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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 not-nonP 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 not-nonP' is true (P and nonP are incompatible), and
'If not-nonP, 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 self-denial.
From this we see that there may be propositions which are logically
self-destructive, and which logically support their own negations. Let us then
put forward the following definitions. A proposition is self-contradictory if it denies itself, i.e. implies its own
negation. A proposition is therefore self-evident
if its negation is self-contradictory, i.e. if it is implied by its own
negation.
Thus, the proposition 'If P, then nonP' informs us that P is
self-contradictory (and so logically impossible), and that nonP is self-evident
(and so logically necessary). Likewise, the proposition 'If nonP, then P'
informs us that nonP is self-contradictory, and that P is self-evident.
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, not-then 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, not-then 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, not-then P' and
'If nonP, not-then nonP' are not meaningful
or logically conceivable or ever true. Note this well, such propositions are
formally false. Since a form like 'If P, not-then 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 not-nonP} is possible').
It follows that the proposition 'if P, then nonP' does not imply the
lowercase form 'if P, not-then P', and the proposition 'if nonP, then P' does
not imply the lowercase form 'if nonP, not-then nonP'. That is, in the context
of paradox, hypothetical propositions behave abnormally, and not like
contingency-based forms.
This should not surprise us, since the self-contradictory is logically
impossible and the self-evident 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 oft-used 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 self-evident 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 self-contradiction; 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 not-nonP or nonP
hence, not-nonP, 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, (not-then not) Q'
or 'Whether P or nonP, (not-then 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 self-evident or
self-contradictory 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,
not-then 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. 'à'
here means implies, and 'ß'
means is implied by. Table
31.2 Corresponding
Definite Hypotheticals.
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