The Radical Implications of Dialetheism

A paper I wrote for a philosophy of logic seminar.

The Radical Implications of Dialetheism

I. Introduction

Those logics are called paraconsistent which allow for true contradictions while attempting to prevent collapsing the system into triviality, that is, without making every statement true. The development and appeal of paraconsistent logics (henceforth PL) can be traced to perhaps three major factors: technical considerations, such as what paraconsistent logic can do as against classical logic; its handling of paradoxes, particularly the Liar; and philosophical considerations relating to the possibility of contradiction.

II. Technical Aspects of Paraconsistent Logic

In PL, sentences are not simply assigned a truth-value, but rather are related to sets of values. That is, a sentence, φ, may be related to values (t, f, p), where t is true, f is false, and p is both truth and false. It is important that the value p be thought of as both true and false, not an in-between and certainly not as an undefined gap. In Priest's version of PL, there are no truth-value gaps. This means that Priest accepts the Law of Excluded Middle (LEM). That is, for any sentence A, it is true that A v ~A. Every sentence is mapped onto one of the three aforementioned value-sets by the evaluation function, v, such that v: P → {t,p,f}, where P is the set of sentences of our propositional language. In section III the question of sentences related to multiple value-sets will be raised, but for now the singly-related scheme is sufficient.

PL, since inconsistent, is semantically closed, containing its own predicates for truth, falsity, &c. Indeed, one of the primary motivations of PL is its relationship to natural language, its lack or even denial of the legitimacy of constructed metalanguages. It skirts Gödel's proof of incompleteness because that proof relies on the language's consistency. Hence, the age-old decision for consistency as against semantic completeness is here turned on its head.

In allowing for sentences to be simultaneously both true and false, questions of how to deal with such sentences immediately arises. Classical logic makes use of what is commonly called the Explosion Principle, which means that everything follows from a contradiction: A & ~A ├ B, for any A and any B. Clearly explosion must be denied if contradictions are to be true, and if the logic is not to be trivial.

However, PL also comes with a number of losses that may at first seem quite significant. In the first place, it is easy to check that Modus Ponens (MP), Reductio ad Absurdum (RaA), and the Disjunctive Syllogism (DS) fail when paradoxical sentences are present. We will examine each of these in turn.

One of either MP or Absorption must be rejected in order to avoid Curry's Paradox. This paradox states that semantically closed theories that have both the Absorption Principle and MP lead to explosion. In particular, where T(x) is 'x is true':

1. T(1) → A
2. T(1) ↔ (T(1) → A)          by the T-scheme
3. T(1) → A                          by absorption
4. T(1)                                   MP
5. A                                       MP

Thus, since any sentence A is provable given MP and absorption, we must dispense of one of the two. Priest decides that Absorption is relatively innocent, and therefore gets rid of MP. While this is not necessary, it is at least a reasonable decision.

Perhaps the most fundamental loss, however, is that of DS. By this I mean that the problems with MP would seem to be reducible to those with DS: p → q may be written as ~p v q, but then given p we cannot conclude q, because we may also get ~p. Furthermore, and perhaps worse, DS and contradiction together lead to explosion:

1. A & ~A                Assumption
2. A 1 S
3. A v B                    2 Add
4. ~A                         1 S
5. B                           3, 4 DS

Finally, RaA is lost for obvious reasons.

It would seem at this point that PL loses too much to be of logical use, particularly in its loss of MP. Indeed, it would not be too great a stretch to consider the conditional of prime logical interest. It is difficult to imagine a conditional without MP, as it is difficult to imagine deductive reasoning without a properly intuitive conditional. There are, however, reasons for taking a different conditional than that commonly used in classical logic. In other words, perhaps the equation of p → q with ~p v q is what is causing the problem. Priest suggests that there may be extensions of PL that can deal with these problems, but for now we will consider the concept of quasi-validity as a solution. In particular, let us assume that MP, RaA, DS, etc. are not valid but quasi-valid, meaning they are valid in cases where the sentences in question are not dialetheias (that is, true contradictions). It can easily be checked that the problems with the aforementioned inference rules are all related to dialetheias. Hence, if we have no reason to believe there is a dialetheia involved in our reasoning, we may help ourselves to quasi-valid rules. When we discover that a dialetheia was indeed present that we did not previously realize, we must simply correct our previous arguments to dispense with the quasi-valid rules. Hence, quasi-valid rules are truth- and validity-preserving in most cases (we assume that paradoxical sentences are a relatively small part of all true sentences).

III. The Liar Paradox

One of the main attractions of PL is its ability to deal with paradoxes, the Liar chief among them. In this section, I will first present the pure Liar and show how PL might deal with it. Then, I will examine attempted constructions of a revenge Liar, and show how they raise difficult but not insoluble problems.

The pure Liar may be given as follows:

(L) (L) is false

Thus, if we evaluate (L) as true, that means it must be false and thus we have a contradiction. Alternatively, evaluating (L) as false leads to its being true, because it said of itself that it was false. Furthermore, it does not help if we admit of truth-value gaps, because then a revenge Liar can be constructed:

(L*) (L*) is not true

This version, if taken as gappy, would seem to imply that it was true, because it says of itself that it is gappy. So then it's true and therefore not gappy, and yet again we have a contradiction.

PL attempts to skirt the structural problems that give rise to the paradox by taking (L) to be simultaneously true and false. Here it will help to translate values into value-sets, a more explicit conception of truth-values in PL. A value-set is here and for the moment defined as the set of all truth-values that a sentence relates to. Thus, t can be written as {1}, f as {0}, and p as {0,1}. That is, since (L) is a dialetheia, it relates to {0,1}. So in this case there is no problem with the contradictory nature of (L), in the sense that the paradox can be contained by denial of both explosion and the LNC. Furthermore, this conclusion would seem quite intuitive, since we can say that (L) is false or that (L*) is not true without the nasty consequences associated with doing so in classical logic, or within an otherwise classical theory allowing for truth-value gaps. In fact, when confronted with (L), we are immediately tempted either to say it is both true and false or that it is neither true nor false—the second option landing us in the aforementioned problem of (L*).

But perhaps a revenge liar can still be constructed—in fact, I believe it can, given a broad enough conception of revenge. Consider the following:

(X) (X) is false only

Of course, the paraconsistent theorist may reply that (X) is false only... and also true. Philosophically, this is a view of evaluation that relies not simply on simultaneity, not on a “once and for all”, but on the ever-present possibility of extending the evaluation without necessarily rewriting what has gone before. Thus, (X) may be both false only and yet also true, where the “true” determination is added information, a revision that does not take back that which is revised. We might question whether the intuitive meaning of “only” has here been captured, but I believe there are philosophical reasons for why this is not a problem (see section IV). However, we get into a more troublesome situation if we translate (X) into the language of value-sets:

(X*) (X*) is related to value-set {0}

If (X*) is evaluated with value-set {1}, which we translate as “true”, that would mean it has value-set {0}, and therefore it has both {0} and {1}, but since each sentence can be related only to one value-set, we have the unfortunate result that 0=1. If (X*) is evaluated as {0}, that means it's false, and therefore what it was saying is true, which means it has {1}. Again, we have that 0=1. It seems likely that a paraconsistent theorist might evaluate (X*) as having value-set {0,1}, in which case it's true, in which case it's value-set is {0}, and therefore again we have the same problem. So, structurally, we may say that the problem arises when the Liar is made to explicitly define it's own evaluation, and it appears that PL does not wholly avoid the problem, ending up not just with contradiction but with wholesale triviality.

However, perhaps there is another, more radical way out of the problem, namely the possibility of a sentence relating to multiple value-sets. There are several ways to denote this, but I shall use the following method: when a sentence relates to sets {0} and {0,1} I shall say it relates to {{0},{0,1}}. In other words, the value-sets related to by a sentence I will put into another set. The implications of this, while not wholly worked out, might be as follows. First, why not just let the {0} in the above example be absorbed into the set {0,1}, so the sentence only relates to that one latter set as defined by Priest? It would appear to me that we cannot do this, because to accurately capture the difference between “false only” and “false (and true)” there ought to be a difference between the two sets. Thus, a sentence relating to {0,1} has a different truth-profile (or some such construction) than a sentence relating to {{0},{1}}. This implies a different conception of sets than the traditional set defined purely by its members. While working this out would be an absolutely enormous undertaking, I think that's the direction in which PL must go. Intuitively, a contradictory theory ought to go with some kind of contradictory or otherwise non-standard set theory.

IV. Philosophical Considerations

In the previous section I alluded to the problem of capturing the word “only”, as in the case “This sentence is false only”. With the above considerations, we have that a sentence may be false only and also true at one and the same time. The obvious question to ask is whether the “also” in fact undermines the meaning of the previous “only”. The dialetheist might say that it does not, because the subsequent “also” simply adds more information rather than somehow canceling out previous information. Put this way, the problem is structurally analogous to the status of contradictions—when we posit something that seems to directly contradict a previous statement, do we not in fact overwrite the older statement in favor of the new one? Can we (rationally or not) accept or even hold in our mind A & ~A, or in this case hold that something is only false and also that it is true as well, which would seem to be explicitly contradictory?

First, it is an evidential claim that people can or cannot hold a contradiction. While I believe it is clear and obvious that people believe many self-contradictory things, a better question would be whether that behavior is rational or not. Following Priest, I am inclined to say that non-contradiction may be one criterion for rationality among many others, and that in some cases it is rational to believe a contradiction, for example where other rational criteria are better fulfilled than by an alternative, non-contradictory belief. However, this is certainly not necessary for a dialetheist to put forward, and in fact Hegel himself believed in the contradiction of everything, propositional or otherwise. In fact, I think Hegel is a good example of a non-trivial dialetheism wherein everything is contradictory (these words have here changed from their previous technical meanings, while still, I hope, capturing the intuitions), though this is far outside the confines of the philosophy of logic.

Second, we may return to the “only” problem and see if we can make any progress. We would not say that dialetheism does not take contradiction seriously because of the “added information” argument, and structurally there is nothing different about the “only” problem, and indeed it is simply a specific linguistic example of a contradiction. What should therefore be said, rather than that dialetheism cannot understand “only”, is that non-dialetheist theories do not treat contradiction seriously, in that it is only dialetheism that takes contradiction as an existing fact not to be rejected. In most cases, “false only” means what the non-dialetheist means by it, but there are also dialetheias, such as the above Liar sentence (X), which must be taken in a contradictory way.

Alternatively, we might say, perhaps even against PL as a whole, that true and false might not in fact be defined purely negatively, that is, against each other such that true is simply not-false and false is simply not-true. Rather, a sentence relating to {0} might in fact have different effects or conceptual implications associated with that 0 than a sentence related to {0,1}. This would make sense epistemologically, because surely the effects or implications (these concepts unfortunately cannot be fully worked out here) are not, at least in some cases, the exact same. While this certainly takes us away from PL as a system, it is a necessary consideration since it does not take us away from contradiction, and attempts to deal with the same problems as does PL, only from a different angle, from a different translation of what would seem to be very similar insights. In any case, this raises many more problems the research of which might be fruitful.

V. Conclusion

PL is certainly not taking an easy way out of the problem of paradoxes. It is certainly not a cure-all for the diseases of traditional approaches to paradox, and providing a set theory appropriate for it is a massive undertaking. However, PL does exemplify the attempt to be comfortable with the universal inability to make everything consistent. In this sense, dialetheism opens the way towards reasoning with what are in the last analysis simply facts of life. In good dialetheist fashion, the rest might follow as so much more “added information”.

Works Consulted

Priest, Graham. In Contradiction: A Study of the Transconsistent. Dordrecht; Boston: Distributors for the U.S. and Canada, Kluwer Academic, 1987. Print.

---. “The Logic of Paradox.” The Journal of Philosophy. Vol 8. No 1 (1979): 219-241. Web. 16 Feb 2014.

---. “What Is so Bad About Contradictions?” The Journal of Philosophy. Vol 95. No 8 (1998): 410-426. Web. 16 Feb 2014.