C. S. HANSEN 59
The contradiction is derived as follows. From (B1) by exis-
tential specification we have aA
, and then from (B4)
follows by universal specification. As-
sume
:Bay aBy
a
:yAya
ba
:ya By
and then deduce. From (B3) follows
and hence by existential specification again we
have . From this follows in conjunction with (B2) and
that and
:ya
By
By
:yb
b
hold. As we also
get from (B4) this implies the contra-
diction. Discarding the assumption, we have
:Bbyb By
a
:aBy
, from
which it follows together with that
holds, and then the use of existential specification
yet again produces
Ba y
:yaBy
b. From this a contradiction can be de-
rived analogously to how it was derived from
a.
What we see is that in addition to (B4) the only thing that is
required is a non-empty set equipped with a transitive and
non-wellfounded order relation. The metric on the reals and the
linearity of the standard order relation on the same do not play
any role in the derivation of the contradiction; their purpose is
only to make the premises plausible. The metric serves to make
it plausible that the traveller can cover the distance from point
A to point B in a finite amount of time. The linearity makes it
plausible that the right hand side of (In) is a given at the time
where god number n must decide whether or not to make the
left hand side true.
What is interesting about the fact that (B1) - (B4) is the “lo-
gical essence” of the paradox is that it reveals that New Zeno
has the same logical structure as Yablo’s Paradox (Yablo, 1993).
This paradox results from this infinite list of sentences:
Y1: For all n > 1 the sentence Yn is not true
Y2: For all n > 2 the sentence Yn is not true
Y3: For all n > 3 the sentence Yn is not true
This set of sentences satisfies (B1) - (B4) when the predi-
cates are interpreted as follows: “Bx” means that x is true,
“
A” means that x is a sentence on the list and “x < y”
means that x is further down on the list than y.
The (semi)formal proof of contradiction above then corre-
sponds to this informal proof: There is some sentence on the list,
call it “a”. a is true iff all sentences further down on the list are
false. Assume that a is true. Then all sentences further down on
the list are false. There is a sentence further down on the list,
call it “b”. All sentences further down than b are false. b is false.
b is true iff all sentences further down on the list than b are
false. Contradiction! So a is false. There is a true sentence fur-
ther down than a, call it “b”. b is true. Contradiction follows
analogously.
The Principle of Uniform Solution
According to Priest’s so-called Principle of Uniform Solution,
paradoxes with the same logical structure should be solved in a
similar way. When it has now been demonstrated that New
Zeno and Yablo’s Paradox have the same logical structure, this
puts a significant restriction on what can be accepted as possi-
ble solutions to these two paradoxes. On the one hand, the de-
nial-of-the-possibility-of-motion solution and the logic-stops-
them solution to New Zeno can not be transfered to Yablo’s
Paradox, and on the other hand, the denial of either semantical
closure, the validity of the Tarskian T-schema or classical logic
—the conjunction of which are normally considered the source
of the semantic paradoxes—will not solve New Zeno.
I claim that constructivism can serve as such a unifying solu-
tion. Three aspects of constructivism, relevant to this paper, can
be distinguished. The first is the thesis that any infinity is
merely a potiential such, in the sense that its elements are cre-
ated in time and at any given time is finite in number. The sec-
ond is mentalism with regard to certain abstract entities, e.g.
mathematical objects. And the third is the rejection of tertium
non datur. It is the first of these three aspects which can unify
the solutions. The other two will become relevant in the fol-
lowing, but do not apply to New Zeno and are therefore not part
of what unifies. The second is a way to explain why the first
applies to abstract entities. And the third, even though often
considered the central thesis of constructivism, is but a side-
effect of the two others.
Constructivism qua the first aspect solves New Zeno as ex-
plained in the first section. Yablo’s Paradox can be solved with
Kripke’s theory of truth (Kripke, 1975)1. And that theory can, I
will argue, be interpreted as a constructivistic theory.
In (Beall, 2007, chapter 1) Beall presents Kripke’s theory
through a metaphor about books. It goes as follows. Imagine a
world initially consisting only of non-semantic facts. In this
world, there is a writer with two very large books. They carry
the titles The True and The False. In the beginning they are
empty, but the writer sets out to fill them so that they accurately
reflect their titles. In the first book, he records every fact of the
world, and in the second, he records every state of affairs that
fails to obtain in the world. For instance, he writes “Snow is
white” in the first book and “Snow is green” in the second.
After having done so, he realises that his work is not complete.
For now there are more facts than when he started. By writing
in the books, he has added facts to the world, namely facts
about what is written in the books, and he did not include these
facts in The True, nor did he include non-obtaining facts about
the books in The False. So in each book, he puts the heading
“Chapter 1” over what he has written so far and starts writing
the more comprehensive chapters 2 of each book. Chapter 2 of
The True is a complete record of all facts about the world out-
side the books as well as about chapter 1 of each of the books.
He uses the predicate “is true” in the meaning “is a sentence
written in The True” and similarly the predicate “is false” in the
meaning “is a sentence written in The False”. So “‘Snow is
white’ is true” and “‘Snow is green’ is false” both appear in this
chapter. Because “Snow is green” is in The False, it is deter-
mined that this sentence will never be in The True, no matter
how many chapters are written, so the writer can put “‘Snow is
green’ is true” in chapter 2 of The False.
There are sentences that the writer puts in neither of the two
chapters 2. One of them is “‘‘‘Snow is white’ is true’ is true’ is
true”. This is because the sentence “‘‘Snow is white’ is true’ is
true” is not in chapter 1 of either of the books, so whether it
will be written in The True is not yet determined. Another is
“This sentence is false”. And for the same reason. The sentence
referred to by “This sentence” namely “This sentence is false”.
itself, is not in chapter 1 of either of the books. After having
written the two new chapters, there are again new facts, so the
writer also compiles increasingly comprehensive chapters 3, 4,
5, etc. Of the two mentioned sentences, the first eventually gets
into one of the books (The True, in chapter 4), while the second
never does. It is “ungrounded”.
According to this metaphor, Kripke’s theory introduces an
element of temporality in semantics; the sentence “‘Grass is
green’ is true” is made true later than the sentence “Grass is
green”. I believe we should take this temporality seriously and
not just metaphorically. Imagine, in analogy to the ideal ma-
1Much criticism can be directed at Kripke’s theory, so whether the paradox
can really be solved by the theory is a controversial issue. I will not go into
that discussion here, but simply assume that Kripke’s theory is adequate.