Two Families of Solutions to the Liar Paradox and Three Aims of Formal Logic

Mathematical logic…has two quite different aspects. On the one hand, it is a section of Mathematics treating of classes, relations, combinations of symbols, etc., instead of numbers, functions, geometric figures, etc. On the other hand, it is a science prior to all others, which contains the ideas and principles underlying all sciences.

– Gödel (1944), 125

1 Science, Paradox, and Teleology

Let us begin with a simple tautology: the Liar paradox is a paradox. This means, among other things, it is a problem of some kind.[1] And if it is a problem, then it is so because it inhibits the attainment of some desired goal.

A situation, event, state of affairs – in short, anything the predicate ‘is a problem’ is sensibly applied to – may be intrinsically problematic if it is the contrary of some state of affairs we desire directly.[2] For instance, one might regard world hunger as problematic in this way, because it is the opposite of a situation, presumably intrinsically desirable, in which everyone’s food needs are provided for.

Something is obliquely problematic if it’s obtaining is sufficient to prevent the obtaining of some other desired state of affairs, but its removal is not of itself sufficient to effect the desired state. There does not seem to be anything intrinsically problematic about Liar paradoxes. But it is manifest that Liars are obliquely problematic.

The Liar paradox is a logical paradox. This means it is a problem for logic. Logic is, in turn, the characteristic activity of the logician.[3] Logical paradoxes, then, do at least this much: they get in the way of what the logician, qua logician, is trying to do.

To know, then, what exactly is paradoxical about a Liar presupposes some knowledge of what the task of the logician as such is. The task of solving this paradox likewise presupposes an awareness of the teleology of logical inquiry itself.

There is nothing unusually spooky about this. Logic is the directed activity of a community of researchers, not the will of a god. The only mysteries here are those concerning how human behavior can be goal-oriented in general, and how the goals of a group are related to the goals of the members of that group. I do not think these are easy questions, but I don’t think their opacity here is any different from any other case.

So in what follows, I take the following to be relatively uncontroversial: that some humans sometimes act in goal-directed ways; that one of the ways humans do this is when they aim to know something; that scientific inquiry aims to know something; that logic is a form of scientific inquiry.[4]

The degree to which logic presents itself as a unified discipline, and not a mere patchwork of problems, depends on there being a unified intention behind logical inquiry. This does not require that most logicians be conscious of this unified intent at any given time. Trivially, it could happen that a majority of the world’s logicians live in the same time zone, and so at 3am in that time zone, are sleeping. Less trivially, a community of researchers can be so absorbed in their own mini-research projects that they fail to actively think about what it is that makes them all members of that community. This is the typical modus operandi for most sciences. For instance, mathematicians are probably more likely to be thinking about how to solve Goldbach’s conjecture than about what makes them mathematicians.

My claims in this essay are a) that there is not one distinct unified intention behind logical inquiry, but rather, several confused ones; and b) that the two families of solutions to the Liar paradoxes each correspond to different ideas about what the logician qua logician is supposed to be doing.

2 Two Families of Solutions to Liar-type Paradoxes

Most attempts at solving Liar-type paradoxes fall into one of two camps – those that retain a single truth predicate, and those that admit several, relativized to a language or context. The flagship for the former was Kripke’s Strong Kleene based solution,[5] but now also includes paraconsistent varieties;[6] while the standard-bearer for the latter remains Tarski’s theory of truth.[7]

Tarski’s theory of truth requires that a truth predicate for a language L can only be expressed in a metalanguage L* essentially richer than L. This limits the expressiveness of L so as to prevent Liar-type sentences from being able to be formed in it. The object language/meta-language distinction, and an ensuing hierarchy of truth predicates, developed as part of Tarski’s attempt to provide a theory of truth adequate for formal languages. Tarski did not pretend to have provided a theory of truth adequate to natural languages: on the contrary, he explicitly denied it.[8] He thought the use of ‘true’ in natural languages was vague, and even conjectured languages as rich as English were semantically inconsistent.

A Tarski hierarchy-of-languages approach probably remains the most familiar way to block Liar-type paradoxes. But this status probably has less to do with any of its own intuitive appeal than it has to do with the fact that gap-based solutions are haunted by problems of “revenge liars,” and paraconsistent approaches – and especially the dialetheist justification with which they are often paired – remain too radical for most to stomach.

3 Three Aims of Formal Logic

Superficially, these are simply two different ways to solve the same problem. But behind these different ways are radically different conceptions of what logic is supposed to do. And as such, the problem they are trying to solve when they solve the Liar is not the same problem at all.

Tarski’s aim was to develop a notion of truth adequate for formal reasoning in a way that ordinary language is not.  The notion of truth is then used as a means of recursively characterizing the meanings and behavior of the functions used in formal deductive systems, and ultimately aims at establishing the formal adequacy of those systems.[9] Let us say, then, that this purpose is that of determining formal meanings for formal languages.

The move toward solutions making use of a single truth predicate, on the other hand, was heavily motivated by the idea that a truth predicate in a formal language ought to faithfully reflect the uses of “true” in ordinary language. As Kripke put it, “Surely our language contains just one word ‘true’, not a sequence of distinct phrases ‘truen‘ applying to sentences of higher and higher levels.”[10] These sorts of solutions demand our formal language be sensitive to the way in which our actual language is used. Let us say here that Kripke-type solutions aim at restoring the material adequacy of formal systems to whatever it is they are supposed to be mapping on to.

Typically, these formal languages have natural language use as their measure. But I’d like the notion of material adequacy I’m using here to be broad enough to apply to other notions of what it is they are supposed to faithfully reflect. For instance, one might also regard them as mapping onto some conceptual language, or even to the ontology of the actual world.

There is, however, an important distinction here. In all of the above cases, it is unclear whether the aim of material adequacy is directed toward an actual language, conceptual scheme, or ontology, or towards any possible one.

4 Actualist Aims and the Well-Formedness of Liars

It seems to me the more modest Tarskian aim has largely been abandoned in favor of a confused mix of the actualist and possibilist aims mentioned above. And this is why Tarskian solutions to Liars and revenge Liars seem more forced upon their adherents than championed by them.

On the other hand, it seems that Kripkean aims, on account of the failure to disambiguate possibilist from actualist goals, suffer from an internal contradiction: on the one hand, they aim for material sensitivity to actual use; on the other hand, by an insistence on a univocal truth predicate, they require an insensitivity to this same use. Explaining this requires us to actually take a closer look at such basic notions as ‘truth,’ and ‘statement.’

The most basic considerations in favor of the well-formedness of Liars in natural languages are by recursion on its parts, and by analogy with similar sentences. ‘This Sentence’ can clearly be the subject of a sentence, ‘is false’ can clearly be the predicate of a sentence, and there does not seem to be anything innately wrong with self-reference. For instance, the sentence,

(1)    ‘this sentence has five words’

is perfectly intelligible, as is the sentence

(2) ‘“It is raining” is false.’

Why, then, should the combination of a falsity predicate (or a truth predicate) and self-reference lead to an ill-formed sentence?

Consider the following famous dictum of Kant:

Being is obviously not a real predicate, i.e. a concept of something that could add to the concept of the thing. It is merely the positing of a thing or of certain determinations in themselves.[11]

 

Notice Kant does not say being is not a predicate: he only says it is not a real predicate. He then goes on to clarify exactly what this means. Call a predicate real iff it adds to the concept of a thing. For instance, ‘Red’ is a real predicate because there is a clear difference in the concept of I have of something when I think it to be red as opposed to when I don’t think it to be red. But when I think something to exist, there is no difference in my concept of the thing. The only difference is, as it were, one of “where” I locate it. The existing red thing is just the red thing’s being in the same milieu as myself.

The same can be said about the T-Schema: To say Tα, where α is a name of a proposition p, is not to predicate a content of p, but to say nothing other than that p is so. But for this reason, what is posited by Tα is not at all the same thing posited when one states Tβ where β is a name of q. The very triviality of the T-schema militates against taking ”true’ to be a real predicate.

Now given ‘true’ does not predicate some formal content to what is expressed by a proposition, neither does it predicate the same content to what is expressed by a proposition. That is, ‘true’ is not a univocal predicate either. The same points can be made for ‘false’ as well.

Now applying this to the truth-teller, the sentence

(T)                ‘This sentence is true’

is equivalent to

(C)                ‘This sentence…’

But this is obviously not a sentence.

Similarly, the statement

(L)                ‘This sentence is false’

just denies

(C)                ‘This sentence…’

But there is no propositional content here, and thus nothing to be denied.

The problem here is that the actualist aims behind the use of univocal truth predicate are motivated by a desire to be sensitive to the structure of natural language, but the form of these solutions is deliberately coarse-grained, and insensitive to that content. The meaning of “true” must be sensitive to what it is that is said to be true in a way that predicates like “red” need not be. Therefore, it is not univocal.

5 …And Possibilist Aims

The above suggests the meaning of ‘true’ is materially sensitive in a way that may jeopardize the very idea of developing a purely formal notion of truth. In particular, it seems to me that in natural languages, the following basic syntactic rule (and similar rules for higher-order predicates) has instances where it fails:

(S)    If a is a name and F is a predicate, then Fa is a sentence

Realistically, though, nobody is going to give this up. If one is going to continue in this direction, then one needs a logic capable of quarantining any problematic sentences that arise on account of accepting these two desiderata.[12] The clearest way to do this is to accept a paraconsistent logic. One can then go on to treat nonsense as if it were capable of having a semantic value without this wreaking havoc on one’s logic. There is no need to informally interpret the logic as entailing the claim that there are true contradictions. In a paraconsistent logic with truth-values V = {0, ½, 1} and designated values D = {½, 1} with 0 and 1 respectively corresponding to the False and the True, one can informally reinterpret ½ to mean ‘is paradoxical or is ill-formed or…’ any number of other informal interpretations. The value ½ just becomes a kind of ‘semantic slum.’ But going in this direction seems to me the only way of preserving the strong universalism inherent in the aim of developing a calculus adequate for any possible language.

6 Conclusion

The two primary ways of solving the Liar – hierarchical solutions and univocal truth-predicate solutions – reflected fundamentally different conceptions of what logic itself ought to do. The motivation behind the development of the latter kind of solution was fundamentally confused between the aims of a) developing a truth-predicate materially adequate to its use in natural languages and b) developing a truth-predicate capable of serving a universal theory of all possible (and perhaps also some impossible) propositional contents.

I’ve suggested that a philosophical investigation of the basic concepts involved in the Liar paradox lead us to the conclusion that it is not only neither true nor false, but it is not even well-formed. But I’ve suggested this is the case not on account of a problem of circularity, but because ‘true’ is neither a real nor a univocal predicate.

In the light of this, it seems the logician is faced with a dilemma: either to ‘soften’ the logic to make certain of its predicates partially depend on what they are applied to for their meaning; or to keep a coarse-grained logic, but to make it coarse enough that it has a way of semantically dealing with nonsense allowed in by the syntax. The clearest way to do this would be to adopt a paraconsistent logic but to informally reinterpret the meaning of the truth value of ½.

[1]This is so even for the dialetheist: even if one should believe there are true contradictions, this doesn’t mean it should to be easy or natural to do so.

[2] Provided one has a robust enough notion of the good, an intrinsically problematic situation could also be characterized as the contrary of one that is intrinsically good, intrinsically valuable, desirable for its own sake, etc. I do not think anything in my account turns on purported differences between these terms, and so I use them interchangeably.

[3]This is not meant as a definition of logic – it is not informative enough to be so. It merely provides what the scholastics called a nominal definition –  a description sufficient for demarcating the domain of ‘logical’ without being informative enough to capture the basic meaning of this predicate.

[4]The sense of ‘science’ I employ here is not restricted to natural science, but broadly intended, as in the German ‘Wissenschaft.’

[5]Kripke, Saul. “Outline of a Theory of Truth,” The Journal of Philosophy 72, 19 (1975): 690-716. Reprinted in Jacquette, Dale, ed. Philosophy of Logic: An Anthology. Malden, MA: Blackwell (2002):70-85.

[6]Priest, Graham. In Contradiction. 2nd ed. Oxford: Clarendon Press (2006), esp. 125-140.

[7]Tarski, Alfred. “The Semantic Conception of Truth and the Foundations of Semantics,” Philosophy and Phenomenological Research 4 (1944). Reprinted in Blackburn, Simon and Keith Simmons, ed., Truth. Oxford: Oxford University Press (1999): 115-43.

[8]See Tarski (1999), 125.

[9]See Tarski (1999), 121-23.

[10]Kripke (2002), 72.

[11]Kant, Immanuel (1998), Critique of Pure Reason. eds & trans. Paul Guyer and Allen W. Wood. Cambridge: Cambridge University Press: 567.

[12] i.e. the syntactic rule S and a univocal truth-predicate that gives us naïve truth.

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