Further Notes on A-Fortiori Argument.
Subjectal and predicatal (or antecedental and consequental) a-fortiori
are sometimes found in tandem,
forming an enthymemic sorites, so that the conclusion of one implicitly serves
as minor premise in the other. For instances:
is more R than B, and B is R enough to be C,
more R is required to be C than to be D;
A is R enough to be D.
the tacit, subjectal result ‘A is R enough to be C’ of the first two premises
serves, together with the third premise, to obtain the final, predicatal
is more R than C,
C is R enough to be A, and more R is required to be A than to be B;
D is R enough to be B.
the tacit, predicatal result ‘C is R enough to be B’ of the last two premises
serves, together with the first premise, to obtain the final, subjectal result.)
sufficiency: X is R “just
enough” to be Y, or
sufficiency: X is R “more than
enough” to be Y.
Thus, “enough” is commonly taken to mean “either just
enough or more than enough”, according to the spread
between the two starting points of Ry and Rx. The “more than enough”
relation may in turn be variously (and more or less precisely) quantified: a
little more, much more, etc. But note well that these subsets of the form are
just two of the many ways, in a broader perspective, that Rx may be included in
Similarly, insufficiency is often expressed in the form:
is R “less than enough” to be Y.
This may be further quantified: a lot less, not-much less, etc. As with
the expressions “just enough” and “more than enough” used to
qualify sufficiency, this expression “less than enough” occurs in
specific contexts, namely where we are dealing with a single, continuous range
of R (starting at Ry and growing beyond it). But where we have discontinuities
in our range, or in other words, several intervals, the language becomes
inadequate. However, we should also note two other variants, which attempt to
verbalize such discontinuities to some extent (implying, respectively, an upper
limit and a lower limit):
through excess: X is “too much” R to be Y, or
through deficiency: X is “too little” R to be Y.
Both excess and deficiency must be taken to imply insufficiency, though
for different reasons. Thus, the relation “not-enough” in our frozen
sense must not be limited to “too-little” (as often in everyday use),
but must range over “too-much” as well. Put differently, insufficiency
(in this broad sense) means either too little or too much. But to repeat, such
expressions, though useful enough in certain commonly encountered contexts,
cannot verbalize all situations. A full analysis of these issues is best carried
out through mathematical logic, using symbolic techniques. I will not even
attempt it – it is not my forte. In any case, these are details which do not
affect the truth of the more generic statements we here make concerning
a-fortiori language and logic.
Lastly, note that valid moods of a-fortiori can be developed, using such
variant forms of suffective proposition in various combinations. To develop
them, we need only take the already validated generic moods as our starting
points, and consider the effect of variations. The conditions under which these
subsets of a-fortiori are valid are the same as those already established for
the main moods from which they derive. Here are a couple of interesting
illustrations of the kind of argument meant:
is more R than Q; therefore:
Q is R just enough to be S, then P is
R more than enough to be S.
note, with the same major premise, ‘if P is R just
enough to be S, then Q is R less than
(i.e. neither more than nor just)
enough to be S’.
R is required to be P than to be Q; therefore:
S is R just enough to be P, then S is
R more than enough to be Q.
note, with the same major premise, ‘if S is R just
enough to be Q, then S is R less than
(i.e. neither more than nor just)
enough to be P’.
The reader is invited to work out all other possibilities and the
I would like to here make some comments concerning the
representation of natural phenomena in mathematical formulae, for readers
unacquainted with the topic which was raised in the context of our discussion of
the dayo principle.
Any two or more phenomena, be they physical or whatever, whose magnitudes
evidently vary together in some way, however complex, can in principle (provided
we are able to measure them precisely) be assimilated into an algebraic
equation. Such equations, in turn, have a geometrical equivalent, in a Cartesian
space where each of the phenomena is represented by a dimension; their
quantitative relationship is then expressed by a straight line or a curve of
whatever shape, or some other figure.
A simple example is the ideal gas equation, “pv/t=k”, where p,
v, t are variables, p=pressure in the gas, v=volume of the gas, t=temperature of
the gas, and k=a constant. The equation is called ideal, because real gases do
not quite behave in this way; but it is a good approximation in ordinary
circumstances. This equation yields a linear relation, in a three-dimensional
Cartesian representation. The relationship could just as well have been
exponential or sinusoidal or whatever; but this is what experimenters found it
to be, by measuring various states of gases and extrapolating the results.
Now, what does this mean in more colloquial terms? An equation like
pv/t=k is a summary of innumerable conditional propositions, concerning all
possible values of the variables. For any given value of p, say, we can predict
by a simple calculation all the correspondences between the values of v and
those of t:
if the gas has pressure p1, then:
if the gas has
volume v1, then it has temperature t1 (=p1.v1/k), and vice versa;
if the gas has
volume v2, then it has temperature t2 (=p1.v2/k), and vice versa;
if the gas has pressure p2, then:
if the gas has
volume v1, then it has temperature t5 (=p2.v1/k), and vice versa;
if the gas has
volume v2, then it has temperature t6 (=p2.v2/k), and vice versa;
…and so on.
Knowing the possible variety and complexity of natural equations, it is
easy to see the reasonableness of the dayo
principle. Two variables may be proportional for part of their course, and then
have a radically different relation, if the equation which links them is
In my early attempts to understand a-fortiori argument, I attempted a
theory which I called the ABCD Format.
Though this may not be applicable to all cases or go to the essence, it may
still have some value, so I will briefly present it here.
‘A‘ stands for the agent, ‘B‘
for what is in between (the means), ‘C‘
for the surrounding conditions, and ‘D‘ for the destination
(goal); note that these terms are used in a broad concept of causality, not
necessarily implying movement or change, nor conscious pursuit of ends. In this
framework, we can conceive of subjectal a-fortiori as follows:
premise (for both
the following moods):
conditions C1, agent A, by means B1, causes D, more
likely than under conditions C2, agent A, by means B2, causes D; therefore:
mood (minor to major):
A in C2 (=Q) does B2 (=Rq) causing D (=S), then
A in C1 (=P) will do B1 (=Rp) causing D (=S).
The children of Israel (A), while Moses is yet alive (C2), do things (B2)
against the Law (D); therefore, they (A), after Moses dies (C1), will probably
do things (B1) against the Law (D).
mood (major to minor):
A in C1 (=P) does not do B1 (=Rp) and not-cause D (=S), then
A in C2 (=Q) will not do B2 (=Rq) and not-cause D (=S).
Joseph’s brothers (A), though out of reach of the Egyptian authorities (C1), did
not keep found money (B1) and thus avoided dishonesty (D); therefore, they (A),
within Egyptian territory (C2), would probably not steal (B2) and thus would
avoid dishonesty (D).
Note that all four factors (A, B, C, D) are involved, if only
particularly and possibly only implicitly, in each of the three propositions.
Note that the relationship between the two clauses of the major premise, which
makes possible our drawing a conclusion from the minor premise, is here
conceived as one of probability, or
effectiveness of causation. In this framework, the quantitative aspect of
a-fortiori is rather incidental, and the argument involved is essentially an apodosis.
Similar constructions can presumably be worked out for predicatal
arguments, positive and negative. And likewise for implicationals.
As already said, it is doubtful that this format is of general or
profound value, except that it shows the causal subtext of some arguments, and
incidentally how the terms of subjectal and predicatal arguments may
occasionally be reshuffled from one form to the other.