A FORTIORI LOGIC
CHAPTER 2 – MORE FORMALITIES
Since, as shown earlier, the propositions used in a fortiori argument can be reduced to simpler forms, it follows that we can formally combine a fortiori argument with syllogism in certain ways. Consider first the positive subjectal mood:
P is more R than (or as much R as) Q (is R),
and, Q is R enough to be S;
therefore, all the more (or equally), P is R enough to be S.
Suppose that the given subsidiary term S refers to a species of a certain genus G. In that case, S is a subclass of G, i.e. ‘All S are G’ is true. The minor premise “Q is R enough to be S” of our a fortiori argument implies that “Q is S.” By a first figure syllogism, we can infer that “Q is G.” It follows that “Q is R enough to be G” is also true. If we now use this as our minor premise in lieu of the preceding, we can infer that “P is R enough to be G,” since the major premise has remained unchanged. We can thus say, with reference to the subsidiary term: if a positive subjectal a fortiori argument is true of a species (S), then it is equally true of any genus of it (G).
It follows from this, by reductio ad absurdum, that: if a negative subjectal a fortiori argument is true of a genus (G), then it is equally true of any species of it (S). That is to say: given (with the same major premise) that “If P is R not enough to be G, then Q is R not enough to be G,” and “All S are G,” we can infer that “If P is R not enough to be S, then Q is R not enough to be S.” (For otherwise, if Q were R enough to be S, then P would be R enough to be S and therefore G.)
The same can be done with positive predicatal a fortiori argument, as follows.
More (or as much) R is required to be P than to be Q,
and, G is R enough to be P;
therefore, all the more (or equally), G is R enough to be Q.
Suppose that the given subsidiary term G refers to a genus of a certain species S. In that case, S is a subclass of G, i.e. ‘All S are G’ is true. The minor premise “G is R enough to be P” of our a fortiori argument implies that “G is P” (i.e. “all G are P”). By a first figure syllogism, we can infer that “S is P.” It follows that “S is R enough to be P” is also true. If we now use this as our minor premise in lieu of the preceding, we can infer that “S is R enough to be Q,” since the major premise has remained unchanged. We can thus say, with reference to the subsidiary term: if a positive predicatal a fortiori argument is true of a genus (G), then it is equally true of any species of it (S).
It follows from this, by reductio ad absurdum, that: if a negative predicatal a fortiori argument is true of a species (S), then it is equally true of any genus of it (G). That is to say: given (with the same major premise) that “If S is R not enough to be Q, then S is R not enough to be P,” and “All S are G,” we can infer that “If G is R not enough to be Q, then G is R not enough to be P.” (For otherwise, if G were R enough to be P, then G and therefore S would be R enough to be Q.)
So much for copulative a fortiori argument. The same can be done with implicational a fortiori argument, except that we would here use hypothetical instead of categorical syllogism. That is, instead of “All S are G” (where S and G are terms), we would say “If S then, G” (where S and G are theses). Although such combinations of a fortiori argument and syllogism are simple and obvious, they are nevertheless sometimes useful and so well to keep in mind.
A case in point is the a fortiori argument suggested in Num. 12:14, viz. “If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.” If we look closely, we see that the subsidiary term in the minor premise “hide in shame seven days” is not identical (as it should be) to the one in the conclusion “shut up without the camp seven days;” in which case the inference is invalid. We can, however, draw a valid conclusion by using the common property (i.e. genus) of these two terms, i.e. what they both imply, viz. “isolation for seven days.” To justify such syllogistic interference, we need the above formal treatment.
Now, it should be obvious that having in our example concluded with “isolation for seven days,” we cannot take that generic conclusion as justification for a more specific conclusion such as “shut up without the camp seven days” which does not exactly correspond to the given species “hide in shame seven days.” This would be fallacious reasoning. In other words, the new “shut up without the camp” specification cannot be claimed to be a conclusion of a fortiori reasoning, but must be regarded as a specification applied to the generic conclusion “isolation for seven days” after the a fortiori argument proper, through some other act of deductive or inductive reasoning or other justification. In this example, the justification would be that God or the Torah has so decreed.
This can be stated in formal terms as follows. Consider a positive subjectal argument, to begin with. As we have just seen, if the minor premise affirms the minor term Q to have species S1 as predicate, then the conclusion must likewise affirm the major term P to have species S1 as predicate, or if we wish it may (via syllogism) affirm the major term P to have some genus G of S1 as predicate. We cannot, however, conclude with affirmation of predication for some separate species of G, say S2 – at least not through our a fortiori argument, though we might arrive at S2 by some other process of reasoning thereafter.
Regarding negative subjectal argument, this rule means (notice the changed positions of P and Q here) that if the minor premise denies the major term P to have genus G1 as predicate, then the conclusion must likewise deny the minor term Q to have genus G1 as predicate, or if we wish it may (via syllogism) deny the minor term Q to have some species S of G1 as predicate. We cannot, however, conclude with denial of predication for some separate genus of S, say G2 – at least not through our a fortiori argument, though we might arrive at G2 by some other process of reasoning thereafter.
In predicatal argument, the equivalent fallacy would consist in passing from a given genus to a not-given other genus. Thus, in the positive mood of predicatal argument, we might start from given genus G1, saying “G1 is R enough to be P;” from that, given that S is a known species of G1, we can legitimately infer (as above shown, through syllogism) that “S is R enough to be Q;” but then we swerve off and illicitly claim that “G2 is R enough to be Q,” where G2 is another genus of S (though not a species of G1).
Similarly, in the negative mood of predicatal argument, it would be fallacious to pass from a given species to a not-given other species. That is, starting with “S1 is R not enough to be Q;” we can, given that G is a known genus of S1, legitimately infer (as above shown) that “G is R not enough to be P;” but we cannot likewise infer that “S2 is R not enough to be P,” where S2 is another species of G (though not a genus of S1).
It should be added that the fallacious reasoning above described is not uncommon. To sum up, we have above, knowing that all S are G, established four rules of transmission of a fortiori argument from species to genera or vice versa:
- If a positive subjectal argument is true of a certain subsidiary predicate (S), then it is also true of any genus of it (G).
- If a negative subjectal argument is true of a certain subsidiary predicate (G), then it is also true of any species of it (S).
- If a positive predicatal argument is true of a certain subsidiary subject (G), then it is also true of any species of it (S).
- If a negative predicatal argument is true of a certain subsidiary subject (S), then it is also true of any genus of it (G).
And likewise, mutatis mutandis, with respect to implicational arguments.
We should also be aware that the middle term R may subtly differ in meaning in relation to the major and minor terms P and Q. In some cases, the meaning of R is identical; but in some cases, the R in relation to P (Rp) and the R in relation to Q (Rq) have the abstraction R in common, but they are each specifically relative to the term they concern. In effect, Rp and Rq are two species of the genus R. The argument, if properly constructed, remains nonetheless valid, because all that matters for its validity is that Rp ≥ Rq and this quantitative condition is here satisfied.
To give an example: “John loves his dog more than Jill does,” could mean that Jill loves John’s dog less than he does (in which case, the middle term is “loves John’s dog”), or it could mean that Jill loves her own dog less than John loves his own dog (in which case, the middle term is the more abstract “loves his/her own dog,” or even just “loves some dog”). In the latter case, the minor premise and conclusion would be: “Given that Jill loves her own dog enough to be classed as an animal lover, it follows that John loves his own dog enough to be classed as an animal lover.” This example is positive subjectal. We may similarly construct examples for negative subjectal, predicatal and implicational arguments with this feature.
This section and the next are very important and should be read carefully.
Closely related to the issue of species and genera, is that of ‘proportionality’. Often, rather than species and genera, what is involved are different degrees or measures of the same term. Thus, in positive subjectal or negative predicatal argument, the distinct species S1 and S2 (neither of which is included in the other, though they have a genus in common) would appear as different degrees or measures of the same genus G; similarly, in positive predicatal or negative subjectal argument, species S would appear as a single degree or measure of two distinct genera, G1 and G2 (neither of which includes the other, though they have a species in common).
For example, in the argument given in Num. 12:14 (see 2.4 below), though the minor premise specifies the quantity “seven days,” we might be tempted (by considerations of proportionality, say) to conclude with another quantity like “fourteen days;” but such reasoning (without additional premises), as we have just shown, would be formally invalid. In purely a fortiori argument, the conclusion can never produce a different quality or quantity than the one given in the minor premise; this is a hard and fast rule based on strictly logical considerations.
Argument a crescendo. A fortiori argument with a ‘proportional’ conclusion is, in itself, by itself, fallacious. The copulative variant has, at least on the surface, the following four forms. The positive subjectal mood resembles that of regular a fortiori, except that, whereas the minor premise predicates a subsidiary term (S) of the minor term (Q), the conclusion predicates a greater subsidiary term (more than S) of the major term (P). It goes like this:
P is more R than Q (is R),
and Q is R enough to be S;
therefore, P is R enough to be more than S.
To avoid confusion between the subsidiary term S in a general sense and its values in the minor premise and conclusion, think of ‘S’ in the former as ‘Sq’ and ‘more than S’ in the latter as ‘more than Sq’ or as ‘Sp’. The corresponding negative subjectal mood has the same major premise, but as usual has the denial of the above conclusion as minor premise and the denial of the above minor premise as conclusion. That is, it argues: Since P is R not enough to be more than S, it follows that Q is R not enough to be S.
The positive predicatal mood resembles that of regular a fortiori, except that, whereas the minor premise predicates the major term (P) of a subsidiary term (S), the conclusion predicates the minor term of a lesser subsidiary term (less than S). It goes like this:
More R is required to be P than to be Q,
and S is R enough to be P;
therefore, less than S is R enough to be Q.
To avoid confusion between the subsidiary term S in a general sense and its values in the minor premise and conclusion, think of ‘S’ in the former as ‘Sp’ and ‘more than S’ in the latter as ‘less than Sp’ or as ‘Sq’. The corresponding negative predicatal mood has the same major premise, but as usual has the denial of the above conclusion as minor premise and the denial of the above minor premise as conclusion. That is, it argues: Since less than S is R not enough to be Q, it follows that S is R not enough to be P.
We could alternatively have, for the positive subjectal mood, ‘less than S’ in the minor premise and ‘S’ in the conclusion; and for the negative subjectal mood, ‘S’ in the minor premise and ‘less than S’ in the conclusion. And likewise, for the positive predicatal mood, ‘more than S’ in the minor premise and ‘S’ in the conclusion; and for the negative predicatal mood, ‘S’ in the minor premise and ‘more than S’ in the conclusion. What matters is that the relative magnitudes be as stated.
In practice, the subsidiary term in the minor premise would always be labeled ‘S’ and the subsidiary term in the conclusion would accordingly be labeled ‘more than S’ or ‘less than S’, as the case may be – for the simple reason that we normally know the minor premise before we get to know the conclusion. I have chosen the terminology above to stress that the negative moods are reducible ad absurdum to the positive ones.
We can likewise construct implicational forms. As these various forms show, ‘proportional’ a fortiori argument is based on the notion that if P is more R than Q is, or more R is required to be P than to be Q, then necessarily a larger amount of S will correspond to P than to Q. But in fact there is no such necessity; it may occasionally be true, but there is no logical reason why it should be. Such arguments, unlike regular a fortiori, simply cannot be validated as they stand.
Looking at the positive subjectal form of ‘proportional’ a fortiori argument, which is prototypical, it is evident that we can equally well refer to it as argument a crescendo (this being a name I invented in the course of my research, having found it useful). This name can be extended to all the other forms. The advantage of such renaming is that it verbally completely distinguishes such argument from strict a fortiori.
Argument pro rata. Argument a crescendo (i.e. ‘proportional’ a fortiori) should not be confused with argument by proportion, which we can refer to as argument pro rata (this Latin name being already well established in the English language), this being understood to mean “at the same rate.” Such argument concerns concomitant variations between two variables, and may be formulated as follows:
Y varies in proportion to X. Therefore:
given that: if X = x, then Y = y,
it follows that: if X = more (or less) than x, then Y = more (or less) than y.
An example of it is Aristotle’s statement: “Every good quality of the soul, the higher it is in degree, so much more useful it is” (Politics 7:1), which intends the argument: given that a certain quality of the soul is good, it is useful; if it is improved, it is still more useful. In practice, pro rata argument is often expressed in the form: “the more X, the more Y; and (by implication) the less X, the less Y.” Note that two variants (which mutually imply each other) are possible: one with “more” and one with “less” – that is, the argument can go either way, increasing or decreasing the quantities involved.
The statement “Y varies in proportion to X” is not an argument but a mere proposition, reflecting some generalized empirical observations or a more theoretical finding. The above argument includes this proposition as its major premise, but requires an additional minor premise (“if X = x, then Y = y”) to draw the conclusion (“if X = more/less than x, then Y = more/less than y”). The conclusion mirrors the minor premise in form, but its content is intentionally different. The quantities involved do not stay the same, but increase or decrease (as the case may be).
Notice that a pro rata argument has no middle term, unlike an a fortiori one. A pro rata argument is thus more akin to apodosis than to syllogism. Its major premise sets a broad principle, of which the minor premise and conclusion are two applications. The argument involved is thus simply inference of one quantity from another within the stated principle. If we found that contrary to expectations X and Y do not vary concomitantly as above implied, we would simply deny the major premise. In other words, this argument is essentially positive in form. A negative mood of it (with the same major premise and denials of the previous conclusion and minor premise) would not make much sense, since its minor premise and conclusion would be in conflict with its major premise.
The above formulas are at least true in cases of direct proportionality; in cases of inverse proportionality, the language would be: “the more X, the less Y; and (by implication) the less X, the more Y;” and the argument would have the following form:
Y varies in inverse proportion to X. Therefore:
given that: if X = x, then Y = y,
it follows that: if X = more (or less) than x, then Y = less (or more) than y.
And of course, in more scientific contexts, we may have access to a more or less complex mathematical formula – say Y = f(X), where f refers to some function – that allows precise calculation of the proportion involved. In other words, the validity of pro rata argument is not always obvious and straightforward, but depends on our having a clear and reliable knowledge of the concomitant variation of the values of the terms X and Y. Given such knowledge, we can logically justify drawing the said conclusions from the said premises. Lacking it, we are in a quandary.
As its name implies, pro rata argument signifies that there is (if only approximately) some constant rate in the relative fluctuations in value of the variables concerned. The variables X and Y may be said to be proportional if X/Y = a constant, or inversely proportional if XY = a constant. In the exact sciences, of course, such a constant is a precisely measurable quantity; but in everyday pro rata discourse, the underlying ‘constant’ is usually a vague quantity, perhaps a rough range of possible values.
Proportionality or inverse proportionality as just defined, which can be represented by a straight line graph, and even when the graphical representation is more curved (e.g. exponential), may be characterized as simple. It becomes complex, when there are ups and downs in the relation of the two variables, i.e. when an increase in X may sometimes imply an increase in Y and sometimes a decrease in Y, it is obviously not appropriate to formulate the matter in the way of a standard pro rata argument. In such cases, we would just say: “the values of X and Y can be correlated in accord with such and such a formula,” and then use the formula to calculate inferred quantities.
Proportionality may be continuous or not. Sometimes, there is proportionality of sorts, but it comes in slices: e.g. from X = 0 to 1, Y = k; from X = 1 to 2, Y = k +1; etc. That is, to each range of values for X, there corresponds a certain value of Y, and the two quantities go increasing (or decreasing, as the case may be). Such proportionality is compatible with pro rata argument. For this reason, it is wise to put the word ‘proportionality’ in inverted commas, so as to remember that it does not always imply continuity.
Note too that proportionality may be natural or conventional. An example of the latter would be a price list: bus fares for children under 16, $1; for adults 16+, $2. However, beware in such case of frequent exceptions or reversals: e.g. unemployed and pensioners, $1. In such cases, any pro rata argument must be stated conditionally: the bus fares are ‘proportional’ to age, provided the adults are not unemployed or pensioners.
It should also be reminded that proportionality (or its inverse), simple or complex, may or may not be indicative of a causal relation (in the various senses of that term). Two variables may vary concomitantly by virtue of being effects of common causes, in which case we refer to parallelism between them, or the one may cause or be caused by the other. Also, of course, such parallelism or causality may be unconditional or conditional. In such cases as it is unconditional, no more need be said. But in such cases as it is conditional, the condition(s) should ideally be clearly stated, although often they are not.
Pro rata argument may occur in discourse independently of a fortiori argument, or in conjunction with such argument. In any case, it should not be confused with a fortiori argument: they are clearly different forms of reasoning. Pro rata involves only two terms, or more precisely two values (or more) of two variables; whereas a fortiori involves four distinct terms, which play very different roles in the argument. Pro rata and a fortiori are both analogical arguments of sorts, but the former is much simpler than the latter.
Having thus clarified the differences between (regular) a fortiori argument, a crescendo argument (i.e. ‘proportional’ a fortiori) and pro rata argument, it is obvious that these three types of argument should not be confused, although many people tend to do so. Although these arguments are far from the same in form and in validity, such people wrongly identify pro rata argument and a crescendo argument with a fortiori argument. The reason for such confusion may be that argument a crescendo appears to be a combination of argument a fortiori and pro rata argument. This could be expressed as a formula:
A crescendo = a fortiori cum pro rata.
Though the latter two forms of argument may occur coincidentally (i.e. they may happen to be both true in certain material cases), it does not follow that they formally necessitate or imply each other. But, we may ask, do they together imply a crescendo argument? That is to say, is argument a crescendo valid in cases where both argument a fortiori and argument pro rata happen to be true? To answer this question, consider a regular positive subjectal a fortiori argument:
P is more R than Q (is R),
and Q is R enough to be S;
therefore, P is R enough to be S.
It informs us that the quantity of R corresponding to P (Rp) is greater than the quantity of R corresponding to Q (Rq), and then argues: since the latter quantity (Rq) is big enough to imply Q to be S, then the former quantity (Rp) must be big enough to imply P to be S. It does not tell us that the subsidiary term S is a variable quantity; S is here clearly intended to have one and the same value in the minor premise and conclusion. And if S happens to be a variable quantity, we cannot automatically suppose that the variation of S is tied to that of R, so that S for P (Sp) is necessarily greater than S for Q (Sq), in concomitance with the variation of Rp with Rq.
Under what conditions, then, can we obtain the conceivable a crescendo conclusion “P is R enough to be more than S” from the above a fortiori premises? That is, what additional information do we need to transform the above valid a fortiori argument into a valid a crescendo argument? Or to put it another way again: having already come to the a fortiori conclusion “P is R enough to be S (i.e. Sq),” how can we proceed one step further and obtain the a crescendo conclusion “P is R enough to be more than S (i.e. Sp)”?
The answer is, of course, that we must obtain the additional information required to construct the following pro rata argument:
If, moreover, (for things that are both R and
S,) we find that:
knowing from the above minor premise that: if R = Rq, then S = Sq,
it follows in the conclusion that: if R = more than Rq = Rp, then S = more than Sq = Sp.
Note well the stipulation “for things that are both R and S.” I have put this precondition in brackets, because it is in fact redundant, since as we saw earlier the minor premise of the a fortiori argument implies anyway that not all things that are R are S, but only those things that have a certain threshold value of R or more of it are S. We should not think of S varying with R as a general proposition applicable to all R (implying that all R are S), but remain aware that this concomitant variation occurs specifically in the range of R where the threshold for S has indeed been attained or surpassed (i.e. where the “R enough to be S” condition is indeed satisfied).
If we know (by induction or deduction from other information) that the major premise of our above pro rata component is true (i.e. that S varies in proportion to R), we can infer its minor premise (viz. if R = Rq, then S = Sq) from the minor premise of the a fortiori component and thence draw the conclusion that “If the value of R for P is Rp (> Rq), then the value of S for P is Sp (> Sq).” This is assuming, as earlier specified, that the proportionality proposed in the major premise is direct and simple.
The desired a crescendo conclusion, viz. “P is R (Rp) enough to be more than Sq (Sp)” can then be confidently drawn. That is, the intermediate conclusions, i.e. the a fortiori conclusion (P, being more than Rq, is R enough to be Sq) and the pro rata conclusion (If R = Rp, then S = Sp) together imply the final, a crescendo conclusion (P, which is Rp, is R enough to be Sp). Note well that Sp, here, means nothing more precise than ‘more than Sq’. Though given a distinct symbol, it is not an exact number, unless we are able to calculate it precisely through some sort of mathematical formula.
Note that we have here taken variations in the value of S to be concurrent with variations in the value of the middle term R. Very often, though, a crescendo thinking is based on the assumption that the values of S are proportional to those of the major and minor terms, P and Q. These two views are not necessarily in conflict, though the former (which we have adopted) is the more essential and more generally applicable. P and Q may be single values, or they may be variable over time – so long as their relative magnitudes or degrees are as specified in the major premise, i.e. that P is always more (more R, to be exact, though R is often left tacit) than Q. Thus, S may well be vaguely thought of as proportional to P or Q, although more precisely perceived the proportionality of S is in fact to R, the common factor of P and Q in relation to which P is greater than Q.
We have thus shown that, under certain circumstances, the formula “a fortiori + pro rata = a crescendo” is indeed true. That is to say, although the putative a crescendo conclusion is not per se valid, it can in some cases be valid if it so happen that the a fortiori conclusion from the same premises can be taken a step further by means of an appropriate argument pro rata. This has just been demonstrated for positive subjectal a crescendo.
A similar two-step argument can, of course, be formulated for positive predicatal a crescendo. In this case, we use the following combination of a fortiori and pro rata arguments:
More R is required to be P than to be Q,
and S is R enough to be P;
therefore, S is R enough to be Q.
If, moreover, (for things that are both R and P
or Q,) we find that:
knowing from the above minor premise that: if S = Sp, then R = Rp,
it follows in the conclusion that: if S = less than Sp = Sq, then R = less than Rp = Rq.
Note well the stipulation “for things that are both R and P or Q.” I have put this precondition in brackets, because it is in fact redundant, since as we saw earlier the major premise of the a fortiori argument implies anyway that not all things that are R are P and not all things that are R are Q, but only those things that have certain threshold values of R or more of it are P or Q. We should not think of R varying with S as a general proposition applicable to all S (implying that all S are R), but remain aware that this concomitant variation occurs (at least) specifically in the range of R where the thresholds for P and Q have indeed been attained or surpassed (i.e. where the “R enough to be P” and “R enough to be Q” conditions are indeed satisfied).
If we know (by induction or deduction from other information) that the major premise of this pro rata component is true (i.e. that R varies in proportion to S), we can infer its minor premise (viz. if S = Sp, then R = Rp) from the minor premise of the a fortiori component and thence draw the conclusion that “If the value of S for Q is Sq (< Sp), then the value of R for Q is Rq (< Rp).” This is assuming, as earlier specified, that the proportionality proposed in the major premise is direct and simple.
The desired a crescendo conclusion, viz. “Less than Sp (Sq) is R (Rq) enough to be Q” can thence be confidently drawn. That is, the intermediate conclusions, i.e. the a fortiori conclusion (Sp, being more than Rq, is R enough to be Q) and the pro rata conclusion (If S=Sq, then R=Rq) together imply the final, a crescendo conclusion (Sq, which is Rq, is R enough to be Q). Note well that Sq, here, means nothing more precise than ‘less than Sp’. Though given a distinct symbol, it is not an exact number, unless we are able to calculate it precisely through some sort of mathematical formula.
Thus, if we can provide an appropriate pro rata argument, we can credibly transform an a fortiori conclusion into an a crescendo conclusion, viz. in the case of positive subjectal argument, “P is R enough to be more than S,” and in the case of positive predicatal argument, “Less than S is R enough to be Q.”
As regards the corresponding subjectal or predicatal negative a crescendo arguments, they would consist of a negative a fortiori combined with the same positive argument pro rata. Referring to the above described positive arguments, keeping the major premise and additional premise about proportionality constant, if we deny the conclusion, we must deny the minor premise, as follows:
P is more R than Q (is R),
and P is R not enough to be more than S,
and S varies in proportion to R;
therefore, Q is R not enough to be S.
More R is required to be P than to be Q,
and less than S is R not enough to be Q,
and R varies in proportion to S;
therefore, S is R not enough to be P.
This is how we would derive the negative moods from the positive ones. But granting that the subsidiary term in the minor premise is thought of first, before the subsidiary term in the conclusion, it is more accurate to present these two arguments independently in the following revised forms:
P is more R than Q (is R),
and P is R not enough to be S,
and S varies in proportion to R;
therefore, Q is R not enough to be less than S.
More R is required to be P than to be Q,
and S is R not enough to be Q,
and R varies in proportion to S;
therefore, more than S is R not enough to be P.
I hope the reader is not confused by this revision. Although the negative forms are validated by reduction ad absurdum to the positive ones, viewed as forms in their own right they would be rather worded as just shown. The comparisons between the subsidiary term in the minor premise and that in the conclusion remain the same – i.e. ‘more than S > S’ and ‘S > less than S’ mean the same, and ‘less than S < S’ and ‘S < more than S’ mean the same. Notice that the revised negative subjectal mood goes from S to less than S, like the earlier positive predicatal mood, while the revised negative predicatal mood goes from S to more than S, like the earlier positive subjectal mood. Similar developments to all those above for copulative argument are possible in relation to implicational argument.
Note well that the relation between R and S changes direction, according as our reasoning is subjectal (or antecedental) or predicatal (or consequental) in form. That is to say, whereas in subjectal argument S varies with R, in predicatal argument R varies with S. That is because, in positive subjectal argument, validation relies on the fact that Rq implies Rs; whereas, in positive predicatal argument, it relies on the fact that Rs implies Rp. In both cases, the R value of the subject precedes the R value of the predicate; but in the former, the subject is Q and the predicate is S, while in the latter, the subject is S while the predicate is P. It is because the subject has a certain value of R that it can be attributed the predicate, for which that value of R happens to be a precondition.
Thus, when reasoning a crescendo in the positive subjectal form, we reason from Rq to Sq, and from Rp to Sp; i.e. when R = Rq, then S = Sq, etc. Whereas, when reasoning a crescendo in the positive predicatal form, we reason from Sp to Rp, and from Sq to Rq, i.e. when S = Sp, then R = Rp, etc. Whence, we must reverse the order of dependence between the middle and subsidiary terms, according as we reason this way or that way, to make possible validation of the argument. Of course, if in a given case the required pro rata argument as above specified is not applicable, then the a fortiori argument (whether subjectal or predicatal, as the case may be) cannot be turned into a valid a crescendo one. All this is equally true of negative arguments, of course, as already made clear.
Validation. It is important to stress that the validity of a crescendo argument depends, as we have above clearly shown, on both its a fortiori constituent and its pro rata constituent. A crescendo is neither equivalent to the former nor equivalent to the latter, but emerges from the two together. The a fortiori element is only able to produce the conclusion “P is R enough to be S” (in the positive subjectal case) or “S is R enough to be Q” (in the positive predicatal case). The pro rata element is only able to produce the conclusion that the S is, parallel to R, greater for P than for Q, or lesser for Q than for P. The a crescendo conclusion is a merger of these two partial conclusions, namely (respectively) “P is R enough to be more than S” or “Less than S is R enough to be Q.” Thus neither element is logically redundant; both are necessary to obtain the final conclusion.
Looking at the above descriptions of a crescendo argument, we see that, while the pro rata conclusion is partly based on information provided by the minor premise of the a fortiori argument, it could conceivably be built up without the latter, since it does not use all the information in it. However, although mere pro rata argument no doubt exists, it remains true that the pro rata constituent could not by itself produce the stated final a crescendo conclusion, since the latter proposition is of suffective form; so the a fortiori constituent is also indispensable. Clearly, the final conclusion is made up of elements derived from both types of arguments; i.e. its semantic charge comes from all three premises. To repeat, then, the a crescendo argument cannot be identified with either constituent alone, but requires both to proceed successfully.
What we have done above is to formally demonstrate that, although drawing a ‘proportional’ conclusion from the premises of a valid a fortiori argument is not unconditionally valid, it is also not unconditionally invalid. Such a conclusion is in principle invalid, but it may exceptionally, under specifiable appropriate conditions, be valid. Formally, all depends on whether a pro rata argument can be truthfully proposed in addition to the purely a fortiori argument. In other words, to draw a valid a crescendo conclusion, the premises of a valid a fortiori argument do not suffice; but if they are combined with the fitting premises of a valid pro rata argument, as above detailed, such a conclusion can indeed be formally justified.
Of course, as with all deduction, even if in a given case the inferential process we propose is ideally of valid form, we must also make sure that the premises it involves are indeed true, i.e. that the content of the argument is credibly grounded in fact. Very often, in a crescendo argument, the process is convincing, but the major premise of the implicit pro rata argument is of doubtful truth; this is obviously something to be careful about. Merely declaring a certain proportionality to be true does not make it true – we have to justify all our premises, as well as their logical power to together produce the putative conclusion.
Another way to stress this is to remind that the concluding predicate Sp of positive subjectal a crescendo argument means nothing more than ‘more than Sq’, and the concluding subject Sq of positive predicatal a crescendo argument means nothing more than ‘less than Sp’. These concluding subsidiary terms are not exact numbers, though in theory they might be exact if we happen to have a precise mathematical formula for their calculation – viz. S = f(R) in the case of subjectal argument, or R = f(S) in the case of predicatal argument. In most cases in practice, however, we do not have such a formula, and the terms we use are correspondingly vague and tentative. It is important to remember this.
Sometimes, unfortunately, rhetoric comes into play here, and albeit the lack of mathematical proof, the conclusion is made to seem more precise than deductive logic allows. We could at best refer to such conclusions as intuitively reasonable, or as inductive hypotheses, partly but not wholly sustained by the data in the premises; but we must realize and acknowledge that they are not deductive certainties. Otherwise, we would be engaged in misleading sophistry. Thus, it is important to keep in mind that, while we have shown that a crescendo argument is in principle, i.e. under ideal conditions, valid – it does follow that every a crescendo argument put forward in practice, i.e. in everyday discourse, is valid. It is potentially valid, but not necessarily actually valid. We have to carefully scrutinize each case.
Thus, to summarize: the expression ‘proportional’ a fortiori argument may be intended in a pejorative sense, as referring to argument that unjustifiably draws a proportional conclusion from only two a fortiori premises; or it may be intended to refer to valid a crescendo argument, consisting of three premises, viz. two a fortiori and one pro rata, yielding a justifiably proportional conclusion. A fortiori argument per se, as such, in itself, by itself, is not proportional; such argument may be verbally distinguished as purely a fortiori. However, when combined with pro rata argument, a proportional conclusion is justified, and we had best in such case speak distinctively of a crescendo argument, or as it is often called in a non-pejorative sense ‘proportional’ a fortiori argument.
Thus, a crescendo argument may be viewed as a special case of a fortiori argument; and it is fair to say that the field of a fortiori logic also deals with a crescendo argument. But strictly spoken, ‘a fortiori’ should be reserved for ‘non-proportional’ argument, and ‘a crescendo’ preferred for ‘proportional’ argument. The former is pure because the major and minor premise suffice for the conclusion; whereas the latter is a compound argument, comprising a pure a fortiori argument combined with a mere pro rata argument. Although people often appear to draw an a crescendo conclusion from a fortiori premises, such inference in fact relies on an unspoken additional pro rata premise, and so is not purely a fortiori.
In any case, it is useful to remember the formula: a crescendo equals a fortiori plus pro rata. This means that if you come across an a crescendo argument that looks valid, you can be sure that underlying it are a valid a fortiori argument and a valid pro rata argument. Inversely, if one and/or the other of the latter arguments cannot be upheld, then the former cannot either.
Alternative presentation. We have thus far considered a crescendo argument as a special case of a fortiori argument where purely a fortiori argument is combined with pro rata argument. Another way we might look upon the relationship between these arguments is to say that all a fortiori argument is a crescendo argument, while purely a fortiori argument is a special case where the pro rata argument involves a fixed quantity instead of a variable, i.e. where the ‘proportionality’ involved is a constant. That is to say, we could regard the general forms of a fortiori argument to be the following (for examples, with regard to positive subjectal and positive predicatal moods):
P is more R than Q (is R),
and Q is R enough to be more than S,
and S is constant (pure) or varies in proportion to R (a crescendo);
therefore, P is R enough to be S, or more than S (as the case may be).
More R is required to be P than to be Q,
and S is R enough to be P,
and S is constant (pure) or R varies in proportion to S (a crescendo);
therefore, S, or less than S (as the case may be), is R enough to be Q.
Note how the additional premise about proportionality is now so broadly stated that it includes both the cases of purely a fortiori argument (where S is constant) and those of a crescendo argument (where S varies in magnitude or degree). The advantage of this approach is that it goes to show that purely a fortiori argument and a crescendo argument are essentially two special cases of the same general form, and so that they can legitimately be referred collectively as forms of ‘a fortiori argument’ in the largest sense.
However, it must be emphasized that this joint formulation is just an ex post facto way of looking at things, a perspective. It remains true, as initially stated, that a fortiori argument is essentially pure, since it can be validated without reference to an additional premise about proportionality. In this more accurate perspective, a crescendo argument is an amplification of a fortiori argument, taking its ‘equal’ conclusion and enriching it by turning it into a ‘proportional’ one (by means of a pro rata argument). When we do not have any additional premise about proportionality, we may logically assume that the subsidiary item remains constant, since that is the minimum assumption of any a fortiori reasoning process. In effect, ‘non-proportionality’ is the default character of a fortiori argument.
We also need to deal with the important issue of hermeneutics. When someone formulates an argument a crescendo, that person presumably believes that its constituent a fortiori argument and pro rata argument both have true premises and are correctly combined to yield a valid and therefore true conclusion, even if all these factors are not explicitly laid out and confirmed. However, how can other persons know what he (or she) had in mind if some of these relevant factors are left tacit? This is the question of hermeneutics – how are we to interpret and judge an incompletely detailed a crescendo argument presented by someone else?
When judging a given concrete argument: (a) if it is formulated in such a way that only an a fortiori conclusion is claimed, we need only test whether the conclusion follows a fortiori from the premises (assuming them true); but (b) if it is formulated in such a way that an a crescendo (i.e. proportional) conclusion is claimed, we must needs test both whether the premises yield an a fortiori conclusion and whether an additional argument pro rata can be presented that makes possible the transformation of the latter conclusion into an a crescendo one, and whether the proportionality is vague or precisely calculable (and ultimately, of course, whether all these premises are true).
Without such additional information on proportionality, if the a fortiori premises are presented (explicitly or by implication) as by themselves yielding an a crescendo conclusion, the argument must obviously be declared invalid. Moreover, even if the need for an additional premise regarding proportionality is admitted, without a precise formula for the proportionality, an exact conclusion cannot legitimately be claimed. We have to assess from the author’s words how well he understands the conditions for valid a fortiori or a crescendo argument, and whether the author considers that he has the required information at hand. If it seems likely that the author is well aware that a fortiori argument cannot logically by itself yield an a crescendo conclusion, and that a precise conclusion requires more information than a vague one, and is tacitly intending the required underlying pro rata argument or mathematical formula, his argument could be considered valid.
Of course, in interpreting a text or speech, we cannot estimate with certainty what its author’s tacit intentions are or are not. We may be able to guess at the author’s general logical knowhow from the context, and give him the benefit of the doubt in the case at hand. Or we may prefer to be strict, and demand explicit evidence that the author intends an argument pro rata in the case at hand, even if he knows the rule (since, after all, people do make mistakes). Thus, validation or invalidation often depends on the general credibility of the author of the text or speech, and on the severity of the interpreter and judge.
To put the problem in more concrete terms: when we read an ancient or modern text, by Aristotle, in the Tanakh, by a Talmudic sage, in the Christian Bible, or the Muslim Koran, or wherever, wherein the author seems to draw an a crescendo (i.e. ‘proportional’) conclusion from a fortiori premises, how should we react? We could possibly say: since the author of this argument has not justified his conclusion by explicitly proposing an appropriate accompanying pro rata argument, we must declare his reasoning fallacious. But this seems a bit rigid and lacking in subtlety, for logicians well know that discourse in practice is rarely if ever fully explicit. Our judgment in each case must clearly hinge on the wider context of the particular statement.
If we know from pronouncements elsewhere of that particular author that he has demonstrated clear understanding of the difference between a fortiori argument and a crescendo argument – i.e. that the former per se cannot yield a proportional conclusion unless it is backed up by an appropriate pro rata argument – then we can reasonably assume that in this particular case, though the author has not explicitly formulated the needed argument pro rata, he left out the missing pieces merely for brevity’s sake. We can in such case give the author the benefit of the doubt and accept his a crescendo conclusion. Of course, even if he has in other contexts demonstrated his theoretical knowledge, or at least his intuitive rationality, it is still possible that in this particular case he unthinkingly made an error of form and/or content; so we can never be absolutely sure. Nevertheless, even if the hypothesis that he knowingly drew an a crescendo conclusion from a fortiori premises cannot definitely be proved, it has inductive support in his overall logical behavior patterns.
A contrario, if we find that there is no reliable evidence that this author has mentally grasped the difference between a fortiori argument and a crescendo argument, we should certainly consider all his a crescendo arguments as fallacious reasoning. This is true even if we find him sometimes drawing a valid a fortiori conclusion and sometimes drawing a doubtful a crescendo one, for he may be drawing different conclusions from similar premises as convenient to his discursive purposes, or without rhyme or reason, and not because of any awareness that there are precise rules to follow. We may also reasonably reject an a crescendo conclusion of his, if we find that the author has elsewhere in his works denied the truth of the proportionality (i.e. the major premise of the pro rata argument) which would be needed to justify this particular a crescendo conclusion.
Thus, judging a concrete a crescendo argument which is not entirely explicit as valid or invalid is not an easy matter. Of course, if the underlying purely a fortiori argument is formally invalid and/or one or both of its premises is/are untrue, or likewise if the required argument pro rata is obviously inappropriate in form and/or content, we can reject that particular a crescendo argument. But obviously such rejection does not always prove the author to be ignorant of the conditions under which an a fortiori argument may yield an a crescendo conclusion. The author may well have in all sincerity believed the implied proportionality to be true, even if we disagree with him and can prove him wrong. In such cases, it is not the inferential process we are attacking, but some premise(s).
In any case, what we must avoid doing is getting entangled in superficial verbal considerations. Usually, people who engage in a fortiori or a crescendo reasoning do so without explicitly labeling their argument as this or that in form. Sometimes, they call their argument ‘a fortiori’, or they use the words ‘a fortiori’ or some similar expression (‘all the more’, ‘how much more’, etc.) within the argument to signal its logical intent. But in the latter case, they make no verbal distinction between a fortiori and a crescendo: firstly, because the latter expression is new (my own invention) and they have no distinctive label for it; and secondly, because the issue of proportionality is vague and uncertain in most people’s mind, if at all present.
What this implies is that we cannot reject an argument as invalid just because it has a fortiori premises and an a crescendo conclusion. Such an argument may indeed be fallacious, or it may merely be an incomplete statement of a valid argument (furthermore, in the latter case, the content may be true or false, of course). We cannot simply refer to the fact that it has not been labeled at all or that it has been labeled incorrectly. We must, as above explained, look into the matter more deeply and try to determine the actual intentions of the argument’s author, even if they are tacit, and judge the matter in all fairness.
A classic illustration of a crescendo argument, and of the hermeneutic difficulties that surround such argument, is the Talmudic reading of Numbers 12:14 in Baba Qama 25a. Without here going into all the details of this example, which are dealt with in the appropriate chapter further on (7.4), I will here just describe the reasoning involved. The Torah passage reads: “If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.” This may be construed as a valid positive subjectal a fortiori argument as follows:
Causing Divine disapproval (P) is a greater offense (R) than causing paternal disapproval (Q).
Causing paternal disapproval (Q) is offensive (R) enough to merit isolation for seven days (S).
Therefore, causing Divine disapproval (P) is offensive (R) enough to merit isolation for seven days (S).
Note that the purely a fortiori conclusion is seven days, since this is the number of days given in the minor premise. The Gemara in BQ 25a (or more precisely, a baraita it quotes), on the other hand, advocates an a crescendo conclusion, namely: “causing Divine disapproval (P) is offensive (R) enough to merit isolation for fourteen days (more than S).” This suggests that the author of this ‘proportional’ conclusion has in mind, consciously or not, the following pro rata argument:
Granting the general principle that the punishment must vary in proportion to the offense, then:
knowing from the above minor premise that: if the offense is paternal disapproval, then the punishment is seven days isolation,
it follows with regard to the conclusion that: if the offense is Divine disapproval (a greater offense), then the punishment has to be fourteen days isolation (a greater punishment).
That is to say, in order to logically end up with the Gemara’s a crescendo conclusion (fourteen days) we have to assume a general principle of proportionality between punishment and offense. Such a principle indeed exists in Jewish tradition – it is the principle of measure for measure (midah keneged midah). The hermeneutic issue here is whether the author of the a crescendo conclusion (i.e. of the Gemara, or of the baraita it relies on) can be reasonably assumed to have reasoned thus (i.e. by means of an argument pro rata) – or whether he believed the a crescendo conclusion to proceed directly from the a fortiori premises, without need of the assistance of the principle of measure for measure.
Another issue in hermeneutics that needs underlining is the issue of the exactitude of the quantity specified in the a crescendo argument conclusion. As we have seen, the formal conclusion is essentially rather vague – that is, the concluding predicate Sp of positive subjectal a crescendo argument means nothing more than ‘more than Sq’, and the concluding subject Sq of positive predicatal a crescendo argument means nothing more than ‘less than Sp’. In most discourse, the subsidiary term used in the conclusion of an a crescendo argument is accordingly vague. But in some cases, a rather precise quantity is proposed (for example, in the above Talmudic illustration, precisely 14 days are specified).
The questions then arise: on the basis of what precise information did the speaker arrive at this specific numerical result? Is he claiming to have a mathematical formula that makes possible its calculation, or at least a generally accepted conventional table? If so, what is it and how reliable is it (merely probable or sure)? Or is he making an unsubstantiated claim, giving the misleading impression that a vague a crescendo argument (or even purely a fortiori argument) can yield such a quantitatively precise conclusion? Is his discourse scientific or rhetorical? Here again, it is only by careful examination of the larger context that we can decide what the speaker consciously or subconsciously intended.
We cannot fully understand the practice of a fortiori argument without consideration of relative middle terms. Two middle terms R1 and R2 may be said to be relative (or antiparallel), if ‘more R1’ is equivalent to ‘less R2’, and vice-versa. Examples of such terms abound: much and little, long and short, big and small, far and near, hard and soft, heavy and light, stringent and lenient, good and bad, beautiful and ugly, hot and cold, rich and poor, and so forth. In principle, any term that varies quantitatively (in magnitude, in direction, in measure or degree of any sort) may give rise to a relative term, although we do not commonly construct relative terms without necessity.
Let us first consider commensurative propositions with relative terms. The two subjectal forms “A is more R1 than B is” and “B is more R2 than A is” may be taken to imply each other, i.e. are equivalent. Such propositions are said to be each other’s reverse (note the reversion of roles of A and B in them). For example, if the relative terms are ‘long’ and ‘short’, then if A is longer than B, it follows that B is shorter than A, and vice versa. Similarly, the predicatal commensurative proposition “More R1 is required to be A than to be B” may be reverted to “More R2 is required to be B than to be A,” and vice versa (again note the reversion of roles of A and B). For example, using the same relative terms: if more length is needed to be A than to be B, it follows that more shortness is needed to be B than to be A, and vice versa.
The formal difference between conversion and reversion is that, in conversion, the major term remains major (i.e. the more), and the minor remains minor (i.e. the less), and the middle term remains unchanged; whereas in reversion, the major term becomes the new minor, and the minor term becomes the new major, and the middle term is replaced by its relative. However, on closer scrutiny we realize that the converse and the reverse of a commensurative proposition are effectively the same. This is obvious, since they are both implicants of the same form. For instance, in the case of the subjectal form “A is more R1 than B,” its converse “B is less R1 than A,” and its reverse “B is more R2 than A,” and indeed the converse of the latter “A is less R2 than B,” are equivalent to it and to each other. Similarly for the corresponding predicatal forms.
Relative terms usually evolve from absolute terms. That is to say: initially, the terms R1 and R2 (e.g. much and little, or whatever) are intended absolutely, so that what is R1 is greater than what is R2. They are conceived as separated at some conventional cut-off value (say, v), such that what is more than v is R1 (e.g. much) and what is less than v is R2 (e.g. little). Then, when we realize that this dividing line v is rather conventional, and may in practice be fuzzy rather than precise, the terms are made relative, i.e. such that the whole range of values under consideration may be viewed as R1 in one direction and as R2 in the opposite direction. In one direction, the values of R1 increase and those of R2 decrease, and in the other direction the opposite occurs. Neither direction is formally preferable to the other. For this reason, such terms may be characterized as antiparallel.
The propositions “A is more R than B is” and “A is less R than B is” cannot both be true at once, but they can both be false. There is an alternative to them, viz. “A is as much R as B.” Note that this third form is applicable to any equal quantity of R in A and B, just as the other two forms are applicable to any unequal quantities. Likewise, when dealing with relative middle terms R1 and R2, we must take into consideration the three alternatives: “A is more R1 (and less R2) than B is,” and “A is less R1 (and more R2) than B is,” and “A is as much R1 (and as much R2) as B is.” These three forms are mutually exclusive, and usually but not always exhaustive.
This brings us to the issue of negative forms. In most cases, the three forms just mentioned are exhaustive, which means that the denial of any two of them implies the affirmation of the third. However, this is not always true. It is quite possible for A to be neither more R nor less R than B, nor as much R as B, for the simple reason that the whole concept of R is not applicable to A or to B. For example, though all objects extended in physical or mental space may be said to be bigger or smaller or equal in size, such characterizations are inapplicable to spiritual and abstract objects; i.e. the latter must be admitted to be neither bigger nor smaller nor equal in size.
Thus, to determine the oppositional relations of given comparative forms, we must first ask whether the concept(s) used as middle term, viz. R (or R1 and R2), is (or are) universally applicable or applicable only within a given sphere. If it is (or they are) universally applicable the said three positive forms (more, less, or equally R – or ditto with R1 and R2) are exhaustive; but if they are applicable only within a circumscribed domain, they may be all three at once denied. Of course, in the latter case, it remains true that the three positive forms are exhaustive contextually, within the sphere of their relevance; so we may continue to think of them as exhaustive provided we keep in mind that this is true conditionally, granting the applicability of the middle term used.
All the above was said for subjectals. It can also be said, mutatis mutandis, for predicatals; and more broadly for implicationals.
Let us now consider the special case of the relativity between a term R and its complement notR. As we saw in the previous chapter (1.4), although these two terms are strictly speaking (by the law of non-contradiction) mutually exclusive, it is possible to conceive of a broader term with the same label ‘R’ which is inclusive of both R in the strict sense and notR, the negation of R in the strict sense. Such broader meaning of R has obviously no negation of its own, note well, since by definition it embraces all conceivable values of the original term R and its negation from plus infinity to minus infinity. However, just as we can construct a broader term R, we can also construct a broader term notR. The latter is not a negation of R in the broader sense, note well, but a term that like it by definition embraces all conceivable values of the original term R and its negation from plus infinity to minus infinity.
Thus, although in their strict senses the terms R and notR are absolutes, and clear contradictories, the broader or looser terms derived from them, also in everyday discourse labeled R and notR, may be viewed as relative terms, which mutually suggest each other, since they both embrace the full range of the strict terms R and notR, although they do so in opposite directions. That is, what is more R is less notR, and vice versa; thus, for instance, “A is more R than B” and “A is less notR than B” are equivalent. For examples: the subjectal forms: “A is more active (R) than B is” and “A is less inactive (notR) than A is;” and likewise, the predicatal forms: “More action (R) is required to be A than to be B” and “Less inaction (notR) is required to be B than to be A.”
As regards eductions, we observed earlier that for any pair of relative terms (R1, R2) the converse and the reverse of a commensurative proposition are effectively the same. This is also true here, with regard to R and notR. For instance, in the case of “A is more R than B,” its converse “B is less R than A,” and its reverse “B is more notR than A,” and indeed the converse of the latter “A is less notR than B,” are four logically equivalent propositions. Similarly for the corresponding predicatal forms.
As regards oppositions, the three sets of propositions “A is more R (less notR) than B is,” “A is less R (more notR) than B is” and “A is as much R (as much notR) as B is,” are not only mutually exclusive but also exhaustive, since here the relative terms are contradictories (so that nothing can be said to lack both R and notR).
The egalitarian forms “A is as much R as B is” and “A is as much notR as B is” are quite compatible; indeed, they imply each other. This may seem odd at first sight, due to thinking in absolute terms. But it is clear that these two propositions do not imply that A and B are both R and notR in absolute terms. They just mean, respectively, that A and B have the same value of R and the same value of notR. And since that value, whatever its magnitude and polarity (positive, zero or negative) is one throughout, the two forms must imply each other.
So for subjectals, and similarly, mutatis mutandis, for predicatals; and more broadly for implicationals.
Some readers may find the above treatment a bit confusing, in view of the different senses of the terms R and notR, as absolute or as relative. For them, I propose a more symbolic treatment, as follows. In this context, let us use the following notation: given the absolute term R and its negation notR, we can conceive of the relative terms R and notR (symbolically distinguished by being underlined).
Whereas the terms R and notR are mutually exclusive, the terms R and notR are inclusive, in the sense that each of them includes both R and notR. Yet R and notR are not identical, because they differ in direction, each being the reverse of the other. That is, whereas R refers to R as a positive quantity and to notR as a zero or negative quantity of R, notR refers to notR as a positive quantity and R as a zero or negative quantity of notR. Thus, R signifies a direction from notR (negative or zero R) to R (positive R), while notR signifies a direction from R (negative or zero notR) to notR (positive notR). We can express these definitions as formulae (where ‘iff’ means ‘if and only if’):
Iff X is R, then R > 0 (i.e. a positive quantity of R).
Iff X is not R, then R ≤ 0 (i.e. a zero or negative quantity of R).
Iff X is not R, then notR > 0 (i.e. a positive quantity of notR).
Iff X is R, then notR ≤ 0 (i.e. a zero or negative quantity of notR).
These formulae imply that ‘R > 0’ = ‘notR ≤ 0’ (since both imply ‘X is R’), and that ‘R ≤ 0’ = ‘notR > 0’ (since both imply ‘X is not R’). Note well that ‘zero R’ and ‘zero notR’ are not the same point, but contradictories, since the former means that R is absent whereas the latter means that R is present. This must be kept in mind to avoid inconsistency. However, the propositional forms involving the terms R and notR being comparative, this issue of ‘zero’ having a different meaning in each of the antiparallel continua has no impact. This will become evident when we consider oppositions and eductions, next.
By definition of ‘more’ and ‘less’, the propositions “A is more R than B” and “B is less R than A” are equivalent (if either is true, so is the other). Likewise, of course, “A is more notR than B” and “B is less notR than A.”
By definition of ‘more’ and ‘less’, the propositions “A is more R than B” and “A is less R than B” are incompatible (only one may be true). Likewise, of course, “A is more notR than B” and “A is less notR than B.”
By definition of R and notR, the propositions “A is more R than B” and “A is less notR than B” are equivalent. Likewise, “A is more notR than B” and “A is less R than B.”
It follows that the propositions “A is more R than B” and “A is more notR than B” are incompatible. Likewise, of course, “A is less R than B” and “A is less notR than B.”
We might define “A is as much R as B” in relation to the propositions “A is more R than B” and “A is less R than B,” as either implying both (i.e. as their intersection) or as denying both (i.e. as an alternative to them). The latter definition seems best, since in accord with actual practice. Similarly, we may take it that “A is as much notR as B” denies both “A is more notR than B” and “A is less notR than B.”
The propositions “A is as much R as B” and “A is as much notR as B” imply each other. For instance, if A and B are both at (say) R = 50, they are equally R (at +50) and equally notR (at –50). This is true even when R = 0 or when notR = 0, i.e. even though a zero quantity of R is a positive quantity of notR and a zero quantity of notR is a positive quantity of R, because each of the propositions “A is as much R as B” and “A is as much notR as B” refers to only one of the relative terms and anyway does not mention any actual quantity.
Whereas, as we have seen, some relative terms R1 and R2 might be both denied (if there exists things to which neither is applicable), in the case of relative complements R and notR, denial of both is impossible. Thus, here, the propositions “A is more R (= less notR) than B,” “A is less R (= more notR) than B” and “A is as much R (or notR) as B,” are always exhaustive (one of them must be true).
So for subjectals, and similarly, mutatis mutandis, for predicatals; and more broadly for implicationals.
Whereas the oppositional relation between the absolute terms R and notR is that they (when predicated of the same subject) are contradictory – i.e. they are incompatible (cannot both be true) and exhaustive (cannot both be false), the relative terms R and notR behave differently. They appear in comparative propositions only, and in that context may be affirmed together (one being more, the other less, or both as much). However, they cannot be both discarded. The peculiarity of such relative terms is that neither of them has a true negation, since both refer to the same full range of existential possibilities from minus infinity through zero to plus infinity (though in opposite directions). That is to say, each of them includes the whole world, as it were (but with a difference in perspective). Everything (not just some X) can be fitted in the continuum R, and simultaneously everything (not just some X) can be fitted in the continuum notR. For this reason, R and notR are not each other’s negation, note well.
Although our introduction of the underlined symbols R and notR for relative complements, to distinguish them from the absolute terms R and notR, does clarify things somewhat, I will not make further use of them here, for the simple reason that I prefer a logic of ordinary language to symbolic language, and in ordinary language we would signify our intention when it is unclear simply by saying of a given term that it is intended as relative (or inclusive). It is just as easy to mentally or out loud say the word ‘relative’ as to say the word ‘underlined’; and the disadvantage of the latter over the former is that one must still add the thought (in words or wordlessly) ‘and underlined means relative’, so one’s thinking is slowed down!
Thus far, we have compared commensurative propositions with relative terms. Let us now compare suffective propositions with relative terms. Two eductions from suffectives need to be investigated: movement from a positive to a negative form, or vice versa, and movement from a subjectal to a predicatal form, or vice versa. Concerning the said changes of polarity, we can do a good deal; but concerning changes of orientation, little can be done. We shall first deal with copulative forms, then with implicationals.
Copulative forms. First, let us interpret the negative forms. As already seen, the positive form “X is R enough to be Y” implies that “X is R” and “X is Y,” as well as “Rx ≥ Ry,” where R is understood as an inclusive middle term, which includes not only R > 0 but also R = 0 and possibly also R < 0. The negation of this form, i.e. “It is not true that X is R enough to be Y,” may colloquially be loosely expressed as “X is R not enough to be Y” or “X is not enough R to be Y” or “X is not R enough to be Y” or “X is R enough not to be Y,” putting the negation in various positions.
However, to avoid ambiguities, we might prefer to write more precisely “X is R [not-enough] to be Y” or “X is [not-enough] R to be Y” or “X is not [R-enough] to be Y” or “X is not [enough-R] to be Y,” adding hyphens as shown. All these forms are equivalent in that they imply “X is R” and “X is not Y,” as well as “Rx < Ry,” note well. They all are contradictory to the said positive form, although they have in common with it that “X is R” (where R is inclusive of notR, remember), because they imply “Rx < Ry” (instead of “Rx ≥ Ry”) and thence “X is not Y” (instead of “X is Y”).
As regards the form “X is R enough to be [not-Y],” with the negation attached to the predicate, it is obviously incompatible with the form “X is R enough to be Y,” since X cannot be R enough to be both Y and not-Y. But are these forms contrary or contradictory? We might think they could both be false, since they have in common that X is R, and this might be false. However, since R is here intended as an inclusive term, “X is not R” is implicitly included in to “X is R;” so it is useless to focus on this factor. We might alternatively compare the form “X is R enough to be [not-Y]” to the preceding three forms, and think that it is not equivalent to them, since it implies “Rx ≥ Rnot-y” whereas they imply “Rx < Ry.” However, albeit the opposite directions of ≥ and < as well as the different suffixes involved, these forms have in common the implication that “X is not Y.”
Wherever the dividing line along the continuum R for Y or for not-Y, once we know on which side of it a subject X falls, the matter is settled. Everything hinges on the result (Y or not-Y), however the condition for it is expressed in a given suffective proposition. The result is what matters, the rest is history. For this reason, whatever their apparent formal differences, i.e. their differences of wording, all suffective forms that imply “X is Y” are equivalent to each other, and all those that imply “X is not Y” are equivalent to each other.
Consider now the ambiguous form: “X is not R enough to be Y,” when its intent is “X is [notR] enough to be Y” (note well the position of the hyphen). Albeit its negative middle term (notR), this is a positive form, which implies that “X is not R” and “X is Y.” Since notR here is also an inclusive term, it does not contradict R but includes it as a possibility among others, and this form is really equivalent to “X is R enough to be Y.” Therefore, we should be careful not to confuse the form: “X is [notR] enough to be Y” with any of the forms: “X is not R-enough to be Y” or “X is R not-enough to be Y” or “X is not-enough R to be Y” or “X is not enough-R to be Y.” They are in fact contradictories.
These interpretations may well appear confusing, because we would not at first sight consider the forms “X is R enough to be Y” and “X is [notR] enough to be Y” to be equivalent. The reason we would not normally equate them is that the ranges R and notR go in opposite directions (the more R, the less notR, and vice versa), whereas the notion of sufficiency is originally in one direction only. Thus, given that “X is R enough to be Y,” we would not naturally simultaneously think that “X is notR enough to be Y.” We would rather wonder whether X is notR enough to be not-Y, associating the negation of R with the negation of Y. But the answer to that question would have to be that “X is not notR enough to be not-Y,” which can also be stated as “X is notR enough to be Y.”
Let me clarify this further. The expression “enough” originally signifies a minimum value, rather than a maximum. Normally, if we say that as of a certain value of R, there is “enough” for Y, we would not say that below that value there is “enough” for not-Y. However, if pressed to the wall, we are forced to say that since there are logically only two choices, viz. Y or not-Y, we must say that “X is R enough to be not-Y” is contradictory to “X is R enough to be Y.” Similarly, as regards forms with notR. It is interesting to note that, although when dealing with commensuratives the difference in direction between R and notR is significant, because more R (less notR) and more notR (less R) are incompatible, when dealing with suffectives R and notR are effectively interchangeable.
In the case of suffectives, since the issue of direction is absent, the terms R and notR in their inclusive senses are just different labels for the same position on the scale. Whether the scale goes from here to there or from there to here, the de facto intended position remains the same. For this reason, the terms are two sides of the same coin, merely differently labeled. In conclusion, though certain constructions are a bit artificial, from a formal point of view we have to accept them and evaluate them as above suggested, so as to be able to anticipate and deal with all possibilities. The same can be done with predicatal suffectives, and with implicational ones, of course.
This understanding for R and its complement notR can obviously be extended to any two terms, R1 and R2, known to be relative. Given, say, “X is R1 enough to be Y,” or any form which like it implies that “X is R1” and “X is Y,” we can readily educe that “X is R2 enough to be Y,” or any form which like it implies that “X is R2” and “X is Y,” because “X is R1” and “X is R2” are equivalent even though in opposite directions. The moment we introduce in either of these forms a negation that affects the predicate Y, or the relational element “enough,” the proposition may be taken as contradictory to the preceding. If on the other hand we introduce a negation that affects the middle term only, making it notR1 or notR2, the relation of X to Y remains essentially unchanged.
Based on this reasoning, we can always interpret suffective propositions involving one or more negative elements. All we need to do is first decide or determine just what each negation is intended to negate. If a negation is aimed at the middle term, assuming it is inclusive, nothing is essentially changed. All other negations are significant; though of course pairs of them cancel each other out. Thus for examples, “X is notR1 not enough not to be Y” is equivalent to “X is R1 enough to be Y;” whereas “X is notR1 not enough to be Y” is equivalent to “X is R1 enough to be not-Y.” The utility of this kind of inference will be seen when we deal with traductions, in the next chapter (3.5).
Second, let us consider the rewriting of subjectal suffectives as predicatals, or vice versa. If we look back at the definitions of these forms, we can see the difficulties that such pursuits present. Consider the components of a proposition of the copulative form “X is R enough to be Y”:
X is to a certain measure or degree R (say, Rx);
whatever is to a certain measure or degree R (say, Ry), is Y, and
whatever is not to that measure or degree R (i.e. is not Ry), is not Y;
and Rx is greater than (or equal to) Ry (whence: “Rx implies Ry”).
In order to obtain a suffective proposition with Y as subject and X (or even its negation) as predicate, we would need to contrapose the conditional propositions concerning X and Y. But though we can do that for Y, we do not have the necessary material for X. Moreover, the quantitative comparison would no longer be appropriate, anyway. So we cannot, as far as I can see, change the orientation of a suffective proposition.
Implicational forms. Let us first closely examine various negative forms. The positive form “X implies R enough to imply Y” implies that “X implies R” and “X implies Y” and “Rx ≥ Ry,” where R is understood as an inclusive middle thesis, which includes not only R > 0 but also R = 0 and possibly also R < 0. The negation of this form, i.e. “It is not true that X implies R enough to imply Y,” may colloquially be loosely expressed in various ways, but it is important here (more so than in the case of conjunctives) to avoid ambiguities. If we want a negative form to closely fit the positive form in all essential respects, we must ensure that it implies “X implies R” and “X does not imply Y” and “Rx < Ry.” The form “X implies R, but not enough to imply Y” would best fit this bill, suggesting that there is a threshold value of R (say Ry) only as of which Y is implied, but that threshold is not reached and therefore Y is not implied; note that the negation here is primarily focused on the relational expression ‘enough’. We should take this as the contradictory form.
As regards the form “X implies R, enough not to imply Y” – which suggests that there is a threshold value of R (say Ry) as of which Y is not implied, and that threshold is reached – although this form still in fact implies both “X implies R” and “X does not imply Y,” it differs from the preceding in that here “Rx ≥ Ry” (though Ry has a different meaning here, note well); that is to say, this form is essentially positive as regards structure, though contrary to the above positive form. As regards the forms “X does not imply R enough to imply Y” or “X does not imply enough R to imply Y,” which are hardly distinguishable, they are also contrary to the positive form, but in a different manner: neither of the propositions “X implies R” and “X does not imply Y” can be educed from them, although presumably they suggest that there is a threshold value of R (say Ry) as of which Y is implied, and that threshold is not reached, meaning that “Rx < Ry;” note that these forms do not clarify whether “X implies R” or “X does not imply R.”
Consider now the form: “X implies notR enough to imply Y.” We should be careful not to confuse this form with any of the preceding four forms: they are in fact antitheses. Albeit its negative middle thesis (notR), this is a positive form, which implies that “X implies notR” and “X implies Y.” NotR here being presumably an inclusive thesis, it does not contradict R but includes it as a possibility among others; therefore, this form is really equivalent to “X implies R enough to imply Y” – from which it follows that their contradictory forms, “X implies notR, but not enough to imply Y” and “X implies R, but not enough to imply Y,” are equivalent to each other.
As regards the rewriting of antecedental suffectives as consequentals, or vice versa, we can as we did for copulatives (and even more so) safely say that it is not formally feasible.
 We could, of course, equally well infer that “P is G” directly from the conclusion that “P is S.” The end result is the same. But the point made here is that for any positive subjectal a fortiori argument in relation to a species S, there is a similar argument in relation to any genus G of S.
 Here again, we could of course equally well infer that “S is P” directly from the conclusion that “G is P.” The end result is the same. But the point made here is that for any positive predicatal a fortiori argument in relation to a genus G, there is a similar argument in relation to any species S of G.
 Though it is nowadays recommended in orthodox Jewish circles to write any name of God, even in languages other than Hebrew, in truncated fashion (e.g. as G-d), so as to avoid the eventual erasing or tearing of such a holy word – I have decided not to practice this restriction anymore, because this makes it difficult for people to search for the word in the Internet or in their computers. Of course, then, every time we change page with a computer or other reader, we effectively erase all the names of God that happen to be in it. This is very unfortunate – but I think inevitable in this day and age. In other words, the recommendation is in the last analysis impractical and impracticable.
 By “separate species of G,” I here mean that S2 is not a genus of S1.
 By “separate genus of S,” I here mean that G2 is not a species of G1.
 ‘A crescendo’, of course, refers to increase in magnitude – and so makes us think especially of positive subjectal argument, which goes from S to ‘more than S’. We could by analogy name positive predicatal argument ‘a diminuendo’, since it involves decrease in magnitude, going from S to ‘less than S’. Negative subjectal, going from not more than S to not S, would then count under the latter heading, and negative predicatal, going from not less than S to not S, under the former. But such multiplication of names would be silly pedantry.
 Which is logically impossible, here.
 Which is here logically possible, but not necessary.
 That is, pro rata argument outside of a crescendo.
 A ‘baraita’ is a statement attributed to an author of Mishnaic times (a Tanna); whereas the Gemara is a collection of later (Amoraic) commentaries.
 For example, we may at first divide women into beautiful and ugly ones, considering that all women fall under one or the other class. Then, we may perceive some cases as doubtful, so that the dividing line is difficult to pinpoint. Moreover, we may realize that individual men sometimes differ in opinion as to which women to class where, though they may agree in some cases. Also, individual men’s viewpoints may vary over time. As a result, we relativize, and avoid laying down absolute rules. An example in physics is temperature: though the sensations of hot and cold are initially thought of as absolute, eventually the terms become relative; note that the location of the zero point and the size of the units varies in different scales (Fahrenheit, Celsius or Kelvin).
 That is, material bodies in physical space or mental images in the inner space of imagination.
 Though analogies remain possible. Thus, we might say that the soul of God is greater than that of humans, or that a generic concept is larger than a specific one. But these analogous ideas would not be used interchangeably in different domains. We would not suggest, for instance, that a soul or a concept has physical size.