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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER
32. DOUBLE PARADOXES.
We
have seen that logical propositions of the form 'if P, then nonP' (which equals
to 'nonP') or 'if nonP, then P' (which equals to 'P'), are perfectly legal. They
signify that the antecedent is self-contradictory and logically impossible, and
that the consequent is self-evident and logically necessary. As propositions in
themselves, they are in no way antinomial; it is one of their constituents which
is absurd.
Although either of those propositions, occurring alone, is formally quite
acceptable and capable of truth, they can never be both true: they are
irreconcilable contraries and their conjunction is formally impossible. For if
they were ever both true, then both P and nonP would be implied true.
We must therefore distinguish between single
paradox, which has (more precisely than previously suggested) the form 'if
P, then nonP; but if nonP, not-then P; whence nonP', or the form 'if nonP, then
P; but if P, not-then nonP; whence P' — and double
paradox, which has the form 'if P, then nonP, and if nonP, then P'.
Single paradox is, to repeat, within the bounds of logic, whereas double
paradox is beyond those bounds. The former may well be true; the latter always
signifies an error of reasoning. Yet, one might interject, double paradox occurs
often enough in practise! However, that does not make it right, anymore than the
occurrence of other kinds of error in practise make them true.
Double paradox is made possible, as we shall see, by a hidden misuse
of concepts. It is sophistry par excellence, in that we get the superficial
illusion of a meaningful statement yielding results contrary to reason. But upon
further scrutiny, we can detect that some fallacy was involved, such as
ambiguity or equivocation, which means that in fact the seeming contradiction
never occurred.
Logic demands that either or both
of the hypothetical propositions which constituted the double paradox, or
paradox upon paradox, be false.
Whereas single paradox is resolved, by
concluding the consequent categorically, without denying the
antecedent-consequent connection — double paradox is dissolved,
by showing that one or both of the single paradoxes involved are untrue,
nonexistent. Note well the difference in problem solution: resolution 'explains'
the single paradox, whereas dissolution 'explains away' the double paradox.
The double paradox serves to show
that we are making a mistake of some kind; the fact that we have come to a
contradiction, is our index and proof enough that we have made a wrong
assumption of sorts. Our ability to intuit logical connections correctly is not
put in doubt, because the initial judgment was too rushed, without pondering the
terms involved. Once the concepts involved are clarified, it is the rational
faculty itself which pronounces the judgment against its previous impression of
connection.
It must be understood that every double paradox (as indeed every single
paradox), is teaching us something. Such events must not be regarded as threats
to reason, which put logic as a whole in doubt; but simply as lessons. They are
sources of information, they reveal to us certain logical rules of concept
formation, which we would otherwise not have noticed. They show us the outer
limits of linguistic propriety.
We shall consider two classical examples of double paradox to illustrate
the ways they are dissolved. Each one requires special treatment. They are
excellent exercises.
An ancient example of double paradox is the well-known 'Liar Paradox',
discovered by Eubulides, a 4th cent. BCE Greek of the Megarian School. It goes:
'does a man who says that he is now lying speak truly?' The implications seem to
be that if he is lying, he speaks truly, and if he is not lying, he speaks
truly.
Here, the conceptual mistake underlying the difficulty is that the
proposition is defined by reference to itself. The liar paradox is how we discover
that such concepts are not allowed.
The word 'now' (which defines the proposition itself as its own subject)
is being used with reference to something which is not yet in existence, whose
seeming existence is only made possible by it. Thus, in fact, the word is empty
of specific referents in the case at hand. The word 'now' is indeed usually
meaningful, in that in other situations it has precise referents; but in this
case it is used before we have anything to point to as a subject of discourse.
It looks and sounds like a word, but it is no more than that.
A more modern and clearer version of this paradox is 'this proposition is
false', because it brings out the indicative function of the word 'now' in the
word 'this'.
The word 'this' accompanies our pointings and presupposes that there is
something to point to already there. It cannot create a referent for itself out
of nothing. This is the useful lesson taught us by the liar paradox. We may well
use the word 'this' to point to another word 'this'; but not to itself. Thus, I
can say to you 'this "this", which is in the proposition "this
proposition is false"', without difficulty, because my 'this' has a
referent, albeit an empty symbol; but the original 'this' is meaningless.
Furthermore, the implications of this version seem to be that 'if the
proposition is true, it is false, and if it is false, it is true'. However, upon
closer inspection we see that the expression 'the proposition' or 'it' has a
different meaning in antecedents and consequent.
If, for the sake of argument, we understand those implications as: if
this proposition is false, then this proposition is true; and if this
proposition is true, then this proposition is false — taking the 'this' in the
sense of self-reference by every
thesis — then we see that the theses do not in fact have one and the same
subject, and are only presumed to be in contradiction.
They are not formally so, any more than, for any P1 and P2, 'P1 is true'
and 'P2 is false' are in contradiction. The implications are not logically
required, and thus the two paradoxes are dissolved. There is no
self-contradiction, neither in 'this proposition is false' nor of course in
'this proposition is true'; they are simply meaningless, because the indicatives
they use are without reference.
Let us, alternatively, try to read these implications as: if 'this
proposition is false' is true, then that proposition is false; and if that
proposition is false, then that proposition is true' — taking the first 'this'
as self-reference and the 'thats' thereafter as all pointing us backwards to the
original proposition and not to the later theses themselves. In other words, we
mean: if 'this proposition is false' is true, then 'this proposition is false'
is false, and if 'this proposition is false' is false, then 'this proposition is
false' is true.
Here, the subjects of the theses are one and the same, but the
implications no longer seem called for, as is made clear if we substitute the
symbol P for 'this proposition is false'. The flavor of paradox has disappeared:
it only existed so long as 'this proposition is false' seemed to be implied by
or to imply 'this proposition is true'; as soon as the subject is unified, both
the paradoxes break down.
We cannot avoid the issue by formulating the liar paradox as a
generality. The proposition 'I always lie' can simply be countered by 'you lie
sometimes (as in the case 'I always lie'), but sometimes you speak truly'; it
only gives rise to double paradox in indicative form. Likewise, the proposition
'all propositions are false' can be countered by 'some, some not', without
difficulty.
However, note well, both the said general propositions are indeed
self-contradictory; they do produce single paradoxes. It follows that both are
false: one cannot claim to 'always lie', nor that 'there are no true
propositions'. This is ordinary logical inference, and quite legitimate, since
there are logical alternatives.
With regard to those alternatives. The proposition 'I never lie' is not
in itself inconsistent, except for the person who said 'I always lie'
intentionally. The proposition 'all propositions are true' is likewise not
inconsistent in itself, but is inconsistent with the logical knowledge that some
propositions are inconsistent, and therefore it is false; so in this case only
the contingent 'some propositions are true, some false' can be upheld.
The Barber Paradox may be stated as: 'If a barber shaves everyone in his
town who does not shave himself, does he or does he not shave himself? If he
does, he does not; if he does not, he does'.
This double paradox arises through confusion of the expressions 'does not
shave himself' and 'is shaved by someone other than himself'.
We can divide the people in any town into three broad groups: (a) people
who do not shave themselves, but are shaved by others; (b) people who do not
shave themselves, and are not shaved by others; (c) people who shave themselves,
and are not shaved by others. The given premise is that our barber shaves all
the people who fall in group (a). It is tacitly suggested, but not formally
implied, that no one is in group (b), so that no one grows a beard or is not in
need of shaving. But, in any case, the premise in fact tells us nothing about
group (c).
Next, let us subdivide each of the preceding groups into two subgroups:
(i) people who shave others, and (ii) people who do not shave others. It is
clear that each of the six resulting combinations is logically acceptable,
since who shaves me has no bearing on whom I can shave. Obviously, only group
(i) concerns barbers, and our premise may be taken to mean that our barber is
the only barber in town.
Now, we can deal with the question posed. Our barber cannot fall in group
(a)(i), because he is not shaved by others. He might fall in group (b)(i), if he
were allowed to grow a beard or he was hairless; but let us suppose not, for the
sake of argument. This still allows him to fall in group (c)(i), meaning that he
shaves himself (rather than being shaved by others), though he shaves others
too.
Thus, there is no double paradox. The double paradox only arose because
we wrongly assumed that 'he shaves all those who do
not shave themselves' excludes 'he shaves some (such as himself) who do shave themselves'. But 'X
shaves Y' does not formally contradict 'X shaves nonY'; there is no basis
for assuming that the copula 'to shave' is obvertible, so that 'X shaves Y'
implies 'X does not shave nonY'.
If the premise was restated as 'he shaves all those and
only those who do not shave
themselves' (so as to exclude 'he shaves himself'), we would still have an out
by saying 'he does not shave at all'. If the premise was further expanded and
restricted by insisting that 'he somehow shaves or is shaved', it would simply
be self-contradictory (in the way of a single paradox).
Further embellishments could be made to the above, such as considering
people who shave in other towns, or making distinctions between always,
sometimes/sometimes-not, and never. But I think the point is made. The lesson
learned from the barber 'paradox' is that without clear categorizations,
equivocations can emerge (such as that between 'shaves' and 'is shaved'), which
give the illusion of double paradox. |
