CHAPTER 30.
LOGICAL APODOSIS AND DILEMMA.

1.
Apodosis.

2.
Dilemma.

3.
Rebuttal.


 

1.
Apodosis.



Apodosis is argument involving a hypothetical proposition as major
premise, and the affirmation or denial, of one thesis as minor premise, and of
the other as conclusion. Needless to say, the two premises must be true, for the
conclusion to follow, as in all argument. There are essentially two valid moods,
as follows:


If
P, then Q

If
P, then Q

and
P

but
not Q

hence,
Q

hence,
not P



We see that the major premise has to be a positive hypothetical, the
minor premise must either affirm the antecedent or deny the consequent, and the
conclusion can only, accordingly, affirm the consequent or deny the antecedent.
Such argument merely activates, as it were, the dormant power of the
hypothetical, when the minor premise is independently found true.


In the first case, the conclusion follows directly; the second case could
be reduced to the first, by contraposition of the major premise. The validity of
these moods can be demonstrated by reference to the definition of the major
premise as ‘the conjunction of P and nonQ is impossible’; it follows that, if P
was true without Q being so, or Q was false without P being so, this
impossibility fail to be upheld.


We can also present as valid moods, and in like manner validate, moods
involving the remaining positive hypothetical forms. These are to some extent
interesting in themselves, showing us still different ways otherwise
inaccessible information might be indirectly arrived at, or that assumptions
might be eliminated. But they are also valuable for the validation of certain
arguments involving disjunctive or compound, derivative forms.


If
P, then not Q

If
P, then not Q

and
P

but
Q

hence,
not Q

hence,
not P


If
not P, then Q

If
not P, then Q

and
not P

but
not Q

hence, Q

hence, P


If
not P, then not Q

If
not P, then not Q

and
not P

but
Q

hence,
not Q

hence,
P



Other moods of apodosis, including those with a lowercase major premise,
such as ‘If P, not-then Q’, have no demonstrable basis or lead to inconsistency,
and so are invalid. The following two are specially noteworthy, because they
represent oft-made errors of judgement.


If
P, then Q

If
P, then Q

and
Q

but
not P

hence,
P.

hence,
not Q.



These are, to repeat, formally invalid. However, note that they are used
for purposes of ‘adduction’, the provision of evidence or counterevidence to
support or discredit theories. The positive mood tells us that the more a theory
P makes successful predictions, such as Q, the more credible it becomes (though
it is not proved); the negative mood tells us that when the presuppositions,
such as P, of a theory Q collapse, it is undermined (though it is not
disproved). This topic is dealt with in detail in a later chapter.


We can interpret the following expectative propositions as abridged,
‘failed’ apodoses of this kind. ‘Even if P were true, Q would be true’ implies
‘if P, then Q’ and ‘P is false, and Q is true’. ‘Though P is true, Q is true’
implies ‘if not P, then Q’ and ‘P is true and Q is true’.


The following invalid moods also typify fallacious attempts at apodosis;
note the negative hypothetical major premises, lowercase forms implying a link
yet too weak to yield any definite conclusion. People often fail to first
establish a bond between the theses, because they hurriedly assume that no
contrary hypotheticals can be put forward.


If
P, not-then Q

If
P, not-then Q

and
P

but
not Q

hence,
not Q.

hence,
P.



In summary, there are 8 valid moods of apodosis. That being out of 8X4X4
= 128 possible moods, the validity rate is 6.25%. Compounds of these premises
would yield compound conclusions.



The above described valid moods of logical apodosis involve a factual
minor premise: P or Q is true or false. The concept can be broadened to involve
minor premises which are modal (this refers of course to logical modality). The
following four valid moods are derivable from the basic two by exposition, as
usual:


If
P, then Q

If
P, then Q

and
necessarily P

but
impossibly Q

so,
necessarily Q

so,
impossibly P


If
P, then Q

If
P, then Q

and
possibly P

but
unnecessarily Q

so,
possibly Q

so,
unnecessarily P



Think of the major premises as modal conjunctions. For example, in the
first of these moods, ‘{P and nonQ} is impossible’ plus ‘P is necessary’ imply
that ‘Q is necessary’, for if nonQ ever occurred in any context, we would be
faced with a contradiction.


The remaining six factual moods can likewise be used to construct another
twelve valid modal moods. The significance of these arguments is of course the
transmissibility of logical modality across the hypothetical relation.
Hypotheticals per se have problematic theses; under the appropriate conditions,
this merely prima facie thinkability is up or down graded to a more definite
logical status.


Modal apodosis suggests that ‘if P, then Q’ implies ‘if Pn,
then Qn‘ and ‘if Pp, then Qp‘, where the
suffixes n and p refer to logical necessity and possibility respectively. However,
note well that the reverse does not follow; the version with nonmodal theses
cannot be inferred from either of the modal-theses versions, since the relation
might conceivably only apply in the collective case or in the indefinite case
without implying a singular and specific equivalent.


Moreover, I see no point in treating hypotheticals with modal theses
independently, and I do not think that we ever do so in practise. Since an
ordinary hypothetical, with nonmodal theses, contains within it all the
requisite information for the solution of problems of a modal nature, we have no
need of these implicit forms, and introducing them would be very artificial.



As already mentioned, the valid moods of apodosis can be reformulated to
describe arguments involving derivative forms. In particular, note the following
examples:


P
and/or Q

P
and/or Q

but
not P

but
not Q

hence, Q

hence, P


P
or else Q

P
or else Q

and
P

and
Q

hence,
not Q

hence,
not P



I would not count the rewriting of valid moods in derivative forms as
yielding additional valid moods. But they become more significant with multiple
disjunctions, which yield more complex conclusions, so long as more than one
alternative have not been eliminated.


P
and/or Q and/or R and/or S

P
or else Q or else R or else S

but
not P

and
P

so,
Q and/or R and/or S.

so,
not Q and not R and not S.



In the mood with inclusive disjunction (left), we are given that at
least one of the theses listed must be true (i.e. they cannot all be false); so
if one is found false, we can conclude that at least one of the remaining ones
must be true. In the mood with exclusive disjunction (right), we are given that
all but one of the theses listed must be false (i.e. only one can be true); so
if one is found true, we can conclude that all the remaining ones must be false.
Note that if both major premises are true, i.e. if the theses are both
‘exhaustive’ and ‘mutually exclusive’, then a conclusion is possible from the
truth or falsehood of any of the theses, as shown in these two moods.


Most of what we have said about apodosis concerns all hypotheticals,
whether of unknown logical basis, normal or abnormal. However, apodosis with a
necessary or impossible minor premise and conclusion (as shown earlier)
obviously concerns abnormal hypotheticals in particular, because the basis is
implied to be not contingent at all. In contrast, apodosis with a possibility or
unnecessity as its minor premise, teaches us the logic specific to normal
hypotheticals, which are contingency-based.


Thus, we have here a foundation for the specialized study of normal or
abnormal hypotheticals, an entry point into the topic; I will not however here
pursue the matter further. The same can be said for disjunctives.


 

2.
Dilemma.



Colloquially, we call a ‘dilemma’, any impossible choice. ‘If I do this,
I’ve had it; if I do that, I’ve had it — so I’ve had it anyway (and it is no
use my doing this or that)’. This is indeed a case of dilemma, but in logic the
expression is understood more broadly, to cover more positive situations. Thus,
often, in action contexts, when we are faced with a choice of means to get to a
goal, we might resolve the dilemma by using all available means, even at the
cost of redundancies, so as to ensure that the goal is attained one way or the
other.


Although dilemmatic argument may be derived from apodosis and syllogism,
it has a certain autonomy of cogency and is commonly used in practise, so it
deserves some analysis. Note well first that the disjunction used in dilemma is
the ‘and/or’ type (not the ‘or else’ type), even if in practise this is not
always made clear.


The hypotheticals which constitute the major premise of a dilemma are
called its ‘horns’; they give an impression of presenting us with a predicament.
The minor premise is a disjunction; it is said to ‘take the dilemma by its
horns’. The conclusion is said to ‘resolve’ the dilemma.


a.
Simple dilemma consists of a
conjunction of subjunctives as major premise, a disjunctive as minor premise,
and a (relative) categorical as conclusion. It normally involves three theses.
Tradition has identified two valid moods.



(i)
The simple constructive dilemma.



If M, then P — and — if N, then P


but M and/or N


hence, P



This is proved by reduction ad absurdum through two negative apodoses, as
follows:



If M, then P — and — if N, then P (original major premise)


and not P (denial of conclusion)


so, not M and not N (contrary of minor).



Alternatively, we could regard the simple constructive dilemma as
summarizing a number of positive apodoses, with reference to the matrix of
alternative conjunctions underlying the minor premise:


If
M, then P — and — if N, then P

(common
major)

but
‘M (and not N)’

or
‘N (and not M)’

or
‘M and N’

(alternative
minors)

whence,
P

whence,
P

whence,
P and P

(common
conclusion).



This shows the essential continuity between the concepts of apodosis and
dilemma, note.



(ii)
The simple destructive dilemma.



If P, then M — and — if P, then N


but not M and/or not N


hence, not P



This is proved by reduction ad absurdum through two apodoses, as follows:



If P, then M — and — if P, then N (original major premise)


and P (denial of conclusion)


so, M and N (contrary of minor).



In contrast, the following two arguments would be fallacious:


If
M, then P — and — if N, then P

If
P, then M — and — if P, then N

but
not M and/or not N

but
M and/or N

hence,
not P

hence,
P



b.
Complex dilemma consists of a
conjunction of subjunctives as major premise, and disjunctives as minor premise
and conclusion. Tradition has identified two valid moods. It normally involves
four theses, though two are occasionally merely mutual antitheses.



(i)
The complex constructive dilemma.



If M, then P — and — if N, then Q


but M and/or N


hence, P and/or Q



This can be proved by reductio ad absurdum, as in simple dilemma.
Alternatively, we may analyze it through a sorites, as follows:



If not P, then not M (contrapose left horn)


if not M, then N (from minor)


if N, then Q (right horn)


therefore, if not P, then Q (transform to conclusion).



(ii)
The complex destructive dilemma.



If P, then M — and — if Q, then N


but not M and/or not N


hence, not P and/or not Q



This can be proved by reductio ad absurdum, as in simple dilemma.
Alternatively, we may analyze it through a sorites, as follows:



If not not P, then P (axiomatic)


if P, then M (left horn)


if M, then not not M (axiomatic)


if not not M, then not N, (from minor)


if not N, then not Q (contrapose right horn)


therefore, if not not P, then not Q (transform to conclusion).



In contrast, the following two arguments would be fallacious:


If
M, then P — and — if N, then Q

If
P, then M — and — if Q, then N

but
not M and/or not N

but
M and/or N

hence,
not P and/or not Q

hence,
P and/or Q



c.
Concerning both the simple and complex valid moods, note that, formally
speaking, we could use as minor premises the equivalent forms ‘not M or else not
N’ and ‘M or else N’, respectively, in the valid constructive and destructive
moods. But this would not reflect the true format of dilemma. The goal here is
only to describe actual thought processes, not to accumulate useless formulas.
However, in view of the similarity in appearance between these valid
substitutes, and the minor premises of the invalid moods, it is well to be aware
of the possibility of confusion.


A special case of complex constructive dilemma is worthy of note,
because people sometimes argue in that way. Its form is:



If M, then {P and nonQ} — and — if N, then {nonP and Q}


but M and/or N


hence, either P or Q.



We may understand this argument as follows: contrapose the left horn to
‘if not-{P and nonQ}, then nonM’; the minor premise means ‘if nonM, then N’;
these propositions, together with the right horn, form a sorites whose
conclusion is ‘if not-{P and nonQ}, then {nonP and Q}’. But we know on formal
grounds, for any two propositions, that ‘if {P and nonQ}, then not-{nonP and
Q}’. Therefore, ‘either {P and nonQ} or {nonP and Q}’ is true, which can in turn
be rephrased as ‘either P or Q’.


Thus, what this argument achieves is the elimination of the remaining two
formal alternatives, {P and Q} and {nonP and nonQ}; the combinations {P and
nonQ} and {nonP and Q} become not merely incompatible, but also exhaustive.
There is no destructive version of this argument, because its result would only
be ‘if {P and nonQ}, then not-{nonP and Q}’, which is formally given anyway.


There is also no equivalent argument in simple dilemma. But note that if
we substitute nonM for N in the one above, we obtain something akin to it: if M,
then {P and nonQ}, and if nonM, then {nonP and Q}; but either M or nonM; hence,
either P or Q. This is not really simple dilemma because the antecedents are not
identical; but there is a resemblance, in that only three theses are involved.
Also, the minor premise here is redundant, since formally true, so the
conclusion may be viewed as an eduction from the compound major premise.


Also note, simple and complex dilemmas may consist of more than two
horns. The following are examples of multi-horned
simple dilemma:



Constructive:


If B and/or C and/or D… is/are true, then A is true


but B and/or C and/or D…etc. is/are true


therefore A is true.



Destructive:


If A is true, then B and C and D …etc. are true


but B and/or C and/or D…etc. is/are false


therefore A is false’.



Similarly with other sorts of arrays. This shows that we can view the
horns of dilemmas as forming a single hypothetical proposition whose antecedent
and/or consequent is/are conjunctive or disjunctive. It follows that simple and
complex dilemma should not be viewed as essentially distinct forms of argument;
rather, simple dilemma is a limiting case of complex dilemma, the process
involved being essentially one of purging our knowledge of extraneous
alternatives.


The commonly employed form ‘Whether
P or Q, R’
is normally understood as an abridged simple constructive
dilemma, meaning ‘If P, then R, and if Q, then R, but P and/or Q, hence R
anyway’. However, we should be careful with it, because in some cases we intend
it to dissociate R from P and Q, meaning ‘If P not-then R, and if Q not-then R,
but R’. Note well the difference. In the former case, the independence is an
outcome of multiple dependence; in the latter case, the independence signals
lack of connection.


Dilemma, especially its ultimate, simple version, is a very significant
form of reasoning, in that it is capable
of yielding factual results from purely problematic theses
(implicit in
hypotheticals or disjunctives). Like the philosopher’s stone of the alchemist,
it turns lead into gold. Without this device, knowledge would ever be
conjectural, a mass of logically related but unresolvable problems.


Note however that the conclusion of a simple dilemma is still, logically,
only factual in status. A thesis only acquires the status of logical necessity
or impossibility, when it is implied or denied by all eventualities; this means,
in dilemma, when the exhaustiveness of the alternatives in the premises is
itself logically incontingent (rather than a function of the present context of
knowledge). The significance of this will become more transparent as we proceed
further, and deal with paradoxical logic.


 

3.
Rebuttal.



The so-called ‘equally cogent rebuttal’ is a special case of dilemma, worthy of
analysis in this context. It happens in debate that a seemingly cogent dilemma
may be rebutted by a seemingly equally cogent dilemma.


a.
With regard to complex dilemma, though the arguments are indeed equally
cogent, the impression of ‘rebuttal’ is illusory, due to a misconception of the
opposition between the conclusions.



If M, then P — and — if N, then Q


but M and/or N


hence, P and/or Q



If P, then M — and — if Q, then N


but not M and/or not N


hence, not P and/or not Q



Clearly, the major premises are compatible; taken together, they signify
two reciprocal subjunctions. The minor premises are also compatible, since they
mean, respectively, ‘if nonM, then N’ and ‘if not nonM, then nonN (i.e. if M,
then nonN)’; taken together, they signify contradictive disjunction between M
and N.


Likewise, for the conclusions: they are not inconsistent with each other,
but taken together mean that P and Q are contradictory. So in fact the two
dilemma do not exclude each other, it is formally quite possible for them to be
both true. If indeed all the propositions involved are true, they merely
together constitute a compound dilemma which is quite valid.


We seem to be faced with equally cogent arguments yielding conflicting
conclusions, but this is an erroneous impression, because in fact the
conclusions are consistent. They may seem to conflict, because they refer to
contradictory theses, P and Q, nonP and nonQ; but the disjunctive way in which
these theses are connected, makes the conclusions complementary, rather than
inconsistent.


Restating the entire arguments in standard hypothetical syllogisms can be
helpful. The conclusions should be viewed as ‘If nonP, then Q’ and ‘If P, then
nonQ’, respectively, to avoid confusion. If the result persists in seeming
unintelligible, the wording may be misleading or there may be a factually
erroneous premise.


The frustration underlying such arguments, why they are experienced as
somehow in conflict — is due to the fact that each party assumed the
contradictory of the other’s assumption to be tacitly included in his or her own
premises. Thus, it is the compound each implicitly assumed, rather than the
explicit elements, which each finds rightly denied by the other.


In some cases, the presumptions are inductively legitimate for the
context each has at hand, following the principle that what is not found
connected may be assumed unconnected, so that the face-off with the rebuttal
view indeed intimates a possible error somewhere in one’s own views. Someone
with an open mind does not feel threatened by such an eventuality, but may give
some attention to the problem without resentment, if the issue is sufficiently
interesting.



b.
With regard to simple dilemma, the rebuttal is, on formal grounds, never
‘equally cogent’, so it should not surprise us that the conclusions are
contradictory.



If M, then P — and — if N, then P


but M and/or N


hence, P



If P, then M — and — if P, then N


but not M and/or not N


hence, not P.



Although the two major premises are formally compatible with each other,
and the two minor premises are formally compatible with each other, the
conclusions are indubitably incompatible with each other. What this tells us is
that the premises, though severally consistent, are taken together inconsistent.
They are not, therefore, equally cogent dilemmas; one or both must contain a
factual error.


In other words, a simple dilemma is not logically valid, if the horns of
the major premise are reversible hypotheticals and the minor premise is a
contradictive disjunctive. The compound propositions ‘Only if M, then P — and
— only if N, then P’ and ‘Either M or N’ cannot coexist. This may be shown as
follows:


The first minor ‘M and/or N’ taken alone allows for the conjunction ‘M
and N’, while excluding ‘nonM and nonN’. The second minor ‘not M and/or not N’
taken alone allows for the conjunction ‘nonM and nonN’, while excluding ‘M and
N’. When these disjunctions are conjoined together, they mean ‘either M or N’
which still allows for ‘M and nonN’ or ‘nonM and N’, but now formally excludes
both ‘M and N’ and ‘nonM and nonN’.


Yet, in the case of M and nonN being both true, the left horn of the
first major and the right horn of the second, would yield conflicting
conclusions: P and nonP; and, in the case of nonM and N being both true, the
left horn of the second major and the right horn of the first, would yield
conflicting conclusions: P and nonP.


Thus, rebuttal of simple dilemma is formally unfeasible with
contingency-based hypotheticals. With an incontingent theses P or nonP, this
paradox is acceptable, because if P is necessary or impossible, the arrival at
its negation does not cause a serious conflict, since then the necessary theses
is implied by its impossible antithesis. Equally cogent simple dilemmas are
therefore feasible in abnormal logic specifically, even though they cannot arise
in normal logic. It follows that in the logic of unspecified-basis
hypotheticals, these are conditionally possible.


The foregoing means that the valid moods of simple dilemma given
initially were not as fully defined and unconditional as they should have been,
in other respects, besides.


For a simple dilemma to be valid, one or both of the horns in the major
premise must be implicitly a subalternation, rather than an implicance (whereas
we left them open as implications); and/or the minor premise must be implicitly
a subcontrariety (if constructive) or contrariety (if destructive) between the
theses in question, rather than a contradiction (whereas we left it open as a
not fully defined disjunction).