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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 36.
NATURAL CONDITIONAL SYLLOGISM AND PRODUCTION.
Syllogism in this context involves three natural conditional
propositions, all having a common subject, and whose three predicates are
positioned in figures analogous to those found in categorical syllogism.
Although the rules of modality, polarity, and quantity are essentially similar,
there are interesting differences of detail in the results obtained.
a.
The premier valid mood of syllogism involving natural conditionals is the
following first
figure singular necessary argument, where M is the middle term. From
this mood all others are derivable.
1/nnn
When this S is M, it must be Q
When this S is P, it must be M
so, When this S is P, it must be Q.
This is validated by exposition: consider any random circumstance in
which this S is actually P; then, by apodosis from the minor premise, it is also
M; and, by apodosis with that from the major premise, it is also Q.
By substituting nonQ for Q, we derive a similar negative-consequent
version:
When this S is M, it cannot be Q
When this S is P, it must be M
so, When this S is P, it cannot be Q.
Next, a potential version may be constructed:
1/npp
When this S is M, it must be Q
When this S is P, it can be M
so, When this S is P, it can be Q.
This mood can be validated by reductio ad absurdum to the previous. If
the conclusion were denied, then 'this S cannot be P and Q' would be true; but
the original major premise implies as its basis that 'this S can be Q'; it
follows that:
When this S is Q, it cannot be P;
but When this S is M, it must be Q,
therefore, When this S is M, it cannot be P.
The connection implied by this result, being 'this S cannot be M and P',
causes the original minor premise to be denied. Ergo, the original conclusion is
undeniable.
The negative-consequent version of this mood is the following:
When this S is M, it cannot be Q
When this S is P, it can be M
so, When this S is P, it can not-be Q.
Needless to say, any modes subaltern to the above are also valid. Thus, nnp
is implied valid, by nnn or npp.
Syllogism in this figure with a potential major premise are not valid.
Consider, for example, the mood below:
1/pnp
When this S is M, it can be Q
When this S is P, it must be M
so, When this S is P, it can be Q.
Although this S is M in all the circumstances relating to this S being P
(minor premise), it remains conceivable that there be circumstances in which
this S is M without being P (as conversion attests); these latter circumstances
may be precisely among the only ones in which this S is Q, as well as M (major
premise); so there is no guarantee that this S can be P and Q together (as in
the attempted conclusion), indeed it may well be that this S must cease to be P
before it is allowed to be Q (in which case, when this S is P, it becomes
Q).
A-fortiori, this invalidation
also applies to the mode 1/ppp. The
argument is essentially that denying the attempted conclusion, by saying 'This S
cannot be P and Q', does not result in the inconsistency of a denied major or
minor premise. Analogous negative-consequent versions are equally spurious, of
course.
We can also construct parallel actual moods. But, the following one might
be regarded as more akin to apodosis than syllogism, though valid:
1/naa
When this S is M, it must (or cannot) be Q
This S is P and M
so, This S is P and Q (or nonQ).
As for the mood below, it concerns the mechanics of categorical
conjunction, and hardly any longer qualifies as conditional argument in the
narrow sense.
1/aaa
This S is M and Q (or nonQ), in actual circumstance,
This S is P and M, in the same circumstance,
so, This S is P and Q (or nonQ).
What we have here, of course, are interface situations, where different
domains of logic meet.
Note that the mode naa is
subaltern to aaa (even though
necessity does not imply actuality here), because we can also infer that 'This S
is M and Q (or not Q)' from the combination of major and minor premise. However,
an actual conclusion from a necessary minor premise (as in 1/nna or 1/ana), and
modes involving a mix of actual and potential premises (ap or pa), are invalid.
This is easily demonstrated.
So much for the first figure. The parallels to categorical syllogism
should be obvious; and indeed, categorical syllogism can be viewed as a special
case of conditional syllogism, where the subject is 'thing' instead of a
specific 'S'.
Note in passing that sorites are possible with natural conditionals, as
with categoricals.
b.
The valid singular moods of the other
figures can easily be derived from those given so far, using the methods
of reduction developed in other contexts. The primary ones are listed below, for
the record, without little further discussion, for the sake of brevity.
For the second figure:
2/nnn
When this S is Q, it must be M
When this S is P, it cannot be M
so, When this S is P, it cannot be Q.
When this S is Q, it cannot be M
When this S is P, it must be M
so, When this S is P, it cannot be Q.
2/npp
When this S is Q, it must be M
When this S is P, it can not-be M
so, When this S is P, it can not-be Q.
When this S is Q, it cannot be M
When this S is P, it can be M
so, When this S is P, it can not-be Q.
Note the change of polarity of the major event, in this figure. Mode nnp
is subaltern to nnn or npp;
but pnp is not valid. Also valid, in the fig. 2, is mode 2/naa;
though not nna, ana. Two actual
premises (aa), with the polarities of
the events as shown above, are naturally impossible, since the middle term would
have mixed polarity; however, if the middle event has exceptionally the same
polarity in the two premises, aaa
becomes feasible, though the minor premise is useless to the inference. Also
invalid, as before, are ap, pa
or pp.
For the third figure:
3/npp
When this S is M, it must be Q
When this S is M, it can be P
so, When this S is P, it can be Q.
When this S is M, it cannot be Q
When this S is M, it can be P
so, When this S is P, it can not-be Q.
3/pnp
When this S is M, it can be Q
When this S is M, it must be P
so, When this S is P, it can be Q.
When this S is M, it can not-be Q
When this S is M, it must be P
so, When this S is P, it can not-be Q.
Subaltern to npp or pnp, is mode 3/nnp;
but mode nnn is invalid. Also valid, in the fig. 3, is mode aaa;
and its subalterns naa and ana, though not nna. Also
invalid, are ap, pa
or pp, as always.
For the fourth figure (significant mood):
4/npp
When this S is Q, it cannot be M
When this S is M, it can be P
so, When this S is P, it can not-be Q.
Note the change of polarity of the major event, in this figure; also, the
mixed polarity of the middle event. Mode nnp
is subaltern to npp; but nnn or pnp are not valid.
Also valid, in the fig. 4, is mode 4/naa;
though not nna, ana. Two actual premises (aa)
are naturally impossible, unless the middle event has exceptionally the same
polarity in the two premises. Also invalid, are ap, pa or pp.
c.
In addition to all the above, we could construct an equal number of valid
moods, whose premises and/or conclusions involve a negative antecedent,
obviously. Such moods are easily validated by substituting the negation of a
term for a term, in various ways. Some interesting results emerge, as the
samples below show.
In figure one, all the primary moods can be reiterated, with a negative
middle term (as in the sample below) and/or a negative minor term.
1/nnn
When this S is not M, it must be Q
When this S is P, it cannot be M
so, When this S is P, it must be Q.
In figure two, all the primary moods can be reiterated, with a negative
major term (as in the sample below) and/or a negative minor term.
2/nnn
When this S is not Q, it must be M
When this S is P, it cannot be M
so, When this S is P, it must be Q.
In figure three, all the primary moods can be reiterated, with a negative
minor term (as in the sample below) and/or a negative middle term.
3/npp
When this S is M, it must be Q
When this S is M, it can not-be P
so, When this S is not P, it can be Q.
In the fourth figure, we may switch the (mixed) polarities of the middle
term, and/or of the major term, and/or insert a negative minor term. We thus
have a total of 8 valid modes of polarity in each of the 4 figures.
These random examples demonstrate that the rules of polarity may
seemingly be by-passed. Thus, for examples, we seem to process a negative minor
premise in the first figure, or to obtain a positive conclusion in the second
figure, or to draw a positive conclusion from a negative premise in the third
figure. But of course, the rules of polarity are still essentially operative,
the changes are illusory.
Still, such moods have practical significance. Without their
clarification, we might miss out on possible inferences from data, or make
errors. The reader is therefore advised to develop a full list of such
syllogisms, as an exercise.
The following table neatly summarizes the results obtained in the
previous section. Note the similarities and differences between the modes of
modality here, and those for categorical syllogism. Table
36.1 Natural
Conditional Syllogisms.
Figure One.
plus 4 with negative minor term. Figure Two.
plus 4 with negative minor term. Figure Three.
plus 4 with negative middle term. Figure Four.
plus 4 with negative minor term.
In the first figure, 2 modal modes, and 1 actual mode, are valid (and
these have 2 subalterns). For 8 polarity modes, this means a total of 24 (+16)
valid moods. Similarly, in fig. 2, there are at least 24 (+8) valid moods, not
counting the special cases of aaa. In
fig. 3, the total is 24 (+24). In fig. 4, it is at least 16 (+8), not counting
the special cases of aaa.
The grand total of primary moods is thus 88 (not counting specials
alluded to in parentheses), of which 56 are modal and 32 are actual; plus 56
subalterns.
All the valid moods listed above are in the singular mode of quantity 'sss',
but they may of course be quantified.
However, the rules of quantity are less stringent for conditional syllogism than
with categorical syllogism.
This is due to sss being here valid throughout, because an individual instance of
the subject, indicated by 'this S', effectively stands outside the syllogistic
procedure as such, and remains recognizable independently of the three
predicates, P, Q, and M which are being manipulated.
It follows that, so long as one
premise is universal, a conclusion can be drawn, having the same quantity as
the other premise; but no conclusion is possible from two particular premises,
and the conclusion cannot be higher than the lower of the two premises.
In other words: uuu, upp, pup,
uss, sus, are all valid,
in all the figures, for all the moods established in sss. The only invalid inferences with regard to quantity, are
therefore upu, ups, puu, pus,
ppp, ppu, pps,
usu, suu, obviously.
Below are the modes of quantity for each figure, with a minimum of
examples, to illustrate some of the deviations from previous rules.
Thus, in the first and second figures, while uuu,
upp, and uss, remain valid, we have additionally pup and sus. For
examples,
1/sus
When this S is M, it must be Q
When any S is P, it must be M
so, When this S are P, it must be Q.
2/pup
When certain S are Q, they must be M
When any S is P, it can not-be M
so, When certain S are P, they can not-be Q.
In the third figure, in addition to upp
and pup, the modes uuu, uss
and sus are valid. For example,
3/uuu
When any S is M, it must be Q
When any S is M, it must be P
so, When any S is P, it can be Q.
In the fourth figure, for the significant mood listed above, instead of
just upp, we also have uuu, pup,
uss, sus. For example,
4/pup
When certain S are Q, they cannot be M
When any S is M, it can be P
so, When certain S are P, they can not-be Q.
The reader is invited to develop a full list of plural syllogisms, as an
exercise.
Production of natural conditionals is their inference from categorical
propositions. This shows us how to construct natural conditionals deductively,
rather than empirically. The structure of the premises follows the model of
categorical syllogism, while the conclusion encompasses all the original terms.
a.
The chief mood of such argument is in the first
figure; it involves a necessary major, a potential minor, and a
necessary conclusion, as follows:
All P must be Q
This S can be P
therefore, When this S is P, it must be Q.
We manage, exceptionally, to reason in the npn
mode, note, because the conclusion, though stronger than the minor premise,
concerns a narrower set of circumstances (SP instead of just S).
This argument can be validated by exposition; for any circumstance in
which this S is actually P, we know that it will also be Q according to the
categorical syllogism 1/AnRR. Note well that we are exceptionally drawing a necessary,
though conditional, conclusion from a merely potential minor premise.
Alternatively, we can use reduction ad absurdum. Denying the conclusion
means either that 'this S cannot be P', which contradicts the minor premise, or
that 'this S can be P and not Q', which implies that, for this S at least, some
P can not-be Q, in contradiction to the major premise. Thus, the conclusion is
indubitable.
Note well that 'When this S is P, it must be Q' does not imply 'All P
must be Q'. Although natural conditionals may be inferred from categorical
premises, it does not follow that that is the only way we can get to reach such
conclusions. Natural conditionals can also be known by induction; so, they do
not logically imply categoricals other than their bases and connections.
The negative version of the above mood is:
No P can be Q
This S can be P
therefore, When this S is P, it cannot be Q.
Note that if the major premise is necessary, and the minor premise is the
actual or necessary 'This S is or must be P', then the conditional conclusion as
such is unaffected; so these are subaltern moods of production.
If both premises are actual, concerning the same circumstances, the
conclusion is a categorical conjunction of all three terms, which represents the
actual form of natural conditional. The positive and negative versions of this aaa
mode, still in the first figure, are:
All P are Q
This S is P
therefore, This S is P and Q.
No P is Q
This S is P
therefore, This S is P and not Q.
We may also have, with the same actual major, a necessary minor 'This S
must be P', without change of conclusion (mode, ana).
Note that the nnn mode is also valid, by subalternation from npn. It is interesting to note, however, that given the premises
'This S must be P and all P must be Q' we would rather draw the categorical
conclusion 'This S must be Q', than the inferior conditional 'When this S is P,
it must be Q'. It shows the essential continuity between categorical and
conditional syllogism. Given that 'Some S can be P' (which is the base of the
minor premise) the conditional conclusion is a subaltern of the categorical one.
Also, two necessary categorical premises, with adequate modality of
subsumption, may also be used to draw an actual conjunctive conclusion (nna).
All the above conclusions of course further imply that 'When this S is P, it can
be Q' or '… nonQ', respectively (as in the subaltern aap mode). However, although npn and nnn are valid, the modes npa or nna are invalid, since a necessary conditional does not imply an actual conjunction. Also, the major premise could not be merely potential, since the middle term P would then not be distributive in respect of modality, even if the minor premise were necessary ( |
