Necessary Assumptions1
GILBERT PLUMER
Law School Admission Council
Abstract: In Evaluating Critical Thinking Stephen Norris and Robert Ennis say:
"Although it is tempting to think that certain [unstated] assumptions are logically
necessary for an argument or position,
they are not. So do not ask: for them."
Numerous introductory logic text authors
and various highly visible standardized
tests (e.g., the LSAT and ORE) presume
that Norris and Ennis are wrong; the presumption is that many arguments have
(unstated) necessary assumptions and
readers and test takers can reasonably be
expected to identify them. This paper
proposes and defends criteria for determining necessary assumptions of arguments. Both theoretical and empirical considerations are brought to bear.
Resume:
Dans leur livre, Evaluating
Critical Thinking, Stephen Norris et
Robert Ennis disent: "Bien qu'on soit
tente de croire que certaines suppositions
[non-prononcees1 sont necessaires
logiquement a un argument ou une position, elles n'en sont pas. Done, n'en
demandez pas." Plusieurs auteurs des
manuels introductoires de la logique et
divers epreuves eta/onnees tres connues
(e.g., Ie LSAT at Ie ORE) presument que
Norris et Eimis ont tort; la presomption
est que plusieurs arguments ont des suppositions
necessaires
[nonprononceesJet que c'est raisonnable
d'attender aux liseurs et ceux qui se
presentent aux epreuves de les cons tater.
Dans cette communication je propose et
defends des criteres pour cons tater les
suppositions necessaires des arguments.
On met en oeuvre les considerations et
theoriques et empiriques.
Keywords: Enthymeme, implicit assumption, missing premise, gap-filler, testing,
principle of charity.
In their book Evaluating Critical Thinking Stephen Norris and Robert Ennis
say: "Although it is tempting to think that certain [unstated] assumptions are
logically necessary for an argument or position, they are not. So do not ask
for them . . . no significant assumptions are logically necessarily made."2
Numerous writers of introductory logic texts,) as well as various highly visible standardized tests, presume without giving much (or any) justification
that the Norris-Ennis view is wrong; the presumption is that many arguments
have (unstated) significant necessary assumptions and that readers and test
takers can reasonably be expected to identify such assumptions. One such
standardized test is the Law School Admission Test (LSA T) (another is the
Graduate Record Examinations (GRE». This paper will propose criteria for
determining necessary assumptions of arguments, criteria that I think are correct in themselves in addition to being appropriate for writers and reviewers of
the kind of test that the LSA T represents: a measure of advanced yet nontechnical reasoning and reading abilities using the format of multiple-choice questions. The account will be defended against the kind of objections Norris and
©In/ormaLLogic Vol. 19, No.1 (1999): pp.41-61.
42
Gilbert Plumer
Ennis raise, among other objections. Both theoretical and empirical considerations will be brought to bear in the course of my argument.
I. Principal Necessary Assumption Criteria
On the LSA T a typical necessary assumption question consists of a short
argumentative passage, a question stem on the order of "The argument depends on assuming which one of the following?", and five answer choices.
Notice that the focus is on the argument, rather than the arguer. Not only is the
former more appropriate for logic (as opposed to psychology), but with no
context other than the brief passage it would be unreasonable to ask the examinee to peer into the mind of the arguer to determine what the arguer is
assuming.
This focus on the argument helps to rule out certain necessary assumption
pretenders. One kind of pretender is that of a presupposition or implication of
a propositional element of the argument, be it a premise or conclusion. The
truth of some proposition p may be a necessary condition for the truth of a
statement s in an argument or a necessary condition for a term in s to have a
referent, etc., but this does not make p a necessary assumption of the argument. One reason is that p is not presupposed by the argument as a whole; a
proposition must be integral to the reasoning or inferential structure of an
argument, not just to a claim made within the argument, in order for the
proposition to be a necessary assumption of the argument (specific examples
will be discussed as we proceed). Another, sometimes related, kind of necessary assumption pretender is that of a general presupposition of rationality.
For instance, that it is possible to do X is not a necessary assumption of an
argument to the effect that X ought to be done, even given that ought implies
can. At best, that it is possible to do X would be a presupposition of a
propositional element of the argument, viz., the conclusion, and the putative
conceptual truth that ought implies can would be a presupposition of something that is so to speak larger than any particular argument, namely, rationality itself (unless, of course, such is the subject matter of the argument). Similarly, neither should we hold that a perfectly general rule of inference such as
Modus Ponens is a necessary assumption of an argument that instantiates
Modus Ponens. If we do, we embark on an infinite regress akin to Lewis
Carroll's (discussed below in §IV). These sorts of considerations function as
some antidote to the view that no proposed set of necessary assumptions of
an argument, if the set is finite, is big enough.4
A criterion that is at least alluded to by virtually every writer on the subject
is the idea that a necessary assumption must somehow fill a 'gap' in the stated
argument's reasoning, thereby making the inference stronger than it otherwise would be. Michael Burke provides a more precise way of stating the
criterion: a necessary assumption is "any proposition which needs to be
Necessary Assumptions
43
true, if the inference is to be justified." However, Burke himself rejects this as
unhelpful on the grounds that there are "infinitely many" such propositions.
"Examples: that there is at least one inference; that there is at least one justified inference; that there is at least one thing that is either an inference or a
polar bear."5 What this fact indicates is not that the gap-filling criterion is
useless; instead, it indicates that the gap-filling criterion needs to be supplemented by other criteria. Because all of Burke's examples are presuppositions
of rationality, or at least of the existence of rationality, none are necessary
assumptions of any particular argument (unless, of course, such is the subject
matter of the argument).
As far as I can see, the most precise and perspicacious way that the subject matter will bear of stating the gap-filling criterion is as follows.
GFC: It could not be the case both that the argument aside from the
proposed necessary assumption is fully cogent and the proposed necessary assumption is false.
In the GFC the central meaning of the clause that 'the argument aside from
the proposed necessary assumption is fully cogent' is that given the truth of
what I will call the received premises (i.e., all of the implicit and explicit
premises, considered apart from the proposed necessary assumption), the
conclusion can be inferred to be true. The standard of an allowable inference
here is to be taken as varying in stringency according to whether the argument
is deductive (in which case validity is the standard) or nondeductive. Of course
a nondeductive argument may have a conclusion that is in fact false, yet it still
be reasonable to infer that it is true from the truth of the premises. Cogency is
perhaps a loose evaluative concept, but I take it to pertain only to an argument's reasoning or logic, not also to the truth value of its propositional elements (unlike the technical concept of soundness). And in no case is cogency
purely a matter of formal validity. For example, adding the stated conclusion
or the contradictory of a stated premise to the stated premises would make
any argument formally valid. But the argument would lack cogency insofar as
it grossly begs the question or engages in self-contradiction. In order to be
fully cogent, it seems an argument must not commit any informal fallacy (that
pertains to its reasoning).
I would hope that the rationale behind the GFC is fairly clear. Suppose the
GFC is violated. That is, suppose that the proposed necessary assumption is
false, and given the truth of the argument's received premises-premises that
might, but need not, include an alternative to the proposed necessary assumption-the conclusion can still be inferred to be true in a fully cogent way. Then
surely, the proposed assumption does not figure in the argument's reasoning;
for the proposed assumption does not have to be presumed to be true in order
to so-infer that the argument's conclusion is true. Hence, it is not in fact a
necessary assumption of the argument.
44
Gilbert Plumer
Notice that the GFC does not deny the logical truth that it could be the case
both that an argument as a whole is fully cogent and an assumption of it is
false. Unlike the GFC, this logical truth involves the notion that if all of the
argument's premises, including the assumption, were true, the conclusion
could be inferred to be true. Similarly, the question stem in Example I (next
section) says, in effect, 'if the argument as a whole is fully cogent, then it
must be assuming which one of the following?, (and Example 3's question
stem is just shorthand for 1's); it does not say 'ifthe argument as a whole is
fully cogent, then which one of the following must be true?'. However, the
point of the GFC is that the most critical and heuristically useful part of what
the question stem means here is 'if the explicitly given argument is fully cogent, then which one of the following must be true?'. The GFC thereby treats
a necessary assumption as an 'inference license'. The general idea is that the
same proposition may be treated both as an implicit element of an argument
that as a whole is fully cogent and as a fact that helps to justify the inference
in a stated argument that itself is fully cogent.6
There is apparently no better way of determining necessary assumptions.
Consider the following criterion:
NAC: A proposition is a necessary assumption of an argument if
and only if it could not be the case both that the argument is fully
cogent and the proposition is not included as a member of the argument's premise set.
While this criterion would not overtly need to be taken in conjunction with
other criteria, unlike the GFC, in fact it does not get us substantially beyond
the question stem in Example 1. The NAC does clearly indicate that cogency
is what assuming the proposition is necessary for. But circularity is a problem
since the members of an argument's premise set are its explicitly stated premises
and its implicit premises (if any), and a premise that is determined to be implicit by logical means just is a necessary assumption.
The reader is warned that there will be little further theoretical treatment of
the concept of cogency itself. A cogent argument is an argument that exhibits
good reasoning, as determined by informal and, in the case of deduction,
formal standards. It is not the point of this paper to elaborate in general terms
on what makes an argument a good one, in this familiar sense. Indeed, it is
hard to see how any mere paper could have this as a topic, since it is the
defining subject matter ofthe discipline of logic. 7 Of course in elaborating and
defending my criteria for determining necessary assumptions I will have to
wield a concept of a good (cogent) argument. Yet I do not think that my
concept contains anything particularly controversial or unusual. Of course if
your concept differs from mine in some reasonable respect, you may make
different reasonable judgments in the application of the criteria to individual
Necessary Assumptions
45
cases. That is to be expected. But what is more important is the structure, not
the details of how it is applied. Besides, logic is hardly distinguished among the
various fields of philosophy and other humanities by its lack of intersubjective
agreement. The very opposite is true.
II. Application to LSAT Examples
Example 1(6191 LSAT)**
Train service suffers when a railroad combines commuter and freight
service. By dividing attention between its freight and commuter customers, a railroad serves neither particularly well. Therefore, if a
railroad is going to be a successful business, then it must concentrate exclusively on one of these two markets.
For the argument to be logically correct, it must make which one of
the following assumptions?
(A) Commuter and freight service have little in common with each
other.
(B) The first priority of a railroad is to be a successful business.
*(C) Unless a railroad serves its customers well, it will not be a
successful business.
(D) If a railroad concentrates on commuter service, it will be a
successful business.
(E) Railroad commuters rarely want freight service as well.
P-H
Example II (6191 LSAT)**
It is even more important that we criticize democracies that have
committed human rights violations than that we criticize dictatorships that have committed more violent human rights offenses.
Human rights violations are always inexcusable, but those committed by governments that represent the will of the people are even
more reprehensible than those committed by dictators. Further, our
criticism is more likely to have an effect on the former than on the
latter.
Which of the following is the proper inference from the passage?
46
Gilbert Plumer
(A) All governments commit some inexcusable and reprehensible
acts.
*(B) Some human rights violations are more reprehensible than other,
more violent human rights violations.
(C) Criticisms of human rights violations is certain to have no effect on a dictatorship.
(D) Human rights violations are more likely to occur in democracies than in dictatorships.
(E) Those who do represent the will of the people are less likely to
be moved by criticism than are those who merely claim to
represent the will of the people.
Example III (2/92) LSAT)**
The brains of identical twins are genetically identical. When only
one of a pair of identical twins is a schizophrenic, certain areas of
the affected twin's brain are smaller than corresponding areas in the
brain of the unaffected twin. No such differences are found when
neither twin is schizophrenic. Therefore, this discovery provides
definitive evidence that schizophrenia is caused by damage to the
physical structure of the brain.
Which of the following is an assumption required by the argument?
(A) The brain of a person suffering from schizophrenia is smaller
than the brain of anyone not suffering from schizophrenia.
* (B) The relative smallness of certain parts of the brains of schizophrenics is not the result of schizophrenia or of medications
used in its treatment.
(C) The brain of a person with an identical twin is no smaller, on
average, than the brain of a person who is not a twin.
(D) When a pair of identical twins both suffer from schizophrenia,
their brains are the same size.
(E) People who have an identical twin are no more likely to suffer
from schizophrenia than those who do not.
**Copyright © 1991, 1992 by Law School Admission Council
Let us consider the examples from actual LSA Ts in more detail. Optiori (C)
in Example I is (I claim) the necessary assumption in a fairly straightforward
case of Hypothetical Syllogism, a deductive pattern. (C) pretty clearly satisfies all the criteria discussed thusfar. And specifically with respect to the
GFC, consider that the conclusioin in the argument claims that if a railroad
Necessary Assumptions
47
combines commuter and freight service (-r), it will not be a successful business (-:-p). Why? The stated grounds are that combining commuter and freight
service leads to a railroad's not serving its customers well (-q). Ifnot serving
its customers well did not in turn mean that a railroad will not be a successful
business, i.e., if(C) were false, then even given the truth of the stated premise,
the conclusion could not be inferred to be true-it could be either true or false.
This is not a matter of claiming that アセイ@
.'. セイ@
is invalid or that MセアIL@
アセイ@
.'. ーセイ@
is invalid, although these claims are true. Nor is it a matter of
claiming that MHーセアI@
logically or strictly implies MHアセイIーL@
which is
a false claim. Rather, the claim is that if (C) were false, there would be no
logically acceptable conclusion-warranting link between a problem that the
argument describes as arising from a railroad's combining commuter and
freight service, viz., not serving its customers well, and the question of whether
a railroad will be a successful business:
In addition to being (I claim) necessary, (C) in Example I is of course also
a sufficient assumption. It wholly fills the gap in the argument's reasoning.
Certain other statements would be sufficient but not necessary, e.g., the broader
statement formed by substituting 'a business' for "a railroad" in (C); such a
statement would go beyond providing everything implicit that the given argument depends on for full cogency. However, nothing in the context provided
by the passage would justify attributing this greater generality to the argument; the passage is about railroads, not businesses in general. Indeed, nothing in the context would justify attributing any assumption other than (C) to
the argument. To take another sort of case, suppose (C) said 'unless a railroad
serves its customers well and stays in business for at least twenty years (s), it
will not be a successful business' セHア@
& s). To attribute this proposition
to the argument as an assumption would be either to make the given argument
lack cogency (which is contrary to the point of determining assumptions) or
to construct a different (albeit cogent) argument, since a requirement for a
railroad's being a successful business, viz., s, would be imported 'out of the
blue' into the context. Perhaps the clearest way of seeing why (C) alone
satisfies the GFC is to consider the fact that every other sufficient assumption
of the argument (with the exception of degenerate cases such as the conclusion or the associated conditional ofthe explicit argument-these will be discussed later) implies (C), but none are implied by (C).8
Intuitively, it seems that option (B) in Example 2 is clearly not a necessary
assumption of the passage's argument. This is hardly because (B) is a presupposition of rationality generally or of a single propositional element of the
argument. Giving an account that at once both supports the sort of intuition
we have in a case like (B) here and supports holding that the credited responses in cases like Examples I and 3 are necessary assumptions is perhaps
the most interesting and vexing problem to be encountered in dealing with
argument assumptions. (B) in Example 2 follows from the conjunction of the
48
Gilbert Plumer
first and second sentences in the passage, which are the argument's conclusion and (first) premise, respectively (the passage's third sentence is a premise
that is independent of the first premise). (B) follows from these two sentences
considered as independent statements, that is, in abstraction from the fact that
they are related as premise and conclusion. This means that (B) has nothing to
do with the argument's reasoning, i.e., its inferring the first sentence from the
rest. (B) rectifies no lacuna in the argument's reasoning; it makes no contribution to the argument's cogency. (It may be worth remarking that the overt
existential commitment of (B) derives from the argument in the passage through
the fact that it could not be important to criticize democracies for certain
actions unless those actions existed.)
A principal virtue of the GFC is that in general it rules out contingent (i.e.,
neither necessarily true nor necessarily false) implications or presuppositions
of any propositional element of an argument or conjunction ofthese elements;
Example 2 (B) is a case in point. (In considering this it is to be understood that
these elements are taken as independent propositions; they are taken in abstraction from any associated implicit or explicit argument-indicator terms
such as 'thus' and 'since'. Otherwise, an implication could very well be a
necessary assumption: the GFC gives a relatively clear meaning to the notion
of being implied by an argument qua argument, i.e., without the abstraction
performed. 9) Since by definition such implications are already part of the content of the argument, the argument could quite well be fully cogent without reintroducing them into the argument individually as further members of its
premise set. Such implications violate the GFC in the following way: Although
if such a proposed necessary assumption is false the conjunction of the argument's conclusion and received premises is also false, this does not rule out
the possibility that if the received premises were true the conclusion could be
inferred to be true in a fully cogent way. Just as it is a logical truth that it could
be the case both that an argument as a whole is fully cogent and the conjunction ofits propositional elements is false, it is a logical truth that it could be the
case both that an argument, considered apart from such an implication, is
fully cogent and the implication is false. (Of course an exception arises from
the curiosity that every proposition logically implies itself. If it is already established of something that it is a necessary assumption of an argument and is
hence a propositional element of the argument, this curiosity does not mean
that the proposition is not a necessary assumption of the argument.)
However, the GFC is not so powerful as to preclude any arbitrarily selected necessary truth from being counted as a necessary assumption; the
same applies, for example, to the proposition that there is at least one fully
cogent argument. So we still need, respectively, the criteria that a necessary
assumption not be an implication of a propositional element of the argument or
a general presupposition of rationality (assuming that not all general presuppositions of rationality are necessary truths). This points to a limitation of the
Necessary Assumptions
49
account: it cannot be used to determine any necessarily true necessary assumptions, as of an argument that has other necessarily true propositional
elements, e.g., a mathematical argument. This is just a c;onsequence of a socalled 'paradox' of logical implication-a necessary truth is implied by any
proposition whatsoever. As far as I can see, the only alternative for determining such assumptions is the NAC (above). As for arguments that have necessarily false propositional elements, insofar as they would lack cogency in
virtue of having such elements, it is not clear that these arguments could even
have necessary assumptions (where cogency is what assuming the proposition would be necessary for).
In Example 3, a clear case of nondeduction, I interpret the causal relationships propounded by the argument to be that damage to the physical structure
of the brain is the common cause of both schizophrenia and the relative smallness of certain parts of the brains of schizophrenics. This is how the argument explains the "discovery," i. e., the apparently perfect correlation of schizophrenia and the relative smallness, after in effect ruling out a genetic explanation. Option (B) contributes to the argument's cogency in an essential way by
ruling out two further salient alternative explanations of this correlation. Ifthe
negation of (B) were the case, then (at least) one of these alternative explanations would be true, and it would fully explain the discovery or evidence
without any need to appeal to a damaging mechanism, contrary to the claim
made through the argument-indicator term "Therefore, this discovery provides definitive evidence that." So the GFC is satisfied. Note that (B) is not
merely an implication of the statements in the passage since these statements,
considered in abstraction from the fact that they constitute an argument (excise "Therefore, this discovery provides definitive evidence that"), are entirely
compatible with the negation of (B): the damage causes schizophrenia, and
schizophrenia or the medications used in its treatment in tum cause the smallness. All of this presumes that the damage and the smallness are not the same
phenomenon; if they were the same phenomenon, then (B) would not be a
necessary assumption ofthe argument. (B) would simply be a presupposition
or implication of the statement in the argument that "schizophrenia is caused
by damage to the physical structure of the brain," i.e., the argument's conclusion.
(B) in Example 3 has the logical form -(t v u). Does the GFC or the
criterion that a necessary assumption not be a presupposition or implication of
a propositional element of the argument mean that -t and -u each are not
necessary assumptions? If -(t v u) is taken to be a necessary assumption,
then of course neither -t nor -u is also a necessary assumption-the argument could quite well be fully cogent without either being included as a further
member of its premise set. Conversely, if -t and -u are each taken to be
necessary assumptions, then of course -(t V u) is not an additional necessary
assumption. We can take either sort of approach in determining necessary
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Gilbert Plumer
assumptions, but we cannot take both. Detennining necessary assumptions
should not be confused with proving validity, which by definition involves the
application of theorems and rules of inference such as those that fonn the
backdrop of the present discussion, i.e., De Morgan's, Simplification, and
Conjunction.
Nor, it seems, is it a necessary assumption of the argument (e.g.) that the
relative smallness is not the result of visitors from outer space picking out
schizophrenics and shrinking parts of their brains with some ray gun. Allowing such propositions as necessary assumptions-propositions that are insufficiently relevant to the evident concerns of the argument (which have to do
with science rather than science fiction)-would to some significant extent
make the argument lack cogency. So they could hardly be necessary for the
argument's cogency. If indeed the scenario envisioned is one of science fiction, then the proposed necessary assumption violates the GFC in that it could
very well be the case both that the argument aside from this proposition is
fully cogent in a scientifically respectable way and yet this proposition is 'false' ,
i.e., the alternative explanation is true only in science fiction. Moreover, the
set of propositions that individually rule out questionably salient alternative
explanations would be open-ended or infinite, and the set would be fonned by
applying a rule of Universal Instantiation to a supposed assumption in the
argument to the effect that no other explanation is true. But this would amount
to treating an argument that is manifestly nondeductive as deductive.
ITI. Narrowness Objection
It might be thought that no putatively necessary assumption (n) of an argument is actually necessary because there is always an alternative candidate
(c)-a narrower version of n, or the explicit argument's associated conditional or even its conclusion-that logically would serve just as well as an
assumption. It might appear that the existence of c makes it the case that n
violates the GFC as follows: n could be false, yet if the received premisestaken as including c-were true, the conclusion could be inferred to be true.
Let us first examine the question of narrowness. Ennis says:
The basic idea is that any incomplete argument can be completed in a
number of different ways. Consider the simple argument, "Since Mike is
a dog, Mike is an animal." To suggest the range of possible gap-fillers, I
shall note two, the one that first occurs to most people [note this admission] and another one:
1.
All dogs are animals.
2.
All dogs whose name begins with "M" are animals.
A person is not logically prohibited from offering the argument, denying
1, and using 2 as a gap-filler. This shows that 1 is not logically necessary
for the offering of the argument. So long as the offered gap-filler is more
general than a statement establishing connection between the offered
Necessary Assumptions
51
premise and conclusion, one can deny the offered gap-filler and derive
the conclusion from the offered premise by using a narrower gap-filler.
The limiting case is the simple statement that the offered premise was
assumed in the circumstances to be sufficient to establish the conclusion. 1O
Though this view has its appeal, I think that it is highly misleading. It is a
principle of rationality that similar cases are to be treated similarly. There is
nothing in the context to indicate that Mike is in any way special with respect
to being a dog. Qua dog, Mike is an arbitrarily selected individual. In logic the
injunction to treat similar cases similarly is the rule of Universal Generalization
(UG). The necessary (and sufficient) assumption here, 1 (x)(Dx-+Ax) , is
determined by the fact that UG applies to the stated argument's inference
(Dm-+Am). Therefore, it does not seem too strong to hold in this case that
both "offering the argument" and "denying 1" would be self-contradictory.
This means that 1 satisfies the GFC: it could not be the case both that the
argument aside from 1 is fully cogent and 1 is false. 1 satisfies the other
criteria proposed above for determining necessary assumptions of arguments
if we presume, as Ennis apparently does (p. 75), that 1 is not a necessary
truth. If I were taken to be a necessary or conceptual truth such that the
concept of animal is involved in the concept of dog, then it would not be at all
clear that the stated argument has any gap that needs filling. II
Notice that the argument "Since Mike is a dog, Mike is an animal" with 2
as the supposedly sufficient assumption (Ennis supposes it 'completes' the
argument) is invalid. It would be valid if it had the more specific or narrower
claim 'Mike is a dog whose name begins with "M'" as the explicit premise
(such dogs constitute a proper subset of all dogs). Indeed, if this were the
explicit premise, 2 would be the (unstated) necessary and sufficient assumption. The general principle operating here is that specificity that restricts the
inferences licensed by the assumption is introduced by the narrower explicit
premise, and the case becomes similar to that of Example 1, as contrasted to
a more general argument that has as a necessary assumption (C) with its "a
railroad" replaced by 'a business'. Yet in the original argument ("Since Mike is
a dog, Mike is an animal") and context the only evidentiary claim made about
Mike is that he is a dog. So what follows from this about Mike (that he is an
animal) follows for any dog. Similarly, as Irving Copi puts it, "In the geometer's proof the only assumption made about ABC is that it is a triangle, hence
what is proved true of ABC is proved true of any triangle."12 Again, this is just
UG at work.
Govier writes: "Consider:
Abortion is wrong because it's killing.
What does this sentence, which encapsulates an extremely simple-minded
argument, entail? Does it entail that all killing is wrong? ... that most killing is
wrong? ... that killing something unborn is wrong? I don't know what the
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Gilbert Plumer
answer is, and if somebody else does, I wish he'd tell me how he worked it
In the absence of any further context other than that provided by the
argument itself, and being faithful to the 'extreme simple-mindedness' of the
argument, the answer is that through UG the argument 'entails' or has as a
necessary assumption that all killing is wrong. If this were not the case, it
would be irrelevant to object to the argument by pointing out that just because
an action involves killing, that does not make the action wrong-after all, for
example, killing a murderous home invader in self-defense does not appear to
be wrong. A cost of adopting an Ennis-like view on necessary assumptions is
a severe restriction on the legitimate use of counterexample and refutation by
logical analogy.
Ennis appears to hold that aside from what he calls "The limiting case,"
there is always a logically viable narrower alternative to any putatively necessary assumption that renders it unnecessary. I have just argued that this view
is false, mainly by arguing that the specificity of the context and explicitly
given argument determines the specificity of the necessary assumption; so if
the former is in certain respects general, the necessary assumption is accordingly general. Let us now consider "The limiting case."
OUt."13
IV. Associated Conditional Objection
Presumably, "The limiting case" is the explicit argument's associated or corresponding conditional; in Ennis' example Dm :.Am it is Dm-?>Am. Ennis indicates that "an arguer is committed to at least this minimal claim" (p. 83),
although such a claim is apparently only an 'insignificant' necessary assumption (as I quoted Norris and Ennis at the beginning of this paper). So the view
I wish to consider, one that is suggested by Ennis' remarks, is that no putatively necessary assumption of an argument is actually necessary, unless it is
already the explicit argument's associated conditional, since that conditional
itself would make the argument fully cogent.
This view has the consequence that every natural language argument that
is enthymematic with respect to its explicit premises is construable as an
instance of pure Modus Ponens, which I think is absurd. Although in formal
logic there are systems that incorporate only one rule of inference such as
Modus Ponens,14 this hardly means that this sort of system could be an adequate representation of fundamental argument structure in ordinary discourse.
In Example I it is difficult to conceive of the whole argument in natural language where the assumption is taken to be the associated conditional of the
argument stated in the passage. It is not at all clear that the whole argument
would be fully cogent; it seems quite trivialized. From my point of view this is
no surprise: the whole argument is a paradigm case of an everyday Hypothetical Syllogism. The same kind of remarks are perhaps even more convincing with respect to Example 3, where there is the added specter of treating an
obviously nondeductive argument (to the best explanation) as deductive. Logic
Necessary Assumptions
53
is in part a study of arguments as they actually appear in ordinary discourse.
Numerous distinct patterns of reasoning or regularities in argument structure
have been identified. To allow this, yet hold that all ordinary arguments that
are enthymematic with respect to their explicit premises could exhibit the
single pattern, Modus Ponens, seems inconsistent. Why should recognizing
the. pattern of reasoning in such an enthymematic argument be so radically
different from pattern recognition in other endeavors where the explicit or
presented pattern is only partial (e.g., continuing a numerical sequence or
recognizing a friend in a photographic profile)? The patterns identified are
hardly all the same in any of these latter endeavors. In such a way as this at
least, the credited responses in cases like Examples 1 and 3 are assumptions
that are 'logically necessarily made'.
Let P stand for the conjunction of an argument's explicit premises; C
stand for the argument's conclusion (taken to be explicit); and I stand for the
conjunction of the argument's implicit premises or necessary assumptions, if
any. That an argument has the form P, I :. C rather than simply P:. C by
itself provides no particular reason for identifying I with pセcL@
i.e., the explicit argument's associated conditional. The only thing that could warrant
this identification would be some aspect of the argument's context, such as
that Mike is somehow a special, nonarbitrarily selected dog in the case of
Dm :.Am. In that case the whole argument would be the pure Modus Ponens
dュセaL@
Dm:. Am, rather than as I am construing it, viz., HゥクIdセaL@
Dm:. AmY But instances of pure Modus Ponens wherein the ュセッイ@
premise
implicit) must constitute the exception and not
is implicit (P:. C, with pセc@
the rule, especially if we take, as many do (see below), the implicit premise
here to simply replicate or be "reiterative" of the implication relationship of the
explicit argument. This reiteration would make the argument self-referential,
and the self-referential case's being exceptional is to be expected in light of the
numerous paradoxes of self-reference (e.g., the liar paradox concerning the
claim 'this statement is false', and Russell's paradox, which concerns the set
property of being self-membered). I!> The rule, on the other hand, would have
to be that it is in fact incoherent to identify I with pセcZ@
That an argument is
enthymematic with respect to its explicit premises means that the conclusion
is not implied by these premises alone; more premise material is needed in
order to properly infer the conclusion-that is precisely how the argument is
enthymematic or "incomplete." This is to say pセc@
only if SI v ... S,,-where
S, v ... S" is the disjunction of the possible sufficient assumptions of the
So if none of
enthymematic argument, none of which is identical to pセcN@
Sr"S" were acceptable, neither would pセc@
be acceptable. (The criteria proposed in this paper are meant to be usable to determine which of SrS" is
necessary, i.e., is identical to I.)
It is therefore at least very misleading to affirm the widespread view that
"we may add the reiterative candidate [i.e., add pセc@
to an argument that has
54
Gilbert Plumer
the form P, I:. C]. This is always unobjectionable, since the acceptability of
the reiterative candidate is a necessary condition of the acceptability of the
argument." (The view does correctly go on to say that this adding is "never
useful, in regard to facilitating evaluation, since it is never easier to evaluate
the reiterative candidate than to evaluate the corresponding inference.")17 We
have just seen the problem with identifying I with P 4C. Yet if these are not
identified, then adding P4C would at least make I superfluous with respect to
drawing the conclusion, so in what sense could I be a set of implicit premises
or necessary assumptions? I think the view is confused about why the
enthymematic argument or arguer is, as Ennis puts it, "committed" to P4C.
It is not because P:. C is stated and P4C just 'reiterates' the implication
relationship of this statement. Instead, as a rule it is because the argument's
being enthymematic means that it has an implicit premise set I(:t:P4C), and
that P, when conjoined with I, implies C «P & 1)4C). By Commutation and
Exportation this is equivalent to 14(P4C), and of course there is commitment to I; as in Example I, p4Q implies (Q4r)4(p4r). (It seems that these
points apply to nondeductive arguments where '4' is read as, e.g.,
'probabilifies'.) However, even though there is commitment to P4C, P4C is
not a further necessary assumption of the argument precisely because it is
implied by what is already, by supposition, a propositional element or conjunction of such elements of the argument, viz., I.
Some may even think that one can freely add the associated conditional
(P4C) to nonenthymematic arguments. If this is done with the justification
that "all arguments depend upon the 'assumption' that you can get from their
specific premises to their specific conclusions,"18 then one has embarked on
Lewis Carroll's infinite regress, which in the example Carroll presents l9 has
the form:
A; (\ix)(\iy)(\iz)«(x = z) & (v = Z»4(X = y»
B: (a = c) & (b = c)
C: (A & B)4Z
D: (A & B & C)4Z
Z: a = b
The vertical margin dots represent an infinite series of recursive iterations in
the manner of premises C and D.
V. Charity
Let us briefly consider how charity enters the picture. To attribute Ennis'
"gap-filler" 2, "All dogs whose name begins with 'M' are animals," to the
argument "Since Mike is a dog, Mike is an animal" would violate any reason-
Necessary Assumptions
55
able principle of charity. This is not merely because this "gap-filler" would
make the argument invalid; it is more generally because the attribution would
come totally' out of the blue'. There is simply nothing in the barren context
presented to justify such an attribution. The attribution of 2 would make the
argument as a whole very strange-how could the character of a dog's given
proper name have anything to do with the dog's being an animal?
It was noted in §II that necessary truths trivially satisfy the GFC. The
other half of the story is that any proposed necessary assumption of an argument that is wholly or irredeemably fallacious will trivially satisfy the GFC,
simply because the argument could not be fully cogent and remain the same
argument. The avoidance ofthis trivial satisfaction constitutes one reason it is
crucial to see that in the very act of attributing a proposition to an argument as
a necessary assumption (in the sense explained above) one presumes that the
argument is cogent, and thereby interprets it charitably, to at least some extent. Cogency is what assuming the proposition is necessary for; the proposition fills a gap in the explicit argument's reasoning, thereby making the reasoning stronger than it otherwise would be (although it could still be weak or
fallacious in other respects). A necessary assumption contributes to an argument's cogency, and a necessary and sufficient assumption completes the
argument so that it is fully cogent. This means that it would be incoherent to
treat as necessary and/or sufficient assumptions of arguments such propositions as the argument's stated conclusion or the contradictory of a stated
premise. For this would be to uncharitably treat the argument as irredeemably
fallacious in grossly begging the question or engaged in self-contradiction,
even though such candidates would make the argument formally valid. Moreover, therefore, one cannot maintain that any putatively necessary assumption
is rendered unnecessary by the fact that such an alternative candidate logically
would serve just as well as an assumption.
It seems that for arguments that are enthymematic with respect to their
explicit premises, the question of which are cogent enough to have necessary
assumptions and which are wholly fallacious is a matter that is determined by
the application of a reasonable principle of charity. It is true that attempts have
been made to determine this matter solely by examining features of the explicit
arguments themselves and not invoking a principle of charity.20 A starting
point is the proposition that a gross nonsequitur does not have any content
expression or nonlogical constant in common between the conclusion and the
stated premises (and such an expression could not be produced by making
definitionally equivalent substitutions). However, the attempts have been limited to deduction, and even there they seem unsatisfactory.2!
Govier presents what I regard as an adequate and insightful theory of
argumentative charity. After reviewing the literature, Govier proposes and
defends in detail a moderate principle of charity (that takes its cue from H.P.
Grice's "Logic and Conversation"):
56
Gilbert Plumer
We presume, other things being equal, that others are participating in
the social practise of rational argumentation. . .. They are operating
within the purpose of the exchange: that is, it is their purpose to communicate information, acceptable opinions and reasonable beliefs, and to
provide good reasons for some of these opinions and beliefs by offering
good arguments. '" The basis for charity is to be found not in ethics,
prudence, or epistemology, but in the nature and purpose of the activity
in which participants are engaging: argumentative discourse. That people are doing this when they appear to be is a rebuttable presumption .
. . , Iffor one reason or another, the presumption would not be appropriate-the people lacking all credibility, or the context being one where
people seek persuasion at any cost-then there is no reason for approaching the discourse charitably-not even moderately charitably.22
In ordinary situations this principle can be effectively applied because the
context of the enthymematic argument is rich enough to determine whether
the arguer is engaged in the social practice of rational argumentation. However, with respect to short passages that appear on an examination like the
LSAT (and other comparably attenuated situations), the matter is often indeterminate. Then it is up to the test writer to somewhat arbitrarily decide whether
to develop the rest of the test item as a necessary assumption question or as a
question that asks the examinee to identify a way in which the argument is
fallacious; Example 3 is a case of this. From the point of view of examinees
the charity issue is already decided for them by which of these two sorts of
questions they confront.
VI. Theory Summary
To summarize, in determining necessary assumptions of arguments one first
applies the principle of charity; only ifthis yields the result that the argument
is not wholly fallacious does one apply the other criteria, viz., that a necessary
assumption must not be a presupposition of rationality generally or of a
propositional element of the argument, and that it must satisfy the GFC. I
think these criteria are appropriately used by writers and reviewers of a test
like the LSA T to determine or check the credited response in a question that
asks for a necessary assumption of an argument. This is not to say, however,
that kinds of cases like what have been identified above as 'necessary assumption pretenders' could legitimately be used as noncredited responses in
such test questions. The ordinary concept of an assumption seems too vague
or amorphous to reasonably permit, for instance, a presupposition or implication of an argument's stated conclusion as a noncredited response.
VII. Empirical Considerations
So much for the theory. How have the principles articulated fared in practice?
It is fairly rare that a philosophical theory can be (even indirectly) tested against
Necessary Assumptions
57
systematic empirical data, but with some qualification this is the case here.
Consider Table 1:23
Percent Rejected
Item
Subtype
Number
Pretested
For Statistical For Content
Reasons
Reasons
Percent
Identified as
Marginal
Assumption
288
6
18
16
Inference
350
8
24
13
Flaw
271
7
18
13
Evidence
388
5
18
9
Table ].
The column farthest left includes names of the three test item subtypes that
are most akin to the Assumption subtype. All four subtypes appear within the
"Logical Reasoning" (informal logic) sections of the LSA T. One qualification
is that the Assumption category includes a minority of sufficient assumption
test items; they ask of a given enthymematic argument, e.g., "The conclusion
above follows logically if which one of the following is assumed?" Test questions of the Inference subtype ask the examinee to identify allowable inferences from given passage material; Example 2 is an item of this subtype.
Flaw questions typically ask the examinee to identify a way in which an argument is fallacious. In Evidence questions the task is to determine the logical
effect of additional or independent evidence--whether it supports or undermines-a given argument, conclusion, or hypothesis, etc.24 The Number Pretested column gives the number of test items of the subtype that were pretested from 6/91 to 2/96. Before any test item is used in a scored section of an
LSA T exam, it appears in an unscored section of a previous LSA T; this is
known as "pretesting." The purpose is to determine the item's psychometric
statistical characteristics so that if these are acceptable, the item can later be
incorporated according to specification into a section that will be scored. The
next column gives the percentage of the test items of the subtype that were
pretested and thereby found to be statistically unusable on the basis of a threeparameter Item Response Theory modeL The three parameters are measures,
roughly speaking, of (a) how well the item discriminates among examinees of
differing ability , (b) how difficult the item is, and (c) the probability of examinees of very low ability answering the item correctly, perhaps by guessing.2S
The next column gives the percentage of the test items of the subtype that
58
Gilbert Plumer
were rejected in a content review conducted subsequent to pretesting (most
content reviews are performed prior to pretesting). Usually, the reason for
rejection is that the item was found to have more than one correct answer on
a certain interpretation or to have no correct answer that is clearly defensible.
The column farthest right gives the percentage of the test items of the subtype
that were pretested and thereby found to be statistically marginal, primarily by
using the various measures of Item Response Theory, as well as Classical
theory26 (e.g., how well performance on the item correlates with performance
on the test section as a whole). There is some overlap between the test items
that were rejected for statistical reasons and those that were rejected for content reasons, and between those identified as marginal and those that were
rejected for content reasons.
The point is, if a view about necessary assumptions like Norris and Ennis'
were correct, one might very well expect that the performance of the Assumption SUbtype would be significantly worse than that of the other subtypeskinds of questions to which they raise no comparable objections. Indeed,
Norris and Ennis say, "the word 'necessarily' [or an equivalent] in the direction would at least slow down some good critical thinkers and force them to
waste time. It might also force them to omit the item or guess wildly" (p.
123). But I have seen no evidence of this in the nine years that I have participated in developing and applying the kind of criteria advocated in this paper to
the LSA T; as Table I indicates, content and psychometric statistical analyses
of necessary assumption test questions yield results that are as good as those
for other comparable kinds of questions designed to assess reasoning abilities. 27
Another consideration is that aside from the case of a special administration of the LSA T, every examinee receives a copy of the LSA T test questions
along with her or his scored answer sheet. This convenience leads to careful
scrutiny of the test questions, if for no other reason than that law schools
assign a great deal of weight to the candidate's test score in the admission
process. There is an established procedure under which examinees can challenge test questions and written defenses of test questions, including asking
for a review by a panel of outside experts. 28 If this panel rules against the test,
then an expensive process of re-scoring and issuing new score reports is
initiated (a relatively rare circumstance in which philosophical theory is concretely and publicly accountable). And developing any standardized test item
is an expensive process anyway. So if necessary assumption test items were
challenged, let alone successfully challenged, proportionately more often than
other kinds of questions designed to assess reasoning abilities, or if they performed significantly worse in content or statistical review, there would be a
strong incentive to simply not use them. But we do. This context of developing a standardized reasoning and reading test 'has provided an organized setting in which actually to engage in the Goodmanian-Rawlsian project of
Necessary Assumptions
S9
endeavoring to achieve a "reflective equilibrium,"29 where theoretical criteria
or principles on the one hand, and particular judgments (about arguments) on
the other, are brought into coherence with one another, and thereby each is
afforded justification.
Notes
Earlier versions of this paper were presented at Law School Admission Council, and at the
Third International Conference on Argumentation at the University of Amsterdam, The
paper has benefited from discussion on these occasions and from written comments by
Rod Bertolet, Anton Bures, Robert Ennis, David Kary, Deborah Kerman, and Brian
MacPherson,
2(Pacific Grove, CA: Midwest Publications, 1989), pp. 122-23. Cf. Ennis' "Identifying
Implicit Assumptions," Synthese 51 (April 1982), pp, 82-83. Ennis is also the principal
author of the Cornell Critical Thinking Tests and is very influential in the field of critical
thinking education,
3For example: Michael Scriven, Reasoning (New York: McGraw-Hili, 1976), e.g., pp. 43,
83, 166; Irving Copi, Introduction to LogiC, 5th Edn. (New York: Macmillan, 1978), p.
242; Robert Fogelin, Understanding Arguments, 3rd Edn. (New York: Harcourt Brace
Jovanovich, 1987), pp. 120-21; David Kelly, The Art of Reasoning (New York: W.W.
Norton & Co., 1988), p. 100.
4This view is apparently held by (e.g.) Trudy Govier, Problems in Argument Analysis and
Evaluation (Dordrecht: Foris, 1987), pp. 92-94; Wayne Grennan, "Are 'Gap-Fillers' Missing
Premises?," Informal Logic 16 (Fall 1994), p. 190.
Interestingly, after the criteria for determining necessary assumptions discussed in this
paragraph were developed, an LSAT examinee challenged a question on the grounds that
the statement "the credited response refers to is not an assumption but something that is
true by definition ... [It is] a given that is inherently true." (The statement in question was
not actually a definitional truth.)
S"Unstated Premises," Informal Logic 7 (Spring & Fall 1985), p. 114.
6Compare A. Sloman's conclusion that "any analysis of a particular argument in terms of a
suppressed premiss may be replaced by an equally general analysis in terms of a rule of
inference and vice versa." "Rules ofInference, or Suppressed Premisses?," Mind 73 (Jan.
1964), p. 88. (Cf. Scriven, op. cit., pp. 83, 178; David Hitchcock, "Enthymematic Arguments," Informal Logic 7 (Spring & Fall 1985), esp. pp. 94-95; Timothy Smiley, "A Tale
of Two Tortoises," Mind 104 (Oct. 1995), esp. pp. 730-32.) However, because such a
proposition must contain at least one content expression (given the criterion that a necessary assumption not be a general presupposition of rationality), insofar as it is a rule of
inference at all, it is a nonformal or specific rule of inference. Contrast, e.g., Modus
Ponens, which as topic-neutral is a formal or general rule of inference.
7But see, e.g., John Slaney, "A General Logic," Australasian Journal of Philosophy 68
(March 1990), pp. 74-88; Trudy Govier, "What is a Good Argument?," Metaphilosophy
23 (Oct. 1992), pp. 393-409.
Note that in this paper I shall also not be directly concerned with the matter of how
one, in the first place,jintiY necessary assumption candidates (that can then be subjected to
my criteria). For a procedure that is designed to help with this and that embodies a version,
albeit a narrower one, of the GFC, see Mike Donn, "Help in Finding Missing Premises,"
Teaching Philosophy 13 (June 1990), pp. 159-64.
I
60
Gilbert Plumer
8Another degenerate case is r. The assumption of r would make the stated argument
formaJly valid. It has actually been objected to this paper that only through an ad hoc
judgment about the cogency of arguments can r be ruled out as a rival to (C). However, the
:. ーセイ@
read with the passage's verbal substitutions is incoherent. This
argument r, アセイ@
entirely superis probably because the assumption of r renders the stated premise HアセイI@
follows formally simply from r),
fluous in deductively drawing the conclusion Hーセイ@
making the argument commit a gross fallacy of irrelevance. I should like to insist that
making such cogency judgments is an everyday occurrence that is well-grounded in the
defining subject matter ofthe discipline oflogic, and hence is hardly ad hoc.
'1C! the view of David Hitchcock. According to Trudy Govier, Hitchcock's view is that in
order to be a necessary assumption a proposition "must follow 'deductively and reasonably immediately' from the combination ofthe stated premises, the stated conclusion, and
the 'drawing ofthe conclusion from the premises'." "What is a Good Argument?," op. cit.,
p. 401, which summarizes Hitchcock's "Methodological Deductivism," presented at the
Eastern Division meetings of the American Philosophical Association in Dec. 1986.
lO"Identifying Implicit Assumptions," op. cit., p. 82.
\lC! Govier, Problems in Argument Analysis and Evaluation, op. cit., pp. 96-97: We need to
distinguish "formal deductive inference" from "substantive deductive inference. (Substantive deductive inference depends on meaning rather than on form.) ... Nor should a
necessary truth which relates the meanings of terms used in an argument already deductively valid without the addition of such a truth ... be regarded as a missing premise." Also
see, e.g., Hitchcock, "Enthymematic Arguments," op. cit., pp. 85-86.
120p. cit., p. 357.
13"What is a Good Argument?," op. cit., p. 404.
14E.g., system R.S. in Irving Copi, Symbolic Logic, 5th Edn. (New York: Macmillan, 1979),
p.228ff.
lsIn my construal, dュセa@
is not also a necessary assumption of the argument since it is an
Recall the
implication of a propositional element of the argument, viz., HvクIdセaN@
point in §II that determining necessary assumptions should not be confused with proving
validity.
16For discussion, see, e.g., Graham Priest, "The Structure of the Paradoxes ofSeif-Reference," Mind 103 (Jan. 1994), pp. 25-34.
17Burke, op. cit., pp. 108, 116. See also Scriven, op. cit., pp. 84, 163-64; Hitchcock,
"Enthymematic Arguments," op. cit., pp. 86, 89; Michael A. Gilbert, "The Enthymeme
Buster: A Heuristic Procedure for Position Exploration in Dialogic Dispute," Informal
Logic 13 (Fall 1991), p. 160; Dale Jacquette, "Charity and the Reiteration Problem for
Enthymernes," Informal Logic 18 (Winter \996), pp. 3, 8. The idea that in a pure Modus
Ponens an implicit major premise simply 'reiterates' the implication relationship of the
explicit argument is subject to the following kind of objection: The implication relationship
between the premises and conclusion of the argument is strict or tautological implication,
whereas the implication between the antecedent and consequent of the major premise may
be something less, such as causal implication or truth-functional (material) implication.
This point raises hoary issues that cannot be addressed here, such as how adequate a
representation of natural language is the purely truth-functional account of conditionality.
18Scriven, op. cit., p. 84.
19"What the Tortoise Said to Achilles," Mind 4 (April 1895), pp. 278-79.
2°Rolf George, "Enthymematic Consequence," American Philosophical Quarterly 9 (Jan.
1972), pp. 113-16 and "Bolzano' s Consequence, Relevance, and Enthymemes," Journal of
Philosophical Logic 12 (Aug. 1983), pp. 298-318; Hitchcock, "Enthymematic Arguments,"
op. cit.
Necessary Assumptions
61
21Tomis Kapitan, "A Definition of Enthymematic Consequence," International Logic Review 9 (1980), pp. 56-59; Hitchcock, ibid.; Govier, "What is aGood Argument?," op. cit.,
p.402n9.
22 Problems in Argument Analysis and Evaluation, op. cit., p. 150.
23Source: Law School Admission Council statistical databases.
24For more discussion of the kinds of questions that appear on the LSAT, see the LSATI
LSDAS Registration and Information Book, which is updated yearly and is available from
Law School Admission Council, P.O. Box 40, Newtown, PA 18940.
23 A standard reference work is F. M. Lord, Applications ofItem Response Theory to Practical
Testing Problems (Hillsdale, NJ: Lawrence Erlbaum, 1980).
26A standard reference work is F. M. Lord and M. R. Novick, Statistical Theories ofMental
Test Scores (Reading, MA: Addison-Wesley, 1968).
27The Norris-Ennis alternative to asking for the identification of necessary assumptions,
since they think that no significant assumptions are actually necessary, is to ask of several
options "which is probably taken for granted or assumed" (p. 124). Such a question would
be seriously unclear: the assumption is probable relative to what? Certainly, the given
argument. But why not also psychological facts about arguers and facts about relevant
aspects of the world in general? Such questions would appear to take a large step in the
direction of achievement (knowledge) testing, as opposed to the intended aptitude testing
for a particular reasoning skill (assumption recognition).
28In a review of a recent challenge and defense of a necessary assumption test question, a
panel member (a philosopher of language and logic) wrote: '''depends on' is hardly a
technical term requiring training in logic ... The Law Services response that (A) is the only
assumption on which the argument depends is correct. (A), and only (A), is such that if we
presume it to be false the argument is undercut." Note the language of the GFC.
29NeJson Goodman, Fact, Fiction, and Forecast, 4th Edn. (Harvard University Press, 1983),
part m.2; John Rawls, A Theory ofJustice (Harvard University Press, 1971), p. 20ff.
Gilbert Plumer, Senior Test Specialist-Logical Reasoning
Law School Admission Council
661 Penn Street, Box 40, Newtown, PA 18940-0040 U.S.A.