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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 29. HYPOTHETICAL SYLLOGISM AND PRODUCTION.
There are several kinds of deductive argument involving hypothetical
propositions or their derivatives. They are distinguished according to whether
they involve only hypotheticals, or hypotheticals mixed with categorical forms.
The main kinds are syllogism, production, apodosis and dilemma. Note that the
valid moods are not here listed in symbolic terms, as we did with categoricals,
to avoid obscuring their impact.
Hypothetical syllogism is argument whose premises and conclusion are all
hypotheticals. It is mediate inference, with minor (symbol P), middle (M), and
major (Q) theses, deployed in figures, as was the case in categorical syllogism.
Its most primary valid mood, from which all others may be derived by direct or indirect reduction, is as
follows. It tells us, as for the analogue in categorical syllogism, that, as
H.W.B. Joseph would say, 'whatever falls under the condition of a rule, follows
the rule'.
This primary mood is valid irrespective of whether the hypotheticals
involved are of unspecified base, normal (contingency-based), or abnormal. That
is generally true for its primary derivatives, too; but subaltern derivatives
are only applicable in cases where both theses are known to be logically
contingent (and not just problematic), because the subalterns require eductive
processes which depend on this condition for their validity.
If M, then Q
if P, then M
so if P, then Q
This is a first figure syllogism. Its validity obviously follows from the
meaning of the operator 'if-then' involved. Although the connection in
hypotheticality is expressed by modal conjunctive statements, 'if-then'
underscores an additional, not-tautologous, sense, occurring on a finer level.
This teaches us a purely conjunctive argument, from which many laws for the
logic of conjunction may be inferred, that:
The premises: {M and nonQ} is impossible,
and {P and nonM} is impossible, together
yield the conclusion: {P and nonQ} is impossible.
This could be written symbolically as 1/H2nH2nH2n,
note.
a.
Figure
One.
(i) From the primary valid mood, we can draw up the
following full list of valid, uppercase,
perfect moods, in first figure, by substituting antitheses for theses in
every possible combination.
(ii) Next, from one of the valid, uppercase, perfect moods,
we derive the primary, valid, lowercase,
perfect mood, by reductio ad absurdum, as follows. Note that the major
premise is uppercase, and the minor premise and conclusion are lowercase.
From this primary mood, we can draw up the following full list of valid,
lowercase, perfect moods, in the first figure, by substituting antitheses for
theses in every possible combination.
(iii) Next, from one of the valid, uppercase, perfect moods,
we derive the primary, valid, imperfect
mood, by reductio ad absurdum, as follows. Note the change in polarity of the
minor thesis in the conclusion, which defines the moods as imperfect, and the
distinct mixed polarity of the middle thesis in the two premises. Note also that
the minor premise is uppercase, and the major premise and conclusion are
lowercase.
From this primary mood, we can draw up the following full list of valid,
imperfect moods, in the first figure, by substituting antitheses for theses in
every possible combination.
(iv) Subaltern
moods. These are valid only
with normal hypotheticals, unlike the preceding, because they are derived from
the latter by subalternating a lowercase premise or being subalternated by an
uppercase conclusion. Their premises are always both uppercase, and their
conclusion lowercase.
The following sample can be derived from moods of type (i) by obverting
the conclusion, or equally well from moods of type (ii) by replacing the minor
premise with its obvertend. On this basis, 8 subaltern moods can be derived in
the usual manner. These are perfect in nature.
If M, then Q
if P, then M
so, if P, not-then nonQ.
The following sample can be derived from moods of type (i) by
obvert-inverting the conclusion, or equally well from moods of type (iii) by
replacing the major premise with its obvertend. On this basis, 8 subaltern moods
can be derived in the usual manner. These are imperfect, since the minor thesis
changes polarity in the conclusion.
If M, then Q
if P, then M
so, if nonP, not-then Q.
In summary, we thus have a total of 3X8 = 24 primary valid moods in the
first figure, plus 2X8 = 16 subaltern valid moods. Or a total of 40 valid moods,
out of 8X8X8 = 512 possibilities.
b.
Figure
Two.
(i) From one of the valid, lowercase, perfect moods, of the
first figure, we derive the primary, valid, uppercase,
perfect mood, of the second figure, by reductio ad absurdum, as follows.
Alternatively, we could have used direct reduction, by contraposing the major
premise, through a valid, uppercase, perfect mood, of the first figure.
From this primary, valid mood, we can draw up the following full list of
valid, uppercase, perfect moods, in the second figure, by substituting
antitheses for theses in every possible combination.
(ii) Next, from one of the valid, uppercase, perfect moods,
of the first figure, we derive the primary, valid, lowercase, perfect mood, of the second figure, by reductio ad
absurdum, as follows. Alternatively, we could have used direct reduction, by
contraposing the major premise, through a valid, lowercase, perfect mood, of the
first figure. Note that the major premise is uppercase, and the minor premise
and conclusion are lowercase.
From this primary mood, we can draw up the following full list of valid,
lowercase, perfect moods, in the second figure, by substituting antitheses for
theses in every possible combination.
(iii) Subaltern moods. These are valid only with normal hypotheticals,
unlike the preceding, because they are derived from the latter by subalternating
a lowercase premise or being subalternated by an uppercase conclusion. Their
premises are always both uppercase, and their conclusion lowercase.
The following sample can be derived from moods of type (i) by obverting
the conclusion, or equally well from moods of type (ii) by replacing the minor
premise with its obvertend. On this basis, 8 subaltern moods can be derived in
the usual manner. These are perfect in nature.
If Q, then M
if P, then nonM
so, if P, not-then Q.
The following sample can be derived from moods of type (i) by
obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived
in the usual manner. These are imperfect, since the minor thesis changes
polarity in the conclusion.
If Q, then M
if P, then nonM
so, if nonP, not-then nonQ.
The following sample can be derived from moods of type (ii) by replacing
the minor premise with its obvert-invertend. On this basis, 8 subaltern moods
can be derived in the usual manner. These are imperfect, since the minor thesis
changes polarity in the conclusion. Note the distinct uniform polarity of the
middle thesis in the two premises.
If Q, then M
if P, then M
so, if nonP, not-then Q.
In summary, we thus have a total of 2X8 = 16 primary valid moods in the
second figure, plus 3X8 = 24 subaltern valid moods. Or a total of 40 valid
moods, out of 8X8X8 = 512 possibilities.
c.
Figure
Three.
(i) From one of the valid, uppercase, perfect moods, of the
first figure, we derive the primary, valid, perfect
mood, with lowercase major premise,
of the third figure, by reductio ad absurdum, as follows. Alternatively, we
could have used direct reduction, by contraposing the major premise, and
transposing, through a valid, lowercase, perfect mood, of the first figure. The
conclusion is of course lowercase.
From this primary, valid mood, we can draw up the following full list of
valid, perfect moods, with lowercase major premise, in the third figure, by
substituting antitheses for theses in every possible combination.
(ii) Next, from one of the valid, lowercase, perfect moods,
of the first figure, we derive the primary, valid, perfect mood, with lowercase
minor premise, of the third figure,
by reductio ad absurdum, as follows. Alternatively, we could have used direct
reduction, by contraposing the minor premise, through a valid, lowercase,
perfect mood, of the first figure. The conclusion is of course lowercase.
From this primary, valid mood, we can draw up the following full list of
valid, perfect moods, with lowercase minor premise, in the third figure, by
substituting antitheses for theses in every possible combination.
(iii) Next, from one of the valid, lowercase, perfect moods,
of the first figure, we derive the primary, valid, imperfect mood, of the third figure, by direct reduction, as
follows. Note the change in polarity of the minor thesis in the conclusion,
which defines the mood as imperfect, and the distinct mixed polarity of the
middle thesis in the two premises. Note also that both premises and the
conclusion are uppercase.
From this primary mood, we can draw up the following full list of valid,
imperfect moods, in the third figure, by substituting antitheses for theses in
every possible combination.
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