we are saying that if P is true then
the disjunction of Q and R is false (the arrow is the main
-> Q) vR here we are saying that
either it's false that P
implies Q or R itself is true (the wedge is the main connective) ~[P v
(Q & R)] here we are denying that either P is true or the
conjunction of Q and R is true (the curl is the main connective)
WHAT ABOUT A TWO-HEADED ARROW?
What if we want to express that idea that something is
sufficient and a necessary condition for a particular result? For
instance, let's suppose that study alone does allow Lee to pass but it
is also what he has to do in order to pass. Here we use a
double-headed arrow (S<->P)
and we have what we call a biconditional, something expressing a
relationship of mutual implication or equivalence. With this
connective, unlike the single-headed arrow, it does
not matter what goes to either side, so we can safely
follow the English word order.
Logic is easy if and only if it is fun. being fun is both a sufficient and a
necessary condition for logic to be easy E <->
F (or F <-> E)
Read your original
sentences carefully. Having just "only if" means we are talking
only about a necessary condition. The biconditional calls for all
four words: "if and only if"
In "translating" from what we
call a natural language such as
English to a symbolic language the most important step is to correctly
identify the key logical relationship expressed. This determines
what will be the main connective as well as how our letters and
connectives are grouped together with parentheses.
expressing a general form it has been traditional to use small letters
with the understanding that any capital letter or any grouping of
symbols could become "substitution instances." In these web
lectures I use only the capital letters, and I refer to them as
variables, although this in a sense somewhat different from what is
meant when we talk about variables later in the course in the sections
dealing with predicate logic.
Logic is easy and it is
fun. we are expressing a
conjunction, so we have E
& F Either logic is easy or it is not
fun we are
expressing a disjunction, so we have E
v ~F If logic is easy then it is both fun
we are expressing implication, so we
-> (F & I)
original statement is a conditional; what matters is that we see
whether or not the logical order when we symbolize will be the same as
the English word order. In this example it is, but we could just
as easily have said "Logic is both fun and interesting if it is easy"
and we would have had to break away from the English ordering of the
ideas. The key is that in expressing a sufficient condition we
look to the lefthand part of the expression (what we call the
antecedent) to convey this meaning while with a necessary condition we
look to the righthand part (what we call the consequent).
It is not true that logic is
fun only if it is both easy and
interesting. we are
expressing the negation of a
conditional, so we have ~(F
-> (E & I))
Most problems appear either either in
failing to recognize the type of
relationship intended or not being careful to rethink a conditional
when we should. Keep in mind that many of the meanings we want to
communicate in an ordinary sentence will not be reflected in its
symbolization. For instance, we intend a contrast when we say
"logic is fun although it is not easy" but in the symbolization we have
just the conjunction (F & ~E). In the same way we lose the
sense of causality when we symbolize "logic is interesting because it
is not easy" as also just a conjunction (I & ~E), and the
temporal sequence in "Alice studied before she passed" (S &
P) is also lost.
The idea here is
that we are interested only in a key set of
truth-functional relationships. What characterizes the last
examples is that we are able to say the entire sentence is false even
though its two parts are not (for instance, logic could be both
interesting and difficult, but it is not the difficulty that creates
Word order does not matter when we
are working with conjunction or
disjunction. For instance, we could say "Alice is happy unless
she didn't pass" or "unless Alice is happy she didn't pass" and we
still come out just with H v ~P.
conditionals we need to be very
careful that in symbolizing a
conditional the antecedent is to the left and the consequent to the
right. How do we know which is which? The antecedent
expresses whatever is a sufficient condition for a result, while the
consequent is either the result or the necessary condition for a result
expressed in the antecedent. (OK on that? What we are
saying, really, is that the expression "if E then F" is logically
equivalent to or has the same truth value as "E only if F" even if the
feeling is different in the two expressions. Saying "if logic is
easy then it's fun" puts the emphasis on logic being enough for it to
be fun. Saying "logic is easy only if it is fun" puts the
emphasis on logic having to be fun for us to consider it easy.)
Something else to
note is that saying "P unless Q" in effect tells us that one of the
terms is a necessary condition for the opposite of the other, so that P v Q has the same truth
value as ~P->Q or ~Q->P. In your
own symbolization, however, play it safe by using the wedge instead of
the arrow since it is is too easy to get the order wrong with a
Often it is necessary to
rethink a conditional relationship in order to arrive at a standard
form of "if .... then ..." with a sufficient condition expressed as an
antecedent and a necessary condition expressed as a consequent.
Rewrite each of the following in this way, then symbolize using P and
Alice will pass if
she studies. 2.
will pass only if he
Carol will not pass if she
does not study. 4.
will pass unless he does
not study. 5.
has to study in order to
Without study Felicity
cannot pass. 7.
Study is enough for George
to pass. 8.
Harry will pass assuming
that he studies.
Symbolize each of the
following using the letters E, F, and I.
Logic is both easy
and interesting. 2.
Logic is not both easy and
logic is either easy or
interesting it will be fun. 4.
Logic has to be both easy
and fun in order to be interesting. 5.
logic is not either easy
or fun then it will not be interesting. 6.
Although logic is not easy
it will be fun if it is interesting.
The red flag aboves are meant to alert you
to the importance of thinking through the type of conditional
relationship expressed in an English sentence since this determines the
correct position of the terms in symbolization. There is more
about this in the additional material you will see on the schedule.
Keep in mind that we do not go by English word order--the most
common mistake made by students.