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.)
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;
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 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.