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The Problems of Indeterminacy and Autonomy of Law

Goedel's theorem implies that law is a logically indeterminate system. Legal determinacy arises however through imperative commands which have no logical value. This underscores a key epistemological point: binary logic is inadequate to describe reality. Indeed, the paradox of material implication arises precisely due to the inadequacy of binary logic. A better logic
would take at least three values, true, false, and indeterminate. Indeterminate values have no truth value. They are neither true, nor false. Current theories on linguistic indeterminacy are inadequate because they are enthymematically bound to binary logic which presumes that values are true or false only and that any statement which is not true is false writes Dr. Eric Engle.

Introduction
The legal indeterminacy thesis argues that law is indeterminate, either due to inherent linguistic ambiguity, or due to the nature of logical argument or because law is rife with antinomies. Parallel to this argument is another related argument, that law is a relatively autonomous discipline with its own inner logic, its own inner development, which functions essentially independently of other considerations such as market forces, political developments or moral considerations.

The theories of legal indeterminacy and the theory of the relative autonomy of legal interpretation are related in that if law were indeterminate then law would necessarily also be autonomous. I argue however that law is not radically indeterminate, and that the indeterminacy thesis is taken too far, generally speaking. Likewise, I argue (contra Teubner and Duncan Kennedy ) that law is not autonomous: law is in an interdependent relation with the economy, a feedback relation where both the law and the economy influence each other but which in the end is determined more often than not by market forces rather than by legal structures. That is, I take a weak and purely descriptive view of law as determined mostly by market forces (secondarily, by moral considerations) rather than the amoral strong prescriptive view of law and economics proposed by Richard Posner.

In my opinion the main cause of the misconception of law as somehow indeterminate is due to erroneous understandings of epistemology which do not see the distinction between theoretical and practical logic. Modernity conflates theoretical and practical logic and thus take the law of identity (p or not-p), a valid rule of theoretical logic, and then misapplies it by using it also as a law of practical logic, essentially arguing that all values must be either true or false. Arguments for legal indeterminacy are also based on misreadings of Goedel and Quine. Those misreadings are a secondary cause of the mistaken view that law is somehow (how?) indeterminate.

Of course, if language were indeterminate or if truth were subjective then law would be indeterminate and thus, in a twisted sense, autonomous. However, language is not indeterminate. Moreover, truth is objective in that it is a reflection of the material world obtained dialectically. Since language and logic are not in fact indeterminate, law is not necessarily, i.e. inevitably, autonomous and justice through law and legal certainty are possible.

Law is a part of the superstructure of society, a construct. This superstructure is extracted dialectically from observations of the material world and then used to try to change that world. The forces of production at the base of society and the superstructural rationalizations that grow out of that base (the relations of production, i.e. rationalizations such as the law, religion philosophy) are dialectically related, mutually influencing each other. Ultimately, the relations of production (superstructure) are more often than not determined by the forces of production (base) rather than the other way around. Law is not autonomous, not even relatively.

And just what is "relative autonomy" -- other than a self contradiction? "It's independent - except when it's dependent!" It's a bit trite to say 'everything is relative' as if that means something deep. Relative to what? Subjective perspective? No, that would be sollipsism. Relative to other facts? That is an equivocation. Of course all objects in the universe are somehow inevitably related to each other. So what? The fact that all objects are related to all other objects does not imply something meaningful like, for example, that thus there is no objective truth. Nor does the fact that everything can be related to anything else, somehow, imply that moral values are subjective. Seeing all experience as relative to the observer is perfect atomism ending in solipsism. Seeing the relatedness of all facts as an argument either for moral relativism or intersubjectivism or outright subjectivity are false consequences drawn from a true but irrelevant premise. Those implications do not follow from the fact that all objects of sense experience are related to each other. The truistic fact that all objects can be related to all other objects is no great revelation of uncertainty or subjectivity of moral choice. "Relative autonomy" seems to be the same thing as "autonomy" with an equivocal weasel word thrown in as a hedge. Law is not autonomous, not even "relatively".

Rather than being "relatively autonomous" law is a (super)structure based ultimately on material reality, i.e. the economy. It is not autonomous, nor is it indeterminate. It is an objective reification from and thus dependant on market forces and conceptions of morality. Moral values incidentally are also in a mutual and dependant relation with market forces.

Truth

A. Kurt Goedel, Indeterminacy and Autonomy

Three erroneous ideas

  • That law is somehow (how?) indeterminate
  • That truth is somehow subjective
  • That law is (thus) autonomous
stem in part from misreadings of the work of the mathematician Kurt Goedel.

Goedel developed theorems about axiomatic systems. Goedel's incompleteness theorem proves, simplified, that an axiomatic system cannot be both complete and consistent: within any consistent formal system there will be statements which are indemonstrable. One can either have an incomplete system or a self-contradictory system but a complete and non-contradictory system does not obtain. The problem is, the very purpose of an axiomatic system is to consistently and completely describe reality in a closed system. Goedel basically proves that to be impossible. The problem is however smaller than one thinks at first. The implication isn't that axiomatization is entirely futile, just that indemonstrable basic propositions (axioms and/or postulates) are inevitable.

Why is this relevant to law? Goedel wrote about axiomatic systems. Law can be formalized and represented as an axiomatic formal system. Thus, Goedel's theorems are relevant to law.

Goedel's theorems, though relevant, do not imply legal indeterminacy. Were law autonomous Goedel's theorems would be a good warrant for the proposition that law be indeterminate. However, Goedel's theorems imply that law is not autonomous -- not that law is inherently indeterminate. Goedel's theorems also imply (contra Teubner) that closure of a legal system from within is not possible, that at some point we ground our theories on postulates or axioms which are indemonstrable and either self evident or admitted by convention and that at other points we must invoke extra-legal arguments to complete legal reasoning in the material world by appealing, for examples, to conceptions of morality and/or market values, i.e. material facts.

In addition to implying that law cannot be autonomous, Goedel's theorems also imply that the problem with our thinking about axiomatic systems isn't a problem of indeterminacy, but rather a problem about the nature of truth. The incompleteness theorem obtains because unprovable statements are inevitable due to the nature of truth. Statements are not always either true or false. And that is why you have inevitably incomplete axiomatic systems. This can be seen in that one could introduce truth/falsehood defiant propositions -- propositions with no truth value -- into a formal system without necessarily altering any of the existing propositions of a given self-consistent axiomatic system.

This shocking but true fact, that our thinking about legal determinacy and autonomy is clouded due to scholastic currents which presuppose that all statements must either be true or false, will be revealed by exploration of the idea of truth, generally, and then some other less tricky and more well known paradoxes.

B. Theories of Truth

It might seem odd that there is a controversy over just what truth is, however there is. Modernity since the scholastics often enthymematically presumes; wrongly, that all statements must be either true or false, only. That can be seen from the logical laws of identity (A = A) and of non contradiction. The enthymeme that all statements must be true or false, only, introduced to modernity by scholastics, is wrong. Not all values are true or false. The results of that error are various forms of relativism and subjectivism.
Just what is truth? Let's look at the competing theories to avoid confusion.

1. The Correspondence Theory of Truth
I wish to argue for the correspondence theory of truth. The correspondence theory of truth argues that truth describes a relationship between our thoughts and reality. That is, a statement is true when its referent obtains -- when there is a correspondence between the statement and the material world. One critique of the correspondence theory is that "facts with which propositions should 'correspond' cannot be identified independent of the sentences that express them'" This may be why Quine takes up a correspondence-coherence theory of truth. Since I reject the linguistic circularity thesis, that language is inevitably circular (it is not, due to material referents) I am not compelled to take up coherence theory, though of course statements which are true are also non-contradictory to other true statements.

2. The Consensus Theory of Truth
According to the consensus theory of truth, truth arises as a socially constructed consensus. The problem with this view is that when large groups of people believe something which does not correspond to the material world the facts of the world do not suddenly conveniently change themselves to suit the wrong majority. In other words, truth exists independently of the observer. Large numbers of people believing something do not make the belief true. True things are true no matter how many or how few people believe them. I specifically reject the intersubjective view of truth proposed by some post-modernists.

3. The Coherence Theory of Truth
The coherence theory of truth is similar to the consensus theory and is not much more tenable. The coherence theory of truth argues a statement, thought, or belief is true if and only if it is a member of a coherent set of propositions. For the coherence theorists, truth is determined by 'its coherence or fit with what is known or at least with what is accepted.'" Thus, coherence theory is really just a reiteration of the consensus view, but with a more rigorous deductive method.

The problem with coherence theory is that a statement may be coherent with all other known statements and facts -- yet still contradict the state of the material world, as when new information is discovered. False beliefs do not become true merely because we are ignorant of those facts which would falsify them. Coherence theory puts the cart before the horse. True statements are of course coherent, i.e. non self contradictory, but their truth does not arise out of their coherence. Rather, coherent statements' truth arises out of the reflective connection of the truth statements to the material world.

The coherence theory is an effort to get around the (supposed) problem of circularity in language. In any language a given term is defined by other given terms, such that A defines B defines C defines A. That is as true as far as it goes but does not go far enough. Language is ultimately rooted in material reality: The material referents of language are in fact objective. This illusory problem of circularity is escaped through the fact that language ultimately refers to sensate experience, to ideas derived from observations of reality. There are material referents in language which permit language to escape circularity of definition.

However, even if we understand that language is not ultimately circular or indeterminate due to its connection to the material world the coherence theory still has some use because, after all, true statements do not contradict other true statements. Thus, Quine argues for a correspondence-coherence theory of truth: "our statements about the external world face the tribunal of sense experience not individually but only as a corporate body." All he is really saying there, in my opinion, is that true statements must both reflect material reality and be non-contradictory to other true statements. Quine may appear to be relativist, but he is not. Just as language avoids circular indeterminacy due to material referents, so Quine avoids indeterminacy by retaining the correspondence theory of truth as can be seen by his rejection of the analytic synthetic dichotomy.

4. The Pragmatic Theory of Truth
The pragmatic view of truth is that true statements are whatever is useful. Pragmatism is instrumentalist. For pragmatists: "an idea is 'an instrument with a particular function. A true idea is one which fulfills its function, which works; a false idea is one which does not.'" Of course, such a flexible definition of truth is a liar's dream come true. To avoid that obvious problem pragmatists usually argue for a pragmatic-coherence theory. That merely displaces the problem to coherence theory where we saw that a coherent set of statements can nonetheless be contradicted by new facts from the material world. A pragmatic coherence theory, when pressed, must collapse into the correspondence theory because being effective and being connected to reality just so happen to coincide.

Another problem with the pragmatic theory of truth is it is difficult to determine what exactly pragmatism is. There are in fact "a number of pragmatisms". That ambiguity is the fault of the pragmatists -- they hesitate to "insist on necessary and sufficient conditions for calling something a pragmatic theory". The pragmatic view seems hardly objective and is potentially unprincipled.

5. Truth Statements are reflections of the material world.
Late modernity considers logic to be one branch of mathematics. Contemporary theories of mathematics, following Hilbert, argue that mathematics is a purely formal system, not a reflection of reality, and that there is no correspondence between mathematical statements and empirical reality. This has allowed mathematics to void itself of metaphysical inquires and focus instead on technicity to explore in a structured fashion all possibilities without concern for possible metaphysical implications or concerns. This has allowed a formalization of logical inquiry to expose and resolve certain higher order abstract problems.

This view of science as a purely formal exercise is taken up by neo-positivists as exemplified in the Vienna school. The most notable example in law is Hans Kelsen's pure theory of law which, if internally consistent, has no predictive or descriptive ability as to the material world. I reject the positivists’ effort at creation of a purely formal logic, a purely formal science of law, and take up Quine's naturalization of philosophy and his rejection of the analytic-synthetic distinction.

A less extreme view of the agnosticism toward the descriptive and predictive capacity of mathematics, logic, philosophy or other formalized systems of rule production is to see a formal system as a model. In this view, the formal system is still not judged experimentally based on its predictive or descriptive power. Rather, the model may, or may not, have some factual correspondence to the material world, but it does not have to, and perhaps cannot due to the fact that sense perception is always only a partial comprehension of realty.

The reasons for taking either of these views on formal systems are to avoid the metaphysical problems presented by such debates as the scholastics' quarrel of universals. It permits the demarcation and limitation of the field of inquiry to a tractable problem because any question can lead to another question.

I wish however to break from that view in the field of legal science. Propositions of law are predictions of and descriptions of reality. Viewing logic as a purely theoretical formal system with no practical predictive use does not advance our understanding of problems such as how to infer factual consequences from legal predictions or to infer one rule of law from another rule or much of anything else in law.

Moreover, seeing logical statements as reflections of the material world allows one to reduce, and possibly entirely avoid, postulates. Postulated statements are decomposed into constituent elements which are sense impressions or inferences from sense impressions. Likewise, propositions escape circularity and pure formalism due to their ultimate connection to some material phenomenon. The linkage between material reality and statements is part of why the legal system is not radically indeterminate.

Seeing logical propositions as reflections of reality allows one to use them heuristically, to argue analogically from one field to another. Though analogical arguments are imperfect in the sense of describing correspondence rather than congruence, and probabilistic arguments are only possibly, i.e. contingently, and not necessarily true, they are nonetheless useful for developing hypotheses for empirical testing. Of course, the purely abstract statements of mathematics may be untestable as pure formalizations.

A less extreme view, which is prudent, is to suggest that rather than being necessary reflections of reality, mathematical propositions are only possibly reflections of reality. Since there are purely speculative propositions, propositions which are no reflection of reality, this is the more correct view. Such propositions are purely formal.

But even if mathematical statements are only possibly reflections of material reality, in law, at least, they are in fact reflections of reality. When we say "if you steal then you will go to jail" we are making a prediction about the material world, a practical (as opposed to theoretical) proposition, a probabilistic one, about what will happen contingent on occurrence of a future event in the world.

So, I take a view of logic and mathematics that was held by antiquity but rejected by late modernity. I argue that mathematical statements can be reflections of the material world, that mathematical statements are not inherently purely formal.

I believe that this problem of the question of whether mathematical propositions have any real existence and/or can or do reflect material reality arose due to a flawed binary epistemology which has tended to predominate in Western thinking since Aristotle. To counteract this and in all events to advance the science of logic, particularly logic as applied to law, I present a system of three and four valued logic with the values necessarily false (0), true (1), either true or false but not both and unknown which (2) and unknowable (3). I argue that Western thought, Constrained to binary logic, tended to try to force real world phenomena into categories that were inapplicable resulting in various aporia.

Logic

A. Practical versus theoretical logic

The confusion about truth statements in law and legal determinacy is partly due to modernity's rejection of Aristotle's distinction between practical and theoretical logic. Theoretical logic concerns only those statements of reality that are necessarily true or false. This is the source of the view that statements must be either true or false. It is also the source of the view that logical statements must be deductive or defective. Both of those views are scholastic and modern gloss. They are distortions of Aristotle's thinking. I describe this distinction in order to make it explicit thereby to avoid enthymemes such as "either true or false, only".

1. Theoretical logic
Theoretical logic concerns only necessary and not possible truths. Consequently, theoretical logic is not defeasible. Likewise, theoretical logic is not a temporal view of the world: statements of theoretical logic are non-defeasible -- if true, they are true in all times and places. Likewise, theoretical logic does not consider such practical issues as: who is speaking and why they are speaking. Identity politics and authoritarian reasoning are not the province of theoretical logic for that reason. It is a formal way of thinking wherein if our presuppositions and forms are correct we are then able to determine necessary implications arriving at true knowledge, necessary universal truths.

2. Practical logic
Practical logic is radically different from theoretical logic, epistemologically speaking. They use similar forms, e.g. syllogistic, but in different ways due to differing presumptions.

Propositions of practical logic are not inevitably true. They are only likely to be true. Probabilistic reasoning, inadmissible in theoretical logic, is permitted to practical reasoning. Practical reasoning's propositions are defeasible and temporal because they are only probably and not necessarily true. Practical logic considers who is speaking and why they are speaking as well as what is being said. Authoritarian reasoning and identity politics are valid within practical reasoning but invalid in theoretical reasoning. Practical reasoning is socially contextualized.

Modernity, since Hume, does not generally recognize or use the distinction between practical and theoretical logic, unfortunately. The consequence of the failure to recognize the distinction is to amplify certain problems and obscure others, to create a world view that is less accurate than it could be.

B. (Qua) Ternary Logic

1. Interpretations (Values) of Statements
Goedel's theorem points us to something which becomes even clearer when we consider logical paradoxes such as Epimenides’s famous liar's paradox (e.g., "This statement is false"). That paradox illustrates the limits of a purely binary logic: it is wrong to see the universe in terms of "either true, or false, only". I wish to argue that statements are either definite (known) or indefinite. Known statements are either true or false (0 or 1, respectively). Indefinite statements are either unknown but knowable (2) or unknown and unknowable (3).

Unknown definite statements are either true or false but which they are is unknown. We do not know the truth value of the unknown definite statement, though we do know that it has a truth value. The unknown definite statement is either true or false; but which is not known. In natural language, unknown definite statements could be represented with the word "maybe, maybe not". For example, "Is it raining in Paris?" "Maybe". We know that in Paris at any time either it is raining or it isn't raining but when we do not know which then we say "maybe". We do not know the truth value of an unknown statement but we could know it - it is knowable, but not known.

Unknowable statements are a bit more complicated. It may help to think of them as the abyss of inescapable doubt, metaphorically speaking. Unknowable statements do not have a truth value. The liar's paradox ("This statement is false") is an example of a statement with no truth value.

What about statements like "Pegasus is flying but the unicorn is not"? Pegasus and unicorns do not exist. Since things which do not exist cannot fly it seems true, that Pegasus is not flying. And if it is false that Pegasus is flying then it seems true that Pegasus is not flying, right? And thing which is not flying is landed. So then Pegasus seems to be landed. But that cannot be - Pegasus does not exist so Pegasus cannot be landed. So the statement Pegasus is flying seems to have no truth value because Pegasus does not exist. But wait -- "Pegasus does not exist" is a true statement about Pegasus. Ok, so Pegasus is not flying, not landed and truly does not exist. How are we to explain this strange conjunction?

Recalling the correspondence theory of truth here might help. Pegasus does not exist. Thus, no statement about Pegasus, other than its non existence, can correspond to the material world. Thus no true statement about Pegasus is possible, aside from Pegasus's non-existence.

What about false statements about Pegasus? "Pegasus is flying" seems to be a false statement. But "Pegasus is landed" also seems to be false. This shows that a false statement is not equivalent to the negation of an untrue statement.

Clearly, some statements such as prayers and commands have no truth value. The liar's paradox also has no truth value. I argue that it is not possible to say anything true about mythical creatures excepting of course existential statements about their non existence. Even if I am wrong about Pegasus and his ilk there are clearly other statements which are neither true nor false. So, the negation of a true proposition is not necessarily the falsification of the proposition.

What about circles? Are they somehow different from Pegasus? Recall, a circle is a set of infinite points equidistant from a central point in a single plane. Circles however do not exist. First, an infinite number of points would take forever to draw. Second, perfectly equidistant placement would likely never occur either. So let's suppose that the unicorn and the circle are both objects which really only exist in our brain. Are truth statements about circles possible? I would argue yes, because, unlike the unicorn the circle is a reflection of material experience. It seems to me that the truths about circles are definitional truths.

Leaving aside the problem of the truth value of statements about circles and mythical beasts there are clearly statements with no truth value (paradoxical liars, religious supplicants, imperious cops). Thus, we can and should distinguish between:

  • Definite statements which are known to be true (truth value: 1), or conversely known to be false (truth value: 0)
  • indefinite statements, which are known to be either true or false, only, but we do not know which value this statement has -- these statements are either true or false but which is unknown -- we do not know their truth value (truth value: 2) and
  • indefinite statements which are not only unknown, but are, moreover, unknowable (truth value: 3). The indefinite unknowable statements are neither true nor false -- they or unknowable -- we cannot know their truth value. With an indefinite and unknowable statement we cannot know its truth value. Examples of unknowable statement include falsidical paradoxes, prayers, interjections, and imperatives, at least. -- they have no truth value and so are unknowable.
2. Truth Functors
Every logical functor can be recast as an unambiguous natural language statement. That fact is consistent with my position that logic is not merely formal and can be a reflection of the material world it. Here I would like to present some natural language versions of functors in order to help people avoid linguistic confusions. A very common confusion is using "or" as a synonym of "and" or vice verse. For example, if I say cruel and unusual punishments are unconstitutional, does that prohibit both cruel punishments and unusual punishments, or only those punishments which are both cruel and unusual? Confusion of "and" with "or" (seen also in Latin with the word vel) is the most common ambiguity of natural language representation of logical propositions but it isn't the only one. Linguistic imprecision exists of course, but is only the result of laziness, a lack of the intellectual rigor that would insist on precision. Linguistic imprecision isn't inherent or inevitable to speech. Here are some common logical functors (operators), recast as natural language statements.

Conjunction: are p and q both true?
Disjunction: are either p or q true?
Exclusive disjunction: are either one, and only one, of p or q true?
If and only if: if p is true then q is true otherwise q is false
Equivalence: p is true if q is true and p is false if q is false and vice verse
Implication: if p is true then q is true otherwise p may be either true or false

The usual description of "if p then q" as "if it is wet then it is raining" contains an enthymeme. It should be recast as: "if it is raining then it is wet and if it is not raining then it may be wet or dry, only." or as "if it is raining then it is wet otherwise we don't know whether it is wet or dry though it must be either wet or dry."

Here are (qua) ternary truth tables for the basic functions:

not p
p-
01
10
22

if p then q and if q then p
p=q
pq=
001
010
100
111
122
202
212
222

if p then q
p=>q ;
p=>q = -p v q

pq =>
00 1
01 1
02 1
10 0
11 1
12 2
20 2
21 1
22 2

pq q iff p

pq iff
00 1
01 0
10 0
11 1
12 0
20 2
21 2
22 2

p=>q
00 2
01 2
10 0
11 1
12 2
20 2
21 2
22 2

p and q
p*q

pq*
000
010
100
111
122
200
212
222

p or q
p+q

pq+
000
011
101
111
121
202
211
222

True Implication

(p=>q) * (-p=>(q*-q))
001 100 0
011 100 0
021 100 0
100 001 0
111 001 1
122 001 2
201 222 2
211 222 2
222 222 2

pq ->
002
012
022
100
111
122
202
212
222

3. Multivariate logic invalidates reductio proofs
Multivariate logic arose due to the inquiry into the universal validity of the law of the excluded middle. In trivalent logic the law of the excluded fourth term takes the place of the law of excluded middle.

A reductio ad absurdum proof tries to prove a proposition negatively, by showing that the opposite argument is absurd. Reductio proofs rely on the law of excluded middle (also known as tertium non datur) which holds that p or not-p. Multivariate logic rejects tertium non datur because multivariate logic admits more values than true or false. A reductio argument only holds true if all other alternatives are absurd. Thus, when logic is seen as multivariate, not binary, reductio proofs become much trickier since the law of the excluded middle (p or not-p) no longer holds and equivocation becomes possible.

C. Puzzles in law

1. Antinomies in Law
To avoid confusion I will use antinomy exclusively to refer to legal self contradiction and paradox to refer to the broader problem of apparent or actual logical self contradiction. I use Perelman's definition of antinomy as an evident contradiction between rules and distinguish that from Quine's definition of paradoxes as either shocking but true or false and misleading which he refers to respectively as veridical paradox and falsidical paradox. I think using Perelman's definition of antinomy and Quine's definitions of paradoxes is the most sensible way forward, for there are distinctions between mere contradictions of competing rules, where a logical solution is possible (the antinomy), as opposed to non-problems, where a contradiction only appears to exist (veridical paradox) and the contradiction where in fact no exit is possible because of some error in our thinking (the falsidical paradox). I am aware Quine uses antinomy as a superset of paradox and that is contrary to Perelman's use of the French cognate term antinomie. but I think it is best to use Perelman's definition of antinomy and obviate confusion thereby.

Antinomies arise in law constantly and their resolution is a regular task judges are confronted with. Antinomies arise in law in part because the law is presumed to be a unitary non-conflicting system. However, though the unitary and non-contradictory nature of law is assumed, and probably a valid presumption within civilian legal systems, rules conflict in practice all the time, and indeed legal dualism argues that each different law-giver can create laws which conflict with the other law-givers. That is, there is no unity of all rules at the global level. We definitely observe conflicting irreconcilable rules of law colliding in legal practice regularly. Even within a single legal system we often see conflicting legal obligations whether issued by the administration, judges, or by the federal or federated state legislature(s). Even though ontologically speaking monism is more valid than dualism, juristically speaking dualism seems more correct than monism. There are a multiplicity of sources of rules which are not necessarily hierarchically arranged.

Goedel's theorem seems to imply that antinomy in law is inevitable. The law is a defeasible set of rules and arguments about rules. In that regard, at least, there is in theory (unlike in practice) no legal certainty or legal finality, only conflicts, between lawyers, before judges about rules issued by the legislature or custom (including the custom of stare decisis). In practice of course final decisions are the usual case. Theoretically speaking however, and consistent with Popper's epistemology, propositions of law are not affirmative statements, true for all time. Rather they are, as yet unfalsified hypotheses, tentative and ever subject to refutation.

a. Conflicts of Law
The most obvious sort of non-unity creating an antinomy in law is the simple conflict between two legal orders -- when for example a French citizen is injured by an Italian in an auto-accident in Germany. The resolution of such conflicts is complicated but not impossible. Conflicts of laws only appear problematic if one believes that the law is a unitary non-conflicting set of rules. That is clearly empirically not the case. Rules conflict constantly. There is more than one source of rules and numerous differing interpretations of those roles.

b. Lacunae
Conflicts of laws occur from "too many laws". The opposite problem, too few laws, also creates antinomies. Lacunae -- gaps in the law - arise due to a lack of legislation, as when a new invention changes the presuppositions of the laws, or when the legislator overlooks something. Again, lacunae are only problematic if one believes the law is a complete and consistent structure of harmonious rules -- a unitary hierarchy -- and not a set of competing and conflicting conditionals that incompletely describe reality.

2. Paradox
The simplest definition of paradox is a belief that goes against common opinion, the literal meaning of the term. A related idea associated with the word paradox is a belief that goes against one's own intuition. However, these sort of paradoxes are not aporia. They indicate only an area where our thought, or generally received opinion is wrong -- not an area where thought is impossible, so to speak. I argue that certain paradoxes arise out of the erroneous enthymematic presumption that all statements must be either true or false, only.

Those paradoxes which go against common opinion yet are true are known by Quine as veridical paradoxes -- they are shocking but true statements. Quine distinguishes veridical paradoxes from falsidical paradoxes. Falsidical paradoxes are not shocking-yet-true -- they are false and misleading. They indicate that we need to rethink some aspect of our conceptual apparatus. For this reason paradoxes are not mere intellectual curiosities, they are heuristically very useful devices for improving one's thought. Paradox, aside from being very useful heuristically is also interesting to lawyers because certain logical paradoxes reappear in law as legal paradoxes, and other paradoxes arise out of the law itself.

Let's consider Epimenides's liar's paradox. The liar's paradox in its various forms comes down to the phrase "This statement is false." Such a statement is a falsidical paradox, in that it is misleading. But I wish to argue that the liar's paradox in fact reveals something fundamental about statements: that not all statements are either true or false, and that as a consequence the negation of a statement does not always imply it's opposite. The liar's paradox demonstrates not systemic indeterminacy but the multivariate character of truth.

Sometimes people argue, or at least seem to believe, that the existence of paradoxes and/or Goedel's incompleteness theorems demonstrates semantic indeterminacy and thus supports relativist or subjectivist views (whether as to ontology, epistemology, or axiology). I disagree with those views. I argue that the existence of paradoxes -- whether falsidical, such as the liar's paradox, or veridical, like Goedel's theorems -- does not demonstrate rampant systemic indeterminacy, supportive of a relativist or subjectivist position. Rather, the existence of paradoxes shows us the scholastic world view, that all statements must be either true or false, is incorrect. Goedel's incompleteness theorems similarly reveal the basic problem of thinking that statements must be either true or false, only. The fact that any axiomatic system is theoretically incomplete does not create random or rampant systemic semantic indeterminacy. As Aristotle noted, statements may either be true, false or have no truth value. The liar's paradox has no truth value. It is not false, nor is it true. It is unknown and unknowable.

What are we to make of statements which have no truth value? Is it possible to further distinguish between statements which have no truth value so as to develop a heuristic wherein we can know: we should pursue that piece of information, or ignore that other piece of information?

I would like to argue that it is in fact possible to refine the understanding of statements which have no truth value further than given by Aristotle. Aristotle seems to regard statements which have no truth value as "non-statements". He uses the example of prayers, commands and interjections as statements which are neither true nor false. The scholastics, in contrast, sought to find in every statement either truth of falsehood, likely following the idea that statements with no truth value are non-statements, pseudo-assertions. But that does not in fact cover the entire class of statements to which we can attribute no truth value. My suggestion, to look at statements of truth as either true, false, unknown or unknowable resolves certain paradoxes and illuminates problems like statements about circles and mythical creatures.

a. Paradoxes of Material Implication Reveal the Inadequacy of binary logic

b. Paradox in Laws
One example of a paradox in law is the paradox of the legislator - 'this prescriptive statement is ineffective'. This is a variant of the liar's paradox and we already know how to get out of those. A more problematic paradox is the paradox of self binding -- how can a legal body logically obligate itself? It cannot, strictly speaking. This brings us to a third and perhaps most difficult paradox: the state is a legal fiction, in fact most, perhaps all, laws are fictions. But if the law is a fiction then how is it different from a mythical creature? How could any logic apply at all to it? I do not present solutions to the paradoxes of self-binding or of legal fictions. Rather I present them as puzzles hoping to inspire some reflection. Toward that end, I wish to examine logical statements concerning intentional entities.

c. Circling the Square: Statements about Pegasus (legal fictions)
What, finally, have we learned about Pegasus? We know that Pegasus does not exist. Does that mean that all other statements about Pegasus are false? Or that they have no truth value? I argue that:
  • Statements about Pegasus, other than the existential statement that Pegasus does not exist, are not true and, moreover,
  • False statements and not-true statements aren't equivalent to each other.
Aside from the epistemology, the analysis about statements of "Pegasus" is relevant because here, "Pegasus" is used as a proxy concept for any legal fiction and also for statements of fact. Truth is of direct importance to the determination of facts and legal inferences therefrom. Of course, statements such as "Hercule's believes Pegasus to have wings" have truth value (presume Hercules exists), this is how we can speak of truths concerning legal fictions.

In binary logic the negation of a negation is an affirmation. Usually it is argued that it should also be so in ternary logic. But this is because of a confusion. Let us consider the proposition:

» A false statement is not true.
» A true statement is not false.
» A statement with no truth value is not false.
» A statement with no truth value is not true.

That reveals the potential for equivocation -- a statement with no truth value is not false, and a true statement is also not false, yet true statements have a truth value and are very different from statements with no truth value. The latent equivocation steps clearly into view when we recognize that not all statements are true or false, and thus not all untrue statements are false. The true statement and the statement with no truth value are both not false and this creates the potential for equivocation and paradoxes.

Is the negation of a statement with no truth value a statement with truth value? Consider the negation of the liar's paradox: "This statement is NOT false". It clearly can have a truth value. But, by way of contrast, consider the negation of a command. The negation of a command is itself a command and thus also has no truth value. Negations of prayers and interjections likewise have no truth value. So, the negation of a statement with no truth value may or may not have a truth value.

This too then underscores the necessity of distinguishing between statements which are true, statements which are false, and statements which are neither true nor false, and then distinguishing between statements which are knowable, but not known (whether Mars has life, for example) and statements which are unknowable (liar's paradoxes, for example).

Let's go back to Pegasus. What about negations of statements concerning Pegasus? Suppose I say "Pegasus can fly". Pegasus does not exist. Only things which exist can fly. Consequently Pegasus cannot fly. So the statement "Pegasus can fly" is not true. However, the negation "Pegasus cannot fly." is also not true. Statements about things which do not exist are, with the exception of existential statements, untrue as is their negation.

Is there a difference between statements about a Pegasus and statements about a circle? What about statements about splorp? Splorp is undefined. It could mean anything, has never existed and never will exist. Is splorp different from Pegasus, and is Pegasus different from circle? Or "France"? Or "springing executory interests"?

Circles and Pegasus alike are different from splorp as they are reflections of empirical experience. Splorp is totally undefined and so it is impossible to say anything about splorp beyond that it is undefined. Pegasus is an imaginary recombination of existing animals and does not exist. A circle is a formal arrangement. If people stopped observing circular material objects the abstraction from those objects would not cease to exist. Pegasus, in contrast, has never existed. Yet, Pegasus does not exist is a true statement about Pegasus.

Discussing the truth value of irreal statements is useful because it allows us to see that nominalist epistemology is correct. Intentional entities have only a reified existence as objects of thought. In contrast, real entities are direct reflections of, and abstractions from, material facts. In contrast, synergies such as economies of scale, specialization of labor, trade, and standardization of parts show that the atomist method is incorrect: the whole is greater than the sum of its parts. Nominalism and holism can together reach cognitivism by way of materialism which is the concluding section of this essay.

Conclusion: Law and Morality

When law and morality diverge it is evidence of social conflict over what is moral. That is evidence that norms are not unitary, contra Kelsen. It also represents a lacuna in the system. To the point of the right relation between law and morality, John Rawls argues that there is a duty or right to disobey unjust laws, i.e. laws which are immoral. How is that possible? It is possible by appeal to values outside the terms of law itself. I would like to close this brief essay with this final puzzle because I think it shows how the open texture of law speaks not to indeterminacy or legal autonomy but to the dependence of law not only on material market forces but also on moral conceptions of ethics. Terms are open to competing interpretations. That is, the law is open textured precisely because law is not a closed system and appeals to teleological values outside of legal texts such as moral values and economics. The open texture of law points not to indeterminacy but to the dependent character of legal argument on extra-legal justifications such as morality.

Law is an intensely moral process which is why it is so hotly contested. Law and morality, at most, are only temporarily separable for analytical purposes. Such analysis of law is however, by definition, incomplete because analysis decomposes objects into their constituent elements and thus ignores holistic insights. Analysis does not and cannot reveal synergies. Ultimately the appeal to morality which underlies any claim to just law explains why law is not relatively autonomous, why it cannot be a closed axiomatic system.

Arguments that law is indeterminate, autonomous, or both are thus really arguments for a purely positivist and voluntarist view of law. They are founded on flawed epistemological assumptions about the nature of truth and morality. Law is a product of market forces and thus is not indeterminate. Law is also, and in my opinion more importantly, dependant on moral and ethical considerations and thus is not purely positivist or voluntarist and can be just. Lex mala - lex nulla.


DR. JUR. ERIC ENGLE is a research aid to Prof. Duncan Kennedy at Harvard Law School. He may be contacted at erengle@law.harvard.edu.


 
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