At the Limits of Thought

Abstract

The inclosure schema has been proposed by Priest as the structure of many paradoxes (Priest in Beyond the limits of thought. Oxford University Press, 2002). The inclosure analysis has many virtues, especially as a step toward a uniform solution to the paradoxes. Inclosure suggests that paradoxes arise at the limits of thought because the limits can be surpassed, and also not; and so dialetheism is true. I explore the consequences of accepting Priest’s proposal. From a thoroughly dialetheic perspective, then, I find that the import of inclosure changes: (i) some limit phenomena cannot be contradictory, on pain of absurdity, and (ii) true contradictions are better thought of as local, not “limit” phenomena. Dialetheism leads back from the edge of thought, to the inconsistent in the every day.

How wonderful that we have met with a paradox.

Now we have some hope of making progress.

– Niels Bohr

Before I studied dialetheism, mountains were mountains and rivers were rivers.

After studying dialetheism for some time,

mountains were no longer mountains and rivers were no longer rivers.

But now that I have found rest, I see

mountains are mountains,

and rivers are rivers.

– adapted from Compendium of the Five Lamps 1252; cf.

(Garfield and Priest 2009)

Appendix: Double Inclosures

There is fun to be had with inclosure structures.

A double inclosure is a pair $\langle \mathcal {I}_0, \mathcal {I}_1 \rangle$ such that

$$\mathcal {I}_0 = \langle W_0, \partial _{W_0} \rangle , \qquad \mathcal {I}_1 = \langle W_1, \partial _{W_1} \rangle $$

are inclosures, where each carrier set is the complement of the other:

$$\overline{W_0} = \{x: x \not \in W_0\} =: W_1$$

If $W_0=\{x: \varphi (x)\}$ then $W_1 = \{x: \lnot \varphi (x) \}$, and $\overline{W_1} = W_0$ by double negation elimination. Basic calculations show that the members of the double inclosure pair share their limit contradictions:

$$\partial (W_0) \in (W_0 \cap \overline{W_0}) = (W_0 \cap W_1) = (W_1 \cap \overline{W_1}) \ni \partial (W_1)$$

So there is an overlapping and non-empty boundary between the two complements. Each of these inclosures can be said to be dual to the other.

Not all inclosures have a dual. Mirimanoff’s paradox, for instance, concerns the set of all well-founded sets; but rather like the truth teller (this sentence is true) the set of all non-well-founded sets does not support an inclosure; see Barwise and Moss (1996). Nevertheless, double inclosures exist. If we take, for example, $W_0$ to be the set of all true sentences, then $W_1$ is the set of all falsities. For any $X \subseteq W_1$, the sentence $\partial _{W_1}(X)=$ “this sentence is in X” is not in X. (If it were, then it would be false, so it is not, by reductio.) And then ipso facto it is in $W_1$. But then $\partial _{W_1}(W_1)$ is “this sentence is false” which is in $W_1$ by usual reasoning. So the dual structure to the liar inclosure is also an inclosure.

What else might this model? Priest has argued that the sorites paradox fits the inclosure schema (Priest 2010) (Sect. 26.4.2.4). Sorites paradoxes concern vague predicates, and if $\varphi$ is vague, then $\lnot \varphi$ is vague, too. Insofar as the inclosure schema matches the sorites, a double inclosure might express it nicely.

Or take an inclosure chain to be a (decreasing) sequence of inclosures

$$\langle \mathcal {I}_0, \mathcal {I}_1, \mathcal {I}_2, \ldots \rangle $$

such that

$$ \begin{aligned} W_0= & {} \{x: \varphi _0(x) \} \\ W_1= & {} \{x: \varphi _0(x) \& \varphi _1(x) \} \\&\!\!\!\!\vdots&\\ W_n= & {} \{x: \varphi _0(x) \& \ldots \& \varphi _n(x) \} \\&\!\!\!\!\vdots&\end{aligned}$$

where each $W_i$ carries an inclosure. Then $W_0 \supseteq W_1 \supseteq W_2 \ldots$ by conjunction–elimination, and the diagonalizer $\partial _{n+1}$ on $\mathcal {I}_{n+1}$ is the diagonalizer $\partial _n$ on $\mathcal {I}_n$ restricted to the carrier set $W_{n+1}$. So each $\mathcal {I}_{n+1}$ is a sub-inclosure of $\mathcal {I}_n$. There will be a corresponding sequence of contradictions at each boundary,

$$ \langle \partial (W_0), \partial (W_1), \ldots \rangle $$

What might this model? Priest has argued that motion and change involve inconsistency (Priest 2006b, Chaps. 11, 12). An inclosure chain could capture Hegel’s claim, that “Contradiction is the very moving principle of the world” [Lesser Logic (1812), par 119]—to which he adds: “and it is ridiculous to say that a contradiction is unthinkable.”

Exercise for the reader on an idle Sunday afternoon. Put these ideas together for a double inclosure chain. [Hint: Such a chain models what happens if you think about inclosures for too long.]