www.TheLogician.net © Avi Sion - all rights reserved
© Avi Sion
All rights reserved
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
The 'scientific method' consists in trying out every conceivable
imaginary construct, and seeing which of them keep fitting all new facts, and
which do not. Those which cease to fit, must be eliminated (or at least
corrected). Those which continue to fit, are to that extent increasingly
probable, until they in turn cease to fit. Whatever theory alone survives this
eliminative process, is effectively proved, since all the shares of probability
have been inherited by it.
In practise, the construction of alternative postulates, and the
discovery of the full implications of each, are both gradual processes. We do
not know these things immediately. Also, the given context is not static, but
itself grows and changes as we go along. This feeds our imagination and insight,
helping theory developments, and stimulating further research.
We may start with one or two partially developed theories, and slowly
find additional alternatives and make further predictions, as events unfold and
the need arises. The extent of our creative and rational powers affects the
exhaustiveness of our treatment.
Several theories concerning some group of phenomena may, at any stage in
the development of knowledge, simultaneously equally fulfill the criteria of
relevance; namely, conceptual meaningfulness, internal consistency, ability to
explain the phenomena in question, and compatibility with all other empirical
givens so far.
In formal terms, this simply means that competing theories T1, T2, T3,…
may, while being contrary to each other, each still logically imply the already
experienced phenomena Q. That is, the hypotheticals 'if T1, then Q', 'if T2,
then Q', etc., are formally compatible, even though 'T1 or else T2 or else
T3…' is true.
The statement that our list of theories for Q is exhaustive, has the form
'If T1 or T2 or T3… , then Q', plus 'one of T1, T2, T3… must be true'.
Although it may be hard to prove that our list is exhaustive, we may
contextually assume it to be so, if every effort has been expended in finding
the alternative explanations.
Each theory contains a number of postulates: T1 = p11 + p12 + p13 +…,
T2 = p21 + p22 + p23 +…, and so on. Some of these postulates might well be
found in more than one theory; it may be, for instance, that p13 = p29 = p36.
But each theory must have at least one distinctive postulate or a distinctive
combination of postulates, which makes it differentiable from all the others.
Also, the phenomenon or group of phenomena labeled Q are already known
empirically, and supposed to be equally embraced by the various theories put
forward. But each theory may have other implications, if we can determine them
through reason, open to empirical testing, though not yet tested.
Each theory has a set of predictions: T1 = q11 + q12 + q13 +…,
T2 = q21 + q22 + q23 +…, and so on. Some of these must be in common,
constituting the given phenomena Q which gave rise to our theorizing in the
first place. That is, say, Q = q15 = q27 = q31.
The rest may likewise be all identical, one for one; or some overlaps may
occur here and there, while some predictions found here are missing there; or,
additionally, some conflicting predictions may occur, so that one or more
theories affirm some prediction that certain other(s) deny.
In principle, it is conceivable that the various theories all make only
the same predictions, in which case they are factually indistinguishable, and we
cannot choose between them on an empirical basis, though we may still refer to
Most often, however, we may eventually find distinctive further
predictions for each theory, or at least some which are not common to all. A
difference in postulates usually signifies a difference in predictions. Here, we
must be careful to differentiate between:
a prediction implied by, say, T1, but neither implied nor excluded by T2,
T3, etc. — if such a prediction passes the test of experience, T1 is
confirmed, but T2, T3,… are neither confirmed nor rejected, though their
probabilities are diminished by the increased probability of T1; whereas if such
a prediction fails the test of experience, T1 is rejected, while T2, T3,…
become more probable by virtue of being less numerous than before; and:
a prediction implied by, say, T1, and logically excluded by T2, T3, etc.
— if such a prediction turns out empirically successful, T2, T3… are
rejected, and (if only T1 is leftover) T1 is proved; whereas if such a
prediction turns out empirically unsuccessful, T2, T3,… are confirmed by their
anticipation of the negative event, while T1 is rejected.
Thus, theory selection depends on finding distinctive predictions, which
can be used in adductive argument or apodosis. These should be empirically
testable predictions, of course.
If one or more theories have an implication which the others lack, though
are compatible with, or if one or more theories have an implication which the
others are incompatible with — we have at least an eventual source of
divergent probabilities, allowing us to prefer some theories over others, even
if we cannot eliminate any of them; and in some cases, we may be able to
eliminate some of them, and maybe ultimately all but one of them.
These methods are of course well known to scientists today. But all this
concerns not only scientists at work, but the development of opinions by
individuals in every domain. It is the 'trial and error' process through which
we all learn and improve our knowledge.
Even if at a later stage we might manage to validate some of our beliefs
more deductively and systematically, this is the method we usually use to
initially feel our way to them and develop them. Knowing the 'scientific method'
explicitly and clearly can help individuals to make their personal thinking on
topics remote from abstract science more scientific.
We have described the ideal pattern of scientific evaluation of theories;
but, in practise things are not always so neat, and we often have to make do
with less than perfect intellectual situations.
For a start, the coexistence of conflicting theories may be viewed less
generously as a source of doubt for all of them; they may each be corroborated
by the delimited data they explain, but their mutual incompatibility is a
significant inconsistency in itself.
We may remain for years with equally cogent, yet irreconcilable theories,
which we are unable to decide between. Our minds are often forced to function
with a baggage of unresolved contradictions.
In such case, we suspend judgment, and make use of each theory for
pragmatic purposes, without considering any as ultimately true as a theoretical
image of reality.
Even as we may give more credence to one theory as the more all-embracing
and most-confirmed, or as the simplest and most-elegant, we may still withhold
final judgment, and not regard that theory as our definite choice, because the
evidence does not seem to carry enough conviction.
Sometimes the available theories only partially explain the given data.
They may embrace some details in common, with comparable credibility, but one
may be more useful than the others in some areas, while another is more thorough
in other respects.
Although this suggests that the theories have distinct implications, they
are each supportable on different grounds, perhaps with the same overall
probabilities. We may not find a way to choose between them empirically, or to
unify them somehow.
In such case, narrowing the field by elimination of alternatives is
hardly our main concern; rather, we are still at a stage where we need a
unifying principle, we effectively do not have a theory in the full sense of the
term. An example of this is the particle-wave dichotomy, and the search for a
unified field theory to resolve it.
Sometimes, we know our list of available theories is faulty, because
their connections to the data are not entirely satisfactory and convincing. In
that case, our 'if-then-' statements are themselves probabilistic, rather than
necessary. Our ideas then had better be called notions or speculations.
Sometimes, no theory at all can be found for the phenomena at hand, for
years. There may be seemingly insurmountable antinomies. We are forced to wait
for an inspiration, a new idea, a new insight, a new observation, which might
lead us to a satisfactory solution.
Because it is in some domains very difficult to develop a meaningful and
consistent conceptual framework, we may be forced to accept one which is
conceptually or logically flawed, as a working hypothesis.
Sometimes, the problem may be shelved, because its impact lies elsewhere,
creating doubts and questions in distant disciplines. For example, Heisenberg's
Uncertainty Principle seems to assault our common-sense conceptions of
determinism for inanimate matter: this might later be resolved by Physics
itself, or might remain an issue for Philosophy to deal with.
In practise, an imperfect tool of knowledge is often better than none at
all. We prefer to have a theory formulated in terms of vague or seemingly
contradictory concepts, with practical value, than to remain paralyzed by a
dogmatic insistence on an elusive ideal.
Thus, sometimes, although a theory may apparently be strictly speaking
felled by hard evidence, and we are unable to pinpoint its mistakes, we may
nonetheless pragmatically hang on to it, if there is no other to replace it. We
simply mentally attach a reservation to it, retain an awareness of its
limitations, and move on cautiously to practical applications.
This is especially justifiable when the reason for its empirical
rejection was an extreme situation, or 'boundary case', not encountered in the
normal course of events. We then recognize the need to specify some limiting
conditions to the theory, without being able to fulfill this need more precisely
at the present stage.
Even when a theory is found empirically wrong, yet has alternatives, we
may avoid outright rejection, and rather first seek to rectify it somehow,
limiting it in scope or shifting some of its postulates slightly. This is
feasible on the ground that there must have been some grain of truth in the
original insight, and we may be able to tailor our assumptions to fit the new
Even if we cannot immediately conceive a correction, we may still choose
to hang on to the original idea in the hope of its eventual redemption. We all
carry a baggage of beliefs through life, which we know lead to contradictions or
have been apparently disproved or rendered very improbable; we keep them in mind
for further verification, anyway. This attitude taken to an extreme is of course
contrary to logic, but within reasonable bounds it has some utility.
The pursuit of truth is not cold and vengeful, as it were, towards flawed
theories, intent on rarefying the alternatives at all costs. Rather, it is a
process of flexible adaptation to changing logical conditions. Our goal is,
after all, to indeed arrive at truth, and not merely to give the impression that
If we manage to modify a theory well enough to fit the new facts, then
effectively we have developed a new theory. It may be a new version of the old,
but still merits consideration as a theory in its own right.
We defined a theory as a number of distinctive postulates together
implying a number of predictions. More loosely, the range of applicability of a
theory might be varied, without radically affecting the substance of its
proposals or its details.
Also, we may distinguish between essential postulates and postulates open
to change. The former may be generic proposals, the latter specifics within them
which we have not yet resolved — postulates within postulates, as it were.
Likewise, we might distinguish between generic predictions, which are necessary
consequences, and their specifics, which may be less firmly bound to the
With these thoughts in mind, we can talk of a theory 'changing', while
remaining essentially the same theory. This may refer to changes in scope or
changes in detail which do not affect the main thrust of a hypothesis. In other
words, a theory may involve logical conditional propositions, as well as
categoricals, leaving room for variations.
Denial of a postulate may mean: either denial of the broadness of the
postulate, without excluding the possibility that a more moderate formulation is
acceptable, or denial of a specific position, which can be replaced by another
specific position with the same generic impact, or radical denial of a generic
position, in the sense that all its possible embodiments are consequently
Denial of a prediction may accordingly either merely cause us to regard
the theory as having a more limited applicability than originally thought, or to
make relatively small corrections in our assumptions, or force us to formulate a
completely new theory.
Thus denial of a postulate or prediction does not necessarily mean
rejection of the whole theory as such, it may be only partly discredited,
requiring a less ambitious or a slightly altered formulation.
Accordingly, a new theory may totally replace an old one, or it may
embrace it as a special case. For example, Einstein's Relativity resulted in our
particularization of Newtonian mechanics to commonplace physical levels; it was
thenceforth seen as inapplicable to more extreme astronomical or sub-atomic
situations, but retained much of its usefulness.
We know from apodosis that affirmation of a postulate implies acceptance
of all its necessary predictions (even those untestable empirically), and denial
of a prediction obliges us to reject (or at least change) the postulates which
Denial of a postulate does not engender denial of its still untested
predictions; it only diminishes their probability. However, empirically
untestable predictions can still be discarded, if we can show them to be
logically exclusive to some empirically rejected postulate(s). The argument is a
Only if postulates p, then predictions q
(implying: if notp, then notq),
but not p,
hence, not q.
Doubt may remain, depending on how sure we are of the postulate's denial,
and especially on the strength of the exclusiveness. Also, what has been said
does not prevent the possibility that a slightly different version of the
predictions still hold.
Likewise, affirmation of a prediction does not in itself prove any of the
postulates giving rise to it, but only confirms them. However, theoretical
postulates can still be established, if we can show them to make some logically
exclusive empirically tested prediction(s).
Only if postulates p, then predictions q
(implying: if notp, then notq),
This too is a valid apodotic argument. Again, such exclusiveness may
often be hard to determine indubitably, but the principle remains valid.
It is not always easy or even possible to find such exclusive
relationships. In such case, we are of course limited to the adductive approach.
Note that, just as necessity is the extreme of probability, so apodosis is the
limiting case of adduction: they differ in degree, not in essence.
Thus, it is not permissible to regard, as some philosophers seem to have
intimated, science as incapable of certitude in disproof of empirical matters,
or of certitude in proof of theoretical constructs. Admittedly, a good deal of
theory selection is based on the processes of adduction and elimination; but
this is only one arrow in the arsenal of the scientific method.
If we regard science as capable of establishing logical (or mathematical)
connections for the purposes of mere confirmation or undermining of theories,
then it is equally capable in principle of establishing exclusive connections
which can be used for the above described demonstration purposes.
All the hypothetical forms are structurally identical, irrespective of
the polarities of their theses. If any one of them is recognized as accessible
to science, then they are all equally so. If we can rely on the 'if p, then q'
of adduction, then we can just as well rely on the 'if notp, then notq' of
There is no intent, here, to underrate the importance of competitive
induction, only to point out that other, more certain, means are sometimes
available to us, though not always. What is at issue here is the suggestion that
we only have a choice of a-priori, axiomatic knowledge versus a posteriori,
There is an in-between alternative: knowledge which is at once
theoretical, and certifiable, and empirical. It is arrived at through the
logical discovery of exclusive relationships between postulates and predictions.
This methodology has the stamp of approval of logical science, and is perfectly
Indeed, all our so-called mind-set concepts, even the axioms of logic,
have such exclusive-empirical grounding, as well as self-evidence (i.e.
self-contradiction of their contradictories). Every particular proposition, for
example, appeals to this reasoning. More generally, any concept which appears as
sole available interpretation or explanation of the experienced phenomena is
justifiable on that basis.