From
Nancy Stanlick, Logic and Language: An
Introduction to the Methods of Analysis, © 1988, 2002.
(2.2)
Additional Translation and Immediate Inference
The three forms of immediate inference
to be discussed are useful in determining the standard form translations of
statements appearing in argumentation as well as in discerning the actual
meaning of statements that have been obscured through the use of negative
terms. In addition to these uses, it is interesting to note the different
ways in which a seemingly simple statement can be worded in such a manner as to
render it more complicated than one may expect.
At any time that one begins adding the
prefixes "non", "in", and "un" to terms, meaning
can be obscured and very difficult to regain without a clear understanding of
the internal structure of the statement. Even a simple statement such as
"this liquid is inflammable" can cause several problems. First,
the prefix "in" sometimes means "not," but in this case it
indicates a capability. This example, of course, shows the capability or
capacity to be set aflame. There are those, however, who take very little
time to understand what they read or hear and mindlessly translate as
"non-flammable" which, of course, has a very different meaning from
"inflammable." In most cases, meaning may be very clear when,
for example, someone is identified as a "non-smoker." If the
concept of "non-smoker" is related to that of being a
"non-drinker" and a new statement is formulated using them, such as
"some non-smokers are not non-drinkers," it may not be obvious that
this statement is equivalent to "some drinkers are not smokers," but
is not equivalent to "some smokers are not drinkers."
There are several charts and methods
for determining the meaning of statements as they are translated to standard
form. Remember that standard form is constituted by A, E, I, and O form
propositions. In order to test properly for validity later in the text,
it is imperative that one be proficient in translation. Propositions out
of standard form are not readily applicable to techniques for testing for
validity.
Conversion
Conversion is a process of standard
translation in which both the quality and quantity of the proposition are
retained, but the position of terms is reversed. As with the other two
types of immediate inference to be presented, it is a simple matter of
equivalent re-statement. In conversion, the same distribution of terms
results; the proposition obtained from the operation is equivalent in meaning
to the given.
Chart
for Conversion[1]
Standard Form Converse
)) DNC
)( )(
() ()
(( DNC
Consider the statement "stoneware
dishes are meant to last a lifetime." Whether this statement is true
makes no difference. The important point is whether there is an
equivalent standard translation. Since the general rule for conversion
consists in reversing the placement of subject and predicate, "all things meant to last a lifetime are
stoneware dishes" is not equivalent to the given. Thus, it has no
standard translation from conversion.
One may argue that there are cases in
which an A form could be converted equivalently. This is true in cases of
definitions or synonymous terms. For example:
All bachelors are unmarried
men
is equivalent to
All unmarried men are
bachelors
but
this truth does not obscure the importance of the lack of equivalence of
statements whose terms are not definitional or synonymous. In general, A
form propositions (of a form that are not definitional), are composed of a
distributed subject and an undistributed predicate. Again turning to the
notion of what constitutes a distributed term, one can see the significance of
the inability to translate an A form proposition. If "stoneware dishes"
is a distributed term, then one is referring to every stoneware dish.
"Things meant to last a lifetime" is, in the A form, undistributed
and, as such, is not referred to completely. If conversion consists of
equivalent re-statement, then the distribution would proceed from
d))u
to
u))d
which
changes the meaning and reference of the terms completely. Thus, A forms
do not convert equivalently.
E and I form propositions, composed of
the same distribution of terms on each side of the copula, do convert equivalently.
"No cats are dogs" is identical in meaning and distribution to
"no dogs are cats." One moves from d)(d to d)(d. There is
no change of distribution or meaning, only a change in the placement of the
terms. The same is true for the I form. To say that "some
horses are thoroughbreds" is the same as "some thoroughbreds are
horses." One moves from u()u to u()u with no change in meaning.
O form propositions, when speaking in
terms of distribution, will not convert equivalently for the same reason that
the A form will not convert equivalently. The distribution of the subject
and predicate is not the same, and to state "some philosophers are not
logicians" is not the same as "some logicians are not
philosophers." One refers only to some philosophers in the given
while in the improperly converted form, one is referring to all of them.
The meaning, therefore, is not identical.
Obversion
It is very common to use the prefix
"non" to describe someone or something not possessing some
quality. Although one usually understands the propositions employing the
use of this prefix, there are times when the meaning of such a proposition is
unclear when it is used in relation to some other sentence or has a place in an
argument.
Chart
for Obversion
Standard Form Obverse
)) )(
)( ))
() ((
(( ()
The term "non", when
appended to any other term, constitutes the formulation of a complementary
class. Complementary classes are constituted by whatever is not included
in the original class. Thus, the complement of "grass" is
"non-grass" and includes everything from engines to
fingernails. The complementary class of grass, however, does not include
"St. Augustine," "zoysia," or "bahia" since
they are all types of grass.
A forms obvert to E forms, Es obvert to
As, Is obvert to Os, and Os obvert to Is, all of which are perfectly
equivalent. Obversion consists of a process of retaining the quantity of
the proposition, but the quality must change since the thing or quality that is
predicated of the subject has received a negative prefix. Consider each
individually using the suggested terms to determine the equivalence of the
propositions. (Remember that "equivalence" means that the two
propositions mean EXACTLY the same thing). For S, substitute
"cats" and for P, substitute "quadrupeds." Remember
again that truth has no bearing on meaning. To make this clearer,
consider the following statement.
The sun
revolves around the earth.
Although
the statement is false, the meaning is clear. Do not let truth and
meaning become a point of confusion. Take the position that meaning is
independent of truth.
Contraposition
Contraposition is a combination of the
inferences conversion and obversion. Propositions that are stated as pure
contrapositives are characterized by complementary classes on each term.
Thus, in the statement "some non-smokers are not non- drinkers", the
standard translation of the proposition is "some drinkers are not
smokers." This can be shown clearly by putting the proposition in
its symbolic form and determining, through conversion and obversion, the
standard translation. I call this method "horizontal" or
"linear" proofs of equivalence. Use "S" for
"smokers" and "D" for "drinkers."
S((D ---> obv ---> S()D ---> conv
---> D()S ---> obv ---> D((S
Rather than to perform all three
operations, i.e., obversion, conversion, and then obversion again, one may
instead use the chart for contrapositives for the O form in which it is clear
that:
S((D
---> contr ---> D((S
For the A form contrapositive,
exemplified by the statement "all non-expensive establishments are
non-French restaurants," one may obtain the statement "all French
restaurants are expensive establishments." The symbolic
transformation of the original is:
E))F ---> obv --->
E)(F ---> conv ---> F)( E ---> obv ---> F))E
OR
)) is
equivalent to ))
through
the chart for the contraposition of an A form proposition.
E and I form propositions will not
contrapose. Take, for example, the following examples as guides.
No
non-signs are non-informative.
Using
conversion and obversion, the reason that the E form will not contrapose should
be clear.
S)(I
---> obv ---> S))I
One
must stop at this point since an A form proposition (represented by S))I) cannot be converted. (Refer back to
"Conversion" if you need to review). Since the A form cannot be
converted, the complementary class appearing in the subject position cannot be
moved to the predicate position in order to apply obversion again. The
same considerations apply to the I form statement represented with complements
appended to both the subject and predicate terms.
In the statement "some non-cubes
are non-circles", the complement in the subject position again cannot be
placed in the predicate position since the result of the first obversion yields
an O form and an O formcannot, of course, be converted. The symbolic
representation appears below:
C)(I
---> obv ---> C((I
CHART FOR
CONTRAPOSITIVES
Standard Form Contrapositive
)) ))
)( DNC
() DNC
(( ((
STANDARD CHART FOR ALL
IMMEDIATE INFERENCES
Standard
Form Conversion Obversion Contrapositive
)) DNC )( ))
)( )( )) DNC
() () (( DNC
(( DNC () ((
EXERCISE IV
The
following problems on immediate inference are meant to familiarize you with the
processes involved in translating a non-standard statement into standard
form. It may have already occurred to you that conversion does not
produce a statement that is non-standard in any way, since "standard"
appears to indicate simply the absence of complementary classes and the
inclusion of standard quantifiers and copulas. But it is not necessarily
the case that a statement is in standard form simply because it contains no complementary
classes. It may be useful at one point or another in working a syllogism
when a statement that can be converted IS converted in order to change the
figure of the argument being considered.
In
each section, use the suggested given proposition as a guide to the order of
propositions in determining standard form translation.
I. Given:
No physicists are carpenters.
1. Some
non-physicists are non-carpenters.
2. Some
non-carpenters are physicists.
3. Some
carpenters are physicists.
4. All
non-physicists are carpenters.
5. All
carpenters are non-physicists.
6. All
non-carpenters are non-physicists.
7.
No carpenters are physicists.
8.
Some non-physicists are not non-carpenters.
9.
Some non-carpenters are not non-physicists.
10. No
carpenters are non-physicists.
II. Use
Symbols ONLY -- Given: P))D
1.
D))P
2.
non-P((non-D
3.
non-P))D
4.
non-D()P
5.
D((non-P
7.
non-P((non-D
8.
D))P
9.
P)(D
10. D))non-P
11. non-P((D
12. non-D)(P
13. P()non-D
14. non-P()non-D
15. D)(P
16. non-D()non-P
17. non-P()D
18. non-P)(D
19. D()non-P
20. non-P))D
21. non-P((D
22. non-P)(non-D
Where
possible, translate the following statements from natural language to symbolic
standard form using whatever method(s) of immediate inference is (are)
appropriate.
1.
Non-existent entities cannot cause changes in physical beings.
2.
No non-authors are writers.
3.
All non-flammable liquids are non-toxic.
4.
All flammable liquids are non-toxic.
5.
All non-flammable liquids are toxic.
6.
No flammable thing is non-toxic.
7.
All except non-believers are moralists.
8.
Almost all physical objects are composed of natural elements.
9.
Only non-members may vote.
10. All
non-members must use the back door.
11. Everything
composed of paper is heavy.
12. All
things composed of atoms are in physical space.
13. Non-material
things are always non-perceptible.
14. Fourteen
cars are in the parking lot.
15. Most
people invited to the meeting attended.
17. Many
of the non-interested parties were able to judge impartially.
18. No
non-metal substances conduct electricity.
19. Non-officers
are prohibited from entering.
20. All
except seeing-eye dogs are prohibited in government buildings.
21. Not
quite all exceptional students attend special classes.