In the previous chapter we have argued for the effective computability of the addition function in (some) heap collapsed models. We argued for this by noting that +M/∼ is defined in terms of +M; based on this fact, we may specify an easy algorithm to compute the collapsed addition in terms of the addition function defined in the original models (provided the original addition function is indeed computable). However, is this really surprising? Couldn’t we have reached the same conclusion already through
a much easier route. It is tempting to think so; after allprima facie it is easily seen that a theory T h(M/ ∼) (closed under LP-consequence) is decidable: the values of atomic formulas may be computed using LP-matrices, and given that the model is finite, the truth-value of quantified formulas are computed using the following equivalences (where n is the numeral for n, and m is the greatest number in the model):
• v(∃xϕ) =v(ϕ(x/0)∨...∨ϕ(x/m)) • v(∀xϕ) =v(ϕ(x/0)∧...∧ϕ(x/m))
Hence, a fortiori, collapsed addition is computable. But things are not so simple. Normal intuitions break down in the inconsistent case, urging great cautiousness with swift conclusions.
Denyer(1995 : § 2) is rather wary of the strength of the argument. The decision procedure for a finite collapsed model works essentially by quantifier elimination: LP-matrices will give an effective procedure to calculate the value of any formula prefixed by any arbitrary finite number of quantifiers that can be eliminated through the above equivalences. But now let us suppose that we apply the procedure to a statement preceded bymmany quantifiers. So, for definiteness, consider a statement: ∀x1, ...,∀xmϕ(with x1, ..., xm free in ϕ). Suppose further thatm is an inconsistent
number in the model; that is,m=m+p(withp6= 0). Sayp= 1 so thatm=m+ 1. By this latter equality, the original statement preceded bym-many quantifiers is, after all, preceded my m+ 1-many quantifiers so that the original statement is equal to ∀x1, ...,∀xm+1ϕ. Now, we may apply the decision procedure to the latter statement
and produce a new statement (an arbitrarily finitely long conjunction) preceded my m-many quantifiers. But then it is clear that the decision procedure didn’t get us any far for there are stillm (andm+ 1) quantifiers to eliminate! A similar worry occurs when the statement is preceded even by only one quantifier. We reduce the statement to, suppose, a finite disjunction withl+ 1 disjuncts: ϕ(x/0)∨...∨ϕ(x/l). Now, since l is the greatest number of the finite model, it will be inconsistent and therefore, say, l=l+ 1; so thatϕ(x/0)∨...∨ϕ(x/l+ 1). But then even when we decide the first of these disjuncts (i.e.ϕ(x/0)) we still havel+ 1 disjuncts to decide. Again, we haven’t really advanced in our decision procedure. Such considerations lead Denyer to claim:
Algorithms that require us to take a magic number of steps are, in short, no more use as decision procedures than algorithms that require us to take infinitely many. [...] But that simply means that the decidability [...] attache[d] to [...] paraconsistent arithmetic [i.e. LP-models] is not quite the decidability which, before the limitative results came along, people had hoped would attach to classical arithmetic. (Denyer, 1995 : 570)
The following arguments are not the final word on the matter. And this because, properly speaking, they are not arguments but mainly argumentsketches. The prob- lem is that Denyer’s arguments merely gesture at the negation of the decidability of paraconsistent models. But for example, if the underlying logic is paraconsistent, it is not clear that Denyer’s arguments follow: the idea that subtracting 1 from m+ 1 equals m only holds for the classical material conditional; LP-conditionals appear only in a non-detachable form. And if we are dealing with non-classical paraconsis- tent models, we may doubt if our reasoning regarding meta-theoretical properties such as the computability of LP-consequence should be guided by classical logic. Either way, these comments should give us pause when entailing decidability from finiteness.
5.4
Summary
As we pointed out before there can be an initial strangeness about an inconsistent effective procedure. This small chapter aimed to show that there is nothing strange about it; or better, that an inconsistent computation is as strange as its underlying inconsistent arithmetic – if the underlying arithmetic admits inconsistent elements, then it is a natural consequence that the computational operations are inconsistent too. In fact, what is problematic is to try to keep a classical computational theory or meta-theory for provability in inconsistent arithmetical settings. Another independent but related problem that we addressed concerned decidability in finite paraconsistent models. If a model is finite, it would then seem that consequence-relation is trivially decidable. However, things are not so simple in the paraconsistent case, and there must be a wider story to be told about such an entailment.
Chapter 6
On Supplementation and
Categoricity
6.1
Introduction
This chapter concerns Conclusion3, what we dubbed ‘Supplementation’, as a possible
answer to theLP-argument. To recall, the conclusion was:
Conclusion3: Supplementation: accept that the argument from Ten-
nenbaum’s Theorem is on the right track. Nonetheless, the intended alien LP-constructions are indicative that supplementation of the computabil- ity requirement with stronger constraints is needed to fully determine the intended models.
Our chapter will be divided in an inductive and deductive part. Regarding the induc- tive part, we will analyse four ways to supplement the computability requirement and see that they are not sufficient to rule out inconsistent LP-models from the class of intended models. Regarding the deductive part we will offer a sketch of a reasoning based mostly on the work of Dummett and McGee, to the extent that no possible criteria can fully rule out inconsistent (intended) numbers.