A Negative Definition of Truth
Consider a tribe that speaks a language containing no truth predicate. They do, however, have a falsity predicate, which they put to good and frequent use, for this is an argumentative tribe. They are forever telling each other that what they are saying is false—false! Their philosophers have naturally given some thought to the meaning of the falsity predicate and they have a theory: they think that “false” expresses negation. When a speaker asserts that pand her interlocutor objects, “That’s false” this is equivalent to asserting that not-p. Thus: it is false that p if and only if not-p. They call this the “negation theory” of falsity. The reason members of the tribe don’t just assert the negation of what someone else has asserted is simply that it is quicker to say, “That’s false”, because then you don’t have to repeat what the speaker said. For a general statement like, “Everything the tribal leader says is false” they offer the paraphrase, “For any proposition p, if the leader says that p, then not-p”.
But being an argumentative tribe they don’t let it rest there: they often respond to an allegation of falsity by denying the allegation—“It’s not false!” they exclaim. In this way they reject an imputation of falsity—they deny a denial. The correct analysis of “p is not false” is “It is not the case that not-p”: a negation of a claim of falsity is equivalent to a double negation. At this point our tribe introduces an abbreviation for “not false” in the form of the word “true”, though somewhat reluctantly given their argumentative ways—they are uncomfortable with a word that expresses commendation. Still, no one ever ascribes “true” outright to another’s assertion: the word “true” is only used in rebuttal of someone else’s imputation of falsity. Someone asserts that p and receives the usual caustic response, “That’s false”. He hotly replies, “No, it’s true” meaning simply “It’s not false”. In the language of the tribe “true” means “not false” and “not false” means double negation, so “true” means double negation in that language.
This suggests a possible theory of the meaning of our word “true”: it means double negation. According to this way of looking at things, falsity comes first, being analyzed as negation, and then truth is the negation of falsity, i.e. of negation. It isn’t that “false” is a mere device of disquotation, since we can’t say, “’Snow is black’ is false if and only if snow is black”—we have to insert a negation sign before “snow is black”. The falsity predicate does not simply disappear on the right hand side; it is replaced with a negation sign. This is a substantive piece of analysis: “p is false” means “not-p’. It is not like the claim that “p is true” means “p”, where “p” contains nothing corresponding to the word “true”: negation is what falsity consists in, its proper analysis. But now “not false” surely means the same as “true” (assuming bivalence), so truth is double negation. We can say: “’Snow is white’ is not false if and only if snow is white”. The difference is purely rhetorical–a matter of sounding more positive. Truth is basically the absence of falsity—the opposite of error. Just as we can think of falsity as the absence of truth, so we can think of truth as the absence of falsity. First we had negation, used to deny what someone else says; then we abbreviated to “false” in order to avoid repetition; then we had negation of falsity; then we arrived at truth. Truth is a logical construction out of negation. The predicate “true” just means “not-not”.
At this point an objection is likely: double negation is simply equivalent to the proposition doubly negated, so if truth is double negation, then it is nothing—it is just the proposition being doubly negated. The double negation theory collapses into the redundancy theory, since doubly negating a proposition just gives the original proposition—formally it is just like disquotation. But this objection conflates logical equivalence with propositional identity: propositions can be logically equivalent without being the same proposition. Clearly adding negation to a proposition changes the proposition (it will go from one truth-value to the other), so it is hard to see how adding an extra negation will return us to the original proposition. We have enriched or extended the proposition when we doubly negate it; we have not left it exactly where it was. That is why it is harder cognitively to process “not-not-p” than “p”: there are more propositional components to go through. All we really have is a mutual logical entailment between the two propositions, but this is a far cry from strict propositional identity. Doubly negating a proposition is adding an extra negation operation to the singly negated proposition, not subtracting the first negation. We generate new propositions every time we add a negation sign, and of course we can do this arbitrarily many times; we do not thereby stand in one place, simply reiterating the original proposition. Very soon the sentences become impossible to process, which they shouldn’t be if we are not moving anywhere in propositional space. Sentences and their double negations are not merely stylistic variations.
The idea, then, is that truth can be analyzed as an operation of double negation without a collapse into a redundancy theory, or equivalently negation plus the falsity predicate. It may seem redundant or merely disquotational because of the logical equivalence of “p” and “not-not-p”, but it is not really so; it is an amalgam of specific conceptual elements—falsity and negation, with falsity itself resolving into negation. The best way to understand truth, therefore, is to begin with falsity, because truth itself gives an illusion of redundancy or vacuity, which falsity does not. Then we construct truth from falsity. It turns out that negation is the underlying logical reality in the analysis of truth. In the beginning was negation, and negation begot falsity, which in turn begot truth. The correct analysis of “p is true” is thus, “It is not the case that not-p”, not simply “p”. If I say, “Everything the pope says is true”, my meaning is best expressed by, “For any proposition p, if the pope says that p, then not-not-p”. When Frege remarked that “it is true that p” and “p” express the same thought, he was strictly speaking wrong, though close to being right; rather, “it is true that p” expresses the same thought as “it is not the case that not-p”. Thus truth is not strictly speaking disquotational, since the double negation of a sentence is not identical to that sentence, i.e. not the result of removing the quotation marks.
It is true that the double negation theory is in roughly the same spirit as the classic redundancy theory, in contrast to the other standard definitions of truth in terms of coherence or correspondence; but it is importantly different, since it gives a real analysis of the concept of truth. The theory allows us to see what is right in the old redundancy theories (as found in Frege, Wittgenstein, Strawson, Tarski, Ramsey, et al) but also to see where it overstates the matter by claiming that “true” is empty of content. Truth does have an analysis (in terms of negation), but it also generates sentences that are logically equivalent to sentences not containing it (nor the concepts used to analyze it), since “not-not-p” entails “p”. The theory also suggests that “true” is not logically a predicate since it reduces to an operator, as “false” reduces to an operator (the same one, but a single application). The reason the theory has not been recognized and favored is that people have tended to investigate truth without investigating falsity, where the role of negation is quite obvious. No doubt there are many things that can be meant by a “definition of truth”, and each may have its value, but the double negation theory is one sort of definition—and quite satisfactory as far as it goes.  It “catches the actual meaning of the word ‘true’”, to borrow Tarski’s phrase. Negation turns out to be integral to its meaning. It should be added to the other standard theories of truth.
One final point: the double negation theory imposes a condition on the possible bearers of truth, namely that they should be logically subject to negation. Any sentence that can be negated is a potential bearer of truth, and none that cannot be negated can be true (or false). Thus moral sentences are capable of truth, given that they can be doubly negated, which they clearly can. But imperative sentences, say, can’t be true if the theory is correct, since you can’t say, “It is not the case that shut the door!” The theory indeed explains how truth distributes over sentences, because it provides a necessary and sufficient condition for sentences to be capable of truth-value: viz. can the sentence be coherently negated.
 The concept of truth can be approached from different directions and different aspects of its significance explored; these different approaches need not be incompatible. Perhaps we shouldn’t be surprised if truth turns out to be multi-dimensional, given its many liaisons.