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The Logician © Avi Sion All rights reserved

FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 31. PARADOXES.
A very important field of logic is that dealing with paradox, for it
provides us with a powerful tool for establishing some of the most fundamental
certainties of this science. It allows us to claim for epistemology and ontology
the status of true sciences, instead of mere speculative digressions. This
elegant doctrine may be viewed as part of the study of axioms.
2.
The Stolen Concept Fallacy.
Consider the hypothetical form 'If P, then Q', which is an essential part
of the language of logic. It was defined as 'P and nonQ is an impossible
conjunction'.
It is axiomatic that the conjunction of any proposition P and its
negation nonP is impossible; thus, a proposition P and its negation nonP cannot
be both true. An obvious corollary of this, obtained by regarding nonP as the
proposition under consideration instead of P, is that the conjunction of any
proposition nonP and its negation notnonP is impossible; thus, a proposition P
and its negation nonP cannot be both false.
So, the Law of Identity could be formulated as, "For any
proposition, 'If P, then P' is true, and 'If nonP, then nonP' is true". The
Laws of Contradiction and of the Excluded Middle could be stated: "For any
proposition, 'If P, then notnonP' is true (P and nonP are incompatible), and
'If notnonP, then P' is true (nonP and P are exhaustive)".
Now, consider the paradoxical propositions 'If P, then nonP' or 'If nonP,
then P'. Such propositions appear at first sight to be obviously impossible,
necessarily false, antinomies.
But let us inspect their meanings more closely. The former states 'P and
(not not)P is impossible', which simply means 'P is impossible'. The latter
states 'nonP and not P is impossible', which simply means 'nonP is impossible'.
Put in this defining format, these statements no longer seem antinomial! They
merely inform us that the proposition P, or nonP, as the case may be, contains
an intrinsic flaw, an internal contradiction, a property of selfdenial.
From this we see that there may be propositions which are logically
selfdestructive, and which logically support their own negations. Let us then
put forward the following definitions. A proposition is selfcontradictory if it denies itself, i.e. implies its own
negation. A proposition is therefore selfevident
if its negation is selfcontradictory, i.e. if it is implied by its own
negation.
Thus, the proposition 'If P, then nonP' informs us that P is
selfcontradictory (and so logically impossible), and that nonP is selfevident
(and so logically necessary). Likewise, the proposition 'If nonP, then P'
informs us that nonP is selfcontradictory, and that P is selfevident.
The existence of paradoxes is not in any way indicative of a formal flaw.
The paradox, the hypothetical
proposition itself, is not antinomial. It may be true or false, like any other
proposition. Granting its truth, it is its antecedent thesis which is
antinomial, and false, as it denies itself; the consequent thesis is then true.
If the paradoxical proposition 'If P, then nonP' is true, then its
contradictory 'If P, notthen nonP', meaning 'P is not impossible', is false;
and if the latter is true, the former is false. Likewise, 'If nonP, then P' may
be contradicted by 'If nonP, notthen P', meaning 'nonP is not impossible'.
The two paradoxes 'If P, then nonP' and 'If nonP, then P' are contrary to
each other, since they imply the necessity of incompatibles, respectively nonP
and P. Thus, although such propositions taken singly are not antinomial, double
paradox, a situation where both of these paradoxical propositions are true at
once, is unacceptable to logic.
In contrast to positive hypotheticals, negative hypotheticals do not have
the capability of expressing paradoxes. The propositions 'If P, notthen P' and
'If nonP, notthen nonP' are not meaningful
or logically conceivable or ever true. Note this well, such propositions are
formally false. Since a form like 'If P, notthen Q' is defined with reference
to a positive conjunction as '{P and nonQ} is possible', we cannot without
antinomy substitute P for Q here (to say '{P and nonP} is possible'), or nonP
for P and Q (to say '{nonP and notnonP} is possible').
It follows that the proposition 'if P, then nonP' does not imply the
lowercase form 'if P, notthen P', and the proposition 'if nonP, then P' does
not imply the lowercase form 'if nonP, notthen nonP'. That is, in the context
of paradox, hypothetical propositions behave abnormally, and not like
contingencybased forms.
This should not surprise us, since the selfcontradictory is logically
impossible and the selfevident is logically necessary. Since paradoxical
propositions involve incontingent theses and antitheses, they are subject to the
laws specific to such basis.
The implications and consistency of all this will be looked into
presently. 2.
The Stolen Concept Fallacy.
Paradoxical propositions actually occur in practise; moreover, they
provide us with some highly significant results. Here are some examples: ·
denial, or even doubt, of the laws of logic
conceals an appeal to those very axioms, implying that the denial rather than
the assertion is to be believed; ·
denial of man's ability to know any reality
objectively, itself constitutes a claim to knowledge of a fact of reality; ·
denial of validity to man's perception, or his
conceptual power, or reasoning, all such skeptical claims presuppose the
utilization of and trust in the very faculties put in doubt; ·
denial on principle of all generalization,
necessity, or absolutes, is itself a claim to a general, necessary, and
absolute, truth. ·
denial of the existence of 'universals', does
not itself bypass the problem of universals, since it appeals to some itself,
namely, 'universals', 'do not', and 'exist'.
More details on these and other paradoxes, may be found scattered
throughout the text. Thus, the uncovering of paradox is an oftused and
important logical technique. The writer Ayn Rand laid great emphasis on this
method of rejecting skeptical philosophies, by showing that they
implicitly appeal to concepts which they try to explicitly deny; she called
this 'the fallacy of the Stolen Concept'.
A way to understand the workings of paradox, is to view it in the context
of dilemma. A selfevident proposition P could be stated as 'Whether P is
affirmed or denied, it is true'; an absolute truth is something which turns out
to be true whatever our initial assumptions.
This can be written as a constructive argument whose left horn is the
axiomatic proposition of P's identity with itself, and whose right horn is the
paradox of nonP's selfcontradiction; the minor premise is the axiom of thorough
contradiction between the antecedents P and nonP; and the conclusion, the
consequent P's absolute truth.
If P, then P — and — if nonP, then P
but either P or nonP
hence, P.
A destructive version can equally well be formulated, using the
contraposite form of identity, 'If nonP, then nonP', as left horn, with the same
result.
If nonP, then nonP — and — if nonP, then P
but either notnonP or nonP
hence, notnonP, that is, P.
The conclusion 'P' here, signifies that P is logically necessary, not
merely that P is true, note well; this follows from the formal necessity of the
minor premise, the disjunction of P and nonP, assuming the right horn to be well
established.
Another way to understand paradox is to view it in terms of knowledge
contexts. Reading the paradox 'if nonP, then P' as 'all contexts with nonP are
contexts with P', and the identity 'if P, then P' as 'all contexts with P are
contexts with P', we can infer that 'all contexts are with P', meaning that P is
logically necessary.
We can in similar ways deal with the paradox 'if P, then nonP', to obtain
the conclusion 'nonP', or better still: P is impossible. The process of
resolving a paradox, by drawing out its implicit categorical conclusions, may be
called dialectic.
Note in passing that the abridged expression of simple dilemma, in a
single proposition, now becomes more comprehensible. The compound proposition
'If P, then {Q and nonQ}' simply means 'nonP'; 'If nonP, then {Q and nonQ}'
means 'P'; 'If (or whether) P or nonP, then Q' means 'Q'; and 'If (or whether) P
or nonP, then nonQ' means 'nonQ'. Such propositions could also be categorized as
paradoxical, even though the contradiction generated concerns another thesis.
However, remember, the above two forms should not be confused with the
lesser, negative hypothetical, relations 'Whether P or nonP, (notthen not) Q'
or 'Whether P or nonP, (notthen not) nonQ', respectively, which are not
paradoxical, unless there are conditions under which they rise to the level of
positive hypotheticals.
Normally, we presume our information already free of selfevident or
selfcontradictory theses, whereas in abnormal situations, as with paradox,
necessary or impossible theses are formally acceptable eventualities.
A hypothetical of the primary form 'If P, then Q' was defined as 'P and
nonQ are impossible together'. But there are several ways in which this
situation might arise. Either (i) both the theses, P and nonQ, are individually
contingent, and only their conjunction is impossible — this is the normal
situation. Or (ii) the conjunction is impossible because one or the other of the
theses is individually impossible, while the remaining one is individually
possible, i.e. contingent or necessary; or because both are individually
impossible — these situations engender paradox.
Likewise, a hypothetical of the contradictory primary form 'If P,
notthen Q' was defined as 'P and nonQ are possible together'. But there are
several ways this situation might arise. Either (i) both the theses, P and nonQ,
and also their conjunction, are all contingent — this is the normal situation.
Or (ii) one or the other of them is individually not only possible but
necessary, while the remaining one is individually contingent, so that their
conjunction remains contingent; or both are individually necessary, so that
their conjunction is also not only possible but necessary — these situations
engender paradox.
These alternatives are clarified by the following tables, for these
primary forms, and also for their derivatives involving one or both antitheses.
The term 'possible' of course means 'contingent or necessary', it is the common
ground between the two. We will here use the symbols 'N'
for necessary, 'C' for contingent
(meaning possible but unnecessary), and 'M'
for impossible. The combinations are numbered for ease of reference. The
symmetries in these tables ensure their completeness. Table
31.1 Modalities
of Theses and Conjunctions.
The
following table follows from the preceding. 'Yes' indicates that an implication and its contraposite are implicit
in the form concerned, while 'no'
indicates that they are excluded from it. '=►'
here means implies, and '◄='
means is implied by. Table
31.2 Corresponding
Definite Hypotheticals.
Normal hypothetical logic thus assumes the theses of hypotheticals always
both contingent, and so limits itself to cases Nos. 1
to 7 in the above tables.
However, the abnormal cases Nos. 8 to
15, in which one or both theses are not contingent (that is, are
selfevident or selfcontradictory), should also be considered, to develop a
complete logic of hypotheticals.
The definition of the primary positive form 'If P, then Q', while
remaining unchanged as 'P plus nonQ is not possible', is now seen to more
precisely comprise the following situations: Nos. 3,
6, 8, 11,
12, 14, or 15,
that is, all the cases where 'P and nonQ' is impossible ('M'), or 'P implies Q'
is marked 'yes'.
The definition of the primary negative form 'If P, notthen Q', while
remaining unchanged as 'P plus nonQ is not impossible', is now seen to more
precisely comprise the following situations: Nos. 1,
2, 4,
5, 7, 9, 10,
or 13, that is, all the cases where 'P and nonQ' is contingent (C), or
'P implies Q' is marked 'no'.
The other six hypothetical forms, involving the antitheses of P and/or Q,
can likewise be given improved definitions, by reference to the above tables.
Notice the symmetries in these tables. In case No. 1,
all conjunctions are 'C' and all implications are 'no'. In cases Nos. 25,
one conjunction is 'M', and one implication is 'yes'. In cases 611,
two conjunctions are 'M', and two implications are 'yes'. In cases Nos. 1215,
three conjunctions are 'M', and three implications are 'yes'. Note the
corresponding statuses of individual theses in each case.
The process of contraposition is universally applicable to all
hypotheticals, positive or negative, normal or abnormal, for it proceeds
directly from the definitions. For this reason, in the above tables, each
implication is firmly coupled with a contraposite. Likewise, the negation of any
implication engenders the negation of its contraposite, so that the above tables
also indirectly concern negative hypotheticals, note well.
We must be careful, in developing our theory of hypothetical
propositions, to clearly formulate the breadth and limits of application of any
process under consideration, and specify the exceptions if any to its rules. The
validity or invalidity of logical processes often depends on whether we are
focusing on normal or abnormal forms, though in some cases these two classes of
proposition behave in the same way. If these distinctions are not kept in mind,
we can easily become guilty of formal inconsistencies.
Paradoxical propositions obey the laws of logic which happen to be
applicable to all hypotheticals, that is, to hypotheticals of unspecified basis.
But paradoxicals, being incontingencybased hypotheticals, have properties which
normal hypotheticals lack, or lack properties which normal hypotheticals have.
In such situations, where differences in logical properties occur, general
hypothetical logic follows the weaker case.
The similarities and differences in formal behavior have already been
dealt with in appropriate detail in the relevant chapters, but some are reviewed
here in order to underscore the role played by paradox.
a.
Opposition.
In the doctrine of opposition, we claimed that 'If P, then Q' and 'If P,
then nonQ' must be contrary, because if P was true, Q and nonQ would both be
true, an absurdity. However, had we placed these propositions in a destructive
dilemma, as below, we would have obtained a legitimate argument:
If P, then Q — and — if P, then nonQ
but either nonQ or Q
hence nonP
Likewise, 'If P, then Q' and 'If nonP, then Q' could be fitted in a valid
simple constructive dilemma, yielding Q, instead of arguing as we did that they
must be contrary because their contrapositions result in the absurdity of nonQ
implying nonP and P.
It follows that these contrarieties are only valid conditionally, for
contingencybased hypotheticals. There are exceptional circumstances in which
they do not hold, namely relative to abnormal hypotheticals (including
paradoxicals).
This is also independently clear from the observation of 'yes' marks
standing parallel, in cases Nos. 8, 14,
15 (allowing for both 'P implies
nonQ' and 'P implies Q', where P is impossible), and in cases Nos. 11,
12, 14 (allowing for both
'P implies Q' and 'nonP implies Q', where Q is necessary).
Similar restrictions follow automatically for the subcontrariety between
'If P, notthen nonQ' and 'If P, notthen Q', and likewise for the
subalternation by the uppercase 'If P, then Q' of the lowercase 'If P, notthen
nonQ' (which corresponds to obversion). These oppositions only hold true for
normal hypotheticals; when dealing with abnormal hypotheticals (and therefore in
general logic), we must for the sake of consistency regard the said propositions
as neutral to each other.
b.
Eduction.
Similarly with the derivative eductions. The primary process of
contraposition is unconditional, applicable to all hypotheticals, but the other
processes can be criticized in the same way as above, by forming valid simple
dilemmas, using the source proposition and the denial of the proposed target, or
the contraposite(s) of one or the other or both, as horns.
Alternatively, these propositions can be combined in a syllogism,
yielding a paradoxical conclusion. Thus:
In the case of obversion or obverted conversion (in the former, negate
contraposite of target):
If Q, then nonP (negation of target)
if P, then Q (source)
so, if P, then nonP (paradox = nonP)
In the case of conversion by negation or obverted inversion (in the
latter, negate contraposite of target):
If P, then Q (source)
if nonQ, then P (negation of target)
so, if nonQ, then Q (paradox = Q)
Thus, eductive processes other than contraposition are only good for
contingencybased hypotheticals, and may not be imitated in the abnormal logic
of paradoxes. This is made clear in the above tables, as follows.
Consider the paradigmatic form 'If P, then Q'. If we limit our attention
to cases Nos. 17, then it occurs in only two situations, subalternating (3)
or implicance (6). In these two
situations, 'P implies nonQ' is uniformly 'no', so the obverse, 'If P, notthen
nonQ' is true; and the contraposite 'Q implies nonP' is also 'no', so the
obverted converse, 'If Q, notthen nonP' is true; 'nonP implies Q' is uniformly
'no', so the obverted inverse 'If nonP, notthen Q' is true; and the
contraposite 'nonQ implies P' is also 'no', so the converse by negation 'If
nonQ, notthen P' is true. With regard to inversion and conversion, they are not
applicable, because 'nonP implies nonQ' and 'Q implies P' are 'no' in one case,
but 'yes' in the other. However, if now we expand our attention to include cases Nos. 815, we see that 'If P, then Q' occurs additionally if P is selfcontradictory and Q is contingent (8) or P is contingent and Q is selfevident (11) or P,Q are each selfevident (12) or P is selfcontradictory and Q is selfevident (14) or P,Q are each selfcontradictory (15). The above mentioned uniformities, which made the stated eductions feasible, now no longer hold. There is a mix of 'no' and 'yes' in the available alternatives which inhibits such eductions.
c.
Deduction.
With regard to syllogism, the nonsubaltern
moods, validated by reductio ad absurdum, remain universally valid, since
such indirect reduction is essentially contraposition, and no other eductive
process was assumed. But the subaltern
moods in all three figures, are only valid for normal hypotheticals. Since
these moods presuppose subalternations for their validation, i.e. depend on
direct reductions through obversion or obverted inversion, they are not valid
for abnormal hypotheticals.
With regard to apodosis, the moods with a modal minor premise provide us
with the entrypoint into abnormal logic. As for dilemma, it is the instrument par
excellence for unearthing paradoxes in the course of everyday reasoning. If
we put any simple dilemma,
constructive (as below) or destructive (mutadis mutandis), in syllogistic form,
we obtain a paradoxical conclusion:
If P, then R — and — if Q, then R
but P and/or Q
hence, R
This implies the sorites:
If nonR, then nonP (contrapose left horn)
if nonP, then Q (minor)
if Q, then R (right horn)
hence, if nonR, then R (paradoxical conclusion = R)
Thus, paradoxical propositions are an integral part of general
hypothetical logic, not some weird appendix. They highlight the essential
continuity between syllogism and simple dilemma, the latter being reducible to
the former.
It follows incidentally that, since (as earlier seen) apodosis may be
viewed as a special, limiting case of simple dilemma, and simple dilemma as a
special, limiting case of complex dilemma — all the inferential processes
relating to hypotheticals are closely related.
The paradox generated by simple dilemma of course depends for its truth
on the truth of the premises. We should not hurriedly infer, from the paradox
inherent in every simple dilemma, that all truths are ultimately selfevident,
and all falsehoods ultimately selfcontradictory. Knowledge is not a purely
rational enterprise, but depends largely on empirical findings.
As already pointed out, simple dilemma yields a categorical necessity or
impossibility as its conclusion, only if all its premises are themselves
indubitably incontingent. Should there be tacit conditions for, or any doubt
regarding the unconditionality of, the hypotheticals (the horns) and/or the
disjunction (the minor premise), then the conclusion would be proportionately
weakened with regard to its logical modality.
Thus, with reference to the foregoing example, granting
the horns of the major premise: in the specific case where our minor premise
is a formally given disjunction — if, say, P and Q are contradictory to each
other (P = nonQ, Q = nonP) — then the R conclusion is indeed necessary. But
usually, the listed alternatives P and Q are only contextually exhaustive, so
that the R conclusion is only factually true.
So, although every logical necessity is selfevident, and every logical
impossibility is selfcontradictory, formally speaking, according to our
definitions, we might be wise to say that these predications are not in practise
reciprocal, and make a distinction between apodictic and factual paradox. The
former is independently obvious; the latter derives from more empirical data,
and therefore, though contextually trustworthy, has a bit less weight and
finality.
Note lastly, the inconsistency of two 'equally cogent' simple dilemmas
can now be better understood, as due to their implying contrary paradoxes.
