**Appendix 1. **

**Further Notes on A-Fortiori Argument. **

**1.** 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:

A is more R than B, and B is R enough to be C,

and more R is required to be C than to be D;

so, A is R enough to be D.

(Here, 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 result.)

D is more R than C,

and C is R enough to be A, and more R is required to be A than to be B;

so, D is R enough to be B.

(Here, 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.)

**2.** We often encounter the following **variant
forms of suffective propositions**; sufficiency, X is R “enough” to
be Y, may mean (and is implied by each of):

Exact
sufficiency: X is R “*just* enough” to be Y, or

Generous
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
Ry.

Similarly, insufficiency is often expressed in the form:

X
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):

Insufficiency through excess: X is “too much” R to be Y, or

Insufficiency 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:

P
is *more* R than Q; therefore:

if
Q is R *just* enough to be S, then P is
R *more than* enough to be S.

Also
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’.

*More* R is required to be P than to be Q; therefore:

If
S is R *just* enough to be P, then S is
R *more than *enough to be Q.

Also
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 validations.

**3.** 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;

· …etc.

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;

· …etc.

…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
sufficiently contorted.

**4.** 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:

__Major
premise__ (for both
the following moods):__ __

Under
conditions C1, agent A, by means B1, causes D, *more
likely than* under conditions C2, agent A, by means B2, causes D; therefore:

__Positive
mood (minor to major): __

If
A in C2 (=Q) does B2 (=Rq) causing D (=S), *then* A in C1 (=P) will do B1 (=Rp) causing D (=S).

e.g. 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).

__Negative
mood (major to minor): __

If
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).

e.g. 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.