Translating Logical Operators
As we've seen, logicians have
devised a simplified set of five basic logical operations—not, and,
or, if, and iff—to make it relatively easy to analyze complex
lines of reasoning. However, plain language has lots of other
operators: also, although, because, before, both, but, either, else,
except, however, instead, likewise, neither, notwithstanding, only,
otherwise, since, so, therefore, thus, unless, until, when, whenever,
and while, to tick off some of the most common. If symbolic logic is to
be of any practical use, we must be able to interpret each of these
operations in terms of our five standard logical operators.
AND
Plain language contains a host of words and
expressions logically equivalent to the logical conjunctive operator:
also, although, besides, both, but, despite, even so, furthermore,
however, in addition, in spite of, likewise, moreover, nevertheless,
nonetheless, not to mention, notwithstanding, regardless, still, though,
yet (and probably others). Each of these carries its own
nuance about a relationship that's certain or chancy, that's crucial or
merely coincidental, that works in concert or at cross purposes.
Yet in the final analysis they all distill to the affirmation of two (or
more) statements.
OR
There are far fewer synonyms for the logical
disjunctive operator, and some of these are simply variations to clarify
grouping of multiple operations: either...or, except when, on the other hand, or
else, otherwise, unless, until. Either...or and or
else sometimes (but
not always) suggest an exclusive sense of OR.
IF
Although a conditional operation is very finicky
about its logical structure, there's a great deal of leeway in how it
can be expressed in plain language. In the real world, it turns
out that
"THEN" is optional; in many cases it's understood from
plain-language context. Also, the order of
a condition's IF and THEN clauses can be swapped (so long as the IF
operator remains with the antecedent): "IF p THEN q," "IF p, q,"
and "q IF p" are all logically equivalent. And in the
only known exception to the "IF stays with with the antecedent" rule, it can also be expressed
as "p ONLY IF q." In this case, notice that
the IF ends up with the consequent of the condition—but only if
the operator is ONLY IF (or ONLY WHEN), because ONLY has the effect of
reversing the usual sense of IF! (Note: ONLY IF should never be confused with
"if only," which usually expresses a wishful sentiment rather than a
logical relationship.)
Common plain-language stand-ins for
IF include because, due to, since, so long as, and when.
(Since is equivalent to IF only when its meaning is similar to
because; when used instead to indicate passage of time—e.g., "He's
lived there since he was discharged from the army"—in such cases
since becomes a synonym for AND, not for IF.)
The trick is deciding which of the
clauses associated with the operator ought to be the antecedent—the IF
clause of the condition—and which the consequent. Any of these
terms might suggest a cause-and-effect relationship, some—because
and due to—more strongly so than others. Of course, when we
think of cause and effect, we usually think of an action and its result,
or consequence. Thus, we might tend to assume that the consequence should always be the consequent
of the condition, and that the cause should therefore always be the
antecedent. However, such an assumption would lead us into error
much of the time. That's because nature runs on the physical
causes of events themselves, whereas logic is often focused on
the causes of our knowledge of events—something else entirely.
Consider: we know that clouds cause
rain.* However, not all clouds produce rain, and those that do
don't produce it all the time; so we can't conclude that it must be
raining whenever there are clouds overhead. On the other hand,
we're quite certain that all rain comes from clouds. Therefore, we
can be just as certain that whenever it's raining there must be clouds
overhead, even if we can't see the sky because of trees or an
overhanging roof. It's this "whenever...must" relationship that
governs the assignment of events to the antecedent and consequent
clauses of a logical condition. Specifically, the "whenever" event
belongs in the antecedent, while the "must" belongs in the consequent,
regardless of which (if either) is the physical cause or which (if
either) happens first.
*That clouds cause rain
isn't strictly accurate. Rain clouds are simply a visual product
of the mass condensation of water vapor into airborne droplets, and rain
is the result of these droplets' falling once they've attained a certain
mass. Still, the example serves our purpose well enough as a
commonplace illustration of a concept.
IFF (IF AND ONLY IF)
A previously noted feature of conditional relationships
logically gives rise to bi-conditional operations. Specifically,
q IF p
is equivalent to p
ONLY IF q.
Ordinarily, the p
and q
clauses of a condition are not interchangeable. But in some cases
it isn't always one that depends on the other. Rather, both are
mutually interdependent; if the truth value of either changes, then the
truth value of the other must change
also. When we come across such a relationship, we can express it as "(p IF q) AND (p ONLY IF q)"—or
to make it a bit more concise, "p IF AND ONLY IF q"—and
to symbolize it: p ↔ q.
There are few (if any) plain-language equivalents for IF AND ONLY IF, so
logicians have adopted IFF as an expedient measure to save pencil lead
and avoid writer's cramp.
But why should we trouble ourselves
at all over an operation if it's seldom used? Actually, IFF isn't
as rare as it might seem. For example, consider: "The president
uses the veto if and only if he thinks a bill is bad." If we split
this bi-condition into two ordinary conditions, we get: "The president
uses the veto if he thinks a bill is bad," and "The president thinks a
bill is bad if he uses the veto." Obviously, the meanings of the
two statements are somewhat different, but they're both valid and
complementary conditions. And if both meanings are intended, then
they can be fully expressed only by the bi-condition, not by either
condition alone. Moreover, if the truth value of either statement
changes, then the truth value of the other changes as well: "The
president doesn't use the veto if and only if he doesn't think a bill is
bad."
One reason we don't hear "IF AND
ONLY IF" very often is because it doesn't accurately represent the
majority of conditional relationships. But another reason is
simply that "IF AND ONLY IF" is quite a mouthful. Most people are
either in too much of a hurry or just too lazy to bother with it; so
they use plain old IF instead, even though it's inaccurate.
Indeed, they're so rushed or so lazy they even use IF when they actually
mean ONLY IF. Consider, "You retain driving privileges IF your
grades are satisfactory." Obviously, driving privileges could also
be denied for reasons (e.g., traffic violations and accidents) other
than poor grades; what is actually meant is "You retain driving
privileges ONLY IF your grades are satisfactory." But while haste
and laziness might seem acceptable motives to butcher plain-language
thinking, they aren't acceptable excuses in logical analysis. If
we don't get the intended true meaning right, the analysis will be
flawed and our efforts wasted.
NOT Just
about any statement that contains a hardly, in no case, can't, don't,
hasn't, isn't, mustn't, needn't, shouldn't, won't, neither, never, no, or none could, with a little manipulation, be
interpreted as the simple negation of a thought. NOT is the only logical
operator that can be applied to one statement alone. However, NOT
can also negate operations and groups of thoughts, and is often used in combination with other operators
to perform a variety of functions.
Hybrid operations
So
much for the stuff that can be directly translated into one of the five
basic logical operators. What about oddball relationships, like
instead of and neither...nor, which don't appear to
correspond directly to any basic operation? These might require a
little head-scratching; and perhaps a truth table would be handy for
testing whether any hybrids we propose are actually accurate reflections
of the corresponding plain-language expressions. Here's a list of
some of the most common expressions that don't have direct counterparts
among the basic operations, along with their symbolic functional
equivalents.
|
PLAIN LANGUAGE |
|
BASIC EQUIVALENT |
|
SYMBOLIC |
|
ALTERNATIVES |
|
either p or q, but not both |
|
NOT IFF p THEN q |
|
~(p ↔ q) |
|
~p ↔ q |
|
p ↔
~q |
|
either p or q,
or neither, but not both |
|
NOT p OR NOT q |
|
~p
˅ ~q |
|
~(p * q) |
|
|
|
neither p nor q |
|
NOT
p AND NOT q |
|
~p * ~q |
|
~(p
˅
q) |
|
|
|
p
instead of q |
|
p
AND NOT q |
|
p * ~q |
|
~(~p
˅
q) |
|
|
|
p otherwise q |
|
IF NOT p THEN q |
|
~p → q |
|
~q →
p |
|
|
|
p
consequently / hence /
so / therefore / thus q |
|
p,
AND IF p THEN q |
|
p * (p → q) |
|
(p → q)
* p |
|
p * (~p
˅
q) |
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