ADDITIONAL
MATERIAL: CONNECTIVES
The
key point is that
capital letters are used to express complete ideas, and then the
connectives express relationships among these ideas. Let's
imagine we want to work with the ideas that logic is easy and another
idea that logic is fun. We'll express "logic is easy" as E and
"logic is fun" as F.
Now let's look at these new sentences:
Logic is not easy: ~E
Logic is easy and it is
fun: E & F
Logic is not easy but it is
fun: ~E & F
Either logic is easy or it is
not fun: E v ~F (that "v" comes from the fact that the
Latin word for "or" is vel)
Unless logic is easy it is not
fun.: E v ~F
If logic is easy it is
fun: E->F (logic being easy is a sufficient
condition for it to be fun, so E is the antecedent)
Logic is easy only if it is
fun: E->F (logic being fun is a necessary
condition for it to be easy, so F is the consequent)
The
antecedent is what is the left of the arrow (or horseshoe) in a
conditional; the consequent is what is to the right.
It's not true that logic
is both easy and fun: ~(E & F)
And what is the reason for all this? If we can express our
statements about facts symbolically, we can test arguments that involve
these facts by seeing whether ot not there is even the possibility that
true premises could allow a false conclusion.
WHY CONDITIONALS MATTER
The most frequent instances of
what
we call deductive reasoning appear when we are working with conditional
relationships--statements telling us that knowing one thing is true
should be enough to know something else is true (for instance: if Jack studies he is
going to pass) or that knowing one thing is a requirement for
something else, so that if it does not happen then the expected result
cannot happen (for instance: Jack
will study only if he does not have a job).
To work more effectively, let's think of how these
relationships could
all be expressed in a standard form of "If X then Y" (or symbolically
as "X -> Y"). We will select key words and use the first
letters as our X and Y in the examples below.
X is a sufficient
condition for Y. Careful, this is not the same as saying that
X is a cause of Y, although that may be true. What we are saying
is that once we know X is true, we can be sure that Y is true as
well.
Jill
will be cut from the team if she
misses one more practice. If M then C. (M -> C)
Hard work ensures success.
If H then S. (H -> S)
A lot of rain means that there is a
danger of mudslides. If R then M. (R -> M)
High grades can be brought about by
consistent study. If S then G. (S -> G)
A
very important thing to note: the order in which we express each
thought is not always the same as the logical order we are concerned
with here. In the first and last examples above what we have in
the X position (what we call the antecedent
in the conditional) was heard after what we have in the Y position
(what we call the consequent
in the conditional).
Y is a necessary
condition for X. What we are saying is that once we know Y
is false we can be sure that X is false as well.
Alice
will do well only if she studies
hard. If
W then S. (W -> S)
Logic has to be interesting in order to be fun. If F then I. (F -> I)
Perfect attendance is a requirement for
a passing grade in this
program. If G
then A. (G -> A)
Something
else to note is that as far as logical relationships are concerned,
saying that X is a sufficient condition for Y and that Y is a necessary
condition for X gives us the same standard form. For instance,
"logic is fun if it is easy" and "logic is easy only if it is fun" are both expressed as "if E then F" (E -> F).
In order to have a valid argument with a conditional as one premise
just one of two things can happen: (1) we can have another premise that
puts the
antecedent as true so that the conclusion will state that the
consequent is true, or (2) we can have another premise that takes away the
consequent as true (states it is false) so that the conclusion
will state that the antecedent must be false also.
 One way of talking
about these patterns is in terms of an inference engine. Imagine
we have X -> Y. A left-hand driven engine is one in which we say X
is true so that Y has to be true. A right-hand driven engine is
one in which we say Y is false so X has to be false.
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These are our two valid patterns, often identified with the Latin words
for putting (ponens) and
taking (tollens) and
abbreviated
as MP (for modus ponens) and
MT (for modus tollens).
If our conditional is seen in standard form, we put to the left or we
take from the right.
If Alice studies then she will do
well. She is studying. Therefore, she will do
well. (MP)
If
Alice studied then she would do well. She has not done
well. Therefore, she must not have studied. (MT)
The most common mistakes in working with these patterns (what we call formal fallacies)
are to deny
the antecedent or to affirm the
consequent.
If Alice studies then she will do
well. She has done well, Therefore, she must have
studied. error: affirming the consequent (there could be
other reasons for her doing well)
If
Alice studied then she would do well. She did not study.
Therefore, she must not have done well. error: denying the
antecedent (as above, there could be other things allowing her to do
well)
What is the practical value of
understanding these rules of MP and MT? The first is a reminder
to avoid the very common fallacies of denying the antecedent and
affirming the consequent (put to the left, take from the right--but not
the other way around). The second is to see what might be needed
for what we may call the pattern-perfect case--a deductively valid
argument. For instance, we have the argument that Alice will pass
because she is studying. By itself, just knowing that she studies
will not guarantee her passing (we could imagine--come up with the
counterexample of--an extremely difficult exam), but the closer we can
come to closing the gap between the information given and the idea that
in fact study would be enough (a sufficient condition) the stronger our
case will be . If we also know the exam is very easy we do come
closer.
The following
three links are to material developed at San Jose State and include
interactive drills.
MORE ABOUT CONJUNCTION AND DISJUNCTION
WORKING
WITH CONDITIONALS (1)
WORKING
WITH CONDITIONALS (2)
RECOMMENDED
PRACTICE TEST ON SYMBOLIZATION
This is an optional exercise that may be helpful in
spotting any problems you may have.
PRACTICE DRILLS
These are programs I
developed quite a long time ago as practice
exercises to be downloaded; they will run on a PC but not a Mac.
For this part of the
course download just the second on working with conditionals.