Defining truth is a harder than it sounds. For example consider the correspondence theory of truth: that truth is a statement corresponding to the way the world really is. This theory works for many sentences, but when applied to statements in mathematics or to the theory itself it fails. This has led some people to theorize that there are different definitions for truth in different endeavors, but in many ways such a solution seems unsatisfactory. Certainly our pre-analytic concept of truth seems unified.
The solution is to accept that truth is relative. Of course I don’t mean relative to people or societies, such a view leads to obvious contradictions; the truth of a statement is always relative to a set of axioms. We call a statement true if it can be proved from the axioms, and false if it can be disproved. This is obviously the case for truths of mathematics and philosophy, but what about empirical truths? The answer is simple: for empirical truths the set of axioms is the totality of facts about the physical world (either at a moment or for all time, depending on the nature of the statement).
Since this definition should seem self evident with respect to truths of mathematics let me focus instead on empirical truths. First let me elaborate more clearly on what the set of physical facts is. Obviously I don’t mean all the true statements about the world (a common use of the word fact), since that would be circular. The physical facts are all the statements about the fundamental constituents of the universe that correspond to the way the world actually is (or that could be deduced from valid observations, I will argue that these formulations are equivalent at a later date). Here a correspondence rule seems to slip back in, but in a more limited form. This however creates another problem: the empirical sentences that we want to “prove” using this massive set of axioms aren’t in the same language. We want to be able to show that statements such as “the cat is on the table” are true or false with respect to them, but the facts contain only information about the smallest pieces of the universe. In addition then we need a set of “translation” axioms, ones that describe what our everyday concepts are in terms of those fundamental pieces. The need for such statements shouldn’t be too surprising, since “the cat is on the table” may be true of cat means feline, but false if you are using the word cat to denote something else (like a cat statue). Such translation statements simply pin down the meaning of words, they don’t add anything new to the statement.
Some people may balk at this, because it is so far removed from our common sense reasoning about truth. Even if we do deduce truth from the physical facts surely those facts are not only about the fundamental constituents of nature, they may argue, since we don’t even know what those are yet, or use them in our common place reasoning. My response to these kinds of concerns is that we aren’t attempting to establish a method to capture the common use of truth, or to be able to deduce true statements from the physical facts. What we are attempting is to find a way to say exactly what truth is, in a way that separates true statements from false statements in way that agrees as much a possible with our conception of truth.
We can use this theory about truth to make some interesting observations. For instance it allows us to make true statements about fictional worlds, as long as we take the set of axioms that such statements are relative to as the physical facts of the fictional world. For example taking Tolkien’s world as the axioms the statement “Frodo took the one ring to Mt. Doom” is true, but “Frodo sold the one ring for hard cash” is false. Secondly, it follows from Gödel’s incompleteness theorem that there are well formed statements that can be neither proved nor disproved given a sufficiently powerful axiom system, and from this it follows that that there might be such statements in less powerful systems as well. Such statements are indeterminate, neither true nor false. For example, the hypothesis that we are really brains in a vat, living a in a perfectly simulated world, is one such indeterminate statement. If the simulation is perfect the physical facts don’t support such a statement, but they don’t deny it either.
Another interesting case to consider is that of truths which may be empirical only in part. For example when considering the statement the truth of “Bob’s action was good” certainly the physical facts must be some of the axioms, since the nature of Bob’s action is surely important. It may not be the case, however, that “good” is can be explained in terms of the physical facts alone (in fact it is unlikely), and may require its own set of axioms (such as “an action is good if …”). This may seem like another of the “translation” axioms I mentioned earlier, but I hesitate to classify it as such because of the controversy surrounding what exactly the ethical axioms are. Given that we need such axioms how could we determine what they were? Are we forced to be relativists about ethics? No, it simply reveals that we need other criteria for picking one set of ethical axioms over the others. What those criteria should be is of course a matter of some contention, it may be the wellness of people or society living under those axioms, or it may simply be how well those axioms fit with our intuitions. In other words we need to do more work in ethics before we can see how ethical truths fit into this system, although I have no doubt that they do.
Finally I should note that this account throws into question the division between a priori and a posteriori. It seems that if we treat the basic physical facts as axioms then empirical truths are as a priori as those of mathematics. It is true that we don’t know all of the physical facts without investigation, but then again we don’t know the mathematical axioms without education and investigation either. One way to defend the distinction would be to note that there are an infinite number of possible sets of physical facts, but that only one is real, and that we don’t know which one it is without investigation. But then again there are an infinite number of possible mathematical axioms, and we don’t know which of those are useful without investigation either. It seems then to come down to if we need out senses or not in order to determine if a statement is true, and is this really what we mean by the a priori / a posteriori distinction?
To conclude allow me to judge this theory by its own standards, which is where many theories about truth fall apart. Is our theory true? Well if the axioms are formulated in terms of the role truth must have (must distinguish two classes of statements, must be unified, must treat mathematics and science as equally true, must not admit error as empirical truth, ect) then I think that it is, although I obviously haven’t constructed a formal proof for it yet. I leave it as an exercise for the reader.