3.
EXPRESSING CONDITIONAL RELATIONSHIPS

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 it.

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 causality.

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 enough.

In
expressing conditional
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
condition.

Study is enough in order for Lee to do well. study is a 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 examples.

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 connective)

~(P -> Q) v R 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)

(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 connective)

~(P -> Q) v R 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 both a 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"

REVIEWING SYMBOLIZATION

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.

In
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 and interesting. we are expressing implication, so we have E -> (F & I)

The
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
the interest).

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.

With 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
conditional.

EXERCISES

I. 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 S.

1. Alice will pass if she studies.

2. Bob will pass only if he studies.

3. Carol will not pass if she does not study.

4. Don will pass unless he does not study.

5. Ed has to study in order to pass.

6. Without study Felicity cannot pass.

7. Study is enough for George to pass.

8. Harry will pass assuming that he studies.

II. Symbolize each of the following using the letters E, F, and I.

1. Logic is both easy and interesting.

2. Logic is not both easy and fun.

3. If 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. If 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.

go to the answer key

I. 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 S.

1. Alice will pass if she studies.

2. Bob will pass only if he studies.

3. Carol will not pass if she does not study.

4. Don will pass unless he does not study.

5. Ed has to study in order to pass.

6. Without study Felicity cannot pass.

7. Study is enough for George to pass.

8. Harry will pass assuming that he studies.

II. Symbolize each of the following using the letters E, F, and I.

1. Logic is both easy and interesting.

2. Logic is not both easy and fun.

3. If 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. If 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.

go to the answer key

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.