### Two objections to Priest's Inclosure Schema

(My student Rein van der Laan will be defending his Bachelors thesis on Priest’s Inclosure Schema this week. It was in the process of supervising him that I developed my current ideas on the topic, which means that the content of this post is basically joint work with Rein.)

In a number of papers (such as this 1994 paper) and in his book Beyond the Limits of Thought (BtLoT), Graham Priest defends the claim that all paradoxes of self-reference can be adequately captured by the Inclosure Schema IS, which he formulates in the following way:

(1) Ω = {y; φ(y)} exists and ψ(Ω)               Existence
(2) if x Ω and ψ(x) (a) δ(x) ∉ x               Transcendence
(b) δ(x) Ω             Closure

The different paradoxes of self-reference would be generated by different instantiations of the schematic letters of the schema (for details, consult BtLoT).

There have been quite some articles discussing IS in the meantime (among others: Abad, Grattan-Guinness, Badici, and responses by Priest and Weber), where a number of interesting objections have been raised against the idea that IS successfully describes all paradoxes of self-reference (the Liar, Russell’s paradox etc). Here I discuss two (not necessarily novel) objections that I think are quite problematic for Priest’s general project with IS -- in particular, that of arguing for the Principle of Uniform Solution: similar paradoxes must receive similar solutions. (Unsurprisingly, he goes on to claim that only dialethism is able to offer an uniform solution to all these paradoxes.)

The over/undergeneration objection. One plausible way to understand what Priest is up to with IS is that it is intended as a formal explanans for the informal notion of ‘paradoxes of self-reference’. If this is correct, then it is legitimate to raise the question of whether IS gets the extension of the informal concept right; it may overgenerate (arguments which we do not want to count as self-referential paradoxes would fit into the schema) and/or undergenerate (it may fail to capture arguments which we do want to count as self-referential paradoxes).

As it turns out, IS seems both to over- and undergenerate. It overgenerates in that a number of reductio arguments which are not paradoxical, properly speaking, seem to conform to IS (as pointed out e.g. by Abad). One example would be Cantor’s diagonal argument for the uncountability of the real numbers (see this earlier blog post of mine for a presentation of the argument). And it undergenerates in that Curry’s paradox, which obviously (?) should count as a self-referential paradox, cannot be accounted for by means of IS. Priest is well aware of this limitation, but retorts:
[Curry] paradoxes belong to a quite different family. [They] do not involve negation and, a fortiori, contradiction. They therefore have nothing to do with contradictions at the limits of thought. (BtLoT, 169)
This seems odd, as the original claim seemed to be that IS was meant to describe paradoxes of self-reference in general, not only those involving a negation. (To be fair, Curry is the hardest of all paradoxes; as Graham himself says, Curry is hard on everyone…) At any rate, if IS both over- and undergenerates as a formal explanans of paradoxes of self-reference (which is at least what the original 1994 paper seems to claim it should be), this is not good news for Priest’s general project. (He may, of course, say that Curry falls out of the scope of IS and thus of the Principle of Uniform Solution, but the overgeneration charge still stands.)

The form/matter objection. A useful and frequently cited definition of paradoxes is the one offered by Sainsbury (2009, 1): what characterizes a paradox is “an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises.” This means that one crucial component of a paradox is the degree of belief an agent attributes to the premises and the cogency of the reasoning, and the degree of disbelief she attributes to the conclusion. The ‘apparently’ clause does not need to entail a relativistic conception of paradoxes, but it does mean that a paradox has a perspectival component. Galileo’s paradox was paradoxical for Galileo and many others, but not for Cantor, who did not see the conclusion of the reasoning as unacceptable.

Now, there is an old but by now largely forgotten conception of the form and matter of an argument according to which the matter of the argument is the ‘quality’ of its premises. In this vein, a materially defective argument is one where the premises are false, while a formally defective argument is one where the reasoning is not valid. (See this article of mine for the historical background of this conception.) On the basis of this idea, we could say that the matter of an argument corresponds to one’s degree of belief/disbelief in the premises/conclusion (again, perspectival), and the form corresponds to the structure of the argument. This would entail that paradoxes come in degrees, in function of the agent's degrees of (dis)belief in the premises, reasoning and conclusion.

With this distinction in mind, we can see why IS fails to capture the extension of the concept of paradoxes of self-reference: it captures only the form of such arguments, but is silent concerning their matter (understood as the degrees of (dis)belief in premises and conclusion). The paradoxical nature of a paradox, however, is crucially determined by the degrees of (dis)belief in the premises and conclusion (as made clear in Sainsbury’s quote). [UPDATE: this sentence has been misunderstood by many people. Notice that I am here using an unconventional understanding of the matter of an argument (introduced in the previous paragraph), not the more familiar schematic notion of form vs. matter.] This is why IS cannot differentiate between a truly paradoxical argument and a reductio argument, intended to establish the falsity of one of the premises rather than being truly paradoxical.

[UPDATE: In BtLoT Priest introduces the restriction that different instantiations of IS must yield true premises for an argument to count as an instantiation of IS, and thus to be an inclosure paradox. This is why the Barber then does not count as an inclosure paradox. This restriction seems to me to be too strong, as often what is under discussion when a paradox emerges is whether the apparently acceptable premises are indeed as acceptable as they seem.]

So ultimately, the conclusion seems to be that IS fails to deliver what Priest wants it to deliver. Nevertheless, I firmly believe that the formulation of IS has been one of the most interesting and important developments in research on paradoxes of the last decades. It forces us to think about paradoxes with a much-needed higher level of generality, and thus leads to a new, deeper understanding of the phenomenon – even if the conclusion must be that IS cannot be the whole story after all.

UPDATE: Some further thoughts on the Inclosure Schema here.

1. I am very surprised by the statement "[Curry] paradoxes belong to a quite different family. [They] do not involve negation and, a fortiori, contradiction.", see also https://plus.google.com/u/0/116887476001511997780/posts/NBvhWqHa4cq

1. In a forthcoming (co-authored) paper, Priest explains in more detail why he claims that Curry is a different beast altogether. In particular, because it does not need to involve a consequent that is false or impossible, there are versions of Curry which do not involve negation in the way you point out. So take C: If C is true, it is sunny today. By the usual means, one can prove that it is sunny today, which happens to be true (where I am at least), but it is still highly suspicious that this gets established in this way.

So now Priest et al say that Curry is not an Inclosure paradox, or a paradox of contradiction. That's fine, but then the original explanandum, which was supposed to be the concept of paradoxes of self-reference, has now become something else.

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3. In my opinion a form (X -> Y) is a generalization of negation. The left hand side of an implication in a positive context forms a negative context, etc.. This is known as sub formula polarity. So I consider Curry a generalized paradoxon of contradiction.

The curry paradoxon symbolizes one trajectory of attack when for example dealing with paradoxes of comprehension. The comprehension schema exists y forall x (x in y <-> A(x)) has its limits. For example using A(x) = ~(x in x), one finds y_0 in y_0 <-> ~(y_0 in y_0).

The set theory attack is to bound the witness, so that the modified comprehension schema would read exists y forall x(x in y <-> A(x) & x in z). The attack symbolized by the Curry paradoxon is to find a logic that doesn't collapse for y_0 in y_0 <-> ~(y_0 in y_0).

The most simple proof of Currys paradoxon in LM (= positive propositional logic) shows that paraconsistent logics such as LM don't help. LM does not have ex falso quodlibet, but nevertheless admits Curry paradoxon. That's how I arrived at the paradoxon.

I stopped then, and started using type theory to bound the comprehension. But I guess it is also possible to tweak the logic, for example by removing contraction. This would not allow the last step in my G+ proof, since there the premisse x -> x -> y is comming from two branches. Right?

So why ban the Curry paradox? Its a real jewel.

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