`This paper is reprinted from Enrichment Study 12, in Ron Yezzi,
`*Practical Logic*` (Mankato: G. Bruno & Co.,
1992). Copyright 1992 by Ron Yezzi`

`The distinction between deduction and induction based upon
whether conclusions follow necessarily or probably from their
premises has achieved nearly universal acceptance among writers
on logic. It has become the standard distinction in introductory
textbooks. Still I find the distinction unsatisfactory for several
reasons.`

`First of all, a disturbing lack of parallelism exists between
deduction and induction by which the standards for establishing
inductive conclusions are more demanding than those for deductive
ones. In deduction we only need to consider the logical form of
the argument; whereas, in induction, we must also add information
about the world to show that the conclusion follows with some
degree of probability. For example, in induction by simple enumeration,
we need to assess the fairness of the sample in terms of our knowledge
of the actual world in order to show that the conclusion probably
follows from the premises. In a modus ponens argument,
on the other hand, we need only consider the logical form present.
This particular lack of parallelism leads to the "`

`Secondly, instructors in introductory logic courses ordinarily
stress the need to evaluate arguments first in terms of the strength
of the conclusion `*relative*` to the premises.
Accordingly, students may be told to assume that premises are
true. This procedure has the beneficial effect that it enables
them to study the basic structure of arguments without needing
to possess expert knowledge about the world. It seems however
that we can seldom assess the strength of inductive arguments
without possessing such knowledge. On the one hand, we seem to
be saying that expert knowledge of the world should not be relevant
to assessing the relation between premises and conclusion; on
the other hand, we seem to be saying that such knowledge is relevant,
in inductive logic. If we respond by saying that we can study
the structure of inductive arguments without assessing the probabilities
of premises relative to conclusion, we are left with the possibility
that there exists some underlying understanding of inductive arguments
aside from the definition in terms of probability.`

`And thirdly, the current distinction between deduction and
induction leads to cases where the same argument should be considered
both deductively and inductively. After all, an invalid deductive
argument may still be a good inductive one, if the premises establish
a high degree of probability for the conclusion. Consider the
following argument that commits the fallacy of an undistributed
middle term:`

All individuals with normal or superior IQs are human beings

John Doe is a human being

_______________________________________________________________

John Doe is an individual with normal or superior IQ

`This fallacious deductive argument would have to be regarded
as a good inductive one, when we assess the degree of probability
attaching to the conclusion, given the knowledge that individuals
with normal or superior IQs constitute 75% of the class of human
beings. Or consider`

All individuals with normal or superior IQs are human beings

All college students are human beings

__________________________________________________________________________

All college students are individuals with normal or superior IQs

`Now this fallacious deductive argument becomes an even better
inductive one with a probability approaching closer to certainty,
given the knowledge that nearly all college academic standards
are sufficiently demanding to require normal or superior IQs.
Although no inconsistency arises in asserting that the same argument
is deductively invalid but still an excellent inductive argument,
the situation is troubling when we consider the degree of stress
we place on students' being able to detect the flaw in syllogistic
arguments possessing an undistributed middle term. We can even
go a step further and produce an apparent inconsistency. Consider`

All human beings are mammals

All Greeks are mammals

____________________________________

All Greeks are human beings

`What happens when we assess the inductive probability of this
invalid deductive argument? If we set up a Venn diagram, then
there are three possibilities for the relation between Greeks
and human beings--namely, the Greeks can be in region 1, 2, or
both. That is why the argument is deductively invalid. But we
now want to proceed `*inductively*`. So we want
to assess the probabilities attaching to these various possibilities.
Now if we think about the meaning of "`*Greeks"
*`and "`*human beings*`," we see
that all the Greeks are inside the class of human beings. That
is, "`*All Greeks are human beings"*`
is an analytic statement. Thus, `*inductively*`,
the conclusion follows necessarily from the premises. But if the
conclusion follows necessarily, we can label it a valid deductive
argument and then face an inconsistency. Note carefully that the
conclusion follows necessarily here through the process of inductive
assessment, not simply because the statement in the conclusion
is analytic. (In other words, if the conclusion here were "`*All
triangles are three-sided figures*`," then the conclusion
would not follow.) It is tempting to deny the inconsistency by
saying that additional information is being introduced that alters
the original argument. But how can the degree of probability in
an inductive argument be determined without introducing additional
information? To eliminate appeals to additional information is
equivalent to eliminating any inductive argument not based upon
some calculable probability set up among the premises themselves.
What all this shows are the difficulties encountered when we define
induction in such a way that standards for assessing the argument
depend upon something other than the logical form of the argument
itself.`

`Likewise, for the fallacy of affirming the consequent, we
may end up with a strong inductive argument. Consider this example
where the fallacy occurs:`

If she won the New York State lottery sweepstakes, then Jane Doe--the law-abiding, Utica, N.Y. welfare mother with two small children, a limp, a lisp, and no other relatives or wealthy friends--has become a multi-millionairess

Jane Doe--the law-abiding, Utica, N.Y. welfare mother with two small children, a limp, a lisp, and no other relatives or wealthy friends--has become a multi-millionairess

__________________________________________________________________

She won the New York State lottery sweepstakes

`Arguments of this form are fallacious because the same consequent
can have more than one antecedent. Suppose however that we consider
the argument inductively by trying to assess the probability that
the conclusion follows. We should recognize that the properties
of Jane Doe very likely exclude the most common, plausible antecedents
other than the one stated in the first premise. For example, given
those properties, it is unlikely that she gained the money through
illegal activities or the signing of a professional contract as
an athlete or entertainer. Consequently, this argument seems to
be inductively strong. Although no inconsistency arises, we can
question whether logic instructors are prepared to stress the
need for detecting the fallacy of affirming the consequent only
to point out, however, that the argument can still be quite strong.
And we can question whether they are prepared to show the ways
in which all the standard, fallacious deductive arguments
are to be evaluated inductively. Given the current definition
of induction in terms of probability, it is not at all clear why
logic texts treat, say, argument by simple enumeration and argument
through agreement in a discussion of induction rather than selecting
the standard fallacious deductive arguments instead. One may appeal
to tradition; but the current definition of induction is not traditional.
So far as I know, these considerations do not appear in any current
introductory logic text. If they did appear, there would be a
significant risk that the distinction between correct and incorrect
reasoning, a basic purpose for logic, would become quite confusing.
At the very least, there would be serious doubts raised about
the importance of deductive logic.`

`I think that there is a way out of these difficulties. We
can establish a parallelism between deduction and induction so
that both kinds of argument can be studied strictly in terms of
their logical form. We can apply the important distinction between
validity and soundness to induction as well as deduction. We can
eliminate any need to consider fallacious deductive arguments
inductively because of the current definition of induction. And
we can redefine deduction and induction in a way more closely
resembling the traditional definitions that go at least as far
back as Aristotle.`

`We can accomplish all this by taking a simple step in induction,
equal in kind with what we do for deduction. We can make an appropriate
assumption that is equal in kind to two assumptions we regard
as legitimate in deduction, when testing for validity, namely,
(1) assuming the truth of the premises and (2) assuming uniformity
of meaning for component terms or symbols.`

`The first assumption requires no explanation here (although
I will add some comments later in a discussion of the definition
of validity). The second assumption, however, (because of the
particular way in which I want to discuss it) does require some
explanation, because it is so obvious that we never state it.
The same term or symbol need not retain the same meaning throughout
an argument. For example, the author of an argument can insert
a rule that, for every successive, even-numbered instance of a
term or symbol, its meaning becomes the negation of its immediately
preceding meaning. According to the rule, `

A implies B

B

____________

A

`becomes a modus tollens argument. Although nothing
forbids insertion of such a rule, it is so silly, confusing, and
unnecessary that we safely assume that the authors of arguments
do not insert it. Accordingly we assume uniformity of meaning.`

`What, then, are the appropriate assumptions that assure the
validity of inductive arguments? (a) In `*enumerative induction,*`
we should assume that the premises constitute a `*fair
sample*`. (b) In `*eliminative induction*`,
we should assume that the premises contains `*all the conditions
possibly relevant*` to the predicate in the conclusion.
(c) In `*analogical argument*`, we should assume
a `*perfect*` analogy in the first premise. Once
we make these assumptions and reduce the three kinds of arguments
to standard form, the conclusions follow necessarily from their
premises. Moreover, in making these assumptions, we are not really
adding external premises since the assumptions are internal to
the meaning of the argument. Thus, in simple enumeration, assumption
of a fair sample is a precondition for the conclusion generated.
For example, scientists would not conclude that "`*Water
boils at 100 degrees C., under standard conditions"*`
unless they presume that the instances of boiling water described
in the premises constitute a fair sample, that is, one that is
representative of the whole of which it is a sample.`

`Making these assumptions, we will assert the validity of some
arguments quite at odds with our knowledge of the world. But we
should not confuse validity with soundness. Just as there can
be silly, but valid, deductive arguments, there can be silly,
but valid, inductive arguments.`

`Making these assumptions, we may seem to be sweeping the problems
in establishing inductive conclusions under the rug. But such
is not really the case. We are merely setting aside problems unrelated
to the logical form of the arguments, just as we set aside the
issue of the actual truth or falsity of premises in testing the
validity of deductive arguments. After we settle the validity
issue, we can assess the degree of approximation to a fair sample,
the completeness of the statement of possibly relevant conditions,
or the degree of approximation to a perfect analogy.`

`We can make this new treatment of induction correspond even
more closely to the validity-soundness distinction by broadening
the concept of `*soundness*` so that it encompasses
`*the practical reliability*` of a valid argument,
not just the truth or falsity of its premises. Accordingly, determining
soundness for the three kinds of inductive argument mentioned
above would include an assessment of (a) the representativeness
of a sample, (b) the completeness of the stated, possibly relevant
conditions, or (c) the strength of the analogy--in addition to
testing for validity and establishing the truth of the premises.
A section dealing with this broadened concept of soundness--that
is, with general ways of establishing soundness--would be a valuable
addition to an introductory logic text or course.`

`This new treatment of induction creates a problem of definition
however since the distinction between deduction and induction
in current use no longer applies. Moreover, we cannot simply reinstitute
the traditional distinction--whereby deduction goes `*from
general to particular*`, and induction `*from particular
to general*`--because of well-known counter examples. Still,
there is a way out, I think. We can retain definitions based on
the direction of arguments, but without injecting the terms `*general*`
and `*particular*`. There is a difference between
`*deductio,*` `*a leading down*`,
and `*inductio,*` `*a bringing in*`.
In deduction, we seem to begin with statements that function as
principles and we argue from them. In induction, we seem to begin
with statements that do not function as principles but rather
lead to them in conclusions. We are dealing here with a fairly
subtle distinction regarding the significance we assign to various
statements in arguments. Within the present context, a `*principle*`
is a basic assertion for purposes of argument and explanation.
(We can easily see how general statements can function as principles.)
In deduction, the premises fit the concept of a `*principle*`
quite well. They are the originating points for the argument and
the argument proceeds from them. In induction, however, a strange
reversal occurs. Although the premises are the originating points
of the argument and the argument proceeds from them, we attach
greater significance to the conclusion. The premises seem to function
merely as tools to produce a conclusion that we can then use as
an originating point for further argument and explanation. That
is, `*deduction*` is argument leading `*from*`
principles; and `*induction*` is argument leading
`*toward*` principles. These definitions avoid the
obvious counter examples that create problems. They can accommodate
the standard deductive and inductive arguments--although, admittedly,
some troubling cases will probably arise. Also, according to these
definitions, mathematical induction and complete, or perfect,
induction would once again be classified as inductive arguments--after
having been labeled as deductive, based on the necessary-probable
distinction.`

`Enumerative induction fits nicely within the new definition
of induction. And eliminative induction does, too, once we recognize
that the argument leads from a state of less definite premises
toward a more definite conclusion. Analogical argument however
does not fit well--since the first premise establishes a definite
analogy from which the conclusion follows. Accordingly, analogical
argument would now be classified as deductive.`

`This new treatment of deduction and induction would also have
further, more far-reaching effects. (1) A unified symbolic logic,
making use of both deductive and inductive arguments, becomes
possible. And (2), the traditional problem of induction dissolves
or, at least, becomes a misnomer, once validity is attributable
to inductive arguments. Just as there is no traditional problem
of deduction, there is not one of induction either.`

`These suggestions would require some substantive, but not
especially intrusive, revision of introductory logic texts. There
is, I think, nothing to be lost and a great deal to be gained.`

`There are several problems arising due to ways of discussing
induction and validity in other logic texts, which warrant some
comment. `

**Objection 1:**` Do we really assume premises
to be true in deductive arguments?`

`According to a number of introductory logic texts, an argument
is valid if and only if it is impossible that the premises are
true and the conclusion false. So an argument can be valid even
though it has false premises. Therefore, apparently, we do not
assume the truth of premises in deduction. This definition brings
the general meaning of validity into accord with the truth-functional
definition of validity in symbolic logic.`

**Reply:**` This definition of validity however
is not universally adhered to. Indeed, when text writers introduce
the concept of validity, their primary concern is not so much
this definition in itself as it is the importance of logical form
in arguments and the need for a necessary connection between premises
and conclusions. They just want to render the actual truth or
falsity of the premises irrelevant in determining the validity
or invalidity of the argument.`

`Although the truth-functional definition of validity serves
a useful function in symbolic logic, we need not regard it as
expressing the real meaning of validity--just as we need
not regard the definition of material implication as expressing
the real meaning of implication. Put differently, I am
just saying that validity need not be defined so that false premises
can produce valid arguments.`

`In traditional logic at least, we can render the actual truth
or falsity of premises irrelevant in considering the validity
of arguments quite well by assuming that the premises are true.
This procedure conforms easily to the basic nature of an argument,
where the premises are regarded to serve an assertive function,
which then warrants a conclusion.`

**Objection 2:**` If it is a characteristic mark
of inductive arguments that the conclusion goes beyond what the
premises assert, how can a conclusion ever follow necessarily?`

`It is often pointed out that conclusions are implicit in their
premises for deductive arguments, so that the arguments produce
no new knowledge. The ampliative character of induction, by contrast,
provides a way to extend our knowledge. This ampliative character
is a primary virtue of inductive arguments.`

**Reply:**` If we make the appropriate assumptions
for induction mentioned earlier in the paper, however, we have
a way of getting conclusions that follow necessarily. If we thereby
deprive induction of its ampliative character, we should not take
this result as a demerit against inductive arguments--given the
well known success and importance of deductive arguments. Even
the `*concept*` of ampliative argument is suspect,
when we remind ourselves that it suggests getting something from
nothing or, more precisely, that its conclusion claims more than
its premises are capable of producing. When we want to extend
our knowledge, it is sufficient to combine the techniques for
establishing soundness (as treated in this Enrichment Study) with
the techniques for arriving at valid arguments. `

**Objection 3:**` If it is a characteristic distinction
between deduction and induction that additional premises added
to a valid deductive argument cannot affect its validity whereas
added premises can affect the strength of inductive arguments,
then how can one say that an inductive conclusion follows necessarily?`

`In induction by simple enumeration we can see how an enlarging
of the sample in the premises can affect the probability that
the conclusion follows. But this cannot happen in valid deductive
arguments because, regardless of additional premises, the conclusion
always remains implicit in the initial premises.`

**Reply:**` Given our new assumptions however,
any problem here disappears. If we `*assume a fair sample*`
in simple enumeration, then no additional premise is going to
deny the necessity with which the conclusion follows from the
initial premises. Any added premise will concur with the initial
ones. If someone asserts that an added premise can establish that
the assumption of a fair sample was incorrect, then we need to
note that, for deductive arguments likewise, an additional premise
can establish that the assumption of true premises was incorrect.
For example, suppose that we add the premise, `*Three human
beings are immortal*`, to this argument:`

All human beings are mortal

All college students are human beings

_____________________________________________

All college students are mortal

`So either both deduction and induction have a problem here
or else neither does. I favor the latter alternative, given the
ways logicians treat validity.`

Comments? yezzi@vax1.mankato.msus.edu

(Note: For years I've maintained a policy of not listening to anonymous telephone callers and of not reading anonymous letters addressed to me. Before reading an e-mail message offering comments, I first satisfy myself that I know the identity of the sender and I then test to make sure that I can send a reply. So if you want me to do more than admire the ingenuity of your e-mail name, be sure to identify yourself and to make a dialogue possible.)

Last updated 12/3/95