3 Formalities of A-Fortiori

We shall in this chapter present, as a propaedeutic, the main formalities of a-fortiori logic and in the next three chapters consider its roots and applications within Judaic logic[1]. All the findings presented here are original.

Let us begin by listing and naming all the valid moods of a-fortiori argument[2] in abstract form; we shall have occasion in later chapters to consider examples. We shall adopt a terminology which is as close to traditional as possible, but it must be kept in mind that the old names used here may have new senses (in comparison to, say, their senses in syllogistic theory), and that some neologisms are inevitable in view of the novelty of our discoveries. (See Diagram 1.)

An explicit a-fortiori argument always involves three propositions, and four terms. We shall call the propositions: the major premise, the minor premise, and the conclusion, and always list them in that order. The terms shall be referred to as: the major term (symbol, P, say), the minor term (Q, say), the middle term (R, say), and the subsidiary term (S, say). In practise, the major premise is very often left unstated; and likewise, the middle term (we shall return to this issue in more detail later).

We shall begin by analyzing "copulative" forms of the argument. There are essentially four valid moods. Two of them subjectal in structure, and two of them predicatal in structure; and for each structure, one of the arguments is positive in polarity and the other is negative.

As we shall see further on, a similar argument with P in the minor premise and Q in the conclusion ("major to minor") would be invalid.

As we shall see further on, a similar argument with Q in the minor premise and P in the conclusion ("minor to major") would be invalid.

As we shall see further on, a similar argument with P in the minor premise and Q in the conclusion ("major to minor") would be invalid.

The expression "all the more" used with the conclusion is intended to connote that the inferred proposition is more "forceful" than the minor premise, as well as suggest the quantitative basis of the inference (i.e. that it is a-fortiori). Note that instead of the words "and" or "yet" used to introduce the minor premise, we could just as well have used the expression "nonetheless", which seems to balance nicely with the phrase "all the more".

The role of the major premise is always to relate the major and minor terms (P and Q) to the middle term (R); the middle term serves to place the major and minor terms along a quantitative continuum. The major premise is, then, a kind of comparative proposition of some breadth, which will make possible the inference concerned; note well that it contains three of the terms, and that its polarity is always positive (this will be demonstrated further down). The term which signifies a greater measure or degree (more) within that range, is immediately labeled the major; the term which signifies a smaller measure or degree (less) within that range, is immediately labeled the minor (these are conventions, of course). P and Q may also conveniently be called the "extremes" (without, however, intending that they signify extreme quantities of R).

Note that here, unlike in syllogism, the major premise involves both of the extreme terms and the minor premise may concern either of them; thus, the expressions major and minor terms, here, have a different value than in syllogism, it being the relative content of the terms which determines the appellation, rather than position within the argument as a whole. Furthermore, the middle term appears in all three propositions, not just the two premises.

The function of the minor premise is to positively or negatively relate one of the extreme terms to the middle and subsidiary terms; the conclusion thereby infers a similar relation for the remaining extreme. If the minor premise is positive, so is the conclusion; such moods are labeled positive, or modus ponens in Latin; if the minor premise is negative, so is the conclusion; such moods are labeled negative, or modus tollens. Note well that the minor premise may concern either the major or the minor term, as the case may be. Thus, the inference may be "from major (term, in the minor premise) to minor (term, in the conclusion)" - this is known as inference a majori ad minus; or in the reverse case, "from minor (term, in the minor premise) to major (term, in the conclusion)" - this is called a minori ad majus.

There are notable differences between subjectal and predicatal a-fortiori. In subjectal argument, the extreme terms have the logical role of subjects, in all three propositions; whereas, in predicatal argument, they have the role of predicates. Accordingly, the subsidiary term is the predicate of the minor premise and conclusion in subjectal a-fortiori, and their subject in predicatal a-fortiori.

Because of the functional difference of the extremes, the arguments have opposite orientations. In subjectal argument, the positive mood goes from minor to major, and the negative mood goes from major to minor. In predicatal argument, the positive mood goes from major to minor, and the negative mood goes from minor to major. The symmetry of the whole theory suggests that it is exhaustive.

With regard to the above mentioned invalid moods, namely major-to-minor positive subjectals or negative predicatals, and minor-to-major negative subjectals or positive predicatals, it should be noted that the premises and conclusion are not in conflict. The invalidity involved is that of a non-sequitur, and not that of an antinomy. It follows that such arguments, though deductively valueless, can, eventually, play a small inductive role (just as invalid apodoses are used in adduction).

"Implicational" forms of the argument are essentially similar in structure to copulative forms, except that they are more broadly designed to concern theses (propositions), rather than terms. The relationship involved is consequently one of implication, rather than one of predication; that is, we find in them the expression "implies", rather than the copula "is".[3]

We need not repeat everything we said about copulative arguments for implicational ones. We need only stress that moods not above listed, which go from major to minor or minor to major in the wrong circumstances, are invalid. The essentials of structure and the terminology are identical, mutadis mutandis; they are two very closely related sets of paradigms. The copulative forms are merely more restrictive with regard to which term may be a subject or predicate of which other term; the implicational forms are more open in this respect. In fact, we could view copulative arguments as special cases of the corresponding implicational ones[4].

A couple of comments, which concern all forms of the argument, still need to be made.

The standard form of the major premise is a comparative proposition with the expression "more...than" (superior form). But we could just as well commute such major premises, and put them in the "less...than" form (inferior form), provided we accordingly reverse the order in it of the terms P and Q. Thus, 'P is more R than Q' could be written 'Q is less R than P', 'More R is required to be P than to be Q' as 'Less R is required to be Q than to be P', and similarly for implicational forms, without affecting the arguments. These are mere eductions (the propositions concerned are equivalent, they imply each other and likewise their contradictories imply each other), without fundamental significance; but it is well to acknowledge them, as they often happen in practise and one could be misled. The important thing is always is to know which of the terms is the major (more R) and which is the minor (less R).

Also, it should also be obvious that the major premise could equally have been an egalitarian one, of the form "as much...as" (e.g. 'P is as much R as Q (is R)'). The arguments would work equally well (P and Q being equivalent in them). However, in such cases it would not be appropriate to say "all the more" with the conclusion; but rather use the phrase "just as much". Nevertheless, we must regard such arguments as still, in the limit, a-fortiori in structure. The expression "all the more" is strictly-speaking a redundancy, and serves only to signal that a specifically a-fortiori kind of inference is involved; we could equally well everywhere use the word "therefore", which signifies for us that an inference is taking place, though it does not specify what kind.

It follows that each of the moods listed above stands for three valid moods: the superior (listed), and corresponding inferior and egalitarian moods (unlisted).

Lastly, it is important to keep in mind, though obvious, that the form 'P is more R than Q' means 'P is more R than Q is R' (in which Q is as much a subject as P, and R is a common predicate), and should not be interpreted as 'P is more R than P is Q' (in which P is the only subject, common to two predicates Q and R, which are commensurable in some unstated way, such as in spatial or temporal frequency, allowing comparison between the degrees to which they apply to P). In the latter case, R cannot serve as middle term, and the argument would not constitute an a-fortiori. The same can be said regarding 'P implies more R than Q'. Formal ambiguities of this sort can lead to fallacious a-fortiori reasoning[5].

A-fortiori logic can be extended by detailed consideration of the rules of quantity. These are bound to fall along the lines established by syllogistic theory. A subject may be plural (refer to all, some, most, few, many, a few, etc. of the members of a class X) or singular (refer to an individual, or to a group collectively, by means of a name or an indicative this or these X). A predicate is inevitably a class concept (say, Y), referred to wholly (as in 'is not Y') or partly (as in 'is Y'); even a predicate in which a singular term is encrusted (such as 'pay Joe') is a class-concept, in that many subjects may relate to it independently ('Each of us paid Joe'). The extensions (the scope of applicability) of any class concept which appears in two of the propositions (the two premises, or a premise and the conclusion) must overlap, at least partly if not fully. If there is no guarantee of overlap, the argument is invalid because it effectively has more than four terms. In any case, the conclusion cannot cover more than the premises provide for.

In subjectal argument, whether positive or negative, since the subjects of the minor premise and conclusion are not one and the same (they are the major and minor terms, P and Q), we can only quantify these propositions if the major premise reads: "for every instance of P there is a corresponding instance of Q, such that: the given P is more R than the given Q". In that case, if the minor premise is general, so will the conclusion be; and if the minor premise is particular, so will the conclusion be (indefinitely, note). This issue does not concern the middle and subsidiary terms (R, S), since they are predicates. In predicatal argument, whether positive or negative, the issue is much simpler. Since the minor premise and conclusion share one and the same subject (the subsidiary term, S), we can quantify them at will; and say that whatever the quantity of the former, so will the quantity of the latter be. With regard to the remaining terms (P, Q, R), they are all predicates, and therefore not quantifiable at will. The major premise must, of course, in any case be general.

All the above is said with reference to copulative argument; similar guidelines are possible for implicational argument. These are purely deductive issues; but it should be noted that in some cases the a-fortiori argument as a whole is further complicated by a hidden argument by analogy from one term or thesis to another, so that there are, in fact, more than four terms/theses. In such situations, a separate inductive evaluation has to be made, before we can grant the a-fortiori inference.

Another direction of growth for a-fortiori logic is consideration of modality. In the case of copulative argument, premises of different types and categories of modality would need to be examined; in the case of implicational argument, additionally, the different modes of implication would have to be looked into. Here again, the issues involved are not peculiar to a-fortiori argument, and we may with relative ease adapt to it findings from the fields concerned with categorical and conditional propositions and their arguments. To avoid losing the reader in minutiae, we will not say anymore about such details in the present volume.

Once examined in their symbolic purity, the arguments listed above all appear as intuitively obvious: they 'make sense'. We can, additionally, easily convince ourselves of their logical correctness, through a visual image as in Cartesian geometry. Represent R by a line, and place points P and Q along it, P being further along the line than Q - all the arguments follow by simple mathematics. However, the formal validation of valid moods, and invalidation of invalids, are essential and will now be undertaken.

The propositions colloquially used as premises and conclusions of a-fortiori arguments are entirely reducible to known forms, namely (where X, Y are any terms or theses, as the case may be) to categoricals ('X is Y', 'X is not Y'), conditionals ('if X then Y', ' if X not-then Y') and comparatives (X > or = or < Y, or their negations; and X É Y, or its negation[6]). Consequently, a-fortiori arguments may be systematically explicated and validated by such reductions. We shall call the colloquial forms bulk forms, and the simpler forms to which they may be reduced their pieces.

Let us first consider the major premises of a-fortiori arguments, whose forms we will label commensurative, since they measure off the magnitudes of the major and minor terms/theses (respectively P, Q) in the scale of the middle term/thesis (R).

This concerns the superior form (briefly put, 'P is more R than Q'). Similarly, for the egalitarian ('P is as R as Q') and inferior ('P is less R than Q') forms[8], except that for them Rp=Rq and Rp<Rq, respectively. Thusly for copulatives; with regard to implicationals (bulk form, 'P implies more/as much/less R than/as Q implies'), the first two pieces take the form: 'P implies Rp' and 'Q implies Rq' and the third piece remains the same.

Again, this concerns the superior form. The corresponding egalitarian and inferior forms[9] differ only in that for them the third piece reads Rp=Rq and Rp<Rq, respectively. Thusly for copulatives; with regard to implicationals (bulk form, 'More/as much/less R is required to imply P than/as to imply Q') there is little difference, except that the first two pieces take the form: 'Rp implies P' and 'Rq implies Q'.

Note that given the first two pieces, the superior, egalitarian and inferior bulk forms are exhaustive alternatives, since the available third pieces are so; that is, if any two are false, the third must be true. Note also the symmetries between subjectal and predicatal forms, after reduction to categorical/conditional and comparative propositions, despite their initial appearance of diversity; their differences are in the relative positions of the terms.

It should be clear that the comparative propositions Rp>Rq, Rp=Rq, Rp<Rq, seem simple enough when we deal with exact magnitudes. But in the broadest perspective, Rp and Rq may each be an exact magnitude, or a single interval, ranging from an upper bound to a lower bound (including the limits), or a disjunction of several intervals; this can complicate things considerably. To keep things simple, and manageable by ordinary language, we will assume Rp and Rq to be, or behave as, single points on the R continuum; when P or Q are classes rather than individuals, we will just take it for granted that the propositions concerned intend that the stated relation through R is generally true of all individual members referred to, one by one.

We need also emphasize, though we will avoid dealing with negative commensuratives in the present work so as not to complicate matters unduly, that the strict contradictory of each bulk form is an inclusive disjunction of its three pieces. For example, in the case of the copulative superior subjectal form, it would be, briefly put: 'Some P are not R, and/or some Q are not R, and/or Rp = or < Rq'; similarly, mutadis mutandis, for the other forms (remembering, for implicationals, that the negation of 'if... then...' is 'if... not-then...', and not 'if... then-not...' which is merely contrary). We may continue to use the same labels (superior, egalitarian and inferior) for negative propositions, even though in fact the meaning is reversed by negation, in order that the intent of the original (positive) forms be kept in mind.

Thus viewed in pieces, the negations of major premises are clear enough; but we must forewarn that the negative versions of the bulk forms are easily misinterpreted. For example: 'What is P is not more R than what is Q' might be taken to mean 'What is P is R as much as or less than what is Q' which is not equivalent to the strict contradictory, since it still maintains the conditional pieces, while denying only the comparative piece. Other interpretations might be put forward. For these reasons, negatives are best expressed by prefixing 'Not-' to the whole positive proposition concerned.

For logicians (as against grammarians) the precise interpretation of variant forms is not so important; what matters is what conventions we need to establish, as close as possible to ordinary language, to assure full formal treatment. We can do this without affecting the versatility of language, because it is still possible to express alternative interpretations by means of the language already accepted as formal.

Let us now consider the forms taken by minor premises and conclusions of a-fortiori arguments, which we will call suffective, since, broadly put, they express the sufficiency (or its absence) of a term/thesis to satisfy some quantitative condition (the middle term/thesis, R) to obtain some result[10]. In subjectal argument the minor premise and conclusion have P or Q (the extreme terms) as subject and S (the subsidiary term) as predicate, whereas in predicatal argument they have S as subject and P or Q as predicate, but otherwise the form remains identical; for this reason, we may deal with all issues using a single paradigm, having X and Y as subject and predicate respectively and R as middle term.

This concerns the copulative form; in the case of the implicational form 'X implies R enough to imply Y', the first two pieces are 'X implies Rx' and 'Ry implies Y', and the third piece is the same.

In the broadest perspective, Ry may be an exact magnitude, or a single interval, ranging from an upper bound to a lower bound (including the limits), or a disjunction of several intervals. Similarly for Rx. Therefore, Rx is "included in" Ry, if and only if every value of Rx is a value of Ry; if only some points overlap, or every value of Ry is a value of Rx but not conversely, then Rx may not be said to be (wholly) "included in" Ry by our standards. However, very commonly, Ry expresses the threshold of a continuous and open-ended range, as of which, and over and above which or under and below which, the consequent Y occurs; while Rx is often a point (for an individual X) or a limited range (for the class of X).

Since negative suffectives (unlike negative commensuratives) are used in the primary forms of a-fortiori argument which we identified earlier, they must be given attention too. The strict contradictory of the above conjuncts of two categoricals and one comparative is an inclusive disjunction of their denials:

This concerns the copulative form; in the case of the implicational form 'X does not imply R enough to imply Y', the pieces are 'X does not imply Rx' and/or 'Ry does not imply Y', and/or Ry does not include Rx.

Here (unlike in the case of commensuratives) we have chosen, by convention - because we must have some practical verbal tool for lack of sufficiency, or insufficiency - to adopt a form with the negation encrusted in it to signify the generic form of negation, namely 'Not-{X is R enough to be Y}'. But it must be kept in mind that this language, which we have frozen to one of its colloquial senses for the purposes of a formal analysis, may in practise be interpreted variously, as 'X is not-R, enough to be Y', or as 'X is R, but not-enough to be Y', or as 'X is not R-enough to be Y', for instances. I will not here say more about such variants, but only wish to give the reader an idea of the complexities involved. In general, absolute precision can only be attained through the explicit listing of the pieces intended, be they positive, negative or unsettled.

Having sufficiently analyzed the propositional forms involved for our purposes here, we can now proceed with reductive work on a-fortiori argument proper. The positive moods here considered are the paradigms of the form; the negative moods are really derivative. The negative moods can always be derived from the positive moods by means of a reductio ad absurdum, just like in the validation of syllogisms or apodoses. That is, we can say: "for if the proposed conclusion is denied, then (in conjunction with the same major premise) the given minor premise would be contradicted".

Validation: translate the bulk forms into their pieces (here, expressed as hypotheticals, for the sake of simplicity; these are, tacitly, of the extensional type, to be precise), and verify that the conclusion is implicit in the premises by well-established (hypothetical) arguments.

· we know directly, from (iv) that "if Rs then S", and from (i) that "if P then Rp"; we still need to show, indirectly that "if Rp then Rs";

· from (iii), we know that Rp implies Rq, if we understand that Rp>Rq signifies that wherever Rp occurs, Rq is implied to have already occurred;

· whence, by syllogism, Rp implies Rs, or in other words, Rs includes Rp. This is true, note well, granting that Rs refers to a continuously increasing open-ended range, for if such a range (=>Rs) includes a number (Rq), it (=>Rs) necessarily includes all higher numbers (like Rp).[13]

One can see, here, why, if the minor premise was with P rather than Q, no conclusion would be drawable (i.e. major to minor is invalid). For then, from Rp implies Rq and Rp implies Rs, there would be no guarantee that Rq implies Rs.

Validation: translate the bulk forms into their pieces (here, again, expressed as hypotheticals, for the sake of simplicity), and verify that the conclusion is implicit in the premises by standard (hypothetical) arguments.

· we know directly, from (ii) that "if Rq then Q", and from (v) that "if S then Rs"; we still need to show, indirectly that "if Rs then Rq";

· whence, by syllogism, Rs implies Rq, or in other words, Rq includes Rs. This is true, note well, granting that Rp refers to a continuously increasing open-ended range, for if such a range (=>Rp) includes a number (Rs), then a longer range, i.e. one with a lower minimum (like =>Rq), necessarily includes that number (Rs).[14]

One can see, here, why, if the minor premise was with Q rather than P, no conclusion would be drawable (i.e. minor to major is invalid). For then, from Rp implies Rq and Rs implies Rq, there would be no guarantee that Rs implies Rp.

All the above is applicable equally to copulative and implicational a-fortiori argument, and (as already stated) the negative moods are easily derived. These dissections make evident the formal similarity and complementarity between subjectal and predicatal arguments. Although on the surface their uniformity is not very obvious, deeper down their essential symmetry becomes clear. And this serves to confirm the exhaustiveness of our treatment. Also note: our ability to reduce a-fortiori argument to chains (known as sorites) of already established and more fundamental arguments, signifies that this branch of logic, though of value in itself, is derivative - a corollary which does not call for new basic assumptions.

In view of the above (and certain additional details mentioned below) the formal definition of a-fortiori argument we would propose is, briefly put: a form of inference involving one commensurative and two suffective propositions, sharing four terms or theses. Which two of the propositions are combined as premises, and what their specific forms are (copulative or implicational), and the respective polarities, quantities and modalities which yield valid moods, and the placement of the terms or theses, are all questions automatically implied in that definition's breadth and the nature of the propositions referred to in it.

a. The arguments developed above can be validated only under the formal limitations initially mentioned, namely that the ranges involved be specifically continuously increasing and open-ended. A-fortiori reasoning remains simple and straightforward only so long as we grant such specific conditions; but if we venture into more difficult situations, with irregular ranges - such as a range with a lower limit or an upper limit or a broken range - the arguments may no longer be automatically relied on and we would have to develop moods with more complicated specifications to ensure inferences. For such reasons, the arguments we have described must be viewed as operative 'under normal conditions', namely the conditions we have already specified in the course of our study. Effectively, these conditions are tacit additional premises.

A larger theory of a-fortiori would require much more sophisticated formal tools - a much more symbolic and mathematical treatment, which is outside of the scope of the present study. I do not want to go into overly picky detail; these are very academic issues. However, we might here succinctly consider the language through which we colloquially express such inhibitions to a-fortiori arguments, signifying thereby that the situation under consideration is abnormal. Conjunctions like 'although... still...' and the like, help to fulfill this function. The following are examples of such statements; they are not arguments, note well, but statements consisting of three sentences which signal an abnormal situation, inhibiting a-fortiori inference from the first two sentences to a denial of the third.

This statement tells us that we cannot draw the normal a-fortiori conclusion from the first two sentences, namely 'P is R enough to be S'. Here, the condition R for S has an upper limit, which Q fits into, yet P surpasses. Similar statements may appear in predicatal form; for example:

We should, however, note that there are similar statements, which do not inhibit a normally valid mood, but positively join sentences which would normally not be incompatible but merely unable to constitute a valid mood; for example:

Finally, it should be clear that we can imagine more complicated cases, where the relation of the range R to S is not continuous, having gaps and/or being wholly or partly inverted. In such cases, the relations between P, Q, R, and S might be such that inferences are not possible, or at least not without access to some contorted formulae. We do not have, in ordinary language, stock phrases for such situations - in practise, if necessary, we switch to mathematical instruments.

b. We will call the form of argument so far considered[15], primary a-fortiori. Such arguments consist of a commensurative proposition as major premise and two suffective propositions as minor premise and conclusion. These forms imply, as we shall now see, a new class of arguments, a host of secondary a-fortiori, which consist of two suffectives as premises and a commensurative as conclusion. Here is how they are derived (we must, in this context, regard P and Q neutrally, without in advance saying which represents a larger or smaller quantity of R):

Let us, to begin with, take the following subjectal (merging two valid moods into a compound argument):

If we deny the conclusion and retain the minor premise, we obtain the denial of the major premise. Thus, the following secondary mood is valid:

Note well that the conclusion here proposed is only valid if it is a given that 'P is R'. For, whereas the major premise guarantees that 'Q is R', if we express the minor premise merely as 'P is not R enough to be S' then that 'P is not R' remains a possibility, and the conclusion has to be a more indefinite negation of the major premise of the root primary argument (i.e. "Not-{P is R more than or as much as Q}"), since we have conceived of the form 'P is less R than Q' as implying that P is R, rather than (as we might have done) including 'P is not R' in it as a zero limit (i.e. viewing 'NotR' as equivalent to 'R=0').

Now, let us transpose the premises, call P 'Q' and Q 'P', and commute the conclusion - and we obtain the following valid secondary mood:

Note, these are analogous to second figure syllogisms (except that the conclusion would be 'P is not Q'). Note also the need to be given that 'Q is R', as in the previous case.

Similarly, we can derive the predicatal moods by ad absurdum from the corresponding primaries; note that here the structure resembles third figure syllogism:

Note well the need to specify in the premises that certain degrees of R are required to be Q or P (as the case may be); otherwise, the conclusion, whose form we have conceived as entailing that R both (as of Rp) implies P and (as of Rq) implies Q', would have to be expressed as a broader negation, namely as "Not-{less R is required to be P than to be Q}". Here, as everywhere, the conclusion must be fully guaranteed by the premises.

Furthermore, strictly-speaking, these two predicatal conclusions are more general than they ought to be. They are true at least for cases of S; assuming them to be true for more would be an unwarranted generalization; one can conceive that in cases other than S, the requirements of R, to be P or Q, are different. In primary a-fortiori, this issue does not arise, insofar as the commensurative proposition is major premise and implicitly given as general; but in secondary a-fortiori, i.e. here, the commensurative is a conclusion and must be carefully evaluated.

Note that in all valid secondaries, the suffective premises are of unequal polarity - this is what makes possible the drawing of a commensurative conclusion, which is never egalitarian.

We may, furthermore, mention in passing the possibility of compound and variant secondary moods, such as the subjectal: 'P is R more than enough to be S; and Q is R less than enough to be S; therefore, P is more R than Q' (similarly, with P just enough and Q less than enough, or with P more than enough and Q just enough). Analogous predicatal: 'S is R less than enough to be P; and S is R more than enough to be Q; therefore, at least for cases of S, more R is required to be P than to be Q' (similarly, with less than enough for P and just enough for Q, or with just enough for P and more than enough for Q).

c. We will now consider the possibility of primary a-fortiori arguments with major premise negative. Such arguments may be shown, most readily, to be invalid, with reference to the secondary arguments which would be derivable from them (by reduction ad absurdum), were they to have been valid. Consider, for instance, the following secondary argument (subjectal) with both premises positive:

The proposed conclusion obviously cannot follow from the premises, because the premises are identical in form for the terms P and Q, and therefore there is nothing to justify their distinction in the conclusion. This is equally true if we try 'P is less R than Q' as our conclusion, or the negation of either of these proposed conclusions. It is clear that these alternatives are, though non-sequiturs, still possible outcomes; and therefore the proposition 'P is as R as Q', or its negation, cannot be necessary conclusions, but likewise are merely possible alternatives. In short, there is no conclusion of the proposed kind with the given premises.

It follows that the primary arguments below, with a negative major premise (commensurative) and negative conclusion (suffective), cannot be valid, either. For if they were, then the secondary argument just considered would have to be valid, too. That is, whether we try major to minor or minor to major form, whether with a superior or inferior (shown in brackets) or egalitarian (similarly, though not shown below) major premise, all such moods are invalid:

Similarly for the corresponding predicatal arguments: the secondary mood given below is invalid:

Are invalid, as well, any other secondary mood with the same positive premises, and any other positive (or negative) conclusion of the same sort, such as 'Less R is required to be P than to be Q'. It follows that primary moods of the kind below are also invalid:

All this is very understandable, because the negative commensurative propositions, which are the major premises of these invalid primary arguments, are all relatively weak bonds between their terms. The situation is similar to that of first-figure categorical syllogism with a particular or possible major premise, or similarly hypothetical syllogism with a lower-case major premise. One can further explore this issue by translating all the propositions involved from their bulk forms into their pieces; negatives, remember, emerge as disjunctions of hypotheticals and comparatives.

d. We might also explore, in a thorough investigation of a-fortiori logic, other irregular forms of the argument. I have done this work, but will not include the results here so as not to overburden readers with relatively unimportant, often trivial, matters. I will just mention certain items as briefly as possible, for the record:

(i) Negative terms/theses, i.e. the appearance of NotP, NotQ, NotR and/or NotS, instead of P, Q, R, S, respectively, in propositions used in a-fortiori, do not in themselves affect the formal properties of the argument - provided they are repeated throughout it. Difficulties arise when combinations of a term/thesis and its negation appear in the same argument; in which case, the oppositional and eductive relations between the positive and the 'negativized' version of each proposition must be carefully studied (translating bulk forms into their pieces), and in particular the compatibility of the premises assured. This is not a problem particular to a-fortiori, but may be found in syllogistic logic. We might in principle hope to find certain combinations of premises capable of yielding new valid moods. However, I can report that I have not found any, because the conceivable premises are always incompatible with each other. For example, given the premises:

we might at first sight think that, by educing from the original minor the following proposition (our effective minor premise):

we could make a negative antecedental a-fortiori inference, and conclude at least that:

(notice that the inference is major to minor, and not minor to major, due to the inherent change of polarity); however, though the educed minor premise is compatible with the given major, the original minor itself is not, so that the whole exercise is futile (I include it here just for purposes of illustration). Similarly, for other ectypical combinations of premises.

It may be that someone discovers valid derivative moods of this sort that I have not taken into consideration, but I doubt it. In any event, any encounter with cases of this kind should be treated with great care: they are tricky. Also, keep in mind that, ontologically, R and NotR, viewed as ranges, are very distinct, their values not having a general one-for-one correspondence. The denial of any given value of R (say, R1) is an indefinite affirmation (in disjunction) of all remaining values of R (R2, R3, etc.) and of all the values of NotR.

(ii) Negative Relationships. The positive forms can also be 'negativized' by negating the relationship they involve, i.e. putting 'is-not' in place of 'is' (for copulatives), or 'does not imply' in place of 'implies' (for implicationals). Some of the primary and secondary valid moods, already dealt with above, involved negative relationships; so that we have incidentally covered part of the ground. However, what interests us here is possible divergences between copulative and implicational arguments, mainly due to the fact that, whereas 'X is-not Y' is equivalent with 'X is NotY' (by obversion), 'X does-not-imply Y' is not interchangeable with 'X implies NotY' (but merely subaltern to it).

Copulative arguments of the sort under consideration are easy to validate, since we merely change predicate, positing a negative instead of negating a positive; for examples:

More R is required not to be P (= to be NotP) than not to be Q (= to be NotQ),

In contrast, in the corresponding implicational arguments (shown below), try as we might to apply the same analytical validation procedure as we used for other implicational arguments (translating bulk forms into their pieces), the proposed inferences are found to be illegitimate, because we cannot syllogistically derive the fourth piece needed to construct the concluding bulk form from the given data[16]:

The major premise entails: P implies Rp; Q implies Rq; Rp implies Rq. The minor premise entails: Q implies Rq; Rq implies Rs; Rs does-not-imply S; and Q does-not-imply S. With regard to the proposed conclusion: we can infer from the given premises that P implies Rp, Rp implies Rs, Rs does-not-imply S; but whether P implies or does-not-imply S remains problematic, so that we cannot infer that P implies R enough not to imply S (though note that if we were given as an additional premise that P does not imply S, we could infer the desired bulk conclusion).

The major premise entails: Rp does-not-imply P and Not(Rp) implies P; Rq does-not-imply Q and Not(Rq) implies Q; Rp implies Rq but Rq does not imply Rp. The minor premise entails: S implies Rs; Rs implies Rp; Rp does-not-imply P; and S does-not-imply P. With regard to the proposed conclusion: we can infer from the given premises that S implies Rs, Rs implies Rq, Rq does-not-imply Q; but whether S implies or does-not-imply Q remains problematic, so that we cannot infer that S implies R enough not to imply Q (though note that if we were given as an additional premise that S does not imply Q, we could infer the desired bulk conclusion).

What the above teaches us, effectively, is that we cannot treat the clause 'does not imply Y' as a conceptual unit, called 'Y1', say, and recast the form 'X implies R enough not-to-imply Y' into the form 'X implies R enough to imply Y1'. Such a artifice, known to logicians as permutation, is acceptable in some domains of logic, as in the case of obversion mentioned above; but in other domains, it has been found unacceptable, as for instances in modal logic (for the modality 'can') and in class logic (where it leads to the Russell Paradox). There is therefore no automatic guarantee that permutation is acceptable, in any given field, and we should not be surprised when, as in the present context, we discover its invalidity.

To sum up the research: implicational a-fortiori, whether antecedental or consequental, involving the negative relationships, were found invalid, using the above mentioned and other methods. The above samples are positive; but it follows that negative versions are equally invalid, since otherwise positive moods could be derived from them by reductio ad absurdum. The same results can be obtained with inferior and egalitarian major premises (even though in the latter case more data is implied). To be precise, I did not prove the various irregular a-fortiori arguments to be invalid, but rather did not find any proof that they are valid. It is not inconceivable that someone else finds conclusive paths of inference, but in the absence of such proof of validity, we must consider the proposed moods invalid.

These findings allow us to conclude that, although the analogy between regular copulative and implicational arguments is very close, there are irregular cases where their properties diverge, and copulatives are found valid while analogous implicationals are found invalid. They are significant findings, in that:

· they technically justify our initial separation of copulative and implicational a-fortiori into two distinct classes;

· they confirm, surprisingly, that our initial list of valid moods is pretty exhaustive (discounting obvious derivatives and variant subsets);

· and they confirm the general lesson of the science of logic that processes which prima facie might seem feasible, often turn out, upon closer inspection, to be illegitimate.

[1] See also, after reading this chapter, Appendix 1, for further notes on a-fortiori.

[2] Such arguments occur quite often in everyday discourse. I give you a couple of examples: "if he can readily run a mile in 5 minutes, he should certainly be able to get here (1/2 a mile from where he is now) in 15 minutes." Or again: "if my bus pass is transferable to other adults, I am sure it can be used by kids."

[3] "Implication" is to be understood here in a generic sense, applicable to all types of modality - we shall avoid more specific senses, to keep things clear and simple.

[4] The logical relationship between "is" and "implies" is well known. X "is" Y, in class-logic terminology, if it is subsumed/included by Y, which does not preclude other things also being Y. X "implies" Y, if it cannot exist/occur without Y also existing/occurring, even if as may happen it is not Y. Thus, if X "is" Y, it also "implies" Y; but if X "implies" Y, it does not follow that it "is" Y. In other words, "is" implies (but is not implied by) "implies"; "implies" is a broader more generic concept, which covers but is not limited to "is", a narrower more specific concept.

[5] For example: Jane is more good-looking than a nice girl; she is good-looking enough to win a beauty contest; therefore, a nice girl is good-looking enough to win a beauty contest.

[6] Note the following. For two magnitudes of something, like X and Y: if 'X is greater than Y', then 'X implies, but is not implied by, Y'; if 'X equals Y', then 'X implies and is implied by Y'; if 'X is smaller than Y', then 'X is implied by, but does not imply, Y'. This merely tells us, for example, that if I have eight apples and you have five, then I have as many apples as you have (plus some): eight implies five. For two classes of something, like X and Y: if 'X includes Y', then 'Y implies X' (notice the reversal of order). Here again, an example: since 'fruits' includes 'apples', then whenever we have apples it follows that we have fruits. Thus, we can elicit conditional propositions from comparative relationships, whether strictly numerical or relating to inclusion (symbol, É ).

[7] Here, the subjects could easily be singular, but to display symmetries with predicatal forms I will concentrate on classes.

[8] Ignoring, in their case, our previous convention that P should represent the larger quantity of R, and Q the lesser.

[9] Ignoring, here again, our previous convention that P should represent the larger quantity of R, and Q the lesser.

[10] I introduced the word 'suffective' for lack of a better one; had I called such propositions 'sufficient' there would have been ambiguity and confusion when the sufficiency of the proposition as such is discussed, in contrast to the sufficiency of one of its terms or theses.

[11] i.e. Rx refers to one or more of the points signified by Ry. Note well the implications of these propositions: What is X, is Y (first two pieces, by syllogism), and What is included in Rx, is included in Ry (third piece, by eduction; we cannot rightly say 'What is Rx is Ry', because we are not dealing with species/genera, but with ranges).

[12] Note well that these three pieces do not imply (nor deny) that: What is X is Y; nor that: If something is included in Rx, then it is included in Ry.

[13] Note incidentally that pieces (ii) and (v), which are the same proposition, if Q then Rq, are not used to draw the conclusion; they are technically redundant.

[14] Note incidentally, here, that pieces (i) and (iv), which are the same proposition, if Rp then P, are not used to draw the conclusion.

[15] We have only thus far dealt with moods involving a positive major premise; those with a negative major premise are discussed further down.

[16] Note that transposing the minor premise and conclusion would not improve matters; the result would remain inconclusive.

JUDAIC LOGIC