Consider the following five sentences:
(1) (1) is false.
(2) (3) is false.
(3) (2) is false.
(4) If (4) is true then φ
(5) If (5) is true then ~φ
All of these sentences involve self-reference, or at least some kind of loopy reference. And all involve the explicit invocation of the is true/is false predicate. Some would claim that these sentences are thus meaningless because of those facts, but I am inclined to disagree. Certainly they seem meaningful enough (for example, we can determine the truth conditions). Of course I do agree that most methods of analyzing these sentences encounter difficulties, but that should motivate us to develop better tools for dealing with them, not to give up.
But before I discuss the alternative analysis that seems most promising let me first say a bit about the standard/intuitive method of analysis. The standard method is to draw up a truth table (in the case of (2) and (3) one truth table for both) and eliminate the lines where the truth value of the whole sentence contradicts the truth value assigned to that sentence as an “input” into the truth table. The remaining “stable” lines are then taken, intuitively, to be the possibilities for the truth of the sentence. And if there is only one possible truth value then it would seem to imply that the sentence has a single determinate truth value. I assume that this process is simple enough to grasp, but I’ll do sentence (4) as an example, marking contradictory lines with an X.
(4) φ (4)→φ T T T T F F X F T T X F F T X
In this example we would thus conclude that (4) is determinately true, and hence that φ is true as well. By this method (1) has no solution (every truth value is contradictory), (2) and (3) taken together seem to require either (2) or (3) to be true, but not both. And (4) and (5) are both determinately true. Obviously then this method of analysis doesn’t work. For (1) it fails to give an answer. For (2) and (3) it says something slightly absurd, that there is a fact of the matter about one of them being true and the other false, despite that there is no fact that could make one of them true and not the other, or pick out the true one. And (4) and (5) are obviously absurd. Not only could we use them to conclude anything we like, we can use them to conclude contradictory things. Now a solution to (1), and possibly to (2) and (3) is to introduce a third truth value (giving us a paraconsistent logic or a logic that permits truth gaps). But this doesn’t resolve the problem of (4) and (5), because we could easily modify them to say “(4) is true or (4) is X, where X is the new truth value, then φ”*. And of course we might also balk at this because it is not so much of an analysis of the truth of these sentences as a redefinition of what we mean by truth, and hence giving a new meaning to the sentences in question, not a step towards understanding them as they are.
This brings us to my proposal. There are two components to my proposal. The first is that we accept a classification of some sentences as having a truth value that is linguistically underdetermined. The property is being linguistically underdetermined is not a new truth value, it is more like the property of being syntactically ill-formed. We wouldn’t say of the sentence “Green” that it is true or false, we would say that it is syntactically ill-formed, and thus not of the right kind to have a truth value. Now saying the a sentence is linguistically underdetermined is not to say that it is not of the right kind to have a truth value, rather it is to say that some facts are missing which are required to determine its truth value. For example, “X is green” by itself is linguistically underdetermined. But if it is preceded by the sentence “X refers to grass” then it is true. Similarly the sentence “the ball is red” out of all context is linguistically underdetermined, but in a specific context it may very well have a definite truth value.
By itself that distinction doesn’t help us much here, which brings me to the second part of my proposal.** I propose that the truth value of logically complex sentences should be thought of as being determined by a “process” which determines the truth of the sentence from the truth of its components, and that before the process is complete the truth of the sentence is linguistically underdetermined. Additionally, that the truth value of the sentence at the end of the process should not be taken to have a retroactive effect and alter the truth of the sentence before the process is over, which means that in some places a sentence may be true or false and in others it may be linguistically underdetermined, and that this is not a contradiction. Remember, linguistic underdetermination is not a claim about the truth value of the sentence. And, while we may want to “redo” the process in which we determine the truth value of a sentence because of a change in the truth value of one of its components (perhaps we discover some new fact), a change from linguistically underdetermined to having a definite truth value does not prompt a “redoing” of the process, because it is not a change in the truth value. (And of course even if we do redo the process the truth value of the sentence from the last evaluation does not carry over, it is still linguistically underdetermined until the process is over.) Of course while we are carrying out the process we can treat “linguistically underdetermined” as a truth value, which works in the same way as a “neither true nor false” truth value in a logic that permits of truth gaps.
Given this analysis we can resolve the apparent problems posed by the five sentences that I began this post with. (1) obviously comes out as linguistically underdetermined. (2) and (3), taken together, also both come out as linguistically underdetermined. To evaluate them both we most engage in a single process, since each sentence incorporates the other. We start the process with the first one, following the conventions by which we understand sentences, and determine that it is linguistically underdetermined, since the truth value of the second sentence hasn’t been fixed by the process yet. Then we move on to the second, and it too comes out as linguistically underdetermined. (4) and (5) are more interesting; since they are both handled the same way I’ll just talk about (4). We can consider two cases, where φ is true and where φ is false. If φ is true then (4) is true (this is the simple case). If φ is false however then (4) remains linguistically underdetermined. Now none of these cases have revealed one of the more subtle nuances of this analysis, so let me introduce a new sentence to highlight it.
(6) (6) is linguistically underdetermined.
By the analysis here (6) is determinately true, and this may seem like a contradiction, since (6) claims that it is linguistically underdetermined. But this is not really contradictory. First of all linguistic underdetermination is not a truth value, as I cannot stress too much, and so it does not “conflict” with the sentence having a particular truth value. And, secondly, linguistic underdetermination is not something that is fixed everywhere, it can vary in different contexts. Within the context of (6) it is linguistically underdetermined, but elsewhere it is true. (6) then simply expresses the fact that within a sentence self-reference is necessarily linguistically underdetermined (although an entire sentence, such as “this sentence is true or grass is green” is not necessarily linguistically underdetermined because it involves self-reference, even though that self-reference within the context of the sentence is linguistically underdetermined).
This concludes my analysis of linguistic underdetermination, and thus of the puzzles resulting from self-reference and loopy reference. However it remains to integrate this analysis with other strong predicates (such as “is known”, which imply truth) and weak predicates (such as “is unknown” which have no implications for truth) which have their own rules for propagation.
* This of course doesn’t work if we are instead adopting a paraconsistent logic that uses double truth valuation instead of a third truth value. But such a logic doesn’t solve the original problem, because it leads us to conclude that either φ is true or φ is to be evaluated as both true and false (there is still no way that φ could be just false). And that is still absurd because it implies that every sentence is at least true, even if it is false as well, and so that there are no sentences that are just false. And that is unacceptable. Surely “I was eighty years old in 1995” is false, and just false, not true in any way.
** Although it could be useful it a context where we are thinking about whether a sentence is meaningless, and whether sentences such as “X is green” are meaningless by themselves because of the problematic nature of finding conditions for their truth. If we were thinking about that problem I would argue that the fact that they are linguistically underdetermined means that if they are meaningless they are not meaningless in the way that sentences which are syntactically ill-formed are, at the very least.