Falsehood and Meaning
In a famous paper entitled “Truth and Meaning” Donald Davidson argues that meaning is constituted by truth conditions. A recursive theory of truth for a language in the style of Tarski is thus a theory of meaning for that language. Understanding a sentence consists in grasping its truth conditions. The meaning of a word is its contribution to determining truth conditions. Truth is the central concept of semantic theory. Davidson says nothing about falsity in relation to meaning; that concept has no place in the theory of meaning. Perhaps the reason is obvious: falsity conditions are not what a sentence means. Suppose we say, evidently correctly, that “snow is white” is false (in English) if and only if snow is not white—the falsity condition is given by inserting negation into the sentence whose meaning is in question. Then clearly it would be wrong to say that “snow is white” means that snow is not white—it means the opposite of that! So falsity conditions don’t constitute meaning. I will return to this point, but at present I merely observe that falsity is not the concept chosen to characterize meaning, by Davidson or by the many others who have seen meaning as residing in truth conditions. I propose to argue that this is a mistake—that falsehood is as closely intertwined with meaning as truth.
The first point to make is that understanding a sentence involves knowing under what conditions it is false. If I understand “snow is white” I know that this sentence is false if and only if snow is not white—just as I know that it is true if and only if show is white. I know its truth conditions and I know its falsity conditions. It is perfectly true that we cannot replace “is false if and only if” with “means that”, but this doesn’t imply that knowing falsity conditions isn’t part of understanding a sentence. For the same thing is true of many sentences in relation to truth: we can’t replace a statement of truth conditions for indexical sentences with a “means that” clause either (“I am hot” uttered by me doesn’t mean that Colin McGinn is hot at the time of utterance), and most sentences of a natural language are at least implicitly indexical. Similarly, a biconditional for “Shut the door!” employing the concept of obedience doesn’t license the proposition that the sentence means such a condition (the sentence doesn’t meanthat the addressee shuts the door in response to the command to shut it). And there is really no reason to suppose that what constitutes grasp of meaning should be susceptible of statement in the “means that” form.  It is just an accident that this holds for truth conditions in the case of context-independent sentences (actually it doesn’t even hold for “snow is white” because of the indexicality of tense). If you say that meaning is use, you are not saying that a given word or sentence means anything about use. In any case, it is not an objection to a claim about meaning that it won’t go over into the “means that” form; and intuitively it is a platitude that to understand a sentence (in the indicative) one needs to know under what conditions it is false. You wouldn’t understand “snow is black” unless you knew that the circumstance of snow being white renders that sentence false. We could test someone’s grasp of meaning precisely by asking her whether the sentence would be true or false under such and such conditions.
But is it possible to give a Tarski-type theory of falsehood analogous to his theory of truth? That was certainly part of the appeal of a truth conditions theory of meaning for Davidson: it permits the employment of Tarski’s powerful and rigorous theory of truth. If falsehood cannot be treated in this way, then it lacks one the most attractive aspects of the concept of truth in semantic theory. To my knowledge neither Tarski nor anyone else has investigated this question, so mesmerized are they by Tarski’s formidable apparatus; but the question is easily answered in the affirmative—falsehood is just as amenable to recursive formal treatment as truth (which is just what we should expect). I will run quickly through the basic clauses for falsity; it is really a routine matter. For any sentence s, s is false if and only if not-p (where p is a sentence of the meta-language translating s). A conjunction “pand q” is false if and only if either p is false or q is false (not if and only if p is false and q is false). A disjunction “p or q” is false if and only if p is false and q is false (not if and only if p is false or q is false). Notice how disjunction is used in the meta-language to give falsity conditions for “and” and conjunction is used to give falsity conditions for “or”, instead of the usual alignment of connectives for truth conditions. A universal quantification “For all x, Fx” is false if and only if something x is not F. An existential quantification “For some x, Fx” is false if and only everything xis not F. Again notice the inversion of the quantifiers compared to the standard clauses for truth. With these clauses we can construct a recursive theory of falsity entirely parallel to Tarski’s construction for truth. The analogue of a satisfaction clause will simply be: an object x counter-satisfies F if and only if x is not F, where “counter-satisfies” means the converse of “false of” (alternatively, “dissatisfies”). We can then speak of “Convention F” which specifies that a definition of falsehood should entail all instances of the schema, “s is false if and only if not-p”; and even define falsehood as “dissatisfaction by all sequences”. There would be F-sentences as well as T-sentences. The apparatus is exactly as for truth but with suitable amendments. Tarski could have written an appendix to his famous 1944 paper  with the title “The Concept of Falsehood in Formalized Languages” and said much the same things as he said about truth. It would be surprising if he couldn’t, given the close connection between the two concepts—it would constitute an important theorem!
So we now add a Tarski-style theory of falsehood to a Davidson-type theory of meaning to produce a theory of falsity conditions for sentences of natural language (or disobedience conditions for the case of imperatives). This will be part of our theory of meaning for the language. It joins with a theory of truth conditions to give (allegedly) a complete theory of meaning. Both theories are necessary and neither is sufficient by itself. A speaker of the language grasps both the truth conditions and the falsity conditions of the sentences of that language. Thus I know that “snow is white” is true if and only if snow is white and that “snow is white” is false if and only if snow is not white. These are separate pieces of knowledge concerning distinct properties and employing different concepts (notably negation in the case of falsity). We can imagine possible beings that embrace one sort of knowledge while eschewing the other—they might be softhearted relativists that reject the notion of falsity altogether or stern skeptics about truth that recognize only falsity—but in our case we have and embrace both sorts of knowledge. Our understanding of sentences includes both truth-conditions knowledge and falsity- conditions knowledge. This implies that a theory of meaning is based around two central concepts, truth and falsehood, not a single concept—which is not what we have been traditionally taught. Word meaning is now geared to two concepts: this is not truth-theoretic semantics but truth-value–theoretic semantics. Truth and falsehood play coordinate roles in the overall theory. Linguistic understanding has two parts or aspects. We could say that a meaning is a location in logical space that comprises both a positive condition and a negative condition: both snow being white and also snow not being white. Meanings are both inclusive and exclusive.
This opens up some interesting perspectives. Suppose you are a hardboiled Popperian: you don’t think truth can ever be established, but you do think falsehood can be. You hold that “all swans are white” cannot be confirmed as true, but can be falsified by observing a single instance of a non-white swan. You believe the concept of truth is irrelevant to science, but you think the concept of falsehood plays an important role. Verification is out of the question, but falsification is the engine of progress. Suppose you even go so far as to believe truth should be eliminated from our conceptual scheme, while retaining falsehood. You accordingly don’t accept that meaning is constituted by truth conditions (any more than you accept that scientific progress is the accumulation of truths) or by verification conditions (there are no such conditions): but you do believe that sentences can be false and can be established to be false. Then you may well find yourself attracted to a pure falsity conditions theory of meaning: the meaning of “all Fs are G” is given by the condition that this sentence is falsified by the fact that an F has been observed not be a G. That is, we understand a sentence by constructing its falsification conditions, which embed its falsity conditions, and truth conditions be hanged. You thus don’t much care for Tarski’s definition of truth—for what use is the concept of truth?—but you do fancy his implied definition of falsity. It enshrines your general “critical epistemology”—your dedication to the notion of falsification. You embrace falsity-theoretic semantics done in the general style of Tarski, as adopted by Davidson. This seems like a coherent position, however radical or misguided it may be. It serves to bring out the change of perspective that results from taking falsehood seriously in semantics. 
Falsity and negation go together—notice how often I used negation in explaining falsity conditions semantics. Similarly for Popperian epistemology: we are always discovering that theories are not true (i.e. false). So negation plays a critical role in the theory of meaning (and in Popperian epistemology): we don’t know the meaning of a sentence unless we know under what conditions it is not true. The concept of negation thus enters into our understanding of any and every sentence, even when the sentence doesn’t contain negation. Hence negation is integral to meaning as such. I doubt that so-called animal languages incorporate negation in this way, even if the animal in question possesses the concept of negation. We might then speak of negation-theoretic semantics—theories that emphasize the role of negation in constituting meaning. This makes a better understanding of negation desirable, and indeed I think negation is an underexplored topic (not counting Sartre’s Being and Nothingness).  Would a good analysis of negation shed light on the nature of meaning?
 It is worth noting that there is a simple rule for deriving assignments of meaning in the “means that” form from a statement of falsity conditions: just drop the negation operator. Also worth noting that falsity is as disquotational as truth, though it clearly doesn’t generate homophonic biconditionals.
 Sartre puts negation at the heart of the mind—consciousness itself is described as incorporating a negative mental act (consciousness “is what it is not and is not what it is”). Wittgenstein in the Tractatus might be construed as implicitly introducing negation into meaning because he takes meaning to divide up logical space, thus excluding certain states of affairs. And even in model-theoretic semantics we employ functions that generate extensions and counter-extensions—what does not fall under a predicate.