ISSA Proceedings 2010 – Proto-Regress Argument Schemas

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1. Two cases
Consider the following two examples of a regress argument.
First example. In discussing what to do about his unreliable wife, Juvenal (1st-2nd century CE) thought hiring guardians would not be a good idea:
But who will guard the guardians? In posing the famous question, the Roman author, Juvenal, was suggesting that wives cannot be trusted, and keeping them under guard is not a solution – because the guards cannot be trusted either. (Hurwicz 2008, p. 577, cf. Juvenal, Satire 6)

Second example. Sorensen addresses the issue whether the following principle holds:
(Access) For any action x, you are obliged to do x only if you can know that you are obliged to do x.

On first sight, this is plausible. Access does not say that we actually have to know our obligations in order to have them, but it is only required that it is possible to know them. By contraposition, Access entails that if you cannot know that you are obliged to do x, then you are not obliged to do x. Yet, a consequence of this is:
(Access*) For any action x, if you eliminate your possibility to know whether you are obliged to do x, then you eliminate your (possible) obligation do x.

Here is the example. I am obliged to donate some of my inheritance to charity only if I can know that I am obliged to donate. So, if I cannot know that I am obliged to donate, I am not obliged to donate. So, if I burn the will before reading whether I am obliged to donate, then (assuming the will was my only access) I eliminate my obligation to donate. In general, the Access principle might be used to get rid of one’s obligations, namely by eliminating one’s possibility to know them (Sider 1995: 277-9). To block such options, a proponent of Access might suggest:
(Block) For any action x, you are obliged not to make it impossible to know whether you are obliged to do x.

But now Sorensen generates a regress:
But now a higher order loophole opens. If I keep ignorant of whether there is an obligation to ascertain my obligations, I can use the access principle to evade those epistemic obligations even if they exist. To close this meta-loophole, the defender of access must invoke a yet higher order principle to the effect that we have an obligation to learn whether we have an obligation to learn our obligations. (1995, p. 255)

(Please note that Sorensen discusses a stronger version of Block: for any action x, you are obliged to learn whether you are obliged to do x. Yet, this stronger version is not needed to get the regress, as we shall see in Sect. 4.)
The remainder of the paper is structured as follows. In Sect. 2, I provide two arguments for the need of regress argument schemas. In Sect. 3, I present the argument schema from Gratton (2010) and my own proposal. In Sect. 4, I apply the two schemas, or proto-schemas as I will call them, to the cases from this section. In Sect. 5, I discuss what is demonstrated by the regress arguments. In Sect. 6, I finally point to a number of further issues that are not addressed in this paper.

2. The need for schemas
In the following I argue for the need of regress argument schemas. Note that in this paper I use the term ‘argument schema’ in a broad sense: argument schemas are general versions of similar arguments. Take for instance: (i) A, (ii) if A, then B, so (iii) B. This well-known schema is a general version of all Modus Ponens arguments. Importantly, argument schemas are to contain letters such that if you fill them out you get specific arguments. Yet, in this paper I shall not consider schemas with such letters, but only proto-schemas, as they may be called, where the lines are only named. A full explication of the schemas will be left for further research (see Sect. 6). The question I want to address here is: why should one want such argument schemas for regress arguments in the first place?

First, just as other philosophical tools like thought experiments, analogies, intuition pumps, contradictions, horned dilemmas, and counterexamples, regress arguments occur in all branches of philosophy. If so, it would be surprising if they have nothing in common (on whatever level of generality). Hence, argument schemas are required, because they are exactly what all those different regress arguments have in common. That is, they are that of which the specific cases are an instance. This is the first argument.

Second, regresses are used to establish all sorts of conclusions. In the texts just cited we have arguments against the use of (i) guardians to guard your wife, and (ii) obligations to secure other obligations. Yet, in both cases it is unclear what the conclusion exactly is, and whether and how it is supposed to follow from the regress. For example, in the obligations case Sorensen seems to assume that the regress undermines Access, whereas Sider denies this (references will be provided in Sect. 5). Now, if there is no argument schema available which rules what premises are to be associated with what conclusions, it is hardly possible to clarify how the argument works. This is the second argument.

Given these two arguments, what is wanted from regress argument schemas is at least the following. They are to consist of lines which are such that: (i) they are what a (preferably large) group of regress arguments from the literature have in common; and (ii) among those lines are a regress and a conclusion.

3. Two schemas
Gratton (2010, p. 4) proposes the following schema (note that the number of regress formulas may vary):

Proto-Schema A
1. Regress formula 1.
2. Regress formula 2.
3. Trigger.
4. Infinite regress:
(a)    Consequence of (3). (a, 1)
(b)    Consequence of (a). (b, 2)
(c)    Consequence of (b). (c, 1)
(d)    Consequence of (c). (d, 2)
etc.
5. Result: There is an infinity of consequences. (4)
6. Further premises.
7. Contradiction. (5-6)
8. ~(1) or ~(2) or ~(3). (1-7)

There are four main inference steps.
Step 1: Regress formulas plus trigger entail a regress.
Step 2: The regress entails a result.
Step 3: The result plus further premises entail that the result is unacceptable.
Step 4: If the regress formulas and the trigger are committed to something unacceptable, then by Reductio Ad Absurdum one of them is to be rejected. (It is worth noting that Black (1996) independently developed an argument schema which is very similar to this one.)

In my dissertation (Wieland, In preparation), I also investigate a somewhat different schema.
Proto-Schema B
1. Problem.
2. Solution.
3. Extra premise: If you apply the solution, then the problem is solved only if another, similar problem is solved first.
4. Infinite regress:
(a)    Problem no. 1. (1)
(b)    Solution no. 1. (a,2)
(c)    Problem no. 2. (a,b,3)
(d)    Solution no. 2. (c,2)
etc.
5. Result: Always yet another problem is to be solved before the initial one is solved. (4)
6. The problem is never solved by the solution. (1-5)

This time there are three main inference steps. Step 1: The regress is entailed by a problem, a solution and an extra premise (which states that the problem is solved by an instance of the solution only if another problem of the same kind is solved). Step 2: The regress entails that there is always yet another problem to be solved in order for the initial problem to be solved (or any other problem of the same kind). Step 3: If there is always yet another problem to be solved first, then it follows the initial problem is never solved by the solution.

In the next section, we shall see that the two examples from Sect. 1 can be spelled out along the lines of either schema. First I will briefly compare the two schemas. The main similarities between the two Proto-Schemas are that they both fulfil the two desiderata from Sect. 2. That is, they are schemas which consist of lines which are such that (i) they are what a group of regress arguments from the literature have in common (this will at least be shown for two cases in Sect. 4); and (ii) among those lines is a regress and a conclusion (this can easily be seen from the schemas themselves). The two schemas also display some significant differences.
First, and most importantly, the rationale of Schema A is that some claims cannot hold all together because they jointly lead, via a regress, to unacceptable consequences (no matter of what kind, the only requirement is that they conflict with something else). The rationale of Schema B is more specific: a certain solution never solves the problem it is to solve because it gets stuck in a regress (namely of problems which are to be solved before the initial one is so).

Second, and related, the infinite regress in Schema A is a series of plain consequences of the trigger statement. The infinite regress in Schema B is a series of problems and solutions.
Third, in case of Schema A, there is a mediate connection between regress and conclusion: you need extra premises to obtain undesirable results and a rejection of the trigger or a regress formula. In case of Schema B, there is an immediate connection: you need no extra premise to obtain the conclusion that the problem is never solved by the considered solution (or at any rate not the kind of premise which is required by Proto-Schema A). (Important disclaimer on validity: in this article I do not discuss whether, and under what conditions, both schemas have valid instances.)

4. Application
In the following the two cases from Sect. 1 will be restated in the format of either schema. Instances of Proto-Schema A will be marked with an ‘A’, and instances of Proto-Schema B with a ‘B’.

Juvenal A
(1) Anyone who is unreliable is guarded by a guardian.
(2) Any guardian is itself unreliable.
(3) My girlfriend is unreliable.
(4) Infinite regress:
(a) She is guarded by a guardian no. 1. (3,1)
(b) Guardian no. 1 is unreliable. (a,2)
(c) She is guarded by a guardian no. 2. (b,1)
(d) Guardian no. 2 is unreliable. (c,2) etc.
(5) There is an infinity of guardians.(4)
(6) There are only 19 guardians in the world (say).
(7) Contradiction. (5-6)
(8) ~(1). (1-7, keeping (2) and (3) in place)

Juvenal B
(1) You should have at least someone guarded.
(2) For any person x, if you should have x guarded, you hire a guardian for x.
(3) For any person x, if you hire a guardian y for x, you do not have x guarded unless you have y itself guarded first.
(4) Infinite regress:
(a) You should have your girlfriend guarded. (1)
(b) You hire a guardian no. 1 for your girlfriend. (a, 2)
(c) You should have guardian no. 1 guarded first. (a, b, 3)
(d) You hire a guardian no. 2 for no. 1. (c, 2) etc.
(5) Each time you should have yet another person guarded first. (4)
(6) You never have anyone guarded if you hire guardians. (1-5)

Sorensen A
(1) For any action x, one is obliged to do x only if one can know that one is obliged to do x. (Access)
(2) For any action x, one can know that one is obliged to do x only if one is obliged not to make it impossible to know whether one is obliged to do x.
(3) I am obliged to donate some of my inheritance to charity.
(4) Infinite regress:
(a) I can know that I am obliged to donate some of my inheritance. (3,1)
(b) I am obliged not to make it impossible to know whether I am obliged to donate some of my inheritance. (a,2)
(c) I can know that I am obliged not to make it impossible to know whether I am obliged to donate some of my inheritance. (b,1)
(d) I am obliged not to make it impossible to know whether I am obliged not to make it impossible to know whether I am obliged to donate some of my inheritance. (c,2)
etc.
(5) I have an infinity of obligations, and can know all of them. (4)
(6) This is beyond human capacities.
(7) Contradiction. (5-6)
(8) ~(1). (1-7, keeping (2) and (3) in place)

Sorensen B
(1) You have to secure my obligation to donate some of my inheritance, given that you are a proponent of Access.
(2) For any obligation x, if you have to secure one’s x, then you appeal to one’s obligation not to make it impossible to know x.
(3) For any obligation x, if you appeal to one’s obligation y not to make it impossible to know x, you do not secure x unless you secure y, given Access, first.
(4) Infinite regress:
(a)You have to secure my obligation to donate some of my inheritance, given Access. (1)
(b)You appeal to my obligation not to make it impossible to know whether I am obliged to donate. (a,2)
(c)You have to secure my obligation not to make it impossible to know whether I am obliged to donate, given Access, first. (a,b,3)
(d)You appeal to my obligation not to make it impossible to know whether I am obliged not to make it impossible to know whether I     am obliged to donate. (c,2) etc.
(5) Each time you have to secure yet another obligation first. (4)
(6) You never secure any obligation if you appeal to obligations not to make it impossible to know one’s obligations. (1-5)

5. Different conclusions
The conclusions in the guardians case are these:
A. It is not so that anyone who is unreliable is guarded by a guardian.
B. You never have anyone guarded if you hire guardians.

The conclusions in the obligations case are these:
A. It is not so that for any action x, one is obliged to do x only if one can know that one is obliged to do x.
B. You never secure any obligation if you appeal to obligations not to make it impossible to know one’s obligations.

Hence, the two schemas give rather different conclusions. In the obligations case, the A argument refutes Access, whereas the B argument demonstrates that Block is of no use to save Access. Let me briefly compare these with Sorensen’s and Sider’s own conclusions. After having described a version of the B argument, Sorensen writes:
One’s prosecutorial enthusiasm for catching the shirker diminishes as one ascends levels. (1995, p. 255)

In other words: the strategy to save Access by appealing to higher-order obligations will not do because it leads to scenarios which are insufficiently serious. Sider is more optimistic about Access:
At best, the regress consists of an infinite sequence of cases, none of which refutes Access. (1995, p. 279)

Here, Sider seems to appeal to the A version of the regress. If so, he is right that the regress itself does not refute Access. The regress merely entails that I have an infinity of obligations, and that I am able to know all of them. Yet, if it is also assumed this is beyond our capacities (or problematic in any other way), then the regress entails something unacceptable and that something is to be rejected. The regress is a threat to Access in that case only. (Indeed, a possible way to resist the A argument is to deny that the regress entails anything unacceptable.)
In any case, neither Sorensen nor Sider draws (or attempts to resist) the B conclusion. Here is whether the regress argument schemas prove useful: they clarify what can be drawn from a regress.

6. Further issues
To sum up, in this paper I provided two arguments for why regress argument schemas are wanted (Sect. 2), presented two proto-schemas which are currently available (Sect. 3), showed for two instances how they can be spelled out along the lines of either schema (Sect. 4), and pointed out that the two schemas establish rather different conclusions: Schema A concludes with rejections, and Schema B with problem solving failures (Sect. 5). It need not be said that all this deserves further attention. There are at least five further issues that I address in my dissertation (Wieland, In preparation).

First, the schemas presented in Sect. 3 are strictly speaking not full argument schemas as all lines are without schematic letters (and quantification), a number of lines are suppressed in full, and all inference rules are missing out. All this is explicit in the dissertation.
Second, Schema B has another variant, and can be presented in different ways. In the present paper, problems are taken as tasks and solutions as actions. But there are other options (i.e. it is also possible to present them as arguments and counterarguments, or as questions and answers).
Third, the examples from Sect. 1 are just an arbitrary selection. There are many more examples of regress arguments in philosophy (ranging from epistemology to ethics), and it is shown for 30 cases that all can be spelled out along the lines of both Schema A and B.
Fourth, if both schemas fulfil the two desiderata listed in Sect. 2, then which schema is the most fruitful? Or are both fruitful in their own right? In my dissertation I approach this issue by introducing extra desiderata for the schemas. For example, one of the issues is whether the instances of either schema can play the same role in a broader dialectical setting.
Last, the reconstruction of regress arguments assumes a rather revisionary take on argument reconstruction. That is, many premises and inferences are to be added and/or modified. Still, an interesting query is whether there are any limitations to revision (or charity, as it is sometimes called). Again, for this I have to refer to my dissertation.

Acknowledgements
I would like to thank Erik Weber and the editors for advice. I am also grateful to Hans Lycke and the participants of the work in progress seminar in Ghent. I am PhD fellow of the Research Foundation Flanders at Ghent University.
Email: Jan.Wieland@UGent.be.

REFERENCES
Black, O. (1996). Infinite regress arguments and infinite regresses. Acta Analytica 16/17, 95-124.
Gratton, C. (2010). Infinite Regress Arguments. Dordrecht: Springer.
Hurwicz, L. (2008). But who will guard the guardians? American Economic Review 98: 577-85.
Juvenal. The Satires. Transl. N. Rudd (1992). Oxford: OUP.
Sider, T. (1995). Sorensen on unknowable obligations. Utilitas 7, 273-9.
Sorensen, R. (1995). Unknowable obligations. Utilitas 7, 247-71.
Wieland, J.W. (In preparation). And So On. Two Theories of Regress Arguments in Philosophy. PhD dissertation, Ghent University.

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