On Philosophy

December 14, 2007

The Logic Of Language

Filed under: Language,Logic — Peter @ 12:00 am

It’s not hard to generate apparent paradoxes by applying logical rules to ordinary uses of language. One famous example of this is the liar, the sentence that says “this sentence is false” (which, if true, is false, and vice versa, and which thus can be neither true nor false, apparently). Another, slightly less famous, example is considering whether the present king of France is bald or not bald. Clearly everything is either bald or not-bald, but the present king of France, as a non-existent entity, can be asserted to be neither. What do such paradoxes tell us? Some have taken them to imply that ordinary language is logically inconsistent. Certainly it might be, however I don’t see much evidence of such inconsistency. Inconsistency, in the kinds of logic usually adopted to determine the implications of these natural language expressions, is a problem because a single inconsistency would imply that every statement was true. But, despite these “inconsistencies”, we have no trouble actually reasoning. It is not the case that, upon coming across the liar, we subsequently become convinced that every statement is true or follows from it. Indeed most people would deny that anything is true as a logical consequence of the liar. This strongly implies that, whatever is going on, language is not inconsistent.

If we don’t want to simply throw our hands up and admit that language is inconsistent there are two possible ways to solve these problems. One is to hold that what is expressed logically by a sentence is more complicated than it seems. This is the solution commonly given to the linguistic dilemma posed by “the present king of France”, as described by Russell. Instead of taking it to be an object to which properties might apply such constructions are taken instead to be a shorthand, expressing something like “there is a single object x such that x is the present king of France and x is bald”. This solves the “paradox” by allowing both assertions that the present king of France is bald and that he is not bald to be false, because they are no longer assert something of the form Bx or ~Bx, of which one must always be true. And a similar solution could be conceivably be given for the liar, where the expression “this sentence” is taken to be a shorthand for something more logically complex. Such solutions may seem appealing initially, but they have their drawbacks. For example, consider reasoning such as “if the wall is less than 5ft high I won’t get hurt, if the wall is not less than 5ft high I won’t get hurt (because of some safety device), therefore, because the wall is either less than 5ft high or not less than 5ft high I won’t get hurt”. This reasoning appears sound, but it simply can’t work if we expand the definite description “the wall” as Russell would have us, because then it wouldn’t necessarily be the case that “the wall is either less than 5ft high or not less than 5ft high”, as both assertions may turn out to be false if the description can apply to more than one object or to no objects. Thus to actually derive the conclusion we must also be entertaining the premise that “there is exactly one object that satisfies the definite description ‘the wall’”. Since we clearly don’t entertain such assumptions when we reason (at least we don’t consciously consider them) then our reasoning must be flawed, that even if the premises we are considering are all true the conclusion may still be false because of the premises we didn’t consider. Which is to ask us to reason differently despite the obvious fact that reasoning as we do works perfectly well.

The other solution is to assume that the logical content of these sentences really is as it appears but that the logical rules of deduction we use to generate the paradoxes are defective, and that the proper rules should more closely reflect how we actually arrive at conclusions, which apparently don’t give rise to contradictions. One promising way of doing that might be to admit partial truth functions. We can think of a partial truth function as an algorithm that operates on objects which, if it yields a value, yields either true or false. However there is also the possibility that the algorithm won’t halt and will fail to produce a truth-value. And this does not effectively introduce a third truth value into the system for reasons that are somewhat complex, but which I can describe in a quick way as stemming from the fact that it is in principle impossible to say whether a partial truth function that hasn’t produced a value will or won’t produce a value at some point in the future. And thus there is no way to talk generally about the function not producing an output. (Or, in other words, the halting problem has no solution.) We could apply this to solve the “paradoxes” mentioned by simply asserting that “is a true sentence” fails to yield a result in certain cases of self-reference, and that most predicates don’t yield a value when applied to non-existent objects. One way to interpret what that means, in ordinary terms, is as a failure to assert, that saying that the self-referential liar sentence is true or false doesn’t really assert anything, nor does attributing properties to things that don’t exist. Of course that still means that when we reason in a normal fashion that we are presupposing facts such as “it is possible to assert things of ‘the wall’”, but that seems like a kind of presupposition that we might actually have in mind, or might endorse our commitment to as a byproduct of actually asserting things about it.

Setting aside how we might fix the problems we encounter when we combine logic and language let us take another look at what the fact that language our reasoning using it actually works indicates. All we can say, without presupposing more than we should, is that evolution knows best. Evolution has produced in people a capability to reason effectively that allows them to correctly deduce facts about the things that they encounter. And, naturally, we can check this capacity of ours for errors by simply examining whether the deductions we make about those things are correct, since they are about things we interact with. But, obviously, evolution doesn’t care about our ability to reason about fictional entities or semantical facts, so long as thinking about them doesn’t interfere with our ability to reason practically. This is why I think it is a bad idea to generalize from logical laws that seem true enough, such as the law of the excluded middle. Sure the law of the excluded middle seems as certain as anything can be, since it holds in every case that we have ever encountered, and thus evolution has led to our minds incorporating it at a very low level, as, given that evolution cares only about the cases we might encounter, there is no reason its validity in other cases to matter in that respect. But why assume, given those facts, that it must hold for non-existent entities as well? Certainly we can’t imagine them in any way except as obeying the law of excluded middle, but that could be simply because of limits on our imagination when thinking about what is not the case. Indeed I can’t even think of a way to settle the matter, even in principle, because given that non-existent things don’t exist there is not way to check. And the same holds for semantic facts in the context of self-referential sentences, because there is no way to check the liar sentence to see whether it is really true or not. Indeed this might even be an argument that we should leave such cases complete aside, after all the fact that we can’t confirm any possibilities regarding them strongly implies that what we say in those cases doesn’t really matter in any significant sense, and thus we can safely ignore them, so long as we firmly resolve to also ignore any similar cases in the future that threaten to give rise to logical paradoxes in language.

So what I am saying then is that, if we must pick some way to deal with such cases, that the choice we make is, to a large extent, arbitrary, so long as it doesn’t interfere with how reason works in normal cases. Which is why I lean towards treating our predicates as partial truth functions, because our normal reasoning is recovered under the simple assumption that those predicates actually have values in the cases under consideration. But no matter how we decide to deal with these cases, or not deal with them, I think there is one position we are certainly justified in resisting, namely the idea that language is in need of repair because of these apparent paradoxes. To endorse this view is seemingly to prefer some particular logic over language, and thus to think that language is deficient so far as it does not conform to that logic. But I think it is easier to argue the opposite, that language and our reasoning with it is extremely productive when it comes to deducing conclusions about the real world (versus artificial mathematical situations), and thus that as far as that logic isn’t a satisfactory model for language and linguistic reasoning, given the contradictions that it taking it as such would give rise to, that it is deficient. Because, while simple, logic is also unnatural and lacks expressive power compared to natural language. Since the claimed inconsistency in natural language doesn’t actually trouble us there is no need to give up its expressive power for a solution to what is essentially a non-problem.

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