Why the doctrinal paradox is stupid shit

According to Kornhauser & Sager, a doctrinal paradox exists when a logically inconsistent result arises when aggregating logically interconnected judgements from n agents.

However, we know that a system is bound to be inconsistent if it is over determined- i.e. there are more equations than unknowns- unless there is linear dependence. In other words, simply by introducing more constraints, you can generate 'paradoxes' though it is obvious that you are going to get overdetermination and therefore inconsistency unless there is linear dependence (which you can forbid by stipulating for 'non-Dictatorship' or 'Independence of irrelevant alternatives' etc.)

Another problem has to do with whether agents have a preprogrammed 'preference' or 'belief'. If they don't and can choose between competing preferences or beliefs then, you are on a slippery slope to infinite agents in which case who can say if there are more unknowns than equations? After all, the set of even numbers has the same cardinality as the set of natural numbers (though the latter must be twice as large!) Thus, infinity brings its own problems. Indeed, its paradoxes are quite genuine- at least at the current state of play. 

Leaving aside talk of infinity, consider the example given by Kornhauser & Sager

A three-judge court has to make judgments on the following propositions:

• $p$: The defendant was contractually obliged not to do action $X$.
• $q$: The defendant did action $X$.
• $r$: The defendant is liable for breach of contract.

According to legal doctrine, the premises $p$ and $q$ are jointly necessary and sufficient for the conclusion $r$.

There is no such 'legal doctrine'. Contractual obligations are defeasible or can be 'read in'. The law is 'artificial reason' and may deem an action to have been performed or not performed according to its own criteria.

Taken together these considerations mean there are far more degrees of freedom- i.e. more unknowns- than there are equations and thus proving inconsistency will be impossible. Thus no paradox can arise. 

Suppose the individual judges hold the views shown in Table 5.

	$p$ (obligation)	$q$ (action)	$r$ (liability)
Judge 1	True	True	True
Judge 2	False	True	False
Judge 3	True	False	False
Majority	True	True	False

Table 5: An example of the ‘doctrinal paradox’

Although each individual judge respects the relevant legal doctrine, there is a majority for $p$, a majority for $q$, and yet a majority against $r$—in breach of the legal doctrine.

Rubbish! There is a majority for 'no liability'. That is the judgment. The ratio can clarify that there were two different ways in which it was arrived at though what is more likely is that it will be 'narrow' and represent some commonality in the majority opinion. Still, a Court may decide that a higher court, or constitutional bench, needs to clarify the law in this matter.

The court faces a dilemma: it can either go with the majority judgments on the premises $(p$ and $q)$ and reach a ‘liable’ verdict by logical inference (the issue-by-issue or premise-based approach); $or$ go with the majority judgment on the conclusion $\left(r\right)$ and reach a ‘not liable’ verdict, ignoring the majority judgments on the premises (the case-by-case or conclusion-based approach). The ‘doctrinal paradox’ consists in the fact that these two approaches may lead to opposite outcomes.

But the Law has its own doctrines. Justice is a service industry. In this case, both parties to the suit are only interested in one thing- are damages payable or not? They don't give a fuck about how the Judges arrive at a conclusion. True, Judges may 'certify a question' - i.e. refer the matter to another, generally higher court, to clarify a point of law. In the UK, appeal can be by means of 'case stated'. In this way, the law has its own method to arrive at 'harmonious construction' and iron out inconsistencies or resolve questions of priority or concurrency.

This is not to say that there is any way to replace 'Dog's law'- which plays catch up- with 'God's law'- which previsions everything. Thus the number of 'unknowns' will always be greater than the number of 'equations'- i.e. rules specifying equivalences. 

List & Pettit have a similar ‘discursive dilemma’ which isn't a fucking dilemma. 

Suppose, for example, an expert panel has to make judgments on three propositions (and their negations):

• $p$: Atmospheric CO${}_2$ will exceed 600ppm by 2050.
• if $p$ then $q$: If atmospheric CO${}_2$ exceeds this level by 2050, there will be a temperature increase of more than 3.5° by 2100.
• $q$: There will be a temperature increase of more than 3.5° by 2100.

These aren't legal judgments. They are predictions. The difference is that legal judgments are protocol bound. In extra-legal contexts, experts are welcome to say 'my gut feeling is that such and such will be the case.' 

If individual judgments are as shown in Table 6, the majority judgments are inconsistent:

Nothing wrong with that. Because of Knightian Uncertainty, some possible future states of the world are not known. 

 despite individually consistent judgments, 

Any judgment by anybody would be consistent because there are far more unknowns than there are equations based on existing knowledge. It is consistent for me to say that in the year 2100 global temperatures will have both risen by ten percent and will have fallen by twenty percent. This is because I subscribe to a particular theory of Physics which predicts an inter-dimensional singularity such that global temperature becomes a cadlag function of a very peculiar type (essentially, a particular range of temperature becomes incompossible). 

the set of majority-accepted propositions, $\{p$, if $p$ then $q$, not $q\}$, is logically inconsistent.

	$p$	if $p$ then $q$	$q$
Expert 1	True	True	True
Expert 2	False	True	False
Expert 3	True	False	False
Majority	True	True	False

Table 6: A majoritarian inconsistency

Note that the patterns of judgments in Tables 5 and 6 are structurally equivalent to the pattern of preferences leading to Condorcet’s paradox when we reinterpret those preferences as judgments on propositions of the form ‘$x$ is preferable to $y$’, ‘$y$ is preferable to $z$’, and so on, as shown in Table 7 (List and Pettit 2004; an earlier interpretation of preferences along these lines can be found in Guilbaud [1952] 1966). Here, the set of majority-accepted propositions is inconsistent relative to the constraint of transitivity.

Fuck off! What matters if these 'experts' think global warming is a real thing. The majority says- 'no'. That's all that matters. Similarly, it does not matter if a guy prefers Trump to Harris because he hates darkies or because he is worried about immigration. All that matters is that he votes for Trump.

	‘x preferable to y’	‘y preferable to z’	‘x preferable to z’
Individual 1 (prefers $x$ to $y$ to $z)$	True	True	True
Individual 2 (prefers $y$ to $z$ to $x)$	False	True	False
Individual 3 (prefers $z$ to $x$ to $y)$	True	False	False
Majority (prefers $x$ to $y$ to $z$ to $x$, a ‘cycle’)	True	True	False

Table 7: Condorcet’s paradox, propositionally reinterpreted

A general combinatorial result subsumes all these phenomena. Call a set of propositions 

there can be no 'set' if some of the propositions are either impredicative (i.e. depend on other members of the set) or are 'epistemic' (because of the 'masked man' or 'intensional' fallacy). Incidentally, in both cases problems of infinity arise.

minimal inconsistent if it is a logically inconsistent set, but all its proper subsets are consistent.

This is fine for Artificial Intelligence research but only so long as the A.I in question is not wholly self-learning. Why? Well, if that were the case, by the Axiom of Choice, there may be an infinite game which is non determinate- i.e. at least one intension can't have a well defined extension ever. On the other hand, if you are using the Axiom of Determinacy, then there can be no nonprincipal ultrafilter on the natural numbers (this can't be constructed even with Zorn's lemma) and this means there is no aggregative Social Welfare Function in the infinite case. 

This means, if finite agents can choose as to what type of Choice function they have- i.e. there is impredicativity or the thing is 'epistemic' (assuming knowledge never ends) - then there is an infinite game which is either not determinate (and thus can't have a deterministic SWF) or is determinate but still can't have an aggregative SWF!

On the other hand, if you have finite agents who have only 'revealed preference' or belong to 'ideal types' (i.e. they lack 'meta-choice' and aren't truly free) then, by something like Szilrajn theorem, there is a complete Social Welfare function with consistent inter-subjective Utility comparisons. But, it this is the case, all decisions should be put in the hands of a Benthamite planner! There would be no need to aggregate preferences because the determinants of utility, and therefore preference, would be independently known or discoverable. In other words, Society has a Plan which people must get with. Contracts are either unnecessary or mischievous.

The point I'm making is simple- at least in so far as the underlying subject can have a mathematical representation and thus criteria like consistency or completeness are non-arbitrary. 

Either determinacy obtains in the Social Sciences, in which case there is category theoretical 'naturality'- i.e. there are objective optimization problems for which everybody would arrive at the same results, or else there is no 'naturality' and thus arbitrary choice sequences underlie all things. This does not mean you can't have consistency, it just means you can't have both consistency and completeness.  But this also means there will be no unicity or uniqueness and no 'naturality' or non-arbitrariness. In other words, for purely mathematical reasons, this entire branch of inquiry is stupid shit because there are no sets, no functions, or anything else with a mathematical representation.

In other words, it is fine if Pundits want to gas on in a rhetorical or sentimental manner but the moment they try to get mathsy, we know they are talking ignorant shite. Yet, not to talk in a mathsy manner is to acknowledge there is no logic or rhyme or reason to your torrent of verbal diarrhoea. That's a good reason for a Socioproctologist not to produce a rigorous proof of this claim.