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JUDAIC LOGIC© Avi Sion, 1995. All rights reserved. Chapter
3.
THE FORMALITIES OF A-FORTIORI
LOGIC.
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.
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). Table 3.1
Classification of a-fortiori arguments.
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. (1) Subjectal moods.
(a) Positive version. (Minor to major.)
P is more R than Q (is R),
and, Q is R enough to be S;
therefore, all the more, P is
R enough to be S.
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.
(b) Negative version. (Major to minor.)
P is more R than Q (is R),
yet, P is not
R enough to be S;
therefore, all the more, Q is
not R enough to be S.
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. (2) Predicatal moods.
(a) Positive version. (Major to minor.)
More R is required to be P than to be Q,
and, S is R enough to be P;
therefore, all the more, S is
R enough to be Q.
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.
(b) Negative version. (Minor to major.)
More R is required to be P than to be Q,
yet, S is not
R enough to be Q;
therefore, all the more, S is
not R enough to be P.
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] (3) Antecedental moods.
(a) Positive version. (Minor to major.)
P implies more R than Q (implies R)
and, Q implies enough R to imply S;
therefore,
all the more, P implies enough R
to imply S.
(b) Negative version. (Major to minor.)
P implies more R than Q (implies R)
yet, P does not
imply enough R to imply S; therefore,
all the more, Q does not
imply enough R to imply S. (4) Consequental moods.
(a) Positive version. (Major to minor.)
More R is required to imply P than to imply Q
and, S implies enough R to imply P;
therefore,
all the more, S implies enough R
to imply Q.
(b) Negative version. (Minor to major.)
More R is required to imply P than to imply Q
yet, S does not
imply enough R to imply Q; therefore,
all the more, S does not
imply enough R to imply P.
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). a.
Subjectals:[7] The bulk form: What
is P is more R than what is Q (is R); its pieces: What is P, is to a certain degree R (say, Rp), What is Q, is to a certain degree R (say, Rq), and Rp is greater than Rq.
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. b.
Predicatals: The bulk form: More
R is required to be P than to be Q; its pieces: What is to a certain degree R (say, Rp), is P, What is to a certain degree R (say, Rq), is Q, and Rp is greater than Rq.
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. a.
Positives: The bulk form: X
is R enough to be Y; its pieces: Whatever is X, is to a certain degree R (say, Rx), Whatever is to a certain degree R (say, Ry), is Y, and Ry includes
Rx.[11]
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: b.
Negatives: The bulk form: X
is not R enough to be Y; its pieces: Some things which are X are not a certain degree of R
(say, not Rx), and/or Some things which are to a certain degree R (say, Ry)
are not Y, and/or Ry does not include Rx.[12]
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". ·
Positive Subjectal (minor to
major):
P is more R than Q (is R),
Q is R enough to be S;
so, P is R enough to be S. 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. Major premise:
(i) if P then Rp, and
(ii) if Q then Rq, and
(iii) Rp > Rq (implying: if Rp then Rq). Minor premise:
(iv) if Rs then S, and
(v) if Q then Rq, and
(vi) Rs includes Rq (implying: if Rq then Rs). Paths of Inference: ·
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; ·
and from (vi) we know that Rq
implies Rs; ·
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] Conclusion:
therefore,
if Rs then S, and
if P then Rp, and
Rs includes Rp. which is the desired result.
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. ·
Positive predicatal (major to
minor):
More R is required to be P than to be Q,
S is R enough to be P;
so, S is R enough to be Q. 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. Major premise:
(i) if Rp then P, and
(ii) if Rq then Q, and
(iii) Rp > Rq (implying: if Rp then Rq). Minor premise:
(iv) if Rp then P, and
(v) if S then Rs, and
(vi) Rp includes Rs (implying: if Rs then Rp). Paths of Inference: ·
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"; ·
from (vi) we know that Rs
implies Rp; ·
and from (iii), we know that
Rp implies 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] Conclusion:
therefore,
if Rq then Q, and
if S then Rs, and
Rq includes Rs. which is the desired result.
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. |
