PARACONSISTENCY AND COMPUTABILITY: A PHILOSOPHICAL EXPLORATION1

João de Fernandes Teixeira2

This paper is rather exploratory in intent. Its aim is to present a few conjectures posed by the

following questions: To what extent are our current conceptions of computation relative to given

frameworks and their underlying logic? Can the adoption of non-classical logic affect orthodox

notions of computation? The standard theory of computation emerged in the thirties when

classical logic could be taken as an absolute presumption for the formulation of mathematical

theories. But what happens once we abandon such an assumption? The most striking result

emerging from the rejection of classical logic as an absolutist paradigm, I submit, is the

possibility of devising alternatives to the Halting Theorem.

An alternative to classical logic is paraconsistency. The choice is not unmotivated:

despite being a non-classical logic, paraconsistent logic is not meant to challenge classical logical

conceptions. Rather, it was originally developed as an attempt to supply alternative tools to

consider some mathematical and logical problems not possibly addressed within a classical

formalism. Furthermore, if the adoption of a paraconsistent logic leads to the rejection of the

Halting Theorem, it should be noticed that such a result does not follow trivially. Paraconsistent

logic is weaker than classical logic, but not as weak as any logic that cannot demonstrate

inconsistency.

Surely the historical development of this kind of non-classical logic raised several issues

concerning its interpretation. Paraconsistent logic can be regarded from two different points of

view: a) as a logic complementary to classical logic, or, b) as a kind of heterodox logic,

incompatible with classical logic, whose aim is to replace the latter in all or some of its

applications.3 For one thing, a) seems to be the case. Since paraconsistent reasoning does not lead

1 This paper benefited from helpful suggestions and criticisms from Prof. D. Dennett (Tufts University U.S.A), Prof. D. Isles (Tufts University U.S.A)) , Prof. G. Sarmento (Universidade Estadual de Campinas, Brazil), Prof. Itala D’Ottaviano (Universidade Estadual de Campinas, Brazil) and Prof. M. Sette (Universidade Estadual de Campinas, Brazil). A preliminary version was presented at the meeting of the Australasian Association for Logic, July 1998, Macquarie University, Australia.2 Department of Philosophy, Universidade Federal de Sao Carlos, Brazil jteixe@terra.com.br3 For more details concerning this discussion, see Da Costa, Beziau & Bueno, (1995).

to trivialization in the presence of contradictions, i.e., since it does narrow the set of inferences

that can follow from a contradiction, such reasoning can be viewed as an attempt to refine

classical reasoning. Furthermore, it would be an attempt to think beyond contradiction and not

just outright reject it. Nevertheless, b) seems also to be the case. Since paraconsistent logic does

deny that from a contradiction anything can plausibly follow, a crucial contrast to classical logic

may emerge. Furthermore, one can also consider that the adoption of a) may also force the

adoption of b): reasoning beyond contradiction would, in this case, be tantamount to

encompassing classical logic in a broader framework. In this case, paraconsistent logic would

contain classical reasoning or a great deal of it. But this is also disputable. Offering a thorough

discussion of both possibilities either from a philosophical perspective or from a technical one is

beyond the scope and limits of this paper. We shall only emphasize that paraconsistent logic can

be the underlying logic of inconsistent non-trivial theories; leaving aside the task of determining

which of these possibilities is the case in general.

However, one can pose a further constraint on the choice of a non-classical logic to

envisage Turing’s Halting Problem (hereafter also referred as THT) by selecting a specific

paraconsistent logic as close as possible to classical logic. This is the case of C1+, recently

developed by Da Costa, Beziau and Bueno. C1+ may be viewed as overlapping with classical

logic in many respects and this is its most strikingly feature. C1+ allows some patterns of

paraconsistent reasoning in the presence of contradictions that, considered from a broader

perspective, overlap with classical reasoning. This is what approximates C1+ to a classical

formalism; i.e., the overall result of C1+ is close to the classical idea that from a contradiction

anything can plausibly follow. However, C1+ differs from classical logic in as much as its

paraconsistent reasoning in the presence of contradictions does not lead to trivialization,

regardless of the fact that it may overlap with classical reasoning.

This proximity to classical logic can be viewed as a relevant criterion to choose C1+

among a broader family of possible paraconsistent logic with the purpose of investigating what

consequences can emerge once Turing’s Halting Problem is conceived from a non-classical

perspective. Now, what happens if THT cannot be derived from C1+? Does it also mean that

Turing’s Halting Theorem is at odds with the classical idea that from a contradiction anything can

plausibly follow? If this is the case, the truth of THT is in jeopardy, even from a classical

perspective. But this is too strong a result, a result which would require a clear characterization of

the relations between classical and paraconsistent logic. Since such a clarification has not yet

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been attained, we shall not discuss this matter here. Rather, we shall show that THT cannot be

derived from C1+. Such a result allows a preliminary rejection of THT since we see no reason

why classical logic ought to be considered an absolute presumption for the theory of computation.

C1+ is enough to question the extent to which THT follows as a reductio in the presence of

contradictions.

I

We shall start our investigation by outlining C1+. For reasons of space, we shall not

present here C1+ in all its technical details. Instead, we shall present two examples of

paraconsistent reasoning allowed by C1+ . Both are provided by Da Costa, Béziau & Bueno

(1995). The first (Reasoning 1) is a typical pattern of paraconsistent reasoning in the sense of

restraining what can plausibly be derived when a contradiction is encountered, i.e., paraconsistent

reasoning challenges the principle that anything follows from contradictory premises. The second

(Reasoning 2) is a specific pattern of reasoning derived from paraconsistent logic C1+ : a pattern

of reasoning which overlaps with classical logic. As we shall see below, the interesting

characteristic of C1+ lies in the fact that both Reasoning 1 and Reasoning 2 can be derived from

it. In this sense, patterns of reasoning allowed by C1+ ultimately do not conflict with classical

logic, except for the fact that they do not follow trivially once a contradiction is encountered.

Let us start by Reasoning 1. A certain Mr. X is sick and goes to Dr. B. who tells him that

he has got cancer. Mr.X. decides to consult another specialist, Dr. P. who asserts that he has not

got cancer. Dr. P does not agree with his colleague on this point but there is one thing they both

recognize:

(1) If Mr. X has got cancer he will die in the next three months.

By using typical paraconsistent reasoning, MrX can make interesting reasonings without

supposing either that Dr. B or that Dr. P is wrong. From the statement of Dr. B, the statement of

Dr. P and the statement they both agree to, paraconsistent reasoning does not allow one to infer

that:

(2) If Mr.X has not got cancer he will not die in the next three months.

In classical reasoning we would have: a = Mr. X has got cancer, a = Mr. X. has not got

cancer, b = Mr.X will die in the next three months. From a, a and b, both (1) and (2) follow.

3

Typical paraconsistent reasoning precludes (2), for in paraconsistent logic (and in C1+ ) it

is not the case that from { a, a, a b } it does follow, as in classical logic, that a b . The

interesting point about this pattern of paraconsistent reasoning is that it does not allow

trivialization in the presence of contradictions. In other words, the advantage of paraconsistent

logic is that we can go on making reasonings without, as in classical logic, supposing either that

one of the terms of the contradiction must be rejected or that from a contradiction we can derive

anything.

Let us now turn to Reasoning 2.

Suppose Dr. B says:

It is not possible that:

Mr. X has not got cancer (a)

And

Mr. X will die in the next three months.(b)

From this statement – and only this one – C1+ allows us to infer, as in classical logic,

that:

(2) If Mr.X has not got cancer he will not die in the next three months.

The interesting feature of C1+ is that, besides Reasoning 1, we can derive a further pattern of

reasoning (Reasoning 2). We can interpret “it is not possible that d” as “there exists such that

(d) = 0”. We have (a b) = 0 and (( a b)) = 1 and the sequent (a b) a b

can be proved in C1+.

Once C1+ precludes (2) in Reasoning 1 and allows (2) in Reasoning 2, we can plausibly

sustain that the overall results of C1+ match with classical logic. However, the difference with

classical logic lies in the fact that there is no trivialization, i.e., the presence of a contradiction

does not “implode” the system in the sense that anything could plausibly follow from a

contradiction. The preclusion of (2) in Reasoning 1 and the possibility of (2) in Reasoning (2) are

not trivial consequences of a contradiction as in classical logic.

Now, what happens if we apply paraconsistent reasoning derived from C1+ to Turing´s

Halting Theorem? Does da Costa´s paraconsistent logic C1+ show that it is possible to ascertain

the possible existence of an algorithm for the problem of non-terminating computations? Our

claim for the possible existence of a Halting Algorithm can be envisaged as a particular

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application of C1+ - an application which suggests that there might be computation beyond the

classical limitations presented by Turing in his paper of 1936.

A caveat may hinder this claim: can THT be treated as a matter of paraconsistency? The

issue crops up in so far as one may sustain that paraconsistent reasoning only applies to worldly

situations from which either agreement or disagreement may emerge. Whether or not Turing

Machines can be conceived as entities in the world or purely mathematical entities we shall

discuss later. The issue as to whether it is a matter of agreement/disagreement if a Turing

Machine halts or not surely can be distinguished from a worldly situation such as medical

diagnosis. Medical diagnosis can be controversial mostly due to lack of conclusive evidence.

However, it may also be controversial due to interpretation of evidence. In this case, the conflict

of interpretation emerges in so far as diagnosis is grounded in mutually exclusive scientific

theories and conceptions. Unless THT is taken for granted, i.e., as an absolute truth that would cut

across any conception of mathematics and independently of any given set of instruments to prove

it, disagreement can also crop up in this field. Nevertheless, the possibly absolute character of

THT as independent of how we conceive of mathematics is not a matter settled once and for all.

For instance, Isles (1981) has pointed out that THT cannot be sustained unless we take for

granted a questionable “intuitive” sequencing of the natural numbers given by the function +1.4

There are two more reasons to sustain that THT at least can be envisaged from a

paraconsistent setting. First, as we said at the outset of this paper, we should consider that THT,

by proceeding by reductio, is a kind of pattern of reasoning in the presence of contradictions. The

underlying intuition of THT is that once trivialization emerges from contradiction, a reductio also

obtains. (Surely, this is a presupposition of classical logic). Secondly, the overlapping between

classical and paraconsistent reasoning as an overall result of C1+. (In C1+ Reasoning 1 and

Reasoning 2 lead to the classical view that from a contradiction anything can plausibly be

derived). If such an overlapping is not just a fortuitous coincidence, (why would it be?) classical

4 Isles questions THT by raising the issue concerning the ordering of natural numbers and proposes a “mitigated” version of THT. However, I do not share his intuitionistic views.

5

and paraconsistent treatment of THT are equally plausible. The issue that remains to be

investigated is whether a paraconsistent treatment though overlapping with a classical one would

still entail the truth of THT.

Let us now explore what consequences may emerge from envisaging THT from C1+ by

initially presenting an intuitive version of the Halting Theorem and of its proof through an

example. Consider a computation on a natural number n. If we call such a computation C(n) we

can conceive it as providing a family of computations where there is a separate computation for

each natural number, 0,1,2,3...i.e., the computations C(0),C(1), C(2),C(3)...C(n) are the action of

some Turing Machine (TM) on the number n, taken as the machine input.

Suppose we have some computational procedure A which, when it terminates provides a

demonstration that a computation such as C(n) does not ever stop. If in any particular case A itself

ever comes to an end, this would provide us with a demonstration that the particular computation

that it refers to does not ever stop. Furthermore, we say that A is sound if it does not give us

wrong answers. For, if A were unsound, then it would erroneously assert that the computation

C(n) does not ever terminate when in fact it does. But if this is the case, the performing of the

actual computation C(n) would eventually lead to a refutation of A.

In order for A to apply to computations generally, we shall need a way of coding all the

different computations C(n) so that A can use this coding for its action. All the possible different

computations C can in fact be listed as:

C0,C1,C2,C3,C4 ...,

and we can refer to Cq as the qth.computation. When such a computation is applied to a particular

number n we shall write:

C0(n),C1(n),C2(n),C3(n),C4(n),....

This ordering can be viewed as a numerical ordering of computer programs. Moreover,

this listing is computable i.e., there is a single computation C• which gives us Cq when it is

presented with q, or, in other words, the computation C• acts on the pair of numbers, q,n (q

followed by n) to give Cq(n).

6

The procedure A can now be thought of as a particular computation that, when presented

with the pair of numbers q,n, tries to ascertain that the computation Cq(n) will never halt. Thus,

when the computation A terminates we have a demonstration that Cq(n) does not halt. Being

dependent on the two numbers q and n, the computation that A performs can be written A(q,n),

and we have:

(1) If A(q,n) stops, then Cq(n) does not stop.

Now let us consider the particular statements (1) for which q is put equal to n. With q

equal to n, we now have:

(2) If A(n,n) stops, then Cn(n) does not stop.

We notice that A(n,n) depends upon just one number, n, not two, so it must be one of the

computations C0,C1,C2,C3.. (as applied to n), since this was supposed to be a listing of all the

computations which can be performed on a single natural number n. Let us suppose that it is in

fact Ck, so we have:

(3) A(n,n)= Ck(n).

Now examine the particular value n=k. From (3) we have:

(4) A(k,k) = Ck(k).

and from (2), with n=k

(5) If A(k,k) stops, then Ck(k) does not stop.

Substituting (4) in (5) we find:

(6) If Ck(k) stops, then Ck(k) does not stop.

From this we deduce that the computation Ck(k) does not in fact stop, for, if it did, then it

does not, according to (6). But A(k,k) cannot stop either, since by (4) it is the same as Ck(k).

Therefore, our procedure A cannot ascertain that this particular computation Ck(k) does not stop

even though it does not.

According to such a presentation the unsolvability of Turing´s Halting Problem is derived

from the second part of Cantor's diagonal slash:

(5) - If A(k,k) stops, then Ck(k) does not stop.

7

Substituting (4) in (5) we find:

(6) - If Ck(k) stops, then Ck(k) does not stop.

Furthermore, we saw that from this we must deduce that the computation Ck(k) does not

in fact stop. For if it did then it does not, according to (6). But A(k,k) cannot stop either, since by

(4) A(k,k)= Ck(k)

it is the same as Ck(k). Thus our procedure A is incapable of ascertaining that this particular

computation Ck(k) does not stop even though it does not. The existence of A is denied for it

implies a contradiction. Since A(k,k)= Ck(k) we can write:

(7) If A(k,k) stops, Ck(k) does not stop.

(8) If A(k,k) does not stop, Ck(k) does not stop.

(7) and (8) can be rewritten in the form:

(9) A is sound,

and

(10) A is not sound.

A cannot exist for it encompasses a contradiction.

Nevertheless, it is agreed on this formulation of Turing´s Halting Problem that

(11) If A is sound A(k,k) stops and Ck(k) does not stop; A(k,k)

does not stop and Ck(k) stops.

Now if we consider such a formulation of Turing´s Halting Problem in the light of

paraconsistent reasoning (and by contrast to classical logic) one cannot infer that

(8) If A(k,k) does not stop, Ck(k) does not stop.

All that is allowed by paraconsistent Reasoning 1 is (7) and not both (7) and (8). From

(7) and (8) and from its corresponding statements (9) and (10) one cannot trivially infer (8). Since

only (7) can be inferred from (9), (10) and (11) we can ascertain that the existence of A is

possible.

Let us now turn to an application of Reasoning 2 to THT. Suppose that from (9) and (10)

one says:

8

(12) It is not possible that

A(k,k) does not stop

and

Ck(k) stops.

However (12)=(8) and, since (12) follows non- trivially from (9) and (10) it means that

both (7) and (8) cannot follow from (9) and (10). Once (7) is to be discarded, THT cannot be

proven since its proof is based on the possibility of deriving both (7) and (8) from (9), (10) and

(11).

The application of both Reasoning 1 and Reasoning 2 shows that from a paraconsistent

setting, the proof of THT as a reductio does not follow. Still, C1+ by allowing both Reasoning 1

and Reasoning 2 does conform to the classical conception that from a contradiction anything can

follow (Reasoning 1 – A is sound, Reasoning 2 – A is not sound). One could that we reached

nothing but a further contradiction resulting from considering both Reasoning 1 and Reasoning 2,

but, if this is the case, we are left with two possibilities. Either classical reasoning is just a

fragment of a broader inconsistent logic, i.e., C1+ encompasses classical logic, or this is just a

fortuitous match between classical and paraconsistent reasoning. The second possibility must be

outright rejected. For, even if this is a coincidence it should be a very superficial one, once

classical and inconsistent logic do not share common grounds – even in the case of non-trivial

inconsistent logics as C1+. The first cannot be outright discarded, but it would ultimately entail a

rejection of proofs by classical reductio reasoning and thus a rejection of THT. In any case a

paradox may arise if we consider that in a paraconsistent setting accepting or rejecting THT

means the rejection of a great deal of classical logic. Such a paradox points to the conclusion that

a Halting Algorithm may exist, and, though we cannot show what it is like as yet, future

investigation may unveil its nature.5

5 The existence of a Halting Algorithm is also suggested by Sylvan & Copeland (1997) who remark that in

a paraconsistent setting the derivation of a contradiction is insufficient for rejecting the assumption that

leads to THT, namely, the assumption that there may exist a Turing machine capable of computing the

halting function. More enthusiastically, Sylvan and Copeland suggest that such a result points to a further

9

II

Several consequences of the result stated above deserve to be discussed at length. To

begin with, there are at least two questions in need of a more detailed consideration: 1) How is it

possible that a weaker logic such as C1+ can solve more problems in the theory of computation

than classical logic? and 2) Does the reasoning developed above mean that any mathematical

proof by reductio can be discarded?

A tentative answer to the first question is to suggest that once one takes paraconsistent

logic as the underlying logic for the theory of computation one also escapes the classical

limitations posed by incompleteness theorems - an assertion that would also be applicable to any

logical system that admit some inconsistency.6 But does that make, for instance, C1+ stronger than

classical logic? Probably not, once there are still more theorems in classical logic than in C1+ .

Furthermore, why should we take paraconsistent logic as a foundation for mathematics and for

the theory of computation? The issue bears not only on the adoption of C1+ ,but on the adoption

of any non-classical logic: once contradiction and diagonal arguments can be rejected, more

unwelcome consequences may crop up, such as, for instance, the collapse of arithmetic ( n would

equal n+1!). As we remarked at the outset of this paper, the price to pay may be too high. Can we

assert he possible existence of a Halting Algorithm without having to pay such a price? We

believe such an alternative is possible and so we shall argue for it in the remainder of this paper.

Such an argument will run as follows:

a) – THT may be conceived as a mathematical truth, but not necessarily as a truth of the theory of

computation in so far as the latter is a chapter of applied mathematics.

horizon for computing theory, i.e., the possibility of developing the new emerging field of paraconsistent

computability theory. Needless to say that, if this is the case, such authors are committing themselves to a

Herculean task.

6 For the completeness and decidability of C1+ see da Costa ,Béziau & Bueno (1995) and Béziau (1995).

10

b) – Mathematical truths and the truths of the theory of computation need not overlap in so far as

the latter deals with specific objects, namely, virtual objects.

c) – Mathematical truths do not necessarily map onto the world. Such an anti-realist assumption

is applicable to THT. Thus, there may exist a thing as a Halting Algorithm though not grasped by

classical logic. Once we abandon the absolutism of classical logic by adopting paraconsistent

logic the existence of such a Halting Algorithm becomes conceivable.

Let us begin by introducing anti-realist claims. An alternative to the sheer adoption of

non-classical logic as the basis for the theory of computation and hence to mathematics can be

figured if we start by questioning the epistemic status of THT. Such a reflection can also point to

an answer to question (2). Surely THT is a mathematical truth if we stick to classical logic, but, to

what extent does it necessarily map to the real world? So posed, the issue has several nuances. To

begin with, there is no reason not to suppose that paraconsistent based computation may continue

beyond the classical limitation which stems from a diagonal contradictory element. In other

words, a computing machine in the real world will not stop when a contradiction is encountered

unless it has a specific instruction to halt. Supposing that the diagonal element is d and that the

instruction is to increase the diagonal by 1, the diagonal element encontered is such that d= d+1.

If d is binary, instead of encountering either 0 or 1, what is encountered is 0 and 1.7 Computation

may continue and the possibility of encountering a Halting Algorithm cannot be downright

discarded. THT can be held, but just as a formal truth which does not necessarily map onto the

real world. Still, this is not to say that classical logic does not map onto the world and that

paraconsistent logic does. To a certain extent, this issue cannot be established only through the

resources supplied by logic and mathematics. Further considerations stemming from philosophy

of science would also be necessary. For the time being we only wish to point out that the

advantage of this view is that if THT does not map onto the world there is no need to dispense

with classical logic and we can plausibly maintain the possible existence of a Halting Algorithm.

7 See Sylvan and Copeland (1997), forthcoming.

11

A further advantage also emerges: we need not expand such a conclusion to the point of having to

reject all mathematical demonstrations which proceed by reductio.

The objection to this view is to hold that there is such a mapping between THT and the

world despite the fact that the elements of the theory of computation are purely mathematical,

and hence there cannot exist such a thing as a Halting Algorithm. The presumption of such an

objection is that the theory of computation can indeed be conceived as a formal game having

nothing to do with real computing machines but still having to do with the computing machines

we can possibly build (!). But if the theory of computation can be viewed just as a formal game

we can plausibly vindicate a non-classical logic for its foundation, once we have no reason to

accept classical logic as an absolute, given framework. The consequences would be devastating

and mostly counterintuitive: arithmetic would collapse, 2 would become rational ....and so

what? A non-classical logic as the underlying logic for the theory of computation would still

allow us to hold the possible existence of a Halting Algorithm even (and perhaps, specially) if

one wishes to envisage C1+ as encompassing classical logic. If this is the scenario, too bad for

orthodox theory of computation ... and too bad for mathematics...!

Both possibilities (namely (1) THT does not map to the real world, and (2) Computation

theory is purely formal and, still, it does map to the world) do jeopardize THT. The first

devastates the relationship between mathematics and the world, the second is insufficient to shun

C1+ as encompassing classical logic (however unsettled this matter may be) and providing a

foundation for mathematics. Still, they both allow the possible existence of a Halting Algorithm.

Rejecting the second is less disadvantageous, but perhaps this is still not a sufficient reason to

accept the first.

The task we face is that of steering between the path of avoiding the collapse of

arithmetic and, at the same time, not succumb to an unquestioned absolutism of classical logic as

a foundation for the theory of computation. However, the difficulty involved in such a task can be

overcome by reflecting upon the status we are to ascribe to the elements involved in the theory of

computation. The anti-realist view of THT allows us to preserve a good deal of classical logic

12

and, at the same time, to hold the possible existence of a Halting Algorithm by pointing to a

partial disentanglement between the objects of mathematics and the objects of the theory of

computation. Such an anti-realism hinges on the extent to which we are to conceive of the theory

of computation as a purely mathematical theory or as a chapter of mathematics.

This is a mostly delicate question, since one could trivialize our criticism of THT by

arguing for the existence of a “natural” gap between a mathematical theory and its application to

the physical world; in the case of the theory of computation such a gap would be, for instance, the

inexistence of an infinite tape for a Turing Machine - an inexistence which does not prevent us

from building actual computing machines. Such a claim is misleading since the issue at stake is

not merely implementational. Furthermore, the disparity between infinite tapes and actual ones

does not seem to affect the results of the theory of computation - not in the way we are

contending here. Physical triangles do not contain 180 but it is still a perfectly good

mathematical truth that Euclidean triangles contain 180.

Surely the theory of computation finds its foundations in an underlying logic and in a set

of mathematical truths. But there should be more to the theory of computation than a sheer

recapitulation of such well-established truths, otherwise it would be of no use. The distinctive

content of the theory of computation must lie in its consequences which ultimately allow to

devise computing machines even when the latter are conceived as abstract or virtual machines. In

this sense, the theory of computation is a chapter of applied mathematics, but a very specific one.

Such a specificity lies in the way the theory of computation maps mathematical and logical truths

to the world - a peculiar mapping to virtual elements. A mapping to virtual elements is very much

peculiar but it is still a mapping into something in the world. But what ontological status are we

to ascribe to a virtual machine? Has a Turing Machine the status of a purely mathematical entity,

i.e., the status of something which has no place in space and time? For one thing we believe that

computation happens in the world, even when performed by a virtual machine: computation

involves time, since the very idea of sequencing (no matter whether or not it is a linear mode of

operation or a cycle of parallel activity) is at the core of any algorithmic process. Turing

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Machines yield specific temporal sequencing for the execution of mathematical operations – a

sequencing without which the solution of certain problems would not be accomplished.8 So

viewed, Turing Machines involve some worldly element, for, no matter how abstract our

conception of time may be, it is still a kind of physical magnitude.9

So viewed, the theory of computation cannot be purely mathematical, but it cannot be

empirical either. Among its elements there are those we are to classify as endowed with the

grayish ontology of virtual elements which, nonetheless, still maintain some reference to the

world - virtual elements which can, in turn, be pressed into service in any number of different

empirical applications. It is this surreptitious instillation of a reference to the world which allows

us to treat virtual machines as something in the world. So THT can be preserved as a classical

mathematical truth despite the fact that we disqualify its realism as well its absolute character by

applying a pattern of paraconsistent reasoning derived from C1+ .THT is a mathematical truth,

but not necessarily a truth of the theory of computation - not when the latter refers to somewhat

real machines in a world of time and space, i.e., machines whose behavior although describable

by mathematical theories ought not to be viewed as entirely predictable by any specific

mathematical framework, let alone classical formalism.

The oddity but also the cogency of our viewpoint can also be supported by a parallel

between classical logic/paraconsistent logic and Euclidean/non-Euclidean geometry. Non-

Euclidean geometries have afforded us to deal with new conceptions posed by contemporary

8 We cannot conceive of a Turing Machine without a reference to time. Even a Turing Machine with just one instruction would require a further one to get the machine stop. Although clock rates are irrelevant to the computation insofar as they do not compromise the reliability of the switchings (the description of the Turing Machine can be nonholonomic) it is unconceivable that two instructions can be realized at the same time. 9 Even subjectivist accounts of time would admit that time does involve a reference to something in the world. Kant, for instance, who held that time is an a priori form of sensitivity says that “Time is not a discursive, or as it is called, a general conception, but a pure form of the sensuous intuition” (emphasis mine). Kant´s first antinomy emphasizes the need to distinguish sensuous and intelligible spheres in dealing with mathematical notions - at least as a means to avoid the raising of pseudoproblems.The same point is noted in an earlier Kant´s work (1770/1967) where he asserts that A=A cannot be considered a purely logical relationship if the equality is mediated by time. (“A enim et non A non repugnant nisi simul (h.e. tempore eodem) cogitata de eodem, post se autem (diversis temporibus) eidem competere possunt” p.60). Such a distinction seems to have been overlooked in current discussions concerning the nature of the elements in the theory of computation.

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physics such as, for instance, space with variable curvature. Similarly, paraconsistent logic may

afford us to deal with peculiarities of the elements of the theory of computation, i.e.,

paraconsistent logic is to replace classical logic in some of its applications such as the main

theorems of the theory of computation. So viewed, the possibility of finding a Halting Algorithm

should neither scare the mathematicians nor force us to give up one of the most thoroughly

studied pieces of mathematics of this century.

BIBLIOGRAPHY AND REFERENCES

Beziau, J.Y. (1990) - "Logiques construites suivant les methodes de da Costa I", Logique et Analyse 131-132 pp.259-272.

Beziau, J.Y. (1991) - "Nouveaux resultats et nouveau regard sur la logique paraconsistente C1"

Logique et Analyse 137-138

Béziau, J.Y. (1995) - “Théorie de la valuation”, Appendix 2 in da Costa, N.C.A., Logiques Classiques et non Classiques, Paris, Masson.

Da Costa, N.C.A. (1963) - "Calculs proposicionnels pour les systemes formels inconsistants" C.R. de l'Academie des Sciences de Paris 257 pp.3790-3793.

Da Costa, N.C.A.& Alves, E. H. (1977) - "A Semantic analysis of the calculi Cn " , Notre Dame

Journal of Formal Logic, 10, pp.621-630.

Da Costa, N.C.A., Béziau, J.Y and Bueno, O. (1995) - "Aspects of Paraconsistent Logic", Bulletin of the Interest Group in Pure and Applied Logic 3 pp.597-614.

Da Costa, N.C.A. (1997) – “Paraconsistent Mathematics” (forthcoming).

Isles, D. (1981) – “Remarks on the notion of standard non-isomorphic natural number series” in Constructive Mathematics: Proceedings of the New Mexico State University Conference, Springer-Verlag Lecture Notes in Mathematics, # 873, pp.111-134.

Kant, I (1770/1967) - De Mundi sensibilis atque intelligibilis forma et principiis - Latin and French version, edited by P. Mouy, Paris, J. Vrin.

Kant, I (1781) – Kritik der Reinen Vernunft - Critique of Pure Reason – Trans. N. Kemp Smith. London: Macmillan, 1929.

Mortensen, C. (1995) – Inconsistent Mathematics. Kluwer Academic.

Sylvan, R & Copeland, B.J. (1997) - “On the relativity of computability” (unpublished).

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Turing, A. M. (1937) - "On Computable Numbers with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society 42 pp.230-265.Turing, A. M. (1939) - "Systems of Logic based on Ordinals" Proceedings of the London Mathematical Society 45 pp.161-228

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